The geometry of an art: the history of the mathematical theory of perspective from Alberti to Monge

Sources and Studies in the History of Mathematics and Physical Sciences

Editorial Board

IZ. Buchwald I Liitzen I Hogendijk Advisory Board

P.I Davis T. Hawkins A.E. Shapiro D. Whiteside

Sources and Studies in the History of Mathematics and Physical Sciences K. Andersen Brook Taylor's Work on Linear Perspective H.IM. Bos Redefming Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction I Cannon/So Dostrowsky The Evolution of Dynamics: Vibration Theory From 1687 to 1742 B. ChandlerlW. Magnus The History of Combinatorial Group Theory

AI. Dale History of Inverse Probability: From Thomas Bayes to Karl Pearson, Second Edition AI. Dale Pierre-Simon de Laplace, Philosophical Essay on Probabilities, Translated from the fifth French edition of 1825, with Notes by the Translator

A Dale Most Honourable Remembrance: The Life and Work of Thomas Bayes P.I Federico Descartes On Polyhedra: A Study of the De Solidorum Elementa B.R. Goldstein The Astronomy of Levi Ben Gerson (1288-1344) H.H. Goldstine A History of Numerical Analysis from the 16th Through the 19th Century H.H. Goldstine A History of the Calculus of Variations From the 17th Through the 19th Century G. GraBhoff The History of Ptolemy's Star Catalogue

A HermannlK. von Meyennfv.F. Weisskopf (Eds.) Wolfgang Pauli: Scientific Correspondence I: 1919-1929 A HermannlK. von Meyennfv.F. Weisskopf (Eds.) Wolfgang Pauli: Scientific Correspondence IT: 1930-1939 C.C. Heyde/E. Seneta, I.J. Bienayme: Statistical Theory Anticipated IP. Hogendijk Ibn A1-Haytham's Completion of the Conics I H0yrup Length, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin A. Jones Pappus of Alexandria, Book 7 of the Collection Continued after the lIIustration Credits

Kirsti Andersen

The Geometry of an Art The History of the Mathematical Theory of Perspective from Alberti to Monge

~ Springer

Kirsti Andersen Department of History of Science The Steno Institute University of Aarhus Denmark

Sources and Series Editor: Jesper Liitzen Institute for Mathematical Sciences University of Copenhagen DK-2100 Copenhagen Denmark

Library of Congress Control Nnmber: 2005927076 ISBN 10: 0-387-25961-9 ISBN 13: 978-0387-25961-1 Printed on acid-free paper. © 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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Contents

Introduction

xix

Key Issues Questions Concerning the History of Geometrical Perspective Questions Concerning Textbooks on Perspective The Word 'Perspective' Other Publications The Period and Regions Examined The Sources and How They Are Used Contexts and Restrictions Conclusions

xix xix xix xx

xxi xxii xxiii

Acknowledgements

xxv

xxi XXI

Colleagues, Students, and Friends Institutions Sources of Funding Libraries

xxv xxvi xxvi xxvii

Notes to the Reader

xxix

Drawings and Notation Concepts Related to the Eye Point and the Picture Plane Concepts Related to Images of Points, Lines, and Planes Orthogonals, Transversals, and Verticals Rabatment Mathematical Terminology, Results, and Techniques Lines and Line Segments Results from the Theory of Proportion Mathematical Techniques The Placement of the Mathematical Explanations Bibliographies Two Bibliographies References, Orthography, and Ordering of Letters Biographies Dates for the Protagonists My Text Quotations and Paraphrases Use of My Earlier Publications

xxix xxix xxx XXXi

xxxii xxxii xxxii xxxiii xxxiii xxxiv

xxxiv xxxiv xxxiv xxxv xxxvi xxxvi xxxvi xxxvii

vii

viii

Contents

Chapter I. The Birth of Perspective

1

11. 12.

1 2

1.3.

1.4.

1.5.

The First Written Account of Geometrical Perspective The Origin of Perspective Four Stimuli Painting a View Representation of Special Lines A Search for Mathematical Rules Inspiration from Optics Brunelleschi Four Possible Techniques Brunelleschi's Conception of Perspective No Conclusion Brunelleschi's Success Perspective Before the Renaissance?

3 3 4 10 10

11 11 13 13 14 15

Chapter II. Alberti and Piero della Francesca

17

Ill.

17 17 18 19

The Two Earliest Authors

112. Alberti and His Work 11.3. 114.

11.5. 11.6. 11.7. 11.8. 11.9.

11.10.

11.11. 11.12.

1113. 11.14.

Alberti's Views on the Art of Painting Alberti's Model Alberti's Two Methods of Producing a Perspective Image Alberti's Construction The Representation of Orthogonals An Open Window A Scaled Unit Placement of the Centric Point The Images of the Transversals Choice of Parameters Alberti's Use of a Perspective Grid Alberti's Theoretical Reflections and His Diagonal Rule The Third Dimension in Alberti's Construction Alberti's Construction in History Piero della Francesca and His Work De Prospectiva Pingendi The Theoretical Foundation of De Prospectiva The Angle Axiom Foreshortening of Orthogonals and Line Segments Parallel to n Piero on Visual Distortion Piero and Alberti's Construction Piero's Rabatment Piero on the Correctness of the Construction Filarete and Francesco di Giorgio Piero's Diagonal Construction Piero's Distance Point Construction The Origin of Distance Point Constructions Piero on the Correctness of His Distance Point Construction The Division Theorem Piero's Treatment of the Third Dimension

21

22 22 23

24 25 25

28 28 29 33

34 34 36 37 37 38 40 40 40 42 43 44

46 46 48 50 50

Contents

ix

11.15. The Column Problem Equidistant Line Segments Columns on Square Bases Cylindrical Columns Piero's Considerations 11.16. Piero's Plan and Elevation Construction The Origin of the Plan and Elevation Technique Piero's Construction 11.17. Piero's Cube Piero's Idea Piero's Illustrations Piero's Heads 11.18. Piero's Anamorphoses 11.19. Piero's Use of Perspective 11.20. Piero's Influence

51 53 54 56 56 59 59 60

Chapter III. Leonardo da Vinci

81

64 66 66 71 71

75 79

111.1. Leonardo and the History of Perspective Leonardo's Trattato Leonardo's Approach to Perspective Outline of This Chapter 111.2. Leonardo's Various Concepts of Perspective Linear Perspective Versus Other Concepts of Perspective Natural Versus Accidental Perspective Composite and Simple Perspective 111.3. Visual Appearances and Perspective Representations IlIA. Leonardo on Visual Appearances of Lengths Leonardo's Axiom and the Angle Axiom The Law of Inverse Proportionality Pacioli and the Law of Inverse Proportionality The Law of Inverse Proportionality and Euclid's Theory Leonardo on the Appearance of a Rectangle The Appearance of the Vertical Boundaries The Appearance of Collinear Line Segments 111.5. Leonardo on Perspective Representations The Perspective Images of Particular Line Segments The Perspective Images of Collinear Line Segments Leonardo and the Column Problem Leonardo's Appeal for a Large Viewing Distance 111.6. Leonardo and Curvilinear Perspective 111.7. Leonardo's Doubts and Their Consequences Perspective and Visual Impressions Fixed Eye Point Leonardo's Use of Perspective

81 82 83 84 84 85 86 87 88 89 89 90 94 95 96 97 98 100 101 102 105 107 107 111 111 111 112

Chapter IV. Italy in the Cinquecento

115

IV.I. The Italian Sixteenth-Century Perspectivists

115

x

Contents

IY.2. The Architectural, Painting, and Sculpting Traditions Gaurico Serlio Sirigatti, Cataneo, and Peruzzi Lomazzo IY.3. A Mathematical Approach to PerspectiveThe Contributions by Vignola and Danti The First Edition of Vignola's Work on Perspective Vignola's Plan and Elevation Construction Vignola's Distance Point Construction Vignola's Comparison of His Two Methods Danti on Convergence Points IY.4. Connection Between Perspective and Another Central Projection - Commandino's Contributions The Context of Commandino's Work Commandino's Constructions Commandino's Influence IY.5. Another Mathematical Approach - Benedetti's Contributions Benedetti's Alberti Construction Benedetti on Pointwise Constructions Benedetti and Convergence Points Benedetti's Influence IY.6. An Encyclopedia on Perspective - Barbaro's Book Barbaro's Sources Barbaro on the Regular Polyhedra IY.7. The Italian Pre-1600 Contributions to Perspective

138 138 141 145 146 146 147 149 152 152 152 155 158

Chapter V; North of the Alps Before 1600

161

Y.I. The Introduction of Perspective North of the Alps Y.2. Viator and His Followers Viator Ringelberg Cerceau Y.3. Cousin Cousin's Introduction of a Distance Point Construction Cousin's Use of Points of Convergence Cousin on the Column Problem Y.4. Durer Durer's Introduction to Perspective Durer's Books Diirer's Plan and Elevation Construction Diirer's Enigmatic Method The Second Method as Described The Second Method as Illustrated The Second Method and Alberti Constructions The Second Method and a Distance Point Construction The Diagrams Illustrating the Second Method Finishing the Image by the Second Method Construction of the Side fg

161 161 162 166 169 172 175 178 182 183 183 188 194 197 199 200 201 202 204 204 205

116 116 116 122 124 125 125 126 128 130 136

Contents

V5.

V6. V7.

Diirer's Programme The Lesson of Diirer's Mistakes Restrictions Induced by Perspective Diirer's Practical Methods Diirer's Diagonal Method Diirer's Influence on the Development of Perspective Diirer's German Successors Perspective Touched Upon by a Painter Perspective Presented by a Count Hirschvogel and Lautensack Ryff Taking Up the Italian Tradition Jamnitzer, Lencker, Stor, and Hass Pfinzing Vredeman de Vries The Sixteenth-Century Non-Italian Tableau

Chapter VI. The Birth of the Mathematical Theory of Perspective Guidobaldo and Stevin VI.1.

Guidobaldo and His Work on Perspective Guidobaldo's Struggle with Perspective The Contents of Perspectivae Libri Sex VI.2. Guidobaldo's Theory of Perspective Line Segments Parallel to the Picture Plane The Main Theorem of Perspective Guidobaldo's Proofs of the Main Theorem Vanishing Lines VI.3. Guidobaldo's Twenty-Three Methods Guidobaldo's Rabatment The Sixth Method The Tenth Method The Twenty-First Method VIA. New Themes in Guidobaldo's Work Untraditional Picture Planes Inverse Problems of Perspective Direct Constructions VI.5. Guidobaldo's Role in the History of Perspective VI.6. Stevin and His Work on Perspective Stevin's Path to Perspective The Contents of Van de Verschaeuwing VI.7. The Foundation of Stevin's Theory The Invariance Theorem VI.8. Stevin's Practice of Perspective Stevin's Basic Constructions Stevin's Rabatment Stevin's Examples VI.9. Stevin and Inverse Problems of Perspective VLlO. Further Issues in Stevin's Work The Contents of Stevin's Appendix The Column Problem

xi 206 207 207 207 210 212 212 213 213 217 222 224 230 230 236

237 237 238 240 241 242 244 246 249 250 251 251 255 256 256 257 259 261 262 265 268 269 270 271 273 273 276 277 279 282 282 284

xii

Contents

A Perspective Instrument An Arithmetical Example VI.ll. Stevin's Influence The Knowledge of Stevin's Work Abroad The Knowledge of Stevin's Work at Home Conclusion

285 285 287 288 289 289

Chapter YD. The Dutch Development after Stevin

291

VII. 1. A Survey of the Literature VII.2. The Theory and Practice of Perspective VII.3. The Work by Marolois Marolois's Theory and Practice of Perspective Marolois's Method of Construction Marolois's Instrument Shadows and Inverse Problems of Perspective The Column Problem Arithmetical Calculations Marolois's Influence VII.4. Van Hoogstraten's Perspective Box The Left-Hand and Right-Hand Side Panels The Bottom Panel The Top Panel The Back Panel VII.5. Van Schooten's Revival of Stevin's Theory Van Schooten's Intention and Inspiration Georg Mohr Abraham de Graaf Hendrik van Houten VII.6. The Problems of Reversing and Scaling The Problem of Reversing The Problem of Scaling Reduced Distance VII.7. 'sGravesande's Essay on Perspective 'sGravesande and His Work on Perspective The Contents of 'sGravesande's Work Camerae Obscurae The Basic Theory The Turned-In Eye Point A Particular Line 'sGravesande's Basic Constructions Oblique Picture Planes 'sGravesande's Examples Shadows Response to 'sGravesande's Work The Audience for Books on Perspective VII.8. Traces of Desargues's Method in Dutch Perspective VII.9. Jelgerhuis and the Choice of Parameters The Parameters of a Picture Jelgerhuis's Choice VII.lO. The Dutch Scene

291 296 297 298 301 302 304 308 309 309 309 313 314 316 317 317 319 323 324 327 328 330 334 336 338 338 339 340 342 343 345 348 351 354 357 359 359 360 363 364 367 367

Contents

xiii

Chapter VIII. Italy after Guidobaldo

369

VIII. 1. Waning Interest VIII.2. Perspective in Textbooks on Architecture Seventeenth-Century Authors: Barca and Viola-Zanini Eighteenth-Century Authors: Amico, Vittone, Spampani, and Antonini The Galli-Bibienas and Piranesi VIII.3. Perspective in Other Textbooks Textbooks on Stage Design - Chiaramonti and Sabbatini A Textbook on Useful Matters for Painters - Zaccolini A Textbook on the Theory of Vision - Diano A Textbook on Mixed Mathematics - Bettini A Textbook on Mathematics - Guarini VIllA. The Prospettiva Pratica Tradition Cigoli Contino Accolti Torricelli Troili Amato Quadri VIII. 5. Pozzo's Influential Textbook Pozzo's Methods Pozzo's Virtual Dome A Vault As Picture Plane VIII.6. A Special Approach to Perspective - Costa VIII. 7. Mathematical Approaches to Perspective Zanotti Stellini Torelli VIII.8. The Later Italian Period

369 370 370 370 371 372 372 372 374 374 374 375 375 377 377 379 381 383 383 386 386 388 389 394 397 397 398 399 399

Chapter IX. France and the Southern Netherlands after 1600

401

IX.l.

401 402 403 403 404 406 407 407 409 410 410 413 415 418 418 420

IX.2.

IX.3.

IXA.

The Early Modem French Publications Perspective and Projective Geometry The Theory of Perspective Taught Aguilon and Mersenne Herigone Bourdin, Dechales, and Tacquet Rohault and Ozanam Ozanam on Measure Points The Encyclopedias The Works of de Caus and Vaulezard De Caus Vaulezard on Cylindrical Mirror Anamorphoses Vaulezard on Perspective The Work of Aleaume and Migon The History of the Book by Aleaume and Migon Introduction of a Perspective Grid

xiv

IX.5.

IX.6.

IX.7.

IX.8. IX.9. IX.lO.

IX.ll. IX.l2.

IX.l3.

Contents

Introduction of an Angle Scale Methods Independent of Vanishing Points Further Issues Treated by Aleaume and Migon Desargues's Perspective Method Desargues's Avoidance of Vanishing Points Theoretical Reflections in La perspective Theoretical Reflections in Aux theoriciens Conclusion on Desargues and Vanishing Points Points at Infinity in Desargues's Work on Perspective Brouillon project and Perspective Cross Ratios Projection of Conics Two Traditions Perspectivists at War - and the Work of Dubreuil Dubreuil Desargues and Dubreuil Dubreuil's Comrades-in-Arms The Work of Niceron Niceron's Construction of an Anamorphic Grid Second Act of the Desargues Drama Desargues's Supporter Bosse Bosse and the Royal Academy of Painting The 1660s and 1670s Huret Le Clerc Bourgoing Perspective and the Educated Mathematician French Eighteenth-Century Literature on Perspective Lamy Bretez, Courtonne, Deidier, and Roy Petitot and Curel Lacaille Jeaurat Taylor's Theory Introduced in France Michel Valenciennes The French Development

422 424 426 427 433 436 437 441 442 445 446 446 447 448 448 449 451 452 454 457 460 460 465 465 466 467 470 471 471 474 477 479 482 482 484 485 485

Chapter X. Britain

489

X.I. X.2.

489 489 490 492 494 494 496 496 498

X.3.

Starting Late British Literature on Perspective Before Taylor Wren, Moxon, and Salmon Ditton Taylor and His Work on Perspective Taylor's Background Taylor's Inspiration Taylor's Aim Taylor's Two Books on Perspective

Contents XA.

X.5. X.6. X.7. X.8. X.9. X.lO. X.ll.

X.l2. X.B. X.l4.

X.l5.

X.16.

X.17.

Taylor's Fundamental Concepts and Results Vanishing Points and Lines The Directing Plane Taylor's Basic Constructions Pointwise Constructions Taylor's Inspiration from 'sGravesande Taylor's Contributions to Plane Perspective Geometry Taylor's Solution to Problem 1 Taylor's Solutions to Problems 2 and 3 Taylor's Contributions to Solid Perspective Geometry Taylor's Examples of Drawing Figures in Perspective Constructions as an Intellectual Experiment A Direct Plan and Elevation Construction Taylor's Treatment of Shadows Taylor on Reflections Taylor on Inverse Problems of Perspective Problems about Determining the Eye Point Problems Concerning the Shape of an Original Figure Determining the Eye Point as Well as the Shape The Immediate Response to Taylor's Work Taylor's Work in History Hamilton's Comprehensive Work on Perspective Hamilton's Background Perspective and Conic Sections Hamilton on Linear Perspective Hamilton's Influence Kirby and Highmore Kirby's Publications on Perspective Kirby's Main Work on Perspective Kirby's Inspiration Kirby on the Theory of Perspective Kirby on the Practice of Perspective Kirby and the Column Problem Kirby's Service to Taylor Highmore The Taylor Tradition Continued Bardwell Protesting Fournier and Cowley Addressing Students at Military Academies Emerson, the Textbook Writer The Scientist Priestley Entering the Field Noble Attempting to Bridge the Gap Between Theory and Practice Malton and Son Clarke Presenting Perspective for Young Gentlemen Wood Writing for Painters Taylor's Influence on the Drawing of Chairs Perspective in Textbooks on Mathematics Martin Muller Wright

xv 502 503 506 508 508 510 511 512 512 515 519 524 524 524 529 534 534 536 537 538 540 541 541 542 542 546 547 548 552 554 554 555 557 561 562 568 568 570 571 573 577 579 584 585 587 588 589 591 591

xvi

Contents

X.18.

British Individualists Halfpenny Hodgson Murdoch Hooper Ferguson Adams British Mathematicians and Perspective The British Chapter

X.19. X.20.

592 592 592 592 594 595 596 597 598

Chapter XI. The German-Speaking Areas after 1600

599

XLI. XL2.

599 599 600 602 603 604 604 605 605 605 609 611 612 614 614 614 614 615 617 617 618 619 619 620 623 623 624 625 626 628 629 631 633

XU.

XL4.

XL5.

XL6. XI.7.

Categorization of the German Literature Perspective Instruments Faulhaber and Bramer Brunn and Scheiner Halt Hartnack Meister and Hoffmann Bischoff and Biirja Anamorphoses Albrecht Kircher and Schott Leupold Mathematischer Lust und Nutzgarten Perspective Presented for Practitioners The Unknown Fiillisch The Philomath Haesell The Painters Sandrart and Heinecke The Architect and Drawer Schiibler The Engraver Werner The Master Carpenter Rodel Gericke and Weidemann, Professors at the Academy of Art The Theologian Horstig Mathematical Works on Perspective The Wolffian Tradition Weidler lena Scholars Hennert and Lorenz Segner and Biirja Kastner's Analytical Approach Kastner's General Theory Karsten's Mastodon Traces of Lambert Perspective in the German Countries

Chapter XII. Lambert XILI. XIL2. XII.3.

Lambert's Special Position Life and Work on Perspective Early Approach to Perspective

635 635 635 642

Contents XIIA. XII.5. XII.6. XII.7. XII.8.

XII.9. XII.IO. XII. I I.

XII.12.

XII.13.

The Contents of Freye Perspektive Lambert's Possible Sources Constructing Polygons in the Picture Plane Oblique Figures Comparing Some of Taylor's and Lambert's Ideas The Applicability of the Theory of Oblique Planes Shadows Reflections Lambert's Room Determination of Areas that Can Be Seen Reflected Reflection in Curved Surfaces Parallel Projections A Precursor of Pohlke's Theorem Inverse Problems of Perspective Lambert's Practice of Perspective An Elliptical Scale Trompe L'ffiiIs Rainbows, Fountains, a Starry Sky, and Perspective Pictures Ruler Geometry The Prehistory The Steiner Circle Perspective Freedom Lambert's Examples Points on a Conic A Specific Application of the Perspective Freedom Constructing a Parallel A Line Through an Inaccessible Point Perspective, Ruler Geometry, and Projective Geometry Lambert's Impact

xvii 647 648 650 655 658 660 661 664 665 669 671 674 678 679 682 682 685 686 689 689 691 691 692 694 695 695 698 702 703

Chapter XUI. Monge Closing a Circle

707

XIII. I. Monge and Descriptive Geometry Creation of Descriptive Geometry Monge's Descriptive Geometry XIII.2. Monge and Linear Perspective Monge's Presentation of Perspective Monge's Influence on Teaching Perspective

707 707 708 709 709 711

Chapter XIV. Summing Up

713

XlVI. Opening Remarks XIV2. Local Approaches to Perspective The Italian Development The French and Belgian Development The German Development The Dutch Development The British Development XIV3. Perspective and Pure Mathematics Innovations in the Mathematical Theory of Perspective

713 714 714 715 715 716 716 716 716

xviii

Contents

Interplay Between Perspective and Other Geometrical Disciplines The Status of the Theory of Perspective XIY.4. The Theory and Practice of Perspective The Practitioners' Appreciation of the Theory of Perspective Communication Between Theorists and Practitioners The Usefulness of the Theory of Perspective XIY.5. The Driving Forces Behind the Theory of Perspective

717 718 719 719 719 720 720

Appendix One. On Ancient Roots of Perspective

723

Optics The Visual Pyramid and the Angle Axiom The Remoteness Theorem The Convergence Theorem Optics and Perspective in Harmony Cartography Ptolemy's Geography Ptolemy's Planisphaerium Scenography Conclusion

723 723 724 725 727 727 727 728 728 730

Appendix Two. The Appearance of a Rectangle a la Leonardo da Vinci

731

The Curves for Three Different Distances The Angle Between the Line Segments

732 734

Appendix Three. 'sGravesande Taking Recourse to the Infinitesimal Calculus to Draw a Column Base in Perspective

735

The First Step The Infinitesimal and Limit Situation The Perspective Image of the Visible Part of the Column Base

736 737 738

Appendix Four. The Perspective Sources Listed Countrywise in Chronological Order

739

Introduction Italy France and the Southern Netherlands Germany, Austria, and Switzerland The Northern Netherlands Britain

739 739 741 742 744 745

First Bibliography. Pre-Nineteenth Century Publications on Perspective

747

Second Bibliography. Supplementary Literature

771

Index

795

Illustration Credits

811

Introduction

Key Issues

E

ver since the late 1970s when Pia Holdt, a student of mine at the time, and Jed Buchwald, a colleague normally working in another field, made me aware of how fascinating the history of perspective constructions is, I have wanted to know more. My studies have resulted in the present book, in which I am mainly concerned with describing how the understanding of the geometry behind perspective developed and how, and to what extent, new insights within the mathematical theory of perspective influenced the way the discipline was presented in textbooks. In order to throw light on these aspects of the history of perspective, I have chosen to focus upon a number of key questions that I have divided into two groups.

Questions Concerning the History of Geometrical Perspective • How did geometrical constructions of perspective images emerge? • How were they understood mathematically? • How did the geometrical constructions give rise to a mathematical theory of perspective? • How did this theory evolve? In connection with the last question it is natural to take up the following themes. • Was there any interplay between the developments of the mathematical theory of perspective and other branches of geometry? • What was the status of the theory of perspective?

Questions Concerning Textbooks on Perspective • What inspired the author of a particular work? • How was the communication between the mathematicians and the practitioners of perspective?

XiX

xx

Introduction

In fact, I touch upon the latter issue so often that part of my book could be seen as a case study of the difficulties in bridging the gap between those who have a mathematical knowledge and the mathematically untrained practitioners who wish to use this knowledge. In addition, for reasons I will come back to, I have found it important to ask: • Were there regional differences in the treatment of geometrical perspective?

The Word 'Perspective' he field now known as 'perspective' got its name from the optical sciences, perspectival being the Latin word Boethius chose as a translation T of the Greek optike (CarterS 1970, 840).2 During the Middle Ages, perspective came to signify a science that has been characterized by A. Mark Smith as "in most respects the bastard offspring of the three basic Greek traditions in optics - the geometrical, the physical and the anatomical" (Witelos Pers, 18). In the fifteenth century, perspectiva was associated with yet another discipline that was also called scenographia and deals with the art of representing spatial panoramas or objects graphically on two-dimensional surfaces. 3 For a time the expressions perspectiva naturalis (or communis) and perspectiva artificialis (or pingendi) were used to distinguish between optics and the geometrical discipline of representation. From the Renaissance onwards, the branch of perspective dealing with representation on a two-dimensional surface was divided into various subdisciplines. The one dealing with the problem of depicting straight lines and lengths was called linear perspective. In general I use the word 'perspective' to mean linear perspective, or, more precisely, the art of using geometry for constructing images obtained by a central projection.

I 'Perspective' comes fromperspicere, which means to look through, into, or at, as well as to perceive clearly. 2To enable readers to survey my primary sources, I have placed them in a separate bibliography, which is included as the first. Other literature is listed in the second bibliography, and references to such works are indicated by an s. 3The word scenographia, or scenography, was most likely taken from Vitruvius, who used it in his De architectura in a meaning that is not clear (Vitruviuss Arch11955, book I, chapter 2, §2, and book VII, preface, §ll; cf. appendix one in the present book, pages 728-729). Some authors used 'scenography' synonymously with perspective because Vitruvius was understood to have hinted at some kind of perspectival representation. A number of other writers, on the other hand, reserved the term scenography for the art of constructing theatre scenes in perspective.

Introduction

xxi

Other Publications

T

he first comprehensive survey of the history of geometrical perspective is found in Noel Germinal Poudra's Histoire de ta perspective ancienne et moderne (History of ancient and modern perspective, Poudras 1864). In this work, written almost one and a half centuries ago, Poudra outlined the mathematical contents of some of the literature on the optical theory of appearance, perspectival representations, and descriptive geometry from antiquity to the publication in 1849 of his own book on descriptive geometry (ibid., 576). Presumably for linguistic reason, Poudra did not pay sufficient attention to literature written in German, but otherwise he provided a helpful survey. Poudra's methodology is, however, outdated. Many excellent books and papers on the history of perspective have appeared since Poudra's publication, not least over the last five decades, but no one after Poudra has composed a comprehensive work focussing on the entire development of the mathematical theory of perspective. With the present book I attempt to fill this gap for the period up to 1800.

The Period and Regions Examined o make my project feasible, I stop my account of the development of the mathematical theory of perspective around 1800. There is no reason to T choose 1800 precisely, but terminating somewhere around that year is fairly natural, for three reasons: By 1759 the mathematical theory of perspective had been fully worked out, following Johann Heinrich Lambert's creation of perspective geometry. In order to describe the impact of this event I find it relevant to include literature from a few decades after 1759. In addition, the development in Britain took an interesting turn in the second half of the eighteenth century. Finally, choosing a year around 1800 allows me to report on how Gaspard Monge incorporated perspective into his descriptive geometry. Continuing into the nineteenth century would - among other things involve a study of how the mathematical theory of perspective was absorbed by projective geometry. This is a very interesting topic, but essentially different from the one I have chosen to focus upon, namely the development of the mathematical theory of perspective as an independent discipline. To be able to characterize the development in a certain region, I have also introduced geographical limitations and only examine areas in which a considerable number of publications on perspective appeared. These were the Italian, German, French, Dutch, and English-speaking parts of Europe.

The Sources and How They Are Used

M

y account is based on more than two hundred books, booklets, and pamphlets on perspective written, and in most cases also published,

xxii

Introduction

before 1800 (and listed in my first bibliography). I am sure I have not found all titles that might be relevant4 - nor have I aspired to. Still, I am confident that my material is so comprehensive that adding further publications would not change my conclusions in any significant way. My intention has by no means been to produce an annotated bibliography. Nevertheless, I have written about each of the primary sources I have seen. s Some of the small comments on rather insignificant publications may seem distracting in relation to the overall objective, yet I have included them because I find them helpful in providing a general picture of how perspective was transmitted in the literature. It goes without saying that I have only read the most important sources in the total material from cover to cover. In studying the other publications I have concentrated on a few select subjects relevant to the cardinal issues: How does the work of the author relate to other literature on perspective? Which perspective constructions did he chose? How did he describe them and, in particular, to what extent was his presentation based on a geometrical background? In addition, I have a few pet topics, the most conspicuous being the question of how to represent a row of columns in a picture, which I call the column problem (and describe in section 11.15). This theme was actually much debated in the considered period, and I find it highly relevant, for it provides an example of some artists revolting against the solution given by the mathematicians, while other artists defended the mathematical approach.

Contexts and Restrictions

T

he history of perspective can be seen in many different contexts. Covering such a long period and such a vast area I cannot, in any realistic way, include social aspects such as patronage, education, vogues in art, or interests in and capacity to buy art. Instead I have mainly chosen to address aspects of the mathematical theory of perspective that are documented by the pre-1800 writings on perspective, and in particular to investigate how the protagonists gained a geometrical understanding of perspective. In addition, I have found it important to look at each text on perspective within the framework of its possible sources of inspiration. As these were, in most cases, local, I have chosen not to organize my book chronologically, but by geographical regions - incor-

4In searching for literature I have been greatly helped by Joness 1947, SchiilingS 1973, and Vagnetti S 1979. 5While undertaking my final revision, I became aware of a number of rare books that had not previously been included in my bibliography. At this point I lacked both the time and the funding to travel to the libraries where these volumes might have been accessed.

Introduction

xxiii

porating a split around 1600 because of a small revolution in the history of the mathematical theory of perspective that took place that year. Among the important questions I have had to leave unexplored is what role the academies of art played in promoting perspective. A few of the books I present were written in connection with academy courses, and they give the impression that perspective was considered important for the artists, but was taught at a rather elementary level. There is, however, much more work to be done in this field (Siebels 1999 offers an interesting case study for the German situation around 1800). Similarly, my primary sources do not give an adequate background for discussing thoroughly the highly pertinent question of the actual use of perspective in paintings, architectural illustrations, and other drawings. I do touch upon the subject a few times, but for the most part I refer to the literature on art history in which the topic has been eminently discussed - and to such a degree, in fact, that I refrain from a general listing of the literature dealing with the question of the application of perspective.

Conclusions

A

lthough I am not writing a suspense novel, I do not wish to reveal any of my conclusions at this premature stage. My answers to the key issues I have addressed will tum up in various chapters, and in the final chapter, XIV, the reader will find some overall conclusions. To those who would like to follow the main line of mathematical development, I can disclose that for this purpose the most relevant parts of the book are those concerning Italy up to and including 1600, France in the seventeenth century and the Netherlands in the seventeenth and early eighteenth century, Britain in the eighteenth century, and the contributions made by Lambert. Finally, let me divulge that in addition to Lambert, the main protagonists in this story are Guidobaldo del Monte, Simon Stevin, Willem 'sGravesande, and Brook Taylor.

Acknowledgements

Colleagues, Students, and Friends

T

his book has benefited from the generous support of numerous colleagues, students, and friends. Henk Bos has followed my writing of the book very closely and has been extremely helpful. He encouraged me when, at times, I found the task insuperable, listened to me as I developed ideas, discussed all aspects of perspective with me, and provided a wealth of insightful and valuable comments to an earlier version of the manuscript. I cannot thank him enough. Two chapters were sent to specialists, and I am grateful to Solve Olsson and Jeanne Peiffer for their support and constructive comments on chapter II and chapter V, respectively. Kate Larsen has put an impressive amount of work and enthusiasm into the book, making my drafts into working drawings and demonstrating endless patience when I changed my mind or found that I had made mistakes. She also detected many typing errors and pointed out unintelligible passages in my manuscript. Rikke Schmidt Kj(J!rgaard has very graciously drawn the figures 1.3, 1.5, and 1.9 and Susanne Kirkfeldt has kindly helped with figure XII.51. Heidi Flegal agreed to Anglicize my written English. She has done so with great professionalism and competence, respecting my personal style and making very instructive comments, while at the same time pointing out statements that seemed illogical. As for developing my photographs, I had the privilege of working with Jens Kjeldsen of the University of Aarhus, who was always willing to do his best with even the most impossible exposures. Similarly, I would like to thank Bent Grondahl for his very kind assistance in providing me with pictures from books at the Danish National Library of Science and Medicine, Copenhagen. Among the many inspiring discussion partners on the history of perspective I have had over the years, I particularly want to thank J. V. Field, Martin Kemp, Marianne Marcussen, Jeanne Peiffer, and Karin Skousboll. In some cases the discussions were more virtual than actual, since it was their papers that induced me to look at a certain matter in a different light. xxv

xxvi

Acknowledgements

One of the things that kept up the momentum in my work was receiving invitations to talk about perspective on numerous occasions, for which I am very thankful. There is no point in listing all the meetings here, but I would like to draw attention to a series of interesting and animated meetings arranged by a French group of ardent historians of perspective, including Didier Bessot, Rudolf Bkouche, Christian Guipaud, Roger Laurent, and Jean Pierre Le Goff, and additionally counting Rocco Sinisgalli from Italy. One member of the group, Roger Laurent, generously provided me with copies of texts that was almost impossible to access otherwise. A number of colleagues have helped me with specific jobs or drawn my attention to particular publications. These kind people I have thanked in footnotes at the specific places where I have benefited from their services. My students have also contributed to my work in various ways. First, as I already noted in the introduction, Pia Holdt opened my eyes to the history of perspective. Later, during courses I taught on the history of the mathematical theory of perspective, many students have posed interesting questions and pointed to fascinating problems. In this connection I would particularly like to mention my thought-provoking talks with Lise Husted Kjelstrom and Rikke Schmidt Kjmrgaard. Additionally, I have learned a good deal from students who contacted me to discuss their theses. In particular I thank Paola Marchi for inspiring dialogues, and Sabine Siebel for an interesting correspondence. They both produced recommendable theses (Marchis 1998; Siebels 1999) - quite independently of me.

Institutions

l

am deeply thankful to the institutions that have housed and helped me while I have been carrying out my research, first of all my home institute, the History of Science Department, the Steno Institute, University of Aarhus, and additionally the following: Accademia di Danimarca, Rome; The Dibner Institute for the History of Science and Technology, Cambridge, Massachusetts; Fondation Danoise, Paris; Herzog August Bibliothek Wolfenbiittel; Institut fiir Geschichte der Naturwissenschaften, Universitiit Miinchen; Department of Mathematics, Utrecht University.

Sources of Funding t is with pleasure and gratitude that I thank the following sources for providing the funding to cover my travel expenses, accommodation during Itravel, and payment of pictures, reproduction fees, and linguistic assistance: Aarhus University Research Foundation, The Carlsberg Foundation, The Danish Natural Science Research Council, Deutscher Akademischer Austauschdienst, The Dibner Institute for the History of Science and

Acknowledgements

xxvii

Technology, Herzog August Bibliothek Wolfenbiittel, Dronning Ingrids Romerske Fond, Ludvig Preetzmann-Aggerholm og Hustrus Stiftelse, and Willers Legat.

Libraries

A

ll the libraries I have visited have received me very kindly and I want to thank the staffs at the following libraries for their help: Bayerische Staatsbibliothek, Munich; Biblioteca Apostolica Vaticana; Biblioteca dell'Universita degli studi di Bologna; Bibliotheca Hertziana, Rome; Bibliotheque nationale de France, Paris; Bodleian Library, Oxford; The British Library, London; Danish National Library of Science and Medicine, Copenhagen; The Harvard Libraries, Cambridge, Massachusetts; Herzog August Bibliothek Wolfenbiittel; Niedersiichsische Staats-und Universitiitsbibliothek Gottingen; Staatsbibliothek zur Berlin; The State and University Library, Aarhus; and University Library Utrecht.

Notes to the Reader

I

n this section I have collected helpful remarks on the various practical solutions I have chosen for presenting diagrams, concepts, mathematical arguments, references, and protagonists.

Drawings and Notation

T

o facilitate the reading of the geometrical figures illustrating perspective projections, I have introduced a practice that applies to all the diagrams I have created and to most of my adaptations of diagrams drawn by the perspectivists.

Concepts Related to the Eye Point and the Picture Plane

T

he reader will often meet the following situation (figure I). Given are an eye point 0 (from oculus), a horizontal plane of reference y- called the ground plane (in former times the geometrical plane) - and a picture plane n. Usually the latter is tacitly assumed to be vertical, but we will also meet situations in which n forms an oblique angle with y. The line of intersection of nand yis called the ground line and denoted GR. The orthogonal projection of 0 upon yis called its foot, and the letter F is used to denote this point. Moreover, the orthogonal projection of 0 upon n is called the principal vanishing point and denoted P, and finally the orthogonal projection of P upon GR, or equivalently the orthogonal projection of F upon GR, is denoted by the letter Q. There is no common name for this point, but I call it the ground point. The line through P parallel to the ground line is called the horizon, and denoted by HZ. On this line, two points D are marked. These are the socalled distance points, defined by PD = OP. They have been given this name because the distance OP between the eye point 0 and the picture plane n is frequently called, simply, the distance - which I write in italics throughout the book. Sometimes it is handy to be able to distinguish between the two distance points, and hence I have introduced the terms right and left distance point, the latter of which, when seen from the eye point, is the point lying to the left of P. xxix

xxx

Notes to the Reader

TJ

r

R

F

FIGURE

1. Concepts related to the eye point and the picture plane.

Following Nicolas Louis de Lacaille, I call the horizontal plane 1] which passes through the eye point 0, the horizontal plane, and name the plane v defined by the points 0, P, and Q the vertical plane.

Concepts Related to Images of Points, Lines, and Planes t is practical to have a term for points and lines (figure 2) situated behind Istructed. such as and whose perspective images are conthe picture plane objects for Inspired by Brook Taylor, I introduce the name 1r,

A

I,

original

such elements - occasionally stressing the fact that object is original by adding a subscript 0, as in A o • My notation for the perspective image of a point A is Ai' and for the perspective image of a line I I use the symbol II" When I cuts 1r, the point of intersection is called its intersection point and is denoted II" To I is also assigned a vanishing point, Vi' defined as the point of intersection of 1r and the line through parallel to the line I. As we shall see later, it is a fundamental result in the theory of perspective that the image Ii passes through VI" For this

°

Notes to the Reader

xxxi

r

F

FIGURE

R

2. Concepts related to the images of a point and a line.

reason many perspectivists, and I with them, also call the point VI the vanishing point of II' Let a be a plane that cuts n (figure 3), Just as two points were assigned to an original line, two lines are associated to a, its intersection ia, and its vanishing line va' The first is the intersection of n and a, and the second is the intersection of n and the plane through 0 parallel to a.

Orthogonals, Transversals, and Verticals

I

n the interest of brevity, I use the word orthogonals to refer to lines orthogonal to the picture plane n, transversals for lines parallel to the ground line, and verticals for lines perpendicular to the ground plane Yo Many readers will presumably keep the book, and hence the figures, in a horizontal position and therefore not always experience the vertical lines as vertical. However, the term 'vertical' is such a handy concept for describing directions in the threedimensional space that I use it quite often, asking the readers to rotate the drawings either in their minds or in reality.

xxxii

Notes to the Reader

r

FIGURE

3. Concepts related to the image of a plane.

Rabatment

S

ince drawing paper is only two-dimensional and a large part of the problems in perspective deal with three-dimensional configurations, different authors have used a variety of procedures for depicting two or more planes on the plane of the paper. For such procedures I use the French term rabalmenlo I myself generally choose parallel projections to illustrate three-dimensional situations.

Mathematical Terminology, Results, and Techniques Lines and Line Segments he mathematical perspectivists I am dealing with kept to a tradition dating back to Euclid: they seldom distinguished between a line and a line T segment. When they wrote the line AB, for instance, they most often meant the line segment AB - defined by the endpoints A and B. When I am sure of

Notes to the Reader

XXXlll

what they had in mind, I write either 'line segment' or 'line', sometimes using the symbol I for the latter. It often happens that my authors have called a line AB and then only later defined the precise position of the point A or B on the line; in a few cases, I do the same. As to ratios, I denote the ratio between two line segments AB and CD as AB : CD. Modern readers may expect a special sign, as in IAB I:ICD I, that shows that the comparison concerns the lengths of the line segments. However, the protagonists followed the classical Greek custom of making 'geometrical calculations' in which they did not assign a number to a line segment, but considered it as a magnitude that could form a ratio with another one-dimensional geometrical magnitude.

Results from the Theory of Proportion

S

tudents trained in classical mathematics learned how to manipulate with ratios and proportions. As this is no longer common knowledge, I have listed below the results referred to in this book. They occur either explicitly as propositions in book five of Euclid's Elements or can be obtained by combining Euclid's theorems. When r : s = t : u, then (r + s) : s = (t + u) : u (1) (2) r : (r + s) = t : (t + u) (3) (r + s) : r = (t + u) : t, and for r > s (4) (r - s) : s = (t - u) : u (5) (r - s) : r = (t - u) : t, whereas for s > r (s - r) : s = (u - t) : u. (6) Again, for r > s, s : (r - s) =m : n implies that (r + s) : (r - s) =(2m + n) : n. (7) Two proportions can be 'multiplied' as follows:

r : s = t : u and s : a = u : b imply that r : a = t : b.

(8)

Finally, when r, s, t, and u are of the same kind in the sense that they can all form ratios with each other, then

r : s = t : u implies that r : t =s : u.

(9)

Mathematical Techniques

M

any of the perspectivists presented in this book complained about how their predecessors treated mathematical techniques. A number of them found that it was presented much too abstractly, and others felt that it was not presented concisely enough. In fact, no solution can satisfy everyone.

xxxiv

Notes to the Reader

This very problem also faces the historian of perspective, who aims to reach readers with different mathematical backgrounds. My approach has been as follows. The mathematical content of the publications considered is extremely important for my investigation, and historiographically I find it essential to describe the contents in the setting of the mathematics the authors had at their disposal. For instance, I only introduce trigonometrical expressions when the authors themselves did so. Actually, the most frequently applied mathematical discipline in the literature discussed is the theory of similar triangles. To the extent that I paraphrase proofs, I repeat the authors' arguments in some detail - sometimes in even greater detail than the authors themselves avoiding to leave the mathematical reasoning as an exercise to the readers. I have done so in the hope that interested readers who are not in the habit of working with geometry may have a chance to follow the arguments. I trust that readers who are familiar with the techniques involved will skip the parts of any proof that bore them. In addition to giving examples of the various authors' treatment of the geometry behind perspective, I have, in a few cases, added mathematical explanations of topics that the pre-1800 literature did not analyse mathematically. An example of this is the construction of perspective boxes (presented in section VIlA).

The Placement of the Mathematical Explanations

W

hen presenting a construction that has been important in the history of perspective, I include a complete description of it in the main text. Quite often, however, I deal with a perspectival technique that is rather special and requires quite extensive mathematical arguments; such presentations are usually placed in figure captions.

Bibliographies Two Bibliographies

A

s indicated in the introduction, this book carries two bibliographies. The first contains all the original works on perspective to which I refer, whereas the second lists supplementary literature: old works on other topics than perspective as well as secondary literature. References to the second bibliography are marked with an s, as in Kemps 1990.

References, Orthography, and Ordering of Letters n some cases, rather than examining the first edition of a source, I have consulted a later one. Yet in order to inform the reader about the first Iappearance, I keep to this earliest date in the references adding, after a slash,

Notes to the Reader

xxxv

the year of the edition I have used, an example being Pozzo 1693/1707. In some cases later editions of books are not listed separately, thus Durer 1538 refers to the second edition from 1538 mentioned under Durer 1525. For books that were only printed long after they were composed, I have introduced an ad hoc abbreviation, using, for instance, the reference 'Piero Pros' for Piero della Francesca's De prospetiva pingendi. When referring to a page number in the present book I write page, otherwise I just write the number, while I have abbreviated "folio" as fo1., and used the symbol § for paragraph. In preparing the two bibliographies, I have come to hold an ever-greater admiration for bibliographers. It is quite challenging, verging on the impossible, to get the places and years of publication of various editions right. Frequently in early modem times, the same print was distributed by two publishing houses, each choosing their own title page for their copy. In principle I have kept the original spelling used in book titles, one exception being that I have changed many us to vs and vice versa. It is however, difficult to copy spelling from a time when no fixed rules existed, so mistakes are unavoidable. As an example I can mention that the title of the book listed as Steiners 1833 contains the German word for constructions, a word I have seen reproduced as Konstruktionen, Constructionen, Construktionen, and Konstructionen. As to the place of publication, I have, in general, included only one town, and I have retained the spelling of the language in which the work was published - although I have translated Latin names of cities. In alphabetized lists I have ordered the German letters i5 and U as oe and ue, respectively, and similarly treated the Scandinavian letter {} as oe.

Biographies

H

aving worked with nigh on two hundred authors who combined span more than three hundred and fifty years and a large part of Europe, I have not aspired to write full biographies I or provide prosopographies. Where it has been possible to find the information, I have mentioned each author's profession. Taken together, my brief remarks on the various authors show, in particular for the early period, that perspective seduced all sorts of people, from artisans to noblemen, with varying degrees of learning and levels of mathematical understanding. To illustrate the important role ecclesiastics

lThe literature on the lives and works of the artists, artisans, and mathematicians who wrote on perspective is so enormous that listing it is major project in itself - and one that has actually been undertaken by Kim Veltman, but not yet published (Veltmans forth). In this book I have only included studies that deal with the topic of this book: the development and use of the mathematical theory of perspective. For bibliographies that cover some of the general secondary literature on perspective, I refer the reader to SchiilingS 1975; VagnettiS 1979; Veltmans 19862; and VeltmanS forth.

XXXVI

Notes to the Reader

played in teaching in early modern times, I have also, when I know it, mentioned an author's affiliation with a religious order. The more remarkable authors have received a few lines of lines biography, 2 and the most influential protagonists rather more. I find it highly relevant to know what motivated the different authors to take up perspective, but unfortunately I have generally been unable to find answers to this question. In a few cases I myself have suggested a motivation.

Dates for the Protagonists

W

hen I have been able to find the dates relating to the authors, I have included the information in the text and in the index. For quite a number of those mentioned there is no consensus as to the exact years of their birth or death. I have made no notes concerning differences of opinion, but often chosen the solution to indicate circa years, while sometimes deciding to trust one source of information over another. For other than the protagonists, whenever possible I have listed their dates in the index, but only for those who lived and died before the twentieth century.

My Text Quotations and Paraphrases

Q

uotations of some length are printed in a smaller type than the rest of the text. All non-English quotations have been translated into English. Sometimes I have used an existing translation, to which I have then naturally referred. The rest are my own translations, which I have tended to leave fairly literal. In a few cases where the original text is not understandable without an extensive explanation, I have made a free translation - and said so in the text. These translations are indented and printed in normal font size. There are similarly some cases in which I have paraphrased a result, and these I have treated typographically like free translations. During the Renaissance it was common to add points both before and after a letter when it referred to a point in a diagram, as in .R., for instance. For the sake of simplicity, in the quotations I have written only R and left out the points.

2For these I have used - without making specific references - the following works: van der Aas 1852; Nouvelles 1852; Allgemeines 1875; DictionaryS 1885; Thieme & Beckers 1907; Dictionnaires 1933; Dizionarios 1960; Wallis & Walliss 1986; Placzeks 1982; Deutschers 1986; Britishs 1990; Indices 1993; Indexs 1993; Biografisches 1997.

Notes to the Reader

xxxvii

Use of My Earlier Publications

P

art of the material in this book has been the subject in some of my earlier publications. 3 Everything has been reworked so as to fit into the context of this book. Nevertheless, I have occasionally recycled some sentences from my previous work.

3Some sections of chapter I were presented at the conference "Higher Artisans": Humanism and the University Tradition, Copenhagen 29-31 October 1998 organized by Jens Hoyrup and Marianne Pade; chapter II and section VII.6 overlap partially with Andersen s 19871' Andersens 19923, and Andersen s 1996; chapter III is a revised version of a paper presented at the meeting Linear Perspective, The First Century, Dibner Institute Workshop, 18-20 May 1995, organized by Jehane Kuhn; the second part of chapter VI and some sections of chapter VII share their themes with Andersen s 1990; chapter IX contains material published in Andersens 1991 and Andersen s 1996; chapter X has some overlap with Andersens 1992 1; and appendix one is an abbreviated and revised edition of Andersens 19872 •

Chapter I The Birth of Perspective

I.1 The First Written Account of Geometrical Perspective

L

eon Battista Alberti was the first to present a perspective construction in writing. He did so in 1435 in his De pictura, in which he also introduced a model for perspective representation. Alberti’s own way of expressing his definition is described in chapter II, but here I find it convenient to translate his definition into modern terms (figure I.1): Given a picture plane p and an eye point O, an object behind p is depicted upon it by mapping each point A in the object upon the point Ai in which the line OA cuts the surface p. This is the same as saying that the object is mapped upon p by a perspective or central projection with its centre in O.

p

O

Ai

A

FIGURE I.1. A perspective projection; the viewer is borrowed from Viator. 1

2

I. The Birth of Perspective

Because Alberti’s description is the oldest – and because I study the development of the mathematical theory of perspective as it can be learned from tracts on perspective – I might well be tempted to begin my account with Alberti. It does not seem right, however, to ignore the early history completely, so I have dedicated the remainder of this chapter to discussing, rather briefly, some events preceding Alberti’s epoch-making presentation of geometrical perspective.

I.2

The Origin of Perspective

T

he challenge of representing three-dimensional objects upon two-dimensional surfaces is an ancient one. Down through the ages, artists and artisans in various civilizations have found different solutions to this problem. There has been a general tendency to adhere to one system for a period within a particular culture. Sufficient material, above all paintings, has survived to allow us to conclude that perspective was born in quattrocento Italy. Doubtless the creation was a result of earlier generations’ struggle to organize objects in space, but we have no documents that reveal the individual steps in this development. The lack of knowledge has inspired many scholars to speculate about what happened. In fact, although perspective is just one technique for representing spatial objects, its emergence has been seen as a major event in the history of mankind and has fascinated a considerable number of art historians and other intellectuals as well.1 Many reasons for the birth of perspective have been put forward. They vary from the extremely simplistic that the central role attached to man in the Renaissance fostered the idea of making a representation that uses a central point to some refined suggestions, which involve aspects of the history of art, science, technology, sociology, philosophy, psychology, and religion. Some suggestions concentrate on the intellectual conception of perspective representation, whereas others focus on the invention of a procedure for performing perspective constructions. However, none of the many discussions offers a convincingly elementary explanation of how and why perspective was conceived – and in my opinion, such an explanation cannot be found. Rather than paraphrasing the various hypotheses or attempting to make a synthesis of them, I deal with some aspects I find important to elucidate the birth of perspective.

1

Works dealing with the problem of the origin of perspective include PanofskyS 1927/1991; EdgertonS 1976; GablikS 1976, 44–45, 66–73; KempS 1978, 134–136, 146–158; AbelsS 1985, 77–94; L. GoldsteinS 1988; KubovyS 1988, 162–173; DamischS 1994; RaynaudS 1998.

3. Four Stimuli

I.3

3

Four Stimuli

A

mong the many stimuli from various fields that led to geometrical perspective I have chosen four that I consider particularly influential, and of which traces can be found in Alberti’s treatment of perspective. The four stimuli concern intellectual as well as technical issues and are, to some extent, interrelated. They are: i. The idea of reproducing an instantaneous view ii. Experiments with depicting particular lines in a composition iii. A search for mathematical rules iv. Inspiration from the theory of optics

Painting a View

F

inding an expression that precisely covers the concept I have in mind is not easy. It is linked to the idea that there should be resemblance between a motif and its pictorial representation. In the language of a later technique one might refer to it as ‘taking a picture’ (KubovyS 1988, 17). Ernst Gombrich has called it ‘the eye-witness principle’ (Gombrich & EribonS 1993, 107). I have chosen the phrase ‘reproduction of an instantaneous view’ by which I mean that everything in the composition takes place at the same time and is seen from one point of view. In using this expression I do not mean an imitation of the visual perception, but rather an imitation that gives the same visual impression as an original situation (see also pages 20, 111, and 727). While avoiding the discussion of what role the reproduction of an instantaneous view played in Greek and Roman art and how it emerged in Italy, I maintain that it was essential in the development of geometrical perspective. There are – at least – two sides to the desire to reproduce an instantaneous view: the one concerns an attempt to depict humans so they resemble living people, and the other concerns the representation of objects delimited by lines and planes, such as architectural elements. I will only deal with the latter here. One reason for wanting to draw a picture so that it reproduces an instantaneous view might be that when we draw, it can be difficult to find room for all the objects we wish to include; if indeed these objects exist within a view, it tells us that they can be depicted on a two-dimensional surface, for instance upon the glass of a window pane. Thus, a reproduction of an instantaneous view provides a method for organizing the images of spatial objects. The performance of this reproduction calls for the development of a system, including certain rules. Once such rules have been established, they can be applied for making drawings of imaginary sceneries as well. The earliest existing description of perspective representation, the account given by Alberti in his De pictura, clearly contains the idea of reproducing an

4

I. The Birth of Perspective

instantaneous view. He formulated it as the requirement that a painting should give the viewer the same impression as he would have looking at the scene from one fixed point through an open window, and also compared it with a mirror image (Alberti 1435/1972, §§12, 19, and 26).

Representation of Special Lines

I

n the following I call lines that are orthogonal to the picture plane orthogonals, and lines that are horizontal and parallel to the picture plane transversals. The idea of reproducing an instantaneous view was already present in Italy – consciously or unconsciously – in trecento, if not earlier. It took the form of various experiments with the representation of space (PanofskyS 1927/1991, 56–59). In an interview, Gombrich described this in Popperian terms as a process of trial and error (Gombrich & EribonS 1993, 106). One of the experiments concerned the drawing of orthogonals. Examples of this are seen in two of Duccio’s paintings from the beginning of the fourteenth century. In his Last Supper (figure I.2) he depicted the orthogonals in the ceiling in a special way that Erwin Panofsky called “fishbone perspective” (PanofskyS 1927/1991, 38). According to this concept (figure I.3), the images of a pair of lines in the ceiling, positioned symmetrically with respect to the

FIGURE I.2. Duccio di Buoninsegna, The Last Supper, c. 1310, Museo dell’Opera Metropolitana, Siena.

3. Four Stimuli

5

FIGURE I.3. Diagram showing some of the orthogonals in the previous figure.

image of the orthogonal at the middle of the picture, are drawn as lines that intersect each other on the line of symmetry. Notice that each pair has its own point of convergence. It was only for lines in the ceiling that Duccio applied convergence points, for instance the orthogonals in the plane of the table are drawn as parallel lines. Duccio’s Announcement of the Virgin’s Death (figure I.4) also shows a distinction between orthogonals in the ceiling and other orthogonals. In this painting, lines through the beams even converge towards one point (figure I.5). The idea of drawing all orthogonals in one plane – but not all orthogonals in the picture – so that they have one point in common was also applied by Ambrogio Lorenzetti in his Annunciation (figure I.6). In this case, the orthogonals of the floor converge towards one point (figure I.7). Later some painters took to drawing all orthogonals with one common point of intersection – the point that much later came to be known as the principal vanishing point. An example of this practice from the mid-fourteenth century is shown in figures I.8 and I.9. Apparently, in some circles it gradually became a rule, which I call the convergence rule, to represent all orthogonals so that they converge in one point. It is unclear when this happened, but it must have been some time before Alberti wrote on perspective in 1435, because in his work the convergence rule occurs as tacit knowledge – and as an important part of his

6

I. The Birth of Perspective

FIGURE I.4. Duccio di Buoninsegna, Announcement of the Virgin’s Death, c. 1310, Museo dell’Opera Metropolitana, Siena.

FIGURE I.5. Diagram showing some of the orthogonals in the previous figure.

3. Four Stimuli

7

FIGURE I.6. Ambrogio Lorenzetti, The Annunciation, 1344, La Pinacoteca Nazionale, Siena.

FIGURE I.7. Diagram showing some of the orthogonals in the previous figure. LarsenS 2003, Tillæg, 23.

8

I. The Birth of Perspective

FIGURE I.8. Detail of a miniature of Petrarch in his study, mid-fourteenth century, Biblioteca Trivulziana, Milan, Codex 905.

construction (section II.4). The rule can actually be deduced mathematically from Alberti’s model, but nobody is known to have done so before Guidobaldo del Monte published a proof of the rule in 1600. Pictures in which orthogonals occur normally also contain transversals. In almost all cases the transversals were drawn as lines parallel to the ground line – that is the lowest horizontal line in the picture.2 The fact that there was a standard for representing the orthogonals may have given inspiration also to create a system for the representation of the equidistant transversals in a floor of squared tiles. From Alberti we know that a rule did exist – one with 2

Exceptions are found in figures III.24-III.26.

3. Four Stimuli

9

FIGURE I.9. Diagram showing some of the orthogonals in the previous figure.

which he disagreed, as we shall see in figure II.12. This rule stated that the intervals between the images of three equidistant transversals should be in the ratio 3:2 with the largest interval nearest to the ground line (Alberti 1435/1972, §19; figure II.12). Similarly, most compositions with geometrical elements contain vertical lines – which I call verticals. These were, as far as I have noticed, always depicted as verticals.

10

I. The Birth of Perspective

A Search for Mathematical Rules

T

he creation of the above-mentioned laws shows that in striving to imitate a visual impression, painters did not exclusively rely upon what they observed. They used observations, for instance, when deciding on various shades of colour or depicting how fabric drapes and folds, but when organizing space they searched for mathematical rules. This development is in agreement with the Renaissance revival of Pythagorean thoughts about the world of mathematics, leading among other things to the idea that mathematics could serve the arts of painting, architecture, and music. Some of those attracted by mathematics wished to find structure and harmony in nature and were fascinated by aspects of mathematics that seemed almost mystical or divine. We find such an attraction in modern times, too; and it has actually been expressed in connection with perspective by the Danish author Karen Blixen – also known as Isak Dinesen:

A new and beautiful world revealed itself to me while I was learning perspective drawing, I was enchanted by the unshakable justice and regularity in the rules of perspective. If I myself acted correctly the outcome could not fail to be correct – but if I permitted the least negligence it invariably and with frightening power took revenge at the conclusion of the assignment.3 [Based upon ThurmanS 1982, 67]

Inspiration from Optics

O

ptics was one of the sciences developed in antiquity that received considerable attention in the late Middle Ages. This discipline provided the early perspectivists with both intellectual and technical stimuli. Intellectually optics contributed to the idea that vision should be treated scientifically, and technically it influenced the formulation of rules. From optics, the theory of perspective took over the fundamental assumption that seeing takes place along straight lines. Much of the other inspiration came from that part of optics which deals with appearance and treats the problem of how the visual impression of an object is dependent on the angle within which the object is seen. A basic hypothesis in this theory is that line segments seen within the same angle appear to be equal,4 and a fundamental result is that parallel lines appear to converge. The theory was restricted to visual appearances, and not all of its results are immediately applicable to the question of how to make representations of objects upon a plane. In some cases an application of the theory of appearances to perspective images can even lead to conclusions that conflict with the theory of

3

En ny og skøn Side af Verden aabenbarede sig for mig, da jeg lærte Perspektivtegning, Jeg blev paa en ejendommelig Maade henrevet af den urokkelige Retfærdighed og Lovbundethed i Perspektivtegningen. Hvis jeg selv bar mig rigtig ad, kunde det ikke undgaas, at Resultatet blev rigtigt, – men tillod jeg mig den mindste Skødesløshed hævnede den sig, ved Afslutningen af en Opgave, ufravigelig, og med frygtelig Kraft! [BlixenS 1950] 4 This is part of the so-called angle axiom (cf. page 38).

4. Brunelleschi

11

perspective (pages 725–726). Nevertheless, some of the results of the theory of visual appearance may have been used – correctly or incorrectly – to deduce useful rules for painting.5 The idea that perspective and optics are connected is, as we shall see, present in Alberti’s work, and also in a large part of the subsequent literature on perspective.

I.4

Brunelleschi

I

t is generally agreed that the famous Florentine goldsmith and architect Filippo Brunelleschi (1377–1446) was the first to paint genuine perspective compositions, and that this happened in the second decade of quattrocento. According to the first biography of Brunelleschi, most likely composed by Antonio Manetti around 1480, Brunelleschi presented his invention on two panels depicting, respectively, the church Santo Giovanni, later known as the Baptistery, and the Palazzo dei Signori in Florence. Unfortunately, these panels are now lost. Manetti also related that Brunelleschi’s presentation of the first panel called for a rather complicated arrangement involving an observer and a mirror (ManettiS Vita, 42–47). Thus, the observer was to look at the mirror image of the panel (figure I.10). Although Manetti did not mention this, my interpretation is that the observer was then supposed to look at the mirror image of the Baptistery and experience the marvel that there was no difference between the reflections.6 The arrangement with the mirror clearly shows that the idea of reproducing an instantaneous view must have been part of Brunelleschi’s creation. Apart from this we know nothing – neither about how he got inspired to use this kind of representation nor about which technique he applied for constructing his panels. Brunelleschi’s method of construction in particular has often been discussed, and various techniques have been suggested.

Four Possible Techniques

I

n the second half of the fifteenth century, Antonio Averlino, known as Filarete, proposed that Brunelleschi had also applied a mirror when he

5 For a more detailed discussion, see AndersenS 19872, 75–82, and appendix one, pages 723–726. 6 My reason for suggesting that the mirror image of the Baptistery, rather than the Baptistery itself, should be considered is that by this procedure both the panel and the Baptistery are mirrored and hence the likeness would be more striking. For interpretations with more technical details, in particular concerning the dimensions of the panels and their eye points, see ParronchiS 1964, 226–295; EdgertonS 1976, 125–129; KubovyS 1988, 32–39; KempS 1990, 11–15, 344–345; for further references, see ElkinsS 1994, 7, note 9.

12

I. The Birth of Perspective p

m P

b

O

B

b bm B

Bm

FIGURE I.10. Brunelleschi’s demonstration. Let p be the picture plane holding Brunelleschi’s painting of the Baptistery – on my diagram only the image b of a point B on the façade of the Baptistery is shown – and let O be the eye point of the painting, P its orthogonal projection upon p, and m a mirror placed parallel to p halfway between O and P. My interpretation is that Brunelleschi’s idea was that the viewer should initially stand with his back to the Baptistery viewing the painting of it by looking into the mirror from the back of the panel through a hole at P, then remove the panel and in his mind compare the mirror image of the panel with the mirror image of the Baptistery – and find no difference. In order to prove this procedure will work, I show that when b is the perspective image of the point B on the Baptistery, the eye at P will experience the mirror image of b as the same point as the mirror image of B. With this aim I let b′ be the orthogonal projection of b upon m, and bm the point on bb′ determined by b′bm = b′b; I call this point the mirror point of b. According to the law of reflection (cf. page 529), the reflected light from b appears to the eye at P to come from bm, and hence from any other point at the line Pbm – the assumption in perspective being that all points on a line through the eye are seen as the same point. A straightforward geometrical consideration shows that the mirror point Bm of B is situated on the line Pbm. This means that the eye at P will see the mirror image of b as the same as the mirror image of B.

constructed his panels (Filarete Arch, 657). Apparently Filarete was of the opinion that not only was copying a mirror image a source of inspiration for Brunelleschi’s creation of a perspective composition, but it was also a means in the construction – and a device in the presentation. On the other hand, about a hundred years later the painter and writer Giorgio Vasari claimed that Brunelleschi applied a plan and elevation technique (VasariS 1550/1987, vol. 1, 136).

4. Brunelleschi

13

In modern times, Martin Kemp has listed six possible ways in which Brunelleschi may have constructed his panels (KempS 1990, 344–345). His favourite solution, which later was supported by Jehane Kuhn, is that Brunelleschi’s perspective compositions originated from his engagement in measuring buildings from a distance. Brunelleschi could have kept a fixed eye point for his enterprise and used measuring rods situated in one plane – corresponding to a picture plane. Besides calculating the dimensions of the buildings from the measures taken, he may also have got the idea to draw a picture based on his measurements (KempS 1978, 144–146; KempS 1990, 345; KuhnS 1990, 118–122). If I understand Kemp and Kuhn correctly, their suggestion is that Brunelleschi applied measuring rods to locate the image points in the picture plane. This technique is similar to the basic technique behind several perspective instruments. In working out his ideas on painters’ use of optical instruments, the painter David Hockney became interested in Brunelleschi’s panels. He reconstructed the situation described by Manetti, reaching the conclusion that it is possible Brunelleschi used a lens for painting his panel of the Baptistery (HockneyS 20011, 286 and HockneyS 20012).

Brunelleschi’s Conception of Perspective

T

he four discussed techniques of constructing a perspective image require different degrees of awareness of what a perspective representation is: a direct construction on a mirror or the use of a lens requires less awareness, and a plan and elevation method the most. Indeed, the latter seems impossible to perform unless it is based on a conception of a perspective image that is equivalent to Alberti’s model. None of the techniques demands the convergence rule (a view also expressed in KempS 1990, 345).7 This is in accordance with Kuhn’s suggestion that Brunelleschi worked independently of the painters’ tradition (KuhnS 1990, 117).

No Conclusion

T

he idea that Brunelleschi applied a technique belonging to his profession appeals to me, and I think that he was familiar with using a plan and an elevation of an object. Nevertheless, I do not claim, as among other authors Hans Schuritz, that Brunelleschi applied a plan and elevation method to construct his panels (SchuritzS 1919, 7), because I do not find that this is in agreement with what happened after the demonstration of the panels. It is unlikely,

7

In order for Brunelleschi’s demonstration arrangement to work (caption of figure I.10) he must have involved the projection of the eye point upon the panel, but that does not necessarily mean he also conceived of it as a convergence point for the images of orthogonals.

14

I. The Birth of Perspective

as noted, that Brunelleschi could have applied a plan and elevation technique without having a precise geometric definition of a perspective representation. If he really had both a definition and a technique, the natural scenario would be that he would have described his findings to some of his artist friends – including Donatello and Masaccio – whom he persuaded to apply his new method of representation, and that Alberti would have heard about it. However, there is no sign that this took place (for another discussion on Brunelleschi’s method see Field, Lunardi & SettleS 1989, 69–71, 77–78). In De pictura Alberti did not mention Brunelleschi, but in 1436 he dedicated his Italian version of the work, Della pittura, to Brunelleschi. This indicates that Alberti wished to acknowledge gaining inspiration from Brunelleschi. Had he learned something of the theory and practice of perspective from Brunelleschi, Alberti would presumably have referred to this, and not merely written, as he did, that in Brunelleschi he recognized “a genius for every laudable enterprise” (Alberti 1972, 33). In this connection it is worth mentioning that Alberti’s method of constructing, presented in section II.4, is only partially based on a plan and elevation technique. Actually, the earliest known description of a plan and elevation method for perspective is written by Piero della Francesca and was presumably composed during the last third of quattrocento – more than half a century after Brunelleschi made his panels. Having dismissed the idea that Brunelleschi developed a precise geometrical construction of a perspective image, I am left with the problem of what he actually did, if not that? The answer is that I do not know, but I find it likely that he applied a technique that was based on some experience. He could have found this by looking in a mirror,8 but I do not think he drew his panel on a mirror. Had that been the case, it is most probable, as Kemp states, that Manetti would have mentioned this (KempS 1990, 345). Nor do I think he applied a lens. It should be noted that although Brunelleschi’s panels were much admired, we cannot be sure they were drawn as exact central projections of the depicted objects.

Brunelleschi’s Success

W

hatever Brunelleschi’s motivation and technique, he initiated a development that would soon have a profound impact on the art world. Within a rather short span of time perspective representation was taken up by a group of Florentine artists. The trend soon migrated to other parts of Italy, and over the following century it travelled North of the Alps. As a result, by the end of the sixteenth century perspectival representation had become a paradigm – and so it would remain for the next three centuries. 8

If one looks in a mirror from the side (to avoid one’s own reflection) it is possible to observe, for instance, the convergence of the mirror images of parallel lines that are not parallel to the mirror.

5. Perspective Before the Renaissance?

I.5

B

15

Perspective Before the Renaissance?

efore leaving the theme of perspective prior to Alberti’s work from 1435, I must mention that not all historians agree that perspective is a Renaissance invention. Some believe that a theory of perspective already existed in antiquity. Two of the advocates of this hypothesis are Samuel Y. Edgerton and John White, who refer to a Renaissance rediscovery of perspective and a rebirth of pictorial space, respectively (WhiteS 1957/1987, EdgertonS 1976). In my opinion there may have been some rules for painting in antiquity, and these rules may have been similar to some of the later laws of perspective, and they may have been partly inspired by optics. However, since we have no traces at all of, or allusions to, an antique, comprehensive theory of drawing based on mathematical principles – like the one presented by Alberti – I am convinced that such a theory was an original Renaissance creation (see also appendix one).

Chapter II Alberti and Piero della Francesca

II.1

The Two Earliest Authors

T

his chapter is devoted to two of the oldest written sources on perspective constructions – and the main sources for quattrocento – composed by Leon Battista Alberti and Piero della Francesca, respectively. These authors were universally gifted men with rather different approaches to perspective. Alberti was a theorist who took a strong interest in painting, architecture, and sculpture, and in the performance of these arts. He seems, however, to have been more attracted to the theory of painting than to the practice of perspective and its geometrical foundation. Piero, on the other hand, was an artist and a mathematician who wanted to connect these two disciplines. He presented many perspective constructions in great detail and speculated about how they were related to mathematics and optics. In 1435 Alberti finished De pictura (On painting), which contains the first known definition of a perspectival representation as well as the first known presentation of a perspective construction. Some four decades later, Piero composed De prospectiva pingendi (On the perspective of painting), which is the earliest work, and the only pre-1500 work solely devoted to perspective. Piero’s opus is remarkable in many respects, one being that it offers a description of four different procedures for constructing perspective images of plane figures, reflecting an impressive development in the emerging discipline.

II.2

Alberti and His Work

E

ven for a Renaissance man, Leon Battista Alberti (1404–1472) had an unusually broad range of interests. He first attended a school in Padua, where he studied Latin and Greek, and then continued his education at Bologna University, where he concentrated on law. He paid more than ordinary attention to the quadrivium of the seven liberal arts becoming particularly engaged in mathematics and optics. In 1432 he chose to earn his living within the administration of the papal state. This job took him on several journeys in Italy. In 1428 a ban that had exiled the Alberti family from Florence was lifted. Some time later, the exact 17

18

II. Alberti and Piero della Francesca

date is unknown, Alberti went to this artistically flourishing town, where he presumably met Filippo Brunelleschi and other famous practising artisans and artists. This very likely inspired his profound interest in the theory and practice of the visual arts. In De pictura, Alberti described himself as a painter, mentioning that he painted for pleasure when his other commitments allowed time (Alberti 1435/1972, §28). None of his paintings seem to have survived, whereas he made architectural designs that can still be enjoyed in Florence, Rimini, and Mantua. Alberti’s diverse fields of interests are reflected in his publications, which include a Latin comedy in the classical style, books on philosophy, ethics, and aesthetics, treatises on practical applications of mathematics, and two works that became classics in painting and architecture. The latter, De re aedifictoria, which Alberti composed during the period 1443–1452, was inspired by Vitruvius, whereas the former, De pictura, was an innovation. Alberti also composed a tract on sculpting, De statu, which dealt more with technical than theoretical aspects.

Alberti’s Views on the Art of Painting

I

n De pictura Alberti presented his thoughts on what makes a good painting, and on how painters should be educated. He praised the art of painting and its status in antiquity, mentioning many painters from this period. He also frequently referred to classical sources – particularly to Pliny. Alberti declared that a good painting can strengthen the faith (ibid., §25). However, all his discussions of pertinent historiae concern themes from classical writers rather than religious motifs. One of Alberti’s aims in writing De pictura seems to have been to give the art of painting a kind of academic position on a par with the liberal arts – which he also thought should be part of a painter’s education:

I want the painter, as far as he is able, to be learned in all the liberal arts, but I wish him above all to have a good knowledge of geometry.1 [Alberti 1435/1972, §53]

Alberti also considered it advantageous for a painter to make “himself familiar with poets and orators and other men of letters”, finding that this would help him create his historia (ibid., §54). Perhaps Alberti advocated a more formal education for painters – similar to the programmes later created within the framework of the art academies. Or perhaps he sought to influence the patrons, wanting them to be impressed with the painters’ skills and generally demand higher standards in the paintings they bought (Kemp in Alberti 1991, 23). At any rate, Alberti hoped that at least some painters would read his work. Of those who were satisfied, he made the following request. 1

Doctum vero pictorem esse opto, quod eius fieri possit, omnibus in artibus liberalibus, sed in eo presertim geometriae peritiam desidero. [Alberti 1435/1972, §53]

3. Alberti’s Model

19

... I would especially ask them as a reward for my labours to paint my portrait in their historiae.2 [Alberti 1435/1972, §63]

II.3

Alberti’s Model

W

hile it is unclear how the history of perspective takes off with Brunelleschi, it seems fairly certain that Alberti’s De pictura not only contains the earliest known introduction to the subject, but indeed the first introduction ever written. Alberti himself actually stated he was the first to write about “this most subtle art” (ibid.). Inspired by classical geometrical optics, he began by describing a model for how to represent a three-dimensional object on a plane (on Alberti and optics, see WheelockS 1977, 26–40). From optics Alberti took over the idea that an eye can be considered as a mathematical point, and the idea that any point in front of the eye is perceived through the straight line, called the visual ray, connecting the point and the eye. He also took over the concept of a visual pyramid (figure II.1)3 – to which Euclid’s notion of a visual cone had been transformed during the Middle Ages. It consisted, according to Alberti, of all the visual rays connecting a given polygon and a given eye point, that is the pyramid that has the polygon as its base and the eye point as its apex. Working with this known material, Alberti conceived the ingenious idea of introducing a precise definition of the image, seen from a given eye point, of a given polygon upon a given picture plane situated between the eye and the polygon. Conceiving of the visual rays as being able to pass through the picture plane (figure II.2) he defined the image of the polygon as the intersection of its visual pyramid and the picture plane (ibid., §12). Alberti himself wrote that “a painting will be the intersection of a visual pyramid” (ibid.). He did not introduce the terms image and perspective image, which I nevertheless use. Alberti’s model – like all mathematical models – contains some simplification. One instance of this is the reduction of vision to something that takes place at one stationary point, when in reality seeing involves two movable eyes and a brain. Several of Alberti’s successors commented upon this problem, but never with an effect that really threatened Alberti’s model. It is still in use, although today it is described in a different language. Instead of defining the perspective image as a section of a visual pyramid, we define it as the central projection upon a picture plane (figure I.1), the centre of the projection being the eye point.

2

... hoc potissimum laborum meorum premium exposco ut faciem meam in suis historiis pingant. [Alberti 1435/1972, §63] 3 None of the known Alberti manuscripts contain diagrams, I have therefore used some drawings by later authors as well as some of my own.

20

II. Alberti and Piero della Francesca

FIGURE II.1. Abraham Bosse’s interpretation of Alberti’s visual pyramids. Bosse 1648, plate 2.

Alberti’s model has been criticized by several later artists, by art historians – most notably Erwin Panofsky – and by the well-reputed mathematician and historian of mathematics Morris Kline for not representing what the eye sees. The latter expressed this in the following way. ... the system of focussed perspective ... does not furnish a faithful reproduction of what the eye sees. [KlineS 1954, 139]

In the opinion of many writers on perspective and several other historians, myself included, this criticism is misplaced. Alberti did not aim to reproduce a visual impression, but to provide an image of an object that appears to the eye to be the same as the original object. He fulfilled this objective by assum-

3. Alberti’s Model

21

FIGURE II.2. A section of a visual pyramid. Dubreuil 1642, plate 6.

ing that the eye cannot distinguish between two different points that lie on the same visual ray (for more on perspective representation versus visual impression see pages 111 and 727).

Alberti’s Two Methods of Producing a Perspective Image

A

lberti’s model settled the question of how to conceive of a representation of a three-dimensional object upon a plane, but at the same time it created a new question: how to produce the representation. Alberti also solved this problem in De pictura, suggesting, in fact, two solutions, one of which was elegantly illustrated by the German artist Albrecht Dürer a hundred years later (figure II.3). Alberti let the picture plane be represented by “a veil

22

II. Alberti and Piero della Francesca

FIGURE II.3. Alberti’s practical method for constructing perspective images in Dürer’s illustration. The artist regards his subject through a net of squares, noticing where a visual ray to a definite point intersects the plane of the net. Then at the corresponding position on his chequered paper he draws the image of the observed point. The 1538 edition of Dürer 1525, fol. Q3v.

loosely woven”, divided into squares by thicker threads (Alberti 1435/1972, §31). This grid of squares served as a kind of coordinate system that could be used for determining where the visual rays between an eye point and an object intersected the picture plane. Obviously this technique required the draughtsman to position himself in front of an existing object, whereas Alberti’s second method, which I call an Alberti construction, can be used for throwing fictitious sceneries into perspective.

II.4

Alberti’s Construction

I

n the method involving a veil, Alberti implicitly worked with coordinates. He did the same in his geometrical procedure, although there he applied another coordinate system, namely a kind of perspective coordinates introduced as the image of a pavement of squares (figure II.4). In modern literature Alberti’s construction and derivations of it are sometimes referred to as the costruzione legittima. However, this term is also used for another construction, so in order to avoid any confusion I prefer to call it an Alberti construction. In the rest of this section, I present Alberti’s original construction in some detail.4

The Representation of Orthogonals

W

hen seeking to follow Alberti, it is useful to be aware of his implicit suppositions concerning the representations of orthogonals, transversals,

4 For a presentation with more details, see AndersenS 19871, and for other discussions of Alberti’s construction, see GraysonS 1964 and EdgertonS 1966.

4. Alberti’s Construction

23

T p L

P D O

G

C B

A R

A

F

FIGURE II.4. Alberti’s “open window”. The rectangle TLGR is the window through which a person looks at a chequered floor – the viewer being borrowed from Pieter de Hooch. To reduce the profusion of lines I have made the window opaque.

and verticals in a vertical picture plane p. Alberti took it for granted that a transversal should be depicted as a line parallel to the ground line, and a vertical as a line perpendicular to the ground line – assumptions that are implied in almost every drawing tradition. He also presupposed, as mentioned in section I.3, the convergence rule according to which all orthogonals should be drawn in p so that they pass through one point, which he called the centric point. He defined this point as the place “where the centric ray strikes” the picture plane (Alberti 1435/1972, §19), by which he meant (figure II.4) the orthogonal projection P of the eye point upon p. This point is instrumental in almost all perspective constructions, and over the years it has been given a multitude of names in various languages. Wishing to keep to one expression throughout this book, I have chosen to use the rather late English term, the principal vanishing point, introduced by Brook Taylor in 1715 (page 503).

An Open Window

A

lberti began his construction by drawing a rectangular frame for the painting “of whatever size”, and he characterized this as “an open window

24

II. Alberti and Piero della Francesca

through which the subject to be painted is seen” (Alberti 1435/1972, §19). In figure II.5 I have reconstructed Alberti’s rectangle and called the ground line GR and the top line TL.5 A comparison of Alberti’s “open window” with the configuration in figure II.4 makes it evident that he first decided on the size of the window, initially leaving its position in relation to the eye and the floor undetermined.

A Scaled Unit

N

ext, Alberti chose a line segment to represent the average height of a person (GH in figure II.5). Alberti assumed that the average height of a man is 3 braccia. This implies that one third of GH represents 1 braccio,6 or, in other words, that it is one scaled braccio. This Alberti expressed by saying that it is “proportional to the measure commonly called a braccio” (ibid.). Alberti also assumed that each side of the squares in his pavement is 1 braccio, and then used the measure of the scaled braccio to divide the ground line “into as many parts as it will hold”. In the interest of simplicity I assume that GR holds a whole number of this length (figure II.5).

FIGURE II.5. The beginning of Alberti’s construction.

5

Hoping to facilitate the reading of my diagrams, I have introduced a system whereby particular letters are used for particular points. Drawings explaining my use of letters are found in the section notes to the reader. 6 Braccio means “arm” and is a unit of measure. Its length varied greatly from place to place (ZupkoS 1981, 40–49), but the length used in Florence seems to have been just over 58 cm (EdgertonS 1976, 195). This appears to correspond fairly well with the fact that Alberti claims that an average man is 3 braccia tall.

4. Alberti’s Construction

25

Placement of the Centric Point

A

lberti then proceeded to the “centric point” P (the principal vanishing point). He wrote that this point can be placed wherever in the rectangle one wishes, although he did find that the most suitable position was near the horizontal line through the point H in figure II.5 (ibid.). He called the horizontal line through P the centric line; it corresponds to what would later be called the horizon (HZ in figure II.5). He did not stipulate where on HZ the point P should be situated, but he may have imagined a central position, located on the rectangular frame’s vertical axis of symmetry. By choosing the position of the centric point, Alberti had decided that the eye point should lie on the normal to p through P. Alberti’s choice of horizon implies that the painter was standing on the ground plane (figure II.4). This can be concluded from the fact that he let the distance between the ground line and the horizon be almost equal to the scaled average height of a person, which means that the person’s head and eyes are near the horizon, and hence that the original person and the painter have their eyes at almost the same horizontal level. Thus, they will also be standing on the same plane. All other persons standing on this plane should similarly be depicted with their eyes near the horizon – a result of which Alberti was fully aware, stating that they would “seem to be on the same plane” (ibid.). To increase the dynamic effect of their compositions, later artists often chose to position the painter’s eye level below or above that of a person standing on the ground plane, producing either a worm’s-eye view or a bird’s-eye view. Having placed the principal vanishing point, Alberti connected it with the points of division on the ground line (figure II.6). He thereby obtained the images of the orthogonals defined by the two sides of the squares. His remaining task was to determine the images of the corresponding transversals. Their positions depend upon the distance between the eye point and the principal vanishing point, hereafter called the distance (OP in figure II.4). Alberti realized this fact and introduced the distance in connection with his construction of the images of the transversals (quote next page).

The Images of the Transversals

T

he ground line GR is the image of the nearest transversal, which can be seen from O through “the open window” GRLT. Let this be the transversal through the point A (figure II.4). To promote simplicity I assume that this transversal also contains the sides of a row of squares. The question is, then, where the transversals through the points B, C, and D should be placed in the picture. Before quoting Alberti’s solution, I demonstrate how the positions of the images of the transversals can be determined by looking at a special diagram. Figure II.7 shows an elevation in the form of the vertical section through the eye point O perpendicular to the picture plane p. The latter is represented by the

26

II. Alberti and Piero della Francesca

FIGURE II.6. One of the steps in Alberti’s construction.

line segment PA′ where A′ is the point of intersection of OA and GR (figure II.4). The point A′ is the image of A, and the points Bi, Ci, and Di, in which OB, OC, and OD intersect PA′, are the images of the points B, C, and D. The images Bi, Ci, and Di define the positions of the images of the transversals containing B, C, and D. Often the line segment AD is too large to be used in a scale of one to one in the construction. Instead, at A′ parallel to AD we can place A′D′, a scaled copy of the line segment AD, scaling it down so that OB passes through B′. The points of intersection of OB′, OC′, and OD′ and p therefore also determine the points Bi, Ci, and Di. The scaling factor for the copy is A′B ′ : AB, and is equal to the one Alberti used when he introduced one scaled braccio.7 This implies that A′B′ is one scaled braccio,8 which is precisely the length into which the ground line GR is divided (figure II.6). Providing neither explanation nor any allusion to an elevation, Alberti presented the determination of the positions of the transversals of the tiled floor as follows (figure II.8, left part). I have a drawing surface on which I describe a single straight line, and this I divide into parts like those into which the ground line of the rectangle is divided. Then I place a point [O] above this line ... at the same height as the centric point is from the ground line of the rectangle, and from this point I draw lines to each of the divisions of the line. Then I determine the distance I want between the eye of the spectator and the painting, and having established the position of the intersection [P] at this distance, I effect the 7

Let F be the orthogonal projection of the eye upon the floor (figure II.7), the so-called foot, making the height of a person OF, while PA′ is the scaled height of a person. A braccio has thus been scaled down with the factor PA′ : OF, and from the similar triangles OA′B′ and OAB, with heights PA′ and OF, we deduce that A′B′ : AB = PA′ : OF. 8 The line segment AB is equal to the sides of the squares, which were supposed to have the length of one braccio; hence the length of A′B′ is one scaled braccio.

4. Alberti’s Construction O

27

P

Di Ci Bi A

F

B

C

D

A

B

C

D

FIGURE II.7. A vertical section of the configuration outlined in figure II.4. intersection with what mathematicians call a perpendicular [PA′]. ... This perpendicular will give me, at the places it cuts the other lines, the measure of what the distance should be in each case between the transverse equidistant lines [the transversals] of the pavement.9 [Alberti 1435/1972, §20 – with base line changed into ground line]

FIGURE II.8. The final steps in Alberti’s construction. 9

Habeo areolam in qua describo lineam unam rectam. Hanc divido per eas partes in quas iacens linea quadranguli divisa est. Dehinc pono sursum ab hac linea punctum unicum ... tam alte quam est in quadrangulo centricus punctus a iacente divisa quadranguli linea distans, ab hocque puncto ad singulas huius ipsius lineae divisiones singulas lineas duco. Tum quantam velim distantiam esse inter spectantis oculum et picturam statuo, atque illic statuto intercisionis loco, perpendiculari, ut aiunt mathematici, linea intercisionem ... efficio. ... Igitur haec mihi perpendicularis linea suis percisionibus terminos dabit omnis distantiae quae inter transversas aequidistantes pavimenti lineas esse debeat. [Alberti 1435/1972, §20]

28

II. Alberti and Piero della Francesca

Because the diagram resulting from Alberti’s instruction is identical to the upper part of figure II.7, I am convinced that Alberti was working with an elevation when he developed his construction of the transversals. Having found the positions of the images of the transversals, Alberti completed the image of the pavement by transferring these positions to his first drawing (figure II.8). It is worth noticing that the distance occurring in Alberti’s construction is not scaled, but the actual distance from which the constructed pavement should be seen. The distance he worked with could be of considerable length. This may explain why Alberti carried out the construction of the positions of the transversals on a separate piece of paper. Alberti himself did not make as clear a distinction between a ‘real’ floor and a small-scale model of a floor as I have done. However, the fact that his presentation starts with an introduction of a scaled braccio, and that he wrote that to him the line segment GR was “proportional to the nearest” transversal “seen on the pavement” (ibid. §19) shows that he was aware of the technique of using scaled line segments. The problem of changing units or scales is not very challenging mathematically, but descriptions of dealing with units tend to become muddled. The majority of Alberti’s successors avoided any confusion by remaining silent on the use of scaled lengths, trusting that practitioners were experienced in working with different scales. In section VII.6 I will return to the question of using scaled copies calling it the problem of scaling.

Choice of Parameters

A

s we have seen, Alberti’s construction involves a number of choices, which he made in the following order: The dimensions of the picture, a scaling factor, the height of the eye above the ground plane, the placement of the orthogonal projection of the eye upon the picture, and the distance. I apply the brief phrase parameters of a picture to denote the lengths that define the relative positions of an object to be depicted, the picture plane, and the eye. These parameters can be – and were – given values in different orders. There is some freedom of choice, but also various restrictions induced by the mathematical relation between the parameters and the requirement that a depicted composition should fall inside the field that one eye can cover in one glance (AndersenS 19871, 112–120; see also page 364).

Alberti’s Use of a Perspective Grid

T

he reason Alberti began his perspective investigations by constructing the image of squares – and the reason we see so many chequered floors in paintings from the Renaissance onwards – is not primarily an architectural one, but a mathematical one. The outcome of Alberti’s construction was a perspective grid that served, as noted, as a kind of coordinate system in the picture plane, and which could help painters organize their compositions. In De pictura Alberti gave a couple of examples of how the perspective grid can

5. Alberti’s Theoretical Reflections and His Diagonal Rule

29

be used. His first example concerns the placement of a rectangular base of a wall (Alberti 1435/1972, §33), as shown in figure II.9. Alberti also described (figure II.10) how the perspective grid can be of use in representing the image of a horizontal circle (ibid., §34).

II.5 Alberti’s Theoretical Reflections and His Diagonal Rule

M

y presentation of Alberti’s construction included an analysis of the ideas behind it and an indication of why it is correct. Alberti himself did not touch upon these points, writing that ... it would be a long, difficult, and extremely involved task to pursue all the mathematicians’ rules ...10 [Alberti 1435/1972, §16]

This statement does not imply that he was uninterested in the foundation of his method: I used to demonstrate these things [concerning the construction] at greater length to my friends with some geometrical explanation. I consider it best to omit this from these books [the three books of which De pictura consists] for reasons of brevity. I have outlined here, as a painter speaking to painters, only the first rudiments of the art of painting.11 [Alberti 1435/1972, §23]

FIGURE II.9. Alberti’s method of drawing perspective rectangles. The perspective image of a chequered floor made up of squares with sides measuring one braccio is used to place two rectangles measuring one braccio times two braccia. This figure also shows how Alberti used the diagonals in a perspective square to determine its perspective midpoint.

10 Etinem longum esset perdifficileque atque obscurissimum ... omnia mathematicorum regula prosequi. [Alberti 1435/1972, §16] 11 ... quas res tamen consuevi apud familiares prolixius quadam geometrica ratione cur ea ita essent demonstrare, quod his commentariis brevitatis causa praetermittendum censui. Hic enim sola prima picturae artis rudimenta pictor quidem pictoribus recensui. [Alberti 1435/1972, §23]

30

II. Alberti and Piero della Francesca

FIGURE II.10. Alberti’s technique for drawing the perspective image of a circle. The circle is inscribed in a grid of squares, and its image is determined as the curve, inscribed in the perspective grid, that cuts this grid at the images of the intersection points of the circle and the original grid.

Intending that his readers should learn how to perform a perspective construction, but not why it worked, Alberti decided not to burden them with mathematical arguments. He thereby initiated a way of presenting perspective that was followed by many others – particularly within the circle of artisans. Indeed, a substantial number of the treatises on perspective are manuals describing recipes for perspective constructions. There was, however, another group of authors who did not want to keep their theoretical understanding to themselves. Before 1600 this group counted as its most remarkable members Piero della Francesca, the architect Vignola, and the mathematicians Commandino, Benedetti, and Danti, all of whose work I will discuss later.

5. Alberti’s Theoretical Reflections and His Diagonal Rule

31

FIGURE II.11. Alberti’s diagonal rule.

Although Alberti did not explain the rationale behind his construction, he did offer his readers a means of checking it (figure II.11) – which I call Alberti’s diagonal rule. This consists in verifying that the images of two squares with one and only one common vertex have a common diagonal. It is difficult to figure out whether Alberti only meant the diagonal rule to test the exactitude in following his prescription, or whether he also saw it as an argument for the correctness of his method. Whatever the case, his rule is important, because it shows that Alberti was aware of the fact that perspective images of straight lines are, in general, straight lines12 – and that he realized this result also applies to lines whose images have not actually been constructed.

FIGURE II.12. A special procedure for representing a chequered floor, criticized by Alberti. The images of the orthogonals pass through P; the images of the transversals are determined by AB : BC = BC : CD = 3 : 2. The prolongation of the diagonal GF in the representation of a square is not the diagonal in the representation BCEF of the adjoining square.

12 If the line to be thrown into perspective passes through the eye point, its image is a point.

32

II. Alberti and Piero della Francesca

FIGURE II.13. Ambrogio Lorenzetti, Presentation in the Temple, 1342, Galleria degli Uffizi, Florence.

Alberti’s rule, which is also valid for the image of a pavement with equal rectangles, is not respected in some of the earlier methods used for depicting chequered floors. One example of this is found in the method, mentioned in section I.3, of drawing the images of transversals in distances that are in the fixed ratio of 3 : 2 (figure II.12). Another is found in Ambrogio Lorenzetti’s Presentation in the Temple (figure II.13). Besides providing a useful method of verification, Alberti’s diagonal rule may have inspired the creation of the perspective techniques later termed distance point constructions, which will be presented in section II.12. Alberti’s

6. The Third Dimension in Alberti’s Construction

33

check may also have had some influence on the way Piero used the diagonal in his constructions – a point to which I will return in section II.11.

II.6

The Third Dimension in Alberti’s Construction

A

lberti introduced an easy method for determining perspective heights at a given position (Alberti 1435/1972, §33). When this position is X (figure II.14) and a given height, say n braccia, is to be represented at X, Alberti argued as follows. The distance XU from X to the horizon represents the height of the eye above the ground level – set as 3 braccia – and hence XV represents n braccia if it is constructed so that XV : XU = n : 3. Alberti mentioned that this construction is based on the assumption that when two heights have a given ratio, their images have the same ratio. It is not difficult to verify this assumption, but since Alberti was not concerned with proofs, he left it to his successors to provide the argument. In section II.9 I present a formula, (ii.2), from which Alberti’s result directly follows. Although Alberti’s method of constructing perspectival heights is quite elegant, it does not seem to have appealed to later authors. At any rate, I have not come across a presentation of it in any of the well-known later treatises on perspective.

FIGURE II.14. Alberti’s construction of perspective heights.

34

II.7

II. Alberti and Piero della Francesca

Alberti’s Construction in History

A

lberti revealed nothing about the history of his construction, such as whether it reflects a procedure already practised in the workshops, or whether it had any relation to Brunelleschi’s method. In his dedication to Brunelleschi in Della pittura Alberti praised Brunelleschi, but not as a master who had taught him perspective (page 14). This leads me to conclude that Alberti was not just presenting a method he had learned from Brunelleschi. The latter may have told Alberti about the techniques he applied for designing his panels, and that may have inspired Alberti to engage in the new form of representation. It was during this process, I think, that he first conceived the idea of giving a precise definition of a representation of a three-dimensional object upon a plane. This definition alone made him very influential, because, as noted earlier, it was never changed. Next, he introduced a precise construction of the image of a pavement with square tiles, and it is my impression that this construction was his own invention. In his procedure he combined the painting tradition of applying the convergence rule with the architectural tradition of working with an elevation (to construct the images of transversals). Alberti himself was also of the opinion that he had created something new, thus remarking:

... in this difficult subject, which as far as I can see has not before been treated by anyone else ...13 [Alberti 1435/1972, §1]

It would be interesting to know whether Alberti’s construction enjoyed success in the workshops. This question cannot be answered, however, as the method used for making a perspective composition cannot be detected in the final result. My guess is that it was not used much. Alberti’s construction, or variations of it, did, however, remain known for more than a century, as treatises on perspective show. Presumably the knowledge of Alberti’s construction was kept alive through oral instructions accompanied by drawings, rather than through artists and artisans reading his directions. The later literature on perspective does not mention Alberti very often, which suggests that his scholarly work on painting was read neither by the artists nor by the mathematicians who wrote on perspective.

II.8

Piero della Francesca and His Work

T

he biographical data on Piero do not provide us with a complete overview of his life. We do know he was born in Borgo San Sepolcro (now Sansepolcro), but not when; nor do we know anything of his first education.

13

... in hac plane difficili et a nemine quod viderim alio tradita litteris materia. [Alberti 1435/1972, §1]

8. Piero della Francesca and His Work

35

He died in 1492, and because he reportedly died old, it is generally supposed that he was born in the second decade of the fifteenth century. It is certain that he was in Florence in 1439 working with (or under) the painter Domenico Veneziano (†1461). Undoubtedly this stay brought Piero into contact with the circle of Florentines interested in perspective. Like Alberti, Piero soon left Florence. He pursued a career as a painter – at least part time – and sojourned, among other places, in Ferrara, Rimini, Arezzo, Rome, and Urbino. At the same time he maintained connections to his hometown and seems to have taken part in its administration during certain periods of his life. Piero wrote three books, the earlier mentioned treatise on perspective, De prospectiva pingendi (abbreviated as De prospectiva), and two volumes on traditional mathematics, Trattato d’abaco (Abacus treatise) and Libellus de quinque corporibus regularibus (A little book on the five regular polyhedra, abbreviated as Libellus). Piero’s motivation for composing the two mathematical books is related to the undocumented part of his life. Of these three books, Trattato d’abaco (PieroS Trat) is supposed to be the earliest. It belongs to the so-called abbaco tradition, in which, despite the title, not the abacus but mathematics on an elementary level was presented for future merchants, artisans, and artists. The books in this tradition dealt primarily with arithmetic, but often also included some algebra and practical geometry as well – which is also true of Piero’s Trattato d’abaco. In addition, Piero treated some more advanced geometrical objects, like the regular and a number of the semi-regular polyhedra. He returned to these solids in Libellus (PieroS Lib), the last of his books. This work was dedicated to Guidobaldo Montefeltro, probably after he had become Duke of Urbino in 1482, so Piero most likely finished Libellus some time between 1482 and 1492 – the year he died. Presumably Libellus was originally written in Italian, then subsequently translated into Latin so it could be suitable for the library of the Duke of Urbino (DavisS 1977, 107–108). In his dedication in Libellus, Piero suggested the book be placed next to De prospectiva. This shows that the latter work was completed before the former, and this is, unfortunately, the only information we have about when De prospectiva was composed. Despite its Latin title, it was written in Italian, probably rather late in Piero’s career. A manuscript of De prospectiva still exists and has been published twice, in 1899 by Constantin Winterberg and in 1942 by Fasola (Piero 1899 and Piero 1974). Giusta Nicco Fasola considered the manuscript to be an autograph whereas Pierre Le Goff doubts that the entire document is written by Piero (Fasola in Piero 1974, 46; Le Goff in Piero 1998, 22). Piero’s text was also translated into Latin, perhaps to be presented to the Duke of Urbino, Federico Montefeltro – the father of Guidobaldo (Fasola in Piero 1974, 46; DavisS 1977, 19). If this really is the case, then De prospectiva must have been composed before Federico died in 1482. Today Piero is much more widely known in the history of art than in the history of mathematics, although his name does occur in the latter (FieldS 19951 and FieldS 19952). It most often crops up in connection with discussions of the

36

II. Alberti and Piero della Francesca

achievements of the mathematician Luca Pacioli who according to Vasari copied Piero and thereby deprived him of becoming famous for his mathematical works (VasariS 1550/1987, vol. 1, 191). It is true that Pacioli himself became famous for two books on mathematics entitled Summa de arithmetica, geometria (The sum of arithmetic, geometry) and De divina proportione (The divine proportion). These were among the first printed books on mathematics (PacioliS 1494 and PacioliS 1509). It is also true that Pacioli treated the same subjects as Piero, and that Piero’s mathematical books fell into oblivion. However, Vasari may not have been the best one to judge whether Pacioli copied Piero or just was inspired by him. I am not aware of a thorough comparative analysis of Piero’s and Pacioli’s texts, and it is outside the scope of my present project to provide such a study. I have looked at one example concerning the construction of a regular tetrahedron, and in this case Pacioli did not copy Piero, but made a construction that is much closer to the one found in Euclid’s Elements than to Piero’s.14 Regardless of how much Pacioli was influenced by Piero, a similarity between the works of the two mathematicians is not surprising because, no doubt, both authors benefited a great deal from earlier tracts – for which the material mainly dates back to Euclid, al-Khwa¯rizmı¯, and Leonardo da Pisa.15

De Prospectiva Pingendi

W

hile able to draw on a tradition for the themes of the Trattato d’abaco and Libellus, Piero was quite the pioneer when he composed De prospectiva. The better part of this work consists of descriptions of various constructions in the following style.

... then I draw Rx, which meets the diagonal Fz in the point K; then draw NP, which cuts Fz in the point L; then draw Sy intersecting the diagonal GZ in the point M. ... then I connect 11 and N, R and 13, 14 and x ...16

Although it tends to become tedious, Piero’s text is useful because it guides the reader through the entire construction, contrary to many other perspective treatises. Piero not only wanted to explain the how of perspective constructions, but – presumably influenced by his mathematical background – the why as well. In other words, he aimed to provide a scientific foundation for his subject. This

14

Euclid’s Elements, book XIII, proposition 13; PieroS Lib, fol. 24r; PacioliS 1509, part one, fol. 8r. 15 On Piero and the algebra tradition, see GiustiS 1993, on Piero and Euclid, see FolkertsS 1996, and for all aspects of Piero’s activities, see FieldS 2005. 16 Poi menerò Rx, che segarà la diagonale Fz in puncto K; poi linearò NP, che tagliarà Fz in puncto L; poi tirarò Sy, segante la diagonale GZ in puncto M; ... poi menerò 11 et N, R et 13, 14 et x ... [Piero Pros, 113].

9. The Theoretical Foundation of De Prospectiva

37

was a novel and demanding task. It is possible, as suggested by James Elkins, that Piero profited from some of the theoretical considerations Alberti did not include in his book, but did explain to his friends (page 29; ElkinsS 1987, 220). By and large, however, Piero had to create his own approach. De prospectiva is divided into three parts; in the first Piero discussed the problem of how to throw plane figures into perspective, applying a total of three different methods: an Alberti construction, a construction I call a diagonal construction, and a distance point construction. In the second part, Piero considered the images of three-dimensional figures, demonstrating how to determine the image of a vertical line segment. The third part of De prospectiva is not structured according to objects, but devoted to a method that applies to both two-dimensional and three-dimensional objects. This method is based on the plan and elevation technique. In the rest of this chapter, I describe the new aspects of Piero’s work in some detail.

II.9

The Theoretical Foundation of De Prospectiva

P

iero began De prospectiva with a number of results that he later used for justifying his perspective procedures. These results were taken from the optical theory of apparent sizes and the mathematical theory of proportions. A few times he even used this theory as a kind of smoke screen behind which he hid the real problem. His ‘proof’ of theorem 24 in the first book is one example of this: with great accuracy he proved that some triangles are similar, whereas he overlooked the fact that the similarity in itself did not prove his original statement (Piero Pros, 87).

The Angle Axiom

A

lberti had related his model of perspective drawing to optics, and Piero retained this link and elaborated on it. He particularly built upon the following assumption, which he had taken from Euclid’s Optics (figure II.15):

FIGURE II.15. According to Euclid’s theory of optics, the line segments AB and CD appear equal to the eye at O because they are seen within the same angle.

38

II. Alberti and Piero della Francesca

All line segments seen within the same angle appear to the eye to be equal, even if they are placed differently.17

Piero applied this result in the proof he presented for his Alberti construction – which I will discuss in the next section. Euclid himself formulated a more general postulate, sometimes called the angle axiom (PanofskyS 1927/1991, 36), which included the assertion that objects seen within a larger angle appear larger – and those seen within a smaller angle appear smaller (Euclid’s own formulation is quoted page 89). Although Piero did not quote this part of the postulate, he nevertheless applied it, as an example in section II.15 will show.

Foreshortening of Orthogonals and Line Segments Parallel to p

T

he theory of proportions served Piero as a means to determine the lengths of perspective images of given line segments. His technical term for throwing into perspective was to “degrade” (degradare), which I translate as foreshorten. To this verb he sometimes added proportionalmente, indicating that the foreshortening of a given line segment is determined by a proportion. In the later literature on perspective, proportions also played a central role, and I therefore find it relevant to discuss two general results before presenting Piero’s own reflections on the issue. Like Piero, I assume in the following that the picture plane p is vertical. First I address the question of how an orthogonal length is foreshortened, or to be more specific, of how a line segment that has one end point on the ground line and is perpendicular to p is foreshortened. For this purpose let us look at a profile (figure II.16) in the vertical plane in which PQ represents p and FA the ground plane; OF = h is O’s height above the ground plane, and OP = FQ = d is the distance. Finally, QA is the orthogonal length, whose foreshortening we want to determine, and QAi is its image. Let the length of QA be a and its foreshortening – the length of QAi – be f(a). The similarity of the triangles AiQA and OFA implies that f (a) : h = a : (d + a).

(ii.1)

This proportion determines f(a) – provided that d, h, and a are given. Second, I consider the foreshortening of lengths parallel to the picture plane. In figure II.17 I have illustrated the case in which the line segment BC is parallel to p, represented by KL, and situated in the horizontal plane through the eye point O. Denoting the distance between p and BC by a, the length of BC by l, and the length of its image BiCi by p(a, l) I deduce from the similar triangles OBiCi and OBC that p(a, l) : l = d : (d + a). 17

(ii.2)

Tucte le base vedute socto uno medessimo angolo, ben che le sieno diversamente poste, s’apresentano a l’ochio equali. [Piero Pros, 66]

9. The Theoretical Foundation of De Prospectiva

39

FIGURE II.16. The foreshortening of an orthogonal line segment.

K

B

Bi

FIGURE II.17. Determining the length of the image of a line segment parallel to the picture plane.

l

p(a,l)

O

Ci d

L

a

C

Further consideration of similar triangles will show that this proportion applies to all line segments that have the length l and are situated in the plane parallel to p at the distance a from it. This implies, in particular, the following result, which will frequently recur. All equal line segments that are parallel to p and have the same distance to it are depicted in equal line segments. (ii.3) Relation (ii.1) figures as a general result – with a proof – in the beginning of De prospectiva (Piero Pros, 75), whereas relation (ii.2) only occurs for special cases, in a form that can be generalized as p(a + 1, l) : p(a, l) = (d + a) : (d + a + 1).

(ii.4)

In fact, Piero considered four equidistant and equal line segments with a distance of 1 braccio and assumed that the first is situated in p. Whether he thought of these line segments as verticals or horizontals – or meant to cover both situations – is impossible to make up. First he let d = 4 braccia and listed a result that corresponds to the following (ibid., 74). p(3, l) : p(2, l) = 6 : 7 = 60 : 70, p(2, l) : p(1, l) = 5 : 6 = 70 : 84, p(1, l) : p(0, l) = 4 : 5 = 84 : 105. He also considered the case for d = 6 braccia and mentioned the corresponding result. He then stressed that the proportions depend on the distance

40

II. Alberti and Piero della Francesca

and formulated a general result for the first two line segments corresponding to (ibid., 74) p(1, l) : p(0, l) = d : (d + 1). Although relation (ii.2) was not explicitly stated by Piero, it seems to have been well known at the end of the fifteenth century. In his Summa, Luca Pacioli also referred to the above-mentioned relations. His aim was not to teach perspective, but the rule of three. In some examples he applied (ii.2) – and in others (ii.1).18 Pacioli emphasized that his problems were part of the theory of perspective, but without explaining how. Only once did he mention the picture plane, calling it an “open window”.

Piero on Visual Distortion

T

he first part of De prospectiva ends with a discussion on what would later be known as visual distortion. Offering an inconsistent argument, Piero reached the conclusion that if the picture plane is seen within an angle of less than 90˚, distortion is avoided (for a detailed discussion of Piero’s text see FieldS 1986). My impression is that Piero first decided upon a maximum angle of 90˚ and then searched for an argument to support this choice. He investigated a special situation in which the image of a line segment becomes longer than the original line segment when the visual angle is more than 90˚. This inspired him to make the ad hoc requirement that the perspective representation of a line segment ought to be, literally, foreshortened, by which he meant that the image of a line segment should be shorter than the original line segment. He was apparently convinced he could obtain an actual foreshortening if the visual angle was less than 90˚. This is not the case, however. In situations other than Piero’s example, a visual angle less than 90˚ does not guarantee an image that becomes smaller than the original.

II.10

Piero and Alberti’s Construction

P

iero’s first perspective construction in De prospectiva was a simplified Alberti construction or – to use Martin Kemp’s well-turned phrase – a condensed Albertian system (KempS 1990, 28).

Piero’s Rabatment

P

iero’s illustration of the method is redrawn, with the letters altered, in figure II.18. Not only did Piero condense the operations carried out in the construction, but he also compressed his diagram so that one of its line

18

PacioliS 1494, first part, fol. 65r –66r; reprinted with English translation in VeltmanS 19861, 426–430.

10. Piero and Alberti’s Construction

41

segments sometimes represents more than one line segment in the threedimensional space. For instance, Piero saw the square GRST in figure II.18 from three different angles. First, he let it represent a horizontal square that has to be thrown into perspective, and in which the side GR lies in the picture plane p while the side GT is orthogonal to p. Second, Piero regarded the square as part of an elevation in which A is the eye point, GT represents p, and GR an orthogonal length. By drawing the line AR, he realized that this orthogonal length is depicted as GB. Third, he considered the square GRST as part of p and marked in it, at the same level as A, the principal vanishing point P – calling this point l’occhio, the eye (Piero Pros, 76). Piero chose P so that its projection upon GR halves this line segment, but he remained open to other choices on the transversal through A. He then proceeded by drawing the lines PG and PR as the images of the orthogonal sides of the square. Finally, to complete the image GRSiTi of the square, he drew a transversal through B and found its points of intersection with PG and PR. Before commenting upon Piero’s construction, let me conclude my remarks on his method of representing three-dimensional objects. As will become clear in later chapters, some of Piero’s successors drew certain mathematical diagrams in perspective, whereas others applied a parallel projection – for more on this subject see FieldS 2004. In illustrating a truncated polyhedron, Piero applied a representation that resembles a parallel projection (figure II.19). In De prospectiva, however, as was customary in his day, he used two-dimensional representations obtained by orthogonal projections. Very often, as in figure II.18, he depicted two or three different projections in the same diagram, thus creating a superposition of various drawings of the object seen from different viewing angles.

FIGURE II.18. Piero’s Alberti construction. I have chosen the letter A to signify an Alberti point – the characteristic point of an Alberti construction. Piero Pros, figure 13 modified.

42

II. Alberti and Piero della Francesca

FIGURE II.19. Piero’s representation of a cuboctahedron, a solid with six square faces and eight triangular faces. From Trattato d’abaco, published in PieroS 1970, 232.

Piero on the Correctness of the Construction

P

iero used the following argument to prove the correctness of his Alberti construction (figure II.18). First he observed that BE and RS appear to be equal, because they are seen within the same angle (from A). He then claimed that the construction is justified if it can be proved that SiTi = BE. By manipulating similar triangles he proved that this is indeed the case. Since SiTi is the image of a horizontal line segment, whereas BE is the image of a vertical line segment (situated together with the eye point A in the vertical plane), it is not immediately clear why it is interesting to know that SiTi = BE. Piero’s argument only makes sense if we assume that he used the result (ii.3): two equal line segments that are parallel to p and situated at the same distance from it have images of equal lengths.19 In Piero’s case it is the segments ST and RS, conceived, respectively, as a transversal and a vertical segment that are equal and situated at the same distance to p. Hence, their images SiTi and EB should also be equal. Piero finalized his proof by establishing that SiTi has the correct length. A later mathematician would regard this result as inconclusive; and it is actually superfluous. Piero’s proof did not provide any reason for drawing the images of the orthogonals as the lines PG and PR through the principal vanishing point. Apparently Piero, like Alberti, took the convergence 19

One could also try to conceive of figure II.18 as a plan, which, however, renders the placement of A problematic: in a plan, A should be placed at the perpendicular bisector of GT in a position corresponding to that of P (see also FieldS 1997, 88–89).

10. Piero and Alberti’s Construction

43

rule for granted and then there is no need to prove that SiTi has the correct length – provided one assumes, as Piero did, that SiTi lies on the line through B parallel to GR. Having given this analysis of Piero’s proof, I hasten to add that I do not think it should be judged by modern standards. It took a long time for the property of the vanishing point to be understood mathematically. Before this understanding was achieved, Piero’s argument that SiTi has the right length could be seen as, if not a final proof, then at least as a strong support for the procedure used in his Alberti construction.20

Filarete and Francesco di Giorgio

P

iero’s extremely detailed treatment of perspective was unique in the fifteenth century. His approach stands in sharp contrast to two not very informative presentations of an Alberti construction, one written by Filarete and one by Francesco di Giorgio Martini. They both took up perspective constructions in their tracts on architecture, which were completed around the 1460s and 1480s, respectively.21 Filarete and Francesco did not define the concepts they were using, nor did they give a precise explanation of how the various steps in their constructions were to be performed. In fact, Francesco’s text is almost incomprehensible. Filarete’s and Francesco’s cryptic instructions are nonetheless interesting because they indicate that perspective was considered an integral part of architects’ education – but an art best learned by practising it in workshops, not through study. This impression is supported by the fact that there are so few written sources concerning the technical aspects of perspective drawing dating from the fifteenth century. Thus, although the art of perspective became widely known among artists and artisans, it was exceptional for them to pass on their knowledge in scripts. Filarete, in his work, stressed that he was dealing with difficult matters: Now you must pay attention and open the eyes of your intellect, for what I have to say is subtle and difficult to understand.22 [Filarete Arch/1965, 302]

Alberti himself had also cautioned his readers: ... he who does not understand at first acquaintance will probably never grasp it, however hard he tries.23 [Alberti 1435/1972, §22]

Piero did not mention any difficulties, apparently expecting that under his guidance, readers would come to understand not only the how of his Alberti construction, but also the why. 20

For other comments on Piero’s proof, see ElkinsS 1987; KempS 1990, 27–29; FieldS 1996, 338–349 and FieldS 1997, 87–89; OlssonS 1999. 21 Filarete Arch/1965, 302–303 or Filarete Arch 650–653; Francesco Arch, 139–140. 22 sì che attendi e apri gli occhi dell’ intelletto, ché questo che s’ha a dire sono cose scabrose e sottili a’ntendere. [Filarete Arch, 650] 23 eadem qui primo aspectu non comprehenderit, vix ullo unquam vel ingenti labore apprehendat. [Alberti 1435/1972, §22]

44

II.11

II. Alberti and Piero della Francesca

Piero’s Diagonal Construction

H

aving presented his Alberti method of depicting the image of a square, Piero proceeded with a clever method for constructing the images of other polygons. For reasons that will soon become clear, I, like Philip Jones, call this method a diagonal construction (JonesS 1947, 36). The basic idea is to describe the location of a polygon with aid of one of the diagonals in a square whose perspective image is known. Mathematicians in later times, starting with Simon Stevin, realized that the cardinal problem in perspective is to construct the image of an arbitrary point – the image of a polygon being determined by the images of its vertices. Piero did not take this approach, but in principle his method is a pointwise construction; the workings of which are illustrated in figure II.20.24 Piero also applied the diagonal method to subdivide a perspective square into smaller perspective squares (figure II.21). FIGURE II.20. Piero’s diagonal method for determining the image of a point. The position of the point K is given in relation to the square GRST, and it is required to construct the image Ki. The image GRSiTi of the square is given, and hence the image GSi of the diagonal GS is also known and so is the principal vanishing point P – the latter being the point of intersection of GTi and RSi . Piero’s procedure was as follows. He drew the line KL parallel to GR cutting the diagonal at L, and drew the orthogonals KM and LN cutting GR at M and N. He then connected the points M and N with the principal vanishing point P and let Li be the point where PN cuts the diagonal GSi. Finally, through Li he drew the line Ki Li parallel to GR cutting PM at Ki and claimed that the point Ki was the image he sought.1 Piero’s solution is based on the idea of determining the image of a given point as the point of intersection of two lines through the point. He thus constructed Ki as the point of intersection of the images of the orthogonal and the transversal through K. To do this, he first found Li as the point of intersection of the images of the diagonal and the orthogonal through L. Adaptation of a section of figure 18 in Piero Pros. 24

The observant reader may have noticed that in figure II.20, the side SR is the right leg of the angle RST, while its image RSi is the left leg of the angle RSiTi. In other words, a kind of reversing or mirroring has taken place. I call this the problem of reversing and deal with it in section VII.6.

11. Piero’s Diagonal Construction

45

FIGURE II.21. Piero’s construction of the image of a chequered floor based on the diagonal method. He assumed that the image GRSiTi of the original square GRST is given, and required that the image be subdivided into perspective squares. He first divided the line segment GR into equal segments at the points C, D, E, and F and connected these points with the principal vanishing point P. Next he drew the diagonal GSi and let the placement of the transversals be determined by the points in which the diagonal meets the lines to P. It can be noticed that the part of the diagram below the line GR only serves a didactic purpose, as it is not used in the construction. Piero Pros, figure 15 redrawn with letters altered.

The application of the diagonal procedure requires the perspective image of a square with two orthogonals as sides, and is independent of how the perspective square is obtained. The procedure can even be performed without a proper method for constructing the image of a square. One could simply draw a trapezium, like GRSiTi in figure II.20, without being concerned about how the placement of the side SiTi is related to the distance of the picture. Heinrich Wieleitner has suggested that before the emergence of the Alberti construction, a diagonal procedure was applied to subdivide a perspective square (WieleitnerS 1920, 254). In support of his hypothesis, Wieleitner submitted only mathematical arguments. As there is no historical evidence of Alberti’s procedure being preceded by a method like the one described by Wieleitner, I am sceptical about his idea. In fact, I believe that Italian artists only came to focus upon the diagonal after Alberti had presented an exact method for throwing a chequered pavement into perspective, and after he had pointed out how the construction can be tested by drawing a diagonal. Wieleitner is right in pointing out that a diagonal method was used without reference to the image of a square. However, that only arose in sixteenthcentury Germany, after Alberti and other constructions had been around for more than a hundred years. Irrespective of the uncertainty about the prehistory of diagonal constructions, it is certainly central to the first part of De prospectiva. Actually, Piero only applied his Alberti construction once; as soon as he had demonstrated how this construction results in the image of a square, he moved on with his

46

II. Alberti and Piero della Francesca

diagonal method. Moreover, he generalized his diagonal construction rendering it applicable to a situation in which the image of a rectangle is known, instead of the image of a square. At this stage a distance point construction turned up in Piero’s work.

II.12

Piero’s Distance Point Construction

ne of the perspective problems considered by Piero was the following.25 Let the perspective image of a rectangle with one side on the ground line be given, and construct the image of the square on this side (Piero Pros, 86). First he assumed that the ratio between the sides of the original rectangle was known, taking as an example the ratio 5:1. Applying a procedure similar to his diagonal method, he solved this problem easily. He then abandoned the assumption about a known ratio between the sides (ibid., 87). This makes the problem equivalent to constructing (figure II.22) the perspective image of a square on a given side GR on the ground line when the principal vanishing point P is given. Piero instructed the reader to find the required image GRSiTi by proceeding as follows. Connect P with G and R, draw the line through P parallel to GR, and make PD equal to the distance; then draw DR and let it cut PG in Ti ; finally draw the line parallel to GR through Ti and let it cut PR in Si. Here we have the earliest known intelligible description of a distance point construction26 – named after the point D, which is so crucial in the construction, and which is called the distance point because its distance to the principal vanishing point P is equal to the distance.

O

The Origin of Distance Point Constructions

F

rom where distance point methods originated is as unclear as the history of the other early perspective constructions. We can be sure that by the sixteenth century distance point constructions became popular and were the dominant method used in Italy and France, but where they came from remains a mystery. My own guess is that the first method was created in a workshop by a perspective constructor experimenting with his lines. By carrying out the diagonal check, as described by Alberti, he may have realized that the point at which the perspective diagonal intersects the horizon – point D in figure II.22 – could be used actively in the construction. Moreover, by measuring he may empirically have come to the conclusion that the length

25

For a thorough discussion of Piero’s text, see FieldS 19951. In the literature the term the distance point construction is most often used. However, as there are several variants of this method, I prefer the non-definite expression a distance point construction.

26

12. Piero’s Distance Point Construction

47

FIGURE II.22. Piero’s introduction of a distance point construction. Adaptation of Piero Pros, figure 23.

PD is equal to the distance.27 As for the name of this experimental creator, I have no suggestions. In cinquecento Vignola and Danti characterized a distance point construction as the rule of Baldassarre da Siena – better known today as Baldassare Peruzzi (1481–1536) – and added that Peruzzi had been preceded by Francesco di Giorgio (Vignola 1583, 68 and 72). In his Trattato di architettura from around the 1480s, Francesco described two methods, one of which, as mentioned, is an Alberti construction. The second procedure could be interpreted as a distance point method (figure II.23), but Francesco’s description is so vague that it is not possible to decide whether he had got the method right (Francesco Arch, 139–140).

FIGURE II.23. Francesco di Giorgio’s illustration for a perspective construction. This diagram may illustrate the beginning of the procedure of a distance point construction. However, neither Francesco’s diagram nor his text allows us to draw any final conclusion about the method applied. Francesco di Giorgio, Trattato di architettura, Ashb. 361, fol. 32v, Biblioteca Medicea Laurenziana, Florence. 27

A similar suggestion has been put forward in SchuritzS 1919, 10.

48

II. Alberti and Piero della Francesca

Similarly, since there is no known presentation of Peruzzi’s rule, we cannot be sure that this was correct either. The question of correctness is indeed relevant, because we are certain an erroneous distance point method existed at that time. Sebastiano Serlio described this method in 1545, also referring, to Peruzzi. The method Serlio presented was actually a combination of the procedures from a distance point construction and an Alberti construction (page 117). Danti often brought up Piero and praised his work, but he did not mention him in connection with a distance point construction. Nor did any of the other Renaissance perspectivists, so apparently Piero’s treatment of the method went unnoticed by his successors. One possible explanation of this is that Piero himself did not stress the power of the method. Instead of introducing it as a general construction, he merely presented the method as a means to solve one specific problem. The fact that a distance point construction only occurs once in De prospectiva has led some historians to doubt the authenticity of Piero’s description and assume it was a later addition to the manuscript (PanofskyS 1915, 36; CarterS 1970, 852). Wieleitner expressed another doubt. For him the question was not whether Piero applied a distance point construction, but whether he applied it correctly without making the same mistake as Serlio later made (WieleitnerS 1920, 255). As long as there is no conclusive argument to the contrary, I prefer to believe that Piero’s text is genuine. Since he only used his Alberti construction once, solving most of his other problems in part one of De prospectiva with the help of his diagonal construction, I do not find it strange that he used distance point construction so sparingly.

Piero on the Correctness of His Distance Point Construction

P

iero’s mastery of distance point construction went no further than its practical performance, which is why he encountered difficulties when he sought to prove its correctness. His argument – in which I have replaced his letters with the ones used in figure II.22 – runs as follows.

... the line starting in the eye D and ending in R divides the line PG in the point Ti so that the point R appears to the eye to be elevated the quantity GTi above G, as was proved in number 11.28

Piero’s mention of section 11 is a problem to which I will return. As for the proof itself, it seems that Piero wanted to demonstrate that GTi appears to have the same length as the sides of the square, that is, as the line segment GR. In arguing for this result he considered the distance point D to be an eye

28 ... la linea se parte da l’occhio ... [D] et termina in ... [R] et devide ... [PG] in puncto [Ti], sì che ... [R] se rapresenta a l’occhio levato più che ... [G] la quantità de ... [GTi] comme per la 11.ma fu provato. [Piero Pros, 87]

12. Piero’s Distance Point Construction

49

point (occhio), and hence considered figure II.22 to be an elevation – which makes no sense. The circumstance that Piero called the distance point l’occhio may have contributed to cause later authors, such as Serlio, to confuse this point with an Alberti point (A in figure II.18). Piero’s reference to section 11 of De prospectiva is not easily understandable, as Constantin Winterberg and Giusta Nicco Fasola have also pointed out. Winterberg suggested the reference should have applied to section 13 – which is Piero’s presentation of his Alberti construction – whereas Fasola thought it should have applied to section 12.29 In the caption of figure II.24 I have demonstrated that the correctness of Piero’s distance point construction can

FIGURE II.24. A proof of Piero’s distance point method. Like Piero’s contemporaries, I take it for granted that the images of orthogonals pass through the principal vanishing point P, and in the proof I use relation (ii.1) – which corresponds to a result in Piero’s section 12. Let GT = RS = a, the distance d, M the point of intersection of Ti Si and the normal to GR through the ground point Q. I have proved that Ti Si is constructed correctly if I can show that MQ is equal to the foreshortening f(a) of a. According to (ii.1), this is the case if MQ : h = a : (a+d ).

(1)

Since MQ is the height in triangle RGTi and MP the height in triangle DPTi and these triangles are similar we have MQ : MP = TiG : Ti P = GR : PD = GT : PD = a : d . Using result (2) page xxxiii, I conclude that MQ : h = MQ : PQ = a : (a+d ), which is indeed the requested relation (1).

29

Winterberg in Piero 1899, XCVI and Fasola in Piero 1974, 87.

(2)

50

II. Alberti and Piero della Francesca

be verified by means he had at his disposal – including a result from his section 12. I do not, however, pretend to be reproducing Piero’s way of thinking. As noted, distance point constructions soon became very popular, presumably because they were easy to perform in practice. Even so, for a long period they remained theoretically less transparent than other constructions. In chapter IV we shall see how Vignola also struggled in his attempts to prove that a particular distance point method was correct.

II.13

The Division Theorem

F

or later use, I would like to emphasize one of the results contained in relation (2) in the caption of figure II.24, namely Ti G : Ti P = a : d.

(ii.5)

This determination of Ti shows that: Given a point T situated at the distance a from the ground line GR on an orthogonal intersecting the picture plane in G, then a necessary and sufficient condition for a point Ti to be the image of T is that Ti lies on the line GP, joining the point of intersection G and the principal vanishing point P, and that Ti divides GP in the ratio a : d. The result is rather simple, but it is nevertheless essential because either directly or in a generalized form, it is the foundation of several perspective constructions that emerged in France and Germany during the seventeenth and eighteenth centuries, as will be shown later. The result is altogether so central to the history of perspective that I find it deserving of a name, and call it the division theorem. Most authors on perspective used a relation similar to (ii.5) without stressing its importance. In the early eighteenth century, Brook Taylor made a thorough analysis of the mathematics behind perspective constructions. He had a gift for pointing out the few results that were fundamental, and the division theorem received a prominent role in his theory – a point to which I will return in chapter X. Piero, a perspective pioneer, did not have the background for a similar feat.

II.14

H

Piero’s Treatment of the Third Dimension

aving explained how to throw polygons into perspective in the first part of De prospectiva, in the second part Piero addressed problems concerning three-dimensional objects. He began with prisms that have vertical edges, assuming that the perspective images of their bases were already known. His next question was then how to foreshorten a vertical segment whose length and placement are given. In section II.7 we saw how Alberti solved this problem by involving a proportion, which meant that a given line segment had to be divided in a given ratio.

15. The Column Problem

51

FIGURE II.25. Piero’s construction of a perspective cube. It is given that P is the principal vanishing point, and that KLMN is the perspective image of a square situated in the ground plane with the side KL parallel to the ground line. It is required to construct the image of the cube on the square. Piero’s procedure was to construct K′ and L′ on the verticals through K and L so that KK′ = LL′ = KL, and then to determine M ′ and N′ as the points of intersection of the verticals through M and N and the lines PL′ and PK′, respectively. Piero’s construction of the points K′ and L′ is based on the result (ii.3) – that two equal line segments, situated parallel to the picture plane and at the same distance from it, have images of equal length. Piero’s technique for constructing K ′ and L′ can be – and was later – applied in any situation in which the image of a line segment on a transversal like KL is given and the task is to construct the image of a vertical line segment that has an endpoint on the original of KL and is equal to it (cf. figure IV.20). Piero Pros, figure 31 modified.

Piero offered two solutions that avoided proportions, and which became standard constructions. In his presentation the first construction may seem somewhat restricted, as he only applied it to a cube with a face parallel to the picture plane (figure II.25). However, as explained in the figure caption, Piero’s method can be – and was – applied far more widely. The second method, explained in figure II.26, is completely universal, also in Piero’s presentation. Although he did not disclose this, it is based on the convergence rule for orthogonals, the convergence point being Piero’s occhio. From simple three-dimensional figures Piero moved on to more and more complicated compositions, such as the house reproduced in figure II.27, and he found several convenient ways of making constructional short cuts.

II.15

T

The Column Problem

he second part of De prospectiva concludes with a discussion concerning the relation between visual angles and perspectival representations. Here Piero discussed a problem concerning perspective images of columns.

52

II. Alberti and Piero della Francesca FIGURE II.26. Piero’s general procedure for foreshortening a vertical line segment. The point P is the principal vanishing point, GR is the ground line, and E is a given image of a point in the ground plane. It is required to place, at E, the image of a vertical line segment of a given length, say l. On an arbitrary point B on the ground line, Piero marked the line segment BC = l perpendicular to GR and connected the points B and C with P. His next steps were to draw the line through E parallel to GR, find the point of intersection F of this line and BP, draw the vertical line FH meeting PC at H, and finally draw HI parallel to GR, meeting the perpendicular to GR through E at the point I; EI, then, is the required line segment. This construction is based on the observation that at any position, a vertical line segment between PB and PC represents the height l.

FIGURE II.27. Piero’s illustration of how to throw a house into perspective. First he constructed the perspective image of the plan of the house. Then he added the appropriate heights according to the technique explained in the caption of the previous figure. Piero Pros, figure 41 with the addition of the letter P.

15. The Column Problem

53

To my knowledge, Piero was the first to treat this problem in detail, but certainly not the last. In fact, the problem was taken up quite frequently in the literature over the next few centuries, though often in a simpler version than Piero’s. I deem the problem so central to the history of perspective that I find it relevant to discuss its technical aspects. Before dealing with Piero’s text, let me present three problems related to the column problem.30 Strictly speaking, the first problem has nothing to do with the actual column problem, but since historically the column problem has been reduced, as we shall see, to the first problem, I include it here.

Equidistant Line Segments

L

et us consider the simple situation (figure II.28) in which the four equidistant points I, J, K, and L lie on a line parallel to the picture plane. For reasons of clarity I assume that the line is situated in the horizontal plane, but the arguments apply to all lines parallel to the picture plane. In a perspective mapping, the four points are depicted as the points Ii, Ji, Ki, and Li, which are also equidistant, meaning that IiJi = JiKi = KiLi.

(ii.6)

In certain situations this result has been seen as problematic by some practitioners of perspective, whereas in other contexts it has been regarded as fully acceptable. For instance, if the line segments IJ, JK, and KL are the sides in a row of tiles, the relation (ii.6) does not seem to have caused any

FIGURE II.28. Perspectival representation of equal line segments situated on a line parallel to the picture plane.

30

For another, and very illuminating, discussion see FrangenbergS 1992.

54

II. Alberti and Piero della Francesca

doubts. If, however, the line segments represent other objects or distances between objects, some have considered the equalities in (ii.6) to contradict Euclid’s theory of apparent sizes. According to this theory, the distance between I and J appears to be smaller than the distance between K and L because the angle IOJ is smaller than the angle KOL. The problem arose because it was believed that – in some cases – the equal distances IJ and KL should be depicted so that the image of IJ not only appears to be smaller, but actually is smaller than the image of KL.31 In the beginning of the seventeenth century the Dutch mathematician Simon Stevin rejected the idea that there was a conflict between the laws of perspective and perception. He pointed out that in the perspective representation, the visual angles are preserved (for instance the line segment IJ is seen within the same angle as its image IiJi), and that therefore the relation between the apparent sizes is also preserved. Thus, an eye placed at O experiences the same relation between the distances IJ, JK, and KL and their images, or in other words, the images of the equal distances deceive the eye in the same manner as the original distances do (Stevin 16051, 87). In his argument Stevin dealt with an example envisioning columns placed at the points I, J, K, and L. His choice of this particular example indicates that he was aware of an ongoing debate concerning the representation of columns. He was generally very sharp at analysing problems, and his observation regarding the preservation of visual angles is exceedingly relevant. By treating the columns as geometrical points he did, however, overlook a vital concern: the real problem connected to depicting columns is not how equal line segments are thrown into perspective, but how two-dimensional sections are thrown into perspective. This is the issue to which I now turn.

Columns on Square Bases

F

irst I look at the problem of finding the perspective images of equal and equidistant columns on square bases (figure II.29). More specifically, I consider a horizontal section through the eye point O, the picture plane (represented by HZ), and four equidistant columns with two sides parallel to the picture plane. The four equal sides AB next to the picture plane are depicted in four equal and equidistant sides AiBi, but these are not the complete images, for the sides B′C ′, B′′C ′′, and B′′′C′′′ can also be seen, and hence their images must also be taken into consideration. The further away a column is from the eye, the larger the image BiCi will be, and the smaller the space to the image

31

We encounter a related problem, although not one connected to perspective representation, in the question of how to determine the size of elevated objects. A solution was chosen for this problem in which vertical intervals that were to appear equal were seen within the same angle. Thomas Frangenberg has called this technique an optical correction (FrangenbergS 1993; see also AndersenS 1996, 4–6; section III.4 and appendix one in this book, pages 99 and 729).

55 15. The Column Problem

D

C  D

C

A

D

Ci Bi

B

Ai

A

Bi

B

Ai

A

H

Ci

C

B

Ai

B i

Ci

D

C

Bi

B

Ai

O

A

FIGURE II.29. Horizontal section of a row of columns with square bases.

Z

56

II. Alberti and Piero della Francesca

of the next column. In the chosen example, there is actually no space at all between the images of the columns after the third column. This version of the column problem will recur several times throughout the book.

Cylindrical Columns

T

he column problem most often discussed in the literature deals with how to depict cylindrical columns, and is more complicated than the previous problem. Again, I look at a horizontal section (figure II.30) through the eye point O, the picture plane (HZ), and four equidistant columns with equal circular bases whose centres lie on a line parallel to the ground line. Before the perspective image of a cylindrical column can be constructed, its visible part must be determined. The latter is given by the arc of the circle limited by the two points in which the tangents from O touch the circle, the points M and N in figure II.30. These points are depicted in the points Mi and Ni , which means that the perspective image of the column covers the interval MiNi. As before, we see that the images of the columns increase with increasing distance to the eye, whereas the distances between the columns decrease; in this example, too, the images of the last two columns overlap. The determination of the visible part of a cylinder is included in Euclid’s Optics32 and could, in principle, have been used by anyone dealing with the problem of throwing cylindrical columns into perspective. However, the authors of treatises on perspective do not seem to have embraced this construction in connection with the column problem. In 1711, ’sGravesande presented a correct construction of a perspective cylinder (page 355), but he did not mention the column problem. In fact, one rarely comes across a post-1500 tract on perspective in which the entire two-dimensional horizontal sections in the column are considered. We saw that Stevin let the sections be represented by points. This was a drastic dimensional reduction; but other authors also reduced the sections, letting them be represented by line segments. Thus, they only considered the diameters of cylindrical columns and the front of square columns in the latter case reducing the column problem to a question of representing equal and equidistant line segments.

Piero’s Considerations

I

n his treatment of the column problem, Piero was more sophisticated than his successors. As I understand him, he had cylindrical columns in mind, and did take the visible parts into account – although he did not construct them correctly. Referring to an existing doubt as to whether columns should be drawn so that they increase in size with increasing distances to the eye, he wrote:

32

In EuclidS Optics it is theorem 28.

57 15. The Column Problem

M H

N 

Mi

M

N

Mi

Ni

M Ni

Mi

N

FIGURE II.30. Horizontal section of cylindrical columns.

Ni

M

Mi

O

N

Ni

Z

58

II. Alberti and Piero della Francesca

... when one is using the true rules, some are amazed that if columns are placed on equal bases, those which are more remote from the eye have a greater thickness than those which are closest. Here I intend to demonstrate that this is so, and what you should do.33 [Translation inspired by VeltmanS 19861, 425]

Acting according to his stated intentions, Piero first constructed the images of the columns (figure II.31). He did not draw the circles, which are horizontal column sections but the diameters perpendicular to the lines joining the eye point and the centres of the circles. This is illustrated in figure II.32 for two columns, one of which is represented by the diameter AB parallel to the picture plane and the other by the diameter A′B′ perpendicular to OC ′. Piero did not explain his construction, but I interpret it as a determination of the visible part of a column. Although this determination differs from the one involving tangents – presented above – the difference between the two solutions is, in most cases, insignificant. Having decided on how the columns appear to the eye, Piero constructed the images Ai′Bi′ and AiBi, seeking to prove that the former is larger than the latter. In his argument he applied several proportions, some of which are erroneous, but he nevertheless reached the correct result, Ai′Bi′ > Ai Bi (Piero Pros, 126). Finally, he argued that the increasing thickness should be accepted because the line segment Ai′Bi′ is seen within the angle A′OB′, which is smaller than the angle AOB within which

FIGURE II.31. Piero’s version of the column problem. Piero Pros, figure 44.

33

... operando le vere ragioni se maravigliano che le colonne più remote da l’ochio venghino de più grossezza che non sono le più propinque, essendo poste sopra de equali base. Si che io intendo de dimostrare cosi essere et doverse fare. [Piero Pros, 125–126]

16. Piero’s Plan and Elevation Construction

59

B C

A

C

B

A

H

Ai

C i

Bi

Ai

Ci

Bi

Z

O

FIGURE II.32. An interpretation of Piero’s treatment of the column problem.

AiBi is seen. Here he implicitly applied the part of the angle axiom declaring that objects seen within smaller angles appear to be smaller (page 89). Piero did not comment upon the images of the intervals between the columns. His own illustration includes more than two columns (figure II.31), but he chose his parameters – among them the viewing angle – so he avoided a situation where one column blocks the view to the next column. Chapter III features an example in which Leonardo da Vinci used overlapping images of columns to question the quality of perspective representations (page 106).

II.16

Piero’s Plan and Elevation Construction

R

eaders completing the second part of Piero’s De prospectiva would have learned techniques that enabled them to construct perspective images of most of the objects used in Renaissance compositions. They were nevertheless offered a third part in which a new perspective construction was presented. This construction is based on the technique of applying a plan and an elevation to describe a three-dimensional object, and I therefore term it a plan and elevation construction. Piero stated he had added this method because of his desire to teach a method that was less abstract yet at the same time more powerful than his previous constructions (Piero Pros, 129). I will later return to these two aspects of his plan and elevation construction.

The Origin of the Plan and Elevation Technique

T

he procedures for drawing a horizontal and a vertical section of a threedimensional object are very old and had been in use for several centuries

60

II. Alberti and Piero della Francesca

before the common era. In his book on architecture Vitruvius introduced the terms ichnographia and orthographia (VitruviusS Arch, book I, chapter 2, §2). The first is a drawing of the ground plane of a building, whereas the second is a drawing of the front. It is not clear whether Vitruvius had mere sections in mind, or whether he imagined that some parts of the building were projected upon the planes of the ichnography and orthography. It is also unclear to what extent Vitruvius and his successors combined the use of the two kinds of drawings to characterize the position in space of points in the objects considered (SakarovitchS 1998, 17–67 and Lefe`vreS 2004). However, even the combined drawings did not yield exhaustive information about an object – unless it was something as simple as a cube. A complete description was obtained by elaborating on the procedure so that, instead of sections, orthogonal projections of the entire object were made upon a horizontal and a vertical plane. No distinction is made between the sections and the orthogonal projections in ordinary language, both being referred to as plan and elevation. In the following I take the liberty of restricting my use of the phrase ‘plan and elevation’ to orthogonal projections upon a horizontal and a vertical plane. We do not know when the plan and elevation technique was introduced. Presumably it evolved as a practical tool that was taught orally for quite some time before it was described in textbooks. Piero’s treatment of the technique in De prospectiva is, as far as I am aware, some of the earliest documentation of its existence. Piero did not, however, treat the subject as a novelty, but assumed that his readers were familiar with the technique. Thus, the concept of plan and elevation must have been around for some time when Piero wrote De prospectiva.

Piero’s Construction

P

iero used the plan and elevation technique to determine what corresponds to the horizontal and vertical coordinates of the perspective image of a point. In this sense his method had the same effect as the practical procedures used for perspective drawings, such as Alberti’s and Dürer’s application of a veil (figure II.3) or a system Vignola illustrated in the early sixteenth century (figure II.33). Compared with these methods, the plan and elevation construction has the advantage that the operations can be performed on a single piece of paper, and that the person drawing does not need an actual object in front of him. It is enough to have the plan and elevation of a configuration consisting of the eye, the picture plane, and either a real or imaginary object to be thrown into perspective. To give a more detailed explanation of the principle of Piero’s plan and elevation construction I have depicted a three-dimensional configuration in figure II.34.34 For the sake of clarity I take the very mathematical approach

34

For another description of Piero’s plan and elevation method and his use of it, see EvansS 1995, 150–158.

16. Piero’s Plan and Elevation Construction

61

FIGURE II.33. Perspective instrument designed by Vignola. This impressive device is based on the same idea as Alberti’s veil (figure II.3). The eye point is fixed by the hole at N. The mechanisms that make the rulers AB and CD move have two functions. First they enable the observer to choose the distance for his composition, in the illustration this is MT. Secondly, he can determine the coordinates of a particular point of the object to be thrown into perspective. These coordinates are then transferred to the drawing paper, which is divided into a grid of squares. Why Vignola placed the draughtsman in a position that does not allow him to view the object from N eludes me. In fact, the drawing he is making does not even seem to have N as its eye point. Egnatio Danti reported finding the drawing, but no comments to it, among Vignola’s papers. Vignola 1583, 60.

of assuming that the only object to be thrown into perspective is the point B. The point O is the eye point, GRLT the picture plane p, and the image of B is the point b, in which the visual ray OB pierces p. The entire configuration is projected upon two planes, a horizontal ground plane g (the plan) and a vertical plane e (the elevation). The latter is perpendicular to both p and

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g and intersects g in the line GH and p in GT. I have marked the projections of the points O, B, and b upon g and e with the indices p and e, respectively. The problem of constructing the point b can be reduced to a problem of determining the points bp and be because the positions of these points define b. This new problem can be solved in the plan and the elevation, respectively, in the following manner. The point bp is the point in which the line GR is cut by the line OpBp, the points Op and Bp being known when O and B are given. Similarly the points Oe and Be are known and the point be is the point of intersection of GT and OeBe. Various authors have used different points of reference to characterize the positions of bp and be on GR and GT. In the plan, Piero used the ground point Q, and in the elevation he used the point of intersection of e and GR – G in the diagram. The configuration can be transformed to two dimensions by turning the plane of the elevation e around the line GH into the same plane as g (figure II.35). The construction of the points bp and be can then be carried out on a piece of paper when the positions ofBp, Be, Op and Oe are known. Finally, the image b (figure II.36) can be drawn by transferring the distances Qbp and Gbe,

T e O e

be p

Be

O b L g

G

H B

Q

Op

bp

Bp R

FIGURE II.34. Plan and elevation.

Oe

T be Be

G

Op

H

Q

bp R

Bp

FIGURE II.35. The elevation turned into the plane of the plan.

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16. Piero’s Plan and Elevation Construction T

L b

be

FIGURE II.36. The image of a point determined by a plan and elevation construction.

G

Q

be

bp R

found in the previous diagram, to the lines GR and GT in the picture plane and then drawing the perpendicular to GR through bp and the perpendicular to GT through be. As an alternative to applying the perpendicular through be, Piero often constructed a second point be on the line RL by making Rbe = Gbe and then connected the two points be. He also suggested that instead of drawing all the many lines, the constructor could use threads fixed at the two projections of the eye point and mark the relevant points of intersection on rulers. He found it best to use silk threads or hairs from horse tails (Piero Pros, 130). It is worth noticing that the method just described is independent of whether B lies in the ground plane or above it. Most other constructions require the perspective image of Bp to be determined before the image of B can be found, whereas this is not the case in a plan and elevation construction. We know as little about the origin of plan and elevation constructions for determining perspective images as we do about the evolution of the plan and elevation technique. Despite a lack of irrefutable arguments I am inclined to believe that Piero himself created this perspective method. In any case, he was certainly the first to write about it. Taking it for granted that his readers knew how to decipher a plan and elevation diagram, Piero regarded his plan and elevation construction more transparent than the other constructions he had presented. He stressed this by describing the method as facile nel dimostrare et nello intendere (easy to demonstrate and understand, ibid., 129). Indeed, a comprehension of the plan and elevation technique makes it easy to see why a plan and elevation construction works. Piero himself considered the method so intuitively clear that he found it unnecessary to justify it – contrary to what had been the case for his other perspective constructions. The first figure to which Piero applied his plan and elevation construction was a square with one side parallel to the picture plane (figure II.37). This example was only meant didactically, for Piero was well aware that the image of such a square can be found much more easily using other constructions. He actually stressed that it is only advantageous to apply a plan and elevation construction to corpi più deficili (more difficult solids, ibid.). Another of Piero’s examples was a mazzocchio (figure II.38), a ring used as part of the headgear worn by higher magistrates in Piero’s days, and which he painted in The Flagellation (figure II.39). In fact, the mazzocchio became one of the popular objects in Renaissance compositions and remained so for a couple of centuries (DavisS 1980).

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II. Alberti and Piero della Francesca

O MN K L KL G K K

MN N

N O

Q M L R L

MN KL

K N

O

M L

M

MN KL

FIGURE II.37. Piero’s application of a plan and elevation construction to throw a square into perspective. He used the same letter to indicate a point, its projections upon the plan and elevation, and its perspective image – and in this example I follow him. The points O represent the eye point, the given square is KLMN, and GR represents the picture plane. The top part of the diagram is an elevation in which the elevations of the images of the square’s vertices are found and marked as KL and MN. The square occurring in the elevation is not really necessary for finding the image of a square, but it would be potentially useful for determining the image of the cube on the square. The middle part is a plan used for determining the projections of the images of the square’s vertices upon the plan. In the lower part of the diagram the image of the square is found by composition. Piero Pros, figure 45 redrawn with letters altered.

II.17

P

Piero’s Cube

iero demonstrated the potential of his plan and elevation method by dealing with a problem that was “more difficult”. It concerns the perspective image of a tilted cube in which none of the edges were horizontal or parallel to the picture plane. The workings of a plan and elevation construction for such a cube are just as straightforward as for other objects – provided the cube’s plan and elevation are known. Indeed, the difficulty lies in determining these figures, not least in Piero’s days when there was no obvious

17. Piero’s Cube

65

FIGURE II.38. A mazzocchio. Piero Pros, part of figure 51.

FIGURE II.39. Piero’s use of the mazzocchio, worn here by the man in the foreground at left. Piero della Francesca, The Flagellation, the 1450s, Galleria Nazionale delle Marche, Urbino.

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geometrical method for this. As we shall see in chapter V, there are indications that some German perspectivists used instruments to measure the distances between the vertices of an object and the planes of the plan and elevation (page 228). Piero, however, devised a theoretical solution so elegant that I find it worthy of a wider audience. Thus I present it, cautioning readers that the next few pages contain quite a bit of technical matter. Piero did not similarly warn his readers, but simply provided them with a meticulous description of what they were to do (Piero Pros, 145–148). Following his instructions is a regular tour de force, for he gave no reasons for his many steps. Therefore, rather than paraphrasing Piero’s text, I explain his diagrams, using my own notation.

Piero’s Idea iero assumed – as he almost always did – that the picture plane p is vertical. As the plan he chose g, the horizontal plane containing the lowest vertex of the tilted cube, and as the elevation he chose e, the vertical plane through this same vertex perpendicular to p. To construct the plan and elevation of the tilted cube he first considered the cube A0B0 C0D0A0′B0′C0′D0′ (figure II.40), which has one face in the plan and two faces parallel to p. By rotating this cube successively around three axes he ended up with an arbitrarily situated cube. The two first rotations are illustrated in figures II.41 and II.42, which for the sake of transparency show only the base of the cube – the rotated cubes being denoted by A0B 0C1D1A1′B1′C1′D1′ and A0 B2C2D2 A2′B 2′C2′D2′. First (figure II.41), Piero turned the cube by an angle, say u, around the edge A0B 0 so that only this edge remained in the plan, and the base became A0B0C1D1. Second (figure II.42), he turned the cube by an angle v around A0D0, after which the cube only had the point A0 in common with g – its base being A0 B2C2D2. The edge A0B2 of the twice-rotated cube was still parallel to p, so to obtain full generality Piero had to perform a third rotation around a vertical axis passing through the point A0; let this rotation be determined by an angle w. Instead, however, Piero rotated the picture plane by an angle w around LR – which yields the same result.

P

Piero’s Illustrations

W

hile it is easy to determine the plan and elevation of the cube in its initial position, the effect of the three rotations is far from obvious. To settle this question Piero characterized the various positions of the cube by projecting it orthogonally not only into the plan g and the elevation e, but also into a vertical plane h parallel to the picture plane. In order to construct the plan of the tilted cube, Piero first looked at the projection of the once-rotated cube, A0B0C1D1A1′B1′C1′D1′, upon e. This elevation is shown in the lower diagram at the left-hand side of figure II.43

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17. Piero’s Cube S e

D0 T p

H

A0

C0 B0 L

G

g C0

A0

B0

R

FIGURE II.40. Piero’s easy cube.

S e

T p

H

D1 D0

C1

G

FIGURE II.41. Stage one of Piero’s rotating process, in which the cube is turned by an angle u around A0B0.

C0

A0 u B0

R

g

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II. Alberti and Piero della Francesca S e

T p

H

D2 D1

C2 C1

g

A0 v G

B2 B0

R

FIGURE II.42. Stage two of the rotating process, in which the cube is turned by an angle v around A0D0 (in the previous figure).

in which UV is the intersection of e and g. If the page is held so the line UV is horizontal and subsequently folded along this line, then the natural position of the elevation of the once-rotated cube is obtained. The orthogonal projections of the vertices of the profile upon UV show on which lines in the plan the projections of the lines A0B0, A1′B1′, D1C1, and D1′C1′ are situated. In figure II.43 these lines appear in the lower diagram at the right-hand side. Next, Piero constructed the profile obtained by projecting the once-rotated cube upon h. This projection is illustrated in the left-hand side of the upper diagram and would reach its natural position if it were turned 90˚ around the point A0, and then folded around the line A0B0 so it became vertical. Presumably, Piero placed the profile as he did because in that position it can be constructed directly from the elevation below it. Piero described the second rotation of the cube in the right-hand side of the upper drawing in figure II.43. The diagram contains the same profile as the one at the left-hand side, which we should imagine first turned to its above-mentioned natural position, and then rotated by an angle v. During the second rotation – around the line A0D0 – each of the lines A0B0, A1B1, D1′C1′, and D1 C1 stays in the same vertical plane, which implies that the plan

17. Piero’s Cube C2

D2 C1

B1 C1

B0

B2

A2

C2

D2

h

h D1

A1

D1 D1

C1 D1

A0

C1

A0

V

D

D u e B1A1 G

B2 v

C

W

C

A0 B0

U

69

A

B g

A A D A

B w CC B B Q D

R

FIGURE II.43. Plan of the tilted cube; the projection of the eye point upon the plan is situated where the lines at the bottom converge. Adaptation of figure 52 in Piero Pros. Piero’s own drawing is reproduced in figure V.26.

of these lines remains unchanged. Hence, the lines perpendicular to UV in the lower diagram determine the plan of the lines A2′B2′, A0B2 , D2′C2′, and D2C2. By marking the points in which these lines intersect the lines parallel to UV through the vertices of the profile of the tilted cube in the upper right-hand diagram, Piero obtained the plan of the twice-rotated cube, shown in the lower right-hand side of the figure – not to drown in indices, I have removed them; logically there should have been an index 2 on all points apart from A, which is still the point A0. The third rotation – around the line through A perpendicular to the plan – has the effect that the entire plan is rotated by an angle w. As already noted, Piero rotated the picture plane instead of the plan. He let the picture plane be represented by the line GR, which makes an angle w with A′B′, and then determined the points A′, D′, A, D, B′, C′, C, and B in which the lines joining the orthogonal projection of the eye upon the plan (lying outside the diagram)

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and the vertices of the plan of the twice-rotated cube intersect GR. These points of intersection are the projections of the images of the cube’s vertices upon the plan. One might fear that constructing the elevation of the tilted cube called for another painstaking procedure, but Piero had the situation fully under control, obtaining most of the required information from the constructions already performed. He first made a diagram (figure II.44) in which GR is the ground line and GH the line of intersection of the plan g and the elevation e. To the right of GH he placed the constructed plan so that it forms an angle w with GR. Each of the vertices in the elevation e lies on a line perpendicular to GH passing through the corresponding vertex in the plan. What remains to be determined, then, is how far the vertices lie from GH, that is, the distances between the vertices of the tilted cube and the ground plane. Piero had already made a drawing establishing the distances between the twice-rotated cube and the ground plane (upper right-hand part of figure II.43). He then used that these distances remain unchanged under the third rotation – around a line perpendicular to the ground plane g. Thus, in the left-hand diagram in figure II.44 he marked the distances determined in the top right-hand part of figure II.43 and obtained the elevation ABCDA′B′C′D′. Finally he determined the elevations of the images of the vertices as the points of intersection of GR and the lines joining the projection of the eye upon the elevation (outside the diagram) and the vertices of the elevation ABCDA′B′C′D′. Having completed the plan and the elevation of the perspective image of the cube, Piero arrived at the final image through the

D C

H

D

D

C

D

C

A

A

C B

B A

A

B C D

BCAD

B

A G

B R

FIGURE II.44. Elevation of the tilted cube. Adaptation of figure 53 in Piero Pros.

18. Piero’s Anamorphoses

C

C D B C A D

FIGURE II.45. Perspective image of the tilted cube. Adaptation of figure 54 in Piero Pros.

B A

C D

D B C

A

71

D

B A

B C A D

B A

A D A D Q BC CB

normal procedure of composition (figure II.45). The level of concentration needed to follow the description of Piero’s construction is just as impressive as his achievement. His capacity for spatial abstraction was so highly developed that he could administrate the various rotations by means of simple drawings.

Piero’s Heads

P

iero also applied the plan and elevation technique, with considerable sophistication, to the problem of throwing heads into perspective (Piero Pros, 173–202). Here, too, the main difficulty is to determine the plan and elevation. The overall irregularity of a head makes it virtually impossible to construct a complete plan and elevation of it. Piero’s solution was to make a plan and elevation of a number of horizontal sections of the head (figure II.46). Piero’s procedure, although feasible, is very time-consuming, involving the determination of the coordinates of more than hundred points in the picture plane (figure II.47). The fact that Piero was prepared to carry out this construction shows a passion for accuracy.

II.18

Piero’s Anamorphoses

G

enerally speaking, an anamorphosis is a drawing that seems distorted unless it is viewed in a special way. The idea of deliberately creating an anamorphosis seems to have been taken up by Western artists and artisans in the second part of the fifteenth century.35 The art form became particularly popular during the Mannerist period, which, among other things, is characterized by a fascination with symbols reflecting a magical or mysterious cosmos.

35 The history of anamorphoses is treated, among other places, in Elffers, Leeman & SchuytS 1975; BaltrusˇaitisS 1977; CamerotaS 1987; Meyere & WeijmaS 1989; and KempS 1990, 208–212. For more literature on the subject see Pérez-Gómez & PelletierS 1995.

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II. Alberti and Piero della Francesca

FIGURE II.46. Piero’s preparation for using a plan and elevation method to construct the image of a head. He let the head be characterized by 128 points distributed over eight different horizontal planes. In the two lower diagrams Piero showed the contour lines of these eight sections – four in each diagram. The sixteen points defining each of the contour lines lie on lines radiating from a central point (which corresponds to using a polar coordinate system). Above the plans, to the left, Piero showed the eight horizontal sections on the head in profile along with the elevations of the 128 points. He connected some of the corresponding points in the plans and the elevations with vertical lines – had he included all the vertical lines his diagram would have been extremely confusing. Together the contour lines and the profile serve as a complete plan and elevation. However, to aid the imagination Piero also showed where his 128 points are situated on a head viewed en face. To illustrate his principle I have marked the representation of one particular point with small circles. In Piero’s drawing this point is denoted as 4 (indistinguishable in the reproduction), and it lies on the top section, which he called A in his text. Piero Pros, figure 64 with a few additions.

18. Piero’s Anamorphoses

73

FIGURE II.47. Piero’s final image of the head. To produce this drawing he first determined the plan and the elevation of the image of each of his 128 points (defined in figure II.46). Since it would be impossible to accommodate all these points on two axes, he assigned a pair of axes to each of his horizontal sections. To find the image of the points on section A in the previous figure, he composed the points on the vertical and horizontal lines called A. Piero Pros, figure 67 with two circles added.

The early examples of anamorphoses were pictures that appear distorted unless they are seen from a certain position. Later came other types that had to be viewed by means of special devices, such as a lens, a cylindrical mirror, or a conical mirror. In the present section I deal only with what I call perspectival anamorphoses: images that, like perspective images, are produced by a central projection. Since a perspective image has a fixed eye point, it might seem natural to have the definition of anamorphoses include perspective pictures as well, and it is in fact difficult to distinguish sharply between these two categories. However, the majority of perspective compositions do not look distorted when seen from another point than their eye point. Decisive to whether a perspective picture has an anamorphic effect is the choice of the relative positions of the eye point, the picture plane, and the object to be thrown into perspective. The choice of objects also plays a role; for instance,

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the perspective image of a row of columns parallel to the picture plane can be anamorphic (cf. section II.15). The procedure most commonly used to create a perspectival anamorphosis was to map a grid of squares upon a grid that seems distorted when not seen from its eye point, and then transfer a drawing from the original grid of squares onto the ‘distorted’ grid (figure IX.35). Although known much earlier, this construction was not thoroughly described until the seventeenth century – a point to which I will return in section IX.8. Yet long before this, Piero had devised another technique for constructing anamorphoses, and he presented three exquisite examples at the end of De prospectiva. Instead of employing a grid of squares, Piero constructed the ‘distorted’ image of an object directly. His technique was actually similar to the one he applied in his plan and elevation construction. His anamorphic examples differ from his perspective problems by having the positions of the object and the picture plane interchanged and the picture plane horizontal, but these two changes do not, in themselves, cause the anamorphic effect – which stems from Piero’s choice of the parameters for the representation. In Piero’s first example the picture plane is the surface of a table. On this surface he wished to paint a picture of a bowl that, when seen from a particular point, makes the viewer believe that a bowl actually is standing on the table. He chose a rotational symmetrical bowl, which means that all its horizontal sections are circles (figure II.48). Piero assumed it to be known that circles parallel to the picture plane (in casu the table) are depicted as circles, and using a vertical section of the bowl he determined the images of a suitable number of the horizontal circles (figures II.49 and 50). In his second example Piero used exactly the same procedure to obtain the projection of a sphere, and concluded his work by addressing a third problem that once more demonstrates his skill in applying the plan and

O

B

G

A

C

Bi

Ci R

FIGURE II.48. A vertical section of Piero’s bowl through the eye point O. It shows the axis of symmetry AB and the plane of the table (GR). The circle with centre B and radius BC is projected upon the circle with centre Bi and radius BiCi.

75

19. Piero’s Use of Perspective O

B

G

C

A

Bi

Ci

R

FIGURE II.49. Piero’s construction of the bowl anamorphosis. The figure includes an elevation of the bowl as well as the final picture. His use of the elevation is explained in figure II.48. The final picture consists of circles with their centres on the horizontal line GR. The eye point for this picture lies in the plane perpendicular to the paper through GR at a position that corresponds to the position of O in figure II.48. Seen from this point, the picture does indeed look like a standing bowl, shown separately in figure II.50. Adaptation of figure 51 in Piero Pros.

elevation technique. He decided on having a horizontal ceiling as the picture plane and a vertical ring, defined by two circles, as his object (figure II.51). This example is more complicated than the two previous ones, because the images of the circles defining the ring are no longer circles, but ellipses. To determine these ellipses Piero used a procedure that later became standard, and which consisted in constructing the images of a certain number of points on the circles. Piero actually selected points that are vertices of two regular hexadecagons. Most of his successors would draw smooth curves between the image points, whereas Piero joined them by straight lines, in reality finding the image of two regular and concentric hexadecagons. Piero’s elegant application of the plan and elevation technique to find the image of a vertex is illustrated in figure II.51, and his result is shown in figure II.52 (for a detailed discussion of Piero’s construction, see AndersenS 19923).

II.19

I

Piero’s Use of Perspective

t would have been most suitable to conclude the sections on Piero by pointing to examples in his painting that demonstrate the application of his profound insights into geometrical perspective. Regrettably, however, this

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II. Alberti and Piero della Francesca

FIGURE II.50. Piero’s bowl anamorphosis. In this drawing the arcs that are invisible have been removed. A final picture of the bowl can be drawn based upon the circular arcs depicted here.

cannot be done, because the question of the interplay between Piero’s engagement in the mathematical theory of perspective and his painting remains unsolved. One problem is that no one knows when Piero seriously embarked upon the mathematical study of perspective, and another is that the matter has not yet been investigated in any great depth. There can be no doubt that in his Flagellation (figure II.39) and in the socalled Perugia Annunciation (figure II.53) Piero used perspective elegantly to create a spatial effect. One thus gets the impression that it is possible to continue to move inwards almost to infinity along the colonnade depicted in the Annunciation. The latter and the Flagellation are, however, built up with the

19. Piero’s Use of Perspective

b p

a

ae

Ap

77

R

Q

P

G Ae A

B

O

FIGURE II.51. A circular ring hangs vertically touching the ceiling in Q (only the outer circle of the ring is depicted here). In the vertical plane, which contains the diameter QB and is perpendicular to the plane of the ring, Piero chose an eye point O. He wished to draw a picture on the ceiling (p) that, when seen from O, gives the impression that the viewer is looking at a ring hanging from the ceiling. This means that Piero had to construct the central projection from O of the ring upon the ceiling. Here I focus upon how Piero determined the image a of a point A on the circle. He did so by applying a special type of a plan and elevation technique. As the elevation, he chose the plane containing O and QB – implying that the elevation of O is also O. The elevation of A is the point Ae on QB, and just as in a normal plan and elevation construction Piero found the elevation ae of the image a as the point of intersection of OAe and p. For later use I like to emphasize that in this particular case, in which O and its elevation coincide, ae is also the perspective image of Ae. As the plan, Piero chose the picture plane – that is, the ceiling. The projections of the points O and A upon this plan are P and Ap. Since the projection PAp of the line OA upon the plan lies in p, Piero could not determine ap in the usual way (that is, as point of intersection of PAp and a line representing the picture plane). Piero solved this difficulty by applying observations corresponding to the following – keeping the observations to himself. The point a lies on the line PAp (the projection of the visual ray OA upon p), and its position on this line can be determined by looking at the line AAe. This line is parallel to the ground line GR (the line of intersection of the plane of the circle and p) and is hence depicted upon the line through ae parallel to GR (ae being the perspective image of Ae). Accordingly, Piero found a as the point of intersection of PAp and the line through ae parallel to GR.

help of orthogonals, transversals, and verticals, and do not reflect any of the more sophisticated results presented in De prospectiva. A characteristic feature in the known Piero paintings is the great care paid to details such as reflections, the lighting of a room, the entry of light through a window, and the draping and folding of fabrics. To get this right in his paintings he had no mathematical theory to aid him, but was

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II. Alberti and Piero della Francesca

FIGURE II.52. Piero’s construction of the anamorphosis of two concentric hexadecagons. To give the illusion of a regular ring, the drawing has to be seen from its eye point. Piero Pros, section of figure 80.

obliged to observe. It may be that initially observations formed the foundation of his spatial compositions as well, and that only later in his career did he work theoretically on perspective. On the other hand, the two processes may well have supplemented each other. Thus, Marie Françoise Clergeau has pointed out that it is likely that Piero applied his perspective method when drawing some of the heads found in his paintings (ClergeauS 1996). At present all that can be concluded is that Piero may have applied perspective to organize space and to draw objects like columns, arches, and probably heads, whereas objects like tilted cube do not occur in his paintings. Perhaps he found the latter too academic for a composition, or perhaps he did not master throwing them into perspective until after he had stopped painting.36 36

For other discussions of Piero’s use of perspective in his compositions, see KempS 1990, 30–35; JanhsenS 1990, 3–34; FieldS 1993; EvansS 1995, 158–167; and FieldS 1997, 80–113.

20. Piero’s Influence

79

FIGURE II.53. Piero della Francesca, Annunciation, from the polyptych Sant’Antonio late 1460s, Galleria Nazionale dell’Umbria, Perugia.

II.20

R

Piero’s Influence

egardless of how Piero used the results presented in De prospectiva, this work is a magnificent achievement. Although not complete in every detail, Piero offered a comprehensive explanation of the perspective technique and careful instruction in carrying out several methods of perspective construction. His accomplishments did not, however, receive the appreciation they deserved. Mathematical arguments of the work were too difficult for practitioners – a circumstance that repeated itself over and over again throughout the history of geometrical perspective. Mathematicians could have read De prospectiva, but they do not seem to have done so – perhaps because they found Piero’s long and detailed descriptions of how to perform a construction too boring. Since De prospectiva did not have much success when it appeared, very few copies of it were made, and then four centuries – during which the text seems to have disappeared – passed before it went through the press (Le GoffS 1991, 186). This meant that later generations did not have easy access to Piero’s work, and hence it did not generally exert much influence. De prospectiva did not go completely unnoticed, however. Pacioli referred to it several times, and Daniele

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Barbaro found it very inspirational, as we shall see in section IV.6.37 One of Pacioli’s references occurs in a letter he wrote to some of his pupils in 1509. In this letter he promised to teach them perspective based on Piero’s treatise, which he claimed to have studied carefully.38 In some of his mathematical problems Pacioli also alluded to perspective, examples having been mentioned in section II.10, but there is no known instances of Pacioli presenting a perspective construction. It is therefore impossible to say how well he had understood Piero, and yet it seems he played quite a central role in making the contents of De prospectiva known.

37

For more examples of references to Piero, see ElkinsS 1987, 221; Le GoffS 1991, 227–235; KempS 1995, 204–220; and FrangenbergS 1996. 38 PacioliS 1509, part one, fol. 23r.

Chapter III Leonardo da Vinci

III.1

Leonardo and the History of Perspective

A

s far as this book is concerned, one of Italy’s most famous painters and great masters, Leonardo da Vinci (1452–1519), presents a serious problem. On the one hand he was too important a thinker to be ignored, and on the other hand his ideas on perspective were so many – and often so perplexing – that a thorough presentation would require a much longer study than can be included here. Fortunately, many facets of Leonardo and perspective have already been discussed by competent historians.1 Keeping the extensive literature and the complications in mind, I have decided to be relatively brief on Leonardo with the somewhat blunt argument that in general his thoughts on perspective only became known after the period I consider here, hence exerting very little influence on developments. In fact, had his ideas been known, he would have contributed much more to raising doubts about perspective than to creating a mathematical understanding of the subject. Leonardo’s theoretical engagement in perspective started in the second half of the 1480s and continued almost until his death. The various investigations he made during this long period are scattered in different manuscripts. At one point he composed a Discorso on perspective as part of a general book on art. The work has since be lost, but we know about it because, among others, Leonardo himself, Benvenuto Cellini, and Giovanni Paolo Lomazzo referred to it.2 The existing comments on the Discorso do not allow us to reconstruct its contents, but one does not get the impression that a geometrical treatment of perspective was central to it. 1

In his 1924–1925 lectures, Erwin Panofsky discussed the discrepancies between the theories of vision and perspective, relating some of his examples to Leonardo’s work (PanofskyS 1927/1991, especially 79–81, 91–93). More recent presentations of Leonardo’s theories are found in AckermanS 1978, 108–114; ElkinsS 19881; KempS 1990, 44–51; FrangenbergS 1992, 10–24. In 1986 Kim Veltman published a detailed investigation of Leonardo’s contributions to perspective, including a generous number of his illustrations (VeltmanS 19861). In addition, he listed the most important studies on Leonardo’s theory of perspective appearing prior to 1986 (ibid., 462–465). 2 Leonardo 16511, §110; SteinitzS 1958, 25–27; Pedretti in Leonardo 1964, 167. 81

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Leonardo’s Trattato

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round 1530 a selection from Leonardo’s manuscripts was compiled by Leonardo’s favourite pupil, Francesco Melzi, who had inherited all his master’s manuscripts. It was given the title Trattato della pittura and is now known as Codex Urbinas 1270. Abbreviated versions of copies of Melzi’s compilation gave rise to the first printed Trattato della pittura, which was edited by Raphael du Fresne and appeared in 1651,3 in a joint volume alongside Italian versions of Alberti’s texts on painting and sculpture. Also in 1651, and perhaps even before Trattato della pittura, a French edition was issued. According to Thomas Frangenberg, Roland Fréart de Chambray, the translator of the work, put much of his own interpretation of Leonardo in the text (FrangenbergS 2004 and FrangenbergS forth). The publishing of Leonardo’s text was a sumptuous French enterprise, and thus the two first editions both appeared in Paris with illustrations by none other than Nicolas Poussin – who was, to put it mildly, chagrined at what the engraving process had done to his drawings. We will meet the French edition again in chapter IX, for it gave rise to fierce discussions about how much – or rather how little – technical perspective should be taught at the Parisian Academy of Painting (page 461). After 1651 Trattato della pittura kept appearing in various languages, whereas Leonardo’s other notes on perspective were not printed until the nineteenth century, therefore failing to influence the development I treat. Like the French edition, the first English translation from 1721 is rather free, so I have chosen to quote John Francis Rigaud’s translation from 1802 – in which the paragraphs are rearranged. Trattato della pittura contains several remarks on perspective, including the following two, the first of which is actually the opening sentence of the book. The young student should, in the first place, acquire a knowledge of perspective, to enable him to give to every object its proper dimensions.4 [Leonardo 1802, §1] Those who become enamoured of the practice of the art, without having previously applied to the diligent study of the scientific part of it, may be compared to mariners, who put to sea in a ship without rudder or compass, and therefore cannot be certain of arriving at the wished-for port. Practice must always be founded on good theory; to this, Perspective is the guide and the entrance, without which nothing can be well done.5 [ibid., §112] 3

For a detailed history of Leonardo’s Trattato della pittura, see SteinitzS 1958, in particular 18–21, 67, 82, 145–147, upon which my account is based. In Leonardo 1964, 3–6, and in Leonardo 1977, vol. 1, 12–43 Carlo Pedretti included helpful surveys. 4 Il giovane deve prima imparare prospettiva, per le misure d’ogni cosa. [Leonardo 16511, §1] 5 Quelli che s’innamorano della pratica senza la diligenza, overo scienza, per dir meglio, sono come i nocchieri ch’entrano in mare sopra nave senza timone o bussola, che mai non hanno certezza dove si vadino. Sempre la pratica deve essere edificata sopra la buona teorica, della quale la prospettiva è guida, e porta: e senza quella nienta si fà bene, cosi di pittura, come in ogn’ altra professione. [Leonardo 16511, §23]

1. Leonardo and the History of Perspective

83

FIGURE III.1. One of Leonardo’s examples of an Alberti construction. In this example the principal vanishing point is centrally placed. Based on Leonardo Manuscript A, fol. 39r – the original is reproduced in VeltmanS 19861, 74.

These strong recommendations were not followed up in the text by a clear introduction to perspective. Had Melzi wanted to include one, he may have searched in vain in his material, because to Leonardo, perspective was a rather extensive concept that defies precise definition, as we will see. It turns out, in fact, that there really was no need to describe precisely what was meant by perspective in Trattato della pittura as the work only treats the subject very superficially. Carlo Pedretti has suggested that Melzi did not wish to include much about perspective because the subject had already been dealt with in the earlier-mentioned Discorso (Leonardo 1964, 167). At any rate, Trattato della pittura contains so little on perspective that it came to play no role at all in disseminating geometrical perspective.

Leonardo’s Approach to Perspective

T

he Leonardo manuscripts show that in his early work on perspective, Leonardo was fairly close to Alberti and Piero. He worked with Alberti’s concept of a section of a pyramid consisting of visual lines, and he used an abbreviated Alberti construction similar to Piero’s. Leonardo made several illustrations of this construction – two of which are reproduced in figures III.1 and III.2. In relation to Alberti and Piero, Leonardo came up with innovations by wondering about how to reconstruct the eye point for a

FIGURE III.2. Another of Leonardo’s examples of an Alberti construction. In this one the principal vanishing point is placed outside the central square. Based on Leonardo Manuscript A, fol. 39r – the original is reproduced in VeltmanS 19861, 74.

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perspective drawing and by introducing picture surfaces that are not plane.6 The first theme later became known as inverse problems of perspective and attracted several mathematicians. In developing his ideas on perspective Leonardo seems to have lost his interest in perspective constructions (an opinion also expressed by Pedretti in Leonardo 1964, 168). Instead his attention was drawn towards the visual process. He saw it as a weakness that Alberti’s model did not take binocular vision into account. He also questioned the soundness of assuming that all visual rays are collected in one point, and this led him to investigate the function of the eye. The latter topic is not discussed in this chapter, and from his theory of vision I only include those aspects that are intimately linked with Leonardo’s ideas on perspective.7

Outline of This Chapter

A

fter presenting Leonardo’s various concepts of perspective, I focus upon how he investigated what he saw as shortcomings of perspective representations. In doing so I deal with rather simple examples, in order to make the basic problems, Leonardo was struggling with, more transparent. In addition, I analyse the problems more mathematically than he did – and was able to do. My aim is to gain some mathematical understanding of his theories of perspective; an aim that can only be partially fulfilled. Leonardo’s arguments can, to some degree, be understood mathematically, but aspects very frequently turn up that render a mathematical interpretation meaningless. In his notes, Leonardo did not distinguish between his reflections on visual appearance and on perspective representations, but I treat the two subjects separately. Moreover, I do not follow a chronological order in describing his ideas – an approach I find defensible thanks to the fact that over a prolonged period, Leonardo returned to the same problem several times, apparently without changing his ideas substantially.

III.2

Leonardo’s Various Concepts of Perspective

O

ne complication in understanding Leonardo is that often it is impossible to decide whether he was dealing with perspective projections of an object or investigating how the object presents itself to the eye. He used the term ‘perspective’ both in the sense of optics and in the sense of what is now known as ‘linear perspective’. He actually introduced the latter expression himself, as we shall see shortly, but it only became common in its present

6

VeltmanS 19861, 60–61, 151–154; KempS 1990, 45–46. For a discussion of Leonardo’s general theory of vision, see LindbergS 1976, 154–168; KempS 1977, KempS 1981, 326–332; AckermanS 1978, 122–145.

7

2. Leonardo’s Various Concepts of Perspective

85

sense after Brook Taylor published two books, both with titles featuring the expression linear perspective (Taylor 17151, Taylor 1719). The various forms of perspective Leonardo considered include: linear (liniale) perspective, perspective of colours (di colore), perspective of disappearance (di speditione), natural (naturale) perspective, accidental (accidentale) perspective, simple (semplica) perspective, and composed (composta) perspective. He did not introduce these concepts in such a way that his commentators have been able to reconstruct distinct definitions – nor have they agreed on how to understand his ideas. In the following I suggest various interpretations, but I am far from claiming to have found any ultimate answers as to what Leonardo meant.

Linear Perspective Versus Other Concepts of Perspective

I

n a note dating from around 1492, Leonardo distinguished between three categories of perspective:

There are three kinds of perspective; the first deals with the reasons of the diminution of objects as they recede from the eye, and it is known as diminishing perspective. The second contains the way in which colours vary as they recede from the eye. The third and last is concerned with the explanation of how the objects should be less clearly delimited as they are remote, and the names are these: linear perspective, perspective of colour, perspective of disappearance.8 [Leonardo 1970, §14 with minor changes]

Leonardo repeated this distinction often.9 In Trattato della pittura readers could learn about the various forms as follows. Perspective, as far as it extends in regard to painting, is divided into three principal parts; the first consists in diminution of size according to distance; the second concerns the diminution of colours in such objects; and the third treats of the diminution of the perception altogether of those objects, and of the degree of precision they ought to exhibit at various distances.10 [Leonardo 1802, §303]

Come sono di 3. nature prospective: la prima s’aste˜de j˜torno alla ragione del diminuire (e diciesi prospettiva diminuitiva) le cose che si allõtanano dall’ochio; la secõda cõtine i˜ se il modo del variare i colori che si allõtanano dall’ochio; la terza e ultima s’aste˜de alla dichiaratione come le cose devono essere meno finite quanto piu s’alontanano e nomi fieno questi. prospetiva liniale, prospetiva di colore, prospetiva di speditione. [Manuscript Ash I, fol. 170v; Leonardo 1970, §14] 9 It is written in almost the same wording in Manuscript A, fol. 98r (VeltmanS 19861, 70), also from around 1492, and three similar formulations can be found in manuscripts dating from the early 1510s (Leonardo 1970, §§15–17). 10 La prospettiva la quale si estende nella pittura si divide en tre parti principali, delle quali la prima è della diminutione che fanno le quantità de’ corpi in diverse distanze. La seconda parte è quella che tratta delle diminutione de’ colori di tali corpi. La terza è quella che diminuisce la notitia delle figure, e de’ termini che hanno essi corpi in varie distanze. [Leonardo 16511, §340] 8

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Leonardo thus separated the problems of dealing with sizes, colours, and contours of objects and gave names to each of these problems. Leonardo related these problems to painting but, as I understand him, his observations concerned the theory of appearance. It is noticeable, for instance, that there is no mention of a picture plane in the last quote. His idea seems to have been that while painting, one should be aware that objects far from the eye are seen with more faded colours and less sharp contours.11 Doubtless he meant that painters should take his observations into account when representing objects – but exactly how is an intricate question, which in the case of linear perspective is the subject of much of the discussion in this chapter. What Leonardo called perspective of colour was referred to by many later writers as aerial perspective. In this case, too, the latter term was actually introduced by Leonardo himself, who seemingly conceived of it as related to, but not identical to, perspective of colour (VeltmanS 19861, 297–302).

Natural Versus Accidental Perspective

L

eonardo introduced two additional terms to specify his ideas on perspective; these were natural and accidental perspective (Manuscript Arundel, fol. 62r and Manuscript E fol. 16r–v; Leonardo 1970, §§107–109). These concepts are quite vague, and I do not think there is one precise interpretation that can be defended as the correct one. To give an impression of Leonardo’s ideas, I nevertheless briefly describe my general comprehension of these two systems of perspective, gained by looking at the geometrical aspects of some of Leonardo’s examples. For the time being I leave out the technical detail (see page 100). Some authors employed the term perspectiva naturalis in the meaning of optics and perspectiva artificialis when they had geometrical perspective in mind (page xx). Leonardo did not follow this distinction. In his terminology, natural perspective covers visual appearance and some perspective representations. As I read Leonardo, perspective images of objects are natural if they are in some kind of correspondence (to be discussed in section III.5) with the visual appearance of the objects. Included in this correspondence seems to be the requirement that the images have to look ‘natural’ from different viewing points. At one point Leonardo defined accidental perspective as “that which is created by art” (quella ch’è fatta dall’arte, Manuscript E, 16r; Leonardo 1970, §107). There can be little doubt that here ‘art’ refers to the geometrical laws of perspective. However, accidental perspective is not identical with geometrical perspective, but rather incorporates those perspective representations

11

For more on this topic, I refer to VeltmanS 19861, 70, 278–320.

2. Leonardo’s Various Concepts of Perspective

87

that do not belong to natural perspective. As examples of accidental perspective Leonardo mentioned the perspective image of a row of equidistant columns (see page 105), and the image of a square constructed with a small distance (figure III.3). The latter indeed gives no impression of being a square when seen from an arbitrary point – and not as a part of a larger composition. Leonardo had a critical and sceptical attitude to accidental perspective, but he also saw its potential value as a basis for creating anamorphoses. Besides his distinction between natural and accidental perspective, Leonardo mentioned a practice of perspective that is a mixture (mistione) of the two (Manuscript E, fol. 16v; Leonardo 1970, §107). Unfortunately I have not been able to find an example that could explain what this concept covers.12

Composite and Simple Perspective

F

inally, Leonardo contrasted the concepts of composite and simple perspective:

Simple perspective is that which is constructed by art on a site which is equally distant from the eye in every part. Composite perspective is that which is constructed on a site in which none of the parts are equally distant from the eye.13 [Translation based on Leonardo 1970, §90 and FrangenbergS 1992, 19]

Since a surface with the property that all its points have the same distance to the eye point is a sphere, Leonardo’s description of simple perspective is geometrically equivalent to making a central projection from the eye point upon a part of a sphere. However, as a spherical canvas is generally rather impractical to position, it is not very likely that Leonardo was prepared to advocate the use of spherical canvases (see also FrangenbergS 1992, 20–22). He may, however, have wished to create a situation approximating it. In the case of a plane picture plane this would mean placing the picture plane far away from the eye point.

FIGURE III.3. An illustration with the image of a square on the right-hand side. Based on Leonardo Manuscript Arundel, fol. 62r – the original is reproduced in VeltmanS 19861, 149. 12 Frangenberg interprets it as being the same as composite or compound perspective – to be presented shortly (FrangenbergS 1992, 13). 13 La s˜eplicie prospettiva è quella che è fatta dall’ arte sopra sito equalmente distante dall’ochio con õgni sua parte, prospettiva conposta è quella che è fatta sopra sito il quale cõ nessuna sua parte è equalmente distante dall’ occhio. [Manuscript G, fol. 13v; Leonardo 1970, §90]

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In fact, some commentators have interpreted Leonardo’s description of simple perspective as meaning precisely the use of a considerable distance, which he indeed recommended a number of times, as we shall see (page 107). Accordingly they have understood composite perspective as having a short distance (PedrettiS 1963, 77; ElkinsS 19881, 193). While agreeing that the practices of using, respectively, a long and a short distance approximate the two perspectives considered by Leonardo, I find it important to keep his theoretical definition, involving a sphere, in mind, because that is presumably closely linked to his ideas on visual appearances. One might wonder why Leonardo called the opposite of simple perspective ‘composite’ perspective. The reason could be, as suggested by Pedretti and Frangenberg, that Leonardo considered it to be composed of natural and artificial, or natural and accidental, perspective (Leonardo 1964, 170; FrangenbergS 1992, 15). In Martin Kemp’s understanding composite perspective, or as he calls it ‘compound’ perspective, relies “upon a combination of foreshortening on the plane and the foreshortening of the plane itself ” (KempS 1990, 49). Some art historians have used Leonardo’s concept of simple perspective to place him in the history of curvilinear perspective (see section III.6).

III.3 Visual Appearances and Perspective Representations

I

n an attempt to gain a better understanding of Leonardo’s ideas on perspective I have selected a number of problems concerning how equal line segments appear to the eye and how the perspective images of such line segments are related (for another discussion on Leonardo’s treatment of visual theory and linear perspective, see ElkinsS 19881). Some of these problems are very central to Leonardo’s own studies, whereas others are selected for illustrative purposes. The solutions are clearly dependent on the positions of the line segments and the eye point, and when the perspectival representations are considered, the placement of the picture plane also plays a role. The basic problems relating to Leonardo’s investigations of visual appearances are the following two. Problem 1. How do two equal and parallel line segments appear to an eye? Problem 2. How do two equal line segments situated on a line appear to an eye that does not lie on the line? For perspectival representations, the relevant problem corresponding to problem 1 is the following. Problem 1′. How are two equal and parallel line segments represented upon a picture plane that is parallel to the line segments? In connection with Leonardo’s work, it is natural to examine the following two perspectival versions of problem 2.

4. Leonardo on Visual Appearances of Lengths

89

m e

f n

FIGURE III.4. Leonardo’s argument for the existence of a visual triangle. To demonstrate that the side mn of a tower and a fixed eye point form a visual triangle, Leonardo gave the following argument. A rod is placed between the eye and the tower; at a certain position the length ef on the rod covers the tower. If the rod is moved nearer to the eye, then mn will be covered by smaller and smaller lengths “and a little farther within the lines must converge in a point” (Manuscript A, fol. 37v; translation from Leonardo 1970, §57). Drawing based on the one reproduced in VeltmanS 19861, 72.

Problem 2′. How are two equal and collinear line segments represented on a picture plane that is orthogonal to the line segments? Problem 2′′. How are two equal and collinear line segments represented on a picture plane that is parallel to the line segments? I first treat the problems relating to visual appearance, then deal with the questions concerning perspective images.

III.4

Leonardo on Visual Appearances of Lengths

I

n chapter II we saw how Alberti and Piero della Francesca were inspired by the classical theory of optics. Both men worked with the concept of a visual pyramid, and Piero based several of his arguments on Euclid’s socalled angle axiom. Leonardo took over fewer of the classical ideas on optics. He accepted the concept of a visual pyramid, but not without reflection, arguing in support of its existence in the case where it was reduced to two dimensions – that is, to a visual triangle (caption of figure III.4). The angle axiom, on the other hand, did not become part of his inherited tools, instead he used his own assumptions as will be shown in the following.

Leonardo’s Axiom and the Angle Axiom

E

uclid’s angle axiom, which in current editions occurs as definition four though it is an axiom, states that:

[Euclid’s angle axiom] Magnitudes seen within a larger angle appear larger, whereas those seen within a smaller angle appear smaller, and those seen within equal angles appear to be equal. [Translated from EuclidS Optics/1959, 1]

Piero used this axiom, but only quoted the part concerning line segments seen within equal angles (pages 38 and 59). Leonardo similarly wrote down this part:

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III. Leonardo da Vinci

The small thing nearby and the large at a distance seen within equal angles will appear to be of equal size.14 [Leonardo 1970, §93]

However, I have found no example in Leonardo’s writings in which he explicitly used this axiom, and my hypothesis is that he did not really believe in it. This hypothesis, which is supported by his treatment of the column problem (page 105), gives me at least a clue to understanding some of his considerations that are otherwise quite enigmatic. Rather than the angle axiom, Leonardo often applied another result, indeed so frequently that I call it Leonardo’s axiom. He formulated it several times, for instance as follows. [Leonardo’s axiom] Among objects of equal size that which is most remote from the eye will look the smallest.15 [Leonardo 1970, §95]

Euclid’s Optics contains a similar statement: [Proposition five] Equal magnitudes at unequal distances appear unequal, and the one nearest to the eye always appear the largest. [Translated from EuclidS Optics/1959, 4]

However, Euclid’s proof, which is based on the angle axiom, does not apply to this general statement. It is likely that Euclid himself, who in most cases was very precise, had included some assumptions about the two objects, and that later copyists left them out. This assumption is supported by the fact that otherwise Euclid’s proposition five would have covered his previous theorem, stating: [Proposition four] Among equal lengths situated on the same line, those which are seen at the largest distance appear the smallest. [Translated from EuclidS Optics/1959, 3].

In its general formulation Leonardo’s axiom (like Euclid’s proposition five) is in contradiction with the angle axiom (caption of figure III.5). But as far as I am aware, Leonardo only applied his axiom for the set-up considered by Euclid in proposition four (collinear line segments) and for the case in which the line segments are opposite sides of a rectangle.

The Law of Inverse Proportionality

L

eonardo quantified ‘his’ axiom by introducing the distance between a viewed object and the eye point in the following rule.

14

La cosa piccola da presso e la grãde da lontano essendo viste dentro a equali angoli apparira˜no d’equale grandezza. [Manuscript A, fol. 8v; Leonardo 1970, §93] 15 Infra le cose d’equal grandezza quella che fia più distante dall’ochio si dimostrerà di minor figura. [Codex Atlanticus, fol. 1r; Leonardo 1970, §95] Among other places, Leonardo also stated this ‘axiom’ in Manuscript A, fol. 38r, Manuscript Forster II2, fol. 63r (Leonardo 1970, §§86 and 97) and Manuscript W, fol. 19115v (VeltmanS 19861, 124).

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4. Leonardo on Visual Appearances of Lengths

B A

C

FIGURE III.5. Contradiction between the angle axiom and Leonardo’s axiom. The line segments AB and AC are equal and the eye point O lies on BC. According to Euclid’s angle axiom AB and AC appear to be equal, whereas according to Leonardo’s axiom AB is seen as smaller than AC because it is further away from O than AC.

O

... of the things of equal size the more remote will demonstrate itself that much smaller to the extent that it is more distant.16 [VeltmanS 19861, 158]

Leonardo’s further calculations show that the phrase “to the extent that it is more distant” implied a law of inverse proportionality: A second object as far distant from the first as the first is from the eye will appear half the size of the first, though they be of the same size.17 [Leonardo 1970, §100] Linear Perspective consists in giving, by established rules, the true dimensions of objects, according to their respective distances; so that the second object be less than the first, the third than the second. ... I find by experience, that, if the second object be at the same distance from the first as the first is from the eye, though they be of the same size, the second will appear half the size of the first; and if the third be at the same distance behind the second, it will diminish two-thirds; and so on, by degrees, they will at equal distances, diminish in proportion ...18 [Leonardo 1802, §115]

As is the case with Leonardo’s axiom, this law of inverse proportionality is not meaningful in its full generality, but if the considered magnitudes are opposite sides of a rectangle it does make some sense. In this situation the law says (figure III.6) that when b and c are the distances between the eye point

16 Delle chose dequalj grãndeza la piu remota si djmosstera tanto mjnore quãnto ella sera piu distante. [Manuscript E, fol. 4r; Leonardo 1977, vol. 1, 147] 17 La cosa 2a che sia lontana dalla prima quãta la prima dall’ochio apparirà la metà minore che la prima benchè infra loro sieno di pari grandezza. [Manuscript A, fol. 8v; Leonardo 1970, §100] 18 La prospettiva lineale s’estende nell’ officio delle linee visuali à provare per misura quanto la cosa seconda è minore che la prima, e la terza che la seconda. ... Trovo per esperienza che se la cosa seconda sarà tanto distante della prima, quanto la prima è distante dall’ occhio tuo, che ben che infra loro siano di pari grandezza, la seconda fia la metà minore que la prima: e se la terza cosa sarà di pari distanza della seconda inanzi à essa, fia minore due terzi, e cosi di grado in grado per pari distanza faranno sempre diminuatione proportionata ... [Leonardo 16511, §322] Manuscript Ash I, fol. 12v contains almost the same text (Leonardo 1970, §99), another example of Leonardo applying the rule occurs in Codex Atlanticus, fol. 130v (ibid., §104).

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III. Leonardo da Vinci D

A c

O

b C

B

FIGURE III.6. The law of inverse proportionality.

O and two opposite sides AB and CD of a rectangle, and s (b,c) denotes the ratio that the sides appear to have, then s (b,c) = c : b.19

(iii.1)

This rule gave Leonardo an answer to the question I presented (page 88) as problem 1 (for other descriptions of the law of inverse proportionality, see DoesschateS 1964, 36–38 and VeltmanS 19861, 91–106). In designing examples of the relation corresponding to (iii.1) Leonardo sometimes chose the values of b and c so that they form nice ratios. As also pointed out by Rudolf Wittkower, Leonardo saw a correspondence between these nice ratios and the ratios occurring in the Pythagorean theory of harmonic musical intervals (Leonardo 1970, §102; WittkowerS 1953, 285–287). It is noteworthy that relation (iii.1) is independent of the lengths of the two sides of the rectangle. In the caption of figure III.7, I have explained that to avoid ambiguities it is natural to assume that the line segments AB and CD are opposite sides of a rectangle. Leonardo himself mentioned that they should be equal, but not that they should form a rectangle. However, they do so in his preserved drawings, such as the one reproduced in figure III.8. In some of the texts where Leonardo discussed the law of inverse proportionality he mentioned a plane,20 which may be interpreted as a plane through a vertical measuring rod or as a picture plane – I will return to the latter 19

Instead of expressing an inverse proportionality as A:B=b:a Leonardo often used the form (A − B) : A = (b − a) : b, which is in accordance with result (5) on page xxxiii. 20 For instance Manuscript Ash I, fol. 12v (Leonardo 1970, §99), Manuscript A, fol. 8v (ibid., §100), and Manuscript Forster II2, fol. 16v (ibid., §103).

4. Leonardo on Visual Appearances of Lengths FIGURE III.7. The reason for assuming that ABCD is a rectangle. In this diagram I have introduced a line segment EF, which is equal to and collinear with AB from figure III.6, and which I would like to compare with DC. The orthogonal distance between EF and the eye point O is b, like the distance between O and AB. Hence, according to (iii.1), EF and AB appear to have the same relation to CD, but this result is not in agreement with Leonardo’s axiom stating that EF appears smaller than AB (AB being nearer to the eye than EF).

93

D A

O

E

b C

B

F

possibility in section III.5. The idea that he worked with a measuring rod fits in well with his remark, quoted above from Trattato della pittura, that he had found the law by experience. If O is the eye point (figure III.9), r a measuring rod, and AB and DC two opposite sides of a rectangle at distances b and c from the eye, Leonardo may have noticed that to an eye at O, the line segment AB will appear to be covered by A′B ′ at the rod, and DC by D′C′. He may further have found by measuring that A¢B¢ : D¢C¢ = c : b,

(iii.2)

a result we could obtain by considering similar triangles. Thus, if Leonardo took A′B ′ : D′C′ as a measure for the ratio between the apparent sizes of AB and CD, (iii.2) is equivalent to (iii.1). It may be that before beginning to measure, Leonardo had discussed the law of inverse proportionality with the mathematician Luca Pacioli with

FIGURE III.8. One of Leonardo’s illustrations of the law of inverse proportionality. I read his drawing as containing two different sections: a vertical one containing the rectangle optr, and a horizontal one through the eye point f, which is looking at the rectangle. In the rectangle Leonardo drew a number of vertical line segments, and in the lower part of the diagram he showed the distances to these segments; the circle he introduced clearly shows that these distances increase in the directions away from the point b, the orthogonal projection of f upon pt. Redrawing of a figure in Leonardo Manuscript E, fol. 4r – the original is reproduced in VeltmanS 19861, 158.

a

o

p

b m

f

r

t

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III. Leonardo da Vinci r

D

A

C

B

D9 O

A9 B9 C9

FIGURE III.9. An eye at O looks at AB and DC.

whom he collaborated for some time. Pacioli himself seems to have considered the law tacit knowledge, simply using it in his calculations without explaining its origins.

Pacioli and the Law of Inverse Proportionality

P

acioli’s application of the law of inverse proportionality occurs in the part of his Summa in which he demonstrated the use of the rule of three – like he did in his problems presented in section II.9. One of Pacioli’s examples deals with the following situation (figure III.10). A person stands at o and looks at three people who are 3 braccia tall and located at the three collinear points d, c, and b. It is given that o’s orthogonal projection upon the line db is a and that oa = 6, od = 10, dc = 6, and cb = 12. After having shown how the remaining distances can be determined by using the theorem of Pythagoras, Pacioli wrote:

Now to find the diminution of the one and the other, you say as follows: If the hypotenuse ob, which is 712, gives me a height of 3, which we said is the height of the men, what will the hypotenuse od, which is 10, give me? Multiplying and 47 178 dividing you will find that it gives you 兹1 苵苵苵 Ⲑ苵苵. That is how large the person at b will 21 present himself to you ... [Translation partly based on VeltmanS 19861, 431]

Pacioli made a similar calculation for the man standing at c. After the first reading the question and answer remain puzzling. The problem does, however, make some sense if we understand it as follows. How tall does a 3 braccia tall man placed at b at a distance 172 from the eye look in comparison with an equally tall man placed at d at a distance 10 from the eye? According to the inverse proportionality law, (iii.1), the answer is s (ob,od) = od : ob = 10 : 172 .

21

Ora per trovare il digradimento de luno e laltro dirai cosi. Se la potumissa ob che e Radix di 712 me da 3 dalteza che dicemo essere la statura de gli homoni che me dara la potumissa od che e 10. multipla e parti e troverai che te darai Radix d’ 1 47 178. e tanto te sapresenta colui che sta nel ponto b ... [PacioliS 1494, Tractatus geometrie, fol. 65v; VeltmanS 19861, 431]

4. Leonardo on Visual Appearances of Lengths

95

FIGURE III.10. Pacioli on appearances. His drawing is not very realistic; ao, which he set to be 6 braccia, is drawn longer than ad, which he assumed to be 8 braccia. PacioliS 1494, Tractatus geometrie, fol. 65v.

This is exactly the relation Pacioli applied, but rather than stopping at the determination of the ratio, he multiplied this ratio by 3, the height of a man, taking this to be the apparent height of the man at d, and concluding that the 47 178 22 man at b appears to be 兹1苵苵苵 Ⲑ苵苵.

The Law of Inverse Proportionality and Euclid’s Theory

A

pparently the law of inverse proportionality contradicts the following theorem in Euclid’s Optics.

[Proposition eight] Equal and parallel magnitudes with unequal distances to the eye are not seen proportional to the distances. [Translated from EuclidS Optics/1959, 6, my emphasis]

Let O be the eye point (figure III.11) and AB and CD the two line segments, and let their distances to O be b and c. In his proof Euclid deduced a result which is equivalent to ∠ AOB : ∠DOC > c : b. D

O

A

B9

C

FIGURE III.11. Diagram to proposition eight in Euclid’s Optics.

22

c b

For another discussion of Pacioli’s calculation, see FrangenbergS 1992, 9–10.

B

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This relation is the real content of Euclid’s proposition eight, implying there is no mathematical discrepancy between this theorem and the law of inverse proportionality (see also BrownsonS 1981, 181–185). The conflict arises from a disagreement about how apparent sizes should be measured (see also MarcussenS 1992, 43–47). This issue must already have been discussed in antiquity, because it was not Euclid’s habit to tell what is not the case. Euclid insisted on using visual angles for comparisons of apparent lengths, whereas the law of inverse proportionality compares line segments. Euclid’s solution is much more sensible, because, as already indicated, the law of inverse proportionality can only be applied in rather special circumstances.

Leonardo on the Appearance of a Rectangle

L

eonardo’s formulation of the law of inverse proportionality, quoted above, occurs together with the following question.

... whether a long-extended rectangular wall with four sides and four angles will appear to the eye with upper and lower boundaries rectilinear or curvilinear.23 [Translation based on Pedretti’s in Leonardo 1977, vol. 1, 147]

In mulling this phenomenon Leonardo seems to have conceived of the rectangle as made up of an infinity of vertical line segments and then considered the apparent diminution of each of these segments. Leonardo’s manuscript does not contain any arguments, but the drawing reproduced in figure III.12 shows that he settled for straight lines. In this drawing, each of the boundary horizontal line segments of the rectangle appears as two straight-line segments. In appendix two, based on Leonardo’s own assumptions, I have proved that the mathematically correct solution is a curve. Furthermore, I have illustrated the curve for some parameters and discussed how well Leonardo’s intuitive solution approximates the mathematical one. The conclusion is that away from the two middle points the curves can be approximated well with two straight lines, although Leonardo’s lines have too deep a slope to yield the ideal approximation.

FIGURE III.12. Leonardo’s suggestion of how a rectangle appears. Redrawing of a figure in Leonardo Manuscript E, fol. 4r – the original is reproduced in VeltmanS 19861, 158. 23

sella pariete paralella dj 4 lati e 4 angholi dj lungha asstensione sidjmossterra allocchio dj termjmj superiori e inferiori rettilinj o churviljnj ... [Manuscript E, fol. 4r; Leonardo 1977, vol. 1, 147]

4. Leonardo on Visual Appearances of Lengths

97

Leonardo also thought about how the angle between the two lines representing a line depend on the distance between the eye and the rectangle, and had some ideas which actually fit in with my calculations (page 734). The fact that Leonardo believed the horizontal line segments appear to be straight lines is another indication that he was not inspired by Euclid’s Optics, which stated in the ninth proposition that a rectangle seen from a distance appears to be rounded.

The Appearance of the Vertical Boundaries

A

n exact solution to Leonardo’s problem requires more than a consideration of the diminution of the rectangle’s vertical line segments, as the law of inverse proportionality applies for any pair of opposite sides of a rectangle, irrespective of their direction. In particular, the apparent horizontal distance between two verticals does not remain constant, but diminishes with increasing distance to the eye – or in other words, vertical line segments do not appear as verticals, which Leonardo seems to have assumed when making the drawing reproduced in figure III.12. Apparently he had a long and not very high wall in mind, and for this shape his assumption that the vertical lines appear straight and the horizontal lines do not, is an acceptable approximation. However, if the ratio between the sides of the rectangle is nearer to one, it would be unacceptable. In a cryptic note alluding to a “bisangular figure with curved sides”,24 Leonardo may have touched upon the problem of representing verticals, but he did not pursue the matter. One of the first inventors of a mechanical calculator, Wilhelm Schickard, also speculated about the appearance of a rectangle, or rather of a square. It is interesting to notice, as also pointed out by Erwin Panofsky, that in 1624 Schickard claimed that a square appears to have the shape depicted in figure

FIGURE III.13. Schickard’s suggestion of how a square looks. The point G is the orthogonal projection of the viewing point upon the apparent shape BDMK. SchickardS 1624, 97.

24

Manuscript G, fol. 32r; VeltmanS 19861, 158–159.

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III.13, in which he seems to have taken into consideration that horizontal line segments also appear to be diminished (SchickardS 1624, 17; PanofskyS 1927/1991, 82–83).

The Appearance of Collinear Line Segments

A

s we saw, Euclid explicitly addressed the question, which I called problem 2, dealing with the appearance of collinear equal line segments (page 90). Based on his angle axiom, Euclid concluded that segments seen from a greater distance appear to be smaller. Leonardo came to the same conclusion, although he reached it in a different way. He considered the situation shown in figure III.14, in which the equal line segments ov and vx are seen from an eye placed in m. Based on his own axiom, he concluded that vx appears to be smaller than ov. It is my impression that Leonardo would have liked to have a means of quantification similar to the law of inverse proportion. It is certain that he speculated about the relation between the apparent sizes of the two line segments ov and vx, for he remarked “the eye hardly detects a difference between them and the reason is that it is close to them” (Manuscript A, fol. 38r; Leonardo 1970, §86). This remark is puzzling and has also caused some confusion.25 I believe Leonardo meant that the eye point m was close to the perpendicular to ox through o. Leonardo’s aim with this example was to point to a discrepancy between the laws of vision and the rules of perspective – the perspectival aspect of the example will be discussed in section III.5, where the role of the circle will also be explained (pages 102–104). Leonardo does not seem to have been very preoccupied with the problem of the appearance of two collinear equal line segments, perhaps because there was so little he could do to solve it. At any rate, I have not found much about this problem in his writings. And yet there is an example, in which he dealt with the n p

m

q r

s t X

25

V

O

FIGURE III.14. Leonardo illustration with two equal collinear line segments. Based on Leonardo Manuscript A, fol. 38r – the original is reproduced in Veltmans 19861, 73.

In his edition of Manuscript A, Charles Ravaisson-Mollien remarked that he thought that a “not” was missing so that the text should read that the eye is not close to ov and vx (Leonardo 1881, 38). Jean Paul Richter found Ravaisson-Mollien’s interpretation “quite inconceivable” (Leonardo 1970, note to §86).

4. Leonardo on Visual Appearances of Lengths

99

a e b

c

FIGURE III.15. Leonardo’s consideration of three equal line segments. Redrawing of a figure from Leonardo Manuscript C, fol. 27v.

n

f

d

following situation (figure III.15). Three equal line segments ab, bc, and cd are situated on a vertical line and observed from the point n lying on the normal to ad through b. In his comments Leonardo concluded from ‘his’ axiom that ab and bc appear to be equal, while cd appears to be smaller (Leonardo 1970, §53). This last example was often taken up in later tracts on perspective. It relates to the problem of how to decide on the size of objects that are to be placed in high positions. Several authors chose a solution that is rooted in antiquity (cf. appendix one, page 729), and of which Albrecht Dürer later made an illustration that became well known (figure III.16).26 Dürer

FIGURE III.16. Illustration from Dürer 1525, fol. Kiv. 26 For a thorough discussion of this problem and a survey of the literature about it, see FrangenbergS 1993.

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considered a case in which letters are to be affixed on a vertical plane above the level of the eye. If he were to use letters of the same size, the ones set higher up would appear to be smaller than the ones lower down. To compensate for this effect, Dürer applied the following procedure. He decided on the position of the eye point and divided the height of the wall into a number of sections that are all seen within the same angle. He then formed the letters so that their heights were equal to the height of the section upon which they were to be placed. It is quite conceivable that others got the idea of applying Dürer’s procedure to tall paintings, using different scales for various sections of a given work, but I have never seen it described. As far as I am aware, in his writings Leonardo did not discuss a problem similar to the one treated by Dürer. Some of his drawings have, however, been interpreted as illustrating the problem (for instance in VeltmanS 19861, 159). Even so, as I doubt that Leonardo was interested in visual angles, I am sceptical about such an interpretation.

III.5

Leonardo on Perspective Representations

T

his section is devoted to Leonardo’s treatment of problems that are perspectival counterparts of the problems considered in the previous section. Before presenting a few of his examples, I find it apposite to remark that I see these as a part of a programme in which Leonardo studied the relationship between the perspective images and the visual appearance of objects. It is my impression that he was only satisfied with those perspective images for which he found some kind of correspondence with the visual appearances, and that in this case he would classify the perspective images as ‘natural’ ones. However, exactly which correspondence he had in mind is unclear. As we shall see, in one special case he found the following agreement. i. The ratio between the perspective images of two lengths is the same as the ratio between the apparent sizes of the lengths.

I am convinced that Leonardo found this example satisfactory (for another discussion of the inverse law of proportionality in the theories of visual appearance and perspective, see PanofskyS 1927/1991, note 16, 92–93). What is more, I tend to believe that he would also accept the following situation. ii. The perspective images of two lengths appear to the eye, placed at the eye point of the perspective composition, to have the same relation as the apparent sizes of the lengths (or in other words, the ratio between the apparent sizes of the perspective images is equal, or almost equal, to the ratio between the apparent sizes of the lengths). It should be noticed that if visual angles are used as measures for apparent sizes, then the ratio between the apparent sizes of two objects is always the

5. Leonardo on Perspective Representations

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p

Di

O

D Ci C

A =A i

FIGURE III.17. The perspective image of a horizontal square.

B = Bi

same as the ratio between the apparent sizes of the perspective images of the objects when seen from the eye point of the perspective projection. However, if it is accepted that Leonardo thought in other terms than those of visual angles, the second situation could have made sense to him – even though the ratio of the apparent sizes is not well defined. My arguments that Leonardo could also have been in favour of the second scenario are indirect. I am not aware of any example in which he dealt explicitly with perspective images appearing to be in the same ratio as the apparent sizes of the original objects, but he does not seem to have expressed misgivings when it occurred. Besides, I find that some of his discussions of discrepancy point towards the second scenario. One example (figure III.17) concerned a horizontal square ABCD and its perspective image ABCiDi in a vertical plane. Leonardo, as noted, presented this example as a case of accidental perspective. His argument was, if I understand him correctly, that AB and CD respect the law of inverse proportionality, and that AiBi and CiDi follow another rule (Manuscript Arundel, fol. 62r; Leonardo 1970, §109). In this connection he did not state that the ratio CiDi:AiBi is different from the ratio between the apparent sizes of CD and AB, but instead commented upon how the images AiBi and CiDi appear (si dimostra).

The Perspective Images of Particular Line Segments

T

he first perspectival question to be considered is problem 1′: How are two equal and parallel line segments represented upon a picture plane p that is parallel to the line segments? In connection with problem 1 – the corresponding problem for visual appearances – I assumed that the two line segments are opposite sides of a rectangle. In the theory of perspective this restriction can be left out, since it does not affect the result, but in Leonardo’s examples it still applies. Here in the perspective case I introduce another limitation, namely that the eye point lies in the plane of the rectangle; it will later become clear why.

102

III. Leonardo da Vinci p

D

A

C

B

Di O

Ai Bi Ci

FIGURE III.18. Perspective images of equal line segments coplanar with the eye.

In discussing Leonardo’s treatment of the law of inverse proportionality I mentioned how he considered a plane that can be interpreted as a vertical plane containing a measuring rod or as a picture plane (page 92). In the latter case figure III.9 becomes figure III.18, while relation (iii.2) becomes Ai Bi : Ci Di = c : b.

(iii.3)

Expressed in words, the lengths of the perspective images of the equal line segments AB and CD are inversely proportional to the distances between the line segments and the eye. This rule is completely similar to relation (iii.1) for apparent sizes, which means that the ratio of the lengths of the images of AB and CD is equal to the ratio expressing the relation between the apparent sizes of AB and CD.

Leonardo did not explicitly state this result, but rather implied it when he wrote about the inverse law of proportionality in such a way that the reader cannot make out whether he had apparent sizes or perspective images in mind. I am therefore convinced that he considered the perspectival representation in figure III.18 to be a case of ‘natural’ perspective. To achieve inverse proportionality for perspective images it is important to assume that the eye point O lies in the plane of the rectangle ABDC, as I have shown in the caption of figure III.19. Hence for practical purposes this case of agreement between the perspective images and the visual appearances is not very interesting, because when it occurs the images AiBi and CiDi lie on the same line, which means they cover each other. It would be interesting to know what Leonardo thought about the perspectival representations of rectangles whose sides are parallel to the picture plane, but not coplanar with the eye point. However, I have not come across any remarks made by Leonardo on this issue.

The Perspective Images of Collinear Line Segments

I

n treating the perspective versions of the second problem, I begin with problem 2′ – which concerns the perspective images of two equal line segments situated on a line perpendicular to the picture plane p. Leonardo

5. Leonardo on Perspective Representations

103

P d O1

f

Ai

A

Hi

H

p Bi B

FIGURE III.19. The relation between the length of an image and its position. In chapter II, the length of an image of a line segment parallel to the picture plane p was related to its distance to p, resulting in relation (ii.2). Now I want to deduce a relationship based on the distance of the line segment from the eye point O. Let P be the orthogonal projection of O upon p, d = OP the distance, AB a line segment parallel to p, OH the normal from O to AB, b = OH, AiBi the image of AB, Hi the point of intersection of OH and AiBi, and finally j the angle between OP and OH. The similarity of the triangles OAB and OAiBi implies that Ai Bi : AB = OHi : OH. A straightforward calculation shows that the latter ratio is equal to d : (b cos j), hence Ai Bi : AB = d : (b cos j). Similarly, another line segment CD, located at the distance c from O, and whose normal through O forms the angle y with OP, fulfils Ci Di : CD = d : cos y. Let CD and AD have the same length, in the case that also y is equal to j, a comparison of the last two relations confirms the rule expressed in (iii.3); it also shows that inverse proportionality requires that cos j = cos y. For situations considered by Leonardo, this only happens when j = y, that is, when O lies in the plane of ABCD.

considered this problem in a note that also dealt with the apparent sizes of the line segments. His treatment of the visual case was presented in section III.4, in which we met his claim (figure III.14) that the line segments ov and vx appear to be almost equal. He proceeded by mapping these line segments upon a picture plane that is orthogonal to the line ox and passes through the point o, obtaining the images or and rq, of which he wrote: And if you carry this out in any place where you can walk round, it will look out of proportion by reason of the great difference in spaces of or and rq.27 [Leonardo 1970, §86]

27 e se tu mettessi questo in opera in qualche loco che vi si potesse andare attorno ti parebbe una cosa discordante per la grã varietà ch’è da lo spatio or e da rq. [Manuscript A 38r; Leonardo 1970, §86]

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It is worth noticing that here Leonardo mentioned how a viewer would perceive the image as particularly awkward if the eye point is moved. This may be taken as a sign of his accepting that perspective images of the line segments achieve the desired effect if they are seen from the eye point of the perspective construction – an issue to which I will return in section III.7. While it is clear that Leonardo was generally dissatisfied with the construction according to the rules of perspective in the preceding example, the precise nature of his objection is not obvious. In continuation of my earlier discussion (i. and ii. page 100), I see two possibilities: i. He required the lengths or and rq to be in the same relation as the visual appearances of ov and vx, that is, he demanded that or and rq be almost equal. ii. He required or and rq to appear to be in the same relation as the visual appearances of ov and vx, that is, he demanded that or and rq appear to be almost equal. The fact that Leonardo stressed the great difference between the lengths or and rq could point to the first possibility. On the other hand, he did not write that or and rq actually are out of proportion, but that they look out of proportion (parebbe una cosa discordante), which could point to the second possibility. Although I am obliged to remain undecided as to what he meant, I find that his remark as such supports my idea that he had situations like i. and ii. in mind. Regardless of the nature of Leonardo’s argument, he found that orq to be an unfortunate picture, and he suggested two procedures for improvement. The first was a procedure he often mentioned, and which I take up at a later stage, namely to make the distance between the eye point and the object larger. He also suggested another possibility (figure III.14): The plane op being always equally remote from the eye will reproduce the objects in a satisfactory way, so that they may be seen from place to place.28 [Leonardo 1970, §86]

Thus, in searching for a solution in which the images of ov and vx in some way respect the relation between the visual appearance of ov and vx, Leonardo chose to project the two line segments upon a circle rather than a straight line. In the three-dimensional situation this would result in a picture plane that is either part of a cylinder or part of a sphere; presumably he was thinking of the latter. As mentioned previously, I do not think Leonardo seriously considered using a part of a sphere as a picture plane (page 87),29 but in 1492 when he wrote this note he may have been wondering about how a projection upon a sphere can be transformed to a mapping upon a plane. The second perspective version of problem two, that is problem 2′′ concerning the perspective images of equal and collinear line segments on a picture plane p parallel to them, does not seem to have bothered Leonardo much. As 28

la pariete op per l’essere sempre equidistante all’ochio a vno modo renderà le cose bene e atte essere vedute da loco a loco. [Manuscript A, fol. 38r; Leonardo 1970, §86] 29 For alternative discussions and views, see WhiteS 1987, 207–215; ElkinsS 19881.

5. Leonardo on Perspective Representations

105

described earlier, some perspectivists concentrating upon visual angles found the perspective representations of the equal line segments unfortunate (page 53). To my knowledge, Leonardo did not express such a view, which is in accordance with my understanding that he had little interest in visual angles. Leonardo’s discussion of the visual aspects relating to figure III.15 was presented earlier (page 99). According to Kim Veltman, Leonardo confirmed by measuring that the three perspective images of the three equal line segments had the same length (VeltmanS 19861, 414), but what he did with this result is not clear. I have the impression that whatever correspondence Leonardo wished to establish between the visual appearance of objects and their perspective images, he did not find this correspondence violated in cases where the objects are equal and collinear line segments parallel to p.

Leonardo and the Column Problem

A

s mentioned in section II.15, the discussion on the perspective representation of line segments parallel to the picture plane has historically often occurred in connection with the column problem. While Leonardo apparently did not bother with the former problem, he gave considerable attention to the latter and took it up several times (VeltmanS 19861, 154–158). One of his diagrams to the column problem is reproduced in figure III.20 and redrawn in figure III.21 showing an eye placed at O looking at the circles AB, CD, and EF, which are horizontal sections in columns with their centres on a line parallel to the horizon HZ. The latter represents the picture plane upon which the images of the circles have to be drawn. In interpreting the remaining part of Leonardo’s diagram I assume that he, like Piero (figure II.32) let the images of the circles be determined as the images AiBi, CiDi, and EiFi of the diameters AB, CD, and EF perpendicular to the lines connecting the eye point and the centre of the circle. In one of his examples, dealing with a situation similar to the one depicted in figure III.20, Leonardo used the column problem to conclude that:

FIGURE III.20. One of Leonardo’s drawings of the column problem. Leonardo Manuscript A, fol. 38r, Bibliothèque de l’Institut de France, Paris.

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III. Leonardo da Vinci B

E C

D

A

H

F Ai

Bi

Ci

Di

Ei

Fi

Z

O

FIGURE III.21. Adaptation of the previous figure.

Natural perspective acts in a contrary way; for at greater distances the object seen appears smaller, and at a smaller distance the object appears larger. [Leonardo 1970, §108]30

The contradiction Leonardo saw was that AB appears to be smaller than CD (which agrees with ‘his’ axiom), and that the images AiBi and CiDi in some sense do not respect this relation. The sense of the matter what he had in mind is, as before, open to interpretation: The line segment AiBi is obviously larger than CiDi, but Leonardo wrote that the eye in O sees (l’ochio vede) AiBi as larger than CiDi (Manuscript E, fol. 16v; Leonardo 1970, §107). His conclusion is different from Piero’s (page 58), and shows clearly that Leonardo did not accept the angle axiom – according to which AiBi appears to be smaller than CiDi because the former line segment is seen within a smaller angle than the latter. In another manuscript Leonardo described how, in the perspective images of a row of columns (figure III.22), ... you will see, beyond a few columns separated by intervals, that the columns touch; and beyond where they touch they cover each other ...31 [Leonardo 1970, §544]

He acknowledged that if the perspective image is seen from its eye point it appears correct, or, to use his own words, it “will be perfect and will deceive the beholder” (ibid.). His worry was that the viewer is limited to one and only one position. Indeed, in his example the perspective image of the columns 30

Il che natura nella sua prospectiva adopera in contrario con ciosiachè nelle maggiori distantie la cosa veduta si dimostra minore e nella distantia minore la cosa par maggiore. [Manuscript E, fol. 16r; Leonardo 1970, §108] 31 ... vedrai infra pochi intervalli d’esse colonne toccarsi, e dopo il toccarsi occuparsi l’una l’altra ... [Manuscript A, fol. 41r; Leonardo 1970, §544]

6. Leonardo and Curvilinear Perspective

107

FIGURE III.22. A row of columns. Based on Leonardo Manuscript A, fol. 41r – the original is reproduced in VeltmanS 19861, 82.

perspective images has an anamorphic effect when seen from any other position than the eye point.

Leonardo’s Appeal for a Large Viewing Distance

A

s mentioned earlier, Leonardo had a means of eliminating the dependence upon the viewing point. This essentially consisted in increasing the distance between the eye and the object to be depicted. In his example with columns he recommended that the eye be “at least twenty times as far off as the greatest height and width of your work” (Manuscript A, fol. 41v; Leonardo 1970, §545). A considerable enlargement of the viewing distance does in fact solve the problem of a fixed station point, but it also introduces a problem pertaining to the viewing angle. Leonardo’s description is too imprecise to enable us to calculate an accurate viewing angle, although we can be sure that it will be very small. For instance, if one looks at a vertical line segment having one end point at eye level and being located twenty times its length from the eye point, the viewing angle is less than 3˚. In the eighteenth century Johann Heinrich Lambert criticized another of Leonardo’s examples; one for which Lambert calculated a possible viewing angle of, at most, 9˚28′, which he found much too small (Lambert 1774, 86). Leonardo did not go so far as to apply an infinite distance, that is, a parallel projection. Yet, in an example of being dissatisfied with the image of a square (figure III.3), he drew something next to it (figure III.23) that may illustrate an alternative solution, and which bears some resemblance to a parallel projection.

III.6

Leonardo and Curvilinear Perspective

B

efore Leonardo pondered whether the horizontal sides of a rectangle appear to be rectilinear or curvilinear, the French painter Jean Fouquet had painted transversals as curves. A quite possible example is shown in

FIGURE III.23. Section of figure III.3.

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III. Leonardo da Vinci

FIGURE III.24. Jean Fouquet, Annunciation of the Death of the Virgin, 1450s, from the Book of Hours of Étienne Chevalier, fol. 9; Musée Condé, Chantilly, France.

figure III.24 (although it cannot be completely ruled out that the carpet had a curved pattern) and an indisputable example in the picture reproduced in figure III.25. It was, however, not Fouquet’s general habit to make transversals curvilinear, as he also painted many tableaux with traditional tiled floors. His motives for depicting transversals as curves a couple of times seem to be more related to the fact that he was fond of including curved elements in his compositions than to any reflections on the theory of vision. There are a number of later examples of artists rendering straight lines as curves, but to my knowledge documented examples of artists using such a procedure

6. Leonardo and Curvilinear Perspective

109

FIGURE III.25. Jean Fouquet, The Arrival of the Emperor Charles IV at the Basilica in Saint Denis, from Les grandes chroniques de France, Bibliothèque Nationale de France, Paris, BN. Fr. 646, fol. 42.

because they found a disharmony between optics and a perspectival representation only occurred much later than the Renaissance.32 In modern times we find an interesting example of curvilinear perspective that as its starting point use the same idea as Leonardo when he considered projection upon a sphere (pages 87 and 104). The twentieth-century painter, engraver, and professor of art Albert Flocon developed the idea of making a central projection upon a sphere further in collaboration with his teaching colleague André Barre. The sphere only became an intermediate stage, as Flocon generally kept to the tradition of using a flat picture plane. He and Barre therefore needed a transformation that could map an image from a sphere to a plane, and chose the so-called Postel projection (Flocon & BarreS 1987, 96–117). The result is a representation in which straight lines are drawn as curves, as shown in the example in figure III.26. In his Birth and Rebirth of Pictorial Space John White claimed that Leonardo’s concepts of perspective involved curvilinear perspective (WhiteS 1957/1987, 207–218). This has given rise to a discussion among a number of art historians.33 The debate is strangely confusing, since it is not based on a

32

Robert Hansen in Flocon & BarreS 1987, xiii–xxi; ElkinsS 19882; KempS 1990, 246–249; ElkinsS 1994, 183–184, 205–216. 33 For an overview of some of the literature on curvilinear perspective, see PedrettiS 1963, note 1; ElkinsS 19881, notes 1 and 2; FrangenbergS 1992, note 12; and for a presentation of the topic, see KempS 1990, 246–249.

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FIGURE III.26. Curvilinear perspective created by Albert Flocon and André Barre. Figure 65 in Flocon & BarreS 1987.

clear idea of what curvilinear perspective is – and, as Socrates would have said, we cannot discuss a concept before we know what it means.34 When curvilinear perspective is used in the sense of representing at least some straight lines as curves, I would not count Leonardo as belonging to its history, for there is no evidence – either in his writings or in his drawings – that he ever experimented with the idea of representing straight lines as curves. On the other hand, if curvilinear perspective is connected to ideas about how straight lines appear, then Leonardo might justifiably be included in the history, because of his reflections on the appearance of the horizontal sides of a vertical rectangle.

34

Various definitions of curvilinear are suggested in ElkinsS 19882, 257 and ElkinsS 1994, 185–186.

7. Leonardo’s Doubts and Their Consequences

III.7

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Leonardo’s Doubts and Their Consequences

Perspective and Visual Impressions

A

major part of Leonardo’s known work on linear perspective concerns examples in which ‘natural perspective’ is at odds with geometrical constructions performed according to the laws of perspective. Leonardo saw such conflicts in cases where the perspective images do not obey what he considered to be the rule of visual appearance. This is, in fact, what happened most often, but in a few cases he seems to have found a correspondence he could accept. I have already pointed out that he found full agreement between his theory of visual appearance and the theory of perspective in one very special case concerning two opposite sides of a rectangle whose plane contains the eye point (page 102). In the 1920s Panofsky advocated a representation that depicts visual appearances faithfully and hence found linear perspective unsatisfactory (PanofskyS 1927/1991, 29–36). Leonardo may have had the same attitude towards perspective as Panofsky later had, but resting on a different foundation, the difference being that Panofsky wanted to see the angle axiom fulfilled, whereas Leonardo, in my opinion, did not. However, I am not so sure that Leonardo always required full agreement between visual impression and representation, but understand, as mentioned earlier, that he would also implicitly accept another situation (item ii. page 100), namely one in which the apparent sizes of the perspective images of two line segments have almost the same relation as the apparent sizes of the line segments. This view actually comes closer to Alberti’s idea of creating a representation of a motif that conveys the same visual experience as the motif itself.35

Fixed Eye Point

L

eonardo’s main worry about perspective may well have been that some perspective compositions look rather distorted when seen from a point at some distance from the eye point of the composition. If this was the case, one could have expected him to investigate the question of when a perspective picture seen from a different point than its eye point looks very distorted and when it does not – and in particular to study how the choice of parameters and motifs influences the distortion effect. This is, however, a complex of problems on which it is very difficult to form a theory – which to my knowledge has never been done. What Leonardo actually did was to study visual appearances and perspective images seen from a fixed eye point. It almost seems as though he thought

35 As stressed earlier, when the angle axiom is accepted, perspective images always fulfil this requirement when seen from their eye points.

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that if pictorial representations approximated visual appearances, then moving the eye point would not spoil the correspondence. It is plausible that Leonardo had hoped his investigations would lead him to a more acceptable form of representation in which movement of the eye point would not spoil the illusion. If so, he had set himself an unsurmountable task.

Leonardo’s Use of Perspective

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eonardo’s views on perspective are likely to have influenced his own art in the sense that he avoided the conflicting situations he had warned against in his writings (see also KempS 1977, 147). For instance, he did not paint a row of columns that are parallel to the picture plane – at least not in the paintings preserved for posterity. It is also remarkable how very few straight lines his paintings contain. Straight lines can be found in his Annunciation (figure III.27), which dates from the early 1470s and was thus created before he started to write critically about perspective. The same applies to his study for the Adoration of the Magi (figure III.28), dating from the early 1480s, in which he used a perspective grid to organize the image of the ground plane. In his later paintings Leonardo seldom allowed straight lines to be noticeable parts of his composition. An exception is his Last Supper (figure III.29) from about 1495 – a motif traditionally painted with a rectangular table and architectural elements. In this painting Leonardo clearly let the images of all orthogonals converge in the principal vanishing point – situated near Christ’s eyes, but otherwise it is far from obvious how the rest of the geometry in the picture is constructed (KempS 1990, 47–48). It may even be that in this painting Leonardo experimented with some of his ideas for avoiding distortion.

FIGURE III.27. Leonardo, Annunciation, early 1470s, Galleria degli Uffizi, Florence.

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FIGURE III.28. Perspective grid in Leonardo’s architectural study for the background of The Adoration of the Magi, early 1480s, Gabinetto dei Disegni e delle Stampe degli Uffizi, Florence.

FIGURE III.29. Leonardo, Last Supper, about 1495, Church of Santa Maria delle Grazie, Milan.

Chapter IV Italy in the Cinquecento

IV.1

The Italian Sixteenth-Century Perspectivists

T

he prominence that perspective enjoyed in Italy during the fifteenth century continued into the cinquecento. More and more people became acquainted with the subject, and the number of those who appreciated it, applied it, and wrote about it, increased steadily. As the century progressed, the mathematical understanding of the rules of perspective also grew deeper. The group applying a general or simplified form of perspective included architects, engineers, painters, stage designers, sculptors, goldsmiths, stonecutters, woodcarvers,1 and engravers. Apprentices in these disciplines were taught at least some rudiments of perspective – presumably often by their masters. Considering how closely perspective is linked to geometry one might expect that the mathematicians’ interest in the subject also increased considerably, but this does not seem to have been the case. While a few mathematicians did write on perspective, their colleagues did generally not indicate any interest in the topic. One notable exception was Luca Pacioli, who praised perspective in his writings.2 Only after 1600, when Guidobaldo del Monte had secured the geometrical foundation of perspective, did mathematicians start to include perspective in books on geometry. The number of authors in the various fields applying perspective was never proportional to the number of active people in these fields. In sixteenthcentury Italy relatively few authors came from the large circle of artists and artisans, while the mathematicians were represented by three authors, and other scholars by two. The authors’ backgrounds naturally influenced their styles – a tendency that became even stronger in the centuries that followed. In presenting the Italian cinquecento publications on perspective, I use style rather than the chronology as my guiding principle.3

1

They used perspective in connection with intarsia, cf. Tormey & TormeyS 1982. In chapter three of his De divina proportione Pacioli argued in favour of including perspective among the mathematical topics (PacioliS 1509). 3 In appendix four I have listed works on perspective in chronological order, country by country. 2

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IV.2 The Architectural, Painting, and Sculpting Traditions Gaurico

T

he very first printed book containing a description of perspective was Gaurico’s De sculptura (On sculpture) from 1504. Pomponio Gaurico (c. 1481–c. 1528) was a humanist and an amateur sculptor who was presumably most experienced in making bronze medals.4 His book takes the form of a dialogue in which the erudite Gaurico initiates two other scholars into the art of sculpting, making numerous references to the classical literature. Gaurico dealt with perspective in a minor section, describing a single technique incompletely (Gaurico 1504) – as Filarete and Francesco had done earlier (page 43). It is therefore impossible to learn how to make a perspective construction from Gaurico’s instructions, and he added no illustrations to promote the reader’s understanding. In fact, his description is so diffuse that it can be, and has been, interpreted in several different ways.5 Following Decio Gioseffi, I take it to be a presentation of a correct construction (GioseffiS 1957, 89–93) that is different from other known methods, and which I have presented in figure IV.1. Like the distance point method, the construction presented by Gaurico was probably developed in a workshop by experimenting with constructions – rather than by an inspiration of theoretical insight. In section IV.3 we meet a construction, presented by Vignola, that has some similarity with the supposed Gaurico construction, and in section V.3 we shall see that the French painter Jean Cousin used a technique that applies an element of this construction.

Serlio

O

ne of the very important authors in the architectural tradition is Sebastiano Serlio (1475–1564). He composed an oeuvre on architecture that includes a Libro di perspettiva (Book on perspective). This work appeared in 1545 as one of the first printed books on the subject that is skilfully illustrated (figure IV.2). Serlio’s treatment of the technical aspects of perspective is, however, less skilful. His description is rather brief, but sufficiently long to allow an unambiguous interpretation (for another discussion of Serlio’s work on perspective see Le GoffS 1991, 235–245). In fact, he presented no less than two faulty methods for constructing the image of a row of squares, situated between two orthogonals. His first method begins with

4

Chastel and Klein in Gaurico 1969, 16. Gaurico’s method is discussed, among other places, in the following works – of which some survey further literature: PanofskyS 1927/1991, 135–138; GioseffiS 1957, 89–93; KleinS 1961; Brion-GuerryS 1962, 46–50; KitaoS 1962, 192–194; Chastel and Klein in Gaurico 1969, 184 and figure 12; KempS 1990, 41. 5

2. The Architectural, Painting, and Sculpting Traditions P

N3 M3

K3

K2

N1 M1

M2 L2

L3

K4

N2

L1

K1

117 D

N M L

K

FIGURE IV.1. An interpretation of Gaurico’s procedure. For constructing the perspective image of a grid of given squares, Gaurico described a procedure (Gaurico 1504/1969, 182–185) that can be understood in the following way. The line segment PK is drawn equal to a given height of the eye above the ground plane. The line through K, perpendicular to PK, is divided in sections KK1, K1K2, ... equal to the side of the squares, and on the line through P parallel to KKi the line segment PD is made equal to the distance. Straight lines are drawn from D to the points K1, K2, K3, ..., and these lines intersect PK in the points L, M, N, ... . Through the points of intersection lines parallel to PD are drawn cutting the first set of lines in the points L1, L2, ..., M1, M2, ..., N1, N2, ... . Finally, the image of the first orthogonal is drawn as the line through the points with index 1, the image of the second as the line through the points with index 2, and so on. These images pass through the point P (the principal vanishing point), but this fact is not applied in the reconstructed procedure. The construction presented here is based on two properties of the point D, namely that it is the vanishing point of one set of the diagonals in the squares, and that it is an Alberti point – when PK is regarded as a profile of the picture. That D is an Alberti point implies that the point of intersection of PK and DKi determines the foreshortening of an orthogonal length that has one endpoint on the ground line and is equal to KKi, for i = 1, 2, 3, ... .

a correct Alberti construction of the first square in the row, but proceeds erroneously, as explained in the captions of figures IV.3–IV.5. His second method is the incorrect distance point construction, mentioned previously (page 48). Serlio’s mistake consists in introducing the distance between the eye and the picture plane in the wrong place in the diagram (figure IV.6). By superimposing Serlio’s two figures upon each other one can see that his two constructions do not lead to the same result, but he himself does not seem to have carried out this test. Both of his methods are correct in the special circumstance where the principal vanishing point is situated vertically above the point G in the figures IV.3 and IV.6.

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FIGURE IV.2. An architectural composition. Serlio 1584, fol. 40r.

FIGURE IV.3. Serlio’s incorrect Alberti construction. Serlio wanted to construct the image of a row of horizontal squares of which the first has its side AG on the ground line GK. The point F is the principal vanishing point. Serlio connected the points A and G with F, drew the line GH perpendicular to the ground line, made HI equal to the distance, and then drew a line parallel to the ground line through the point of intersection B of IA and HG; this line intersects FA in A′ and FG in B′. Thus far, Serlio’s procedure has provided the correct image AA′B ′G of a square. His next step was to connect I with A′ and to draw a transversal through the point C in which the line IA′ intersects GH. Let this transversal meet FA and FG in A′′ and C′. Serlio claimed that A′A′′C′B′ is the image of the second square, but this is not the case, as will be explained in the caption of figure IV.5. Analogously he presented an incorrect construction for the succeeding squares. Serlio 1584, fol. 19r, with some letters and apostrophes added.

2. The Architectural, Painting, and Sculpting Traditions F

119

H

I

S T R

A

G

FIGURE IV.4. A correct Alberti construction of Serlio’s second square. A valid procedure for constructing the next square would have been to let the point R on the ground line be given by AR = AG and then let the next transversal be defined by the point of intersection, S, of GH and IR, rather than the point C in the previous figure.

F

H

I

S A9 T R

A

B9

B G

FIGURE IV.5. Comparison of the constructions of the second square presented in the two previous figures. Let T be the point of intersection of IR and AF. If T was equal to A′ (from figure IV.3), the two constructions would give the same result. However, as I will show in following, the points T and A′ only coincide in the special case when the points F and H are identical. The similarity of the triangles TRA and TIF implies that AT : TF = AR : FI, and since AR = AG, then AT : TF = AG : FI.

(1)

By considering the pairs of similar triangles FA′B ′, FAG; GB′B, GFH; and BGA, BHI, I conclude that AA′ : A′F = GB′ : B′F = BG : BH = AG : HI, that is, AA′ : A′F = AG : HI.

(2)

A comparison of (1) and (2) shows that only when HI is equal to FI, that is, when the points F and H coincide, do the two points A′ and T divide AF in the same ratio, and hence only in this case is it correct to use the point A′ in the construction of the next square. In the Serlio example where HI is long compared to AG, the distance between the points A′ and T is very small.

Serlio’s books became a standard work on architecture and maintained this position for the better part of the next two centuries. Originally it was printed with parallel texts in Italian and French, then went through many editions and was translated into Dutch, English, German, Latin, and Spanish (SchülingS 1973, 42–47). It was also plagiarized by Walther Hermann Ryff,

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FIGURE IV.6. Serlio’s incorrect distance point construction. As in figure IV.3, the point F is the principal vanishing point and the task is to throw a row of squares on the line segment AG into perspective. In the construction Serlio used the point I as a distance point, but rather than making FI equal to the distance, he let GK be the distance. In fact the point I in this diagram is the same as the point I in the figure IV.3. Serlio 1584, fol. 19r.

who included parts of it in a German translation in his book on mathematics for architects without any reference (Ryff 1547). The numerous Serlio editions taught two incorrect methods of perspective constructions to a large group of people, but Serlio’s mistakes were in general not repeated in the literature on perspective. The fallacy of Serlio’s second method was explicitly pointed out by Danti in 1583 (Vignola 1583, 18); two years later Benedetti complained about an incorrect distance point method without mentioning an author (Benedetti 1585, 119); and as we shall see in chapter VI, Stevin also commented upon the method (page 283). The faulty step in Serlio’s second method does not have serious consequences in practice, since the result of his construction is a perspective composition with a larger distance than he claimed to have used. For the purpose of perceiving the composition this is unimportant – and in fact, most perspective pictures are recognized as such even when not viewed from the correct distance. In an interesting passage of Libro di perspettiva, Serlio dealt with the construction of a theatre stage and referred to his own experience in creating a theatre in Vicenza. He was not very explicit about his use of perspective in general, but he did give some details concerning his design of the stage floor (figure IV.7). This floor consisted of two parts: a horizontal part on which squares were drawn unforeshortened, and an oblique part (with slope 1:9) with foreshortened squares, whose orthogonals converge towards one point. Serlio stressed that the best effect is obtained if this point was positioned on a vertical plane situated behind the end wall of the stage (Serlio 1584, fol. 48v). Although not formulated as such by Serlio, his example concerns the mathematically interesting problem (figure IV.8) of throwing horizontal lines perpendicular to the ground line GR into perspective on an oblique plane a. According to the theory developed later, the vanishing point of these lines is the point of intersection, V, of a and the line through the eye point O parallel to the perpendiculars. Serlio intuitively conceived of the convergence point as

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FIGURE IV.7. A stage set. Serlio 1584, fol. 50v.

p

a

V

G O

R

F

FIGURE IV.8. Determination in an oblique floor of the vanishing point of horizontal lines orthogonal to GR.

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being the principal vanishing point in a vertical picture plane p that lies sufficiently far from the ground line. This is indeed true when p is taken to be the vertical plane through V parallel to the ground line. Serlio did not comment upon how he had constructed the perspective images of the transversals on the oblique floor. Since his construction of the images of transversals was incorrect in the usual case of a vertical picture plane, it is highly unlikely that he solved this more demanding problem relating to an oblique plane in a mathematically correct way.

Sirigatti, Cataneo, and Peruzzi

H

alf a century after the appearance of Serlio’s work on perspective, the architect Lorenzo Sirigatti also devoted an entire book, La pratica di prospettiva (1596, The practice of perspective), to the discipline. He presented many examples from the world of architecture and included a section with very elaborate drawings. The book became quite popular and its good reputation was so long-lived that an English translation was published no less than a hundred and sixty years after the Italian original appeared. The procedure, Sirigatti presented was a plan and elevation construction, in which he made shortcuts for symmetrical objects (figure IV.9). Some of his successors praised him much for this method, but since it predates Sirigatti, the credit ought to go to another person, but I do not know to whom. In fact, Sirigatti’s technique – like several other perspective methods – cannot be

FIGURE IV.9. Sirigatti’s construction of the perspective image of a symmetrical hexagon. He applied a plan and elevation method to throw half of the hexagon into perspective and then drew the rest of the image symmetrical to the part constructed. Sirigatti 1596, chapter 7.

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123

traced back to a definite origin. We can be sure, however, that the technique was published at least three decades before Sirigatti’s book appeared, as it occurs (figure IV.10) in the second edition of a general work on architecture written by the architect Pietro Cataneo (c. 1510–c. 1571). About this method Cataneo volunteered no other information than he wanted to be clearer than Serlio (Cataneo 1567, Book 8).

FIGURE IV.10. Cataneo’s example of throwing two squares into perspective. He applied the same method as the one presented in the previous figure. Cataneo 1567, 185.

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FIGURE IV.11. Peruzzi’s Sala delle prospettive, c. 1516, Villa Farnesina, Rome.

The perspectivists in cinquecento often referred to the painter and architect Baldassare Peruzzi, and his name was, as we saw in section II.12, associated with the distance point method. Whether he dealt with other methods is difficult to say, since we know of no written presentation of perspective constructions by Peruzzi. It seems safe to assume, however, that he was instrumental in spreading knowledge of the subject. That he was a master of perspective constructions is still evident in the interior decoration of Sala delle prospettive (The perspective hall), which he designed for the Villa Farnesina in Rome (figure IV.11) – for this and other trompe l’œils in form of painted architecture see MilmanS 1986.

Lomazzo

A

lthough the architects in sixteenth-century Italy presumably applied perspective less than the painters did, the discipline was described by more architects than painters. Only one painter, Giovanni Paolo Lomazzo (1538–1600), is known to have treated perspective in a publication (Lomazzo 1584). He did so in a work dealing with painting, sculpture, and architecture in general, parts of which were taken from an unpublished treatise by Bartolomeo Suardi, better known as Bramantino (RobertsonS 1996, 654). Even so, Lomazzo only touched briefly upon perspective techniques (for more on the contents and the aim of Lomazzo’s work, see PeifferS 2002, 112–116).

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IV.3 A Mathematical Approach to Perspective – The Contributions by Vignola and Danti The First Edition of Vignola’s Work on Perspective

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he Florentine architect Giacomo Barozzi da Vignola (1507–1573) is known, among other things, for a book on the five classical orders of architecture, published in 1562. He also composed a manuscript on perspective left unpublished at the time of his death. His son kept the manuscript for some years before asking the Dominican Egnazio Danti (1536–1586) to edit it. It is unclear when Vignola composed his manuscript and how long he worked on it. Danti indicated that it was a work from Vignola’s youth, whereas Vignola’s son said that his father had been engaged in writing it until his death (KitaoS 1962, 173, note 1). In 1583 the work appeared under the title Le due regole della prospettiva pratica con i commentarii del Egnatio Danti (The two rules of practical perspective with the comments by Egnatio Danti). Around that time Danti was summoned to Rome to serve as cosmografo et matematico pontificio, following a long career as a mathematical adviser and teacher of mathematics. Danti’s edition is rather unevenly structured, being a mixture of Vignola’s original text and Danti’s own comments. Vignola was interested in the mathematics behind perspective and included some material on this matter, but he did not intend for his book to be heavily loaded with mathematical arguments. His editor, however, was a mathematician who was enchanted by the possibility of treating perspective as a geometrical discipline, and he therefore elaborated quite extensively on the theoretical parts of Vignola’s manuscript. As a reader it is not easy to distinguish between the texts of one contributor and the other. Danti explained that all the material printed in carattere grosso (large type) was written by Vignola, and the rest by himself (Vignola 1583, 1). The understanding of this statement is complicated by the facts that first of all, the book was typeset with more than two font sizes, and second, that the largest letters occur in two styles, as shown in figure IV.12.6 Presumably Danti meant that Vignola’s passages are the ones printed in the largest regular letters. If this interpretation, which agrees with the understanding of several other commentators, is correct, then only slightly more than one sixth of the text is written by Vignola. This leaves us to wonder why Danti did not simply edit Vignola’s text, and write his own book on perspective. One possible explanation is that Danti wanted to have his text together with Vignola’s illustrations, which are far more sophisticated than those he himself drew. In any case, it became a fairly popular book, which went through several editions and was reissued as late as 1743. Danti organized the book so that it contains a long introduction, consisting almost entirely of his own text, and two chapters – one for each of the

6 Later editions of Vignola’s Le due regole made it even more difficult to get an idea of who wrote what, since the original changes in font size were not retained.

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FIGURE IV.12. Samples of the text in Danti’s 1583 edition of Vignola’s work, all reduced in the same scale. They show why Danti’s statement that Vignola’s contribution is printed with a large type requires some interpretation. My understanding is that only the text printed in the regular, and largest, font is by Vignola, or in other words that definition one is by Vignola, whereas the rest including definitions two and five are by Danti. Vignola 1583, 1, 2, and 4.

two rules mentioned in the title of the book. Vignola’s two rules were, in reality, two methods, namely a plan and elevation construction and a distance point construction. In the following I first deal with Vignola’s material, then look at a few of Danti’s many additions.7

Vignola’s Plan and Elevation Construction

V

ignola did not spend many words on describing his illustrations of various constructions. In fact, his readers have to find out about Vignola’s methods by studying his drawings rather than his text. Hence my presentation of his procedures relies strongly upon my interpretation of his diagrams.

7

For another discussion of the relations between the theoretical and practical parts of Le due regole, see PeifferS 2002, 107–112.

3. A Mathematical Approach to Perspective

127

e

B

H

g

p Be

P b be

G I O

bp Q

R

F

FIGURE IV.13. The basis of Vignola’s plan and elevation construction. Here p is the picture plane, O the eye point, F its foot, P the principal vanishing point, and Q the ground point. The point B is situated in the ground plane g, its orthogonal projection upon GR is I, and its image b has to be constructed. As plan Vignola used g and as elevation the vertical plane e. The plan of the point B is the point itself, and its elevation is Be. As in other plan and elevation constructions Vignola determined the distances Qbp and Qbe. In some cases he determined b by composition, and in others he abbreviated the procedure by involving the principal vanishing point P and applying that b is situated on the image IP of the orthogonal BI. He then needed only one of the points bp and be. His procedures are explained in the caption of figure IV.14, and examples can be seen in figures IV.15–IV.17.

Vignola’s examples of applying a plan and elevation construction look different than Piero’s. There are two reasons for this. The first is that in combining the plan, elevation, and picture plane in one drawing, Vignola applied a different rabatment than Piero did – his choice is explained in figure IV.14. The second reason is that Vignola sometimes modified the plan and elevation construction by involving the principal vanishing point or a distance point. In the captions of figures IV.13 and IV.14, I have explained the principle of Vignola’s plan and elevation construction and how his modified method involving the principal vanishing point works. The latter method is in fact very similar to an Alberti construction. Figure IV.15 illustrates how Vignola applied a traditional plan and elevation construction to throw a circle into perspective. To construct the perspective image of a cube, Vignola applied the modified method involving the principal vanishing point as shown in figure IV.16. By incorporating a distance point he achieved a method – of which an example can be seen in figure IV.17 – that to some extent resembles the supposed Gaurico procedure.

128

IV. Italy in the Cinquecento O

P b

G

F

I

be

Q

Be

R

bp

B

FIGURE IV.14. Vignola’s two plan and elevation constructions, for which he applied a special rabatment. This can be understood by imagining (figure IV.13) that the picture plane p has been turned around PQ into the plane of elevation e and that the ground plane g has similarly been rotated around the line FQ into e. When considered as part of p, GR is the ground line; when considered as part of the plan, the ground line is Qbp which is then also p’s plan. The elevation of p is the line Qbe. In his version of a traditional plan and elevation construction, Vignola found the image point b as the point, on the line through be parallel to the ground line GR, determined by bbe = Qbp. In his abbreviated form of a plan and elevation construction involving the principal vanishing point P, Vignola determined the image b as the point of intersection of IP and the transversal through be. The operations in this construction are the same as in an Alberti construction, in which the point bp is not needed.

Vignola’s Distance Point Construction

V

ignola’s presentation of the distance point construction is the first unmistakable introduction to this method in Italy.8 In fact, he presented a generalization of earlier techniques. In Piero della Francesca’s work the distance point construction occurred as a way of solving a particular problem concerning the image of a square. Vignola presented it as procedure which in principle can be used for constructing the perspective image of any point given in the ground plane, namely as the point of intersection of the images of the orthogonal and a diagonal 9 through the given point. To introduce the method Vignola used a perspective representation of the three-dimensional geometrical configuration (figure IV.18), thereby producing a kind of a double-perspective diagram. Vignola’s illustration of how he applied the distance point construction to determine the perspective image of a circle is shown in figure IV.19. To be 8 9

The occurrence of distance point methods outside Italy is treated in chapter V. Hereby I mean a horizontal line forming an angle of 45˚ with the ground line.

129 3. A Mathematical Approach to Perspective

FIGURE IV.15. Vignola’s example of throwing a circle into perspective by means of a plan and elevation procedure. He constructed the perspective circle by determining the images of a number of points on the circle and then drawing a smooth curve connecting the image points. He did not mention that the image is an ellipse. Vignola’s construction method is the same as the first procedure presented in the caption of the previous figure, but to avoid confusion as a result of too many lines, he did not include all the instrumental lines in his drawing. The points C and G are the plan and the elevation of the eye point, and with help of these points, he found the image of a given point, such as 10, as follows. He constructed the point of intersection of C10 and AE, say X, obtaining the plan of the image. Furthermore he projected the point 10 upon the line AD obtaining its elevation, which he denoted 10 as well. He then connected the latter point 10 with G, obtaining the elevation of the image as the point of intersection, say Y, of G10 and AB. Finally, he drew a transversal through Y and determined the image – which he also called 10 – as the point on the transversal for which Y10 = XA. Vignola 1583, 77 with the numbers 10 enlarged and the letters X and Y added.

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FIGURE IV.16. Vignola’s illustration of how to obtain a perspective cube by a plan and elevation construction. Vignola used this figure to introduce the parameters of a perspective construction, but made no remarks on the actual performance of construction. It is, however, evident that he incorporated the principal vanishing point C. He chose his parameters so that the three lines Ogg, ggt, and qt coincide, which will not generally be the case. Vignola 1583, 65.

able to move on to three-dimensional figures he needed a method for constructing the image of a vertical line segment, but he did not treat this topic separately. Danti recognized this omission and presented two methods (figure IV.20) that Piero had also applied (figures II.25 and II.26). One of these, namely the method presented last in the caption of figure IV.20, had also been mentioned by Filarete (Filarete Arch, 655). Vignola probably applied a combination of the latter method and the distance point construction for throwing a cube into perspective (figure IV.21).

Vignola’s Comparison of His Two Methods

V

ignola’s ambitions went beyond presenting the two methods of perspective construction that presumably were the most popular at his time. He also wanted to compare them, both with regard to the practical aspect of how they were to perform, and with regard to their theoretical foundation. Vignola’s attitude to the plan and elevation construction was very similar to Piero’s. Vignola thus claimed that the construction was better known and more easy to grasp, but involved longer and more tedious operations,10 and 10

piu nota, & piu facile a conoscersi; ma piu lunga & piu noiosa all’operare. [Vignola 1583, 52]

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FIGURE IV.17. Vignola’s special way of throwing a grid of horizontal squares into perspective. For this particular configuration he applied a method combining elements from a plan and elevation construction and his distance point construction. His rabatment is as explained in figure IV.14. The point B is the principal vanishing point, G is a distance point as well as an Alberti point, and CA is the ground line in the picture plane. In the plan he drew a row of three equal squares, whereas he included nine squares in the perspective image. In this construction Vignola did not use B as the point of convergence of the orthogonals. Instead he applied that G is both a distance point and an Alberti point. Vignola determined the positions L, K, and H of the transversals as the points of intersection of AB and the lines that connect G with the points R, P, and Q, using that G is an Alberti point. By using G as a distance point – that is, the vanishing point of one of the sets of diagonals in the squares – Vignola was able to finish the construction. This technique greatly resembles the one Gaurico may have applied (figure IV.1), the difference being that in the latter method the right distance point was used, rendering the points R, P, and Q superfluous. In his drawing Vignola involved the plan to determine the lengths of the foreshortened transversal sides of the squares (for instance, the side LO is equal to the length Add ). For the construction of the perspective squares, these lengths are not really necessary. Vignola 1583, 69.

accordingly he found the distance point construction was more difficult to understand than the plan and elevation construction, but more easy to carry out.11 In other words, Vignola found that for practical purposes the distance point construction was best – and according to Danti it was also Vignola’s preferred method (Vignola 1583, 97). Vignola’s impression of the two methods is confirmed by his own examples, for instance in a comparison of figures IV.15 and IV.19, or figures IV.16 and IV.21.

11

piu difficile a conoscere, ma piu facile ad esequire. [ibid.]

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FIGURE IV.18. Vignola’s distance point construction. The principal vanishing point in the picture plane ACNK is the point L, whereas the principal vanishing point of the diagram itself is symbolized by the eye at the top. Vignola 1583, 105.

While comparing the two methods Vignola claimed, “many have said that in perspective there is only one true rule [method]”.12 This remark is extremely interesting because it suggests that some Italian practitioners of perspective had debated the correctness of the methods they were applying. The view that there could only be one true method seems, in particular, to reflect doubts about the correctness of the distance point construction. Historically it took a long time before the mathematical theory of perspective was developed so far that it was possible to grasp the geometry behind this method. We already have seen how Piero struggled, largely in vain, with the problem of proving the correctness of the method (page 48). Vignola himself took up the challenge of proving that the distance point construction is correct – not by interpreting the various steps of the method, 12

molti habbiano detto, che nella Prospettiva una sola regola sia vera. [ibid.]

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FIGURE IV.19. Vignola’s example of throwing a circle into perspective by applying his distance point method. The point A is the principal vanishing point and the point B the right distance point. Vignola chose twelve points on the circle (shown at the righthand side of the diagram) so that the chords 12, 11; 1, 10; 2, 9, etc. are diagonals and so that the chords 2, 12; 3, 11, etc. are orthogonals. He determined the image of the point 2 as the point of intersection of the perspective orthogonal 2A and the perspective diagonal 2B, and treated the other images similarly. As in the example in figure IV.14, he then connected the twelve image points with a smooth curve. Vignola 1583, 111 with the numbers 2 enlarged.

FIGURE IV.20. Danti’s two methods for constructing perspective heights. He assumed that the perspective image of a grid of squares is given, and that F is the principal vanishing point. At the point Q, which is the image of a point in the ground plane, he constructed the image of a vertical line segment equal to eight times the side of the original squares. One of his ways was to determine the point of intersection B of the line FQ and the ground line, draw the vertical line segment BA equal to eight times the sides of the original squares, connect A and F, let the vertical through Q cut AF in R, and conclude that QR is the requested line segment (cf. the caption of figure II.26). His other method was to determine the point P so that QP on the transversal through Q is equal to eight times the side of the perspective squares, and then make QR equal to QP (cf. (ii.3) page 39). Vignola 1583, 93.

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FIGURE IV.21. Vignola’s construction of a perspective cube by applying his distance point method. This diagram shows that he not only considered the distance points D and B of the horizontal diagonals, but also the distance points C and E; these are the vanishing points of the diagonals in the two vertical faces of the cube that are perpendicular to the picture plane. Vignola 1583, 107.

but by showing that they led to the same result as the plan and elevation construction. An establishment of the equivalence of the two procedures would, for Vignola, secure the distance point construction since he, like Piero and probably many others as well, believed it was evident that the plan and elevation construction was exact (ibid., 99). Vignola thought that the equivalence of the two methods could be asserted by proving that they produce identical images of a square. While his attempt to prove that the two images in question are the same was unsatisfactory from a mathematical point of view (ibid., 99–100), from a historical point of view his endeavours into the

3. A Mathematical Approach to Perspective A

D

H

K G

B

P

L

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Z

M C

R

FIGURE IV.22. Danti’s theorem three. In an abbreviated and modernized version with altered letters, the result is as follows. Given two parallel lines GR and HZ, the line segment BC on GR, and the point P on HZ. Let BH be the common normal to GR and HZ through the point B, and let the points A and D on HZ have the property that AH = DP (1) Let the line AC intersect BH in K, and the line DC intersect BP in L, then the line KL is parallel to GR and HZ. Danti’s proof is based on the following considerations. The line segment BH is divided by AC in the ratio BK : KH, and the segment BP by DC in the ratio BL : LP. From the similarity of the pairs of triangles KBC, KHA and LBC, LPD combined with relation (1), it is seen that BK : KH = BC : AH = BC : DP = BL : LP. The result BK : KH = BL : LP applied in triangle BHP shows that the line KL is indeed parallel to the line HZ. At first glance Danti’s theorem does not seem to deal with perspective, and Danti himself did not explain how his theorem three can provide the result that Vignola wanted to prove. His ideas are presumably covered by the following argument. On the horizon HZ, let P be the principal vanishing point, D and A a distance point and an Alberti point, respectively (in agreement with (1)), and BC a given line segment on the ground line. It has to be proved that the perspective images of the square on BC obtained by a plan and elevation construction and by a distance point construction are identical. When the convergence rule is taken for granted – as people generally did in Danti’s days – it only has to be proved that the two constructions produce the same image of the fourth side of the square (the one parallel to BC). The determination using a plan and elevation construction of the elevation of the fourth side’s image is indistinguishable from the one applied in an Alberti construction. In the latter method, the position of the fourth side is the transversal through K (the point of intersection of BH and AC). In a distance point construction, the image is found to be LM determined by the transversal through L (the point of intersection of DC and BP). According to Danti’s theorem, the line KL is parallel to the ground line; hence the two methods result in the same image of a square. Adaptation of Danti’s illustration in Vignola 1583, 18.

foundation of the distance point construction are more interesting than his failure. As far as is known, no one succeeded in proving the distant point method to be correct until Guidobaldo created a new mathematical theory – which was published in 1600, and which we shall meet in chapter VI. Danti did some polishing on Vignola’s proof, subsequently remarking that Vignola’s claim could also have been obtained directly from a result Danti had presented as “theorem three” (ibid., 100–101). I have explained Danti’s idea in the caption of figure IV.22.

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Danti on Convergence Points

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he above-mentioned theorem by Danti occurs in the introductory part of Le due regole della prospettiva, in which he devoted more than fifty pages to presenting various definitions concerning perspective, as well as some geometrical theorems with more or less obvious ties to perspective. In studying how a late sixteenth-century mathematician like Danti understood the properties of a perspective projection, I find it particularly interesting to discuss four of his definitions. They show some measure of original thinking and are worth quoting: Definition five Perspective parallel lines are those that will meet at the horizontal point [a point on the horizon]. Definition seven A distance point is that at which all the diagonals arrive. Definition ten Principal parallel lines are those which will all converge in the principal point of perspective [principal vanishing point]. Definition XI Secondary parallel lines are those which will be united at the horizontal line at their particular points apart from the principal point.13

Phillip Jones has correctly characterized this as an awareness of the existence of vanishing points of other sets of horizontal parallels than the two usual ones, namely the orthogonals and the diagonals (JonesS 1947, 147). Paola Marchi has given a historically even more precise description, which she arrived at by distinguishing between a vanishing point and a convergence point (MarchiS 1998, 23, 37). The former is the point in the picture plane assigned to a three-dimensional set of lines; the latter she connects to the images of parallel lines and calls an operative point in the construction. It is indeed this quality of being functional in a construction that Danti, and his contemporaries, had in mind when they considered meeting points of the images of parallel lines. En passant, I would like to call attention to the interesting fact that rather than referring to the perspective images of parallel lines, Danti used the

13

Definitione quinta Linee parallele prospettive sono quelle, chè si vanno à congiugnere nel punto orizontale. Definitione settima Punto della distanza è quello, dove arrivano tutte le linee diagonali. Definitione decima Linee parallele principali son[o] quelle, che vanno à concorrere tutte insieme nel punto principale della Prospettiva. Definitione XI Linee parallele secondarie sono quelle, che vanno ad unirsi fuor del punto principale nella linea orizontale, alli loro punti particolari. [Vignola 1583, 4–5]

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expression “perspective parallel lines”. The idea of studying properties – such as parallelism – perspectively is found occasionally in the works of later authors as well. It became a key idea in the work by Johann Heinrich Lambert, as we shall see in chapter XII. Reading Danti’s definitions one could also get the impression that he believed all sets of parallel horizontal lines – apart from those parallel to the picture plane p – are depicted as lines converging at a point on the horizon. However, the illustration he made to accompany definition XI, reproduced as figure IV.23, seems to contradict this impression. In his diagram Danti let one pair of the images of the parallel sides of the horizontal square P converge on the horizon, whereas the point of intersection of the images of the other pair does not lie on the horizon. He may, of course, have made a drawing error, but if so the diagram shows that it was not deeply embedded in him that the images of parallel horizontal lines not parallel to p should meet on the horizon. It is remarkable that Le due regole contains a drawing by Vignola (figure IV.24) that is similar to the Danti figure just discussed. In Vignola’s diagram the images of both pairs of the parallel sides in the square L converge on the horizon – although the sides 1, 4 and 2, 3 are not prolonged to the horizon, their point of intersection does lie on it. There are no other illustrations in the book allowing us to conclude any more about how Vignola drew images of parallel lines that are not orthogonals or diagonals. However, Vignola did make another striking illustration (figure IV.21) in which the diagonals of the images of two of the vertical sides of the cube occur. He drew them as lines

FIGURE IV.23. Danti’s diagram to his definition XI. Vignola 1583, 5.

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FIGURE IV.24. Vignola’s construction of the image of a square with no lines parallel to the ground line GH. Vignola 1583, 115.

that converge on the line through the principal vanishing point perpendicular to the horizon. It thus seems that Vignola himself worked – at least implicitly – with converging points for more than the two usual directions, and even for non-horizontal lines. The introductory part of Le due regole della prospettiva also contains descriptions of some instruments for making perspective constructions, a theme that became popular after Albrecht Dürer published his Underweysung der Messung in 1525. Earlier we saw an example of an instrument drawn by Vignola (figure II.33); figure IV.25 shows Danti’s version of one of Dürer’s instruments.

IV.4 Connection Between Perspective and Another Central Projection – Commandino’s Contributions The Context of Commandino’s Work

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ome decades before Danti began to write his comments on Vignola’s book, another Italian mathematician, Federico Commandino (1509–1575), had been working on perspective. Commandino is best known for his scholarly editions of classical Greek works, among them are Ptolemy’s Planisphaerium (1558) and Apollonius’s Conics (1566). Actually Commandino’s contribution

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FIGURE IV.25. Danti’s version of an instrument first presented by Dürer (cf. figure V.49). The screw eye G represents the eye point, and the string stretched from it a ray of light. The image of the point L is determined as the point of intersection N of the string GL and the picture plane. This point of intersection is marked by two strings that can move within the picture frame. It must be accidental that one of the strings has the position of the diagonal. When the point N has been marked, the string GL is removed and the mounted paper (on which some part of the image is already drawn) is swung into the frame, and the point N transferred onto the drawing – indeed a time-consuming process. Vignola 1583, 56.

to perspective is part of a commentary he wrote on the former Greek work applying a result from the latter. The Planisphaerium has as its theme the stereographic projection (figure IV.26), which is a central projection with its centre at the celestial south pole. Ptolemy used it for mapping points on the celestial sphere upon the plane of the equator, applying that all the circles of the sphere – apart from great circles passing through the poles – are projected

B

Bi Ai

A S

FIGURE IV.26. A stereographic projection. It projects points on the celestial sphere upon the plane of the equator, with the south pole S serving as the centre of projection.

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upon circles. He did not prove this result, probably assuming that his readers would know how to derive it from Apollonius’s work (HeathS 1921, vol. 2, 292). By the 1550s such an assumption was no longer reasonable, so Commandino provided a proof (Commandino 1558, fol. 20r); this was presumably very close to the one Ptolemy had in mind. Commandino’s proof is based on Apollonius’s concept of a subcontrary section to a given conic section, in casu to a circle. Let AB (figure IV.27) be the diameter of a circular section in the cone with vertex S, and let a plane a perpendicular to ABS cut it at CD. When the triangles SAB and SCD are similar, Apollonius called the section of the cone determined by the plane a subcontrary to the circle with diameter AB. He proved that the subcontrary sections to a circle are circles as well (ApolloniusS Con, 9–10). The stereographic projection of a given circle on the sphere is the curve in which the plane of the equator intersects the cone with the south pole as its vertex and the circle as its base. Commandino proved that this curve is subcontrary to the given circle, and hence a circle (figure IV.28).14 While working with the stereographic projection and conic sections, Commandino remarked that the sections can also be considered sections in visual cones or pyramids, in other words, as perspective images. This inspired him to take up a study of perspective and to devote the first nineteen folios of his comments on Ptolemy’s Planisphaerium to the subject (Commandino 1558). It is worth noting that Commandino’s realization of the conceptual relation between a stereographic and a perspective projection did not bring him any technical advantages in treating the two subjects. Thus, the central problems studied in connection with the projection of the stars and with perspective were so different that a result gained in one field was of no interest in the other – an exception being the question of when a circle is projected into a circle. In later chapters we shall see that A B

D

C

S

14

FIGURE IV.27. A subcontrary section. The triangles SAB and SCD are similar, the angles at A and D being equal, as are those at B and C.

For a presentation of Commandino’s arguments, see SinisgalliS 1980.

4. Connection Between Perspective and Another Central Projection FIGURE IV.28. A proof that the stereographic image of a circle is a subcontrary section to the circle. Let the circle SDI represent a celestial sphere with S as its south pole and the circle with diameter DI as its equator. Further, let AB represent a circle with diameter AB, and A′B′ represent its stereographic image in the plane of the equator. It has to be proved that the triangles SAB and SB′A′ are similar. Since the two triangles have the angle at S in common, it is sufficient to prove that ∠BAS = ∠A′B′S. To do this I introduced the auxiliary line BC parallel to DI. The point S – being the south pole – halves the large arc CB, and hence the angles at the periphery ∠BAS and ∠CBS are equal. The latter angle is also equal to ∠A′B′S – the lines DI and CB being parallel. Thus, ∠BAS is equal to ∠A′B′S.

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A C

D

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A9

B9

I

S

under other circumstances where perspective was related to another central projection, it brought no new results to perspective, either.

Commandino’s Constructions

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n his section on perspective Commandino presented a couple of perspective constructions and proved that they are correct. His section is remarkable in that it is the first example of an entirely geometrical and rigorous approach to perspective. Thus, he involved not a single argument from optics in his proofs and he did not assume the convergence rule – nor did he apply it. Commandino kept to the tradition of relating a perspective projection to figures rather than to points. The figures he treated were quadrangles, circles, and segments of circles. For each type of figure he went through many special cases, ending with the situation in which a figure is located arbitrarily in relation to the picture plan p – and he even included the possibility that the figure is located in front of the picture plane. To determine the perspective image of an arbitrary quadrangle, Commandino began with a horizontal rectangle with one side on the ground line, continued with a horizontal rectangle with two sides parallel to p, a

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horizontal quadrangle with one side parallel to p, an oblique rectangle with one side on the ground line, and so on (ibid., fol. 2v–6r, 8v–12v). In retrospect, this seems unnecessarily complicated, particularly because his procedures were, in principle, pointwise constructions. However, throughout the history of mathematics the first solutions to problems have generally been rather knotty – the fundamental ideas crystallizing only after a period of investigation. Commandino presented two methods for determining the images of horizontal figures, the second of which is a plan and elevation construction. His first construction is unusual – not to say downright awkward. In fact, there is no intuitive way of imagining the three-dimensional configuration from which he deduced it. In presenting this construction I treat it as a pointwise construction (for another discussion of Commandino’s constructions, see FieldS 1985, 75–78 and FieldS 1997, 150–161). In figure IV.29, I have illustrated the three-dimensional configuration in which the image Bi of the point B, situated in the ground plane g, has to be determined. As usual, the eye point is O, its orthogonal projections upon g and the picture plane p are the points F and P (implying that P is the principal vanishing point), and the orthogonal projection of P upon the ground line GR is the point Q. Commandino characterized the point B by its orthogonal projections C and I upon the lines FQ and GR, respectively. After having constructed the image Bi – I later come back to how he did this – Commandino presented a long proof for the correctness of his procedure (Commandino 1558, fol. 3r–4r). His proof implicitly contains the result that I have called the division theorem (page 50), namely that when OP = d and IB = a, then the image Bi of B is the point on PI determined by IBi : Bi P = a : d.

(iv.1)

As noted, the division theorem is fundamental to understanding some of the early perspective constructions, however, the Renaissance and early

p P g O

Ci G Q

F

C

Bi

I

B

R

FIGURE IV.29. The idea behind Commandino’s special rabatment.

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modern writers did not point out that their constructions were inspired by this result. This also applies to Commandino, who went through many calculations before arriving at (iv.1). Nevertheless, I believe that precisely this result may have guided him to devise his special construction. Before a study of Commandino’s construction it is appropriate to be familiar with his quite special rabatment (figure IV.30), which includes the ground plane g, an elevation, and the picture plane p. The upper left-hand part of the diagram represents the vertical plane OPQF (figure IV.29), the lower right-hand part represents the ground plane with IQ as part of the ground line, and the upper right-hand part the picture plane in which QC is the ground line. This rabatment can be understood by imagining the following two rotations (figure IV.29). First the rectangle QCBI is rotated 90˚ downwards around QC so that it lies in the vertical plane OPQF, and next the latter plane is turned around PQ into the p. In Commandino’s own illustration, reproduced in figure IV.31, the three involved planes actually overlap each other. Commandino’s procedure of representing g, p, and an

P

O

E

FIGURE IV.30. An adaptation of a part of one of Commandino’s diagrams (cf. figure IV.31). I have mirrored his drawing and used different letters.

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Q I

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FIGURE IV.31. Commandino’s own illustration of how to find the image of the rectangle abgh. Commandino 1558, fol. 2v.

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elevation in one drawing plane was actually similar to the procedures applied by Piero (figure II.18) and Vignola (figure IV.14). Let us now turn to Commandino’s determination of the image Bi of the point B (figure IV.30), and let us assume that the two lengths determining B are given by QC = IB = a and QI = CB = b. In the part of the diagram representing the picture plane he constructed the points K and S on the line FQ so that QK = QI and KS = IB = a,

(iv.2)

and then let T be the point of intersection of PK and OS. With the division theorem (iv.1) at our disposal, we can see straightaway that T is the image of B: The equality QK = QI in (iv.2) yields that PK in figure IV.30 corresponds to PI in figure IV.29; the similarity of the triangles TPO and TKS, together with the second equality in (iv.2), implies that KT : TP = KS : OP = a : d – which means that the point T is so constructed that it divides PK in the correct ratio. Commandino’s own proof is much more complicated, among other reasons because, as noted, he did not take the convergence rule, and hence not the result that T is situated on PK, for granted. Commandino defined the image of the line segment BC (figure IV.29) as the intersection of p and the triangle OBC and introduced the point of intersection, E, of OC and PQ (figure IV.30). After having involved many similar triangles and a considerable number of proportions he came to the conclusion that his constructed line segment ET is identical to the above-mentioned intersection, that is, the image of BC. In dealing with a given point outside the ground plane Commandino used a procedure that corresponds to regarding the horizontal plane through the given point as the ground plane, and then applying the method just described. Having been motivated to study perspective by considering projections of circles, Commandino naturally paid much attention to the perspective images of circles, listing the conditions under which these would be either circles or ellipses (ibid., fol. 6r–6v, 12v–13r, 15v–16r, 17v). He also looked at segments of circles and included cases in which they would be mapped as parabolas or hyperbolas (ibid., fol. 6v–8v, 13v–14r, 16r). Although Commandino’s main concern was to throw plane figures into perspective, he also touched upon methods for constructing the perspective image of a polyhedron such as a pyramid or a cube (figure IV.32). Commandino’s attitude to the plan and elevation construction was different from the views held by Piero and Vignola in the sense that he thought that a proof is needed to establish its mathematical correctness. He provided a proof that is more elegant than the one he gave for his first method, but it is difficult to follow his arguments visually because of his special rabatment (ibid., fol. 4r–4v).

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FIGURE IV.32 Commandino’s examples of how to throw a pyramid and a cube into perspective. Commandino 1558, fol. 18v.

Commandino’s Influence

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ince Commandino’s contribution to perspective was part of an academic work on an astronomical projection, written in Latin in a mathematically demanding style, its natural fate would have been to exist unobserved by practitioners of perspective, unknown to all but a handful of mathematicians. However, this does not seem to have happened. I have not noticed other mathematicians than Danti mentioning the work (Vignola 1583, preface) whereas several mathematical practitioners did mention it – a phenomenon that can probably be attributed to Barbaro. In his Pratica della perspettiva (1568), which will be presented in section IV.6, he referred scholars to Commandino’s work and warned non-scholars that it was difficult reading (Barbaro 1569, 3). The architects and artists presumably took Barbaro’s warning seriously and did not read Commandino, but they still mentioned him. They may have found it reassuring that a mathematician had proved at least some perspective procedures to be correct.

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IV.5 Another Mathematical Approach – Benedetti’s Contributions

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n 1585, Giovanni Battista Benedetti (1530–1590), a well-esteemed court mathematician with the duke of Savoy in Turin, published the treatise De rationibus operationum perspectivae (On the reasons for the operations in perspective), opening it with these words:

Since no-one so far (that I know) has given a perfectly correct account of the reasons underlying the operation of perspective constructions, I thought it worthwhile to give some discussion of them.15 [FieldS 1985, 80]

He was not the first to seek to prove the correctness of perspective constructions, as we have seen, but he related his proofs more clearly than his predecessors had to the three-dimensional model defining a perspective projection.16

Benedetti’s Alberti Construction

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enedetti first presented an Alberti construction, and to prove that it is correct he referred to the diagram reproduced in figure IV.33. To achieve a more familiar viewing angle and use the familiar letters, I have shown it in a rotated version with altered letters in figure IV.34. The point O is the eye point, OF is perpendicular to the ground plane g, AB is a given line segment in g parallel to the ground line GR (which in this particular example is a line segment forming a rectangle together with AB), the plane through GR perpendicular to g is the picture plane p, and the line through F perpendicular to AB cuts GR at Q and AB at K. Benedetti defined the point P (the principal vanishing point) in a rather complicated way, involving the two oblique faces of the prism, but in the end he obtained P as the orthogonal projection of O upon the picture plane p, and called it oculus (the eye). Finally, L is the point of intersection of OK and PQ. Benedetti defined the image AiBi of the line segment AB as the intersection of p and the triangle OAB, and then used his prism to show that his Alberti construction did indeed produce this image. Like Euclid, Benedetti left no details to the reader’s imagination, so he needed several pages for his proof, which basically consists of showing that AiBi lies on the horizontal line through L, and that Ai and Bi lie on PG and PR, respectively. The idea of involving a prism in mathematical considerations about perspective images was later taken up by Simon Stevin (figure IV.35). 15

Cum nullus adhuc (quod sciam) veras internas; causas operationis perspectivæ perfectè docuerit, operæprecium existimavi aliquá de ijs disputationem suscipere. [Benedetti 1585, 119] 16 For more on Benedetti’s consideration of the three-dimensional space and for other presentations of Benedetti’s work on perspective, see FieldS 1985; FieldS 19871; and FieldS 1997, 161–170.

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FIGURE IV.33. Benedetti illustrating the three-dimensional situation related to his Alberti construction. Benedetti 1585, figure A, 120.

P p

A

Ai G

K

L

O

Bi

g

B

Q R F

FIGURE IV.34. A rotated version of figure IV.33 with altered letters.

Benedetti on Pointwise Constructions

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aving established the correctness of his Alberti construction, Benedetti gave some examples of how to apply it, the first few of which deal with rectangles. He was, however, aware of and emphasized the fact that mathematically, the essential problem is to be able to construct the image of an arbitrary point (Benedetti 1585, 124). He actually extracted a pointwise construction of his Alberti procedure for a point

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FIGURE IV.35. Stevin’s prism. Stevin 16051, 18.

O

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Si U

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FIGURE IV.36. Benedetti’s Alberti procedure for constructing the perspective image of a point. Above the ground line GR, Benedetti drew the picture plane (at right, containing P, Q, and R) and (at left) the line segment OF, which can be seen as the OF from figure IV.34 rotated into the picture plane around PQ, implying that O serves as an Alberti point. Below GR he placed the ground plane, in which the point S is given and T is its orthogonal projection upon GR. Benedetti let the point U on GR be determined by QU = ST, found V as the point of intersection of OU and PQ, and found the image Si of S as the point of intersection of TP and the transversal through V. Benedetti used cutting planes to prove that his construction is correct. We can verify this by noticing that the point V determines the position of the transversal that is located at the distance OU = ST from the ground line, and that TP is the image of the orthogonal ST. Adaptation of figure F in Benedetti 1585, 126.

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FIGURE IV.37. Benedetti’s diagonal construction. The point i is the principal vanishing point and qd the ground line. The point b is given in the rectangle quad, the perspective image of the latter is known to be qerd, and the image of b is the point sought. Benedetti solved this problem as follows. Through b he drew two lines passing through two of the vertices of the rectangle – in this case through q and u. He let g and f be the points in which qb meets ua and ub meets qd, and let the point n on qd be determined by qn = ug. He then drew the line ni meeting er at c, proved that c is the image of g (which fits in with the fact that gn is mapped upon ni), and concluded that qc is the image of qg. He similarly found that fe is the image of uf, and hence the point of intersection t of fe and qc is the required image of b. Figure K in Benedetti 1585, 129.

given in the ground plane (figure IV.36). He proceeded to determine the image of a point above the ground plane, given by its foot in the ground plane and its height (ibid., 126–128). Benedetti also presented a pointwise construction that is similar to Piero della Francesca’s diagonal construction in presupposing the point to lie in a rectangle whose image is already known. However, Benedetti used other lines than Piero to characterize the position of a given point (figure IV.37).

Benedetti and Convergence Points

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o draw the perspective image of a horizontal rectangle that has no sides parallel to the ground line, Benedetti applied yet another procedure. His illustrations of this are interesting, because they show that Benedetti, like Danti and Vignola, had at least a partial understanding of the general concept of a convergence point. Benedetti’s own illustrations are reproduced in figure IV.38, and in figure IV.39 I have shown the three-dimensional

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FIGURE IV.38. The “figura corporea” and the “figura superficialis” D in Benedetti 1585, 123.

configuration from another viewing angle. In the ground plane g Benedetti let the rectangle quad, which has no sides parallel to the ground line gr, be given and sought to determine its perspective image. The picture plane is p, o is the eye point, its orthogonal projection upon g is p, and oi and px are parallel to

i p

g

o

a u g

d X dg

q

p qg r

FIGURE IV.39. The basic situation in the previous “figura corporea” seen from another angle.

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151

the sides ad and qu of the rectangle. The points dg and qg (which in Benedetti’s figures are marked with d and q) are the points in which ad and uq meet the ground line. It is clear from figure IV.38 that Benedetti was well aware that the images of the lines ad and uq meet at the point i. He actually stressed that this point is not the oculus (the principal vanishing point), but a “punctum perspectivae” (point of perspective, Benedetti 1585, 122–123). However, he did not similarly notice that the images of the other pair of parallel lines au and dq also have a convergence point on the horizon. Thus, in his drawing (figure IV.38 ) their images ez and fm do not meet on the horizon it. Despite the fact that Benedetti’s drawings in figure IV.38 are not made according to the rules of perspective known today, his description of the construction is correct. Actually the construction itself is also interesting, because implicitly, it contains a pointwise construction that was presented by Guidobaldo del Monte fifteen years later as one of twenty-three methods of construction, and which Stevin chose a few years later as one of his main constructions. The principle behind the construction is explained in figure IV.40.

p

i

g

o s m

g x

q p

w qg r

FIGURE IV.40. A Benedetti result. Benedetti’s proof that the construction of the image of the point q in the ground plane g (figures IV.38 and IV.39) is correct contains the following observation. Let o be the eye point, p its orthogonal projection upon g, p the picture plane, and gr the ground line. Let any line – not parallel to p – through q in g cut gr in the point qg (in Benedetti’s example this line was given as a side in a rectangle, but it could have any direction). Let i be the point of intersection of p and the line through o parallel to qqg, let qp cut gr at w, and let the perpendicular to gr through w cut iqg at m. This point m is the image of q. Benedetti used cutting planes to prove this result, and we can verify the result by reasoning that since pq is the orthogonal projection of the visual ray oq upon g, the image of q lies on the vertical through w. That it also lies on iqg follows from a theorem we will meet later – I call it the main theorem – which applies the fact that i is the vanishing point of the line qqg. Benedetti himself did not create a general construction based on his result.

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Benedetti’s Influence

B

enedetti may not have realized all the subtleties of perspective, but he showed more understanding of the discipline than most of his predecessors. As pointed out by Heinrich Wieleitner and even more eminently by J.V. Field, Benedetti is indeed a remarkable figure in the history of perspective (WieleitnerS 1918, FieldS 1985 and FieldS 19871). He came to the conclusion that theoretically, a perspective construction is a question of determining the image of an arbitrary point, and he incorporated more than the theory of similar triangles from the body of geometrical knowledge in his proofs. In particular he drew upon the results from three-dimensional geometry. With this approach his natural audience would have been his fellow mathematicians, but I have not found any examples of them referring to Benedetti. This does not necessarily mean that they did not read him, for it seems to be a rule – almost without exception – that mathematicians writing on perspective made no references to other mathematicians. I have the distinct impression, although I cannot prove it, that Guidobaldo and Stevin read Benedetti’s work and found it inspirational.

IV.6

An Encyclopedia on Perspective – Barbaro’s Book

D

aniele Barbaro (1514–1570) (figure IV.41), whose name has already cropped up a few times, apparently took an enthusiastic interest in perspective without gaining a deeper mathematical understanding of the subject. He was a well-educated scholar from the Venetian nobility and had a distinguished diplomatic career, first within the political administration – spending part of this period as ambassador to England – and later in ecclesiastical circles – earning the title Patriarch of Aquileia and bringing him to the Council of Trent. His many activities included commenting on, and supervising the publication of, an edition of Vitruvius’s work on architecture. In 1568 Barbaro published La pratica della perspettiva (The practice of perspective). On the title page of the work he claimed it was beneficial reading for painters, sculptors, and architects. He apparently wanted to share with them all the wonderful things he had read about perspective, choosing Italian as his language, contrary to the other cinquecento scholars writing on perspective, namely Commandino and Benedetti (not counting Danti, who commented upon a treatise written in Italian). Barbaro was well acquainted with the perspective literature, and covered almost all the aspects of the discipline known at the time, and in a way that makes his book almost encyclopedic (for another presentation of Barbaro’s book see PeifferS 2002, 104–106).

Barbaro’s Sources

B

arbaro acknowledged being inspired by other authors, particularly Piero della Francesca, Dürer, and Commandino (Barbaro 1569, fol. A2r). He

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FIGURE IV.41. Tiziano Vecellio, known as Titian, Daniele Barbaro, 1545, Museo Nacional del Prado, Madrid.

did not reveal, however, that this inspiration often led him to copy parts of his sources. I have not tried to trace the origin of all Barbaro’s material, but I am convinced that most of his figures and much of his technical text can be found in earlier works. We will meet some examples of Barbaro’s use of Dürer’s figures, but there are others as well. Similarly, a number of his other diagrams are taken from Piero. He also directly copied Serlio’s two drawings of theatre stages (ibid., 156–157). The source for a substantial part of Barbaro’s theoretical considerations is Piero’s De prospectiva pingendi (ibid., 31–39). There is even an example of

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Barbaro reproducing a copying mistake in a manuscript.17 As indicated above, I suspect that Barbaro’s reason for remaining so faithful to his sources was that he had not quite understood the theoretical arguments. Barbaro’s examples of throwing objects into perspective deal with plane figures, regular and semiregular polyhedra, architectural elements, and shadows. For this he took many of his illustrations from Dürer – figure IV.42

FIGURE IV.42. Barbaro’s version of Dürer’s cube (cf. figures V.36 and V.37). Barbaro 1569, 178. 17

In one of his geometrical arguments Piero deduced a relation that in modern notation is the following (BE + FH) : CG = HG : AG (Piero Pros, 76). Some manuscripts, however, render the right-hand side as AG : HG, which is also what Barbaro did (Barbaro 1569, 32).

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FIGURE IV.43. Barbaro’s use of Dürer’s illustration of how to throw a lute into perspective. The original by Dürer is reproduced as figure V.49. Barbaro 1569, 191.

being one example – but in general Barbaro had more text than Dürer. Additionally Barbaro described some instruments for making perspective drawings, among them one of Dürer’s devices (figure IV.43). Dürer presumably also inspired Barbaro to include a non-perspective section on proportions in human bodies while Commandino gave Barbaro the impulse to take up the subject of stereographic projections, and somewhere he got the idea of touching upon the theme of constructing sundials. Finally, Barbaro also included a section on anamorphoses in his book (ibid., 159–161), calling it “the beautiful and secret part of perspective”. Apparently he had not found a clear text around which he could build his presentation. The result was that he gave impression that the art of making anamorphoses is in itself a secret.

Barbaro on the Regular Polyhedra

O

ne of the largest sections of La pratica della perspectiva concerns regular and semiregular polyhedra. Barbaro’s motivation for devoting so much space to these solids most probably springs from the fact that during the Renaissance many people found them really fascinating. These polyhedra were regarded as part of a mathematical world that was both divine and magical, and they became favoured decorative objects in Renaissance art – and even more widespread in manneristic art.

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s

l

d r

FIGURE IV.44. A regular pentagon. In his determination of the plan and elevation of a dodecahedron, Barbaro involved line segments defined by the regular pentagon that constitutes the faces of the dodecahedron. I have denoted the lengths as follows. The length of a side by s, of a diagonal by d, of the radius of the circumscribed circle by r, and the distance between a vertex and the opposite side by l.

FIGURE IV.45. Barbaro’s preparations for making a perspective dodecahedron. At the bottom is the plan of the dodecahedron, and at the top all the vertices are shown in a special representation. For the plan Barbaro applied the result that it is composed by two regular decagons, and that the radii of their circumscribed circles are in the same ratio as the diagonal d and the side s in the pentagon presented in the previous figure. This is the ratio later referred to as “the golden section”. To construct the elevation Barbaro applied several results that are not quite obvious, but nonetheless correct figure IV.46). Barbaro 1569, 51.

157

6. An Encyclopedia on Perspective – Barbaro’s Book FIGURE IV.46. An elevation of the dodecahedron. It is obtained by projecting the dodecahedron upon the plane that is perpendicular to the edge fh (figure IV.45) and passes through the point b. To determine the three levels of the vertices above the plan, Barbaro involved the line segments, presented in figure IV.44, and the radius r of the circumscribed circle to the largest decagon in the plan of the dodecahedron (figure IV.45) Barbaro let the first horizontal level be determined by the third side in the right-angled triangle with the hypotenuse s and the horizontal side r-r. He constructed the second level in a similar fashion using the right-angled triangle with the hypotenuse l and the horizontal side r – (l – r). Rather than determining the third level by using that its distance to the second level is equal to the distance between the first level and the plan, Barbaro characterized it as having the distance r + r to the plan. His results are correct, but not obvious.

l s r r − (l − r)

d,k

b

f,k

q,s

o,u

m

FIGURE IV.47. An elaboration on the previous elevation of the dodecahedron. Here I have marked the projection of all the vertices shown in Barbaro’s upper diagram in figure IV.45. From the plan and elevation Barbaro produced the perspective dodecahedron reproduced in figure IV.48.

r−r

l

p,t

n,x

c,l

e,i

r

g

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FIGURE IV.48. Barbaro’s final diagram showing the perspective dodecahedron. Barbaro 1569, 52.

Barbaro applied the plan and elevation construction to throw polyhedra into perspective. As mentioned in section II.16, this procedure works straightforwardly when the plan and elevation are known, but in Barbaro’s day determining these projections could pose quite a mathematical challenge. The plans and elevations of the regular polyhedra, apart from the dodecahedron, can be obtained without much trouble from Euclid’s description of these solids in book thirteen of his Elements. Euclid’s introduction of the dodecahedron, on the other hand, is not very helpful for solving the problem of determining its plan and elevation, and I am not aware of any other publication available at the time that would have been useful. Barbaro nevertheless presented a correct solution. Considering how the rest of his book fails to demonstrate any mastery of mathematics on his part, I doubt that Barbaro solved the problem himself, and I continue to wonder about his source. In figures IV.44–IV.48 are presented the principles Barbaro used for his construction.

IV.7

A

The Italian Pre-1600 Contributions to Perspective

lthough perspective was applied and described in France, Germany, and the Low Countries in the sixteenth century, as we shall see in the coming

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FIGURE IV.49. Donatello, Miracle of the Ass, mid-to-late 1440’s, Basilica di Santo, Padua.

chapters, Italy played the dominant role in the history of the subject during the Renaissance. Italian artists and scholars introduced perspective and guided its development until it had become a well-established discipline. The process leading up to this situation took place in artists’ workshops and scholars’ studies – with some interaction. Not all is known about the work of the scholars, but at least we have some of their texts, whereas our knowledge of what happened in the workshops is frustratingly incomplete. It is, however, beyond any doubt that the workshops were instrumental in promoting the awareness and use of perspective techniques, and in developing new methods of constructions. It is striking that one of the popular practical applications was never described in books on perspective, namely how to construct the perspective composition applied in relief sculptures, like the one by Donatello shown in figure IV.49. In the development of perspective constructions and the understanding of their mathematical foundation the Italian Renaissance is especially important. There are countless ways of performing a perspective construction, but relatively few constructions are easy or convenient and most of those were actually developed in pre-1600 Italy. Initially the perspectivists’ theoretical understanding was not deep, whereas their intuition was good and their mistakes surprisingly few. Alberti declined to describe the theoretical foundation of his construction, his excuse being that he was writing as “a painter speaking to painters” (page 29). However, he may also have had some difficulty making his understanding explicit. Piero took up the challenge of explaining why certain perspective techniques work. He was formidable at keeping complicated three-dimensional configurations under control, yet he still did

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not manage to explain why the distance point construction is correct. Gradually the level of theoretical insight became more profound, and by the end of the sixteenth century the seeds for a complete understanding were scattered around in various writings. The time had grown ripe for the emergence of a general theory of perspective, and such a theory was indeed created by Guidobaldo del Monte. With his book Perspectivae libri sex (1600) he became, as I argue in chapter VI, the father of the mathematical theory of perspective. This accomplishment would have been unlikely, if not unthinkable, without the preceding fruitful development in Italy.

Chapter V North of the Alps Before 1600

V.1

The Introduction of Perspective North of the Alps

T

here can be no doubt that Italian sources brought about the introduction of perspective constructions North of the Alps: a connection to Italy can be established for all of the early influential non-Italians who published works on perspective. However, we do not know much about the channels through which the knowledge of perspective spread. As was the case in Italy, the texts from the North do not allow us to follow a continuous development, but indicate that essential steps were taken in workshops, or other places, without being recorded. In this chapter I deal with the non-Italian publications on perspective that appeared before 1600. This brings us to the parts of Europe where French, German, and Dutch were spoken. As for Britain, no book on perspective was printed there before the seventeenth century. It happened that a Northern perspectivist found his inspiration in a book from abroad, as we shall see. I nevertheless treat the tracts by French and German authors separately, because they are related to two different developments – of which the French was the most advanced. Indeed, the literature does not give much impression of interactions between the presentations of perspective in the French and German books. Along with the French treatises I describe one from the Southern Netherlands, and I conclude the chapter with a description of a contribution from the Northern Netherlands.

V.2

Viator and His Followers

I

n sixteenth-century Italy we met humanists, painters, architects, and mathe maticians writing on perspective. During the same period in France, one person from each of the first three categories composed a book entirely devoted to perspective, while I have found no traces of the subject having been included in other French books. The three authors were the scholar Jean Pélerin – known as Viator – the painter Jean Cousin, and the architect Jacques Androuet du

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Cerceau. The work of Cerceau was based on the same ideas as Viator’s, as was, to some extent, the contribution from Southern Netherlands written by Fortius Ringelberg.

Viator

T

he first printed book dealing exclusively with perspective was Viator’s De artificiali perspectiva (On artificial perspective). It was also the first known work on perspective written outside Italy. Viator’s work was published in 1505 in Toul with parallel, and rather brief, texts in Latin and French. Only four years later a new edition appeared in which Viator had augmented his text – but still kept it to less than seven pages – and replaced some of his drawings with new compositions. A third edition with just a few changes was published in 1521.1 Before commenting upon the contents of De artificiali perspectiva, let me look into Viator’s sources of inspiration. A fair amount is known about the life of Jean Pélerin, alias Viator (c. 1440–1524), but we lack information that can shed light on how his ideas on perspective developed. He was an ecclesiastic who studied law and then worked as a diplomat in the circles around King Louis XI; later Viator held several administrative posts and became a canon in Toul during the 1490s. Despite his many duties, Viator found time to cultivate a wide spectrum of interests. He studied Ptolemy’s Geography and was involved in architectural projects, which included designing parts of the cathedral of Toul and a number of elaborate tombstones (Brion-GuerryS 1962, 363, 365, and 399). His professional life led him to travel extensively, but whether that is what earned him the name Viator or whether it is some kind of translation of Pélerin remains unclear (ibid., 414).

FIGURE V.1. Viator’s construction of a perspective square. He used both distance points and the principal vanishing point. The drawing contains the initial steps in the construction of a second perspective square. Viator 1505, fol. Avr.

1

For a comparison of the three editions, see Brion-GuerryS 1962, Table de Concordance.

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FIGURE V.2. The image of a pavement of square tiles obtained by a distance point construction. Viator 1505, fol. Avir.

Viator’s travels brought him to several French estates where he had ample opportunity to get acquainted with the world of art and artists. He himself named some twenty artists (ibid., 415) – but no well-known perspectivists. In the second edition of De artificiali perspectiva he wrote that he had learned perspective by studying books and works of art and by discussing with savants, but in this connection he did not mention any names (Viator 1509, fol. Aixv). To trace Viator’s sources for perspective it would be helpful to know whether he also visited Italy. According to Liliane Brion-Guerry, there is no certain answer to this question (Brion-GuerryS 1962, 356, 362, and 414). Although it remains an enigma where and by whom Viator was schooled in perspective, we can be reasonably sure that what he learned had its roots in Italy. He himself wrote that in pictorial art the Italians took the lead (les Italiens tiennent la palme, Viator 1505, fol. Fir). During his studies on perspective Viator probably became acquainted with more than one way of making perspective constructions. He presented only

FIGURE V.3. Viator’s demonstration of how his distance point construction can provide the image of a system of squares whose sides are diagonals. Viator 1505, fol. Avir.

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one method, however, namely a distance point construction, based on the construction of the image of a square (figures V.1–V.3). Since this method had not been thoroughly presented in the literature that preceded Viator’s book – at least not in the treatises that had been published – it is likely that Viator learned it from someone who applied it in practice. In the version of a distance point construction, Viator was taught, the distance between the principal vanishing point and a distance point was not distinctly related to the distance. Thus Viator stressed that the distances between the principal vanishing point (point principal ) and the two distance points (tiers points – third points) should be equal, but he did not explicitly set these distances equal to the distance. He only mentioned that the distance points should be located closer to the principal vanishing point when the view is near, and further away when the view is distant (ibid., fol. Aiir and fol. Fiv; Viator 1509, fol Aii). As an introduction to perspective constructions Viator’s text is very short, whereas his illustration material is rich, so naturally there are many details in his drawings that he did not explain. He did not, for instance, give much guidance to the foreshortening of heights in a perspective composition. Nor did he explain the use of the scale on the ground line that can be found in several of his illustrations, such as the one reproduced in figure V.4. This and other diagrams suggest that Viator used the scale not only for measuring lengths of line segments situated on the ground line, but also for making a (mathematically faulty) determination of perspective images of horizontal line segments situated on lines making an angle of 45° with the ground line – the type of

FIGURE V.4. The perspective image of an oblique row of tiles. The diagram is not constructed very precisely, but a reasonable reconstruction leads to the hypothesis that the sides of tiles are diagonals. The ground line has indications of a scale for measuring lengths, which appears to have been used in the construction – perhaps in the same way as explained in figure V.7. Viator 1505, fol. Biiir.

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FIGURE V.5. An example of Viator’s architectural compositions. Viator 1505, fol. Dv.

lines I call diagonals. I shall be more explicit on this point shortly, when I turn to Ringelberg, who described a procedure I assume is the same as the one Viator used. The contrast between Viator’s brief text and his generous number of illustrations – which is not uncommon for his time – may reflect the opinion that perspective was better learned from looking at drawings than from reading explanations. There are solutions in Viator’s drawings that do not fit in with the general laws of perspective, formulated about a hundred years later. Viator himself

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foresaw the possibility of mistakes in his book, taking the precaution of pleading old age – he was in his sixties when De artificiali perspectiva appeared – and the engravers’ inexperience in reproducing perspective drawings (Viator 1505, fol. Fiiiv). Nevertheless, his selection of drawings is quite impressive. His motifs, he recounted (ibid.), were partly taken from artists’ paintings, partly created from his own imagination, and partly based on buildings he had seen, as shown in figure V.5. Viator’s drawings were noticed outside France, not least in Germany. In his encyclopaedia for young people, Margarita philosophica nova (The new philosophical pearl), the prior of the Carthusian monastery in Freiburg, Gregor Reisch (c. 1467–1525), incorporated parts of De artificiali perspectiva (Reisch 1508). The Margarita was a successful enterprise that went through many editions and made Viator’s drawings well-known – but not his name, for Reisch did not give credit to his source. The German engraver Jörg Glockendon was similarly attracted by Viator’s book without feeling any need to acknowledge his inspiration; in fact, he published it as his own work in 1509 with the title Von der Kunnst Perspectiva (On the perspective art). These German plagiarisms of Viator’s material may have inspired some artists and practitioners in their work with perspective, but Viator’s approach to perspective was not followed up in later German tracts on the subject.

Ringelberg

P

resumably inspired by Viator, but far from completely dependent on him, Joachim Fortius Ringelberg (†1556) presented perspective in his Opera, published in Lyon in 1531. Ringelberg came from Antwerp and began his career as an engraver attached to the court of Maximilian I. He studied at various universities and composed many tracts in the fields of the mathematical sciences and mysticism, as well as astrology. In his Opera he devoted some twenty pages to perspective under the heading of optics. Ringelberg’s background as an engraver – perhaps combined with the influence from Viator – may have convinced him that the way to learn perspective was to see it performed, rather than to read about it. The contemporary North European painters applying the art of perspective, such as Jan Gossaert, also called Mabuse (figure V.6), may have had the same attitude. At any rate Ringelberg’s section on perspective consists of fifty short lines of text and thirty two illustrative drawings, which demonstrate better than Viator’s compositions the single steps involved in a construction. Ringelberg opened with a distance point construction of a perspective square, and contrary to Viator he stressed that the distance between the principal vanishing point (oculus – eye) and a distance point (tertia puncta – third point) should be equal to the distance (Ringelberg 1531, 459). He elaborated his distance point construction to apply to a rectangle whose sides are diagonals.

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FIGURE V.6. In this painting, Mabuse clearly demonstrated an awareness of the convergence rule. He also seems to have used the distance points as convergence point for the diagonals. Mabuse (Jan Gossaert), St. Luke Painting Maria, c.1515, National Gallery, Prague.

Ringelberg’s procedure, which may be similar to Viator’s earlier-described method, runs as follows (figure V.7). To construct the image of a rectangle with the sides 15 and 9 he used the scale on the ground line to mark two line segments that have a point in common and the lengths 15 and 9. He then connected the end points of these line segments with the distance points, and did indeed obtain the image of a rectangle whose sides lie on diagonals. However, the sides have lengths 15 2 2 and 9 2 2 rather than 15 and 9 (figure V.8). For the pictorial effect the factor 2 2 is of no consequence, since it is difficult

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FIGURE V.7. Ringelberg’s construction of the image of an oblique rectangle. It is explained in the caption of the next figure. Ringelberg 1531, 462.

anyway to judge the lengths of original line segments in a perspective composition. Ringelberg applied the method in several examples, including some in which he threw three-dimensional objects into perspective (figure V.9). His construction of height was correct – not involving the factor 2 2 – so in practice he applied different units for horizontal and vertical lengths.

FIGURE V.8. Explanation of Ringelberg’s construction illustrated in the previous figure. The points D1 and D2 are distance points, and correspond to the points marked 12 by Ringelberg in figure V.7. The quadrangle KLMN is obtained in the following way. On the ground line GR, the line segments AK and KB are made equal to given lengths – Ringelberg set these as 15 and 9. The points L, M, and N are the points of intersection of the lines connecting the points A, K, and B with D1 and D2. Ringelberg claimed that his bold quadrangle (corresponding to the quadrangle KLMN) is the image of a rectangle with sides equal to AK and KB, whereas I claim it is the image of a rectangle with sides of the lengths AK 2 2 and KB 2 2 . In order to prove this, I note that AL and KL are images of diagonals, and that AK is situated on GR. This implies that ALK is the image of an isosceles right-angled triangle, whose hypotenuse lies on the ground line, and hence also implies that LK is the image of a side that has a length of 15 2 2 . The result for KN is obtained similarly.

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FIGURE V.9. An example in which Ringelberg added heights to a perspective plan. Ringelberg 1531, 466.

Cerceau

B

ack in France it took 55 years before a successor to Viator’s De artificiali perspectiva arrived. This work, written by Cousin, shows that much development had taken place in the interim. I will soon turn to this, but first treat the last book in the Viator style which is Cerceau’s Leçons de perspective positive (Lectures on positive perspective) from 1576. The title itself indicates inspiration from Viator, who called one of his sections Abregie de perspective positive (Summary of the positive perspective, Viator 1505, fol. Fir) – although neither of the authors explained what they understood by “positive perspective”. Jacques Androuet du Cerceau (c. 1515–c. 1585) was an architect and engraver who was first and foremost known for publishing drawings of a large number of French buildings, several of which no longer exist. He presented the buildings by ground planes, façades, and perspective views. According to Naomi Miller, Du Cerceau’s primary aim was to disseminate the heritage of Classical art and propagate the style of the Italian Renaissance by making works accessible without foreign travel and by providing practical handbooks for architects, painters, sculptors, designers, craftsmen, cabinetmakers, goldsmiths and jewellers. [MillerS 1996, 350]

Cerceau followed Viator in basing many of his constructions on a distance point construction of a square. Moreover, like Viator, he placed much more emphasis on the drawings than on the text – the 60 Leçons in his book being so brief that the text taken together covers less than 15 pages. On the other hand, each is supported by excellent illustrations, like the ones reproduced in figures V.10 and V.11. In choosing his motifs Cerceau worked independently of Viator, and with his strong interest in architecture he naturally included some buildings, one of which can be seen in figure V.12.

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FIGURE V.10. Cerceau’s illustration of how the horizon of a picture changes with the position of the drawer. At the top he has shown a bird’s-eye view, and at the bottom a worm’s-eye view. Cerceau 1576, Leçon 1.

Cerceau’s illustrations show that he was fond of using a diagonal method (figure V.13) – a method upon which he did not comment at all. Looking at this method with Piero’s procedure in mind (figure II.20), one first gets the impression that Cerceau’s method was faulty.2 However, a closer examination 2

William M. Ivins wrote that the method contains “a crucial error” (IvinsS 1975, 40). See also note 19.

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FIGURE V.11. Cerceau showing how to add heights to a plan in perspective. The horizon of the composition lies in the same horizontal plane as the eyes of the people depicted on the ground plane. Cerceau 1576, Leçon 36.

leads to the conclusion that Cerceau’s diagonal construction is correct (figure V.14) and also avoids the problem of reversing – which will be discussed in section VII.7. As we shall see later in this chapter, the method applied by Cerceau was published by Dürer in 1538. It is unclear whether Cerceau learned it from Dürer, or whether the two perspectivists had a common unpublished source.

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FIGURE V.12. One of Cerceau’s architectural compositions. Cerceau 1576, Leçon 57.

V.3

J

Cousin

ean Cousin (c. 1500–c. 1560) was active in surveying, fortification, making designs and drawings for tapestry, embroidery, weaving, glass painting, etching, engraving, and woodcarving; besides which he supposedly produced a handful of paintings (ZernerS 1996). Perspective compositions were often featured in his designs, and according to Henri Zerner Cousin borrowed some of his motifs from “Italian prints and from Sebastiano Serlio’s architectural treatise, but in a fully informed way” (ibid., 67). Towards the end of his life, Cousin published a book that demonstrated how well-versed in perspective he had become.

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FIGURE V.13. Examples of Cerceau applying a diagonal method. Cerceau 1576, Lec,on 14. FIGURE V.14. The diagonal method used by Dürer and Cerceau. Given are the principal vanishing point P, a square ABCE, and this square’s perspective image AiBiCiEi, situated so that AiBi lies on the ground line. To find the image of a point K, Dürer and Cerceau did as follows. They drew the lines through K parallel to AB and AE intersecting the diagonal BE at L and the side EC at M, and then drew the line through L parallel to AE intersecting EC at N. Let R and S be the points at which the lines KM and LN meet AiBi. The two perspectivists used that the lines KM and LN are depicted in RP and SP, found the image Li of L as the point of intersection of SP and BiEi, and found Ki as the point of intersection of RP and the transversal through Li. In their drawings Dürer (figures V.52 and V.53) and Cerceau (see previous figure) let EC and AiBi coincide, which could give the false impression that the line EC was situated on the ground line, and hence that there was something wrong with the method.

P

Ei

Ai E

Ki R M

K A

Ci Li S N

Bi C

L B

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FIGURE V.15. The frontispiece from Cousin’s Livre de perspective, fol. Aiir. One part of it shows an image in a vertical plane, whereas another part illustrates a perspective composition in a horizontal plane.

Cousin’s Livre de perspective (Book on perspective, 1560) is a very remarkable and intriguing work. It is remarkable because it reflects a knowledge of perspective that is more advanced than anything found in any other published text from this period. And it is intriguing because this knowledge is not stated explicitly, but found between the lines and in the illustrations. In interpreting this phenomena I see two possibilities. One is that Cousin himself had made new discoveries, but had difficulty expressing his thoughts clearly. The other is that

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he repeated insights he had heard from others without quite grasping the details. I am inclined to support the second explanation, as I find Cousin’s material rather heterogeneous, giving the impression that he picked it up from different sources without coordinating the different approaches – an example of which we shall meet shortly. Cousin’s sources for the novelties are unknown to me, and could either consist in oral information or unpublished manuscripts. At any rate, the new ideas he presented merit a closer examination.

Cousin’s Introduction of a Distance Point Construction

C

ousin first piqued his readers’ curiosity with an impressive frontispiece (figure V.15) and then turned to a “rule or figure by which the source and origin of the art of perspective can be affirmed” (translated from Cousin 1560, fol. Avr). The figure he used to illustrate this (figure V.16) is extremely interesting because it combines elements from an Alberti construction, a distance point construction, and the construction I presented as Gaurico’s (page 117) – and because it implicitly contains a partial proof for the correctness of his distance point construction. Cousin’s concern was to present a construction of the perspective image of a square. His figure consists of two diagrams, the upper showing the picture plane and the image of the square. The lower diagram to some extent resembles the one Piero used for his presentation of an Alberti construction (figure II.18). The square to be thrown into perspective is shown by its plan abcd. To this is added a profile in which the line ge (connecting the midpoints of ad and bc) represents the ground plane and the line ag represents the picture plane. It is new that the ground line in the profile is not represented by the lowest horizontal side of the square (dc). Cousin’s diagram also goes against tradition by including the two diagonals and the line connecting the midpoints of the remaining two sides of the square. On the ground line in the upper diagram, Cousin has drawn the line segment aidi equal to ad and its midpoint gi;3 they are images of the corresponding elements in the plan. He has furthermore chosen K as the principal vanishing point – which he called the point principal. From the profile he deduced that a line segment with an end point on the ground line and the same length as the sides of the square, in casu ge, is foreshortened as gf, and the segment gh is foreshortened as gi. He then transferred these line segments to the upper diagram by placing them on the line segment giK. He finished his perspective square by connecting the points ai and di with K, and by drawing the line through fi parallel to aidi. This procedure is a modified Alberti construction. The modification consists in letting the perpendicular, Kgi, through the principal vanishing point, represent the elevation of the picture plane instead of letting the line perpendicular to the ground line through ai do so. In this particular respect, Cousin’s procedure is similar to Gaurico’s (figure IV.1). 3

The indexes i are my addition.

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n

K

fi ii

bi

ai

gi

ci

di b

a

f i g

e h

d

c

FIGURE V.16. Cousin’s “figure by which the source and origin of the art of perspective can be affirmed”. I have enlarged his letters and changed some of those in the upper diagram to indicate that they are images of corresponding points in the lower diagram. Cousin 1560, fol. Avr.

Having constructed the perspective image of the square, Cousin went on and prolonged the diagonals aici and bidi so they meet the horizon. The points of intersection m and n are what we call distance points. Cousin introduced them as third points (tiers poincts), then stopped his commentary, leaving the rest of his method to be inferred from the figure. He expected the reader to realize that instead of involving the length gi f, one could use m and n to determine bi and ci, namely as the points in which mdi meets aiK and ain meets diK – which is the crucial step in a distance point construction. Cousin did not stress that the distances between the principal vanishing point K and the third points m and n are equal to the distance; like Viator he merely stated that mK = Kn. His drawing does show, however, that mK in the upper diagram is equal to the distance in the lower diagram. If one accepts the convergence rule of orthogonals, Cousin’s illustration also implicitly proves that his distance point construction of a square (having a side ad on the ground line) is correct in the special case where the principal vanishing point lies on the perpendicular to the ground line through the midpoint g of ad. Or in other words, it can be concluded from his diagrams that the determination of the points bi and ci with the aid of the distance points is

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correct. This can, for instance, be done as follows. The lower diagram shows that i is the image of the point h – the point of intersection of the diagonals – so the diagonal db is depicted on the line diii, which is the same as the line dim. The last result implies that the image bi is indeed the point of intersection of mdi and aiK; similarly it follows that ci can be constructed with the aid of a distance point. Although Cousin did not comment upon the details in his diagram, arguments similar to mine must have been on his mind – or the mind of the person who inspired him. This I conclude from the fact that Cousin claimed, as we saw, that he was going to present a “rule or figure by which the source and origin of the art of perspective can be affirmed” and the fact that he did not just start directly with a distance point construction. Cousin applied his distance point method to construct the images of a pavement of squares and various other horizontal figures. A couple of times he presented constructions involving the image of a square whose sides are diagonals. He also presented the construction (figure V.17), which was used by Ringelberg, and which provides the images of sides that are 2 2 times smaller than claimed (figure V.8). In other of his examples, for instance the constructions reproduced in figures V.18 and V.20, this problem does not occur. The circumstance that Cousin applied two constructions giving different results supports my impression that he benefited from various sources of inspiration for various parts of his book. For throwing a number of other horizontal figures into perspective, Cousin applied the same diagonal method as Piero had used. This means that he put the figure inside a square and used the perspective square and its diagonals to determine the images of relevant points of the figure. One of his examples of this procedure (figure V.19) deals with a spiral – which he called a volute. Cousin also knew how to apply a distance point construction in a pointwise manner. Among other things, he determined the image of the rectangle abcd in figure V.20 in this way, as explained in the figure caption.

FIGURE V.17. Cousin applying the same method as Ringelberg for constructing the images of squares whose sides are diagonals. Cousin 1560, fol. Biiir.

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FIGURE V.18. Another of Cousin’s construction involving squares whose sides are diagonals. In this case he proceeded correctly. Cousin 1560, fol. Giir.

Cousin’s Use of Points of Convergence

F

igure V.20 further illustrates that Cousin performed constructions that are based on the concept of an “accidental point” (poinct accidental, Cousin 1560, fol. Fiiir). This is a vanishing point different from the principal vanishing point and the distance points. In his diagram Cousin determined

FIGURE V.19. Cousin’s construction of a perspective spiral. Cousin 1560, fol. Eiiiv.

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di ai

ci

b

c a

d

FIGURE V.20. Cousin’s use of a pointwise distance point construction. In the ground plane is given the rectangle abcd, whose image Cousin found in the following way. The image ci of the point c is constructed as the point of intersection of the image of the orthogonal through c and the image of the diagonal through c; the images of the points a and d are found similarly. An elevated three-dimensional figure is drawn vertically above aibcidi, but is not placed directly on aibcidi – presumably to avoid obstructing the view of this perspective rectangle. Cousin 1560, fol. Iivv with some letters added.

the vanishing points for each pair of the parallel sides of the rectangle as the point of intersection of their images. He then used these vanishing points for making a pattern in the rectangle, and for constructing the images of lines that are situated above the ground line and are parallel to the sides of the rectangle. The new vanishing points introduced by Cousin lie on the horizon, which, for us, is in accordance with the fact that they are vanishing points of horizontal lines. For Cousin this was most likely a result of tacit knowledge – rather than a result of a very precise construction. In fact, we shall shortly see how, in a more complicated example, he was unable to obtain the correct positions of a vanishing point by constructing it.

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FIGURE V.21. Cousin’s construction of perspective stars. Cousin 1560, fol. Fiiir with two numbers enlarged.

Cousin also worked with vanishing points of lines situated in a vertical plane, as can be seen in figure V.21. His objective was to throw a star prism into perspective, the main problem being to construct the image of a vertical star perpendicular to the picture plane p. I have chosen to concentrate on this part of his construction – the details of which I have explained in the caption of figure V.22. In my figure I have reconstructed Cousin’s vanishing points as the points V1 and V2, applying my knowledge that they lie on the vanishing line of the vertical planes perpendicular to p, that is, on the line through the principal FIGURE V.22. Explanatory diagram relating to the previous figure. I have redrawn the part of Cousin’s illustration that concerns throwing a vertical star into perspective, and I have altered his letters. The line GR is the ground line, P the principal vanishing point, and D the left distance point. Cousin’s description was rather brief, so part of the following is a reconstruction that is in accordance with Cousin’s own drawing. Cousin’s first step was to construct the perspective image HIiJiKiLiMi of the six vertices of the horizontal star HIJKLM, whose vertices span a regular hexagon. He proceeded by determining the image of this star turned into vertical position around

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D

P

Ji  Ki Ki Li G

Mi S M

i

H

Hi

Ti L

V2 Ii

Mi

N

R

I

L T

Ji

Ii

J K

FIGURE V.22. (caption continued ) the line LM. However, he chose to construct the image on the perspective orthogonal PN rather than on LiMi. Thus, he moved Li and Mi to Li′ and Mi′. The effect of rotating the star is that the transversals through the vertices become verticals whose images are also verticals. Let S and T be the points in which the line LM cuts the ground line and the transversal through K. The rotated vertices then lie on the verticals through S, M, L, and T, and their positions on these lines can be determined as follows. The rotated H lies on the vertical through S at the distance SH from the ground line. Hence its image – moved to the right – is the point Hi′ on the vertical through N determined by NHi′ = HS. Before discussing how the image of the rotated K can be determined, let us look at the moved image of the point T. Since T is the point of intersection of the transversal through K and the orthogonal through S, its moved image Ti′ is the point of intersection of the transversal through Ki and the perspective orthogonal PN. The rotated K – being the point of intersection of the vertical through T and the orthogonal through the rotated H – is then depicted at the point of intersection of the vertical through Ti′ and the line Hi′P. The moved images of the rotated points I and J can be determined as the points on the verticals through Mi′ and Ni′ whose perspective heights over these points are equal to MI. Cousin described the four vertical lines, but not how the points Hi′, Ii′, Ji′, and Ki′are determined on these lines. He also mentioned that the vanishing point V1 is obtained as the point of intersection of Hi′Ji′ and Mi′Ki′ and V2 in the same way as the point of intersection of Ii′Ki′ and Hi′Li′. He used these vanishing points in a further construction in which he added a third dimension to obtain a star prism (figure V.21).

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vanishing point P perpendicular to the horizon. In Cousin’s own diagram the latter is the poinct principal and the other vanishing points considered are the points marked 13 and 14, which do not lie on the mentioned line. Geometrically there is no significant difference between the horizon and the vertical line through P, since the latter can be considered as the horizon turned 90˚ around the principal vanishing point P. When Vignola constructed his perspective image of a cube, reproduced in figure IV.21, he seems to have been aware of the fact that lines lying in a vertical plane perpendicular to p could be treated similarly to horizontal lines. This argument was not common knowledge among sixteenth-century perspectivists, however, and had not reached Cousin. Otherwise he would undoubtedly have used this insight in his construction, just as he used a result concerning horizontal lines in figure V.19. It is not surprising that his construction did not empirically show that the points 13 and 14 were to lie on the vertical line through P, for it requires a meticulously precise construction to get these vanishing points situated on that line. One of Cousin’s main aims was apparently to describe how elaborate objects can be thrown into perspective. Thus, his book includes several examples that are quite as complicated as the star prism. In addition he included the perspective representations of architectural objects, as well as the five classical orders of columns. He rounded off his book with one of the popular topics – the perspective images of the five Platonic solids.

Cousin on the Column Problem

C

ousin also participated in the never-ceasing debate on how to throw columns into perspective (cf. section II.15). Cousin described a case involving columns with square bases and two sides parallel to the picture plane. However, he disregarded the orthogonal sides (figure V.23), and so he actually treated a simplified version of the problem in which the question is how to represent a series of equal and equidistant vertical faces. Cousin mentioned that some solved the problem by drawing the intervals between the faces of the columns so that they become smaller the farther away they are from the eye point. He considered this solution wrong, stating that it gives rise to an artificial foreshortening (Cousin 1560, fol. Civv–Dir). He agreed that a row of equidistant and equal columns, and the intervals between them, are seen as decreasing with increasing distance to the eye, but he did not want this fact to influence the construction of their perspective images. His argument was that when the columns are drawn equal, and with equal intervals between them, they give the same visual effect as the originals. It is remarkable that in treating the column problem, the painter Cousin settled for the solution that has historically been most strongly supported by mathematicians (for another presentation of Cousin’s ideas see FrangenbergS 1992, 26–28). The examples given should be sufficient to support my claim that Cousin’s Livre de perspective contains much material that is fairly advanced for its time. From a mathematical point of view the book’s most remarkable particulars are the implicit proof of the correctness of a distance point construction

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FIGURE V.23. Cousin’s version of the column problem. Cousin 1560, fol. Civv.

and the occurrence of general vanishing points of horizontal lines – plus Cousin’s example of vanishing points for lines situated in a vertical plane.

V.4

Dürer

A

lbrecht Dürer (1471–1528) grew up in Nuremberg, one of the German artistic and intellectual centres during the Renaissance. He chose to become a painter, and after completing his apprenticeship he followed the tradition of going on a bachelor’s journey (Wanderjahre) visiting various artists, engravers, and printmakers over a period of four years. Then in the autumn of 1494, he took a step unusual for his day and travelled to Italy to learn from the masters there. He became fascinated not only by Italian art, but also by its connections to mathematics.

Dürer’s Introduction to Perspective

I

n 1506 Dürer returned to Italy for a lengthy stay in Venice, where in October he wrote to his friend Willibald Pirckheimer that he would soon travel to Bologna to meet a person who had agreed to teach him the “secret perspective” (DürerS Nachlass, vol. 1, 59). Dürer had previously experimented with converging lines in his drawings, but he had obviously not learned the rules of perspective, as is evident from the drawing reproduced in figure V.24.4 The fact that Dürer arranged to have a tutor confirms my impression, expressed earlier, that generally speaking, perspective was not an art non-scholars would learn by reading books, but by getting it demonstrated. 4

See also SchuritzS 1919, 22–38; Strauss in Dürer 1977, 27; and KempS 1990, 53.

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FIGURE V.24. Woodcut by Dürer illustrating Saint Jerome in his study. In this composition the images of some parallel horizontal lines converge, for instance two in the alcove, and two in the door of the lectern. The points of convergence do not, however, lie on a horizontal line. Frontispiece to Liber epistolarum sancti Hieronymi, Basel 1492, with some lines added.

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Dürer also wrote to Pirckheimer that he planned to stay in Bologna for eight to ten days – which was a short period for his undertaking. It is unknown whether Dürer actually reached Bologna and really became acquainted with perspective there, or whether he learned it later. At any rate, by 1510 he had still not become acquainted with the more subtle aspects of the discipline – for instance that according to the rules of perspective, haloes should most often be drawn as ellipses (figure V.25). He also had problems, as we shall see, in understanding one of the perspective methods he encountered. By analysing his drawings and his writings I have come to the conclusion that his knowledge of perspective constructions consisted in an ability to apply a plan and elevation method, and the awareness of convergence rule.

FIGURE V.25. Dürer remaining faithful to the old tradition of depicting haloes as a kind of “plates” behind the saints. Albrecht Dürer, Saint Veronica between Saints Peter and Paul, woodcut 1510.

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FIGURE V.26. Piero della Francesca’s box method, which is explained in section II. 17. Piero Pros, part of figures 52 and 53.

FIGURE V.27. Dürer’s box method. To illustrate rotations of a part of the human body, for instance a head, Dürer circumscribed it with a box and then described the movement in terms of rotations of the box along its three main axes. To represent the rotations graphically Dürer used plans and elevations. The diagrams describe the rotations of a box with base 1, 2, 3, 4 and top a, b, c, d, defined by the plan and elevation shown in diagram O. In the original position the edges 14 and 23 are parallel to the plane of the elevation. The diagram A consists of a plan and elevation of this cube when

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FIGURE V.27. (caption continued ) it has been rotated over an angle x around the line 12. The diagram B shows the plan and elevation of the box when it has once again been rotated over an angle z around a vertical axis; the diagram C describes the same procedure, but for another angle z. Diagram D is a continuation of B and illustrates the situation when the box has also been turned over an angle y around the line 14 in its original position; the diagram E is an analogous continuation of C. Basically Dürer’s technique is similar to the one Piero della Francesca applied for describing a cube in an arbitrary position – shown in figure V.26. Dürer’s description is, however, incomplete, which the following considerations show. The diagrams A and B (or C) can be constructed directly from the plan and elevation of the box in its original position (diagram O) and the angle x. The diagram D (or E) cannot similarly be obtained from B (or C). To overcome this problem, Piero involved a third plane of reference, namely one parallel to the third side of the box in its original position, whereas Dürer skipped this step. Vier Bücher von menschlicher Proportion (DürerS 1528), fol.Zir – supplemented with my diagram O and with the angles x, y, and z added to Dürer’s drawing.

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Naturally, various historians have tried to identify the person Dürer planned to meet in Bologna, but thus far without success. We can be sure, though, as already pointed out by Erwin Panofsky in 1915, that Dürer must have picked up his knowledge of perspective from sources close to Piero’s work (PanofskyS 1915, 36–46). One of Dürer’s manuscripts contains a nearverbatim translation from a Piero manuscript (ibid., 42–43), and some of the notes and sketches Dürer made from 1508 onwards show that he was greatly influenced by Piero – also in matters other than perspective. To the evidence given by Panofsky I add the observation that Piero’s graphical description of a rotation of a box also seems to have inspired Dürer. The similarity of their drawings can be observed by comparing the two illustrations reproduced in figures V.26 and V.27. There is, however, an essential difference in their descriptions, and presumably also in the profoundness of their understandings. Piero’s treatment gave a complete determination of the rotation, whereas Dürer’s did not do so, as I have explained in the caption of figure V.27. Concerning other sources of inspiration, Walter L. Strauss has argued that Dürer’s approach to perspective was influenced by the appearance in 1509 of Jörg Glockendon’s pirated edition of Viator’s book on perspective (Strauss in Dürer 1977, 28). I find it very likely that Dürer was familiar with this work. It may also have inspired his use of perspective, but certainly not his choice of construction methods or examples.

Dürer’s Books

D

ürer’s contemplations on perspective were part of a broader programme that was in accordance with the neo-Platonic ideas of his time. He attempted to grasp beauty by using mathematical concepts such as geometrical forms and proportions (SteckS 1971, 260). In pursuing his ideas, Dürer took up a study of mathematics during which he read classical sources including Euclid and works from his own time – presumably including the few German books on mathematics for mathematical practitioners and some, if not all, of Pacioli’s work.5 Dürer wished to share his new-found knowledge with young colleagues and planned to include sections on proportions and perspective in a very comprehensive book he worked on from the beginning of the sixteenth century to around 1513 where he gave it up in its original form.6 His project had different working titles, among them Speiss für Malerknaben (Nourishment for painter’s apprentices), and as figure V.28 shows, it was meant to cover all aspects of the aspiring painter’s education, including his eating habits, lodging, and sex life. It is remarkable that Dürer included Latin in his proposed curriculum for painters. Presumably, he wanted them to be able to enter the world of erudition that he himself had worked so hard to reach. As time passed Dürer seems to have become less interested in the social and religious sides of the education of painters and more interested in the mathe5

Strauss in Dürer 1977, 12–32; Peiffer in Dürer 1995, 53–95. SchlosserS 1924, 232; Rupprich in DürerS Nachlass, vol. 2, 83–90; Strauss in Dürer 1977, 8–9; Peiffer in Dürer 1995, 39–43. 6

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I. Introduction 1. Choice of an apprentice, with attention given to his temperament. a. Consideration given to his birth sign. b. His body structure and proportions. c. Initial instruction. d. Consideration of whether the apprentice is to be treated with kindness or severity. e. Retention of the pleasure of learning. f. When there is a danger of melancholy caused by overwork the apprentice shall be cheered up by practising music. 2. Education of the apprentice. a. Fear of God; prayer for the grace of acumen and veneration of God. b. Modesty in eating, drinking, and sleeping. c. Pleasant and quiet living quarters. d. He shall be prevented from improper dealings with women. e. He shall be taught reading, writing, and the Latin language. f. He should be able to do this for his payment and if necessary get medical help. 3. On the use, pleasure and delight of painting. a. Painting is a useful art because it is created by God and serves a holy edification. b. Useful, because it prevents much evil. c. Useful, because it gives rise to so much pleasure. d. Useful, because through it is obtained glory and remembrance. e. Useful, because God gets praise through its works and one gains the friendship of the wise men. f. Useful, because it makes it possible for a poor man to acquire goods. II. The practice of painting 1. On the generosity of painting 2. On the theory of proportion and the requirements of painting a. Proportions of man. b. Proportions of the horse. c. Proportions of buildings. d. Perspective. e. Light and shade. f. The theory of colour. 3. On the visible things and their selection in painting: a–f. III. Conclusion a. On the domicile in which the artist shall live to perform his activity: a–f. b. On the remuneration for the artist’s accomplishment: a–f. c. Praise and thanks to God: a–f.

FIGURE V.28. Hans Rupprich’s reconstruction of the content of Dürer’s book for young painters. DürerS Nachlass, vol.2, 84; translation by Walter L. Strauss in Dürer 1977, 8–9, with a few changes.

matical aspects. He returned to the latter in the early 1520s. By 1523 he had a manuscript on the proportions of the human body ready for print, but he did not hand it over to the printer until 1528 – and then in a revised form. It appeared posthumously the same year under the title Vier Bücher von menschlicher Proportion (Four books on human proportions)7. From Dürer’s 7 Dürer and his successors used the expression the proportion of humans; I have translated it as proportions.

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FIGURE V.29. Two suggestions by Dürer for constructing a capital D. Dürer 1525, fol. Kiiiv.

dedication to Pirckheimer we learn that he had postponed the publication because he thought his instructions would be understood better if he let them be preceded by a general introduction to geometry (DürerS 1528, fol. Aiiv). Dürer’s book on geometry appeared in 1525 under the title Underweysung der Messung mit Zirckel und Richtsheyt (Instruction in measurement with compass and ruler, abbreviated as Underweysung der Messung). He addressed this work to painters, goldsmiths, sculptors, stonemasons, and carpenters (Dürer 1525, fol. Aiv). In the last group he may have been especially mindful of those who manufactured intaglio. As Jeanne Peiffer so elegantly expressed it, the work is in itself “a geometry at the crossroads of practical, scholarly, and artistic traditions” (translated from Dürer 1995, 51). In it, besides perspective, Dürer treated a variety of elementary applications of mathematics, such as how to construct the letters of the alphabet (figure V.29). However, a large part of it is devoted to rather academic topics. Some of these were of use to mathematical practitioners, including how to construct the Platonic solids and how to design sundials, whereas others had no obvious practical applications and seem to have been included because Dürer found them fascinating – his treatment of the trisection of an angle being one example. In fact, I have the impression that in deciding upon the contents of his book, Dürer did not target a specific group of readers but wrote the text book he himself would like to have had at his disposal. For more on the contents of Underweysung der Messung, I refer to Jeanne Peiffer’s valuable study which places Dürer and his work in a wider historical context (Dürer 1995, 19–128). Before turning to Dürer’s work on perspective, let me touch upon his general way of doing geometry, since that provides a good background for under-

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FIGURE V.30. Dürer’s representation of a tetrahedron. He represented the polyhedron in two ways, namely by unfolding it (left) and by drawing its plan (right bottom) and elevation (right top). He straightforwardly illustrated an unfolded tetrahedron, but had problems depicting its elevation. He correctly drew the plan of the base as the equilateral triangle bcd in a horizontal plane and the plan of the top vertex as the centre a of the circle circumscribing bcd. A circle with the same radius occurs in Dürer’s elevation, which also contains another equilateral triangle. His two figures indicate that he chose a vertical plane parallel to the edge bc as the plane of the elevation. It is correct, then, that the edge bc has the same length in the elevation as in the plan, and that the point d is projected upon its midpoint. The edges ab and ac, on the other hand, should have been drawn smaller than bc. The two circles in the drawing are puzzling. Dürer may have seen a representation of the tetrahedron that included the plan and elevation of the circumscribed sphere – the latter indeed being two circles with the same radius – and wanted to draw this situation. If this is the case, he got almost everything wrong: the radius should have been greater than ab, and hence the lower circle circumference should not have contained the points b, c, and d; whereas the top circle circumference should only have contained the point a. Dürer 1525, book 4, figure 29.

standing him as a perspectivist. Although he took over themes from mathematicians, Dürer did not use their style. His aim was not to supply his readers with a geometrical explanation of why a construction works, but to teach them how to perform the construction. Sometimes it is impossible to reproduce Dürer’s constructions due to the paucity of the information given. It seems that in his enthusiasm Dürer had become too ambitious, seeking to cover themes that he himself did not fully comprehend. His treatment of the plans and elevations of the Platonic solids is an illustrative example of inadequate information – and even of incorrect information, as Hermann Staigmüller pointed out long ago (StaigmüllerS 1891, 33). In figures V.30 and V.31 I have reproduced two of Dürer’s plans and elevations and explained their shortcomings in the captions.

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FIGURE V.31. Dürer’s representation of a dodecahedron. It includes a correct illustration of how the polyhedron looks when unfolded, and a plan and an elevation. Dürer did not describe how he had constructed the latter. Within the precision of measuring, his plan is correct, whereas his elevation is problematic and reflects the same misconception as the previous figure. (For a description of a construction of the plan and elevation of the dodecahedron see the captions to figures IV.45–IV.47). Dürer 1525, book 4, figure 33.

In dealing with Dürer as a geometer it is natural to compare him with Piero della Francesca, with whom he shared interests in both painting and mathematics, and in whose work he found much inspiration. Their mathematical works reveal that Piero had a considerably deeper understanding of the discipline than Dürer had. This is partly due to the difference in their talents for mathematics, but it certainly also played an essential role that Piero grew up in Northern Italy, where there was a much stronger tradition for studying mathematics than in Nuremberg. A comparison of their work also shows that Dürer was much more keen on showing examples of applications of mathematics than Piero. With its broad spectrum of themes, Dürer’s Underweysung der Messung is a unique work in the history of geometry. It was noticed in its own time, even outside the German-speaking region and was translated – somewhat contrary to Dürer’s intention of reaching mathematical practitioners – into Latin rather than into vernacular languages. The first Latin edition appeared in Paris in 1532.

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FIGURE V.32. Dürer’s representation of a helix by means of its plan and elevation. Dürer 1525, book 1, figure 17.

Generally speaking, Underweysung der Messung has not received as much attention in the history of mathematics as it deserves, but one section has become famous. In this Dürer described a few skew curves, for instance a helix (figure V.32), by means of their plans and elevations. The fact that he dealt with skew curves, which only became a common object of study in the eighteenth century, has fascinated historians of mathematics. The additional fact that he described the skew curves by applying two orthogonal projections has caused some to characterize Dürer as an important forerunner of Gaspard Monge, the creator of descriptive geometry, while they fail to mention Piero della Francesca in this connection. Considering Piero’s mastery of the plan and elevation technique, and the great likelihood of his ideas inspiring Dürer, I find that Piero merits as much recognition as Dürer. It is worth noticing that in his presentation of applications of plans and elevations, Dürer, like Piero, assumed his readers were already familiar with the technique.8 8 Hans Rupprich suggested (DürerS Nachlass, vol. 2, 76) that Dürer learned the technique of describing an object by its plan and elevation from stonemasons, because he explicitly referred to their art in introducing this method (Dürer 1525, fol. Oiiiir). Rupprich also pointed out that the technique is described in textbooks for stonemasons and mentioned in particular RoriczerS 1486. This book does indeed treat the plan and elevation method, but in a way that can only be understood by readers already familiar with it. For more on Dürer and the use of the plan and elevation technique, see Peiffer in Dürer 1995, 59 and PeifferS 2004, 248–258.

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V. North of the Alps Before 1600 o

P

O

p 1 4

d

R

3 f

e a

p

h

2

b c g G

FIGURE V.33. The configuration considered by Dürer. It includes the cube 1234abcd, the platform efgh, a source of light at the point o –whose projection upon the ground plane is the point p – the picture plane p, the principal vanishing point P, and the eye point O.

Dürer’s Plan and Elevation Construction

W

hen presenting perspective methods, Dürer’s predecessors usually began by finding the perspective image of a square that is situated in the ground plan and has one side on the ground line. Most then continued with more complicated problems. Dürer took his own approach by starting with a three-dimensional object that has no line segment on the ground line. He only considered this one figure and determined its perspective image by two different methods, namely a plan and elevation construction and a variant of an Alberti construction. Figure V.33 shows the configuration Dürer was dealing with: a cube standing on a platform that has a square base and sides parallel to the sides of the cube. A picture plane p is placed parallel to one side of the platform without touching it, and there is also a source of light at the point o. Dürer not only treated the problem of throwing the platform and the cube into perspective, but also included the perspective image of the shadow cast upon the platform by the cube. He was the first to publish on perspectival shadows, and his treatment of this theme remained unique in the literature on perspective for more than a century. As we shall see, some of Dürer’s successors dealt with the construction of shadows of three-dimensional objects, but in general they did not explain how a shadow should be drawn in perspective – exceptions being authors who copied Dürer, such as Barbaro and de Caus.9 By constructing a plan and elevation of the entire configuration (figure V.34), Dürer completed his perspective image without difficulty (figure V.36). 9 For more on the history of shadows, see Da Costa KaufmannS 1975, in which Dürer’s contribution is treated on pages 273–275.

4. Dürer

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FIGURE V.34. Dürer’s plan and elevation corresponding to the situation depicted in the previous figure. To make the principle of Dürer’s construction clearer I have created a diagram based on two of Dürer’s illustrations. In the upper left-hand section of the drawing, Dürer drew the elevation of the platform, the cube, the source of light o, and the light rays from o to the top vertices of the cube and their continuations to the ground plane. Dürer used these rays to determine the shadow cast by the cube. In the lower left-hand section, he constructed the plan of the same objects. In the middle of the drawing he let the straight-line parallel to fg represent the elevation and the plan of the picture plane. The two eyes on this line are the elevation and plan of the principal vanishing point. In the right-hand side of the diagram he showed the elevation and plan of the eye point. The diagram also contains the plan and the elevation of the line from the eye point to the point 2 on the cube. (One of Dürer’s original diagrams contains similar lines to all the other projections of the vertices.) For later reference, the lines from the eye point in the plan to the points f and g are also included. A combination of figures 52 and 56 in Dürer 1525, book 4, with some lines removed.

The plan and elevation gave him what corresponds to the coordinates of each of the image points. He measured these lengths from another point than Piero (figures V.35 and II.36). Leaving out all the line segments that are not part of the final picture, Dürer obtained the image shown in figure V.37. Dürer drew his source of light as a sun and assumed that it sends light rays in all directions. Later writers taking up the problem of perspective shadows would make a construction similar to Dürer’s if the source of light was a burning candle, whereas they would consider sun rays to be parallel because

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V. North of the Alps Before 1600 Pe

2e

Oe

2e

Pp

Op

2p 2p

A

P B

2

FIGURE V.35. Dürer’s procedure explained for the point 2. On the right-hand side, the elevation and the plan of the eye point are marked with Oe and Op; the line in the middle is the elevation and plan of the picture plane with the elevation and the plan, Pe and Pp, of the principal vanishing point; finally, on the left-hand side are shown the elevation and the plan, 2e and 2p of the point 2. The point 2′e, in which Oe2e meets the line PePp, is the elevation of the image 2′ of the point 2. Similarly 2′p is the plan of this image. To construct the image of the point 2 (the lower diagram), Dürer first decided on the position of the principal vanishing point P in the picture, which he marked as an eye (figure V.36). Through P he drew a transversal and a line perpendicular to this, on the transversal he made PA equal to Pp2′p, and on the other line he let PB be equal to Pe2′e. He determined the image 2′ as the point of intersection of the line through A parallel to PB and the transversal through B. For measuring lengths like Pp2′p in the plan, Dürer suggested using one pair of compasses, then using another for measuring lengths like Pe2′e in the elevation. Dürer 1525, fol. Piiiv.

of the great distance between the earth and the sun. This means that rather than working with the sun as a point, they would introduce the vanishing point of the sun rays (for more details on shadows, see section VII.7). In his example Dürer chose an unconventional position for the eye point, namely one in which the principal vanishing point – which Dürer called the eye – is close to one of the edges of the picture. It was not only in his presentation of perspective constructions that he chose this position for the eye; it is also to be found in his well-known engraving Saint Jerome in His Study from 1514 (figure V.38). However, in most of his other perspective compositions the principal vanishing point is situated more traditionally, an example of which can be seen in a woodcut on the same theme completed three years earlier (figure V.39).

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FIGURE V.36. Dürer’s illustration of the result of his plan and elevation construction. He marked the principal vanishing point as an eye. Dürer 1525, book 4, figure 57.

Dürer’s Enigmatic Method

H

aving explained how the image reproduced in figure V.37 is obtained by a plan and elevation construction, Dürer proceeded by describing a “shorter way” to construct the image of the same configuration (Dürer 1525, fol. Pivr). This method is incorrect and has given rise to many comments by historians of

FIGURE V.37. Dürer’s final perspective drawing of a cube on a platform and the shadow of the cube. Dürer 1525, book 4, figure 58.

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V. North of the Alps Before 1600

FIGURE V.38. An example of Dürer placing the principal vanishing point near the righthand side edge of the picture. I have indicated the approximate position of this point with a white circle. Albrecht Dürer, Saint Jerome in His Study, engraving 1514.

FIGURE V.39. An example in which Dürer chose a more traditional position for the principal vanishing point. To indicate this position I have marked several orthogonals. Albrecht Dürer, Saint Jerome in His Cell, woodcut 1511.

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art and mathematics alike.10 All commentators point out that Dürer made a mistake, but they differ in their descriptions and analyses of his error. I therefore find it relevant to discuss the method thoroughly, although doing so is a challenging task. What makes the task so difficult is that Dürer’s presentation is unclear, and that on some points it does not correspond with his illustration. To deal with the latter complication I look at his second method as two methods: one being the second method as described, and the other being the second method as illustrated. I comment upon them separately, rather than attempting to come up with a single interpretation of his second method.

The Second Method as Described

D

ürer’s second method as described is as follows (figure V.40). He began by telling his readers to draw the edge fg of the platform and the vertical face through it. Although this construction is not straightforward, Dürer left it to his reader to find out how to perform it. To avoid treating too many issues at once, I postpone the discussion of this part of the construction until the end of this section, initially following Dürer in his assumption that the line segment fg is constructed. So for now I concentrate on how Dürer constructed the image of the square efgh (figure V.33). He first marked the principal vanishing point, which I, as usual, have denoted by P, and which Dürer drew as an eye (figure V.41), now calling it a near eye. Joining f and g with P (figure V.40), he obtained the images of the two lines ef and gh. To determine the position of the image of eh, he placed what he called a second eye on the horizontal line through the principal vanishing point P, so that the distance between the two eye points equals the distance – I denote this second point with an A.11 His further description runs as follows. I

hi g

c

P

A

ei f

FIGURE V.40. My interpretation of Dürer’s second method as described. To make it easier to distinguish the various points, I have made the distance between the principal vanishing point P and the line perpendicular to fg through f larger than in Dürer’s drawing (figure V.41). 10 Presentations of Dürer’s incorrect method can among other places be found in StaigmüllerS 1891, 45–48; PanofskyS 1915, 30–36; SchuritzS 1919, 40–42; IvinsS 1975, 35–40; Peiffer in Dürer 1995, 106–110. 11 Actually, the distance between A and P is not quite equal to the distance that Dürer introduced in connection with his plan and elevation construction. I will come back to a possible reason for this in note 15.

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V. North of the Alps Before 1600

P

A

FIGURE V.41. Dürer’s second construction as illustrated. Dürer 1525, book 4, figure 59, with the letters P and A added.

From this eye draw two straight lines to both ends of the given line [fg]. Then make a perpendicular line aa.bb, which touches the foremost vertex.12

Dürer’s instruction seems to be that the second eye A in figure V.40 should be connected with the points f and g, and that a line perpendicular to fg should be drawn. In the following I call this line the perpendicular and have drawn it as the line fI – interpreting “the foremost vertex” to be f. Dürer denoted the point of intersection of Ag and the perpendicular with a cc, whereas I leave it at a single c. Dürer’s final step was to draw the line through c parallel to fg and to let this determine the position of image eihi of the edge eh.

The Second Method as Illustrated

I

n his second method as illustrated (figure V.41 and figure V.42), Dürer did not follow the procedure presented in the quotation. He only drew the line Ag leaving out the line Af, which actually has no function in his construction. He also placed the perpendicular differently, positioning it so that it passes (or almost passes) through the principal vanishing point P. Retaining the

FIGURE V.42. The image efgh from the previous figure reproduced in the same scale as figure V.36. 12

Ausz disem aug zeuch zwo gerad linien an bede ort der fürgelegten linien. Darnach reysz ein aufrechte lini .aa.bb. die das forder eck anrürt. [Dürer 1525, fol. Piiijr]

4. Dürer P ei

hi g f

201 A

c b

FIGURE V.43. My diagram to Dürer’s second method as illustrated (cf. figure V.41). I have changed Dürer’s bb and cc to b and c, and I have moved the point b to the line fg –which has no theoretical consequences. As in figure V.40, I have enlarged the distance fb.

positions of the points P, A, f, and g from figure V.40, I have drawn Dürer’s perpendicular as the line Pb in figure V.43. In his illustration, Dürer proceeded by determining the point c or cc as the point of intersection of Pbb and Ag, and then finished the construction as he did in the described method by drawing a transversal through c.

The Second Method and Alberti Constructions

B

oth versions of Dürer’s second method are mathematically incorrect, but each contains components of a correct construction, as I will show by comparing the method as described with a well-known Alberti construction, and comparing the method as illustrated with an unusual Alberti construction. The well-known construction is Piero’s version of the Alberti construction. As in Dürer’s described method (figure V.40), Piero’s used the perpendicular through f. Piero then introduced a point A, which I have called an Alberti point and which Dürer called the second eye. When I is the point of intersection of the perpendicular through f and the transversal through the principal vanishing point P, Piero’s Alberti point was determined by making IA – and not PA as Dürer did – equal to the distance. Piero finished the construction, as Dürer described it, by placing the image of eh on the transversal through the point of intersection c of the perpendicular If and Ag. Thus, Dürer’s second method as described did get most things right, but not the position of the second eye – the Alberti point. In his second method as illustrated, Dürer applied a rarely seen perpendicular, namely one passing through the principal vanishing point (the lines Pbb and Pb in figures V.41 and 43, respectively). This perpendicular occurs in the previously presented interpretation of Gaurico’s method, published in 1504, and also appeared in 1560 in Cousin’s method (figures IV.1 and V.16). When the perpendicular Pb represents the picture plane in a profile, the second eye A must be placed as Dürer did: so that PA is equal to the distance. The images of the orthogonals through f and g should also be determined in the same way Dürer did: by connecting P with f and g. The determination of the position of the image eihi of eh, however, requires special attention (figure V.44).When Pb represents the picture plane, the

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V. North of the Alps Before 1600 P ei

hi

d

g k

A

f

b

FIGURE V.44. An Alberti construction applying the perpendicular Pb.

position of the side eh should be determined by a point on the ground line, say k, whose distance to Pb, bk, is equal to side length of the square, that is to fg. Hence, the correct last step of the construction is to find the point of intersection d of Ak and Pb, and then let the transversal through d determine eihi. In his second method as illustrated, Dürer thus had the point A in the correct position, but connected it with a wrong point on the ground line, choosing g instead of k. In Dürer’s example, the perpendicular through his near eye P lies so close to f that the differences between the construction he described, the one he illustrated, and a correct one are only interesting from a theoretical point of view. Had he chosen a more conventional position for the eye, however, the differences would have been noticeable. Dürer might then also have realized that his second construction led to a different result than his plan and elevation construction. As noted, Dürer’s confusing presentation of his second method for throwing a square into perspective prevents me from reconstructing the method he wanted to teach his readers. However, I find that my analysis permits the conclusion that he was mixing up steps in various procedures he had seen – or procedures he himself was trying to develop.

The Second Method and a Distance Point Construction

I

n wondering about Dürer’s second method it is natural to discuss whether the method contains elements of a distance point construction, as some commentators claim.13 That this could be the case is supported by two particulars. The first is that Dürer’s second eye A (figure V.41) is a distance point (because Dürer claimed that PA is equal to the distance). The question is, therefore, whether Dürer saw his second eye as a distance point and had a distance point construction in mind. This would involve letting the image of e, which he also denoted e, be constructed as the point of intersection of Pf and Ag. Before answering this question, I will turn to the second, and seemingly stronger, argument for Dürer having considered a distance point method – which is that he actually illustrated it (Peiffer in Dürer 1995, 348, note 394).

13 See for instance StaigmüllerS 1891, 46–47; SchuritzS 1919, 41–42; SteckS 1948, 79; HofmannS1971, 149; IvinsS1975, 38; Peiffer in Dürer 1995, 108.

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4. Dürer P

FIGURE V.45. Dürer’s image of the cube obtained by his second method. It can be noticed that although Dürer claimed to have used the diagonal eg, the line ac does not really pass through g. It is difficult to say whether this is due to inaccuracy on Dürer’s part or a result of his attempting to position the cube as in figure V.36. Dürer 1525, book 4, figure 60, with the letter P added.

Dürer did so in a diagram (figure V.46) meant to show how shadows are constructed according to his second method. The figure contains an eye point, which I have called E, situated on the diagonal ge. Compared with his other diagrams, Dürer’s illustration reproduced in figure V.46 is rather confusing, because he has marked the point where ge meets Pbb as cc – whereas in figure V.41 cc is the point of intersection of Ag and Pbb, and is not lying on ge. If the diagram in figure V.46 is understood as an illustration of how the efgh is constructed, it depicts a correct distance point construction, since e is the point of intersection of Af and Eg. However, I do not think this is the way Dürer constructed the diagram. Rather, I see it as a result of an inverse process in which Dürer prolonged the diagonal ge in the already constructed perspective square efgh in figure V.41.14 As for the function of the prolonged diagonal, I unfortunately cannot tell. Dürer did not comment upon the point E nor did he use it in his construction of shadows. Still, I do believe that Dürer prolonged the diagonal ge and in that connection misplaced the point cc – in the sense that it is not situated in the same position as in his other drawings. It is precisely the recurrence of the point cc and the line Dürer called the perpendicular (the line aabb) in all his illustrations that fuels my argument to dismiss the idea that Dürer consciously applied elements of a distance point construction in his second method. 14

One of my arguments for conceiving the point E in figure V.46 as being constructed with the aid of the diagonal ge, rather than being constructed independently of the perspective square efgh, runs as follows. If Dürer had started from scratch, it would have been natural to construct the distance PE equal to the distance he used in the plan and elevation construction, but PE is actually shorter than the distance. The reason for the latter could be that in placing the point E on the diagonal ge, he was not overly conscientious about making PE parallel to gf.

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V. North of the Alps Before 1600

P

E

FIGURE V.46. Dürer’s construction of the perspective shadow according to his second method. Dürer 1525, book 4, figure 61, with the letters P and E added.

The Diagrams Illustrating the Second Method

T

he above reflections on Dürer’s second method presupposed that he started from scratch when illustrating the procedure. However, I strongly doubt that his illustrations to the second method are based on an actual performance of this construction. In fact, I believe he produced these diagrams by taking the image of the platform he had constructed with the aid of a plan and an elevation (figure V.36), and then adding some lines to the image. My hypothesis is based on the observations that the two perspective squares in figures V.36 and V.42 are identical in the sense that they can cover one another, even though the latter is claimed to have been obtained by a method we know should not give the correct result. Most of Dürer’s other drawings illustrating the second method similarly correspond to the diagrams he produced using the plan and elevation construction.15 Only in two cases, which I will present and explain later, did he derive slightly different results using his second method.

Finishing the Image by the Second Method

H

aving described how the perspective platform can be obtained using his second method, Dürer proceeded by constructing the image of the cube standing upon it. This he did in two steps, first throwing its base and then the five remaining faces into perspective. In placing the base of the cube on the platform, he gave the new information that its diagonal ac lies on the diagonal eg of the top of the platform – which is actually also the case in figure V.34.

15

The hypothesis that Dürer started with the perspective square efgh and an argument similar to the one presented in the previous note can explain why PA in figure V.41 is not quite equal to the distance (cf. note 11). Thus, PA is not really parallel to fg, and has therefore become shorter than the distance.

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205

Dürer claimed he found the images of the vertices a and c of the base abcd (figure V.45) as the points of intersection of the perspective diagonal eg and the perspective orthogonals of the base square. From the plan (figure V.34) he determined the points in which ab and cd meet fg, specifying them as x and y in the picture plane (figure V.45),16 then drew the perspective orthogonals Px and Py. Drawing the transversals through a and c enabled him to finish the perspective square. Finally, he obtained the perspective cube by letting the points 2 and 3 be determined by b2 = c3 = bc, letting the point 1 be the point of intersection of the vertical through a and the line P2, and defining the point 4 similarly. The resulting cube is not identical to the one he had constructed in figure V.36. This is most likely due to the fact that his determination of the points x and y is not precise, as explained in note 16. To repeat the entire construction performed by the plan and elevation construction, Dürer still had to determine the shadow of the cube. In principle he needed to construct the images of two arbitrarily situated points, namely the image of the point of the light source – given by its plan and elevation in figure V.34 – and that of its orthogonal projection upon the plane of the square efgh. However, this was not Dürer’s approach. He simply told his readers to place a point o above the eye and a point p vertically below it in the picture plane (Dürer 1525, fol. Qir). In his illustration, reproduced in figure V.46, he marked them as o and p. Although presenting figure V.46 as a continuation of figure V.45, Dürer returned to figure V.3617 and – not surprisingly – took the position of o from that figure, which effectively means he took the position he had constructed using the plan and elevation method. Figure V.35 does not contain the projection p of o upon the platform, so Dürer had to make a choice. An analysis shows that he placed p (figure V.46) on the vertical through o a bit higher than the position defined by the plan and elevation. Finally, he constructed the shadow correctly by finding the points of intersection between lines from o and p to the relevant vertices of the perspective cube. Dürer’s choice of the point p had the effect that the shadows in figures V.36 and V.46 are not exactly the same, which means that neither are they congruent in the final pictures V.37 and V.47.

Construction of the Side fg

I

n presenting Dürer’s second method, I have thus far ignored his determination of the image of the side fg of the platform. As Dürer’s plan (figure V.34) shows, there is a small distance between the face through fg and the picture plane, and both his elevation and other drawings show that fg lies

16 In his description of how to construct x and y Dürer did not take into account the fact that the line segment fg in figure V.44 does not have the same length as the line segment fg in figure V.34 – the first being the perspective image and the second a plan of a line segment that is not lying on the ground line. 17 This can be seen by comparing the cubes in figures V.36, V.45, and V.46.

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V. North of the Alps Before 1600

FIGURE V.47. Dürer’s final image based on the previous figure. The shadow differs from the one in figure V.37. Dürer 1525, book 4, figure 62.

above the ground plane. It was possible at Dürer’s time to construct the image of the line segment fg by other procedures than the plan and elevation construction, for instance by applying an Alberti construction and a procedure for determining perspective heights. However this was presumably far too difficult for Dürer – and for most of his contemporaries. Therefore I am convinced, as noted earlier, that rather than attempting to construct the image of fg in figures V.41 and V.42, Dürer copied it from figure V.36.

Dürer’s Programme

A

fter presenting his second method Dürer wrote:

In the same manner as I have foreshortened a cube in a painting, so one may bring into a painting all the solids whose plan and elevation can be drawn.18

The requirement that one should be able to draw the plan and elevation of an object is, of course, quite essential. Dürer tried to throw some of the other Platonic solids into perspective, but without much success (StraussS 1974, 2879). One central reason for his problems could be that he generally lacked the skill to determine the correct plans and elevations of the Platonic solids, as mentioned in connection with figures V.30 and V.31. 18

In gleycher weyss wie ich den cubus in ein abgestolen gemel gebracht hab/also mag man alle corpora die man in grund legen und aufziehen kan/ durch söliche weg in ein gemel pringen. Dürer 1525, fol. Qiv.

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The Lesson of Dürer’s Mistakes

T

he reason I have paid so much attention to Dürer’s mistakes is this: I want to illustrate that as long as there were no clear expositions of the various methods of perspective construction, and of how and why they work, it was demanding even for a gifted mathematical practitioner like Dürer to find his way through the jungle of possible constructions. The case of Dürer also shows it was challenging to find out how to apply the methods to new problems. Although this was not his only difficulty, Dürer had complicated matters by trying to find the perspective image of a square situated above the ground plane at some distance from the picture plane by another method than a plan and elevation construction. In fact at his time the standard constructions, not based on a plan and an elevation, concerned how to determine the image of a square that has one side on the ground line. He had also set himself a demanding task when seeking to construct a perspective shadow using his second method. Piero’s claim that for more difficult perspective problems it was advantageous to use a plan and elevation construction remained true for a long time. However, this construction required more work than an Alberti or a distance point construction and gradually lost popularity in tracts on perspective. This in turn caused authors to avoid treating some of the more difficult problems. This may partly explain why it took so long before new methods of constructing shadows occurred.

Restrictions Induced by Perspective

D

ürer’s Saint Jerome illustrations show another effect of the circumstance that the early period of geometrical perspective was characterized by a limited understanding of the subject. The effect concerns the choice of line directions. In the Saint Jerome compositions Dürer made after he had learned perspective, reproduced in figures V.38 and V.39, the dominating directions are horizontals, orthogonals, and verticals, whereas in his earlier picture of the same scene, shown in figure V.24, he used many more directions. As Dürer had not learned how to throw arbitrary lines into perspective, the new art to some extent restricted him – and other artists as well.

Dürer’s Practical Methods

D

ürer concluded Underweysung der Messung with two beautiful illustrations of how to make perspective compositions with the aid of instruments rather than geometrical constructions (figures V.48 and V.49). In the second edition he added two more illustrations. The first of these demonstrates the method described by Alberti and is reproduced in figure II.3, whereas the second (figure V.50) depicts a variant of the method shown in figure V.48.

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V. North of the Alps Before 1600

FIGURE V.48. A perspective apparatus, consisting of a frame and an instrument to position the eye point. The choice depicted by Dürer implies that the principal vanishing point is close to the right edge of the frame – whereas the principal vanishing point of the illustration itself lies between the frame and the drawer. Dürer 1525, fol. Qijv.

Dürer’s illustrations were often copied – as exemplified in figure IV.25. I am unsure whether this meant that his practical methods really were widely used, or whether others took over Dürer’s drawings because they so pedagogically illustrate the concept of perspective projection. A problem like the one depicted in figure V.49 – throwing a lute into perspective – is unquestionably difficult to handle if geometric methods are used. In fact, not even the plan and elevation method is suitable for dealing with curved surfaces, as we saw in connection with Piero’s procedure for throwing heads into perspective (figure II.47). However, would two men really go through the long, arduous process portrayed in Dürer’s illustration? or did his drawing mainly served a didactic purpose? I cannot decide, but I am fairly certain that Dürer’s proposed method was not followed, although de Caus suggested so (figure V.51) if Dürer’s lute was replaced by a rectilinear figure such as a cube – which de Caus’s example even has one face parallel to the picture plane.

4. Dürer

209

FIGURE V.49. A string method for producing a perspective composition. The idea of the illustrated technique is explained in the caption of figure IV.25. Dürer chose a viewing angle that makes it difficult to see what the man with his hands in the frame is doing, but presumably he is manipulating strings similar to those shown in figure IV.25. Dürer 1525, fol. Qiijr.

FIGURE V.50. Another string method that also involves a sighting instrument. Dürer referred to the painter Jacob Keser as the inventor of this way of producing perspective images. Compared with the method illustrated in figure V.48, this one has the advantage that the distance can be longer than an arm’s length. Dürer 1538, fol.Q3v.

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V. North of the Alps Before 1600

FIGURE V.51. Dürer’s string method illustrated by Salomon de Caus. De Caus 1612, figure 6.

Dürer’s Diagonal Method

B

esides new illustrations of mechanical methods, the second edition of Underweysung der Messung contains a diagonal method (figure V.52) that Dürer also used in some of his manuscripts (figure V.53), and which he was the first to publish. This method, requiring a square and its perspective image, was presumably inspired by Piero’s diagonal method (figure II. 20). In representing the original square Dürer chose a rabatment (figure V.52) different from the one used by Piero, and this seems to have made some commentators think Dürer’s method was wrong.19 The technique in Dürer’s diagonal method was presented in figure IV.14, and was also applied by Cerceau in his work from 1576, as we saw in figure V.13. As noted, it is unclear whether Cerceau was inspired by Dürer. If Cerceau saw Dürer’s illustration of the diagonal method, it must have been in the second edition of Underweysung der Messung, because the method was not included in the Latin edition. In chapter VII we shall meet a diagonal method similar to Dürer’s again in a book by Samuel Marolois from 1614 (figure VII.11). 19

For other presentations and interpretations of Dürer’s diagonal method, see SchuritzS 1919, 43; IvinsS 1975, 40–41; Strauss in Dürer 1977, 427; and Peiffer in Dürer 1995, 111–112.

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FIGURE V.52. Dürer’s diagonal method. This method is explained in the caption of figure V.14. Dürer 1538, fol. Q1r.

FIGURE V.53. Examples of Dürer applying his diagonal method. Sächsische Landesbibliothek, Dresden R 147, fol.184v.

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V. North of the Alps Before 1600

Dürer’s Influence on the Development of Perspective

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ürer’s work on perspective was not directly continued in the German areas. There is, however, no doubt that his engagement in perspective was an inspiration to later generations of German artisans and artists in their dealings with the discipline. Moreover, he also sent new impulses to the world outside the German-speaking communities. The theme of Dürer’s general influence as an artist lies outside the scope of this book. It is worth noting, however, that among his hundreds of drawings, relatively few show any striking use of perspective. Thus, his art as such can hardly have contributed to making perspective pictures fashionable.

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Dürer’s German Successors

he earliest examples of perspective constructions published in the German countries appeared, as noted previously, in the pirated editions of Viator’s drawings published by Gregor Reisch in 1508 and Jörg Glockendon in 1509. It is difficult to tell whether these works had any influence on the further development of German perspective. As we shall see, some of the sixteenth-century German authors presented a method that resembles Viator’s, but with the crucial difference that his “third points” (distance points) were not involved (figures V.56 and V.62). This method may have been a simplification of Viator’s procedure, but it may just as well have come from somewhere else, and I am actually inclined to believe the latter. German authors did not present Dürer’s plan and elevation method until the end of the sixteenth century, but seemingly the appearance in 1525 of his Underweysung der Messung stimulated the idea of applying geometry in art. It is perhaps an exaggeration to talk of a ‘Dürer school’, yet we do find a number of German authors who took up some of Dürer’s topics – mainly the elementary ones – and gave their books titles that were apparently inspired by Underweysung der Messung (appendix four lists the German books chronologically). These authors also took over Dürer’s style which was to speak directly to the reader. Thus, expressions like musst du (you must) occur very frequently in the writings of Dürer and his successors. It is striking to see how many of those publishing on perspective after Dürer were engaged in two or more of the fields of drawing, engraving, etching, goldsmith’s work, graphic arts, painting, and woodcarving – precisely the types of people to whom Dürer had addressed his book. His assumption that the artisans needed some knowledge of geometry and perspective proved to be correct, for in the last decades of the sixteenth century German decorative art was greatly influenced by a general preoccupation with mathematical symbols that were supposed to reflect a magical or mysterious aspect of the cosmos. The perspective examples occurring in the German literature from this period deal much more with images of geometric figures, not least the

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Platonic solids and other polyhedra, than with architectural settings; and the authors mainly came from other circles than those of painters and architects. Basically, Dürer only presented one example of a perspective composition, making his description rather brief, and keeping any theoretical considerations he may have had to himself. His one example probably was sufficient to explain the principle of the plan and elevation construction, but his description of his second method – irrespectively of its incorrectness – hardly encouraged readers to create new applications. Although his German successors provided many more examples, they added little in the way of explanation. Many of them were actually much more concise than Dürer; those who included a lengthy text spent it on detailed descriptions of how a pair of compasses and a ruler should be used when carrying out a construction, not on a presentation of the concepts involved, and certainly not on providing a geometrical understanding of the principles behind the construction. It is particularly remarkable that when the authors presented other methods than the plan and elevation procedure, they did not treat the essential problem of how to foreshorten a vertical length. While the mathematical level in the tracts is far from impressive, their sheer number is. Besides the second edition of Dürer’s Underweysung der Messung (1538) another ten titles in German, entirely or partly devoted to perspective, were published over the last three quarters of the century. Thus, the first period of German literature on perspective shows that the discipline was alive as a practical art (see also PeifferS 2004, 258–267). Since there is not much new material in these tracts, my description of them is fairly brief.

Perspective Touched Upon by a Painter

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ürer’s pupil Erhard Schön (c. 1490–1542), whose Unnderweissung der Proportzion unnd Stellung der Possen (Instruction in proportions and positions of models20) appeared in 1538, focussed especially upon the use of proportions for describing human beings. He mentioned perspective (Schön 1542, fol. Fir), but did not explain how perspective constructions were performed. Nor did he comment upon his own perspective diagrams (figures V.54 and V.55) – on the whole including very little text in his book. He is still known for some of his anamorphoses (BaltrusˇaitisS 1977, 11–15), but it is not very likely that he related these images to perspective.

Perspective Presented by a Count

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ll the German authors whose works are presented in this chapter were artists or artisans, apart from one, namely the author of Eyn schön nützlich Büchlin und Underweisung der Kunst des Messens (A nice useful

20 The German word Possen formerly meant a relief-like figurative image – and Julius Schlosser suggested it is related to Bossen – roughed-out figure (SchlosserS 1924, 245). Schön applied Possen to polyhedric models of human bodies, like the ones depicted in figure V.54.

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FIGURE V.54. Schön’s illustration of how to place models on a perspective grid. He added four lines of explanation, which basically said: look at the drawing, then you will know how to proceed (so wayst du dich darnach zu richtenn). Schön 1538, fol. Ciir.

booklet and instruction in the art of measurement). This small volume appeared in 1531 with a preface by the printer Hieronymus Rodler, who stated that he found the perspective methods presented by Dürer too difficult for practitioners, for which reason he had decided to publish a book by an author who knew how to address this group. The identity of Rodler’s chosen author remained unknown for centuries, but in 1991 Werner Wunderlich solved the riddle by pointing to the palsgrave Johann II von Simmern (1492–1557) as the author (WunderlichS 1991, 25–27).21 Having observed that the initial letters of the first 54 paragraphs in the book form the following sentence: JOHANS PFALTZGRAF BEJ REJN HERTZOG JN BEYRN UND GRAF ZU SPANHEJM.

Wunderlich took this to be a display of the author’s name. The letters could also have been a hidden dedication, but Wunderlich found additional support 21

I am thankful to Eberhard Knobloch for having made me aware of this publication.

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FIGURE V.55. Schön’s perspective construction. This drawing is obviously meant to demonstrate how a room and its floor with square tiles are thrown into perspective, but Schön left it to the reader to draw his own conclusions simply by looking. The drawing shows the principal vanishing point and contains an indication that one set of diagonals converge in a point. Schön 1538, fol. Fiv.

for his hypothesis in Hans Lencker’s Perspectiva from 1571, in which Lencker stated that Count Johann had published a book on perspective.22 What the Count wanted to teach his readers was not any of the perspective methods presented by Dürer, but a simple procedure for drawing a grid of squares in perspective. His technique is one of those mentioned earlier, which resembles Viator’s method, but which does not involve the horizon or the distance (figure V.56). Having described how the images of the orthogonals should converge in one point, Count Johann let his readers freely choose the direction of the image of one of the diagonals. He then used this perspective diagonal to make the first nine rows of perspective squares (Johann 1531, fol. Aviv; see also PanofskyS 1927, 131–133). Like Viator (figure V.1), Count Johann applied two sets of perspectively parallel diagonals but without specifying how the second set of diagonals should be chosen. In his instruction he did not involve the distance points on the 22 In dedicating his Perspectiva to the son of Count Johann II, Lencker reminded him about his father’s book (Lencker 1571, fol. A2v).

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FIGURE V.56. Count Johann’s perspective grid. Johann 1531, fol Aviv.

horizon, and in fact, his diagonals do not meet on the horizon (figure V.57). Having organized the image of the floor, he turned to the determination of the heights of people appearing in a perspective composition (figure V.58). Based on his simple perspective method, Count Johann made some quite convincing illustrations (figures V.59 and V.60). His Büchlin also contains an illustration of the practical method described by Alberti (figure V.61). Count Johann’s illustration may not be as impressive as the one found in the second edition of Dürer’s Underweysung der Messung (figure II. 3), but then again, it was published seven years before Dürer’s figure appeared.

FIGURE V.57. The previous figure with a horizon added and the diagonals prolonged.

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FIGURE V.58. Count Johann’s illustration of how to construct the heights of people. Johann 1531, fol. Gviv.

Counts do not typically write books on perspective, but there are other authors from the higher circles, namely Commandino and Barbaro, as well as Guidobaldo, whom we will meet later. There are also several examples of members of this group taking a strong interest in perspective. A number of books are – according to their authors – the results of courses on perspective given to princes, among them Stevin 16051, De Caus 1612, and Kirby 1761.

Hirschvogel and Lautensack

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few years after Count Johann had published his Büchlin, Augustin Hirschvogel (1503–1553) presented a perspective method similar to his, but not identical. Hirschvogel was active in many fields of craftsmanship, including etching, glass painting, and cartography. In this last field he was so successful that he was summoned to the emperor’s court in Vienna. Through his work Hirschvogel became convinced of the importance of knowing geometry, and in 1543 he published his own introduction to the discipline,

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Figure V.59. A room with a Rechenmeister’s tools (a counting board and pebbles) drawn in perspective. Johann 1531, fol. Bvv.

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FIGURE V.60. Count Johann illustrating how to show part of a castle in perspective. Johann 1531, fol. Evv.

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FIGURE V.61. Count Johann’s interpretation of the Alberti idea of equipping the picture frame with a grid. As Lawrence Wright has remarked, the draughtsman in the picture is not reproducing what he sees through the window, but what we see (WrightS 1983, 314). Johann 1531, fol. Hiiv.

Ein aigentliche und grundtliche Anweysung, in die Geometria (A proper and thorough introduction to geometry), opening it with these lines: The book of geometry is my name, all liberal arts first from me came. I bring architecture and perspective together.23

Among several other subjects, this book deals with the simple perspective method (figure V.62), referred to a couple of times. 23

Das Buch Geometria ist mein Namen all freye Kunst aus mir zum ersten kamen ich bring Architectura und Perspectiva zusamen. [Hirschvogel 1543]

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FIGURE V.62. Hirschvogel’s method of making the perspective image of a horizontal square covered with square tiles. Like Alberti, he assumed that the side of the tiles is one third of the height of the eye. He let the image of the remote horizontal side of the square be determined by the same third. A straightforward calculation shows this implies that the distance is equal to two times the side CD of the square. However, he did not involve the distance and the distance points. In his example he chose to have a row of seven tiles, but stressed that the same method could be used for any other number of tiles. To subdivide the perspective square into images of squares, he used the perspective diagonals. Hirschvogel 1543.

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Like so many of his countrymen, Hirschvogel included the Platonic solids and some semiregular polyhedra in his work. As explained in connection with Barbaro’s treatment of the dodecahedron (page 158), it was a challenge in early modern times to determine this figure’s plan and elevation, and we saw that Dürer was far from getting it right (figure V.31). In illustrating a dodecahedron, Hirschvogel devised his own approach. He showed the dodecahedron in two projections (upper and lower left hand images in figure V.63), referring to both as a Geometria (a word most often used to denote a plan), and unlike Dürer, he described how they had been constructed. The first of them (upper left hand) is a plan, and a correct one at that, although a few points are missing (caption of figure V.63). The second representation (lower left hand) is puzzling. It is unlikely that Hirschvogel meant it to be an elevation, for the following reason. A correct elevation of a dodecahedron has its vertices lying in four horizontal levels, something Hirschvogel could easily have ascertained by looking at a dodecahedron. In this connection it is worth noting that although Dürer’s elevation is wrong, it only involves four levels of vertices (figure V.31). In Hirschvogel’s second representation of the dodecahedron the vertices are distributed on nine levels. Hence my suggestion is that rather than aiming for an elevation, Hirschvogel wanted to show the dodecahedron in a representation that makes it look three-dimensional, an idea supported by the fact that he showed his two representations in shaded versions as well (upper and lower right hand). The construction he used for his second representation (described in the caption of figure V.63) may have been a general rule of thumb. The goldsmith and painter Heinrich Lautensack (1522–1590) followed Hirschvogel’s style in his 1564 work Des Circkels unnd Richtscheyts, auch der Perspectiva, und Proportion der Menschen und Rosse, kurtze doch gründtliche Underweisung, des rechten Gebrauchs (Brief yet thorough introduction to the correct use of compasses and ruler, and of perspective, and proportions in humans and horses). Lautensack stressed the importance of knowing geometry and illustrated its use in, among other things, perspective constructions. He applied a simple method similar to Hirschvogel’s. He also illustrated how the image of a pavement of square tiles can be used as (to apply a modern term) a coordinate system in the picture plane (figure V.64).

Ryff Taking Up the Italian Tradition

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ürer’s presentation of perspective had been inspired by the Italian way of treating the subject. Count Johann’s simple method came from another tradition; as noted, it may have been a simplification of Viator’s procedure. Alternatively it may have been created independently by someone who had seen a perspective grid used, but not learned enough – or cared enough – about its construction. Hirschvogel’s method is a kind of combination of Count Johann’s method and Alberti’s idea of involving the height of a person. Most of the German sixteenth-century literature on perspective is in the style applied by Count Johann and Hirschvogel, but 1547 saw the

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FIGURE V.63. Hirschvogel’s projections of a dodecahedron. He constructed the plan in the upper left hand by first inscribing the regular decagon 1a2 ... , in a circle with centre f, next constructing a circle with radius 1a and centre f. A calculation shows that the ratio between Hirschvogel’s two radii is the correct one, namely the one that was later called ‘the golden section’ (cf. the caption of figure IV.45). Had he wanted to make a complete plan, he should have included five more vertices by also drawing a decagon in the inner circle, as Barbaro later did (figure IV.45). By not including the last five points, Hirschvogel made his plan look very three-dimensional, which is particularly noticeable in its shaded version (upper right hand). He constructed the figure in the lower left hand (with a shaded version to the right) as follows. In a circle with centre n and apparently the same radius as the top one, he inscribed a regular hexagon 123456, and then divided each of the arcs into three equal arcs 1a, ab, b2, ... .* Next he constructed a second circle with centre n and radius ma; where the line in intersects the last circle, he put the point o, constructed the points r and q so that the three points divide the circle into three equal arcs, then connected the points as shown in the figure. Part of figures 9 and 10 in Hirschvogel 1543, rearranged. *Hirschvogel presumably had an approximate construction in mind, since an arc of 60° cannot be trisected with a ruler and compasses.

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FIGURE V.64. Lautensack’s use of a perspective grid to determine the heights of people. Lautensack 1564, fol 28v.

publication of a German-language presentation in the Italian tradition. Its author was Walther Hermann Ryff (born 1500), who was also known as Rivius. He studied medicine and worked as a doctor, but had many other interests and published on several subjects. It seems that architecture came to attract him ever more strongly, and as technical background for this discipline, he took up perspective in Der furnembsten, notwendigsten, der ganzen Architectur angehörigen mathematischen und mechanischen Künst (The most elegant, most necessary mathematical and mechanical art belonging to the entire architecture, 1547). This work consists of extracts from various authors, including Alberti, Gaurico, and Serlio – as far as I have noticed, without source references (see also SchülingS 1973, 21, 26, and 46). Ryff also dealt with proportions in human bodies and illustrated Vitruvius’s ideas (figure V.65) in roughly the same way as Leonardo da Vinci’s well-known and ever-popular drawing. Ryff followed up his work on mathematics applied in art with a book on architecture based on Vitruvius.

Jamnitzer, Lencker, Stör, and Hass

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rom the late 1560s onwards, a number of books appeared featuring Latin titles, German subtitles, very little text (in German), and an abundance of perspective compositions. The most famous of these was Perspectiva corporum regularium (Perspectives of the regular solids) from 1568, written by one of Nuremberg’s leading goldsmiths, Wenzel Jamnitzer (c. 1507–1585). In his preface Jamnitzer announced that he was planning a second volume, but no such volume is known. An example of his impressive compositions is shown in figure V.66. Besides designing and making showpieces, including bowls that were sold to royal courts, Jamnitzer was an energetic draughtsman, etcher, and

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FIGURE V.65. Proportions of a man. Ryff 1547.

engraver responsible, for instance, for the illustrations in Rivius’s edition of Vitruvius. Jamnitzer is also said to have designed mathematical instruments. According to Paul Pfinzing, one of Jamnitzer’s instruments was an apparatus, which he had created by elaborating on some of the ideas shown in Dürer’s drawings (Pfinzing 1599). Jamnitzer’s instrument is shown as illustrated by Jost Amman and Pfinzing in figures V.67 and V.68, respectively. In the latter rendition the eye point is situated on the pin C and can be moved like the eye point in figure V.48. The picture plane D can be turned down upon the table, the device E substitutes the man at the frame in figure V.49, and instead of the object, a plan (Geometria) and a pointwise elevation (marked on F ) are used. Pfinzing also reported that Hans Hayden improved Jamnitzer’s instrument. In Hayden’s version (figure V.69), the picture plane D lies behind the object. The picture drawn on D in this position is an enlarged version of the one produced using Jamnitzer’s instrument.

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FIGURE V.66. Jamnitzer’s imaginative polyhedra. Jamnitzer 1568.

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FIGURE V.67. Jost Amman, Portrait of Wenzel Jamnitzer with a perspective machine, c. 1565, Rijksmuseum, Amsterdam, RP-P-1976-26.

From the Renaissance onwards many artisans and mathematicians were fascinated with the idea of creating perspective instruments and described them in their books on perspective or in separate tracts.24 At first two kinds of instruments were described: some that like Dürer’s, depicted in figure V.49, were meant to be used for drawing the perspective image of a concrete object, and some that like Jamnitzer’s instrument only required a plan and elevation of the object. Both kinds of devices had the limitation that the perspective image had to be constructed pointwise. The seventeenth and eighteenth centuries witnessed the design of drawing machines that would produce a complete

FIGURE V.68. Jamnitzer’s instrument. Pfinzing 1616, fol.13r. 24

For a description of a number of perspective instruments see DawesS 1988; HamblyS 1988, 144–155; KempS 1990, 167–188.

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FIGURE V.69. Hayden’s version of Jamnitzer’s instrument with decoration. Pfinzing 1616, fol.14r.

perspective image (examples are shown in figures X.1 and XII.7). The constructions these instruments were meant to perform are largely correct, but most often it is doubtful whether the machines would work in practice and how many of them were ever actually built. Jamnitzer’s instrument required a plan and an elevation, but constructing these is not always easy, as noted earlier. This art is particularly difficult if the object considered is positioned obliquely in relation to the ground plane and the picture plan, as indeed the objects in some of Jamnitzer’s examples were. Therefore I find it likely that some of the so-called perspective instruments had the function of measuring the distances from points on a real object to the planes of the plan and the elevation. Polyhedra, for instance, could be constructed from a diagram showing them unfolded, examples of which we saw in figures V.30 and V.31. By holding the actual three-dimensional object in a certain position, one could then construct its plan and elevation from measurements.

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Like Jamnitzer, the goldsmith Hans Lencker (1523–1585) published a collection of perspective drawings. He called his book Perspectiva literaria (The perspective of letters, 1567), and rightly so, for it mainly consists of perspective images of the letters of the alphabet, supplemented by the regular polyhedra and a few more figures. Four years later Lencker supplied more textual information concerning the performance of perspective constructions in his Perspectiva (Perspective, 1571). Like several of his predecessors, he carefully described how to use a ruler and compasses. He particularly presented this technique in connection with an instruction on how to make plans and elevations. Finally, he showed how these projections could be used for throwing objects into perspective. Another book (figure V.70) that can be grouped with Jamnitzer’s work was published by the German painter Lorenz Stör. His work, entitled Perspectiva

FIGURE V.70. Lorenz Stör’s title page with polyhedra. Stör 1567.

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et geometria (Perspective and geometry, 1567), contains a collection of perspective compositions that was meant to provide motifs for artisans making intaglio. The only text it had was on the title page. Finally, a similar work, for once bearing a German title – Künstlicher und zierlicher newer vor nie gesehener funffzig perspectifischer Stück (Fifty artistic, delicate, new and never before seen perspective pieces) – was published in 1583 by the Viennese court cabinet maker Georg Hass.

Pfinzing

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he above-mentioned artist Paul Pfinzing (1554–1609) belonged to the group of etchers and drawers who were fascinated by perspective. In 1599 he had Ein schöner kurtzer Extract der Geometriae unnd Perspectivae (A beautiful and brief extract from geometry and perspective) printed, most probably as a private edition. In 1616 it appeared in a new version under the completely new title of Optica, das ist gründtliche doch kurtze Anzeigung wie notwendig die löbliche Kunst Geometriae seye inn der Perspectiv (Optics, that is a thorough yet brief announcement of how necessary the laudable art of geometry is in perspective). His booklet shows how impressively well versed he was in the literature on perspective – both German and non-German publications. Within some thirty pages, Pfinzing reviewed what he considered to be the development of the subject. He referred to Pacioli, Piero della Francesca, Dürer, Eyn schön nützlich Büchlin (by Count Johann), Cerceau, Lautensack, Lencker, Jamnitzer, and Sirigatti. Pfinzing’s technical treatment of perspective is dominated by his descriptions of the apparatus, mentioned in the previous section.

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Vredeman de Vries

he history of Dutch literature on perspective apparently began with the publication in 1553 of a Dutch translation of Serlio’s book on perspective, accomplished by the famous Flemish painter Pieter Coecke van Aelst (Serlio 1553). Serlio’s presentation of perspective constructions is, as we saw, far from clear (page 117), but it nevertheless influenced a Dutch painter, namely Johan Vredeman de Vries (c. 1526–c. 1616), whose first name often is abbreviated to Jan or Hans and who is depicted in figure V.71. Serlio’s work inspired Vredeman de Vries to draw perspective illustrations with great enthusiasm, which is partly how he “found his way and became an excellent painter of architecture and perspective” (translated from Molhuysen & BlokS 1911, vol.7, 1292). Vredeman de Vries initiated a particular Dutch style of architectural painting and became known for his mannerist decorations (WinckelS 1996, 724). An example of his style can be seen in figure V.72, which shows a painting similar to some of his drawings (figure V.73). Vredeman de Vries first published a book in 1560 with samples of his perspective compositions. He continued to add new plates to his collection, and

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FIGURE V.71. Drawing of Vredeman de Vries by Hendrik Hondius. Vredeman de Vries 1604, fol. a2r.

selections of these plates appeared regularly under various Latin titles. His books contained a few lines of text in Dutch, the remainder consisting of instructive diagrams. Looking at Vredeman de Vries’s drawings one almost gets the impression that the art of perspective can be learned from studying these diagrams (figures V.74, V.75, and V.76).

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FIGURE V.72. Palace architecture with walkers by Johan Vredeman de Vries and his son Paul, c. 1596. Kunsthistorisches Museum, Wien.

FIGURE V.73. One of Vredeman de Vries’s architectural drawings. Vredeman de Vries 1605, figure 14.

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FIGURE V.74. Vredeman de Vries’s use of perspective squares for determining lengths. Vredeman de Vries 1604, figure 28.

FIGURE V.75. Illustrative drawing. Vredeman de Vries 1604, figure 10.

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FIGURE V.76. Vredeman de Vries showing the effect of looking down five galleries from a balcony. Vredeman de Vries 1604, figure 39.

Until the publication of his first book Vredeman de Vries spent most of his time in Friesland, but after that he became a frequent traveller, his destinations including Antwerp, Aachen, Wolfenbüttel, Hamburg, Danzig, and Prague. In his later years he settled in the Hague, publishing another two selections of his plates from there in 1604 and 1605 under the title Perspective. These volumes became very popular and were frequently reedited, in Dutch and in translations. They differ from his earlier collections by containing descriptions of each drawing. It is not easy to see whether Vredeman de Vries himself or his publisher, Hendrik Hondius, wrote the text, so Perspective is often attributed to Hondius. Hondius, a publisher and engraver, collaborated with the mathematician Samuel Marolois, who also edited Vredeman de Vries’s Perspective. Marolois’s first edition appeared in 1614 in his Opera mathematica and in several later reprints. Although Vredeman de Vries’s drawings are enlightening, they are not always constructed in accordance with the geometrical laws of perspective. For one thing, Vredeman de Vries placed all vanishing points on the horizon (figure V.77). Marolois pointed out some of Vredeman de Vries’s mistakes and changed some of his plates with drawings that are mathematically correct, as in the example shown in figure V.78. Undoubtedly Vredeman de Vries was instrumental in awakening an interest in perspective in the Northern Netherlands. As noted, much could be learned from his drawings, but not everything. Those who wanted to know

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FIGURE V.77. Drawing in which Vredeman de Vries placed all vanishing points on the horizon. Vredeman de Vries 1605, figure 3.

FIGURE V.78. Samuel Marolois’s correction of Vredeman’s understanding of vanishing points as illustrated in figure V.77. Vredeman de Vries 1615, figure 3.

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more of the technical details were soon offered two excellent opportunities in the form of publications by Simon Stevin and Marolois, which appeared in 1605 and 1614, respectively. I return to these two important contributions to perspective in chapters VI and VII.

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The Sixteenth-Century Non-Italian Tableau

he things shown in this chapter include the following. The knowledge of perspective constructions reached France, some of the German states, and the Netherlands in the sixteenth century. In these regions a number of men – mainly from the circles of artists and artisans – decided to pass on this knowledge through publications in which they focussed on reproducing perspective compositions rather than describing how perspective constructions should be performed. It is therefore fair to conclude that the Northern publications did not contribute to the development of an understanding of the geometry behind perspective constructions. Even a book like Cousin’s, which contained much interesting theoretical material, did not stress the new insights or pass on any wisdom the authors might have gained.

Chapter VI The Birth of the Mathematical Theory of Perspective Guidobaldo and Stevin

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or reasons soon to be explained, I consider Guidobaldo del Monte to be the father of the mathematical theory of perspective and hence pay considerable attention to his work. It is natural to take up Simon Stevin’s work on perspective in connection with Guidobaldo, since Stevin’s contributions are a direct continuation of some of Guidobaldo’s accomplishments. In fact, it was through reading Guidobaldo that Stevin came to see the fundamental problems of perspective, and he was such a gifted reader that he understood the master’s ideas better than the master himself.

VI.1

Guidobaldo and His Work on Perspective

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n essential part of the scientific activity in the sixteenth century – not least in Italy – sought to gain a deeper understanding of the classical Greek sources. Having come to comprehend the ancient sciences, scholars guided their thoughts along new, fruitful paths leading to what has been termed a scientific revolution in astronomy and physics and to the creation of several new mathematical disciplines including analytical geometry, calculus, and theory of probability. This rise of the exact sciences was only possible because the well-known scientists of the day, such as Galileo Galilei, Johannes Kepler, Christiaan Huygens, and Isaac Newton, were surrounded by many intellectuals, who were fascinated by the study of nature and the mathematical sciences. One of them was Guidobaldo Marchese del Monte (1545–1607). He was a pupil of Commandino and took part in his project of making the classical sources available by seeing a manuscript of Pappus’s Collection, left by Commandino, through the press (1588). Guidobaldo took a particular interest in statics, mechanics, and astronomy publishing five books on these subjects, but he was also engaged in mathematics and as we shall see perspective. When Guidobaldo is mentioned, it is most often in connection with Galileo, to whom he was an important patron and friend. The two scientists 237

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discussed many of Galileo’s ideas and conducted experiments together (RoseS 1974). Guidobaldo does, however, deserve to be better known for his own accomplishments, especially for his contributions to the mathematization of the foundation of perspective constructions.

Guidobaldo’s Struggle with Perspective

I

n 1600 Guidobaldo published his results on perspective in Perspectivae libri sex (Six books on perspective, figure VI.1) which is a major work and signifies a turning point in the history of the mathematical theory of perspective.1 As we saw in chapter IV, before Guidobaldo, the mathematicians Commandino, Danti, and Benedetti aspired to achieve an understanding of the geometry behind perspective, and they were successful in proving the correctness of certain constructions – although Danti did not start from scratch, but assumed the convergence rule. Guidobaldo took a new approach in which he based his considerations on general laws. He was the first to realize the importance of considering the perspective images of sets of parallel lines and created the concept of a general vanishing point. His accomplishment turned out to be so fruitful that I find it appropriate to designate him the father of the mathematical theory of perspective. We do not know how and when Guidobaldo became interested in perspective, but he may have been inspired by his teacher Commandino’s work on the subject. One Italian and one Latin composition on perspective by Guidobaldo that precede Perspectivae libri sex have survived and been thoroughly described by Paola Marchi in her tesi di laurea (graduation thesis, MarchiS 1998). Marchi finds it likely that these manuscripts date from the period 1588–1592, and has shown that the elder manuscript, the Italian one, reflects many of Commandino’s ideas (ibid., 72, 74–81). The younger manuscript in Latin contains new ideas (ibid., 20) that seem to have guided Guidobaldo to take his innovative approach to perspective and to publish a book on the subject. Thus, in January 1593 while struggling with the subject, he wrote the following to Galileo. My Perspective is half asleep and half awake, because – to tell you the truth – I have so many engagements that I can scarcely breathe. For these matters I need to be free of all concerns. Still, I do want to finish it. ... However, I have not yet discovered everything, and above all I would like to hear your assessment.2

Whether Galileo reacted to Guidobaldo’s letter is unclear – as in so many other, similar cases in which a scientist asked for Galileo’s opinion. By 1

For another presentation of Guidobaldo’s work, see GuipaudS 1991. La mia Prospettiva mezzo dorme e mezzo vegghia, ché, a dir il vero, io ho tante le occupationi, che non mi lasciano respirare; e per queste cose bisognarebbe esser libero da ogni fastidio: pur la voglio finir ...; ma non ho ancor trovato ogni cosa: e prima di ogn’altra cosa ci vorrò poi il suo giuditio. [GalileiS Opere, vol. 10, 54]

2

1. Guidobaldo and His Work on Perspective

239

FIGURE VI.1. The title page of Guidobaldo’s Perspectivae libri sex. With help I have come to read the sentence in the ribbon as “without deception we are deceived”.* *I am extremely thankful to Pernille Harsting for this suggestion.

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September, Guidobaldo thought it would be possible to finish his work during the winter and to have it published within a year (GalileiS Opere, vol. 10, 62). Even so, it took another six years before it appeared in print. Guidobaldo did not mention which books on perspective he had studied, but he seems to have been well acquainted with most of the Italian literature on the subject, because he included most of his predecessors’ constructions in Perspectivae libri sex. Considered separately, hardly any of the earlier treatises could have inspired him to take a new approach, but together they may have had that effect. In trying to unify the various results he may, in particular, have been stimulated to base his arguments on results from three-dimensional geometry and to concentrate on perspective representations of lines.

The Contents of Perspectivae Libri Sex

A

lthough he did take a mathematical approach to perspective, Guidobaldo did not follow Commandino and Benedetti in treating the subject as a purely geometrical discipline, but borrowed some of his scientific arguments from optics, as several of his predecessors had done. In fact, with a reference to Euclid’s Optics he let the first book of Perspectivae libri sex begin with a section on visual angles and apparent sizes. Here he displayed a style he often used: including much more material than necessary to make his points clear. In connection with his treatment of optics he also took up some geometric problems that had no relevance to perspective, but which he apparently found interesting (Guidobaldo 1600, 24–30). The rest of his first book is devoted to perspective images of lines. The better part of the second book deals with the problem of throwing plane figures situated in a ground plane into perspective. Guidobaldo was very generous with solutions to this problem, presenting no less than twenty three different methods. In the third book he went on to determine perspective images of some three-dimensional figures, in particular polyhedra to which he applied the method of constructing the images of the base and the heights. He also dealt with examples in which objects are projected upon other surfaces than a plane. Presumably intending to help his readers apply the common technique of constructing a perspective image based on a plan and an elevation, Guidobaldo devoted the first part of book four to orthogonal projections and determinations of plans and elevations of polyhedra – also in cases where the base of a polyhedron is not parallel to the plane of the plan. Another essential part of this book deals with the perspective images of circles. Guidobaldo was especially intent on determining the conditions under which circles are depicted as circles. This led on to considerations concerning subcontrary sections similar to those published by Commandino. In book five Guidobaldo turned his attention to another kind of central projection than perspective, namely the shadow cast by an object upon a plane when the light source is located in a single point. As we have seen,

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Dürer took up the special case in which the object is a cube. Using his plan and elevation method, Dürer not only determined the shadow, but also showed how it can be drawn in perspective. In Dürer’s example it is very easy to determine the shadow itself, but very often it is not such a simple matter. Guidobaldo taught his readers how to construct the shadows of polyhedra, cylinders, cones, and spheres. In one example only – dealing with a cylinder – did he also demonstrate how to construct the perspectival image of the shadow (ibid., book 5, proposition 6). In the sixth and final book of Perspectivae libri sex Guidobaldo treated theatrical stage design, a topic previously treated by Serlio, albeit in a way that was rather unclear from a mathematical point of view (page 120). It is very likely that Serlio’s description inspired Guidobaldo to show how the question of drawing set pieces in perspective could be solved geometrically. The theme of designing theatre scenes became quite popular among the Italian writers on perspective in the two centuries that followed, as we shall see in chapter VIII. In all probability, this development had far more to do with Serlio’s influence than with Guidobaldo’s. The following sections discuss some of Guidobaldo’s achievements in greater detail. In accordance with my aim of tracing various methods of construction and the mathematical understanding behind them, I concentrate on how Guidobaldo founded the theory of perspective and on his choice of perspective constructions. To be able to point to improvements introduced by Stevin, I also give an impression of Guidobaldo’s style and consequently deal with quite a number of technicalities. In addition, I present some examples of how Guidobaldo not only cleared up existing theoretical problems, but also took up new, mathematically interesting ones. Finally, I look at Guidobaldo’s role in the history of perspective.

VI.2

G

Guidobaldo’s Theory of Perspective

uidobaldo’s text shows that he conceived of the image of a point as the point of intersection of a visual ray and the picture plane (ibid., 6–7). However, he was so steeped in tradition that he began with Alberti’s definition of a perspectival representation as a section of a visual pyramid – but de facto he restricted himself to sections of visual triangles. The reason was that he wanted to justify the definition by means of the angle axiom (page 89), and this applies only for two-dimensional angles. Thus, he considered (figure VI.2) the visual pyramid, defined by a polygon and the eye point A, and decomposed it into visual triangles, one of which is ABC. Letting the intersection of ABC and the picture plane be the line segment MN, he used the angle axiom to conclude that MN appears to be equal to BC (and presumably he also meant that they appear to coincide). He therefore found it reasonable to define the image of the line segment BC as the line segment MN – calling the latter linea apparens (the apparent line segment). By treating the other visual triangles in the visual pyramid similarly, he introduced the image of a polygon

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Figure VI.2. Guidobaldo’s visual pyramid. Like Vignola (figure IV.18), Guidobaldo drew some of his mathematical illustrations in perspective. Guidobaldo 1600, 5.

(Guidobaldo 1600, 5). Turning to a polyhedron, he looked upon this as being composed of polygons and applied his definition of the image of a polygon. Also, when considering curvilinear figures he kept to visual pyramids and introduced a sort of a generalized visual pyramid, for instance one that has a sphere as its “base” (ibid., 8).

Line Segments Parallel to the Picture Plane

T

he first perspective problem Guidobaldo investigated was how to determine the image of a line segment parallel to the picture plane (figure VI.3). His conclusion is part of the following general theorem for which Willem ’sGravesande later gave a very simple reductio ad absurdum proof. Result 1. A line AB parallel to the picture plane p is depicted in a line AiBi that is parallel to AB (’sGravesande 1711, §4).

’sGravesande’s proof runs as follows. Assume that the line AiBi is not parallel to the line AB, which – since they both lie in the plane OAB – implies that they intersect in a point C. Because AiBi lies in p, the point C is also a point of intersection of AB and p, which contradicts the fact that AB is parallel to p.

2. Guidobaldo’s Theory of Perspective p

243

A Ai

O G Bi R

B

FIGURE VI.3. The image of a line segment parallel to the picture plane p. Guidobaldo actually considered three parallel line segments that are parallel to p and then began by proving that their images are parallel. In a corollary he proved that the images of the line segments are also parallel to the line segments themselves (Guidobaldo 1600, book 1, proposition 24). His argument runs approximately as outlined below – where the brackets refer to the theorems he applied from Euclid’s Elements. Guidobaldo imagined that AB is situated in a plane parallel to p, then used the result that when the two straight lines OA and OB are cut by two parallel planes, they will be cut in the same ratio (XI.17). This means that OAi : AiA = OBi : Bi B. By applying this relation to the triangle OAB, Guidobaldo concluded that AB and Ai Bi are parallel (VI.2).

As can be seen in the caption of figure VI.3 Guidobaldo’s proof is less elegant. The reason could be that he did not think along the same lines as ’sGravesande, but he might also have preferred a direct proof and only wished to involve the image of the line segment AB instead of the entire line AB. During the early period of perspective representations, transversal line segments were depicted as line segments parallel to the ground line, and vertical line segments as line segments perpendicular to the ground line without any supporting arguments. Guidobaldo found that these practices needed proofs, which could have been provided by his conclusion corresponding to result 1. Strangely enough, however, he only applied result 1 to argue that line segments parallel the ground line GR have images that are parallel to GR (Guidobaldo 1600, book 1, proposition 25). In order to conclude that verticals are depicted as verticals – provided the picture plane is vertical – he constructed a new proof (ibid., proposition 26).3 In dealing with perspective images Guidobaldo, as we shall see, used vanishing points as an effective tool. Lines parallel to the picture plane p have no vanishing points, so Guidobaldo proved that their images are parallel to the lines themselves. Later in the seventeenth century, Girard Desargues introduced points at infinity to reach the same result. 3

In considering a vertical line segment, Guidobaldo applied that the plane, defined by the line segment and the eye point, cuts a vertical picture plane in a vertical line.

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Desargues also noted that it is not only a set of parallel lines, parallel to p, that have parallel images, but that this also applies to a pencil of lines if the line connecting the vertex of the pencil and the eye point is parallel to p (page 442). Guidobaldo did not work with pencils of lines, but he was aware that in a vertical picture plane p some line segments in the ground plane g have vertical – and hence parallel – images. Thus, he proved (figure VI.4) that if the plane a, determined by a given line segment AB and the eye point O, is perpendicular to g, then the image AiBi of AB is also perpendicular to g, and in particular to the ground line GR (ibid., proposition 27). The reason is that since both a and p are perpendicular to g, so is their line of intersection, on which AiBi lies. Guidobaldo’s theorem implies Desargues’s result for the pencil of lines that has its vertex in the foot of the eye point. It also implies: Result 2. A horizontal line passing through the foot F of the eye point is depicted as a vertical line. Although he did not formulate this result explicitly, Guidobaldo applied it in several of his proofs (an example can be found in the caption of figure VI.6).

The Main Theorem of Perspective

T

his section is devoted to Guidobaldo’s fundamental result within the mathematical theory of perspective. To place him in a wider historical context I occasionally compare his ideas with those of the English mathematician Brook Taylor, because Taylor was the first to clearly formulate some thoughts that originated with Guidobaldo. Among other things Taylor introduced the term vanishing point (figure VI.5). He defined the vanishing point of a line l, that is not parallel to the picture plane p, as the point Vl in which the line through the eye point O parallel

p O

a

Bi B

G Ai A g

R F

FIGURE VI.4. A non-vertical line segment AB being depicted in the vertical line segment Ai Bi.

2. Guidobaldo’s Theory of Perspective

245

to l intersects p (Taylor 17151, 3). In his constructions, however, he also called Vl the vanishing point of l’s image li (for instance Taylor 17151, 21). In other words Taylor started by assigning the vanishing point to a line and implicitly connected it to the image of the line as well. This custom has generally been respected – also in this book. Historically, things happened in the reverse order, though. Guidobaldo, who was the first to define a concept corresponding to a vanishing point, started with a point related to the image of a line, or to be more precise, he introduced a convergence point, naming it punctum concursus (meeting point) in connection with the images of a set of parallel line segments that are not parallel to p. Not until later did he relate it to the original lines themselves. Having explained the different ways in which a vanishing point and a punctum concursus were introduced, I take the liberty of translating the latter term with the former. It is worth noting that the way Guidobaldo conceived of a punctum concursus carries on the tradition, mentioned in chapter IV, of considering a convergence point in the picture plane as an operative point in a construction (page 136). A vanishing point turns up in the following important result, which I call the vanishing point theorem (figure VI.5). The perspective image li of a line l that is not parallel to p passes through the vanishing point Vl of l, and so do the perspective images of all lines parallel to l. The British mathematician Humphry Ditton characterized the vanishing point theorem as “the main and Great Proposition” (Ditton 1712, 45). Taylor realized that even more fundamental was the result obtained by combining the vanishing point theorem with the simple observation that the image li of a line l contains the point of intersection Il of l and p. He thereby came to the following result, which I call the main theorem of perspective. p Vl l li O G Il g R

FIGURE VI.5. Illustration for the definition of the vanishing point of a line l, the vanishing point theorem, and the main theorem of perspective.

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The image li of any line l that is not parallel to the picture plane is determined by the vanishing point Vl and the intersection Il.4 In his New Principles of Linear Perspective Taylor strongly stressed the importance of this theorem (Taylor 1719, 14 see also page 503). Although he was presumably the first to do so, the theorem was fundamental in all mathematical theory of perspective developed after 1600 – hence my choice in naming it. I have formulated the vanishing point theorem and the main theorem for lines, but actually, in Guidobaldo’s days it was more common to consider line segments than lines. He and his contemporaries followed the Euclidean tradition of not making a linguistic distinction between the two concepts – although Guidobaldo did sometimes use the expression linea infinita (infinite line) when he meant a line (for instance Guidobaldo 1600, 54).

Guidobaldo’s Proofs of the Main Theorem

H

aving given the linguistic and conceptual background for a vanishing point, I turn to Guidobaldo’s role in the history of the vanishing point theorem and the main theorem. Guidobaldo applied the main theorem when he showed how to determine the image of a line and of a line segment (for instance ibid., book 2, propositions 2 and 11). As we shall see, he actually proved the main theorem as well, but he did not formulate it explicitly. Instead he presented the vanishing point theorem, or rather, he presented five results that are special cases of it (ibid., 35–44). One reason Guidobaldo included so many versions was that he did not treat a two-dimensional and three-dimensional set of parallel line segments as one type. Besides this, he distinguished between the situations involving parallel line segments lying in a horizontal plane and in an oblique plane. Not only did Guidobaldo formulate different versions of the vanishing point theorem, he also proved the various versions differently, starting with a special proof and ending with a general one that actually covered all cases. His entire treatment of this theorem seems to reflect a process in which he gradually penetrated to the core of the problem of determining the image of a line. This understanding is confirmed by Paola Marchi’s study of Guidobaldo’s Latin manuscript, mentioned in section VI.2 (MarchiS 1998, 97–113). Here she puts forward a very plausible hypothesis about how he was steered towards the concept of a vanishing point while working with a rather complicated configuration to prove that the images of a set of parallel lines converge (which is actually the proof described in the caption of figure VI.6).

4

To be precise, the half-line behind p (as viewed from the eye point O ) with its end point Il in p, is depicted upon the line segment IlVl, and Vl is only an image point if the point at infinity is added to the line l.

247

2. Guidobaldo’s Theory of Perspective D γ

a Di

C Ci

O

C 9i

π

Bi

Ai

B 9i

A

V B

G S E F

R

FIGURE VI.6. Guidobaldo’s first proof of a version of the vanishing point theorem (Guidobaldo 1600, book 1, proposition 28). His own drawing is reproduced in figure VII.8. Let p be a vertical picture plane, GR the ground line, O the eye point, F its foot in the ground plane g, and AB and CD two parallel line segments (Guidobaldo included a third) situated in g so that they are not parallel to GR. Guidobaldo’s aim was to prove that the images Ai Bi and Ci Di of the two line segments, if prolonged, intersect in a point V whose distance VR to g is equal to OF. In his search for a proof, Guidobaldo transformed the problem so he was able to apply his theorem corresponding to result 2 (page 244). He assumed that the two lines AD and BC pass through the foot F – when the theorem is proved for this case it is easy to obtain the general result. From result 2 he concluded that the images Ai Di and BiCi are vertical and hence parallel. His further strategy was to prove that the two segments Ai Di and BiCi have different lengths, from which it follows that the lines Ai Bi and CiDi indeed intersect each other. To establish that AiDi ≠ BiCi, Guidobaldo determined the point of intersection G of the ground line and FA, drew the line GS through G parallel to AB intersecting BC in S, introduced the vertical picture plane a that had GS as ground line, and considered the image AiBi′Ci′ Di of the trapezium ABCD in a. By means of his construction, he had obtained an image that is a rectangle (result 2 implies that AiDi and Bi′Ci′ are vertical, and the fact that AB and CD are parallel to the ground line GS implies that AiB′ and C′Di are transversals); this means, in particular, that AiDi = Bi′Ci′. Looking at the triangle BOC, in which BiCi and Bi′Ci′ are parallel, Guidobaldo concluded that BiCi ≠ B i′Ci′ = AiDi , and hence that the sides Ai Bi and Ci Di intersect in a point V. Guidobaldo finally applied the theory of similar triangles to prove that VR = OF. It is notable that in this proof Guidobaldo did not make use of the fact that OV is parallel to AB and DC.

One may well wonder why in editing his text Guidobaldo did not limit himself to presenting the general version of his theorem and its general proof. It is difficult to decide which explanation is more plausible: that he was not keen on deleting proofs he had created, or that he thought it would be helpful for

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his readers to be guided towards the general situation step-by-step (for another interpretation see BkoucheS 1991, 252–253). Whatever his reasons were, it is interesting to follow the development of Guidobaldo’s ideas concerning the vanishing point theorem. Let me give an impression of this by describing the first and last stages in some detail in, respectively, the caption of figure VI.6 and here in the main text. Guidobaldo’s final version of the vanishing point theorem – which contains no requirement that the picture plane be vertical – runs as follows. If the eye sees equidistant [parallel] lines which, extended if necessary, intersect the picture plane, then the images of these lines converge in one point that is as high above a plane equidistant to the parallel lines as the eye.5

Let p be the picture plane (figure VI.7), O the eye point, and AB a line segment intersecting p in A. Presumably, to be able to consider a convergence point, Guidobaldo involved more than one line segment in his formulation of the theorem, and in his proofs he first considered three coplanar parallel line segments and then three parallel line segments that do not lie in the same plane. However, the essential argument in his proof concerns the image of one line segment, say AB – about which, without any loss of generality, he assumed that its end point A lies in p. For AB his theorem states that its image, ABi, if prolonged, passes through a point, V, whose distance to any plane parallel to AB is equal to O’s distance to the same plane. In formulating his final version of the vanishing point theorem, Guidobaldo concentrated on V’s and O’s distances to certain planes – which he had also done in his initial version of the theorem (caption of figure VI.6). In his proof of the final version, however, he introduced V as the point of intersection of p and the line through the eye point O parallel to the line

p V B Bi O

a

A

5

FIGURE VI.7. Illustration for Guidobaldo’s final version of the vanishing point theorem.

Si oculus aequidistantes videat lineas, quae cum sectione convenire possint, lineae in sectione apparentes in unum punctum concurrent aequealtum supra planum lineis parallelis aequidistans, ut oculus. [Guidobaldo 1600, book 1, proposition 32]

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segment AB, which is in fact similar to Taylor’s later introduction of a vanishing point. While Guidobaldo’s main techniques in his earlier proofs of the vanishing point theorem involved looking at similar triangles, he went directly to the relevant intersections in his last proof. According to his definition of a perspective image, ABi is the intersection of p and the visual triangle OAB. Let a be the plane determined by this triangle. Using proposition 7 in book XI of Euclid’s Elements, though without referring to it, Guidobaldo concluded that since OV and AB are parallel, the point V lies in a, and hence also on the line of intersection of p and a. Since this line also contains the image ABi, the prolongation of the line segment ABi passes through V. The second part of the theorem – that the distance between V and a plane parallel to the line AB is equal to the distance between O and this plane – follows from the fact that OV is parallel to AB. Because he considered line segments, Guidobaldo did not focus upon the points of intersection of these segments and p when formulating his fundamental result. However, since he chose a line segment with an end point A in p, his proof covers the main theorem – stating that the image of the line AB is determined by the point of intersection A and the vanishing point V.

Vanishing Lines

A

hundred years later, Taylor generalized the concept of a vanishing point and introduced a vanishing line. He assigned the latter concept to a plane a that is not parallel to the picture plane p and defined it as the intersection of p and the plane through the eye point parallel to a. Taylor was the first to extensively apply the concept of a vanishing line, but not the first to consider it. Guidobaldo’s text implicitly contains the general concept of a vanishing line, as I will shortly argue, but explicitly only one particular vanishing line occurs, often referred to as the horizon – although not by Guidobaldo. He introduced this line by considering several sets of parallel lines that lie in the ground plane. He stressed that the vanishing points of these sets of parallel lines lie as high above the ground line as the eye point, concluding that they lie on a line parallel to the ground line (Guidobaldo 1600, book 1, proposition 33, corollary 2). He also mentioned that all horizontal lines have their vanishing points on this line (ibid.). Guidobaldo’s implicit consideration of a general vanishing line is connected to a theorem that is rather strangely formulated, its content being – freely translated – the following.

In a picture plane there can be an infinite number of collinear vanishing points that have different distances to the ground plane.6

6 In eadem sectione infinita esse possunt puncta concursus in eadem recta linea existentia, quae supra subiectum planum inaequales altitudines habeant. [Guidobaldo 1600, book 1, proposition 35]

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In chapter V, we saw how Vredeman de Vries thought that all vanishing points ought to be placed on the horizon (figure V.77). Indirectly, Guidobaldo’s theorem maintains that this is not the case. In his formulation he paid attention to the distances between the vanishing points and the ground plane, noticing that the property of equality between the distances was lacking – which is presumably a result of how he began his work on vanishing points. The interesting part of his theorem is, however, the statement that the considered vanishing points are collinear – or rather this result becomes interesting when it is known how these vanishing points are defined. In our terminology they are vanishing points of lines situated in a non-horizontal plane a that is not parallel to the picture plane p. Guidobaldo proved that these vanishing points lie on the intersection of p and the plane through the eye point parallel to a – which is indeed the same as Taylor’s vanishing line of a. In the final proposition of the first book of Perspectivae libri sex, Guidobaldo stated that when a set of parallel lines and various eye points situated on a line parallel to the parallel lines are given, then the parallel lines have the same vanishing point with respect to the different eye points. After his proof he added that the result might seem paradoxical, but that it is nevertheless true being verified by a proof (Guidobaldo 1600, 49). This remark, coupled with the fact that it had been such a struggle for Guidobaldo to get the vanishing point theorem established, indicate that his investigations had led him to achieve results that surprised himself. However, when he had passed the hurdle of presenting and proving the vanishing point theorem, he was able to fully appreciate its potential.

VI.3

Guidobaldo’s Twenty-Three Methods

B

ased on his theory of perspective, Guidobaldo developed various constructions of perspective images. He first addressed the question of constructing the image of a plane figure, once again displaying his fascination with problem solving. He formulated and solved the following problem no less than twenty-three times (Guidobaldo 1600, 61–104):

Given the eye point and a rectilinear figure in the ground plane, draw the image in a given plan perpendicular to the ground plane.7

In his various propositions Guidobaldo considered different polygons. The differences in the propositions do not, however, concern the figures to be thrown into perspective, but the methods of construction. Although all his methods are presented in connection with polygons, they are in principle pointwise constructions and I present them as such – assuming, like Guidobaldo did, that the point to be thrown into perspective lies in the ground plane. 7

Oculo dato, dataque in subiecto plano rectlinea figura, in proposita sectione subiecto plano erecta figuram apperentem describere.

3. Guidobaldo’s Twenty-Three Methods

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Guidobaldo’s Rabatment

T

o be able to perform his constructions on a piece of paper Guidobaldo almost always applied a particular rabatment. Its qualities and drawbacks will be discussed in section VI.9, here I simply present it. In figure VI.8 I have outlined both the three-dimensional and the two-dimensional situation for the case in which the object is just one point (figure VI.10 shows one of Guidobaldo’s own examples). The left diagram displays the ground plane g, the eye point O and its foot F, the picture plane p with ground line GR and the horizon HZ, and a given point A in g. Guidobaldo’s rabatment corresponds to turning p down into g around GR towards A, so that the area in which the image of A has to be constructed is situated above GR, as can be seen in the right diagram. Below GR lies the foot F, besides which a point O′ is marked. The latter is situated on the line through F parallel to GR, so that FO′ = FO and has the function of indicating the distance between O and the ground plane.

The Sixth Method

G

uidobaldo characterized his various constructions by means of certain points he chose for his operations. In a number of cases these points were two vanishing points V1 and V2 on the horizon, and the foot F of the eye point (figure VI.9). He described a general version of this situation in his sixth method, which I first present and then comment upon. In this method Guidobaldo included the orthogonal projections of V1 and V2 upon the ground line GR, U1 and U2, and used the fact that the lines connecting U1 and U2 with F define the directions of the lines that have V1 and V2 as vanishing points. He thus constructed the image Ai of A, by first drawing the two lines through A that are parallel to FU1 and FU2. Letting these lines cut

p H

H

Z

A OG

Z

G

R

A g F

R F

FIGURE VI.8. Guidobaldo’s rabatment.

O'

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VI. The Birth of the Mathematical Theory of Perspective

p H

V1

V2

G

A

Ai

U1

Z I2 g U2

R

I1 F

A

H

V2

V1

Z

Ai

G

U1

I2

U2

I1

R

F

FIGURE VI.9. Guidobaldo’s sixth method.

GR in the points I1 and I2, Guidobaldo drew the lines I1V1 and I2V2 and claimed that their point of intersection is the image Ai of A. To prove the correctness of this method, Guidobaldo applied the main theorem (Guidobaldo 1600, 72–73). From this he deduced that I1V1 is the image of AI1, and similarly I2V2 the image of AI2, and hence their point of intersection is the image of the point of intersection of AI1 and AI2 – that is, of the point A. Figure VI.10 shows how Guidobaldo illustrated the method.

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FIGURE VI.10. Guidobaldo’s own illustrations of his sixth method. The upper diagram is the three-dimensional situation, and the lower the two-dimensional one. The eye point is A, V and X are the two given vanishing points, the ground line is BF, and in the ground plane is given the triangle CGH, whose image LMH he constructed. Guidobaldo 1600, 72–73.

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Guidobaldo’s last argument is based on the assumption that incidences between lines are preserved under a perspective projection, or in other words, he assumed that the image of the point of intersection of two lines is the point of intersection of the images of those lines. Most of the pre-nineteenthcentury perspectivists considered this result so obvious that they did not bother to comment upon it. The only exception I know of is John Lodge Cowley, who actually proved the result (Cowley 1765, 32). Guidobaldo similarly took the projectively dual result – that the image of a line joining two points is the line joining the images of the two points – for granted, whereas a few mathematicians, including Stevin and Taylor, explicitly mentioned it (Stevin 16051, 16; Taylor 1719, 13). Guidobaldo’s sixth method is neat because it is both general and simple. It would therefore be natural to expect that the method would become popular among Guidobaldo’s successors, but this did not happen. As far as I have noticed, Frans van Schooten was the only perspectivist who decided to use this method as a basic construction. His illustration of the method is shown in figure VI.11 (see also page 320). Guidobaldo himself did not comment upon the qualities of his various methods, but his sixth one appears to be central in his work because it gave rise to a number of other methods. He obtained these by letting one or both of the vanishing points V1 and V2 belong to the particular set consisting of the principal vanishing point and the two distance points – the vanishing points of orthogonals and diagonals. In this way he obtained, among other methods, a distance point construction.

FIGURE VI.11. Frans van Schooten’s version of Guidobaldo’s sixth method. Here the given vanishing points are O and N, V is the foot of the eye point, and the distance between the two last points is given as the line segment marked S. Applying the sixth method, van Schooten constructed the image X of W. Van Schooten 1660, 529.

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The Tenth Method

I

n another group of constructions, Guidobaldo only involved one vanishing point on the horizon and then the foot of the eye point as well. He further then incorporated the line joining the foot and the given point in the ground plane to be thrown into perspective. The most straightforward of these procedures, his tenth method, is similar to one of Benedetti’s constructions and, as noted, became Stevin’s basic construction (page 274). Guidobaldo’s original version is illustrated in figure VI.12. Grouped together with this elegant method are some rather cumbersome constructions. p V

O

Ai G

A

U I

g

B F

R

V A Ai

G

U

I

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FIGURE VI.12. Guidobaldo’s tenth method. Let V be a vanishing point situated on the horizon, U its orthogonal projection upon the ground line GR, O the eye point, F its foot, and A a given point in the ground plane g. To construct the image Ai of A, Guidobaldo proceeded as follows. He drew the line AF meeting GR in I, the line through A parallel to FU intersecting GR in B, the line BV, and the normal to GR at I. He then claimed that the intersection of the last two lines is Ai. That Ai lies on BV follows from the main theorem, and that it lies on the normal to GR at I follows from the fact that FA is the orthogonal projection upon g of the visual ray joining the eye point O and the point A.

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Dismissing the second vanishing point as well, but retaining the line through the foot and the given point whose image has to be found, Guidobaldo developed some methods that are variants of a plan and elevation construction.

The Twenty-First Method

G

uidobaldo also presented some constructions in which he determined the distance between the image of a point and the ground line with the aid of a proportion rather than an elevation. As in other cases, he gave simple examples, as well as rather complicated ones, of how to apply his idea. The simple methods include one – the twenty-first – that I find very elegant and have presented in the caption of figure VI.13. This method appealed to Marolois, who presented it in 1614, and to Lambert more than a hundred years later, as we shall see (pages 301 and 643). Guidobaldo himself also seems to have been quite satisfied with his twenty-first method, for he applied it often in his work. The last two of Guidobaldo’s procedures involve points or lines whose perspective images are known. Having treated the problem of constructing the perspective image of polygons in much greater detail than most readers would ever wish for Guidobaldo proceeded to look at polyhedra. Once again, his basic technique was in principle a pointwise construction, in which he first constructed the image of the orthogonal projection of a point A upon the ground plane g, and then foreshortened the distance between A and g. Guidobaldo gave many examples of this technique. He did, however, refrain from illustrating it in connection with all his twenty three methods only returning to about a third of them.

VI.4

I

New Themes in Guidobaldo’s Work

n this section I deal with three of Guidobaldo’s new themes, the first of which concerns the construction of perspective images in other surfaces than a vertical plane. The second theme is known as inverse problems of perspective – meaning problems in which a perspective image is given, while certain information about the original configuration is sought. The third theme is related to one of the most interesting aspects of the further development of the theory of perspective; one that resulted in what Lambert called perspective geometry. This deals with how constructions similar to the usual Euclidean constructions can be performed directly in the picture plane without involving any points from a plan or an elevation.

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p

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FIGURE VI.13. Guidobaldo’s twenty-first method. Before presenting this method I describe the observations upon which it is founded. In the upper diagram are given a point A in the ground plane, the ground line GR, the eye point O and its foot F. The point O′ is constructed on the line through F parallel to GR so that FO′ = FO. Finally the lines AF and O′A cutting GR at the points I and C are drawn. Let the image of A be Ai. By considering the two pairs of similar triangles AAiI and AOF, and AIC and AFO′ we see that AiI : OF = AI : AF = IC : FO′. Since, as we saw in the caption of figure VI.12 Ai lies on the vertical through I, this relation, together with the fact that FO′ = OF, show that Ai is the point on the vertical through I determined by AiI = IC. Guidobaldo used Ai I = IC in the following way (lower diagram). First he determined the points I and C as described above, then on the perpendicular to GR at I he constructed the line segment IAi equal to IC. This is a very simple method of obtaining the image, because all it requires is drawing the lines AF and AO′ and rotating the line segment IC by an angle of 90˚.

Untraditional Picture Planes

I

n giving up the requirement that a perspective composition should be drawn on a vertical plane, Guidobaldo first considered an oblique picture

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plane and then went on to more untraditional screens, such as one consisting of three vertical surfaces, another composed of a vertical and an oblique rectangle, and a number of curved surfaces, such as the surfaces of cylinders, cones, spheres, conoids, and spheroids (Guidobaldo 1600, 141–170). In one case he even dealt with a picture plane formed by the combined surfaces of a cylinder, a sphere and a cone (ibid., 164). Guidobaldo’s problems about curved surfaces were not as difficult as they may sound, since he only took up fairly simple examples. His treatment of the construction of images in oblique planes was based on the vanishing point theorem, which he had formulated so generally that it also includes oblique planes. However, it requires patience for the reader to follow his description of how to construct a vanishing point of a horizontal line in an oblique plane (ibid., 140). Guidobaldo introduced his method for constructing images in a ‘broken picture plane’ by stating a rather puzzling theorem the content and function of which I have explained in the caption of figure VI.14. His own illustration of how to throw a horizontal triangle into perspective on a surface that consists of three vertical rectangles is reproduced in figure VI.15. To make his principle more transparent I have illustrated it in figure VI.16.

G

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FIGURE VI.14. Guidobaldo’s puzzling proposition 29 in book 3 of Perspectivae libri sex. The proposition asks for the solution to the following problem (in which I have introduced my own symbols). Throw a plane figure into perspective when the height of the eye is given as FO′′ on a line that is not parallel to the ground line GR. In his twenty-first method (cf. figure VI.13) Guidobaldo had involved the point O′ directly in the construction – O′ being the eye point turned down into the ground plane around the line through F perpendicular to GR. For that procedure it was essential that FO′ be parallel to GR. In the present problem, Guidobaldo sought a method similar to his twenty-first method covering the situation in which FO′′, rather than FO′, is given. One may well wonder why he did not just construct FO′ = FO′′ on the parallel to GR. The answer seems to be that for a construction in a ‘broken plane’ – the ground line being a ‘broken line’ – Guidobaldo did not want to work with several representations of the eye, but preferred a single operative point.

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FIGURE VI.15. Guidobaldo’s example with a ‘broken plane’. The ground line is the ‘broken line’ EFGH and the object to be thrown into perspective is the triangle BCD. Its image is folded down in the ground plane as three pieces, namely HGYZ, GRKF, and FPLI. Guidobaldo 1600, 152.

Inverse Problems of Perspective

I

n general, inverse problems have more mathematical than practical relevance since a perspective projection is chosen when the task is to give an impression of how an object looks, and not when the task is to give a precise p

A Ai

G

X Xi

O

A9i

g H

A9 p9

F R

FIGURE VI.16. A ‘broken picture plane’. The basic idea in Guidobaldo’s construction is the following. Let p and p ′ be picture surfaces whose intersection with the ground plane g make up the broken line GHR, O the eye point, F its foot, and AA′ a line that is situated in g and has to be thrown into perspective. Guidobaldo drew the line FH cutting AA′ in X, and then determined the image of the two line segments AX and XA′ in p and p ′, respectively.

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representation of the shape of an object.8 For the latter purpose it is better to apply a parallel projection or construct a plan and elevation of the object. There is, however one inverse problem of perspective that does have some practical interest, namely the following. Where should a person standing in front of a perspective image place his or her eye to properly perceive the scene created by the artist? From a mathematical point of view, answering this question means reconstructing the eye point that the artist used for making his perspective drawing. Since an important part of the visual process takes place in the mind, and since we have got used to interpreting perspective images, it is not always crucial to find the exact eye point. Still, in some cases the three-dimensional effect becomes much stronger when a perspective picture is seen from its eye point. There are also perspective compositions in which some objects seem to have unnatural dimensions when looked upon from a position other than the eye point. I have particularly noticed this effect in connection with tables whose longer sides are perpendicular to the picture plane – as exemplified in figure VI.17.

FIGURE VI.17. A composition that affords a far more striking spatial experience when viewed from the eye point. The distance of the picture is about two-third of the picture’s length, and the principal vanishing point lies in the point of intersection of the two long sides of the table. Niels Larsen Stevns, Mary Magdalene anointing Christ’s feet, 1907, Nordjyllands Kunstmuseum, Aalborg, Denmark.

8

In the mid-nineteenth century the theory of inverse perspective underwent a revival and was developed as the discipline photogrammetry in connection with photographic surveying.

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The general problem of finding the eye point for a perspective composition is indeterminate, unless some information is given or some assumptions made. This might typically involve assuming that a tiled floor is the image of a floor with square tiles. As we shall see, after Guidobaldo, inverse problems of perspective developed their own academic life in which practical problems played no role. Instead it became a question of what kind of knowledge was required to solve the problems. Among the mathematicians investigating this issue were Stevin, Taylor, and Lambert; they were – without using these exact words – attempting to find various ways of characterizing a perspective projection. The kind of inverse problems treated before 1800 fall naturally into two groups: one in which assumptions about the original figure are made and the eye point is to be determined, and another in which the eye point is assumed to be known and the original figure is to be found. Guidobaldo presented problems from each of the two categories (Guidobaldo 1600, 110–112, 126–128). Rather than being inspired by practical questions or driven by a wish to gain mathematical insights, Guidobaldo seems to have taken up inverse problems because he wanted to use their solutions as auxiliary results for perspective constructions, an example of which is shown in the caption of figure VI. 19. In dealing with the determination of the eye point, he showed that this can be done when two points in the ground plane and their perspective images are given. His solution consisted in inverting the steps of his twenty-first method of constructing the image of a point given in the ground plane. In working with a given eye point, Guidobaldo took up, among other questions, the following simple problem. Given a point A in a vertical picture plane, determine the point Ao in a given ground plane g that is depicted in A. As so often before, he offered two solutions, and again he found them by inverting the steps of his construction procedures in his sixth and twenty-first methods.

Direct Constructions

I

n later chapters we shall see how some of Guidobaldo’s successors treated a subject I call direct constructions which covers constructions that are performed directly in the picture plane without involving auxiliary drawings such as a plan. In particular, the later perspectivists systematically took up the perspectival versions of the most common Euclidean constructions. Guidobaldo dealt with two of the fundamental problems, namely those corresponding to the two following usual constructions (Guidobaldo 1600, 113–114). 1. Through a given point, draw a line parallel to a given line. 2. Through a given point on a given line, draw a line that makes a given angle with the given line. The perspectival equivalent of the first problem is special by being one that is easier to solve in the picture plane than in the Euclidean plane. Guidobaldo

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Z B

M

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M

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FIGURE VI.18. The problem off constructing a line that is perspectively parallel to a co given line.

considered a version (figure VI.18 ) in which the horizon HZ, the line segment BC, and the point M are given in the picture plane. It is known that BC is the image of a horizontal line segment, and the task is to construct the line passing through M that is the image of a line parallel to the original of BC. Guidobaldo solved this problem by determining the point of intersection V of HZ and the prolongation of BC, and by drawing the line VM – which is obviously the required line since V is a common vanishing point for VM and the line BC. In treating the second problem, Guidobaldo yet again offered his readers two solutions. Actually, only one of these follows the principle that the construction should be performed directly in the picture plane (ibid., book 2, proposition 35). The second solution, presented in the caption of figure VI.19, provides an example of the above-mentioned procedure of involving the solution of inverse problems of perspective. Besides these two fundamental problems, Guidobaldo dealt with the problem of dividing a perspective rectangle (figure VI.20).

VI.5

W

Guidobaldo’s Role in the History of Perspective

hat Guidobaldo achieved in Perspectivae libri sex is really quite impressive. He realized that the key to understanding perspective constructions was to look at vanishing points. He also demonstrated that the mathematics inherited from the Greeks was sufficiently rich to provide a geometrical foundation of perspective. Finally, he opened new paths in the theory of perspective. He was indeed the father of the mathematical theory of perspective. As impressive as his accomplishments are, his ineptness in presenting them was equally astounding. In September 1593 he wrote to Galileo that he was

5. Guidobaldo’s Role in the History of Perspective

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C

B A

Ci

Bi Ai G

F

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FIGURE VI.19. Guidobaldo’s first solution to the problem of constructing the image of a given angle (Guidobaldo 1600, 112). In a vertical picture plane p are given the ground line GR and the image AiBi of a line segment situated in the horizontal plane through GR. The eye point is given by its foot F and the length FH equal to the distance, and an angle v is also given. It is required to construct the line AiCi so that it forms an angle with AiBi that is the image of v. Guidobaldo began by using an earlier solution to determine a horizontal line segment AB that has AiBi as its image. He then constructed the line AC, forming the angle v with AB, and finally constructed the image AiCi of AC.

B R

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FIGURE VI.20. Guidobaldo’s perspective division of a rectangle. It is given that ABCD is the perspective image of a vertical rectangle, and that X is the vanishing point of the line AD. The task is to construct the line parallel to AB dividing ABCD into two areas that are images of rectangles in a given ratio. Guidobaldo’s solution was to divide the line AB at the point E in the given ratio, draw the diagonal AC and the line through E, which are perspectively parallel to AD. Through the point of intersection F of these two lines, he then drew the vertical GH and proved that this is the solution. Adaptation of the figure on page 234 in Guidobaldo 1600.

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cutting and abbreviating his manuscript as much as he could, because he found it too long (GalileiS Opere, vol. 10, 62). However, Guidobaldo seems to have been so attached to his material that he could not restrict himself to the really important issues. The result was that his brilliant ideas drowned in a sea of irrelevant propositions. This makes the reading of Perspectivae libri sex a tedious and confusing experience. Apparently Guidobaldo also confused himself, for instance during his treatment of oblique picture planes. First he presented all the necessary theory in a fairly clear fashion, but then in his examples he failed to use all his insights and gave rather clumsy solutions. Jean Étienne Montucla expressed the following opinion on Guidobaldo’s Perspectivae libri sex in his Histoire des mathématiques (1758). Moreover, the work by Guidobaldo suffers from the usual fault of its time; the matter it presents in a multitude of theorems could have been expressed far more neatly in fewer pages.9

I agree with Montucla’s observations on Guidobaldo’s style, but I would not call it typical for his time. As I have already indicated and will demonstrate shortly, not long after the publication of Perspectivae libri sex, Stevin presented the basic ideas of the work in a few pages. Rather than seeing Guidobaldo’s approach as a result of his time, I would regard it as an example that the first formulations of concepts or ideas are often more complicated than later presentations – as well as a result of Guidobaldo’s personal style. In analysing Perspectivae libri sex I have concentrated on Guidobaldo’s presentation of the theory of perspective. One of his successors, Claude François Milliet Dechales, was not too dissatisfied with the work, but still found that Guidobaldo had paid insufficient attention to the practice of perspective: The theory of this work is solid and geometrical, yet the method is somewhat on the difficult side. As a result, no one could learn perspective from this book alone, because it appears that it does not go sufficiently down into its practice. However, if one is moderately versed in perspective, one can gain much enlightenment from the book.10

It is true that Perspectivae libri sex is not a work for non-mathematicians, and certainly not a book that can easily teach anyone to perform perspective constructions – but being written in Latin and mainly containing examples dealing with geometrical figures, it was presumably not meant to be either. As

9

Au reste, l’ouvrage de Guido-Ubaldi a le défaut ordinaire de ceux de son temps; ce qu’on y trouve exposé en une multitude de propositions, pouvoit être dit avec plus de netteté en peu de pages. [MontuclaS 1758, 636] 10 Doctrina hujus operis solida est & geometrica, tamen methodus paulo difficilior; ita ut nullus in eo solo libro possit perspectivam addiscere, videtur enim non satis ad praxin descendisse, qui tamen in perspectiva mediocriter esset versatus posset ex eo multum lucis haurire. [Dechales 1690, vol. 1, 68]

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I will also stress in later chapters, Guidobaldo was actually the last mathematician who did not claim to be writing for practitioners. Although highly theoretical, Guidobaldo’s work does also contain material that can be useful to practitioners performing perspective constructions, not at least the concept of vanishing points – which practitioners actually adopted very quickly. Considering the effort it takes to read Perspectivae libri sex, I suspect that the number of people who ever studied the work thoroughly is small. Yet, all the further developments of the theory of perspective have their roots in this work. Guidobaldo’s ideas were spread to a wider audience by a number of perspectivists, among whom Stevin, Aguilon, and Marolois seem to have been the most important. The contributions of the two mentioned last will be discussed in later chapters, and Stevin’s contribution in this chapter. Throughout the seventeenth century authors on perspective used many of Guidobaldo’s expressions, which they picked up either directly from him or through other authors. We will see examples of this later. In short, it can be concluded that Guidobaldo’s achievements were extremely important for the development of the mathematical theory of perspective and had a number of consequences for its practice, but it was his results rather than his presentation that inspired his successors. Nevertheless, Perspectivae libri sex maintained its reputation as an important book on perspective for quite a long time.

VI.6

T

Stevin and His Work on Perspective

here are many similarities between the lives of Guidobaldo del Monte and Simon Stevin (1548–1620), whose portrait is shown in figure VI.21. They were born in the same decade and actively took part in the revival of science in their countries. Guidobaldo’s first book treated mechanics, and in particular statics, as did one of Stevin’s first important publications, Beghinselen der weegconst (Principles of the art of weighing, 1586). Moreover, the two shared a keen interest in applied science. There are also differences in the life stories of the two scientists. Unlike Guidobaldo, Stevin was not a man of independent means, but had to earn his living. He presumably began his career in the financial administration in the part of the Netherlands that now belongs to Belgium. From around 1582 he worked in the Northern Netherlands as an engineer, advancing to the position of quartermaster around 1604. Stevin’s engagement in civil and military administration and in engineering are reflected in his writings. His very first publication was a table of interests (1582), and his next, De Thiende (1585), was an introduction to calculating with decimal fractions that greatly influenced the teaching of arithmetic in and outside the Netherlands. Stevin’s other publications on applied science and mathematics include introductions

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FIGURE VI.21. Portrait of Simon Stevin. Artist unknown. Leiden University Library, Icones 40.

to bookkeeping, constructions of mills and sluices, house building, fortification, geography, and navigation (MinnaertS 1976). Another difference between Guidobaldo and Stevin is that the latter wrote precisely and concisely. Stevin often found his material in other books, but he

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presented it more pedagogically than his sources, stressing the important points of his topic. He also made independent contributions and is still well known in the history of science for some of his clever observations, the most notable being his deduction of the law of equilibrium of bodies on an inclined plane, presented in Beghinselen der weegconst. Besides mechanics, his publications on science concern astronomy, geometry, algebra, the theory of tuning, and perspective. Guidobaldo and Stevin also differed in their choice of publication languages: Guidobaldo followed the scientific tradition and wrote in Latin, whereas Stevin published the greater part of his works in Dutch, hoping to reach the group among his countrymen who were aspiring scientists and not well versed in foreign languages. Stevin shared the idea of publishing in the vernacular with several scientists writing in the late sixteenth and early seventeenth centuries. Galileo, for instance, published his works in Italian, and René Descartes in French. Stevin faced the problem that the existing Dutch vocabulary lacked many scientific terms. He had, however, a talent for coining new words and enriched the Dutch language with numerous expressions, several of which have survived to the present day. In the beginning he only filled out lacunae, but this habit soon developed into an almost fanatical purism and he exchanged foreign words already in common use with Dutch neologisms. Geometry became meetconst (the art of measuring) and mathematics became wisconst (the art of certainty) – the “art” element was later exchanged for “learning”, so today the Dutch speak of wiskunde (on Stevin and the Dutch language see DijksterhuisS 1943, 298–320). Through his work Stevin came into close contact with Prince Maurice of Orange, who as stadtholder and military leader played an important role in the northern provinces’ struggle for independence towards the end of the sixteenth century. During his lifetime Maurice was actually in contact with three engineers publishing on perspective, and we will meet the other two, Marolois and Jacques Aleaume, in chapters VII and IX. Maurice had a strong interest in mathematics and used Stevin as both a mathematical adviser and a tutor. Stevin collected the lectures he gave to the prince and published them as Wisconstighe Ghedachnissen (Mathematical memoirs), one part of which appeared in 1605 under the title Van de deursichtighe (figure VI.22). The very word deursichtighe was Stevin’s translation of perspectiva, which he applied in the broad sense that includes some optical theories as well. Stevin’s original plan was that Van de deursichtighe should consist of three books, one on verschaeuwing,11 one on spieghelschaeuwen (catoptrics), and one on wanschaeuwing (refraction). Ultimately, however, Van de deursichtighe only came to contain Van de verschaeuwing and a brief section on catoptrics.

11 Verschaeuwing is Stevin’s Dutch translation of the word scenography, which he used it in the sense of linear perspective (cf. note 3 page xx).

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FIGURE VI.22. The title page of Stevin’s book on optics.

Stevin’s Path to Perspective

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ccording to his own account, Stevin first took up the study of perspective while composing a work on architecture (Stevin 16051, 4) – which he never finished (HeuvelS 19942 and HeuvelS 2005). Stevin’s return to perspective was motivated by Prince Maurice, who was very interested in the subject. Stevin claimed that the prince had learned perspective from the “ablest masters of painting that could be enlisted”, and had been taught methods in

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which “the foreshortening of lines and the change of angles was obtained by sight or by guessing” (translation by Struik in Stevin 1958, 801). Maurice was not satisfied with this and asked Stevin to teach him exact constructions and the mathematical reasons behind them. Recalling his reaction, Stevin wrote: ... I perused and examined, more fully than before, several writers who deal with this subject and made a description thereof in my own style. After his Princely Grace had looked it through, helped to correct the imperfections that are commonly found in first attempts, had also profoundly understood the common rule of throwing any given figure into perspective, and practised it to his satisfaction, I included this description among the Mathematical Memoirs ...12 [Stevin 16051/1958, 801 – with minor adaptations]

Stevin did not reveal whose works he “perused and examined”. There can be no doubt, however, that Guidobaldo’s Perspectivae libri sex was his main source of inspiration and the work he really examined, for many of Stevin’s expressions, theorems, and basic ideas in proofs and constructions are similar to Guidobaldo’s (see also Struik in Stevin 1958, 790, and SinisgalliS 1978, 125–134). Irrespective of any stimulus Stevin may have found in Guidobaldo’s treatment of perspective, he did, however, develop his own style – as he himself stressed in the quote – and did create a work of remarkable originality – as Dirk Struik stressed (Stevin 1958, 790). Besides Guidobaldo, Benedetti may have given Stevin some ideas on how to treat perspective. For one thing, as we have seen Stevin, like Benedetti, introduced a prism in an argument, and we will soon see other similarities.

The Contents of Van de Verschaeuwing

I

n setting up his work on perspective Stevin followed the tradition of classical Greek mathematics and began by setting out definitions and postulates. From these he deduced six theorems, which constitute the foundation for his solutions of perspective problems. His problems fall into two categories, one of which concerns proper perspective constructions. The other concerns inverse problems of perspective, which deal with the determination of the eye point when some properties of the original of a perspective figure are given. His chapter on inverse problems of perspective is followed by a short section called fautmercking (detection of errors) containing five rules of

12

... soo heb ick breeder dan daer te vooren, deursien en ondersocht verscheyden Schrijvers van dese stof handelende, en na mijn stijl van dies een beschrijving ghemaeckt: Welcke nadien se sijn Vorstelicke Ghenade oversien hadde, en helpen verbeteren de onvolcomentheden die ghemeenlick in eerste vonden sijn, oock grondelick verstaen de ghemeene reghel om all voorghestelde verschaeulicke saeck te verschaeuwen, en dat hij tot sijn vernoughen dadelick verschaude: Soo heb ick dese beschrijving onder sijn Wisconstighe Ghedachtnissen vervought ... [Stevin 16051, 4]

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perspective, useful for either avoiding or detecting errors in perspective constructions. They are all elementary applications of his theory, such as: If in an image we see lines which we know must be the images of different sets of parallel lines, which are also parallel to the floor, but non-parallel to the glass [picture plane], while one set is non-parallel to the other; if all the meeting points of those different sets do not fall in a straight line, we can infer from it that the drawing has been performed in a defective manner ...13 [Stevin 16051/1958, 955]

Van de verschaeuwing concludes with an appendix in which Stevin briefly discusses eight other matters connected to perspective.

VI.7

The Foundation of Stevin’s Theory

A

lthough Stevin got the idea of treating perspective together with other optical theories, he did not mix the theory of perspective with the theory of appearances – contrary to most of the early theorists. To Stevin perspective was a purely geometrical discipline whose foundation he described in sixteen definitions and two postulates. In mathematizing the situation of a viewer looking at an object that has to be depicted on a surface, he defined the eye as a point (definition 6) and the surface as a transparent and infinite plane, which he called the glass (definition 9). In his introduction to the concept of a perspective image, Stevin’s mathematical approach becomes strongly evident. Most of his predecessors had considered sections in visual pyramids or triangles. For Stevin it was sufficient to introduce the perspective image of an arbitrary point – an approach also taken by Benedetti (page 147). According to Alberti’s model, the image of a point is the point in which the visual ray between the point and the eye pierces the picture plane. Stevin made this assumption his first postulate, stating that a point, its image in the picture plane, and the eye point are collinear. He was well aware that he was working with an idealized notion of vision, and he explained that his postulate was necessary because the eye functions in a complicated manner (Stevin 16051, 14). To cover the case in which the point to be thrown into perspective already lies in the picture plane, Stevin added a second postulate, claiming that objects situated in the picture plane serve as their own images. Later in the seventeenth century van Schooten considered Stevin’s second postulate a corollary of the first (van Schooten 1660, 511). Stevin condensed the theory of perspective into six theorems – or in reality, into five, as his fourth theorem is a corollary of his third. In the first he

13

Als wy in eenighe schaeu linien sien, die wy weten schaueuwen te moeten sijn van verscheyden partien van verschaueulicke evewijdeghe linien, die mette vloer oock evewijdich sijn, maer mettet glas onevewijdich, en d’een partiee onevewijdich van d’ander: Soo al de saempunten dier verscheyden partien, niet in een rechte linien vielen, men can daer uyt oirdeelen de verschaeuwing qualick ghedaen te weten ... [Stevin 16051, 82]

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presented a result referred to earlier, namely that the straight line connecting the images of two given points is the image of the line joining the two given points (Stevin 16051, 16). In his second theorem he dealt with images of lines parallel to the picture plane. Stevin’s third theorem is in effect Guidobaldo’s important vanishing point theorem (page 245), stated in complete generality. Taking up an idea from Guidobaldo’s general proof, Stevin proved this theorem elegantly. He did not formulate the main theorem explicitly, but he used it repeatedly and has presumably considered it an obvious consequence of the vanishing point theorem. He called a vanishing point a saempunt – a Dutch word undoubtedly inspired by Guidobaldo’s term punctum concursus. Taken in its entirety, Steven’s treatment of the fundamental result concerning vanishing points is a concise version of Guidobaldo’s theory. However, in his further treatment of the theory of perspective Stevin worked independently of Guidobaldo.

The Invariance Theorem

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ith his fifth theorem Stevin provided a foundation for performing constructions in oblique picture planes. In principle Guidobaldo’s theory covers this problem, but as noted earlier, his determination of vanishing points in oblique planes was not very suitable for actual constructions. Stevin chose an approach in which he reduced the problem of constructing images in an oblique plane to a construction in a vertical plane. His solution was based on an ingenious observation pertaining to the invariance of a perspective image. More specifically (figure VI.23), Stevin’s problem was one of projecting a point A in the ground plane g from the eye point O upon an oblique plane p with the ground line GR. First he considered the usual situation (figure VI. 24), in which A – seen from the eye point O′ – has the image Ai′ in a vertical plane p ′. Next he imagined that p ′ and the vertical line O′F ′ are rotated simultaneously, p ′ around GR and O′F ′ around the line through F ′ parallel to GR, so that O′F ′ stays parallel to p ′. He then posed the interesting question of what happens with the image Ai′ during the rotation? His answer, given in his fifth theorem, was the following. Result 3. The image Ai′ remains the image in the rotated picture plane of the point A seen from the rotated eye point (Stevin 16051, 29). Based on this result, he was able to construct the image of the point A in p in a fairly straightforward fashion (caption of figure VI.24). Stevin’s proof of his rotation theorem is uncomplicated, involving nothing more than arguments concerning similar triangles. His accomplishment was thus not the proof itself, but the idea of looking at rotations. An admirer of this idea, Gino Loria claimed that Stevin had discovered “the fundamental theorem of the method of central projection” (LoriaS 1908, 587) and decided to name a more general theorem after Stevin (LoriaS 1907, 131).

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FIGURE VI.23. An oblique picture plane.

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FIGURE VI.24. Stevin’s use of his rotation theorem. To construct the image Ai in p of the point A seen from the eye point O, Stevin applied his result concerning rotation (Stevin 16051, 33). He assumed that the angle u between p and p ′ is given, along with the position of O (implying that the length OF ′ is given), and straightforwardly determined the eye point O ′, which belongs to a vertical plane p ′, and which after a rotation falls in O when p ′ falls in p. Afterwards he constructed the image in p ′ of A seen from O′. According to his theorem, this is also the image in p of A seen from O.

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FIGURE VI.25. Stevin’s procedure for determining the image of a point above the ground plane in an oblique picture plane.

In his sixth theorem Stevin also worked with rotations (figure VI.25), now focussing upon the image of a point C′ that lies at the distance A′C′ from the ground plane. His idea was basically the same as in the previous theorem: as before, the eye point O′ and the picture plane p′ are turned, and additionally the vertical line A′C′ is turned around a line parallel to GR through A′ so that it stays parallel to p′. Stevin then proved the following generalization of result 3. Result 4. Seen from the rotated eye point, the rotated image of C′ will remain the image of C′ in the rotated picture plane. His application of this result was similar to his use of theorem 5, the difference being that besides constructing the eye point O′, which will fall in a given eye point O, he also had to construct the point C′, which will fall in a given point C (Stevin 16051, 34).

VI.8

Stevin’s Practice of Perspective

Stevin’s Basic Constructions

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n accordance with his view that it was sufficient to define the perspective image of a point, Stevin considered the basic problem of perspective construction to be the following. Given an eye point O and an arbitrary picture plane p (not containing O), determine the image of an arbitrary point A that

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lies on the opposite side of p from O. Starting from the situation in which p is vertical and A is situated in a given ground plane, Stevin successively covered all possibilities. Many constructions solving Stevin’s problem already existed – Guidobaldo alone offered twenty three. As noted, Stevin chose Guidobaldo’s tenth method, which was also one of Benedetti’s methods, as his basic construction (pages 151 and 255). We cannot know what made Stevin prefer this construction, but it was convenient for his treatment of inverse problems of perspective, as its steps can easily be reversed. In showing how to apply the method, he actually presented two procedures, one of which he called “mathematical”, the other “mechanical” (ibid., 22–25). The “mathematical” procedure is similar to that of Guidobaldo (figure VI.12), whereas in the “mechanical” one Stevin constructed parallel lines using a technique that deviates from the exact Euclidean way. In an actual performance Stevin’s “mechanical” construction requires fewer steps than his “mathematical” one, and the latter is in practice as precise as the former. The “mathematical” construction can also be abbreviated, however. For practical purposes it is, for instance, unnecessary to go through all the steps required by Euclid to draw a line parallel to a given line. It is therefore an open question whether Stevin really had practical application in mind when he introduced his “mechanical” construction, or whether he simply wanted to display a clever construction. Stevin presented a geometrically elegant proof of his “mathematical” construction, but it is not so easy to follow his arguments relating to the threedimensional configuration on his diagram (figure VI.26). He seems to have been aware of the problem and suggested that the reader should imagine the paper to be folded. He provided the diagram reproduced in figure VI.27 to support the

FIGURE VI.26. Stevin’s illustration of Guidobaldo’s tenth method (figure VI.12). The line BC is the ground line, D is the foot of the eye point, and DE is equal to the height of the eye above the ground plane. Guidobaldo in his diagrams drew the latter line parallel to the ground line, whereas Stevin sometimes drew it, as here, perpendicular to the ground line, and at other times followed Guidobaldo’s custom. Stevin 16051, 22.

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FIGURE VI.27. Stevin’s attempt to facilitate the imagination of a three-dimensional configuration. Stevin 16051, 23.

impression of the triangles AHI being horizontal and the rectangle T being a vertical picture plane. It is doubtful that this drawing would be of any help to a reader imagining the three-dimensional situation. In other words, Stevin’s many gifts did not include the ability to make illustrative drawings. It may have been to compensate for this that he made a few of his figures three-dimensional in the sense that a number of triangles were glued in separately and could be made to stand perpendicular to the page by pulling threads attached to the figure. Stevin also described a modification of Guidobaldo’s tenth method that he devised himself. Stevin’s version is based upon his considerations concerning rotation, and it contains a mathematically very neat observation that I have presented in the caption of figure VI.28.

FIGURE VI.28. Stevin’s modification of Guidobaldo’s tenth method. To understand the principle, let us assume that the turning process described in connection with figure VI.24 is continued until the picture plane and the eye point have reached the ground plane, the eye point falling in O′′. Stevin’s result, namely that the rotated Ai′ is still the picture of A, combined with a continuity argument implies that Ai lies on O′′A. Thus, this line can be used as one of the two lines that determine the position of Ai. In his first application of Guidobaldo’s tenth method (figure VI.12), Stevin determined Ai as the point of intersection of BV and the vertical through I. In his own modification, Stevin let Ai be the point in which BV meets the line AO′′. He constructed O′′ by drawing the perpendicular to GR through F, and on it cutting off FO′′ equal to the height of the eye above the ground plane.

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FIGURE VI.29. Stevin’s generalization of Guidobaldo’s tenth method (figure VI.12). This time the point M, given by its orthogonal projection A and its height AM above the ground plane, has to be thrown into perspective. In the upper diagram, illustrating the three-dimensional situation, the line through M parallel to FU is drawn intersecting the picture plane p in N – implying that BN = AM. Stevin applied the fact that the image Mi lies on NV (according to the main theorem) and on the normal to GR at I (since orthogonal projection of OM upon the ground plane is FA). He thus constructed Mi as the point of intersection of these two lines. The lower diagram illustrates how Stevin performed the construction on a piece of paper, the only change from the determination of the image of A being that instead of BV he drew the line NV, N being determined by making BN = AM perpendicular to GR.

In throwing points above the ground plane into perspective, Stevin – which Guidobaldo had not done – expanded Guidobaldo’s tenth method to cover this situation as well (figure VI.29). Stevin’s next step was to rotate the picture plane. In the previous section we saw how he reduced problems of constructing images in oblique planes to problems of determining images in a vertical plane – viewed from a rotated eye point. Stevin also studied a situation in which the picture plane is parallel to the ground plane. Although his general theory covered this case as well, Stevin’s solution for constructing images in ceilings is neither particularly elegant, nor very practical (Stevin 16051, 35–36).

Stevin’s Rabatment

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o combine a plan of a figure and the picture plane in a plane drawing, Stevin followed Guidobaldo (figure VI.8) and chose a solution that

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corresponds to rotating the picture around the ground line into the plan of the object to be thrown into perspective, or equivalently, turning the plan into the picture plane. This resulted in the plan of the object and the image of the object both being situated above the ground line. This solution is rather inconvenient, since it causes the final perspective drawing to get mixed up with the plan – as can be seen in figure VI.15. Most other authors applied a technique that situates the plan below the ground line, reserving the space above it for the perspective construction. Some of the procedures based on this rabatment do not, however, produce the same image as the one obtained by Guidobaldo’s and Stevin’s rabatment, but a reversed image. In section VII.7 I will return to this problem – which I have given the name the problem of reversing – and describe its technical aspects in greater detail. Some of Stevin’s successors, for instance ’sGravesande and Taylor, paid attention to the problem of reversing, but most authors did not mention the topic. As far as I am aware, none of the perspectivists preceding Stevin touched upon the problem of reversing, nor did Stevin. Even so, he may have been aware of the issue and deliberately avoided it by choosing Guidobaldo’s turning procedure. At any rate, he achieved a representation that is mathematically satisfactory, but rather inconvenient for practical application.

Stevin’s Examples

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aving solved the problem of finding the image of a point regardless of how the picture plane is situated, Stevin had, in principle, solved all problems of perspective. He stressed this by demonstrating how his procedures could be applied to construct the image of a rectilinear solid figure given by its

FIGURE VI.30. Stevin’s image of a tower. In this example Stevin supposed the picture plane to be vertical. In other diagrams Stevin showed how the images of the tower look when the picture plane is oblique, slanting either towards or away from the eye, and when the picture plane is horizontal. Stevin 16051, 36.

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FIGURE VI.31. A cube thrown into perspective. Stevin 16051, 55.

plan and elevation. One of his examples is reproduced in figure VI.30 – which, like other of his diagrams, shows that Stevin’s imagination was better suited to formulate mathematical arguments than to produce illustrative drawings. Stevin did not end his chapter on perspective constructions with the examples of pointwise constructions of images of rectilinear figures. He knew that this solution was not the most refined, and he therefore added a number of examples to illustrate how an application of his theory led to shortcuts in the actual performance of the constructions. His examples mainly concern the construction of perspective images of parallelograms and parallelepipeds (figure VI.31) lying in various positions relative to the picture plane. In general Stevin avoided throwing curvilinear figures into perspective, but he did pay attention to how circles are depicted, and especially to the question of when a circle is depicted as a circle. As we have seen, Commandino and Guidobaldo had previously dealt with this question and had used the result that when a section in a cone is a circle, then its subcontrary section is also a circle (pages 140 and 240). Stevin also applied this result. He considered a circle situated in the ground plane and the cone defined by it and the eye point, and then characterized the direction of the picture planes that cut the cone, producing subcontrary sections (Stevin 16051, 58–59). He looked, in particular, at the situation in which the ground line is a tangent to the original circle, and using his own procedure of constructing images in oblique planes, he determined the diameter of the image circle (figure VI.32). Working with perspective circles inspired Stevin to take up a new topic within perspective, namely tangency. He took it for granted that tangency is

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9. Stevin and Inverse Problems of Perspective FIGURE VI.32. Stevin’s example of a circle depicted as a circle. The circle with the diameter AB lies in the ground plane and touches the ground line GR in A. Given is an eye point whose projection upon the ground plane lies on the line through A perpendicular to GR. Stevin determined the slope of the picture plane p so that p produces a subcontrary section in the cone defined by the eye point and the circle with the diameter AB. He constructed the image Bi of the point B, and concluded that the image of the circle is the circle with the diameter ABi. Redrawing with letters altered of Stevin’s diagram in Stevin 16051, 59.

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preserved under a perspective projection and suggested this property be used to test the correctness of some constructions. In the special case (figure VI.32) in which the circle with diameter ABi has been constructed as an image of the circle with diameter AB, Stevin prescribed the following method for checking the construction. Around the original circle he circumscribed the square KLMN and constructed its image KLi Mi N. In order for the construction of the image circle with diameter ABi to be correct, the quadrangle KLi Mi N must circumscribe this circle, and the points of tangency Bi, Ci, and Di in the perspective image must be the images of the points of tangency B, C, and D in the original configuration. Stevin went even further in his mathematical reasoning (ibid., 60) and inversely concluded that a circle inscribed in the image KLiMiN of the square KLMN must necessarily be the image of the circle inscribed in KLMN. Similarly, he remarked that a circle inscribed in the image of a given rectangle circumscribing an ellipse is the image of that ellipse. These reflections are interesting instances of Stevin looking upon conic sections from a projective point of view.

VI.9

Stevin and Inverse Problems of Perspective

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tevin introduced his treatment of inverse problems of perspective by pointing to the fact that, precisely like viewers of anamorphoses, viewers looking at a perspective composition will get the best impression if their eyes are situated near the eye point:

... it is known that pictures are made which, when seen from in front, seem very faultily made, not resembling that which they are intended to depict, but when the said

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pictures are seen from one side, through a small hole provided for the purpose, indicating the place of the eye, they make a very nice impression. And in the same way it should be understood that other images, which have been made perfectly in accordance with the art, have such a place that when the eye is placed there, the picture is seen in its perfection. If such a small hole were given for every picture or perspective drawing, it would not have to be sought. But since this is not the custom, we will write down what occurs to us on this subject ...14 [Stevin 16051/1958, 911]

Stevin’s examples cannot be applied by a museum visitor seeking the eye point of a painting, however, as they are too much sophisticated for that. Despite his comment on the convenience of having eye points marked, Stevin does not seem to have been interested in describing solutions to simple cases, such as determining the eye point for a painting of which one knows that a row of quadrangles are images of squares. Instead he was intrigued by the thought of detecting a pattern in the necessary assumptions. It is likely that he had asked himself what information is needed to obtain a unique solution to an inverse problem. He did not find the answer to this question, but he did show that to secure a unique solution, at least four different points in the picture plane must be given – as well as some information concerning the shape formed by the original points (Stevin 16051, 77–80). Stevin pursued the issue of finding the necessary conditions by starting with relatively easy cases, and then gradually building up increasingly generalized problems (ibid., 61–77). In presenting his steps I have reversed his process by first formulating his two most general inverse problems of perspective, and then indicating how he reduced them to simpler cases. Both problems concern a perspective polygon – by which I mean the perspective image in a plane p of an original polygon situated in a plane g having a ground line GR in common with p. They are as follows. Problem 1. Given are a perspective polygon with at least four vertices, the angle between p and g, the shape of the original figure, and that the polygon has at least one line segment – side or diagonal – parallel to GR. Find the eye point (ibid., 69). That the shape of the original polygon is known means that its angles and the ratios between its sides are given.

14 ... tis kennelick dat men schilderyen maeckt, welcke van vooren ghesien seer mismaeckt schijnen, niet ghelijckende t’ghene sy beteyckenen moeten, maer de selve schilderie van ter sijden gesien deur een seker gaetken daer toe veroirdent, anwijsende de plaets des ooghs, sy ghelaten seer hupsch: En alsoo salmen verstaen ander schaeuwen die volcomelick na de const ghemaeckt sijn, sulcken plaets te hebben, alwaer het oogh ghestelt, de schilderye in haer volcommenheyt ghesien wort. Nu soo an alle schilderyen of schaeuwen sulcken gaetken ghestelt wierde, men soude dat niet behouven te soucken: Maer t’selve inde ghebruyck niet sijnde, wy sullen schrijven t’ghene ons van dies nu te vooren comt ... [Stevin 16051, 60]

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Problem 2. Given are a perspective polygon with at least four vertices, the angle between one of its sides and GR, the angle between p and g, the shape of the original figure, and that the latter contains at least one pair of parallel line segments. Find the eye point (ibid., 73). In all his examples Stevin first considered the case in which the angle between p and g is 90˚, and then reached the general situation by applying his result concerning the rotation of the picture plane. Stevin’s solution to the first problem applied the fact that the perspective polygon has a line segment parallel to GR. By constructing another line parallel to GR through one of the vertices of the perspective polygon, and prolonging sides if necessary, Stevin first reduced the problem to one of determining the eye point for a perspective trapezium – whose original shape is known. Next, he reduced the latter problem to one in which the eye point for a perspective parallelogram with a pair of sides parallel to GR has to be found (figure VI.33). Finally, he solved this problem by inverting his method of perspective construction (figure VI.34). Stevin also reduced the second problem to one of constructing the eye point for a perspective parallelogram. The reduced problem caused him some difficulties, however. His solution is correct, albeit confused (figure VI.35). At a certain stage in the procedure he confessed – with surprising frankness – that since no mathematical solution for constructing a particular point had occurred to him, he chose a mechanical construction (Stevin 16051, 71).

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FIGURE VI.33. Stevin’s reduction of an inverse problem of perspective. Let it be given that AiBiCD is the image of a trapezium that has the shape ABCD and two sides parallel to the ground line – without any loss of generality, Stevin assumed that the line DC lies on the ground line. He reduced the problem of determining the eye point for the trapezium to one of determining the eye point for a parallelogram in the following way. He drew AE parallel to BC, obtaining the parallelogram ABCE. From the fact that E lies on the ground line and thereby in the picture plane, he concluded that the line AiE is the image of AE, which means that he now knew a perspective parallelogram and the shape of the original figure. Redrawing with letters altered of a figure in Stevin 16051, 64.

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FIGURE VI.34. One of Stevin’s examples of determining the eye point. The quadrangle A iBiCD is given to be a vertical image of a horizontal parallelogram of a given shape, and it is required to determine its eye point. Stevin first constructed the vanishing point V of AiD and BiC – as their point of intersection – and then the horizon HZ. Next he determined the foot F of the eye point by inverting Guidobaldo’s tenth method (figures VI.12 and VI.26). His construction of the foot runs as follows. On CD he constructed a parallelogram ABCD with the given shape of the original parallelogram. Since the image and the original have the side DC in common, Stevin concluded that Ai is the image of A. Hence, to invert the tenth method, he drew the lines through Ai and V perpendicular to GR cutting it in I and U, and finally the line AI and the line through U parallel to AD. The point of intersection F of the last two lines is the foot, which, together with the horizon, determine the position of the eye point. Redrawing with letters altered of figure in Stevin 16051, 63.

VI.10

Further Issues in Stevin’s Work

The Contents of Stevin’s Appendix

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o illustrate Stevin’s further interest in perspectival matters, let me describe the themes he took up in the eight-section appendix to Van de verschaeuwing. In the first section Stevin stated that just as a mason has to practice before he can build a house, it requires practice to make a perspective composition (ibid., 83). In the next two sections (ibid., 84–85), Stevin commented upon other authors’ choices of terminology – and he was not pleased. In this connection

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FIGURE VI.35. Another of Stevin’s inverse problems of perspective. It is given that AiBiCDi is a vertical image of a horizontal parallelogram whose shape is known, and it is required to find its eye point (Stevin 16051, 71). Stevin first found the vanishing points V1 and V2 of the two pairs of parallel sides of the parallelogram and then constructed the horizon. Without any loss of generality, he considered the line through C parallel to V1V2to be a ground line GR, and constructed the points I 1and I2 in which AiBi and AiDi meet GR. His idea was to reconstruct the parallelogram ABCD depicted in AiBiCDi, and the crucial step was to reconstruct the point A. According to the main theorem the sides AB and AD intersect the ground line in the points I1 and I2. This implies that A lies on the circular arc from which the line segment I1I2 is seen within an angle equal to the given angle DAB, since A similarly lies on the circular arc from which the line segment CI1is seen within an angle equal to the given angle CAB, the point A is a point of intersection of the two arcs. Each of these arcs can be constructed by applying proposition 33 in book 3 of Euclid’s Elements. Stevin admitted only to be able to remember how such arcs are constructed when the visual angle is right (ibid.) and instead obtained the point A by involving a “mechanically” construction. The point A gave Stevin the directions AB and AD, and using them he reconstructed the foot F of the eye point. The foot, together with the horizon, determines the eye point.

he presented an Alberti construction and pointed out how Serlio had got it mixed up with a distance point construction. Stevin devoted the fourth section to a discussion of the choice of parameters for a composition, and in the fifth section he presented Alberti’s method of circumscribing a figure with a grid of squares and using this grid to construct the perspective image. As an example Stevin showed a fortification (figure VI.36). The last three sections concern the column problem, an instrument, and an arithmetical solution to a perspectival problem, all of which I will treat in some detail below.

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FIGURE VI.36. A perspective image of a fortification. Stevin 16051, 86.

The Column Problem

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tevin introduced the column problem with the remark that this problem had caused some experts in perspective drawing to doubt the general validity of the rules of perspective (ibid., 87). He wanted to put these doubts to rest, but chose to present a simplified form of the problem. Although Stevin’s version has already been discussed in section II.15, I repeat here that he represented the columns as points (figure VI.37), thereby focussing upon the aspect of the

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FIGURE VI.37. Stevin’s version of the column problem. The diagram illustrates a horizontal section in which C is the eye point, HI the picture plane, A, D, etc. columns and K, L, etc. their images. Stevin 16051, 83.

problem that involves the preservation of visual angles, and leaving out the aspect that concerns the visible portions of the columns. Had he made a more realistic drawing, he would surely have been able to solve the latter problem. Stevin’s treatment of the column problem came to the attention of the Dutch natural scientist Isaac Beeckman. In his journal from July 1623, Beeckman referred to a discussion of the problem and citing Stevin as an authority, he claimed that columns should be painted according to the rules of perspective15 (BeeckmanS Journal, vol. 2, 248–249; van HeuvelS 19941, 72).

A Perspective Instrument

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lthough the aim of Van de verschaeuwing was to explain the mathematics behind perspective constructions, Stevin also touched upon the use of perspective instruments. In fact, he designed an instrument for Prince Maurice, who later had it built (figure VI.38). Stevin acknowledged having found inspiration for his instrument in a work he thought might have been by Dürer (Stevin 16051, 88). Perhaps his source of inspiration was the instrument reproduced in figure V.48. According to Stevin, the Prince took great pleasure in working with the instrument, and in the process detected some mistakes in the manuscript for Van de verschaeuwing. It continues to puzzle me what kind of mistakes Stevin had made that could be spotted by using the instrument.

An Arithmetical Example

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tevin concluded Van de verschaeuwing with an example of calculating the sides and angles of a perspective square (figure VI.39). Prince Maurice also got credit for this example. He apparently asked Stevin whether it was possible to calculate the solution to a construction problem. By way of an example in which he computed the sides and angles in a perspective square, Stevin showed

15

I am thankful to Charles van den Heuvel for having made me aware of this example.

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FIGURE VI.38. Stevin’s perspective instrument. The eye point is at D, and A is the picture plane. The distances between the two can be regulated, and so can the distance between the eye point and the floor. Stevin 16051, 89.

the prince that this could indeed be done. Some later perspectivists took an arithmetical approach to perspective in which they calculated the foreshortening of certain lengths, but they did not determine sides and angles as Stevin had. As we shall see in chapter XI, the arithmetical treatment of perspective

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FIGURE VI.39. Stevin’s arithmetical example. It is given that ABCD is a square with a side length of 2 feet and DC situated on the ground line. The orthogonal projection of the eye point upon the ground plane is the point F, given by the lengths CE = 3 feet and EF = 4 feet. Moreover, the distance between the eye point and F is given as 5 feet. It is required to determine the angles and sides in the perspective image DCIK of the square. By trigonometric calculations, Stevin found that ∠IDC = 45˚, ∠DCK = 120˚58′, ∠CKI = 59˚2′, ∠KID = 135˚, CK = 34 /9 , KI = 11⁄3, and ID = 50 /9 . Stevin 16051, 91.

never became a success, since geometrical constructions are much more convenient for solving perspective problems than calculations are – or at least they were before the age of computers.

VI.11

G

Stevin’s Influence

uidobaldo’s and Stevin’s contributions meant that besides being an art, perspective became a science. In the history of this new science it was to be expected that Stevin – who had solved many of the fundamental problems of perspective so elegantly – came to figure prominently. As it turned out, however, this was not the case. Outside the Netherlands no significant references were made to Stevin’s work on perspective until 1837, when Michel Chasles wrote in his Aperçu historique:

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S’Gravezande and Taylor are often mentioned, and rightly so, for having treated perspective in a new and scientific manner. However, we are surprised that Stevin is passed over in silence. A century earlier he had also innovated the subject, which he treated as a thorough geometer and, seen from the theoretical aspect, perhaps more completely than any other.16

I share Chasles’s surprise, and wonder about the reason for explicit recognition of Stevin’s work coming so late. In the following I search for the answer by looking at how Stevin’s work was received, first outside the Netherlands, and then at home.

The Knowledge of Stevin’s Work Abroad an de verschaeuwing had the potential to become influential outside Dutch circles, for within the year of its publication, translation appeared in both French and Latin – provided by Jean Tuning and Willebrord Snellius, respectively (Stevin 16052 and Stevin 16053). We cannot know how widely these translations were studied, but we can be sure that only a few traces of Stevin influencing foreign perspectivists are known – and all the ones I know of are from France. As we shall see in chapter IX, the French literature on perspective flourished during the seventeenth century. The majority of French authors worked independently of Stevin and his approach. Many of them composed books entirely devoted to perspective and addressed them to practitioners. These authors were probably unfamiliar with Stevin’s contribution because they naturally read books in the same genre as their own rather than Stevin’s Mémoires mathematiques, in which his work on perspective was published. The only exception I am aware of is Bernard Lamy, who appears to have found significant inspiration in Stevin for his book on perspective, published in 1701 (page 472). Stevin’s Mémoires mathematiques were more likely to be noticed by scholars who themselves composed general works; and I have in fact found the three following examples of the use of Stevin’s work on perspective in French textbooks. In 1637 Pierre Hérigone published a Cursus mathematicus, which includes a section on perspective. Hérigone based this section on Stevin’s presentation, taking over his idea of considering the picture plane as a glass (vitre). Hérigone did not go into detail with the theory, but presented the main results as axioms. His first axiom is similar to Stevin’s first postulate, and some of his later axioms contain material from Stevin’s theorems (Hérigone 1637).

V

16

S’Gravezande et Taylor sont cités souvent, et à juste titre, comme ayant traité la perspective d’une manière neuve et savante: mais nous nous étonnons que l’on passe sous silence Stevin qui, un siècle auparavant, avait aussi innové dans cette matière qu’il avait traité en profond géomètre, et peut–être plus complétement qu’aucun autre, sous le rapport théorique. [ChaslesS 1837, 347]

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In 1644 we once again find traces of Stevin’s work, this time in a book by Marin Mersenne on geometry. Mersenne just barely touched upon perspective, and wishing to be brief, he copied a few of Stevin’s results without repeating the proofs and without providing drawings (Mersenne 1644, 541–544). The third and last example of Stevin’s influence occurs in another Cursus mathematicus, this one published by Claude François Milliet Dechales in 1674. Contrary to Hérigone, Dechales included all the theories necessary for deducing the basic rules of perspective. His presentation of this theory is quite close to Stevin’s, whereas he chose constructions that differ from Stevin’s (Dechales 1674). Of the three authors mentioned above, only Mersenne acknowledged being inspired by Stevin. And since Mersenne’s treatment of perspective was insignificant, his mentioning Stevin cannot have had much influence. Dechales characterized Stevin’s work on perspective in a general survey of mathematical books. In accordance with his own use of Stevin’s work he wrote that it contained good demonstrations, but no method of construction that was adequate for practitioners (Dechales 1690, vol.1, 69) – which is actually more or less the same thing he wrote about Guidobaldo’s work (page 264). Dechales’s remark could hardly have contributed to making Stevin’s work famous either.

The Knowledge of Stevin’s Work at Home

I

n the Netherlands Van de verschaeuwing was well known, presumably because Stevin’s mathematical works became part of the Dutch mathematical heritage. As I show in more detail in chapter VII, the Dutch academic development was, influenced by Stevin. However, neither of the two mathematicians who truly benefited from his work, Frans van Schooten and ’sGravesande, ever referred explicitly to Stevin.

Conclusion

I

n summing up, I claim that Stevin’s work had a very small impact in France, but was largely responsible for the development of a Dutch academic approach to perspective. There were, however, no important authors who pointed to Stevin as the creator of an excellent theory of perspective. Hence it is not so surprising after all that the discovery of Stevin’s achievements was left to scholars with an interest in history. Part of Stevin’s accomplishment was to lift the theory of perspective, inspired by Guidobaldo’s work, to a higher level. In fact, he did such an excellent job that more than a century passed before mathematicians made any essential improvements of the theory – apart, that is from the introduction of an angle scale, which the reader will become acquainted with in chapter IX.

Chapter VII The Dutch Development after Stevin

VII.1

A Survey of the Literature

I

n previous chapters we have seen how the early Dutch literature on perspective was much influenced by Italians. Serlio’s work caught Vredeman de Vries’s interest, and Guidobaldo’s book was an extremely important source of inspiration for Stevin. This chapter deals, among other things, with how these impacts influenced the further development in the Northern Netherlands. Vredeman de Vries’s style of presenting perspective was followed by the engraver Hendrik Hondius (1573–c. 1650) in his Onderwijssinge in de perspective conste (Instruction in the art of perspective, 1623), which, like Vredeman’s publications, mainly consists of a collection of perspective plates. As noted, Hondius also collaborated with the mathematician and engineer Samuel Marolois in editing Vredeman de Vries’s work. Marolois himself took up Vredeman’s ideas, as well as Guidobaldo’s, and published his own book on perspective in 1614. This work, which combined a practical approach to perspective constructions with some theoretical considerations, became very popular and seems to have dominated the Dutch scene for the next four decades. The mathematician Frans van Schooten brought Stevin’s academic approach to perspective back to life in 1660, about half a century after it had been developed. When yet another half century had passed, in 1711 to be precise, there was a second revival by the mathematician and natural philosopher Willem ’sGravesande. The line of Stevin-inspired publications also includes a work from 1676 by the mathematician Abraham de Graaf, and one from 1705 by Hendrik van Houten. Dutch society flourished during the seventeenth century, and so did Dutch painting. There was a market for specific motifs – the most interesting pieces in connection with perspective depict townscapes and interiors of homes and churches – and a veritable industry sprang up to meet the demand for the various genres. Of the many who specialized in church paintings, Pieter Saenredam is particularly well-known (figures VII.1 and VII.2). Domestic scenes were depicted by several other gifted Dutch painters, among them is Pieter de Hooch (figure VII.3). These two painters, like many of their 291

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FIGURE VII.1. The interior of St. Bavo Church in Haarlem by Pieter Saenredam. It is most likely a preparatory drawing for the painting reproduced in the next figure. Until 1945, Kupferstichkabinett, Berlin.

colleagues, used perspective constructions for creating their compositions, but did not always follow all the rules of perspective. In Saenredam’s case, this is well documented by Robert Ruurs (RuursS 1986; see also KempS 1986, 242–251). The Dutch painters generally wrote very little about their art, and next to nothing about perspective constructions. The latter point can be illustrated by two influential books on painting published by the painters Samuel van Hoogstraten and Gerard de Lairesse (Hoogstraten 1678 and Lairesse 1707). They both highly recommended the use of perspective, but neither gave any introduction to the subject. In that respect they showed the same attitude that characterized Leonardo da Vinci’s Trattato della pittura (page 83). Without

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FIGURE VII.2. Pieter Saenredam, The interior of St. Bavo Church in Haarlem, 1633, Kelvingrove Art Gallery and Museum, Glasgow.

going into detail, Lairesse simply mentioned how a room can be thrown into perspective (figure VII.4) and touched upon perspective a few times (among other places in Lairesse 1707/1740, vol. 2, 140, 156). Van Hoogstraten advised his readers to study the works of Dürer, Vredeman de Vries, Guidobaldo, Marolois, and Desargues (Hoogstraten 1678, 276). It is noticeable that while he included two of his Dutch predecessors, Vredeman and Marolois, he did not mention Stevin. I take this as a confirmation of my earlier claim that Stevin was only known among mathematicians (page 288). That van Hoogstraten also referred to Dürer, Guidobaldo, and Desargues does not necessarily imply that he had studied their works, but rather that he had met their

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FIGURE VII.3. Pieter de Hooch, The Bedroom, about 1660, National Gallery of Art, Washington.

names quite often in other books on perspective. Perhaps as an excuse to include so little on perspective, van Hoogstraten made the promise that if he found the time and felt the inclination, he would present a quick way to make perspective constructions, but apparently neither the required time nor the inclination ever materialized. His ability to make extraordinary perspective compositions and play with them is demonstrated in his creation of a perspective box – to which I will return. Another example is shown in figure VII.5, which belongs to his treatment of shadows. The sketch is drawn in perspective, but van Hoogstraten did not comment upon how he had performed the construction. In fact, his chapter on shadows precedes his treatment of perspective. It is also striking that no architects are found among the authors of Dutch books on perspective, whereas this profession was well represented in Italy and France, particularly in the sixteenth and seventeenth centuries. Desargues was, as I just indicated, quite well known among seventeenthcentury perspectivists. His method of perspective construction – to be presented in chapter IX – was published in 1636, and explained in detail in 1648 by the engraver Abraham Bosse, whose book influenced the Dutch literature in the eighteenth century. Two men of the same profession as Bosse, Dirk

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FIGURE VII.4. Lairesse’s showing how to draw a room in perspective. Although he elaborated his room less than Vredeman de Vries (figure V.74), de Vries may have inspired him. Lairesse 1738, figure 64.

Bosboom and Caspar Jacobszoon Philips, took up Desargues’s method in works that appeared in 1705 and 1765, respectively. The last author to be mentioned in this survey is Jacob de Vlaming, who in 1773 published the booklet Kort zaamenstel der perspectief op eene geheele nieuwe wyze afgeleid (Brief compendium on perspective derived in a completely new manner). The work contains a presentation of a plan and elevation construction, and is in itself unremarkable, yet it is worth noting that by 1773 the construction itself, which is one of the oldest in the history of perspective, could be presented as a new method. This circumstance indicates

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FIGURE VII.5. Perspectival shadows cast upon a horizontal plane. Hoogstraten 1678, 269.

that the method had been out of use in workshops in the Northern Netherlands for quite some time. If this is indeed the case, it stands in contrast to the situation in Germany, where plan and elevation constructions were taken up regularly during the eighteenth century, as we shall see in chapter XI.

VII.2

B

The Theory and Practice of Perspective

efore going into more detail about the Dutch literature on perspective, I would like to comment upon an aspect that can be observed in the Northern Netherlands, but which applies to other countries as well. Several perspectivists gave their books titles that included the expression “the theory and practice of perspective”. By “theory” they usually meant the geometrical theorems upon which the laws of perspective are based, and by “practice” they normally understood perspective constructions. This division became quite common, but there was great variety in how the authors presented these two elements and what they considered to be the main points in teaching perspective. The mathematicians, with varying degrees of detail, included the geometrical arguments that were necessary for deducing the fundamental laws of perspective, and used these arguments to carefully derive some basic constructions. In general, they were inclined to treat the practice of perspective rather geometrically, and by and large restricted their applications to constructing perspective images of geometrical figures. Practitioners of perspective, on the other hand, tended to make their theoretical sections very brief, sometimes even skipping the theory completely and leaving the rules of perspective unexplained. In treating the practice of

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perspective, they included constructions that corresponded to what artisans might encounter in their work, such as a room, a staircase, or an exterior thrown into perspective. The circumstance that Guidobaldo and Stevin had been able to find a sound theoretical foundation for perspective constructions actually widened the gap, in existence since the sixteenth century, between how nontheoreticians and mathematicians approached perspective. The art historian Laurence Wright has characterized the latter as “dry-as-dust geometricians to whom drawing is an exact science” (WrightS 1983, 158). The gap between the two groups was often deplored. Although various attempts to bridge this gap were made – some of which we will meet later – these efforts met with little success.

VII.3

O

The Work by Marolois

ne of the most read Dutch perspectivists, if not the most read, was Samuel Marolois (c. 1572–c. 1627). Marolois’s father was a Protestant who had been exiled from France. He was engaged by the Prince of Orange and presumably paved the way for his son’s position at the court of Prince Maurice as a mathematical practitioner. Samuel Marolois seems to have taken a vivid interest in perspective and made himself familiar with a large part of the literature before he composed La perspective contenant la theorie et la practique d’icelle (Perspective including both its theory and practice), which was published in 1614 in French as well as Latin as part of his Opera mathematica. His perspective treatise was frequently reissued, appearing in separate French, Latin, Dutch, and German versions in a rather chaotic fashion. In fact, it is not unusual to meet copies in which the language of the main text is not the same as that occurring on the title page or in the figures. One of his impressive title pages can be seen in figure VII.6. Marolois admitted he had benefited from his predecessors’ publications, but only referred explicitly to three perspectivists, namely Dürer, Serlio, and Lencker (Marolois 1614/1638, 27–28). He thus followed a trend, which became almost a habit, of omitting references to fellow mathematicians. It is particularly remarkable that he did not mention his compatriot Stevin, who, like Marolois himself, had moved in the circle around Prince Maurice – they may even have inspected a fortification together in 1611 (DijksterhuisS 1943, 17). However, neither in style nor in content does Marolois’s La perspective show any trace of inspiration from Stevin. On the other hand, it does contain many signs of an extensive use of Guidobaldo’s work on perspective, as I will demonstrate shortly. I find it difficult to imagine that Marolois would have studied Stevin without at times preferring his elegant solutions to Guidobaldo’s more long-winded ones. I am therefore tempted to conclude that Marolois never read Stevin – or at least not thoroughly.

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FIGURE VII.6. Title page of a French edition of Marolois’s work on perspective.

Marolois’s Theory and Practice of Perspective

M

arolois belonged to the group of perspectivists who divided their books into a theoretical and a practical part. Since he was experienced in both theoretical mathematics and its applications, one may have expected that his book would, to some degree, bring the mathematicians’ and practitioners’ two different approaches closer to each other. He does not seem to have had this goal, however, because in using earlier material from the theory and

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practice of perspective, he copied the various approaches taken by others instead of creating a homogeneous style. Marolois’s section on the theory of perspective includes a few optical considerations and consists of seven theorems. The first five of these are taken from Guidobaldo, whereas the sixth is a reformulation of one of Guidobaldo’s other results – even retaining Guidobaldo’s expression si oculus videat (note 5, page 248). His last theorem is a lemma not directly connected to perspective. In proving his theorems, Marolois also found his arguments in Guidobaldo’s work. Thus, the proof he gave of the vanishing point theorem is almost identical to the first and most cumbersome of Guidobaldo’s various proofs (caption of figure VI.6). When it came to choosing illustrations, however, Marolois worked independently of Guidobaldo, providing diagrams that are far more instructive, as can be seen by comparing figure VI.2 with figure VII.7, and figure VII.8 with figure VII.9. The practical part of La perspective is a mixture of what was described in the previous section as mathematicians’ and practitioners’ interpretations of the practice of perspective. Marolois presented some geometrical constructions, explaining all the details. In doing so he was clearly inspired by, though not completely dependent on, Guidobaldo. He also included numerous examples from the practical tradition, taking over not only the designs, but also the habit of providing very little description. For instance, he did not expend many words on explaining how the rather complicated composition, reproduced in figure VII.10, was to be constructed. Altogether, La perspective is very lavishly illustrated, containing no less than 275 figures – including a number of engravings by Hondius.

FIGURE VII.7. Marolois’s illustration of the perspective model. Engraving by Hendrik Hondius, Marolois 1614, figure 7.

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FIGURE VII.8. Guidobaldo’s diagram to illustrate the vanishing point theorem. Guidobaldo del Monte, 1600, 35.

FIGURE VII.9. Marolois’s diagram to illustrate the vanishing point theorem. Marolois 1614, figure 22.

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FIGURE VII.10. A staircase in perspective. Marolois 1614, figure 178.

Marolois’s Method of Construction

M

arolois shared Stevin’s idea that the fundamental problem in relation to perspective constructions was determining the image of a point, but contrary to Stevin, he only dealt with the case in which the given point lies in a ground plane. To throw such a point into perspective, Marolois chose as his basic construction one of Guidobaldo’s most elegant methods, namely the twenty-first. He presented a few other constructions, among them a distance point construction, a plan and elevation construction, and a kind of a diagonal construction. This last method he referred to as “Lencker’s method”, which I find puzzling, as I have not noticed it in Lencker’s work on perspective (Lencker 1571). Actually, I suspect Marolois got the method from Dürer, for it is very similar to Dürer’s diagonal method as explained in the caption of figure VII.11. Marolois applied his pointwise constructions to throw objects such as curves and, in particular, circles into perspective (Marolois 1614/1638, 24). In discussing this theme he stayed close to Guidobaldo, and like him, he too discussed the question of when a circle is depicted as a circle (ibid., 28). In treating three-dimensional objects, Marolois worked quite traditionally. Thus, he first determined the perspective image of a plan of an object

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FIGURE VII.11. Marolois’s diagonal method. In his diagonal method (figure V.14), Dürer included a diagonal in the square at the bottom. Instead of using the diagonal gh, Marolois went about it as follows. To construct the perspective image of the point p, he drew p11 parallel to gf and let the point 2 be determined by g2 = g11. The rest of Marolois’s procedure is identical to Dürer’s. Marolois 1614, figure 81.

and then added the perspective heights – an example of which is shown in figure VII.12. In some cases he applied a plan and elevation construction to determine the image of a three-dimensional object, as can be seen in the illustrations reproduced in figures VII.13 and VII.14.

Marolois’s Instrument

L

ike several of his predecessors, Marolois also offered his readers the possibility of using an instrument rather than geometrical constructions. Earlier I mentioned that I suspected some of the German perspective instruments had a dual function, namely to determining the plan and elevation of objects for which this was not obvious, and to throwing objects into perspective based on their plans and elevations (page 228). As for Marolois, we can be sure he designed an instrument that was meant to operate in both ways (figure VII.15). This instrument seems to be his own invention, though whether it was ever produced is unclear. He described in great detail the construction of the instrument, its dimensions, and the materials from which it was to be made – the

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FIGURE VII.12. A three-dimensional cross in perspective. Marolois 1614, figure 109.

board, for instance, was to be made of pear wood or some other fine-grained wood (ibid., 37–38). His instruction in the use of the instrument, on the other hand, is very brief. Readers wishing, for instance, to reproduce the drawings shown in figure VII.16 were left to their own devices. Yet another example of

FIGURE VII.13. Plan and elevation of a cupola. Marolois 1614, figure 191.

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FIGURE VII.14. A perspective image of the cupola defined in the previous figure. Marolois 1614, figure 192.

how, when dealing with practical matters, Marolois took over the style of the mathematical practitioners and addressed himself to readers who were experienced in applying instruments and accustomed to short explanations.

Shadows and Inverse Problems of Perspective

M

arolois also touched upon the theme of shadows in La perspective. In his mathematical treatment of this subject he followed Guidobaldo and dealt with shadows of objects in the three-dimensional space, but not with how they are thrown into perspective. Nevertheless, his illustrative material does contain examples of perspectival shadows (figures VII.10, VII.12, and VII.17). Like Guidobaldo and Stevin, Marolois also treated inverse problems of perspective, but in an independent manner. He examined both types of inverse problems: those concerning the determination of the eye point, and those dealing with the reconstruction of an original figure from its perspective image (ibid., 60–65). He first considered a problem in which the ground line of a picture plane, a point in the ground plane, and its image are given, the task being to find the principal vanishing point and a distance point (whereby the position of the eye point is determined). He concluded that this problem is indeterminate. He proceeded to consider three points, or equivalently a triangle, in various positions, and a few examples of other polygons. He also presented a number of examples in which the perspective image of a figure and the figure itself, but not its position are given. In treating the second kind of inverse problems Marolois assumed that the eye point, a ground line, and a perspective figure are given, the requirement

3. The Work by Marolois

e

305

f b s c

k d

q

p

o

s n

m

FIGURE VII.15. Marolois’s instrument. It consists of board abcd (the two upper diagrams) and a number of sliding elements: 1. The holes e and f (upper left-hand diagram), which can slide along the line ab, and into which two threads are fixed. 2. The pair of rulers m and op (diagrams at left) connected at n. The ruler m can slide along ik and the ruler op along m; moreover, the height of op is adjustable. 3. The pair of rulers vx and ye (diagrams at right) that can be adjusted to fix the position of the eye point 3 (upper right-hand diagram). 4. The pair of rulers s and q (diagrams at right). The latter can slide along cd whereas ruler s is fixed to q, but it can be vertical as well as horizontal, and the element s can be moved. 5. At n and s are attached drawing pens, which Marolois called “cursors”, shaped as shown at 6 (at right). Marolois did not provide a precise description of how the instrument was to be used, nor did he check that all the letters in his text corresponded to those on his drawing. As I understand him, he had designed the instrument for three functions. Firstly, he meant for the board, together with the rulers m and op, to be used for determining the plan and elevation of existing objects. Secondly, Marolois’s examples show that the ruler s in horizontal position is to be applied together with the two threads fixed in e and f to perform a distance point construction. This procedure is discussed in connection with figure VII.16, in which Marolois unfortunately introduced new letters. Thirdly, his description indicates that the ruler s in a vertical position is meant to function along with the eye 3, but says nothing of how. Presumably s is supposed to determine the position of the image of a given point, above the ground plane, by being put in such a position that a thread from the given point through s to 3 represents a straight visual ray. Marolois 1614, figure 112 with most letters enlarged.

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I J

K

FIGURE VII.16. One function of Marolois’s instrument. My understanding is that he determined the image of a three-dimensional object (consisting of two boxes) from its plan and elevations as follows. In diagram CXV, Marolois showed a plan of the object, and he began by throwing this into perspective. To explain how, I look at the point I. To this point Marolois assigned two points 1, the one farthest to the right, which is the orthogonal projection of I upon the ground line, and the one farthest to the left (near to one of the points 2). The latter 1 is constructed by making the distance between it and the former point 1 equal to the distance between the point I and the ground line. Similarly, he assigned two points 1 to the point J, the one farthest to the right and the middle one. Having mapped all the vertices of the plan of the object twice upon the ground line, Marolois placed the latter on the board of the instrument as shown in diagram CXVII. He moved the points A and B so that B becomes the principal vanishing point and A the right distance point, then found the image of the point I as the point of intersection of the two threads connecting B with the right-most point 1 and A with the left-most point 1, respectively. Following this, he plotted the image by moving the ruler qe to the point of

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FIGURE VII.16. (caption continued ) intersection of the threads and placed one of the ink cursors r at this point. In diagram CXVII the lower cursor is at the position of the image of the point J from diagram CXV. To determine the perspective heights, Marolois applied the elevation shown in diagram CXVI. He projected the vertices of the elevation orthogonally upon a line perpendicular to the ground line and marked these points of projection on the board in diagram CXVII. The projection of the point K from diagram CXVI is thus marked by f, whereas z is the corresponding point on the ground line. Next, Marolois chose an arbitrary point o on the horizon AB and stretched threads from o towards z and f. The distances between these threads, measured along lines perpendicular to the ground line, define at any position the perspective distance between the ground plane and a horizontal plane through K. In his diagram CXVII, Marolois indicated how the ruler qe can be applied to transport this distance to the perspective composition: Keeping the lower cursor r at the position of the image of the point J, he moved the ruler towards the right to the position in which this r is the point of intersection of the ruler and the thread oz. He then placed the upper cursor in the point of intersection of the ruler and of and moved the ruler back, thereby determining the image of the point defined by J and K in the plan and elevation. Marolois 1614, figures 115–117 with the points I, J, and K added.

FIGURE VII.17. A chair and its shadow drawn in perspective. Marolois 1614, figure 223.

being to construct a figure in the ground plane that is depicted in the given figure. He solved these problems straightforwardly by inverting the steps in a usual perspective construction. It is interesting to notice that in one of these examples he used the expression perspective plane when referring to the picture plane (ibid., 64).

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FIGURE VII.18. Marolois’s treatment of the column problem. He assumed that the distance between the eye point and ab is 16 feet and the distance between the centres of the columns is 2 feet. The most observant readers may have noticed the writing error left of the central line: 3˚34′35′′ should have been 3˚34′53′′. Marolois 1614, figure 259.

The Column Problem

T

owards the end of La perspective Marolois took up the column problem, already a virtual classic. His illustration is reproduced in figure VII.18, where f is the eye point, the line ab a horizontal line that intersects the front of a row of columns – the centres of the intersections being a, i, b, and so on. Above the line ab is a vertical section through half a column, which in its entirety presumably had a square base. By considering only the centres of the columns Marolois, like Cousin and Stevin, simplified the problem considerably, making it a matter of depicting equidistant points on a line (cf. figures V.23 and VI.37). Marolois argued very concretely about the angles defined by the eye point and two consecutive centres of the columns. He did not simply conclude that these angles decrease with increasing distances between the eye and the centres, but actually calculated the angles for a particular example. He made it clear that according to

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the rules of perspective, the image of an angle between the eye point and the images of two centres should have the same size as the original angle. Marolois stressed that he took up this example because many painters were of the opinion that equidistant columns should be drawn so that the distances between them are not only seen within smaller angles, but actually are smaller. Thus far – apart from the calculations – Marolois’s treatment of this problem is similar to Cousin’s (page 182). Marolois, however, added the following remark, which is new (ibid., 69). If painters advocating columns to be drawn with decreasing distances took their argument seriously, they should similarly conclude that columns of equal height should be drawn with decreasing heights, since the angle within which the heights are seen also decreases with increasing distance to the eye. Marolois was convinced that no painter would want to do this.

Arithmetical Calculations

I

n the last section of La perspective Marolois presented twenty rather repetitive problems concerning appearances and solved them arithmetically. In the first problem he assumed that two heights of 24 and 28 feet, are seen as equal from a point situated 120 feet from the smallest height and collinear with both heights. He sought to determine the distance between the two heights and found the answer to be 20 by looking at similar triangles.

Marolois’s Influence

S

trictly speaking, Marolois’s work was unimportant in the development of a mathematical theory of perspective, but it did play a role in spreading an awareness of the theory. It contributed much to the understanding of the concept of a vanishing point, both within and outside the Netherlands. More than a century and a half after its publication, the English painter John Joshua Kirby became quite enthusiastic about Marolois’s work. He claimed that Marolois’s La perspective and Vignola’s Due regole (1583) made up the most important pre-Taylorian literature on perspective, remarking on the former: “This work ... though tedious in its Operations, is nevertheless a very curious Performance” (Kirby 1754, second book, 81, note †).

VII.4

Van Hoogstraten’s Perspective Box

I

n general this book only deals with the part of the history of perspective that can be derived from publication on perspective constructions. In the case of Samuel van Hoogstraten’s perspective box, however, I make an exception, since it enables me to demonstrate an important point: the construction of this device was a fascinating application of elementary perspective theory, which was known to most seventeenth-century practitioners of perspective.1

1 I am grateful to Claus Jensen for awakening my interest in van Hoogstraten’s box. This section is partly a result of stimulating discussions we had about the geometry behind the box – a theme Jensen has also written about (JensenS 2004).

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FIGURE VII.19. Van Hoogstraten’s perspective box. Its inner dimensions are (width, length, height) 51.2 cm × 77.9 cm × 52.4 cm (JensenS 2004, 164). The front of the box is opened to let in light. National Gallery, London. Reproduced from BrusatiS 1995 with the kind permission of the author and the National Gallery.

The idea of making perspective boxes seems to have emerged in the midseventeenth century in a circle that included van Hoogstraten. Only six boxes are known to be preserved: three in the National Museum in Copenhagen, one in the Bredius Museum in the Hague, one in the Detroit Institute of Arts, and one in the National Gallery in London.2 They have different shapes, but are all constructed so the inner sides of the box show an interior that must be viewed through a peephole in the box – light being let into the box by various means. Only the box in the National Gallery is signed (figure VII.19). Indeed, van Hoogstraten identified himself as its creator by including a letter addressed to himself in his composition. The panels of this box depict a main room 2

For literature on these boxes, see HulténS 1952; KoslowS 1967; ElffersS et al. 1975, 82–84; BrownS et al. 1987 (in which David Bomford discusses the geometry of the box, 65–77); BrusatiS 1995, 169–217; BomfordS 1998; and JensenS 2004.

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with views into adjoining rooms. Van Hoogstraten probably brought the box with him in 1662 when he crossed the Channel to try his luck in London, leaving it behind when he left some time after the Great Fire in 1666. In his introduction to the art of painting, which he published a dozen years after his return to the Netherlands, he touched upon perspective boxes: The knowledge of this science [perspective] also enables one to produce the perspective box, which is a wonder to behold. If it is painted correctly and with skill, a person the height of a finger will appear life-sized.3 G

K

J F

H

V

L ES

M

I N B

A

U

C

DR

FIGURE VII.20. A diagram of van Hoogstraten’s box. The peepholes have centres in I and N, respectively. Let the orthogonal projections from N upon the left, the back, the bottom and top panel be the points I, M, R, and S, which are then the principal vanishing points of the perspective projections upon each of these four panels. Similarly, the principal vanishing points of the four projections from the peephole centred in I are the points N, J, U, and V. For later reference it is also relevant to know the right distance point for the projection from N upon the left panel. The distance of this projection is NI. Hence, if K is determined so that IJ + JK = NI, then K is the right distance point for the composition upon the left panel when JK is rotated into this panel. In fact, K is also the left distance point for the projection from N upon the back panel, because this projection has the principal vanishing point M and the distance NM which is equal to MK.* When the point L is determined by ML = JK, a symmetry argument proves L to be the left distance point for the projection from I upon the right panel, and the right distance point for the projection from I upon the back panel. 3

Door de kennisse van deeze weetenschap maekt men ook de wonderlijke perspectyfkas die, als ze regt en met kennisse geschildert is, een figuur van een vinger lang als leevensgroot vertoont. [Hoogstraten 1678, 274–275] *Because MK = JM - JK = NI - JK = IJ = NM.

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It is remarkable that he mentioned the size effect rather than the sense of experiencing a three-dimensional scene. Still, there can be no doubt that his requirement of painting correctly – which he did not elaborate on – is related to the rules of perspective. The rest of this section deals with the mathematical principles of van Hoogstraten’s perspective box. In figure VII.20 I have made a schematic drawing of the box and introduced some letters. I refer to the inner panels as the bottom (ABCD), the left (ABGF ), the back (BCHG ), the right (CDEH ), and the top (EFGH ). The left panel contains a peephole with centre I, while the right panel has a peephole symmetrically placed with centre N. The definition of the remaining points is explained in the figure caption. By deciding to have two peepholes, van Hoogstraten introduced a level of sophistication unmatched by any of the other five preserved boxes – as well as a problem, to which I return, that has no geometrically correct solution. Figure VII.21 shows van Hoogstraten’s five panels placed in the plane of the back panel and supplied with my letters. (Even though turning four of the panels into the back panel causes some points to change position, the letters have been retained.)

F

F

E

G

H

E

N

A

B

C

A

D

D

FIGURE VII.21. The five inner panels of van Hoogstraten’s box folded into the plane of the back panel, with letters added. Reproduction from a paper model of the box on which, for practical reasons, the peepholes are in a different scale than the panels.

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313

If the box had only one peephole, for instance the one at N (figure VII.20), it would have been fairly easy to construct a perspective composition inside the box. The difference between this exercise and making an ordinary perspective drawing is that instead of merely projecting a motif from N upon one picture plane, projections upon four planes has to be performed, in casu upon the left, the back, the top, and the bottom panels. The projection upon each of these planes has a principal vanishing point and a distance – introduced in the caption of figure VII.20. Van Hoogstraten did not publish any instructions on how to paint the panels. I find it likely that he made some construction drawings which he afterwards transferred to the panels. Before I begin to identify relevant vanishing points in his panels, let me remark that rather than searching for mathematical points, I look for small circles. In fact, the experience of trying to make more than two lines meet in any one point has taught me that this calls for considerably greater precision than one could reasonably expect a painter to apply.

The Left-Hand and Right-Hand Side Panels

V

an Hoogstraten’s left panel (figure VII.22) is painted in accordance with the theory presented above: One set of diagonals in the tiles are perspective orthogonals and should therefore converge towards the principal vanishing point I – which they actually do. Similarly, one set of the sides of the tiles should converge towards the distance point K – which they also do. These observations support my suggestion that van Hoogstraten first drew the

FIGURE VII.22. van Hoogstraten’s left panel with a letter added.

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composition on a sheet that is large enough to contain K, as this point lies outside the panel and would not have been easy to reach in a construction performed directly on the panel. It can be noticed that the adjoining room has a different tile pattern than the main room, and that the ortogonals in the former room also converge towards I. Had he settled for the one peephole at N, van Hoogstraten could have constructed the back, the top, and the bottom panels much like the left panel. Opting for two peepholes forced him to include the eye point at I in his considerations. For the right panel (figure VII.23) – which can only be seen from I – van Hoogstraten designed a composition based on the same principles as the ones he used for the left panel. The three remaining panels, the back, top, and bottom, presented geometrical problems, since here van Hoogstraten had to create perspective compositions that can be seen from two eye points – situated far apart. This can only be done in a mathematically correct way if the objects that are drawn in perspective are parallel to the picture plane. For these three panels, van Hoogstraten devised three different ways of camouflaging the impossibility.

The Bottom Panel

F

or the bottom panel (figure VII.24) van Hoogstraten chose a motif mainly consisting of elements that are parallel to the picture plane: a

FIGURE VII.23. The right panel in van Hoogstraten’s box.

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315

F V

E

G

H

B

C

U A

R D

FIGURE VII.24. Van Hoogstraten’s top and bottom panels, with letters added. The paintings to the right in the top panel have sides that are perspectively orthogonal to the top panel. These sides converge towards the point V, the principal vanishing point of the projection from I upon the top.

tiled floor. Seen from either of the peepholes the tiles look correct, and so do the horizontal parts of the furniture standing on the floor. Van Hoogstraten also placed a few objects so close to the left-hand side of the box that they are only noticeable from the peephole at N. These include two legs of a chair (most of which is projected upon the back

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panel, as seen in figure VII.21). Since the originals of these legs are orthogonal to the bottom panel (figure VII.24), their perspective images should converge towards the principal vanishing point R of the projection from N upon the bottom – which they actually do, as do the two front legs of the sitting dog. At the far right of the panel, van Hoogstraten placed a chair whose one leg similarly converges towards U – the principal vanishing point for the projection from I. Near the opening he included another leg converging towards U. Seen from I the latter leg is perfect, whereas seen from N it does not make much sense. It is, however, unlikely that this phenomenon would be noticed by anyone other than a perspectivist meticulously analysing the box.

The Top Panel

F

or the top panel van Hoogstraten created an ingenious and quite fanciful solution (figure VII.24). Seen from I, he has elongated the right wall of the main room by drawing the images of one complete painting and most of another (the rest of which is projected upon the back panel). Moreover, in the ceiling he has depicted beams that run parallel to the picture plane. Observed from the peephole at I, this composition works fantastically, and

FIGURE VII.25. The back panel in van Hoogstraten’s box. One set of diagonals in the tiles and other perspective orthogonals converge towards J, which is the principal vanishing point of the projection from I upon this panel. This projection has L as its right distance point (caption of figure VII.20), towards which the images of one set of tile sides converges. Seen from N, the tiles do not respect the rules of perspective.

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viewed from N the beams appear to constitute a ceiling just above the upper line in the left panel. In other words, van Hoogstraten managed to give the main room two heights, one for each peephole.

The Back Panel

I

n designing the back panel (figure VII.25), van Hoogstraten’s solution was not quite as elegant as those for the bottom and top panels. He could have avoided some problems if he had decided to paint only the back wall of the main room and views into the other rooms on this panel. However, he does not seem to have considered this solution sufficiently imaginative, for he also included part of the image of the floor on the panel. This floor and most of the other objects on the panel function perfectly when seen from I, whereas an attentive viewer at N will find the floor distorted. Another noteworthy object on the back panel is the chair at the left. Seemingly van Hoogstraten, when drawing this chair, was not guided by geometrical laws, as I argue in the caption of figure VII.26. Altogether van Hoogstraten’s box is an impressive technical accomplishment, in which he very creatively played with the rules of perspective – and with the possibilities of breaking these rules. Not only is it highly entertaining to notice the details in his various panels. It is also enjoyable to look into the box through the peepholes, which offer quite a convincing impression of spatial relations. I imagine van Hoogstraten took great pleasure in designing the box, and that he found it particularly amusing to deal with the objects he projected upon more than one plane.

VII.5

W

Van Schooten’s Revival of Stevin’s Theory

ith two such remarkable works as Stevin’s Van de verschaeuwing (1605) and Marolois’s La perspective (1614), the Northern Netherlands were blessed with better perspective literature than any other country in early seventeenth century. Perhaps as a consequence of this, the two books, together with the publications of Vredeman de Vries’s perspective plates, constituted the Dutch literature on perspective for almost half a century. A letter from 1645 illustrates the general awareness of these sources and also indicates that Dutch mathematicians had come to consider perspective as a part of their field. The letter was sent by Jan Stampioen de Jonge to his student Christiaan Huygens – then in his mid-teens – with the purpose of outlining how the young Huygens could pursue his studies of mathematics. Stampioen de Jonge included perspective in his proposed curriculum and recommended Huygens read the works by Marolois, Stevin, or Vredeman de Vries (HuygensS Œuvres, vol. 1, 6). Huygens seems to have settled for Stevin and his mathematical approach. Thus, twenty three years later, when he was consulted by his brother Constantijn Huygens about the literature on perspective, he wrote:

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X

FIGURE VII.26. Van Hoogstraten’s problematic’ chair. Assuming that this chair, like the others, has a rectangular seat, I expect to find two pairs of lines in it that are images of two pairs of parallel lines perpendicular to each other. The pair defined by the longer sides of the seat I call the first pair, and the other I call the second pair. The first pair of lines converges towards M, the principal vanishing point for the projection from N upon the back panel. With respect to N, these lines are thus drawn as perspective orthogonals, and one would therefore expect the second pair to be transversals – which they obviously are not. This pair of lines, and other lines perspectively parallel to them, are drawn as parallels (or near-parallels). It is thus clear that while drawing the second set of lines for the chair, van Hoogstraten violated the rules of perspective, which dictate that the images of a set of parallel horizontal lines are either transversals or converge in a point on the horizon. He may have done so unconsciously, but there is also a possibility that he did so deliberately, falling between two stools, or rather in this case between two eye points: Although far from conspicuous from the peephole at I, the chair can with some efforts be seen from this point. With respect to eye point I, the point M is a vanishing point corresponding to a certain direction and to it corresponds a vanishing point X of the lines that are perpendicular to the lines having M as vanishing point.* Hence when I is the eye point, the lines in the second pair should, to be geometrically correct, converge in the point X – which they do not do. Perhaps in drawing the chair van Hoogstraten made a compromise between how the chair is seen from N (the second pair consists of transversals) and from I (the second pair of lines is converging in X ). Concerning perspective, I have read no other authors than Stevin, which is why I cannot tell you who is the best.4

Christiaan Huygens was a very gifted student, and reading Stevin’s treatise most likely enabled him to quickly pick up the essential problems of perspec*Since J is the principal vanishing point for the eye point I, the line segment IJ is the height in the right-angled triangle IJX implying that the point X is determined by JX.JM = IJ 2. 4 Pour la perspective je n’ay veu aucun autheur que Stevin, c’est pourquoy je ne puis pas vous dire qu’il est le meilleur. [Huygens Œuvres, vol. 6, 216]

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tive. He did not give credit to Stevin, however, for he continued his letters as follows. This science, being understandable through one or two rules, presents so little difficulty, that I am sure you can discern it all by yourself. Tell me what you find most difficult about it, and I shall explain it to you.5

Christiaan Huygens’s highbrow attitude to perspective must be the reason he failed to mention the most relevant work he could have referred his brother to, namely Tractaet der perspective (Treatise on perspective), published eight years earlier – in 1660 – and written by Huygens’s former mathematical mentor, van Schooten. Huygens’s ignorance of this treatise is particularly striking considering the fact that it appeared in van Schooten’s Mathematische Oeffeningen (Mathematical exercises) alongside one of Huygens’s own works, Van rekeningh in spelen van geluck (On calculating in games of luck), which earned him a prominent place in the history of probability theory. Frans van Schooten (c. 1615–1660) is especially well known for his dedicated efforts to make the richness of René Descartes’s La géométrie accessible to a wider audience. He was generally very active in disseminating mathematical knowledge as a lecturer, commentator, and writer. He included perspective among the subjects he thought would benefit from being better understood from a mathematical point of view, and probably for this reason he lectured on the subject (Schooten 1660, 543). His interest in perspective was presumably not entirely academic, but may also have been fuelled by a personal interest in drawing. Even today, one of his drawings – a portrait of Descartes – is often reproduced (figure VII.27).

Van Schooten’s Intention and Inspiration

V

an Schooten’s aim with Tractaet der perspective was to provide a mathematical understanding of the basic rules of perspective. He structured his presentation around six theorems and examples of how to construct the images of various lines and points. Like the works of several other mathematicians, Tractaet der perspective contains no references to other writers on perspective. As noted, however, it is clear that he was greatly influenced by Stevin. He took over Stevin’s set up and most of his terminology; he used his definitions, postulates, theorems, and problems, and he also applied his rabatment. The result was no mere plagiarism, but rather a second, and generally improved, version of the introductory part of Van de verschaeuwing.6 Van Schooten had probably also cast more than a passing glance at Guidobaldo’s Perspectivae libri sex. He thus began five of his theorems with 5

Il y a si peu difficulté en cette science, qui se peut comprendre dans une ou deux regles, que je ne doute pas, que vous ne puisserez trouver tout par vous mesme. Dites moy ce que vous y trouvez de plus difficile, et je vous l’expliqueray. [ibid.] 6 For a more detailed discussion see AndersenS 1990, 41–42, 57.

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FIGURE VII.27. Van Schooten’s drawing of Descartes. Van Schooten intended to include the picture in his Latin edition of Descartes’s work on geometry. Descartes himself, however, asked van Schooten not to do so (DescartesS Œuvres, vol. 5, 338). Van Schooten respected this wish at the time, but did publish the portrait in the second edition of the book, which appeared after Descartes’s death. DescartesS 1659.

the phrase “so het ooch ... siet”,7 corresponding to Guidobaldo’s “si oculus videat” (note 5, page 248). Likewise, in solving the problem of throwing a point into perspective, van Schooten did not follow Stevin, but chose to present a construction similar to Guidobaldo’s sixth method, as shown in figure VI.11. Guidobaldo’s sixth method is admittedly a straightforward application of the main theorem, so van Schooten could have developed his construction independently of Guidobaldo, just as he could have been inspired to use the phrase “so het ooch ... siet ” by reading Marolois (page 299). 7

Schooten 1660, 514, 516, 518, 520, 525.

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Nevertheless, I am inclined to think that van Schooten actually studied part of, if not all of, Guidobaldo’s work. There is one theme that van Schooten treated differently from both Stevin and Guidobaldo – more or less out of necessity – namely the construction of perspective images in oblique picture planes. For this problem Stevin had presented a solution that was based on his theorems concerning simultaneous rotations of the picture plane and the eye point. Keeping to the more elementary aspects of the theory of perspective, van Schooten did not introduce these rotations and was therefore unable to use Stevin’s solution for constructing images in oblique planes, and Guidobaldo’s work offered no attractive solution. Van Schooten’s procedure is quite elegant, though not completely transparent. In the captions of figures VII.28 and VII.29 I have explained his ideas and shown how he constructed the image of a point situated in the ground plane. Despite his intention to concentrate on the fundamental geometrical problems, van Schooten could not resist the temptation to include considerations that have more mathematical appeal than relevance for practical perspective constructions. In particular, and presumably inspired by Guidobaldo, he

p B Bi E

O

C Ci

G

90

∞u

u Q

g

R F

FIGURE VII.28. Van Schooten’s considerations in connection with an oblique picture plane. To describe the idea behind van Schooten’s solution of how to construct images in an oblique picture plane, I first consider the three-dimensional situation. The point B lies in the ground plane g and its image in picture plane p forming the angle 90˚–u with g has to be determined. O is the eye point, F its foot in g, and Q the ground point. C is the point in which FQ meets the line through B parallel to GR, and E is the point of intersection of the visual ray OC and the vertical line through Q. The image Ci of C in p can be determined as the point of intersection of OC and the line in the plane OFC, which passes through Q and forms the angle u with EQ. When Ci has been found, the image Bi of B can be determined as the point of intersection of the visual ray OB and the line through Ci parallel to GR.

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VII. The Dutch Development after Stevin B

C

B i99

Ci99 I u

G

Q

F

Ci9 E9 R

O9

FIGURE VII.29. Van Schooten’s construction of the image in an oblique picture plane. He applied the characterization of the point Bi given in the caption of figure VII.28 (Schooten 1660, 533). In doing so he considered the situation in which p is rotated into g around GR, in addition to which the triangle OFC is turned down into g around FQ making the points O, E, and Ci coincide with the points O′, E′, and Ci′. These points van Schooten constructed straightforwardly. When p is rotated into g, the image Ci falls in a point Ci″ on FC, which van Schooten determined by making QCi′′ = QCi′. He introduced the point I, in which the line through Ci′ parallel to GR cuts O′B, and determined the point Bi″ by making Bi″Ci″ parallel to GR and equal to ICi′. He then claimed that Bi″ is the required image of B. That the constructed Bi″ really is the image of B – rotated into g – can be seen in the following way. The similarity of the triangles OBiCi and OBC (figure VII.28) and the triangles O′CiI and O′BC implies that BiCi : BC = OCi : OC and ICi′ : BC = O′Ci′: O′C = OCi : OC. Hence ICi′ = BiCi, and since by construction Bi″Ci″ = ICi′ then also Bi″Ci″ = BiCi. Thus Bi″Ci″ has the correct length, and it is clear that Bi′′ should lie on the transversal through Ci″.

examined the situation in which a point to be thrown into perspective lies between the eye point and the picture plane. Similarly, he also considered points that lie below the ground plane. In treating the problem of projecting a point outside the ground plane upon an oblique plane, van Schooten actually distinguished between eight different cases, letting the picture plane be inclined towards or away from the eye, and letting the point lie behind or in

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front of the picture plane and above or below the ground plane (Schooten 1660, 535–539). All his solutions follow the same principle as the one shown in figure VII.28. Van Schooten did not treat the practice of perspective in his Tractaet der perspective, although he did lecture on the topic. Rather than preparing his ideas for print, he referred his readers to notes his brother Petrus van Schooten had taken during his lectures (ibid., 543). Unfortunately I have been unable to trace these notes, so I cannot tell whether Frans van Schooten followed the mathematicians’ tradition of treating the practice of perspective, or whether, perhaps inspired by Marolois, he took up some more practical examples. In his introduction van Schooten gave the impression that he hoped to reach practitioners. The likelihood that he succeeded in this endeavour is remote, for like other mathematicians with the same goal, he kept his presentation in the style of abstract geometry. However, by leaving out several of Stevin’s mathematical observations, van Schooten had made it easier for his readers to grasp the main principles required to gain a fundamental mathematical understanding of perspective. That he managed so well was undoubtedly due in part to the fact that he had a work as excellent as Stevin’s to build upon.

Georg Mohr

V

an Schooten translated most of his Mathematische Oeffeningen into Latin, but not his treatise on perspective. His clear introduction to the topic was therefore reserved for readers who knew Dutch – and, as I have argued, mathematics. This seriously limited the potential target group of Tractaet der perspective. Actually, the contemporary readers of the work probably formed an exclusive circle, to which belonged the Danish mathematician Georg Mohr (1640–1697). Mohr spent several years in the Netherlands, where he worked out the idea of performing traditional Euclidean constructions without the use of a ruler. He presented his results in the booklet Euclides Danicus (The Danish Euclid), which appeared in both a Dutch and a Danish edition in 1672. Although Mohr’s Euclides did not attract much attention at the time, in the twentieth century Mohr gained no small measure of fame for being the first to take up the enterprise of constructing with a compass alone. Mohr’s name became known after Johannes Hjelmslev described the contents of his work (HjelmslevS 1931). Mohr is only loosely connected with the history of perspective, but in showing how constructions can be performed with a compass alone, he in fact treated four problems relating to perspective constructions (Mohr 1672, part two, problems 19–23). In the eighteenth century Lambert made a direct connection between perspective and the theme of constructing with restricted means, as we shall see in section XII.12. In Mohr’s case, perspective seems to have served the sole purpose of providing examples – different from the usual Euclidean

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FIGURE VII.30. Top to bottom: Stevin’s, van Schooten’s, and Mohr’s Dutch formulations of a problem presented in connection with figure VII.31. Stevin 16051, 21; Schooten 1660, 529; Mohr 1672, 32.

constructions – of demonstrating that he did not need a ruler. He undoubtedly found these examples in van Schooten’s Tractaet der perspective, and in fact, as illustrated in figure VII.30, he followed van Schooten very closely in formulating his problems. The solutions, however, were Mohr’s own creation, in which an essential part was the construction of a fourth proportional. An example of this is given in figure VII.31.

Abraham de Graaf

A

nother of van Schooten’s readers was the Dutch mathematician Abraham de Graaf (1635–c. 1717). In 1676 he published a work bearing the ambitious title De geheele mathesis (The entire mathematics). This book apparently became fairly popular, being regularly reissued until the appearance of the

*Meaning the height of the eye above the floor (the ground plane). † Corresponding to the ground line.

325

5. Van Schooten’s Revival of Stevin’s Theory p O

Ai

Ai

G Q F

A

C B g

R

G Q

C

A B R

F

FIGURE VII.31. One of Mohr’s constructions. Given are two points G and R on the ground line, the point A in the ground plane, the foot F of the eye point O, and the length OF. It is required to construct the image Ai of A in a picture plane that is perpendicular to the ground plane. Because Mohr only allowed the use of a compass, all lines occurring in his construction are – as he himself expressed it – “imagined”. His solution contains various steps, all of which he had treated earlier demonstrating how they could be performed using only a compass. Mohr’s idea (left-hand diagram) was to determine the point of intersection C of AF and GR, and then to construct the point Ai on the vertical through C at the correct distance from C. First he constructed the orthogonal projections B and Q of A and F upon GR. To construct C he looked at the similar triangles QFC and BAC and obtained the relation FQ : AB = QC : BC, from which he deduced (cf. relation (1) page xxxiii) (FQ + AB) : AB = BQ : BC. This shows that the length BC can be constructed as a fourth proportional. Knowing the length BC Mohr, could construct C on GR. Analogously Mohr concluded from another pair of similar triangles that the length AiC is determined by (OF + AB) : AB = OF : AiC. Having constructed this length as a fourth proportional, Mohr cut it off on the perpendicular to GR through C, thereby obtaining the image Ai. The right-hand diagram showing the construction in the picture plane is based on Mohr’s figure 19 in part two of Mohr 1672.

FIGURE VII.32. De Graaf’s illustration of the main theorem. Graaf 1676, figure 5. Leiden University Library, 2008 D25.

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seventh edition in 1737. Wishing to deal with all the mathematical disciplines in a single volume, de Graaf had to be concise, and so he spent no more than a dozen pages on perspective. On these few pages, however, he actually managed to give a good presentation of the theory of perspective as well as some basic constructions. He began with seven theorems that are, by and large, van Schooten’s six theorems rearranged. Contrary to van Schooten, de Graaf formulated the main theorem explicitly, and he illustrated it with the diagram reproduced in figure VII.32. He moreover observed that a point on a line situated at an infinite distance from the picture plane is depicted in the vanishing point of the line (Graaf 1676, 214). In deciding which constructions to display, de Graaf worked independently of van Schooten and in fact made some untraditional choices, like the one shown in figure VII.33. De Graaf recognized the value of being generous with examples and provided a wealth of illustrations, but he economized with his space and made them tiny. In the 1694 edition, he added even more illustration on perspective, bringing the total number to 117, two of which are shown in figures VII.34 and VII. 35. A

H P

G

O9

P

Z

A9 Ai A9 A

Q g

H

Z

O

O9

B F

R

G

Q

B

R

FIGURE VII.33. One of de Graaf’s constructions. Given a point A in a ground plane (left-hand diagram), determine its image in a vertical picture plane p, O being the eye point, P the principal vanishing point, and B the orthogonal projection of A upon the ground line GR. To solve this problem, de Graaf imagined A is rotated into p around GR so it falls in the point A′, and similarly that O is rotated into p around the horizon HZ so it falls in the point O′. When A and O are given, the points A′ and O′ can easily be determined in p. de Graaf constructed the image Ai of A (right-hand diagram) as the point of intersection of O′A′ and BP. The correctness of this construction follows from the main theorem and the division theorem. According to the first theorem, Ai must lie on BP, and according to the second it must divide the line segment BP in the ratio AB : OP = A′B : O ′P – which the constructed Ai indeed does (because the triangles PO′Ai and BA′Ai are similar).

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FIGURE VII.34. A cross thrown into perspective. Graaf 1676/1694, figure 67. Leiden University Library, 542 D16.

comparison of the drawing reproduced in figure VII.34 with figure VII.12 shows that de Graaf was inspired by Marolois.

Hendrik van Houten

T

he Stevin-line in the Dutch perspective literature was carried on by Hendrik van Houten whose Verhandelingen van de grontregelen der doorzigtkunde (Treatise on the fundamental rules of perspective) was published

FIGURE VII.35. De Graaf on perspectival shadows. Graaf 1676/1694, figure 93. Leiden University Library, 542 D16.

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FIGURE VII.36. Van Houten illustrating how to determine heights in a perspective composition. Houten 1705, diagram for his example 39, Leiden University Library, 547 C 18.

in 1705. I have not been able to find any biographical information about the author,8 but there can be little doubt that van Houten was acquainted with both the theoretical and the practical tradition in the Netherlands. His introduction to the theory of perspective is copied almost verbatim from de Graaf’s six theorems.9 Moreover, van Houten applied the rather special de Graaf construction presented in figure VII.32. Nevertheless, van Houten’s illustration material is independent of de Graaf’s and is quite good, as is evidenced in figures VII.36 and VII.37.

VII.6

The Problems of Reversing and Scaling

B

efore continuing the description of how Stevin’s ideas influenced the treatment of perspective in the Northern Netherlands, it seems appropriate to bring up two rather special issues that our next protagonist, Willem ’sGravesande, also touched upon. This is contrary to the majority of authors on perspective, who avoided these issues – as I, too, have largely done thus far. With that, let us turn to the problem of reversing and the problem of scaling, briefly mentioned earlier.

8

Hendrik van Houten’s book is mentioned in van der AaS 1852, 409, but associated with the theologian Cornelis van Houten, who is said to be the possible author of a unusual book entitled Grondregelen der Doorsichkunde en perspectief. 9 For instance, de Graaf formulated his first theorem as follows. Indien het Voorwerp een rechte lijn is, zoo is de Aftekening mede een rechte lijn (When the object is a straight line, then the image is also a straight line). [Graaf 1676, 213] Van Houten did not change this wording much in his first theorem: Indien het voorwerp een rechte lijn is, zo zal mede de aftekening een rechte lijn wesen (When the object is a straight line, then the image will also be a straight line). [Houten 1705, 3]

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329

FIGURE VII.37. Hanging objects drawn in perspective. Houten 1705, diagram for example 30, Leiden University Library, 547 C 18.

It is understandable that these problems, which will shortly be described in greater detail, are not often treated in tracts on perspective, for introducing them requires explanations that most novices would find more perplexing than elucidating. On the other hand, since these problems influence the correspondence between an object and its image, I find it remarkable that before 1800, they were almost universally ignored. I have seen no thorough treatment of the problem of reversing, and come across only one instance in which the problem of scaling was clearly presented. This is found in a book from 1794 by Bernhard Friedrich Mönnich, whom we will meet in chapter XI. Some authors may have neglected these problems because they themselves had not given them much thought, if any at all. Yet other authors were well aware of the problems and still made only vague allusions to them. This latter category includes ’sGravesande. In the following I combine a description of the two problems with a presentation of ’sGravesande’s treatment of them – the rest of his work being presented in the section VII.7. Before treating the problems of reversing and scaling, I would like to mention that one of my reasons for including them here is that I want to use them to indicate that in general, authors on perspective did not consider it their task to exhaustively detail the practice of making perspective constructions. Authors with a practical background presumably found these problems too speculative and supposed that with practice their readers would find ways to deal with the issues that arose. On the other hand, mathematicians writing on perspective treated the topic in a more abstract manner, and would not find the problems of

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reversing or scaling interesting enough to deserve many words. The circumstance that some of the practical problems were left out of the literature can also be seen as a sign that the aim of perspective drawing was not to make an exact reproduction of a given motif, but to design a harmonious imaginary composition – a view Martin Kemp has also expressed (KempS 1990, 15).

The Problem of Reversing

T

he problem of reversing occurs in connection with rabatments that bring the picture plane p and the ground plane g into the same plane. Figure VII.38 illustrates a three-dimensional situation in which, as usual, O is the eye point, F its foot in g, GR the ground line, P the principal vanishing point, and Q the ground point. Additionally, ABC is a triangle in g, and ABiCi its image in p. Looking at a similar configuration, ’sGravesande suggested two possibilities for bringing ABC and ABiCi into the same plane: to turn p around GR into g either away from the foot F, or towards it (’sGravesande 1711, §20). The first of these two turning procedures was used by Guidobaldo and Stevin (pages 251 and 276). It has the advantage that it does not involve the problem of reversing but it also has the drawback, observed earlier, that the figure and its perspective image overlap.10 Using precisely this latter argument, H P

p

C

Z

Ci O

G Bi

Q A g

B

R F

FIGURE VII.38. Introduction to the problem of reversing. The diagram shows a triangle ABC situated in the ground plane g, and its perspective image ABiCi in the picture plane p. 10

Some of the early perspectivists avoided the problem of reversing as well as the problem of overlapping by making two separate drawings. An example of this is found in a diagram by Vignola, reproduced in figure IV.24.

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’sGravesande dismissed the first procedure and recommended the second (ibid., §21). This second procedure (figure VII.39) has the advantage that the ground plane is kept on one side of the ground line GR and the image on the other, but the disadvantage, that, rather than the image ABiCi, its reverse side ABi ′Ci′, is constructed. We notice, for instance, that in the original triangle the line segment AB is a right leg, whereas in ABi′C i′ its image ABi ′ is a left leg. To achieve a more traditional orientation in figure VII.39, which ’sGravesande also did, I revolve the diagram an angle of 180˚ around the normal to the ground plane through A. As shown in figure VII.40, this allows the figure to be thrown into perspective to be situated on the same side of the ground line as the foot. Here the elements that were lying to the right of PQ in p (figure VII.38) lie to the left of it, and vice versa. In other words, the picture is mirrored in the line PQ, which ’sGravesande expressed as follows: ... the Perspective Plane lying down upon its other Face; what ought to be on the Right Hand appears on the Left and that on the Left appears on the Right; producing exactly the same Effect, as looking thro’ the Back-side of a Paper, at a Picture drawn thereon.11 [’sGravesande 1724, §21] C

g B

Q G

R

A Bi9 Ci9

P

FIGURE VII.39. The rabatment suggested by ’sGravesande. He let the picture plane p from the previous figure be turned down into the ground plane g, causing P to lie on the same side of the ground line GR as the foot. 11

... le Tableau étant couché sur son autre face, ce qui doit être à droit dans ces Représentations est à gauche, & ce qui doit être à gauche est à droit; cela faissant le même effet que si après avoir fait un dessein, on le regardoit par derriére. [’sGravesande 1711, §21]

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To compensate for the effect of his rabatment, ’sGravesande suggested two techniques, the first of which concerns the figure in the ground plane and runs as follows: ... in drawing upon the Geometrical Plane,12 we need but place that on the Right Hand which we have a mind should appear on the Left; or if the Geometrical Plane be drawn upon Paper, it may be oil’d or dipp’d in Varnish, which will render it transparent; and then the Back-side of the Paper may be thrown into Perspective.13 [’sGravesande 1724, §21]

This solution of using the reverse side of the paper corresponds to working with the reverse side of figure VII.40, shown as figure VII.41. As an alternative ’sGravesande recommended reversing the constructed perspective image of the plan (’sGravesande 1711, §21). Resolving the problem in this way requires some attention to the position of the figure in relation to the eye point, and in particular to the line PQ. When using the non-reversed P

Ci9 Bi9

R

A

G

Q

B

C

FIGURE VII.40. The previous figure rotated 180˚ to show ’sGravesande’s rabatment with the plan of the figure under the ground line GR.

12

By this ’sGravesande meant ground plane or plan. ... en traçant son Plan Géométral, on n’a qu’à mettre à droit ce qu’on veut représenter à gauche: ou si le Plan Géométral est tracé sur du papier, on peut le frottant d’huile ou de vernis le rendre transparent, & mettre ensuite en Perspective le revers du Papier. [’sGravesande 1711, §21]

13

6. The Problems of Reversing and Scaling

333

P

Ci Bi

G

Q

A

R

B9

C9

FIGURE VII.41. The reverse side of the previous figure.

triangle ABC (figure VII.40) to obtain the correct image as the reverse side of ABi′ Ci′, one must remember to place the point A on GR to the correct side of Q. In the present example this means placing it on the left-hand side (although when GR is part of p (figure VII.38), A is to the right of Q). The configuration in figure VII.41 could have been produced more simply than described by using the following procedure, which occurs implicitly in many publications on perspective. The object in the ground plane (figure VII.38) is turned around GR into p so that it lies below GR. This rabatment gives the same result as the one shown in figure VII.40, causing the reverse side AB′C′ of triangle ABC to lie under GR. In other words, the way to avoid ending up with a reverse image of ABC is to place its reverse side under the ground line, and then throw this into perspective by a standard construction, such as the distance point method. This yields the ‘true’ image ABiCi of the original triangle ABC. Having told his readers how to compensate for the effect of the problem of reversing, ’sGravesande, like his predecessors, ignored the problem in the rest of his book, with the result that his illustrations contain examples of reversing, as in figure VII.50. It is therefore highly doubtful that his description of the problem of reversing made his readers aware of the more subtle implications of bringing g and p into the same plane. Although I have explained the issue in much greater detail than ’sGravesande, I presume my exposition, too, is difficult to follow. As I said by way of introduction, this is a topic that can easily cause

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confusion.14 Still, as already noted, it is amazing that so many mathematicians have treated perspective without mentioning the problem of reversing. In keeping silent about this problem they have also failed to introduce the adjustments that can be made to remedy it. And that is why the textbooks are full of examples in which the image of a right leg in an angle has become a left leg – actually, earlier chapters in this book also offer several examples of reversing not mentioned. I would point to figures IV.37 and V.20, to mention but a few. A lack of compensation for the problem of reversing is indiscernible in finished perspective compositions, since they are not shown alongside their plans, and more often than not, it does not matter that a picture is shown in reverse – except, of course, in the case of well-known sceneries.

The Problem of Scaling

T

he problem of scaling occurs in connection with the use of plans in a perspective construction. As ’sGravesande pointed out, it is impossible for painters wishing to draw an image of a landscape to work with a plan in scale one to one (’sGravesande 1711, §20). He described the procedure applied by painters, who knew to solve this problem, as follows:

At the Foot of their Perspective Plane, they place a Plane, upon which are drawn in Miniature the Bases of Houses and Trees, which are in the Country to be represented ... always observing that there be the same Disposition between the Objects and their different Parts, upon this new Geometrical Plane, as the Objects truly have in the Country to be represented.15 [’sGravesande 1724, §20]

Stevin had published a similar brief remark. He imagined a painter who was three hundred palms away from a building with a height of one hundred palms and had placed his picture plane (in Stevin’s terminology “the glass”) parallel to the front façade at the distance three palms from his eye. Stevin continued:

14

One way to better grasp the problem of reversing is by carrying out this practical exercise: Take a piece of transparent paper and fold it so it corresponds to the threedimensional situation shown in figure VII.47 – that is, a horizontal ground plane g, a vertical picture plane p, and a horizontal plane h containing the eye point O placed in front of p. Draw, in g, a line l cutting the ground line in the point Il, then determine in h the vanishing point Vl of l by drawing the line through O parallel to l. The points Il and Vl determine the image li of l in the side of the picture plane p that turns towards O. The rabatment used by ’sGravesande corresponds to unfolding the paper so that the side containing l is facing upwards with the consequence that the upward facing side of li is the reverse. The alternative rabatment, mentioned earlier, corresponds to unfolding the paper so that li and the reverse side of l are facing upwards. 15 Pour cet effet ils placent au bas de leurs Tableaux, un Plan, dans lequel ils tracent en petit les bazes des Maisons, & des Arbres qui sont dans la Campagne ... en conservant dans ce nouveau Plan Géométral, aux objets & à leur diverses parties, la même disposition qu’elles ont véritablement entr’elles dans la Campagne. [’sGravesande 1711, §20]

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We imagine we have before us a small physical model whose front façade, standing in the glass, is ... one palm high, and the rest according to requirement, and if this model is taken for the object figure, and a ground-plan and a vertical plan are made of it, while for the perspective drawing the rule is followed, then the required image is obtained ...16 [Stevin 16051/1958, 871]

The remarks by ’sGravesande and Stevin presumably did little to help the untrained practitioner of perspective, but they did make him aware of the problem, which is more than most authors did. The way to handle the problem of scaling is closely linked to the way in which parameters of a perspective composition are selected, but I will not discuss this issue here.17 I thus envision a painter (figure VII.42) who has come so far in his decisions that he has fixed the eye point O, made the distance OP equal to d, and placed his canvas so that its bottom line GR has the distance h to the horizon HZ. I further assume this painter wants to depict an object whose real plan is situated in a horizontal plane in which Fo is the

H P Z

G O

A

B BO R

F

GO

AO

g

QO FO

RO

FIGURE VII.42. The problem of scaling.

16 Men beelt sich selfs in, al oftmen voor sich had een cleene lichamelicke bots, diens voorghevel int glas staende, hooch ... een palm, en de rest na den eysch, welcke bots ghenomen voor verschaueulicke form, en daer* af grontteyckening med stantteyckening ghemaeckt, en daer me int verschauewen de reghel ghevolght, men heeft t’begheerde ... * ichnographia cum ortographia [Stevin 16051, 40] 17 AndersenS 19871, 112–114, contains a detailed description of the problem, which is also outlined in section VII.9 of the present volume.

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foot and OFo = k. h. To concentrate on the basic idea, I will let the plan be as simple as the line segment AoBo, and to save space I have chosen a scaling factor k that is smaller than the ones generally needed in practice. What the painter must do is draw a plan of the object in a scale of 1 : k (one unit in the scaled plan corresponds to k times this unit in the real plan) and place it in the horizontal plane through GR. In figure VII.42, AB is the scaled plan. When the painter uses his scaled plan – and a correspondingly scaled elevation – for his construction, he will end up with the same image he would have obtained using a plan and elevation in a scale one to one. This sounds easy enough, but there is one thing the painter must be aware of, namely the placement of the real plan and the scaled plan in relation to the picture plane. To be more specific, I let GoRo be the intersection of the picture plane and the plane containing the real plan, assume that A lies on GR, and let Ao be the point of intersection of OA and the real plan. The latter point is situated at a certain distance, say a, from GoRo. To determine a I use the similarity of the triangles OFA and OFAo to obtain OF : OFo = FA : Fo Ao = d : (d + a). Since OF : OFo was given to be 1 : k, I can conclude that a = (k − 1)d.

(vii.1)

This relation shows that the transversal in the real plan, which is depicted on the chosen ground line, is situated in the distance (k − 1)d from the picture plane. Placing the canvas in relation to the eye point corresponds to choosing Alberti’s “open window through which the subject to be painted is seen” (pages 23–24). This metaphor was retained by several of Alberti’s successors, but in illustrating the perspective model they usually changed Alberti’s “window” to a door. Thus they drew the ground line of the picture and the ground plane at the level of the viewer’s feet. In this way they avoided introducing the problem of scaling. Most authors were aware that the height of the observer plays no role in perspective constructions, and that the important quantity is the distance between the eye point and the lowest horizontal plane that occurs in the perspective composition. This fact was seldom spelled out, however, and when it did come up, it was not in connection with the use of scaled plans and elevations, but in the context of a worm’s-eye or bird’s-eye view.

Reduced Distance

L

et me round off this section by briefly touching upon an issue that is related to the problem of scaling, and which ’sGravesande took up, namely the problem of what to do when the distance is large. In most practical situations the distance is so large that a very large sheet is required to apply a construction that involves one of the distance points. This problem was solved by ’sGravesande in a manner that several earlier perspectivists also may have used. However, I have not been actively tracing its history and know only that ’sGravesande was preceded by Grégoire Huret in his use of the procedure in question (Huret 1670, 116–124). Let P be the principal vanishing point (figure VII.43), D the left distance point, A a

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point in the ground plane to be thrown into perspective, B its orthogonal projection upon the ground line, and C the point on the ground line to the right of B determined by CB = AB. In an ordinary distance point construction the image Ai is determined as the point of intersection of the lines BP and CD. When PD is too large to be placed on the sheet used for the construction, ’sGravesande suggested that the point Ai should instead be constructed in the following way. On the horizon he introduced what could be called an auxiliary distance point Da determined by PDa = (1/n)PD, n being chosen so large that Da falls within the confines of the sheet. Next he determined the point E on the ground line so that EB = (1/n)AB and constructed Ai as the point of intersection of BP and EDa. He did not prove that this modified distance point construction is correct, but the result is obviously based on the division theorem according to which the image Ai is the point on BP that divides this line segment in the ratio AB : PD, that is BAi : AiP = BA : PD. Since the latter ratio is equal to BE : PDa, the construction involving Da and E is correct. It is worth noting that the reduced distance, PDa, is only introduced as an auxiliary tool, the composition itself keeps PD as its distance. This is contrary to the effect of a procedure used by a number of French seventeenth-century perspectivists, who worked with a de facto reduced distance that was scaled down by the same factor as the plan of the object to be thrown into perspective. The result of this method is that the constructed image is also scaled down – by the same factor as the plan. In other words, if the authors wanted to construct the image they had originally defined, they had to enlarge their constructed image – a fact they considered tacit knowledge. H

D

Da

P

Z

Ai

G

B

E

C

R

A

FIGURE VII.43. Working with too large a distance. Based on a section of figure 7 in ’sGravesande 1711.

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VII.7 ’sGravesande’s Essay on Perspective ’sGravesande and His Work on Perspective

I

n the history of science, Willem Jacob ’sGravesande (1688–1742) is known as a scholar who was influential in introducing Newtonian philosophy to continental Europe. He began his career by practising law for about ten years after graduating in the discipline at the age of nineteen. His lively interest in Newton’s ideas was awaked while he was staying in England in 1715, on which occasion he was also elected Fellow to the Royal Society. In 1717 he was appointed a professor of mathematics and astronomy at Leiden University. Seventeen years later, his professorship was enlarged to include philosophy as well. After his first appointment he gave an inaugural lecture on how other sciences can benefit from applying the exactness of mathematical thinking (Allamand in ’sGravesande 1774, XXII). In fact, he had previously promoted the idea that an exact mathematical understanding is useful, and quite early on, namely in his juvenile work Essai de perspective, which was published in 1711 but composed before that time. According to his biographer, Jean N. S. Allamand, ’sGravesande composed his book on perspective during lectures he found uninteresting, finishing the work before he was nineteen, but showing he “had the wisdom” not to publish it before some years later (translated from Allamand in ’sGravesande 1774, XI).18 Before completing his own work, ’sGravesande had studied the literature on perspective. Like so many others, he too was dissatisfied with what he read, and he joined the ranks of those bemoaning that authors of theoretical books paid too little attention to practice, and vice versa. He additionally criticized the authors writing on the practice of perspective for being overly conservative, claiming that they who

... might be thought to have more carefully treated of the practical Part of Perspective, do indeed at first lay down some general Rules ... but [which] are nothing the easier for having pass’d thro’ so many Hands; and that indeed, because they have not endeavour’d to make them so. They thought that all Objects might be thrown into Perspective by these Rules, and therefore it would be useless to search after others.19 [’sGravesande 1724, iii]

His own presentation of perspective, ’sGravesande addressed to painters who would only need, he maintained, a short time to understand his treatise. They 18 David Bierens de Haan claimed that ’sGravesande published a book called Proeve over de doorzigtkunde (Essay on perspective) in 1707 (Bierens de HaanS 1883, 111). I have not been able to verify the existence of this book, nor is any manuscript in Dutch of any early work on perspective by ’sGravesande known. After consultations with Jan van Maanen and Cees de Pater – both of whom I gratefully thank for their kind help – I have come to the conclusion that this reference is based on a misunderstanding, perhaps caused by Allamand’s information about ’sGravesande’s early engagement in perspective. 19 Les autres Auteurs ... avoir traité la pratique avec plus de soin, en donnent d’abord quelques régles générales ..., & qui pour avoir passé par tant de mains n’en sont pas devenuës plus aisées; aussi n’ont ils pas travaillé à les rendre telles. Ils ont crû que tous les objets pouvant se mettre en Perspective par ces moyens là, il seroit inutile d’en chercher d’autres. [’sGravesande 1711, preface 5v–6r]

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would soon recoup the time spent, for their new knowledge would enable them to make their constructions much more quickly (ibid., preface 8r). He also claimed he had discussed the contents of his book with a painter (ibid., 12r–12v). Generally ’sGravesande’s preface gives the impression that his intention was to write the textbook for those applying perspective. Even so, when it came to putting this idea into practice, ’sGravesande, like so many other mathematicians, was deceived by his own profound understanding of mathematics. In some places he made an effort to reach readers without much mathematical background; for instance, instead of formulating just one general theorem, he sometimes treated a number of special cases as well. More often, however, he allowed himself to get carried away by the intriguing mathematical possibilities his topic offered. Thus, Essai de perspective contains several examples that apply mathematical techniques far beyond what most practitioners would be able to grasp – a point to which I will return.

The Contents of ’sGravesande’s Work ssai de perspective consists of nine chapters and a separate section on the camera obscura (dark chamber) – which I will deal with shortly. In the first chapter ’sGravesande introduced the mathematical model for a perspectival representation and the associated concepts. Contrary to Stevin and van Schooten, he did not include a formal postulate, but simply claimed that the central projection of an object from the eye point upon a picture plane has the same effect upon the eye as the object itself (’sGravesande 1711, §1). Based on this assumption he developed the theory of perspective, which he then used to describe and prove the correctness of a number of constructions. ’sGravesande’s material on applying the theory of perspective to constructions is very rich, and indeed so rich that not all of it can be covered here.

E

FIGURE VII.44. A problem relating to sundials. ’sGravesande 1711, figure 65.

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’sGravesande’s last chapter is devoted to showing how the theory of perspective can be used to solve the following rather special problem about sundials (figure VII.44). Given a horizontal sundial with hour lines corresponding to a gnomon with a given pointed top and let there also be given a direction of not horizontal planes; construct the hour lines on a sundial whose gnomon has the same top as the gnomon of the given sundial and which has the given direction.

Camerae Obscurae ssai de perspective contains a 38-page description of two designs ’sGravesande made of a camera obscura (’sGravesande 1711, part two). One is formed like a sedan chair (figure VII.45), whereas the other is much

E

FIGURE VII.45. A camera obscura designed by ’sGravesande. The drawing paper is placed upon the table A. The apparatus that guides the light rays from an object onto A is situated above the sedan, and includes a lens in the cylinder C and two mirrors denoted by H and L. In the illustration one side of the camera obscura has been removed, revealing the interior. ’sGravesande 1774, Chambre Obscure, plate 1 with the L changed to white.

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more handy and easy to transport (figure VII.46). The basic idea of a camera obscura is to project a scenery or an object upon a piece of paper placed in a dark box into which light rays enter through a small hole. This device has been used by astronomers since antiquity for observing solar eclipses. In later and more sophisticated versions of the instrument, the hole was equipped with a lens, and one or more mirrors were added to direct the light to the desired position within the chamber (for more on the history of this instrument, see LindbergS 1968 and HammondS 1981). The idea of applying the camera obscura for making perspective drawing of existing motifs had already been mentioned during the Renaissance, for example in Barbaro’s La pratica della perspettiva (Barbaro 1569, 192–193). However, as far as I can discern, ’sGravesande was the first to go into any

FIGURE VII.46. A second design of a camera obscura by ’sGravesande. The drawing paper is placed upon G, and the light-guiding apparatus is built into Y. ’sGravesande 1774, Chambre Obscure, plate 3.

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detail about how the instrument functions relative to a particular perspective projection. While leaving some decisions to the constructors of his camerae, he gave several quite specific instructions – stressing, for instance, that the sedan must be provided with ventilation ducts that would allow no light to enter (’sGravesande 1711, part two, §§4–14 and 39–46). In treating the geometry of the apparatus, on the other hand, he left no questions unexplored. Among other things, he described how to place and tip the mirrors depending on whether the final result should correspond to a perspective projection upon a vertical, oblique, or horizontal picture plane. He also supplied his claims with proofs. Although ’sGravesande became involved with many other disciplines after composing Essai de perspective, he apparently did not lose interest in perspective. Shortly before his death in 1742 he returned to perspective and had plates made for a new and revised edition of his book (Allamand in ’sGravesande 1774, XI). No manuscript for this edition has ever been found, which Allamand attributed to ’sGravesande’s custom of keeping his work in his head and only writing things down when they were finished.

The Basic Theory

T

he theorems ’sGravesande considered fundamental can be described very briefly, as they contain familiar material. He himself was aware of this, stating that the theory of perspective can be reduced to three well-known theorems (’sGravesande 1711, preface, 8r–9v). In the first of these, ’sGravesande elegantly proved (ibid., §4), as we saw in chapter VI, that the image of a line segment parallel to the picture plane is a line segment parallel to the original line segment (page 242). His second theorem establishes that the ratio between a line segment parallel to p and its image is equal to the ratio between the distance from the eye point to the vertical plane through the line segment and the distance (ibid., §8). The third theorem is the main theorem (ibid., §13). Like most of the mathematicians preceding him, ’sGravesande did not refer to colleagues. However, there can be little doubt that in choosing his fundamental theorems he found inspiration in Stevin’s and van Schooten’s treatises on perspective. Besides the three basic theorems, ’sGravesande’s theoretical chapter contains three other results that are rather special. Two of them concern invariance of the image of a line in connection with movement of the eye point. Together they state that whether a line l is parallel to the picture plane p (ibid., §12) or not (ibid., §18), its image does not change if the eye point O moves on the line through O parallel to l. He applied the latter theorems in an example in which the principal vanishing point falls outside the picture (ibid., §78). The last of his special theorems (ibid., §19) was included to provide the theoretical background for certain exceptional constructions (ibid., §§71, 87). An analysis of the techniques ’sGravesande applied in his constructions shows that his presentation of the theory is not complete. He thus used

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Stevin’s result concerning the invariance of a perspective image under simultaneous rotations of the picture plane and the eye point (page 271) without any comments (ibid., §§82–83). Considering that otherwise he did not apply results for which he had not argued, I wonder why he did not include Stevin’s theorem, or at least refer to it. One possibility is that originally – while trying to write for practising perspectivists – he intentionally omitted the topic of oblique picture plane and Stevin’s rotation theorem because he found the proof of the latter too difficult for his target group. As he progressed with his writing, he became less concerned about what he could expect from his readers, as previously noted. In this connection he may have decided to deal with oblique picture planes based on Stevin’s theorem, forgetting or ignoring the fact that he had not presented it.

The Turned-In Eye Point

A

chieving a deeper understanding of ’sGravesande’s diagrams calls for an introduction to how he represented the eye point. In the previous section we saw how he worked with a technique (figure VII.41) that corresponds to placing the reverse side of a figure in the ground plane under the ground line, and the picture plane p above the ground line. In order to have the eye point O represented in this configuration as well (figure VII.47), he turned the horizontal plane h through O around the horizon HZ into p. The result is (figure VII.48) that the rotated eye point, which I denote Op and sometimes call the auxiliary eye point and other times the turned-in eye point, lies above the horizon. In my illustrations the picture plane is assumed to be vertical, but ’sGravesande’s technique of rabatment also works for oblique picture planes.

H p

P g

h

Z

O G

A Q

I R

F

J

FIGURE VII.47. Preparation for showing how ’sGravesande rotated the eye point into the plane of the drawing.

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I

H

G

Fp

J

P

Z

R

Q

A

FIGURE VII.48. ’sGravesande’s rabatment of the configuration in the previous figure.

In some of the diagrams containing the auxiliary eye point, ’sGravesande included a line he called the geometrical line. This is the line IJ (figure VII.47) through the foot, F, of O parallel to GR, which he turned into p around GR so that it lies above GR (figure VII.48). As demonstrated in the next section, he used this geometrical line in one of his constructions. As we will see shortly, it is very handy to have the eye point represented by the point Op in the picture plane. Actually ’sGravesande was the first of the influential perspectivists to apply this procedure consistently, although it had been used earlier by the French mathematician Ozanam (Ozanam 1693, 45).20 The latter introduced the turned-in eye point in connection with a special, and for his time rather unusual, construction of the image of a point. This construction, which will soon be presented as a visual ray construction, was among ’sGravesande favourites. I find it very plausible that it was by reading 20

In presenting the rabatment involving the turned-in eye point, Heinrich Wieleitner and Rudolf Bkouche have mentioned that the same procedure is to be found in Philippe de La Hire’s Plani-coniques (WieleitnerS 1913; WieleitnerS 1914, 324; BkoucheS 1991, 264, 276; the work in question is published in La HireS 1673, 75–94). It may be that Ozanam was inspired by de La Hire, but I find it just as likely that here we are looking at an interesting case of parallel developments in mathematics.

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Ozanam, ’sGravesande became aware of the elegance of the visual ray construction and the advantage of introducing the point Op. One gain in using Op is that this point makes it very easy to construct the vanishing point Vl of a horizontal line l and then apply the main theorem to construct the image of l (figure VII.49). ’sGravesande did not present this as a separate construction, but applied it frequently, for instance in §26 of his Essai de perspective. The theme of constructing images of polygons and other rectilinear figures had traditionally been popular in textbooks on perspective. With ’sGravesande’s method, these problems became so uncomplicated that he saw no reason to expend much energy describing them. In fact, he gave only a couple of examples, one of which is reproduced in figure VII.50.

A Particular Line

T

he point Op also offers an advantage for constructing the image of a point by procedures I call visual ray constructions. These are based on the following observation, which ’sGravesande did not formulate as a separate result, but included as part of his proof of the correctness of his visual ray construction (’sGravesande, §27).

Op

H

Vl

Z

li

FIGURE VII.49. Construction of the image of a line by ’sGravesande. Given are a line l in g and it intersection Il with the ground line GR. He determined the vanishing point Vl of l as the point of intersection of the horizon HZ and the line through Op parallel to l. Thereafter he constructed the image li of l by joining Il and Vl.

Il

G

l

R

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FIGURE VII.50. One of ’sGravesande’s examples of throwing a rectilinear figure into perspective. Given are a rectangle ABCD in the ground plane and the eye point turned into the picture plane O – which is the same point as my Op. ’sGravesande constructed the vanishing point F of lines parallel to AB as the point of intersection of the horizon and the line through O parallel to AB. Similarly, he constructed the vanishing point of lines parallel to AD. He then drew the images of the various lines as the lines joining the intersections of the lines and their vanishing points, finally determining the image of a relevant point, such as A, as the point of intersection of the images of two lines passing through A. ’sGravesande 1711, figure 17.

Observation 1 (figure VII.51). When Op is the auxiliary eye point in the picture plane p, A a point in the ground plane turned into p, and Ai its image, then the points Ap, Ai, and Op are collinear. Ozanam had applied this result, but not proved it. The proof ’sGravesande gave ran along the following lines. Let the line OpAp cut the horizon HZ in C and the ground line GR in B. If the plane above HZ and the plane below GR

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Op

H

C

Z

Ai

G

B

R

A

FIGURE VII.51. Illustration to observation 1.

are turned back to the three-dimensional situation (figure VII.52), the line OC is parallel to AB. Hence, C is the vanishing point of the line AB, and since its intersection is B, the image Ai – according to the main theorem – lies on the line BC and therefore also on the line OpAp. Most likely inspired by ’sGravesande, Taylor later applied a line corresponding to OpAp (Taylor 1719, 22). Taylor also introduced the term visual ray which is not really a fortunate one, since it could give the impression that OpAp is the visual ray OA turned around Ai into the picture plane.21 Nevertheless, I have respected Taylor’s choice and refer to constructions involving OpAp as visual ray constructions. In the following I retain the notation Op for the auxiliary eye point, but omit the index p for points in the ground plane turned into the picture plane.

21

One way of relating Taylor’s ‘visual ray’ OpAp with OA is to conceive of the first as the image of the latter under the parallel projection, which has direction OOp.

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Op

H C O

Z

Ai

A

G

g

B R

Ap

FIGURE VII.52. The three-dimensional configuration corresponding to the previous figure.

’sGravesande’s Basic Constructions

L

ike his Dutch academic predecessors, ’sGravesande considered the fundamental problem of perspective to be that of constructing the image of an arbitrary point. He did not follow van Schooten in considering points in front of the picture plane p, but worked traditionally with points lying in or behind p. Most constructions of the image of a point are based on defining the point as the intersection of two lines – the choice of these two lines being what makes the various methods differ. ’sGravesande presented five constructions, the first of which is the well-known distance point method. His remaining four procedures involve somewhat untraditional choices of the intersecting lines. ’sGravesande’s second method is a visual ray construction. In his own diagram (figure VII.53), the ‘visual ray’ is the line AO – O being the auxiliary eye point – that cuts the ground line and the horizon at B and C. He let the points D and E on the horizon and the ground line be determined, respectively, by CD = CO and BE = BA, and then chose DE as the second intersecting line. That the image a of A lies on OA follows from observation 1. To prove that a also lies on ED, ’sGravesande remarked that the construction of the points D and E

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FIGURE VII.53. One of ’sGravesande’s visual ray constructions. ’sGravesande 1711, figure 8.

implies that OD and AE are parallel, and that the result therefore follows from the main theorem (’sGravesande 1711, §27) – which indeed it does, since D is the vanishing point of the line AE, implying that this line is mapped upon ED. An anonymous review of Essai de perspective praised ’sGravesande highly for this visual ray construction (’sGravesandeS 1711). As it happens, ’sGravesande should have shared this praise with Ozanam, who published a visual ray construction earlier, as noted above. There is, however, no doubt that it was only after ’sGravesande had shown how elegantly constructions can be carried out with the help of ‘visual rays’ that this sort of constructions received any serious attention. Figure VII.54 presents one of ’sGravesande’s refined applications. In his fourth construction, ’sGravesande also applied a ‘visual ray’, while choosing a rather special line as the second line to pass through the point he wished to throw into perspective (’sGravesande 1711, §§31–32). As indicated, later Taylor also had ‘visual rays’ play a key role in his procedures. He generalized ’sGravesande’s two methods, letting the lines AE and OD in figure VII.53 be any pair of parallel lines through O and A. Moreover, he developed his method for the general case in which the angle between the ground plane and the picture plane is arbitrary (though not 180˚). In his two remaining methods, ’sGravesande applied other lines than ‘visual rays’. His third construction (figure VII.55) involves circles, and his last makes use of the line he had introduced as the geometrical line. This procedure, too, caught Taylor’s attention, and in order to demonstrate in chapter X just how much inspiration Taylor found in ’sGravesande’s work, I have presented ’sGravesande’s version of this construction in figure VII.56. He himself called this method “the sixth”, since as his fifth he presented a procedure for constructing the image of a point when the image of another point is given (ibid., §§35–36). In chapter eight of Essai de perspective,

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FIGURE VII.54. An example of ’sGravesande applying ‘visual rays’ to divide the image of a line segment. It is given that O is the eye point turned into the picture, and that ab is the image of AB. It is required to divide ab into four sections that are perspectively equal. ’sGravesande’s solution was to divide the line segment AB into four equal sections, and then find the points of intersection of ab and the lines from O to the points of division on AB. A section of figure 14 in ’sGravesande 1711.

Op

H

T

Z Ai

G

S

A

R

FIGURE VII.55. ’sGravesande’s third method for constructing the image of a point in the ground plane. He determined the image Ai of A as the point of intersection of two common tangents to the circles having their centres in Op and A and touching HZ and GR, respectively. To establish the correctness of this construction, he first proved the following result. If a common tangent to the two circles cuts GR in S and HZ in T, then OpT and AS are parallel. From the main theorem he then concluded that Ai lies on ST; analogously Ai lies on the other tangent. Based on ’sGravesande 1711, figure 9.

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Op

FIGURE VII.56. ’sGravesande last method for constructing the image of a point in the ground plane (’sGravesande 1711, §39). In this construction he involved his “geometrical line” IJ (figures VII.47 and 48). He first drew two arbitrary lines through A cutting GR at B and C, and IJ at D and E. Then he drew the line BL parallel to OpD and the line CK parallel to OpE, and finally determined the image Ai of A as the point of intersection of BL and CK. To establish the correctness of this construction, he proved that when K and L are the points in which the intersecting lines cut the horizon HZ, then OpK is parallel to AC and OpL parallel to AB. The result then follows from the main theorem.

I

D

E

J

Fp H

K

L

Z

Ai G

R B

C

A

’sGravesande returned to some of his basic constructions, showing how they can be performed by using sliding rulers and threads. In choosing constructions of the image of a point above the ground plane, ’sGravesande also went his own way. Most authors had applied the orthogonal projection upon the ground plane of the point and the image of the projection. ’sGravesande presented a new method that does not involve the projection (ibid., §§51–52), accompanied by three constructions based on the projection (ibid., §§56–59). Two of these methods are unusual – and to me they seem overly complicated.

Oblique Picture Planes

H

aving solved the fundamental problem of throwing a point into perspective in a vertical plane, ’sGravesande turned to picture planes that are oblique or parallel to the ground plane. He followed Stevin’s idea of reducing the problem of determining the image of a point in an oblique picture plane to the problem of constructing its image in a vertical picture plane – seen from another eye point. When applied to a point in the ground plane, ’sGravesande’s solution (ibid., §82) is close to the one Stevin gave. ’sGravesande’s treatment of the situation in which the given point lies above the ground plane, offers a very illustrative example of what has often happened in the history of explaining perspective constructions: ’sGravesande himself had gained a complete understanding of how the construction can be performed, and in presenting it he assumed that his readers had achieved the same level of understanding (’sGravesande 1711, §83). To document this point, and to indicate that it was still possible for his successors to introduce new simplifications and greater elegance to some perspective constructions, I have described the details in figures VII.57–VII.59.

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VII. The Dutch Development after Stevin H p

S E L

O

Bi

G

v

g

Q

Z

B A

C

R

F

T

FIGURE VII.57. The first preparatory diagram for a presentation of ’sGravesande’s determination of the image of a point in an oblique picture plane. In this diagram p is an oblique picture plane that forms a given angle, say v, with the ground plane g, GR is the ground line, O the eye point, F its foot in g; Q the ground point, and HZ the horizon. In addition, S is the point of intersection of HZ and the line through O parallel to FQ, and T the point of intersection of OF and SQ and therefore also the vanishing point of lines orthogonal to g. Finally, B is a point above g given by its orthogonal projection A upon g and the height AB, and L the point of intersection of the line AB and p. To determine the image Bi of B, ’sGravesande searched for two lines containing it. He first introduced C as the point in which FA cuts GR, and E as the point of intersection of TC and HZ, and then claimed that Bi lies on CE.

(1)

His argument for this was rather long (’sGravesande 1711, §83). Alternatively, the result can be obtained by involving the image of the vertical line AB. According to the main theorem, this image is TL, and thus Bi lies on TL. Since the orthogonal projection of TL upon g is FA, the point C lies on TL, which implies that the lines TL, TC, and CE are identical. This proves relation (1). ’sGravesande’s second characterisation of Bi is presented in the caption of figure VII.58.

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H p9

Z

O9

Bi G

B9

Ai9 A9

Q K g

R

F9

FIGURE VII.58. The second preparatory diagram for a presentation of ’sGravesande’s determination of the image of a point in an oblique picture plane. As noted, ’sGravesande took over Stevin’s theorem concerning the invariance of the image of a point above the ground plane under simultaneous rotations of the eye point, the picture plane, and the point (cf. result 4 page 273). This means he took it for granted that given a vertical picture plane p′, there exists an eye point O′ as well as a point B′ that produces the same image in p′ seen from O′ as B (figure VII.57) does in p seen from O. To find a second line through Bi, he let A′ be the orthogonal projection of B ′ upon g and F ′ the foot of O ′. He then introduced the point of intersection K of GR and F ′A′ and claimed that Bi lies on the vertical line through K. (1) His proof runs along the following lines (’sGravesande 1711, §83): Since the line F ′A′ passes through the foot F ′, it is depicted as a vertical line through K.* This implies, in particular, that the image A′i of A′ lies on the vertical line through K, and since the image of B′ lies vertically above A′i, the point B′ is also mapped upon the vertical line through the point K.

’sGravesande’s construction of the image of an oblique line segment in a vertical picture plane similarly provides an example of a theme that could be made more transparent (’sGravesande 1711, §§70–73). At the same time it demonstrates progress, because compared to Guidobaldo’s method of determining the vanishing point of an oblique line the one by ’sGravesande is crystal clear (ibid., §69). *This is one of Guidobaldo’s theorems, cf. result 2, page 244. It follows from the fact that the vanishing point of F ′A′ lies vertically above K.

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Z F Bi

G

Q

C

K

R A9

A

T

FIGURE VII.59. ’sGravesande’s construction of the image of a point B above the ground plane in an oblique picture plane. In this diagram the points correspond to those shown in figure VII.57 and figure VII.58, but some have been rotated into the vertical picture plane p ′ shown in figure VII.58: QT (figure VII.57) around GR so that it lies below this line, and QF′ (figure VII.58) around GR so that F and F ′ lie above GR. To construct the required point Bi, ’sGravesande determined C as the point of intersection of AF and GR, E as the point of intersection of TC and HZ, and K as the point of intersection of A′F′ and GR. Combining the results (1) from the two previous captions, he found Bi to be the point of intersection of CE and the line through K perpendicular to GR. Although complicated to substantiate, the construction itself is easy to carry out. The most demanding part is actually constructing the points T, F′, and A′. The determination of F′ was explained in the caption of figure VI.24 (the angle u occurring there being equal to 90˚–v), the point A′ is constructed similarly (figure VI.25), and the determination of T can be seen in figure VII.57. Adaptation of figure 45 in ’sGravesande 1711.

’sGravesande’s Examples

I

n figure VII.53 we saw one of ’sGravesande easy examples. Another of his straightforward examples concerns the image of a circle situated in the ground plane, as reproduced in figure VII.60. Meanwhile, ’sGravesande did not

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FIGURE VII.60 ’sGravesande’s construction of the perspective image of a circle. Like other perspectivists, he applied a pointwise construction. He drew a number of parallel chords in the circle, constructed the vanishing point of these chords, and then applied ‘visual rays’ (not included in the diagram) to determine the images of the endpoints of the chords. ’sGravesande 1711, section of figure 16.

content himself with having found the image of the circle (’sGravesande 1711, §50). His mathematical inquisitiveness made him take up the issue of how to construct two conjugate diameters in the image – which is an ellipse – even though this has no obvious practical uses. He did not prove that his construction was correct, but referred to a book by Philippe de la Hire on conic sections (La HireS 1685). In treating the perspective image of a sphere, ’sGravesande similarly used the theory of conic sections (’sGravesande 1711, §64). Most of ’sGravesande’s other examples are rather tricky. He seemingly decided to leave simple applications to the readers and treat problems that required mathematical observations of the sort his readers themselves would not so easily have made. Some of these deal with the determination of visible parts of three-dimensional objects. While presenting the column problem in section II. 15, I mentioned that to achieve an accurate solution one must find the visible parts of the columns before throwing them into perspective (page 56) – and the same applies to other solids. As far as I am aware, ’sGravesande was the first to write about this procedure. He did not discuss the column problem, but he did demonstrate how to determine the visible part of a cylinder (’sGravesande 1711, §60). He similarly constructed the

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visible part of a cone (ibid., §§54–55), and finally took up an extremely intricate problem involving the visible part of a column base (ibid., §§65–68). This last issue was so difficult that ’sGravesande resorted to the infinitesimal calculus in his solution. He used this relatively new discipline to analyse the problem, and was led to a construction that is no more elaborate than many of his other constructions. The analysis itself, however, is far more sophisticated than the rest of the material in Essai de perspective. From a historical point of view ’sGravesande’s analysis is fascinating because it is so different from any other early eighteenth-century example of applying the infinitesimal calculus. Actually, I find ’sGravesande’s solution so remarkable that it deserves to become better known, which is why I have described it in appendix three. ’sGravesande treated the problem of visibility in connection with the column shown in figure VII.61, the cylindrical component of which blocks the view of part of the base – an effect that ’sGravesande also took into account.

FIGURE VII.61. A column base. For details, see appendix three. ’sGravesande 1774, plate 10.

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Shadows

S

hadows in perspective compositions was another theme that caught ’sGravesande attention. Thus far we have seen that Dürer presented a method of determining the perspective shadow when light comes from a point, and that he applied a plan and elevation method to construct this shadow. It has also been mentioned that some perspectivists, including Guidobaldo and Marolois, dealt with the problem of how to determine shadows cast by given objects, but were very brief on throwing such shadows into perspective. Unlike his predecessors, ’sGravesande treated this topic quite thoroughly (’sGravesande 1711, §§103–109). He examined the two traditional cases: One in which the light comes from an object (typically the sun) so far away that its rays can be considered parallel, and one in which the source of light is assumed to be a point. In the first case the shadow of an object upon a plane is a parallel projection of the object upon the plane, and in the second case it is a central projection. In mathematizing the question of perspective shadows, ’sGravesande considered the fundamental problem to be determining the perspective image of the shadow of an arbitrary point cast upon a plane g, and he generally assumed that this plane was the ground plane. Irrespective of whether the light rays form a pencil of lines or a bundle of parallel lines (figures VII.62 and VII.63), the shadow a of a point A upon a plane g is the point of intersection of the light ray through the point and its orthogonal projection upon g. When these configurations are transferred to the picture plane they become more alike, in the sense that in both cases a particular point is used for constructing the image of the shadow point – namely either the image of the light point L or the vanishing point of parallel light rays (cf. figure VII.64). ’sGravesande presented two methods for throwing a shadow point into perspective. One was first to determine the shadow as described in figures VII.62 and VII.63, and then construct its perspective image with the aid of one of his procedures for determining the image of a point. The other method L* A

L' A' a g

FIGURE VII.62. Determining a shadow when the source of light is the point L. Let the point A cast the shadow a upon the plane g, and let the orthogonal projections, A′ and L′, of A and L upon g be known. The point a can then be constructed as the point of intersection of LA and L′A′.

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A

A9

a

γ

FIGURE VII.63. Determining a shadow when the light rays are parallel. Let A cast the shadow a upon the plane g. The latter point is the point of intersection of the light ray through A and its orthogonal projection upon g.

involved transferring the process of determining shadows to the picture plane. His procedure for handling a case in which the light rays are parallel – but not parallel to the picture plane p – is shown in figure VII.64. ’sGravesande told his readers that only for rectilinear figures is it advantageous to determine a shadow mathematically, stating that for other objects it is so difficult to determine them [the shadows] Geometrically that it is much better to examine those which are daily observed, and so imitate them.22 [’sGravesande 1724, §102] V

H

Z U

Ai9 Ai ai

FIGURE VII.64. A perspectival shadow construction by ’sGravesande. He assumed that the light comes from the sun and that the rays are not parallel to the picture plane, and he wished to construct the perspective image of the shadow cast by the point A upon a horizontal plane g. He let the images Ai and A′i of the point A and its projection A′ upon g be given – or already constructed. He then determined the vanishing points V and U of the sun rays and their projections upon g, respectively, thereby obtaining the perspective shadow point ai as the point of intersection of the lines VAi and UA′i. Adaptation from ’sGravesande 1711, figure 58. 22 quant aux ombres des autres corps, il est si difficile de les déterminer Géométriquement, que le meilleur c’est d’examiner celles qu’on voit tous les jours, pour se former une routine de les imiter. [’sGravesande 1711, §102]

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He gave the same advice to any painter wishing to illustrate the effect of light entering a room through several windows (’sGravesande 1711, §102). This is indeed a difficult problem that some of the painters from the golden era of Dutch painting were engaged in, as exemplified in figure VII.2.

Response to ’sGravesande’s Work

A

s already noted, Essai de perspective was reviewed positively and anonymously in Acta eruditorum (’sGravesandeS 1711).23 This review was most likely written by Johann Bernoulli, for it strikes the same tone as a letter he sent to ’sGravesande in March 1714 along with a book he had just published (BernoulliS 1714). Among other things, Bernoulli wrote: I beg you to accept it [the book] as coming from a person who has great respect and esteem for your merits in and knowledge of mathematics. I have seen sufficient proof of this in your excellent treatise on perspective. ... In it I found several highly ingenious and very handy rules that are not to be found anywhere else.24

This was high praise indeed coming from one of the leading mathematicians of the time, not least because Bernoulli was not always generous in his reviews, which will be made amply clear in chapter X. I know of no other contemporary reactions, but later historians of mathematics, including Jean Étienne Montucla, Michel Chasles, Noël Germinal Poudra, and Gino Loria, expressed a similarly positive attitude.25 What I have written thus far on ’sGravesande does not indicate that his work exerted any major influence, but that is because I have only looked at the praise explicitly expressed. As noted earlier, I am convinced – as I will argue in chapter X – that Taylor found quite some inspiration in ’sGravesande’s ideas. In fact, I see Taylor’s work in Britain as a continuation of the Dutch academic approach to perspective. Besides, ’sGravesande’s book was also imported directly into Britain in the form of an English translation published in 1724.

The Audience for Books on Perspective

I

t is noticeable that the acknowledgements of ’sGravesande’s Essai de perspective came from mathematicians, and not from the practitioners of perspective to whom he had addressed the work. Among his admirers I have actually been able to find only one with a practical background, namely the

23

The contents of Essai de perspective were also outlined in ’sGravesandeS 1712. Je vous supplie de l’accepter comme venant d’une Personne qui a beaucoup d’égard & de considération pour votre mérite & savoir dans les Mathématiques, dont j’ai vu une preuve suffisante par l’excellent Traité sur la Perspective ... J’y ai trouvé plusieurs régles fort ingénieuses & très commodes pour la pratique que l’on ne trouve pas par tout ailleurs ... [’sGravesande 1774, XI] 25 MontuclaS 1758, 636; ChaslesS1835–37, 347; PoudraS1864, 491, 495; LoriaS 1908, 595. 24

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English draughtsman Thomas Malton. In 1775 he published some thoughts on ’sGravesande’s work that are discussed in chapter X (page 580). Through his mathematical analyses ’sGravesande had managed to make some perspective constructions more transparent, but only to such a degree that mathematicians could see through them. Artisans and artists who were willing to accept a mathematical style may have benefited from his treatment of how to throw a horizontal figure into perspective in a vertical plane, but for most of them the rest of Essai de perspective was probably too exclusive. It is in a way surprising that ’sGravesande himself did not admit that a large part of his book was a mathematical work. This might be because he misjudged practitioners’ command of mathematics, but it could also have something to do with mathematicians’ views on perspective. We saw how Huygens expressed the opinion that there is no intellectual challenge in the subject (page 319). And ’sGravesande in his preface expressed the same view: the tedious Particulars [of perspective] ... will always hinder Genius’s capable of great Matters, from undertaking a Subject so little worthy of their Endeavours, and so barren of great Discoveries.26 [’sGravesande 1724, iv]

Although Bernoulli had become aware of ’sGravesande’s mathematical gifts through his work on perspective, the subject itself was apparently not considered a relevant mathematical research area for its own sake – that is, separated from its applications. Guidobaldo had written his work on perspective for mathematicians, but he was the last scholar to do so for a long time. ’sGravesande’s assertion that perspective would not appeal to geniuses, and that it did not give rise to great discoveries is worth discussing. If he had mathematicians of the likes of Isaac Newton, and discoveries like differential calculus in mind, he was right. The two more than able mathematicians, Brook Taylor and Johann Heinrich Lambert, also became attracted by perspective, however, and although they cannot lay claim to great discoveries, they certainly made mathematically clever contributions to the theory of perspective. Among many other things, they were able to cover constructions in oblique planes in an elegant manner – an accomplishment that eluded ’sGravesande. Despite their very mathematical approach, the works of Taylor and Lambert were also meant for those who applied perspective, as we shall see in chapters X and XII.

VII.8 Traces of Desargues’s Method in Dutch Perspective

T

wo of the Dutch books on perspective written by practitioners of the discipline are particularly interesting because they show that Desargues’s

26

les détails ennuyeux ... ne permettroient jamais aux genies capables de plus grandes choses d’entrer dans une carriére peu digne de leurs efforts, & inaccessible aux grandes découvertes. [’sGravesande 1711, preface 6v–7r]

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method found an audience in the Netherlands. This method, discussed in detail in chapter IX, is based on the idea of applying perspective scales. Desargues’s method did not become known in the Netherlands through his own description, which is very brief, but through a comprehensive presentation, Maniere universelle, published by Abraham Bosse in 1648. This book appeared in a Dutch translation in 1686 and gave rise to general admiration of Desargues’s ideas, although several found they were not adequately presented. In the words of the engraver Dirk Bosboom (1641–1707), one of Bosse’s Dutch colleagues: Is it not a pity that such a wonderful invention is not described in a simple, orderly, and understandable way without any detours?27

Bosboom’s own publication Perspectiva of doorzicht-kunde (Perspectiva or perspective) from 1703 is a nicely illustrated manual in which he took great pains to introduce Desargues’s perspective scales more clearly than Bosse had done. The knowledge of Desargues’s method in Bosse’s presentation was seemingly kept alive within the circle of Dutch engravers for quite a long time. At any rate, another Dutch engraver, Caspar Jacobszoon Philips (1732–1789), whose Uitvoerig onderwijs in de perspectiva (Thorough instruction in the perspective) appeared in 1765, commented upon the Dutch edition of Maniere universelle in much the same way as Bosboom had done sixty years before: It is known that Desargues has invented this [the perspective scale] and that Bosse wrote a considerable work about it in French. But it is no less known how the Dutch translation of this work has turned out to be so incomprehensible that one can hardly make any use of it.28

Although apparently dissatisfied with the translator’s linguistic abilities, Philips is probably criticizing Bosse’s pedagogical skills more than the language. Like Bosboom, Philips was an avid proponent of Desargues’s method, claiming that having taught perspective for some years, this method, in his experience, was the best (Philips 1765, 3). In fact, he did provide a good introduction to the method in his Uitvoerig onderwijs, which was richly illustrated with informative drawings – as exemplified in figure VII.65. The writing of his first book on perspective seems to have stimulated Philips a great deal, for he devoted four more volumes to the subject. As shown in figure VII.66, one dealt with the difficult question of throwing reflections into perspective (Philips 1775) – a theme I revisit in chapters X and XII. Another of Philips’s volumes combined the teaching of basic

27 Is het niet jammer dat zulk een heerlijke uijtvinding, niet eenvoudig, ordentelijk, verstaanlijk en zonder eenige omwegen beschreven is? [Bosboom 1703, 66] 28 ’t Is bekend dat de Heer Desargues dezelve uitgevonden, en dat de Heer Bosse daarover een aanzienlijk Werk in ’t Fransch geschreeven heeft; maar ’t is ook niet minder bekend, hoe hetzelve in ’t Nederduitsch vertaald en zo onverstaanbaar uitgevallen is, dat men’er genoegzaam geen Gebruik van maken kan. [Philips 1765, voorrede, 4r]

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FIGURE VII.65. One of Philips’s perspective compositions. Philips 1765, plate 17, Leiden University Library, 2314 E 10.

astronomy with a demonstration of a solution for drawing perspective images of shadows (Philips 17861). His next publication, Zeemans onderwijser in de tekenkunst (Seaman’s instruction in the art of drawing, 1786), is a textbook in elementary geometry and perspective drawing for sailors. In this book he did not apply Desargues’s method, but used a distance point construction, additionally replacing the traditional objects to be thrown into perspective with maritime objects (Philips 17862). In the last of his books on perspective, Philips took up the theme of choosing parameters for a perspective drawing (Philips 1788), perhaps inspired by the work presented in the next section.

9. Jelgerhuis and the Choice of Parameters

363

FIGURE VII.66. Philips showing reflections in perspective. Philips 1775, plate 9, Leiden University Library, 168 C 52.

VII.9

T

Jelgerhuis and the Choice of Parameters

he engraver and painter Rienk Jelgerhuis (1729–1806) became so absorbed with Philips’s Uitvoerig onderwijs that four years after the work appeared, he published a collection of “Precise comments” to it (Jelgerhuis 1769). In his preface Jelgerhuis initially showered Philips with praise, but then proceeded to list examples he believed to contain mistakes. These concern Philips’s treatment of the theory of vision and some of his choice of parameters resulting in visual angles that Jelgerhuis found

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unrealistic. Before describing Jelgerhuis’s view I will briefly present the mathematical problem linked to the choice of parameters and its relation to visual angles. I have previously published detailed calculations related to this problem (AndersenS 19871, 112–120), but here I confine myself to the main points of the problem.

The Parameters of a Picture

L

et us first assume (figure VII.67) that a rectangle RSTU with height a and width b has been chosen as the picture area, and moreover that the principal vanishing point P is centrally placed in the picture, which means it coincides with the point of intersection of the diagonals. Let the distance be d, z = EPOS, and the length of the diagonals SU and RT be t implying that t = a2+ b2 . From triangle POS I find that t . 2 tan z Since the visual angle 2f has to be of a reasonable size, these relations induce some restriction on the choice of the parameters. Let fmax and fmin be, respectively, half of the maximum and minimum allowed for the visual angel. The parameters should then be chosen so that d=

t t . #d# 2 tan zmax 2 tan zmin In the pre-1800 literature on perspective I have met various advice on the choice of parameters, but no explicit calculations. I assume that several authors made some calculations, but kept it to themselves. This view is supported by Edme Sébastien Jeaurat, who stated that he did not include his calculations because it would be impossible for his readers to understand them (Jeaurat 1750, 48). S T P φ O R U

FIGURE VII.67. The relation between the parameters of a perspective composition and the visual angle.

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Several authors, such as Guidobaldo, Marolois, Jean Dubreuil, and Lambert, claimed that the maximum visual angle is 90˚.29 In deciding on the minimum visual angle the authors differed, but generally chose an angle between 50˚ and 60˚. Marolois, for instance, settled for a visual angle that should not be much smaller than 60˚, arguing that otherwise it would not contain enough light rays to render the object visible (Marolois 1614, Maxime I). The English painter John Joshua Kirby let the minimal visual angle be 60˚ precisely (Kirby 1754, first book, 67). When the principal vanishing point is situated in the middle of the picture, as in figure VII.67, then 2fmin = 53˚, 13 and 2fmax = 90˚ give the neat relation 1 t ≤ d ≤ t. 2

(VII. 2)

FIGURE VII.68. The limit of a picture for a given distance. Jeaurat 1750, plate 16, figure 23. 29

Guidobaldo 1600, 8–9; Marolois 1614, Maxime II.; Dubreuil 16421, 15; Lambert 1759, § 70.

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When the principal vanishing point is not centrally placed, the relation between the parameters and the visual angle becomes more complicated. Presumably no one tried to perform exact calculations for this case, but rules of thumbs were developed, one of them being that the distance should not be smaller than the distance from the principal vanishing point to that vertex of the picture that is farthest away. For a central vanishing point, this in accordance with (VII.2). This rule was among others stated by Kirby (Kirby 1754, second book, 10). He, like later Lambert, also advised their readers generally to place the principal vanishing point on the vertical line of symmetry in the picture (ibid.; Lambert 1759, § 76). As for the maximum distance, I have found no mention of the limit in (VII.2) requiring that the distance be smaller than the length of the diagonal of the picture. Yet I have noticed another rule, formulated by Kirby, recommending readers “to take the distance, in general, equal to the utmost dimension of the picture, whether in breadth or height” (Kirby 1765, 92). He credited Andreas Pozzo with formulating this rule, which was indeed published by Pozzo in 1693 and later (for instance in Pozzo 1707 as accompanying text for figure 1).

FIGURE VII. 69. Jelgerhuis’s vision of the ideal choice of parameters. In the diagram G is the foot of an observer whose height is 6 feet, like the person at E. The picture is the square 1234 with a side length of 12 feet* and the distance is likewise 12 feet. In the square 1234, Jelgerhuis inscribed the circle I as the limit for visible things. Jelgerhuis 1769, part of figure 3.

*These are foreshortened, as the composition is drawn in perspective.

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Thus far I have discussed the situation in which the parameters a and b are chosen first and the distance d second. If d is chosen first, then there was a rule of thumb telling to keep the picture inside a circle that has its centre in the principal vanishing point and the distance as its radius. This rule is illustrated by Jeaurat in figure VII.68. The choice of parameters a, b, d, and the vertical position of the eye point of O is related to one’s selection of the motif to be thrown into perspective – an aspect I leave out here (but have discussed in AndersenS 19871, 114–115).

Jelgerhuis’s Choice

W

hile criticising Philips’s choice of parameters, Jelgerhuis presented what he considered the ideal choice. This involved placing the principal vanishing point on the vertical line of symmetry in the picture. He let the latter be a square, such as 1234 in figure VII.69. He then let the distance be equal to the side of the picture. While in my calculation I let the visual angle be defined by the diagonal of the picture, Jelgerhuis chose to let it be determined by the side of the square picture. His choice corresponds to setting the ideal visual angle equal to 53˚, 13 – the number mentioned earlier in connection with the minimum visual angle. In his picture Jelgerhuis inscribed a circle with the distance as its diameter and claimed that what lay outside this circle could not be seen clearly (Jelgerhuis 1769, text for figure 3).

VII.10

A

The Dutch Scene

ll in all, Dutch perspectivists were important in the history of their discipline. There existed a sufficient number of textbooks of good quality to support the subject inside the Northern Netherlands, a few of which were translated and made some impact abroad. In the history of the mathematical theory of perspective, the Dutch are prominently represented by Stevin and ’sGravesande. The former pointed to the core of Guidobaldo del Monte’s theory, while the latter carried the theory further in such an engaging way that his work caught Taylor’s attention, thereby providing inspiration for further development of the theory.

Chapter VIII Italy after Guidobaldo

VIII.1

Waning Interest

T

his chapter is relatively brief since, as stressed earlier, I have chosen to devote most of my attention to the episodes in the history of perspective that contributed to the development of a mathematical understanding of perspective procedures – and from this point of view there is not much to write about Italian perspective in the seventeenth and eighteenth centuries. Italy, which had completely dominated the field up until 1600, lost her leading position due to a drastic decline in the Italians’ interest in the mathematical aspect of perspective. As baroque painting evolved as a style, the influential painters seem to have turned away from perspective. Many of the most admired compositions had a dominant group of people in the foreground, whereas the background was rather vague. The challenge lay in depicting persons seen at close range, which requires experience and talent. Perspective schemes could theoretically be of help, but for the purpose in question they are too time-consuming to be a practical tool. Just recall the intricacies involved in Piero della Francesca’s constructing the perspective image of a single head (page 71). One might have expected some mathematicians to carry on Guidobaldo’s ideas, published in 1600, but I have found no traces of this. In the considered period, I am only aware of one Italian mathematician, namely Zanotti, who published on perspective, but that was as late as 1766, and his book was a follow-up on works by Guidobaldo’s non-Italian successors rather than on Guidobaldo’s own contributions. Guidobaldo’s basic ideas may have been presented in some Italian textbooks treating mathematics or geometry in general. My sources, however, do not mention such examples, and in relation to my present project an independent search into the relevant literature would involve a disproportionate amount of work.1 Undoubtedly, there would be many interesting stories to tell if a more detailed study of the context of Italian perspective were made, but such an

1

To find Italian books on perspective from the seventeenth and eighteenth century I have mainly relied on JonesS 1947 and VagnettiS 1979, who in turn have used the standard bibliographical works including RiccardiS 1870. 369

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investigation falls outside the scope of this book. Hence this chapter must content itself with merely enumerating a profusion of authors and their works, and so may seem tedious. I would nevertheless not want to leaving it out, because in surveying the history of the literature on perspective from Alberti to Monge I find it important to also document the absence of mathematical interest in the field. The circle that kept the Italian tradition of writing on perspective alive mainly consisted of men with links to architecture, in particular stage designers. In fact, such authors are responsible for some fourteen of the twenty or so publications presented in this chapter. The theatre perspectives are related to illusionistic paintings and trompe l’œils, sometimes called quadratura, which were still fashionable in seventeenth-century Italy. The painters creating these must have acquired some knowledge of perspective, but they themselves did not provide new textbooks – apart from one outstanding example written by one of the most eminent illusionistic painters, Andrea Pozzo. In presenting the literature I start with textbooks in which perspective is treated together with other matters. These I have divided into two groups, the distinction being whether they are books on architecture or not. I continue with textbooks entirely devoted to the practice of perspective, and then finally turn to the few books dealing with the theory of perspective.

VIII.2

Perspective in Textbooks on Architecture

Seventeenth-Century Authors: Barca and Viola-Zanini

T

he two architects Pietro Antonio Barca (1606–1639) and Giuseppe ViolaZanini (c. 1575–1631) each published a textbook on architecture in the 1620s. Barca’s Avvertimenti ... circa l’architettura (Lessons ... on architecture) was rather short and elementary – explaining such concepts as the triangle. He also devoted three pages to a description of a plan and elevation construction of a perspective image (Barca 1620, 25–27). In his Della architettura (On architecture), Viola-Zanini briefly outlined how to construct the image of a grid of squares using a distance point method (Viola-Zanini 1629, 29). Despite his brief treatment of the topic, he claimed to have dealt with its principles and foundation. He further remarked that some painters remained unacquainted with perspective because they did not find it a necessary foundation for their art (ibid., 30), a view that indeed supports one of my introductory observations.

Eighteenth-Century Authors: Amico, Vittone, Spampani, and Antonini

T

he Italian authors presented in this section spent more pages on perspective than those above, namely about twenty pages each, but the methods

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of construction they treated were the same. The Sicilian theologian and royal architect Giovanni Biago Amico (1684–1754) presented a plan and elevation construction in his comprehensive three-volume work, L’architetto pratico (The practical architect, 1726–1750). He treated perspective in a part he called Compendio della prospettiva pratica (Compendium on practical perspective), which shows signs of being inspired by Pozzo (Amico 1750). The architect Bernado Antonio Vittone (1702–1770), on the other hand, settled for a distance point construction in his general introduction to architecture for young students (Vittone 1760), a work in which he also included sections on geometry, arithmetic, and algebra. While still students, Giovanni Battista Spampani and Carlo Antonini (†1832) published a textbook on architecture containing a separate part on perspective, Prospettiva pratica di M. Giacomo Barozzi da Vignola (Practical perspective of Giacomo Barozzi da Vignola, Spampani & Antonini 1770). This part of the work consists of Vignola’s original text, edited and supplemented by the authors. Vignola himself had presented a plan and elevation method and a distance point method, and his commentators also confined themselves to these two methods. One of their additions was an introduction of the concept of a general vanishing point. This shows that at least some of Guidobaldo’s theory had reached Italian architects. The editors also included a section on perspectival shadows.

The Galli-Bibienas and Piranesi

S

tage design seems to have played an important role in keeping the interest in perspective compositions alive in Italy. Peruzzi had already painted theatre perspectives early in the cinquecento, but the demand for this art grew in the seventeenth century and persisted into the eighteenth century. Ferdinando Galli-Bibienas (1657–1743) founded a school of celebrated stage designers. He himself had been apprenticed to Giulio Troili, whom we will meet in the next section. Galli-Bibiena’s fame brought him to the court of Charles, who in 1711 became Emperor Charles VI. In 1711 Galli-Bibiena published a textbook on architecture, which was dedicated to Charles and presumably therefore very luxurious (F. Galli-Bibiena 1711). The text later appeared in a more affordable edition addressed to young students of architecture (F. Galli-Bibiena 1731). The work contains no theory of perspective, but the second edition was nevertheless republished posthumously under the title Direzioni della prospettiva teorica (Instructions in theoretical perspective, F. Galli-Bibiena 1753). In his first edition Galli-Bibiena devoted some forty pages to perspective, presenting a distance point construction as well as a plan and elevation construction. He also included the abbreviated version of the latter used to throw a symmetric object into perspective – the method we saw in connection with Cataneo and Sirigatti (pages 122–123). Naturally enough, Galli-Bibiena also included a section on theatre perspective (F. Galli-Bibiena 1711, 129ff). At the

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beginning of his book, Galli-Bibiena listed some earlier perspectivists, among them his master Troili, Sirigatti, Dürer, Vignola, Accolti, Niceron, and Chiaramonti – whom we will meet shortly. Galli-Bibiena passed on his craft to his son Giuseppe (1696–1756), who also made quite a career. He published a book, entitled Architettura & prospettiva (Architecture and perspective, 1740), and like his father dedicated his work to Charles, who by then had become emperor. The publication consists of impressive perspective drawings with very little text, and no explanation of how the drawings are made. It would be interesting to know more about Giuseppe’s theoretical background, for among other things he seems to have been well versed in how to throw oblique lines into perspective. Another Italian involved with stage design was Giovanni Battista Piranesi (1720–1778), an influential theorist and all-around artisan, who was also an active architect, engraver, etcher, and designer. Like Giuseppe Galli-Bibiena, he also published a collection of perspective drawings without accompanying text (Piranesi 1750).

VIII.3

Perspective in Other Textbooks

Textbooks on Stage Design – Chiaramonti and Sabbatini

F

ollowing up on the theme from the previous section, let me begin by mentioning that the scientist Scipione Chiaramonti (1565–1652) composed a work entitled Delle scene, e teatri (On stages and theatres) in the early seventeenth century. The book was published posthumously in 1675, and according to G. Benzoni, this book includes a presentation of perspective that relates to Guidobaldo del Monte’s work on theatres in Perspectivae libri sex (BenzoniS 1980, 542). The first book that appeared on the theme of theatres and perspective was Pratica di fabricar scene e machine ne’ teatri (The practice of producing stages and machines for theatres, 1638). It was written by the architect and theorist Niccolò Sabbatini (1574–1654), who also designed stage sets. Sabine Eiche describes Sabbatini’s work as “a codification of the theatre practices of his time” (EicheS 1996). Sabbatini spent about fifty pages explaining how to construct a stage set (figure VIII.1), deciding on a principal vanishing point, a distance, and so forth. The work is a very detailed manual in which no attempts are made to explain to readers why they must follow the instructions given. Figure VIII.2 is an example of Sabbatini demonstrating how to divide a wall on the stage set perspectively.

A Textbook on Useful Matters for Painters – Zaccolini

A

round 1600 Matteo Zaccolini (1574–1630), a lay-brother of the Theatine Order, studied optics and perspective with Chiaramonti. About twenty

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FIGURE VIII.1. Sabbattini on how to light a stage set. In his discussion of this theme he opted for the solution shown here, that is, for light coming in from one side. Sabbatini 1638, 23.

FIGURE VIII.2. A wall of a stage. To paint architectural elements on the stage sets, Sabbattini recommended first making a perspective grid. Sabbatini 1638, 31.

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years later he wrote his own work in four parts. This work was never printed,2 but it nevertheless seems to have been quite well known and widely copied – by Nicolas Poussin, among others (BellS 19962). Its parts are called De colori, Prospettiva del colore, Prospettiva lineale, and Della descrittione dell’ Ombre (On colours, Colour perspective, Linear perspective, and On tracing shadow, KempS 1990, note 156, 352). I have only seen a copy of the third part, which treats perspective projections. It mainly deals with optical matters such as the theory of apparent sizes and refraction, but does also contain a section presenting a distance point construction (for a further description of Zaccolini’s work, see ibid., 132–134).

A Textbook on the Theory of Vision – Diano

T

he mathematician and theologian Ferdinando di Diano, also known as Polienus, published a book with the enigmatic title L’occhio errante dalla ragione emendate, prospettiva (The wandering eye for emending reason, perspective, Diano 1628). Diano used prospettiva in the broad sense including the theory of vision. In a rather long-winded style he commented upon the works of Dürer, Benedetti, Sirigatti, and Accolti, among others. Although Diano devoted almost thirty pages to perspective constructions, he did little more than discuss how a square is thrown into perspective.

A Textbook on Mixed Mathematics – Bettini

M

ario Bettini (1582–1657), a Jesuit professor of mathematics in Parma, published a general textbook on mixed mathematics with the title Apiaria universae philosophiae mathematicae (Beehives of universal mathematical philosophy). He devoted one apiary to what he claimed was optics and scenography (Bettini 1642), though in reality it dealt with anamorphoses and enlarging drawings with a pantograph. This last theme will be taken up in connection with Troili in section VIII.4.

A Textbook on Mathematics – Guarini

T

he famous baroque architect Guarino Guarini (1624–1683) was, like Zaccolini, a member of the Theatine Order and for a period taught mathematics and philosophy at a couple of the order’s schools. Perhaps it was this experience, combined with an interest in astronomy, that moved him to compose his Euclides adauctus et methodicus mathematicaque universalis (The

2

Biblioteca Medicea-Laurenziana, Florence, has a copy of the entire manuscript (Ms. Ashburnham 1212). For Zaccolini and the Leonardo da Vinci tradition, see Pedretti in Leonardo 1977, vol. 1, 36–43.

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methodical and augmented Euclid and universal mathematics). In one section Guarini dealt with stereographic projections, and in this connection touched upon some problems concerning perspective (Guarini 1671).

VIII.4

The Prospettiva Pratica Tradition

V

ignola’s work on perspective was called Le due regole della prospettiva pratica, as mentioned earlier. This choice seems to have led to the coining of the term prospettiva pratica, which was taken up by several seventeenth-century and eighteenth-century Italian authors. Outside Italy we find several titles, as noted in section VII.2, alluding to “the practice and the theory of perspective”. By contrast, the Italian books in the prospettiva pratica genre do not include any theory of perspective – Zanotti’s work being an exception (cf. section VIII.7). In general, a prospettiva pratica would contain a presentation of some geometrical figures relevant for perspective, one or more methods of making perspective constructions, a number of examples, and often a section on perspective instruments – there being no intention of providing a geometrical understanding of the topics treated. The favourite objects for perspective exercises were polygons, circles, some polyhedra (not least the regular polyhedra), crosses, columns, arches, vaults, and simple rooms with a few windows; frequently a section on scenography in the sense of constructing perspectival theatre stages was also included. There is at least one prospettiva pratica to which I have been unable to gain access. It was written by the architect, painter, and professor of art Baldassare Orsini (1732–1810) and was published in 1773 (Orsini 1773).

Cigoli

O

ne of the earliest seventeenth-century books in the prospettiva pratica tradition was written by the painter Lodovico Cardi, called il Cigoli (1559–1613), who was a favoured painter at the Medici court in the 1590s (Profumo in Cigoli 1992, 9). In the history of science Cigoli is known for his friendship and correspondence with Galileo (PanofskyS 1954, 5; ChappellS 1996, 312). Although plans for issuing Prospettiva pratica were made shortly after Cigoli’s death, it only appeared in 1992 (Cigoli Pros). Had Cigoli’s treatise been published in the beginning of the seventeenth century it would have been one of the most interesting textbooks for practitioners at the time.3 Cigoli divided his Prospettiva pratica in two books of which the first dealt with geometry needed for perspective. In this book Cigoli began by presenting very elementary

3

For another presentation of Cigoli’s work, see KempS 1990, 97–98 and 177–180.

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geometrical concepts and ended with a much more thorough introduction to the plan and elevation techniques than perspectivists normally gave. In his second book Cigoli turned to perspective proper and introduced three “rules” – an expression he doubtless took over from Vignola. In fact Cigoli’s two first “rules” are the same as those presented by Vignola, that is, a plan and elevation construction and a distance point construction. In treating the plan and elevation method, Cigoli included the shortcut in constructing the images of symmetric figures earlier applied by Cataneo and Sirrigatti (pages 122–123). Presumably inspired by Guidobaldo, Cigoli began his presentation of a distance point construction by introducing the general concept of a vanishing point for any horizontal line – that is not parallel to the horizon (Cigoli, Pros, 83). Many of Cigoli’s examples are also inspired by Vignola, among them a perspective cube (figure VIII.3) which is very similar to Vignola’s reproduced as figure IV.21, but again Cigoli had added something – in this case the distance circle (defined page 484). Whereas Cigoli’s two first “rules” are clear it is difficult to see what he meant by his third “rule”. In the section presenting it he first reminded his readers about the definition of a vanishing point (for a vertical picture plane and a horizontal line) and then introduced a principal vanishing point for oblique picture planes. In the surviving manuscript, however, he apparently did not use the latter concept turning to the use of an instrument for determining vanishing points in vertical picture planes and to methods for designing stage sets. The theme of perspective machines really seem to have fascinated Cigoli, thus he described not less than five different instruments (KempS 1990, 177–180).

FIGURE VIII.3. Cigoli’s perspective cube. (Cigoli Pros, 93.)

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Contino

T

he first Prospettiva pratica published in the seventeenth century appeared in 1645, but it must have been written much earlier, for its author, the architect, sculptor, and etcher Bernardino Contino, died in the late sixteenth century. He presented a distance point method, but exactly how he meant it to be performed is unclear from his text. Phillip Jones interpreted him as having made the same mistake as Serlio (JonesS 1947, 181). Since Contino’s book was obviously not published because of its lucid explanations, its appeal may have lay in its drawings – which are quite excellent. Contino’s work was actually found so attractive that it was republished in 1684.

Accolti

T

he first book to be both written and published in the seventeenth century, and including prospettiva pratica in its title, came from the architect, drawer, and painter Pietro Accolti (1579–1642). He called his work Lo inganno degl’occhi, prospettiva pratica (Deception of the eyes, practical perspective, 1625), and claimed that he had written it on request from the Accademia del Disegno (Academy of drawing) in Florence (ibid., preface) – an institution founded in 1562. The contents of Accolti’s prospettiva pratica fit my general characterization of the genre, although the work is more erudite than most, and full of references to classical writers.4

FIGURE VIII.4. One of Accolti’s constructions. The point H is the principal vanishing point, and E represents the eye. Accolti did not disclose how he had chosen E on the transversal through H. If he made HE equal to the distance, then the diagram shows a correct distance point construction but if BF is equal to the distance an incorrect one is presented. Accolti 1625, 17. 4

For other descriptions of Accolti’s work, see KempS 1990, 134–136 and BellS 19961.

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FIGURE VIII.5. Another construction by Accolti. In this diagram he let the line HL represent the picture plane in profile. If HE is equal to the distance, the diagram shows a correct Alberti construction of a square on AE (see the caption of figure IV.36). The square on BL is a bit misleading: the only thing he needed to do was draw BL = AE. Acccolti 1625, 19.

As the title Lo inganno degl’occhi indicates, Accolti treated visual perception as well as perspective. He referred, among others, to Witelo, Aguilon, Danti, Benedetti, and Guidobaldo del Monte, but his acquaintance with works by these authors does not seem to have helped him in gaining an understanding of perspective constructions. The first method he described is unclear (figure VIII.4) and may actually, like Contino’s construction, have

FIGURE VIII.6. Accolti’s anamorphosis of an ear. Accolti 1625, 49.

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FIGURE VIII.7. Torricelli throwing a floor pattern into perspective. Torricelli Pros. Biblioteca Nazionale Centrale di Firenze, mss. Gal.134.f. 5bis v.

fallen victim to Serlio’s mistake. (For a certain misunderstanding, by Accolti, see FieldS 1997, 30.) Accolti also presented a presumably correct Alberti construction (figure VIII.5), and besides he, like many of his contemporaries, engaged in anamorphoses (figure VIII.6).

Torricelli

H

ad you been planning to teach Italian artists perspective in the mid-1640s and surveyed the books available (cf. appendix four), you may well have come to the conclusion that none of them were suitable. The very gifted scientist and mathematician Evangelista Torricelli (1608–1647), known first and foremost for his barometric experiments, may have found himself in precisely this situation. In 1641 he moved to Florence to stay with Galileo, and the following year, after Galileo’s death, Torricelli took over his position as a mathematician and philosopher for the Grand Duke of Tuscany. Apparently the plan was that Torricelli would teach perspective at the Accademia del Disegno, and he may even have done so for a while. Actually it seems that Galileo himself, who was consulted in 1593 about perspective by Guidobaldo (page 238), had applied for a teaching position in geometry and perspective at the accademy in 1588 (EdgertonS 1991, 224). As for Torricelli, he started writing a prospettiva pratica designed as a dialogue in twenty parts between two characters called Alessio and Conti (Torricelli Pros). After some small talk about Conti having gone walking in the rain in his summer clothes, Conti asks why one uses the term prospettiva pratica, and he gets the following reply. Because with the aid of rules and methods one can trace any object on a surface, a canvas, or a wall, without knowing anything else than the principle which, because it does what it does, is used with confidence by practitioners. They do not wish to become acquainted with the further theoretical foundation, the principle being learnt easily and in a short time, and the foundation being difficult and time-consuming. The principle

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FIGURE VIII.8. Torricelli sketching how people should be drawn in perspective according to their positions. Torricelli Pros. Biblioteca Nazionale Centrale di Firenze, mss. Gal.134.f. 5bis v.

investigates drawing alone, whereas the foundation, besides drawing, investigates mathematics – a long and limitless study, which is fatiguing to convey to the memory.5

It is interesting that one of the leading European geometers of the time should decide to present perspective without the underlying geometry. He is the only mathematician among the authors I deal with who did this. Torricelli’s attitude reflects the fact that the only book on perspective, he left, was non-mathematical, namely Sirigatti’s La practica di prospettiva (Torricelli Opere, vol. 4, 103). All in all, not much can be said about his plans, because he did not carry his project very far. The manuscript contains 11 pages of text, covering the first lecture, and twenty-two drawings, some of which are for the later sections. His first lecture mainly deals with orthogonal projections upon a

5

Ò perchè per mezzo di regole e modi si descrive in superficie, o carta tela, o muro qual si voglia oggetto, senza sapere altro, che la regola, che, perchè la fa quello che fà, è usata con fede dalli Pratici, senza volerne notizia di altro fondamento di Teorica, essendo quella facile e breve a impararsi, e questa difficile e lunga; quella ricerca il solo disegno, questa oltre al disegno le Matematiche, studio lungo e senza fine fastidiosi a mandarsi a memoria. [Torricelli Pros/Opere, vol. 2, 314]

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FIGURE VIII.9. Torricelli showing a tilted cube in perspective. Torricelli Pros. Biblioteca Nazionale Centrale di Firenze, mss. Gal.134.f. 5bis v.

plan, and is presumably meant as a preparation for throwing solid figures into perspective. Among his drawings is a rather traditional floor (figure VIII.7), along with a scheme for deciding on the perspectival heights (figure VIII.8), and a tilted cube in perspective (figure VIII.9) – a theme earlier treated by Piero della Francesca (figure II.45). We do not know how interested Torricelli was in perspective, but we can say not so much that the topic took a prominent place in his correspondence: In October 1643 he wrote to Jean François Niceron, who five years earlier had published a well-received book on perspective, but the letter says nothing of perspective (Torricelli Opere, vol. 3, 146). Niceron had incidentally spent a few years in Rome at a convent with the Order of the Minims at Trinità dei Monti, leaving a trace in the form of an anamorphosis – to which I will return in section IX.8.

Troili

G

iulio Troili (1623–1685), also known as il Paradosso, was one of the few seventeenth-century Italian painters to publish on perspective. He settled in Bologna around 1650 and specialized in perspective paintings (SouthornS 1996). Around that time he edited a booklet on the pantograph (Scheiner 1653). This instrument had been invented by the German astronomer Christoph Scheiner for copying a plane figure in a different scale, but Scheiner also suggested applying it to draw in perspective (figure VIII.10). Troili illustrated how the pantograph can be used to enlarge a picture (figure VIII.11). In his example he chose a portrait; I assume, however, that the instrument was often used

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to transfer a perspective composition constructed on a piece of paper to a larger canvas, although I have not come across any descriptions of this procedure. Troili was especially interested in using the pantograph for making perspective compositions, which he did in a slightly different way than Scheiner (figure VIII.12). Thirty years after his edition of the book on the pantograph had appeared, Troili published a prospettiva pratica bearing the unusual title Paradossi per pratticare la prospettiva senza saperla (Paradoxes of practising perspective without knowing it, 1683). It falls into three parts, the last of which is called Paradossi overo fiori e frutti di la prospettiva pratica (Paradoxes or the flowers and fruits of the practice of perspective). Troili based his constructions

FIGURE VIII.10. Scheiner’s idea of using a pantograph to draw in perspective. A pantograph has two pins, at least one of which is also a pen. Usually one of the pins is manipulated so that it traces the lines of an existing diagram, which the drawing pen then reproduces it in a different scale – and on a separate piece of paper – as shown in figure VIII.11. Here, Scheiner considered a situation in which K is the eye point, the plane defined by VXNO the picture plane, and an object including the points Y, Z, a, b, and g is to be thrown into perspective. He let the tracing pen at P follow the contours of the object as seen in the picture plane. The drawing pen simultaneously reproduces these contours in the picture plane, but not so that the image of a point is the intersection of the picture plane and the line connecting the point and the eye point. Hence, if K is kept as the eye point, the finished picture must be moved a certain distance in a certain direction, depending on the parameters of the pantograph. Scheiner 1631, figure 40.

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FIGURE VIII.11. Troili’s demonstration of a pantograph in use. Scheiner 1653, figure 1.

on a distance point method, and his work is generally typical for the practical tradition, but more elegantly illustrated than most, as shown in figure VIII.13.

Amato

L

ike Amico, Paolo Amato (1634–1714) was a Sicilian ecclesiastic and architect. In 1701 he began writing a textbook on perspective (HillsS 1996), but it only appeared posthumously thirty-five years later under the title La nuova pratica di prospettiva (The new practice of perspective, 1736). Amato was well acquainted with the earlier literature, referring to a dozen of his Italian predecessors – including Guidobaldo – and also to a considerable number of foreign perspectivists, among them Dürer, Cousin, de Caus, Marolois, Niceron, Dechales, and Lamy (Amato 1736, 4–10). Unfortunately, the only copy of Amato’s book I have been able to peruse had lost its drawings. I am therefore unable to really describe its contents. My impression is that the work is a rather traditional prospettiva pratica, though there are signs that Amato had some new ideas bearing in the direction of what later became descriptive geometry. Any elaboration on this point would, however, require an examination of his figures.

Quadri

T

he last prospettiva pratica to be mentioned here is La prospettiva pratica delineata in tavola a norma della secondo regola di Giacomo Barozzi da

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FIGURE VIII.12. Troili’s idea of using a pantograph to draw in perspective. Contrary to Scheiner (figure VIII.10), Troili let the tracing pin also serve as the pen that draws. The putto moves the pen so that the eye point, a point in the object, and the other pin are collinear. For a large picture this process could require a longer arm than the putto actually has. Scheiner 1653, figure 2.

4. The Prospettiva Pratica Tradition

385

FIGURE VIII.13. Troili illustrating the three principal positions of the horizon. In the upper diagram the drawer was situated on the same level as the persons depicted, in the middle diagram situated under them, obtaining a worm’s-eye view, and in the lower diagram situated above them, which gives a bird’s-eye view. Troili 1683, 115.

Vignola (Practical perspective for delineating in a painting according to the second rule of Giacomo Barozzi da Vignola). It was published in 1744 by the architect and engraver Giovanni Lodovico Quadri, who had learned perspective from Francesco Galli-Bibiena, a brother of the above-mentioned Ferdinando Galli-Bibiena (Quadri 1744, 123).

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After a short presentation of a plan and elevation construction, Quadri concentrated, as his title indicates, on the second rule of Vignola, that is, a distance point construction. He applied this to throw a multitude of polyhedra into perspective – including the Platonic solids, and some of the truncated versions of them – after which he turned to architectural themes.

VIII.5

Pozzo’s Influential Textbook

A

ndrea Pozzo (1642–1709) was a Jesuit Brother who created many architectural designs and perspectival paintings for the Jesuit Order in the time of the Counter-Reformation. He was also interested in stage design. In much of his work perspective was a central element, and he decided to share his experience in the field with other architects and painters by writing Perspectiva pictorum et architectorum (Painters’ and architects’ perspective – for more on this work see MarryS 1998 and MarryS 2002). The first volume of this work appeared in 1693 and the second in 1700, both with parallel Latin and Italian texts. The first volume immediately generated widespread attention and was published with English and Latin texts as early as 1707. The book later appeared in more than thirty editions, and in at least nine languages (KerberS 1971, 267–270). There are even two Chinese editions from the period when the Jesuits brought western science to China. Although Pozzo’s work probably had limited influence there, it otherwise became a classical textbook for students of architecture. In his Perspectiva Pozzo used a style that resembles that of a prospettiva pratica, for instance in its choice of elements to be thrown into perspective. However, his explanations were less detailed than was customary, while he included many more – and far more impressive – illustrations than usual (figure VIII.14). As for Pozzo’s rather brief style, his English translator, John James, wrote ... the Brevity or Silence of our Author [Pozzo], (who, writing in a Country where the Principles of this Art are more generally known than with Us, had no need to insists so long on some things, as may be thought necessary to Beginners) ... [Pozzo 1693, 8]

If this remark is more than an excuse for the text being difficult to understand, it rests on an overestimation of Italian readers’ knowledge of perspective. Evidently Pozzo himself became aware that some more text and an introduction to the basic principles were needed, because he added some of this in his second volume.

Pozzo’s Methods

P

ozzo chose, as his fundamental method for throwing plane objects into perspective, a distance point construction, which in principle is a pointwise construction, although he illustrated it for polygons (figure VIII.15). He

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FIGURE VIII.14. A perspective composition by Pozzo. He described it as a “Theater representing the Marriage of Cana in Galilee, erected in the Jesuits Church at Rome, in the Year 1685; for the Solemnity of exposing the Holy Sacrament”. Pozzo 1707, figure 71 and accompanying text.

decided on a special procedure, rather than one in which the figure is placed under the ground line (as in figure IX. 34), and thereby avoided the problem of reversing (cf. section VII.6). Whether he himself was aware of this fact I cannot tell. As regards his method, he wrote that he had decided to present “this common and easy rule”, continuing:

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VIII. Italy after Guidobaldo

D A C E

E

O

B

C

D

F

G D

C

B

A

FIGURE VIII.15. Perspectival rectangles by Pozzo. He applied a distance point construction in which O is the principal vanishing point and the two points E are the distance points. At left he constructed the image of a rectangle with BC as the length of the transversal sides and CD as the length of the orthogonal sides. Similarly, the rectangle he threw into perspective at right has sides equal to CB and CD. The folded plans placed at the ground line function solely to mark the distances CD, for which a compass could also have been used. Pozzo 1707, figure 3 with new letters.

But if it please God to give me Life and Health to compose another Book, I shall then shew the Method of putting Works into Perspective ...6 [Pozzo 1707, 12]

As already revealed, it did please the Lord to let Pozzo write the second volume and present the “other method” he had mentioned, which turned out to be a plan and elevation construction. In his first volume Pozzo applied a technique for adding the third dimension that I have not seen in any of the books by earlier authors, namely one that involves the perspective image of an elevation. In chapter X we shall see that Brook Taylor also occasionally used this method (page 524). In figure VIII.16 I have reproduced one of Pozzo’s easy problems concerned with drawing a three-dimensional object in perspective.

Pozzo’s Virtual Dome

P

ozzo also offered far more complicated examples. One of these became well known, namely the problem of throwing the inside of a dome into perspec-

6

Deinde, si tempus & vires ad aliud Opus conscribendum Bonitas Divina dederit, projectiones quascunque absolvemus regulâ ... [Pozzo 1693/1707, 12]

5. Pozzo’s Influential Textbook

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FIGURE VIII.16. Pozzo’s method for throwing boxes into perspective. At right he showed the plan and elevation of the boxes. In the top drawing he constructed a number of perspectives of the plan and of the elevation, and in the bottom drawing he completed the perspective boxes by composing the perspective plans and elevations. Pozzo 1707, figure 6.

tive on a plane surface (Pozzo 1693/1707, 197). His motif resembles his design for the virtual cupola with a lantern on a flat ceiling in the Sant’Ignazio church in Rome, completed in 1685 (KerberS 1971, 55). Figure VIII.17 shows the painting located in the church, and figure VIII.18 Pozzo’s drawing from Perspectiva. To explain his construction, he added the diagram reproduced in figure VIII.19. In the caption of this figure I have presented the construction, including the use of the perspective elevation. In this context I would like to comment upon Pozzo’s surprising choice of the point O in the diagram as the principal vanishing point. Initially it might seem natural to place this point near the centre I of the outer circle so that spectators standing under the centre of the virtual dome would be deceived into thinking they were under a real dome. Pozzo, however, sought to give viewers another impression, which is indeed far more spectacular. Thus, Pozzo’s painted dome looks real when one is walking towards it, whereas seen from other locations than near the eye point, it has an anamorphic effect.

A Vault As Picture Plane

N

ext to Pozzo’s trompe l’œil dome in the Sant’Ignazio church there is a (truly) cylindrical nave vault that he also played with by painting a most impressive perspective composition in honour of Saint Ignatius (figure VIII.20). Having made the flat surface at the crossing of the church look

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FIGURE VIII.17. Pozzo’s virtual dome in Sant’Ignazio, Rome 1685. It is painted on a canvas with a diameter of 18 metres (BöselS 1996, 414), and still amazes visitors to the church.

curved, he chose to ignore the curvature of the vault in the sense that he drew the images of straight lines on it, and having set an eccentric eye point for the dome, he chose one centrally situated for his painting on the vault. While Pozzo performed geometrical constructions of the images of certain elements in his virtual dome, he used another technique for the vault, drawing his composition in perspective on a plane surface, equipping this with a grid of squares (figure VIII.21), projecting the vertices of this grid from the eye point upon the semi-cylindrical surface, and then connecting the images of the vertices to a grid. He wrote that theoretically a lamp placed at the eye point could be used to project the grid, but that in practice the light was not strong enough to provide distinct images, so that threads had to be used instead for projecting the vertices (Pozzo 1693/1707, 214). The next chapter recounts how Abraham Bosse had previously presented the method of projecting a grid, which Pozzo described (figure IX.45). Because Pozzo never revealed his sources, it is impossible to say where he got the method. It may have been a technique various authors independently took over from workshops, because it seems to have been the only functional

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391

FIGURE VIII.18. Pozzo’s illustration of his virtual dome. His comments included the following. Some Architects dislik’d my setting the advanc’d Columns upon Corbels, as being a thing not practis’d in solid Structures; but a certain Painter, a Friend of mine, remov’d all their Scruples, by answering for me, That if at any time the Corbels should be so much surcharg’d with the Weight of the Columns, as to endanger their Fall, he was ready to repair the Damage at his own Cost. Pozzo 1707, figure 91 and accompanying text.

method. An alternative method would have been to construct the perspectival grid directly on the picture surface. This was sometimes done in connection with anamorphoses, as we shall see, but I am not aware of any such construction when the picture plane was a vault – which is understandable because this would mathematically be quite demanding. In Pozzo’s case, the difficulty is that the images of the lines, parallel to the diameters of the semicircles at the ends of the semicylinder, are different elliptical arcs that cannot be determined in any straightforward manner. Although Pozzo’s technique for producing perspectival grids on vaults is the one most often described and seems the most natural, there was a description of another method that we

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VIII. Italy after Guidobaldo O

C

D

L

N Q

3

S

4 H

M

I P 2

T

R

Y

1 F

B

E

G

Z

A

FIGURE VIII.19. Pozzo’s construction of the perspective image of a cupola topped by a lantern. His picture plane, the surface of a flat ceiling, is horizontal. The point I is the centre of two outer circles that are supposed to lie in the ceiling, hence they are their own images. The points O and D are, respectively, the principal vanishing point and the right distance point. At right in the diagram is a perspective elevation of half the cupola and lantern (as Pozzo assumed the cupola to be symmetric, making one half suffice). The elevation itself (which Pozzo did not include in his diagram) lies in the vertical plane through AD (the picture plane being horizontal) and is bounded by the two vertical lines through A and H, whose perspective images are AO and HO. Pozzo determined the image Y in the perspectival elevation of a point lying on AD at the distance a from AD by applying a distance point construction as follows. On the line AB he marked the point Z so that AZ = a, and then constructed Y as the point of intersection of AO and ZD. Reversing Pozzo’s procedure, we can conclude that the height of the imagined dome is about 1.6 times its diameter at the bottom (18 metres), that is 28.8 metres. Similarly, a comparison of the height and the distance gives a ratio of about 1.4. Pozzo built up his image from images of horizontal sections in the virtual dome. Since these sections are circles parallel to the picture plane, they are depicted as circles, so Pozzo’s only problem was to determine the images of their centres and radii. For this he applied the perspective elevation which immediately gave him the radii of the image circles. For instance, the image of the circle represented in the perspective elevation by LM has a radius equal to LM. To find the centre of this image circle, Pozzo drew the line through L parallel to the ground line AB intersecting the line OI in N – this point being the centre he was looking for. In his drawing he only included the part of the circle that can be seen. Pozzo 1707, figure 90, with the letters Y and Z added and the original letters enlarged.

FIGURE VIII.20. Pozzo’s illusionistic painting, The Apotheosis of St. Ignatius, on the nave vault of Sant’Ignazio in Rome, c. 1690–1694.

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G F

E

H A

L

B

N

O

O

O

FIGURE VIII.21. Pozzo’s demonstration of how to map a flat grid of squares onto a cylindrical vault. At left, the principle of the projection is shown in perspective: The eye point is O and the horizontal grid of squares N is projected from O upon the vault, the projections being illustrated for two points. Pozzo 1707, part of figure 10 with enlarged letters.

shall see in connection with Kirby – actually a method that is as complicated as the one of constructing the perspective grid directly (pages 556–557).

VIII.6

T

A Special Approach to Perspective – Costa

he Venetian architect, stage designer, painter, and etcher Giovanni Francesco Costa (†1775) published a book in 1747 bearing the rather traditional title Elementi di prospettiva per uso degli architetti e pittori (Elements of perspective for the use of architects and painters). Its examples were also fairly conventional, whereas its method of construction was not. In fact, Costa modified the plan and elevation construction in a way I have noticed nowhere but in his work. He did not refer to other authors, and he did not argue the cause for his modification. His underlying idea seems to be that in adjusting perspective images to the theory of appearances it would be more natural to make a central projection upon a sphere than upon a plane – an idea we have also seen in connection with Leonardo da Vinci (pages 87 and 104). However, if this was the case, it must have been academically too challenging for Costa to translate this idea into a mathematically consistent rule. Whatever his reason, Costa decided to introduce circular arcs to represent the picture plane in a plan as well as in an elevation. According to traditional plan and elevation methods (like the one shown in figure II.35), the picture plane is represented as one straight line, and if the paper is folded around the line that separates the plan and the elevation, the divided straight line becomes the ground line and a vertical line in the picture plane p (cf. figure II.34). It was these two parts of the line representing the picture plane that Costa replaced with two circular arcs.

395

6. A Special Approach to Perspective – Costa

X

D

3 1 4 2

E 3 1 2

C

4

3

2

B

4

1

1 2

1 2

A

3

3 4 4

FIGURE VIII.22. Costa’s special plan and elevation method. To compose the plan and the elevation of the cube, Costa first took the following step. He used the dividing line between the plan and elevation as a ground line and chose an arbitrary point 1 upon it as the orthogonal projection of two of the vertices in the perspective cube upon the ground line – thereby ignoring to some extent the relationship between the position of the eye point, the picture plane, and the vertices to be thrown into perspective. Having decided on the position of the point 1, he constructed the point 2 by making the distance 12 equal to the chord 12 in the lower part of the diagram. Analogously he constructed the points 3 and 4 on the ground line. His next step is explained in the caption of figure VIII.23. Costa 1747, table 12 with symbols changed.

To be more specific, let us look at his example of throwing a cube into perspective (figure VIII.22). As his planes for the plan and for the elevation, Costa chose a ground plane and the vertical plane so that the plan of the eye point, C, coincides with the foot of the eye point, and its elevation, D, coincides with the eye point itself. The figure A is the plan of the cube, and B is its elevation (the cube having no sides parallel to p). With centres in C and D, Costa drew two circular arcs with the same radius and let them represent the picture planes in the projections. He then drew lines from the vertices 1, 2, 3, and 4 in the plan of the cube to the point C, likewise marking the points of intersection of these and the arc as 1, 2, 3, and 4. Similarly he connected the vertices of the elevation with the point D and again used 1, 2, 3, and 4 to mark the two pairs of points of intersection with the upper arc. His final step was – as in a traditional plan and elevation construction – to compose the perspective image from the points of intersection. His procedure for doing so is explained in the captions of figures VIII.22 and VIII.23. Costa illustrated many more examples of applying the described method, one of which is shown in figure VIII.24.

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VIII. Italy after Guidobaldo 3i

1i

4i

3 1 4 2

E 1i

3 1 4 2

B

2i C

4

3

2

1

0

FIGURE VIII.23. Costa’s use of the elevation. To explain Costa’s procedure I have added some numbers, apostrophes, and indexes to his drawing. It is sufficient to look at how he constructed the images 1i′ and 1i′′ of two of the vertices of the cube, situated respectively at the base and top – as the rest of the images are constructed in a similar fashion. On the perpendicular to the ground line through the point 1 Costa constructed the line segment 11i′, equal to the chord 01′, and the line segment 11i′′, equal to the chord 01′′. Upper part of previous figure, with the symbols changed and some added.

FIGURE VIII.24. Costa applying his special method for drawing a dodecahedron in perspective. Costa 1747, table 17.

7. Mathematical Approaches to Perspective

VIII.7

397

Mathematical Approaches to Perspective

Zanotti

A

mong the Italian books on perspective published after 1600, only a single one treated the theory of perspective. This was Trattato teorico-pratico di prospettiva (Theoretical and practical treatise on perspective, 1766), a book written by Eustachio Zanotti (1709–1782), who was a member of several learned societies, a professor of astronomy, and the director of the observatory in Bologna. I have not found out how Zanotti became interested in perspective, but he had already treated the subject eleven years earlier in the paper “De perspectiva in theorema unum redacta” (Perspective reduced into one theorem) published in the journal of the Bologna Academy of Sciences (Zanotti 1755). This is one of the few occasions after the publication of Guidobaldo del Monte’s Perspectivae libri sex (1600) on which perspective was presented for a circle of academic readers. The theorem referred to in the title is actually not a theoretical statement, but a presentation of a perspective construction whose correctness Zanotti proved. The construction provides the perspective image in an oblique picture plane of a point given above a ground plane, and the proof is based on the division theorem. In his book Zanotti pursued the approach expounded in his paper, but spelled his ideas out much more clearly, because he wanted to reach a much broader audience – including practitioners of perspective (for another presentation of Zanotti’s work, see LoriaS 1908, 603–606). Zanotti was particularly concerned that this group not only learn the rules of perspective, but also understand the reasons for these rules. He repeated the often-stated complaint that practitioners did not care enough about the theory of perspective, and that theorists ignored the practice of it (Zanotti 1766, introduction). Zanotti also followed the tradition of not referring to fellow scientists who had written on perspective, but for reasons I will come back to, I am fairly convinced that his sources of inspiration included Brook Taylor’s New Principles of Linear Perspective, which had actually appeared in an Italian translation a decade before Zanotti published his book (Taylor 1755). Stylistically, however, Taylor’s New Principles and Zanotti’s Trattato are very different. Taylor structured his exposition like a classical textbook in mathematics with definitions, axioms, and theorems with proofs. In addition, Taylor treated a general situation in which the picture plane is supposed, from the very outset, to form any angle with the ground plane. Zanotti chose a more informal style in which he argued his conclusions, but without formulating them as traditional mathematical proofs, and he used several steps to reach a general situation. In this respect his form of presentation is similar to that of Johann Heinrich Lambert in his La perspective affranchie de l’embaras du plan géometral from 1759. Zanotti also treated many of the topics Lambert had dealt with, yet I still do not think Zanotti was directly influenced by Lambert,

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because I have not noticed any instances in Zanotti’s work where he chose a solution that resembles one given by Lambert.7 Zanotti’s first construction involved throwing a point in the ground plane into perspective on a vertical plane. For this he used the division theorem from which he elegantly deduced a distance point construction (Zanotti 1766, II, §§1–2). He naturally proved the division theorem by looking at similar triangles, and he used the same technique for proving how a vertical line segment is foreshortened (ibid., III, §1). Not until section four of Trattato teorico-pratico di prospettiva did Zanotti turn to more general theoretical considerations – such as the introduction of a general vanishing point, and how a set of parallel lines and a pencil of lines are depicted depending on their positions in relation to the picture plane and the eye point. In the rest of his book Zanotti treated perspectival shadows – which he treated in greater detail than most of his predecessors, images of curvilinear figures (not least circles), the Platonic solids, the use of a ceiling as a picture plane, stage design, inverse problems of perspective, and direct constructions in the picture plane. His examples included an elegant construction of the image of vertical circle that is not parallel to the picture plane (ibid., IX, §19). Most of the themes Zanotti dealt with, including direct constructions, were also treated by Taylor, although not in such detail. This is why I think Taylor’s New Principles was an important source of inspiration for Zanotti. It is also worth noting that Zanotti, like Taylor, used the expression ‘linear perspective’ that was somewhat unusual on the continent (prospettiva lineare, Zanotti 1766, I, §3). As indicated in chapter VII, and as argued further in section X.5, Taylor on his side got some ideas from ’sGravesande, so one cannot rule out the possibility that Zanotti saw some of the issues common to the works of the three scientists on perspective in ’sGravesande’s Essai de perspective rather than in Taylor’s New principles. Among these common issues, which are not simply standard fare in treatises on perspective, is the mentioning of the problem of reversing (cf. section VII.6), which can occur when the plan of an object is applied in the construction of a perspective image (Zanotti 1766, II, §3). All in all, Zanotti’s book is a well-turned and not overly difficult presentation of the basic theory and practice of perspective. It did not seem to exert any immediate influence on the way perspective was treated in textbooks in Italy, but there may have been some later effects.

Stellini

Z

anotti was not the only Italian who showed an interest in the mathematical theory of perspective in the eighteenth century. Giacopo Stellini (1699–1770), who is best known for his work on ethics and pedagogics and who became a professor of ethics in Padua in 1739, also studied the topic. According to Antonio Evangel, the editor of Stellini’s works, Stellini began

7

Gino Loria wondered whether Zanotti might have found inspiration in Lambert’s work on perspective, while Luigi Vagnetti believed he had done so (LoriaS 1908, note 1, 606; VagnettiS 1979, 457).

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399

to translate Taylor’s New Principles in the 1730s, and by 1754 he had initiated the process of having it printed in Venice. He gave up publishing the book, however, when another Italian translation appeared in 1755 (StelliniS 1782, vii). The latter (Taylor 1755) was the work of François Jacquier, a Frenchman living in Rome (for comments on this translation, see LoriaS 1908, 601–602). It is slightly surprising that a man with Stellini’s background would become engrossed in such a technical piece of geometry as Taylor’s New Principles. But Stellini apparently saw it as a pedagogical challenge to present Taylor in a lucid form, for according to Evangel, he aimed to explain all Taylor’s examples carefully (StelliniS 1782, viii). Nevertheless, Stellini did not get far in this enterprise and eventually decided in 1754 only to include a few notes in his translation which first appeared in volume three of his posthumous Opere varie in 1782. As with Zanotti’s work, the translations of Taylor’s work had no noticeable immediate impact in Italy.

Torelli

T

he last work to be mentioned in this chapter bears the Latin title Elementae perspectivae libri duo (Elements of perspective in two books, 1788) indicating a scholarly work on perspective. This work, which does not really fit into any section in this chapter, was written by Giuseppe Torelli and published posthumously. Torelli graduated in law, but later devoted his time to literature and science (LoriaS 1908, 617). The style of his book is very mathematical, with theorems and proofs, but it is extremely difficult to see the function of these theorems, many of which deal with pyramids and spheres, and occasionally with projections. In analysing Torelli’s book Noël Germinal Poudra concluded that it did not treat perspective as the discipline was commonly understood in the eighteenth century, and Gino Loria observed that one of Torelli’s aims was to prove perspective was known in antiquity (PoudraS 1864, 536; LoriaS 1908, 617).

VIII.8

A

The Later Italian Period

s stressed in the introduction to this chapter, as far as the development of the mathematical understanding of perspective is concerned, this chapter does not have much to offer. In fact, several of the authors, presented here, seem to have shared Torricelli’s view (page 380): that an understanding of the foundations of perspective was too demanding a study. During the seventeenth and eighteenth centuries less was published on perspective in the Italian countries than in the other large areas discussed in this book, and the knowledge of perspective was mainly transmitted in Italy by illusionists, among whom Pozzo was the most remarkable. In addition, many of the authors had – like Pozzo – taken holy orders. Despite the lack of Italian interest in publishing on the mathematical theory of perspective, the field did not fall completely into neglect, as the work by Zanotti and the two Italian translations of Taylor’s book show.

Chapter IX France and the Southern Netherlands after 1600

IX.1

The Early Modern French Publications

D

uring the Renaissance and the first decades of the seventeenth-century Italian scholars dominated the fields of pure and applied mathematics. In the 1630s French mathematicians became as active as their Italian colleagues, and in the following decades they took over the leadership, championing an immense growth in pure mathematics. The history of the mathematical theory of perspective generally leads us to other people than the main protagonists in the history of mathematics. Nevertheless, when it comes to writing on perspective, there was a contemporary shift from Italy to France. The seventeenth century had begun with Guidobaldo’s significant contribution to the theory of perspective. In Italy there was no response to his work, as we saw in the last chapter. In the Northern Netherlands, Stevin followed up on Guidobaldo’s ideas and gave rise to further development there, although this did not start until the 1660s. It was in the Southern Netherlands and, most particularly, in France that Guidobaldo’s and Stevin’s works first had any real impact. At the same time, several French perspectivists initiated a new approach to the discipline by investigating what forms the usual Euclidean constructions take in the picture plane. Guidobaldo’s work actually contains the earliest traces of this idea (page 261), which culminated with Johann Heinrich Lambert’s creation of a perspective geometry published in 1759. The two developments described were to a large extent separate. The presentations of Guidobaldo’s and Stevin’s ideas occurred in general textbooks on mathematics or optics, usually written in Latin by academics and addressed to other academics and aspiring scholars. The new approach, on the other hand, is found in books that exclusively dealt with perspective and were written in French for practitioners of perspective. The authors of these books were mathematicians and savants with a keen interest in mathematics. Among the seventeenth-century French publications, the most noticeable and influential were written by Niceron and Dubreuil, respectively. These two authors gave clear instructions on how to construct perspective and anamorphic 401

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IX. France and the Southern Netherlands after 1600

images applying the new mathematical understanding of perspective – without aiming at transmitting this understanding to their readers. The prefaces to a large number of the tracts on perspective contain remarks about how important the discipline was for painters and architects. Undoubtedly, Frenchmen belonging to these groups learned perspective, but not with such enthusiasm that they became inspired to compose works of their own on the subject. In this chapter I deal with more than forty titles, so needless to say, it is impossible to describe them all thoroughly. It is not even possible to present all the new methods of perspective constructions that occur in these publications. I devote most of my attention to the French development in the seventeenth century, because this was a very creative period in the history of perspective – and about three-quarters of the titles covered in this chapter are actually from the seventeenth century. The story of what happened in France in the eighteenth century is less exciting, and is therefore told in some brevity. I comment on all the titles of which I am aware, but do refrain from treating Claude Mydorge’s brief mention of perspective in his work on recreational mathematics (Mydorge 1638).

Perspective and Projective Geometry

I

n general, perspective does not receive much attention in the history of mathematics, but it is common to link the developments of perspective and of projective geometry in France in the seventeenth century. Projective geometry itself became a successful mathematical discipline in the nineteenth century. One of the main contributors to the theory was Jean Victor Poncelet, who developed his new approach to geometry while being held as a prisoner of war in Russia from late 1812 to the middle of 1814. He published his findings in Traité des propriétés projectives des figures (Treatise on projective properties of figures, PonceletS 1822), remarking that some of his ideas already occur in a work by Desargues, often abbreviated as Brouillon project (Rough draft, DesarguesS 1639).1 In the seventeenth century this work was not considered particularly remarkable, and was almost forgotten. With the revived interest in projective geometry, however, Desargues and his work became famous in the history of mathematics. Scholars realized that in 1636 Desargues had published a method of perspective constructions, and it was assumed that this had inspired him to take a new approach to geometry – the approach presented in Brouillon project. Some historians of mathematics have therefore tended to see Desargues’s achievement in projective geometry as one of the most important events in the history of perspective. It is against my historiographical attitude to regard the climaxes in the development of perspective as contributions to pure mathematics. What is more, I do not

1

PonceletS 1822, xxxviij–xl and §178.

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believe there was any strong link between Desargues’s work on perspective and projective geometry. To be able to argue for this point of view, I will treat Desargues’s perspective method in detail later in this chapter. Poncelet’s work on projective geometry was born out of a French geometry tradition whose roots partly go back to another important episode in the history of geometry, namely the creation of descriptive geometry in the eighteenth century. This theme is dealt with in chapter XIII.

IX.2

The Theory of Perspective Taught

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uidobaldo’s and Stevin’s contributions to perspective had the consequence that the subject came to be included among the mathematical disciplines, and was hence often treated in general books on mathematics. These books did not bring many novelties, yet they still play an important role in the history of the subject, in that they kept the knowledge of the new theory of perspective alive.

Aguilon and Mersenne

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he first presentation of perspective I discuss here did not occur in a textbook on mathematics, but in an extensive work on optics, Opticorum libri sex (Six books on optics, 1613). Its author was the Jesuit mathematician and physicist François Aguilon (1546–1617), who was from the Southern Netherlands. Aguilon had been involved in organizing the teaching of science there with special regard to its practical applications (MorèreS 1970). Since all theoretical presentations of perspective predating Stevin’s work had associated the discipline to the theory of optics, it is at first sight not surprising that Aguilon included a chapter on perspective in his opus on optics. A closer look shows, however, that his reason for writing this chapter is related less to optics than to the fact that he decided to devote the entire sixth book, covering more than 200 pages, to projections. Aguilon dealt with both parallel and central projections. He remarked that a parallel projection can be considered as a central projection having the eye located at an infinite distance (Aguilon 1613, 503). He liked to play with the variety of parallel projections, and showed how it is possible to have a cube projected as a regular hexagon (figure IX.1). He gave several examples of applications of projections, of which perspective was one, and the construction of sundials another. The circumstance that Aguilon grouped perspective representations together with other projections did not influence his approach to the subject. Like Commandino (page 140), he did not apply any general results concerning central projections, but treated perspective as a self-contained theory. Aguilon’s style was actually very similar to Guidobaldo’s – to whose work he also referred.

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FIGURE IX.1. Aguilon’s example of mapping a cube into a regular hexagon by means of a parallel projection. Aguilon 1613, 557.

While it is clear that Aguilon was inspired by Guidobaldo, it is hard to tell what role Aguilon’s own work played in the history of perspective. Partly due to sparse references, it is at all difficult to settle the question of the independence, or interdependence, of the various books in the Guidobaldo–Stevin tradition. It is, however, my impression that Aguilon’s work on perspective had very little impact, although it did inspire a few later authors. We find traces of Aguilon’s ideas in the work of Andreas Tacquet, who, like Aguilon, was a Jesuit from the Southern Netherlands, and as late as 1712 Aguilon’s influence is apparent in a British book written by Humphry Ditton (page 492). Regardless of their role in the history of science, Aguilon’s Opticorum libri sex later became known because Peter Paul Rubens was responsible for some of the illustrations in the books (figure IX.2). In their general works, the French mathematicians usually presented perspective and optics as different applications of geometry. However, one writer, the Minim scientist and distinguished organizer of scientific communication, Marin Mersenne (1588–1648), treated the two subjects together by including his brief presentation of perspective in a section devoted to optics (Mersenne 1644). As noted, Mersenne’s exposition mainly consists of references to Stevin’s work (page 289).

Hérigone

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ike several other seventeenth-century scholars, the French mathematician Pierre Hérigone († c. 1643) gave private tutorials (StrømholmS 1972). His teaching resulted in a general textbook entitled Cursus mathematicus (Mathematical course) featuring parallel texts in Latin and French. Hérigone’s course was the first of the books in this genre to give a thorough treatment of perspective (Hérigone 1637). Hérigone let his section on perspective be an integrated part of his work in the sense that in proving theorems on perspective, he applied results from earlier chapters. For his Cursus, he developed his own peculiar system of abbreviations and symbols (for instance using “2⁄2” for “equal to” and “3⁄2” for “larger than”), and following

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FIGURE IX.2. Peter Paul Rubens’s composition introducing the sixth book of Aguilon’s Opticorum libri sex, which deals with projections. Aguilon 1613, 452.

his presentation of perspective therefore requires some preparation. Readers familiar with his style were given a concise introduction to perspective – including proofs. As noted in section VI.11, Hérigone presented the foundation of perspective in terms of axioms whose formulations hint at inspiration from Stevin. To construct the image of a point in the ground plane, Hérigone chose a specific version of Stevin’s method, namely one that involves the principal vanishing point. Hérigone also presented a distance point construction and described the idea of performing it mechanically by using gliding squares and threads furnished with weights (Hérigone 1637, 201). The problems he treated included the one of determining the set of plane curves that are reflected as a set of squares in a cylindrical mirror with its base in the plane of the curves (Hérigone 1637, proposition 9). This question had been treated in detail seven years earlier by Vaulezard – as we shall see in the next section. Hérigone returned to perspective in a Supplementum to his Cursus (Hérigone 1642). This second treatment is by and large similar to his first, but contains some additions, for instance a reference to Desargues’s method, which was published in 1636. Hérigone skipped any presentation of the method, arguing that Desargues had not proved the correctness of his method, and that Desargues’s own publication still was available. The first reason presumably carried most weight, for it does require quite a bit of

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explanation to cover the entire foundation of Desargues’s method, as will be shown in section IX.5. Another addition in the Supplementum was a demonstration of how a sector can be applied in performing a perspective construction. Hérigone claimed that the technique had been invented three or four years earlier by a certain father Besson (Hérigone 1642, 110). The idea of using a sector for perspective constructions actually occurred earlier in France, as we shall see in section IX.2, but apparently Hérigone had no contact with the circle in which the idea first came up. Finally, Hérigone also discussed a special instrument for perspective constructions, but his description of the instrument is not clear (ibid., 114).

Bourdin, Dechales, and Tacquet

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he group of people spreading the knowledge of mathematics included many Jesuits. One of them, the Frenchman Pierre Bourdin, briefly touched upon perspective in his Cours de mathématique (Bourdin 1661). Another French Jesuit and mathematician, Claude François Milliet Dechales (1621–1678), was particularly well known for his ability to explain mathematics (Schaaf S 1971). He published his expositions in a comprehensive Cursus seu mundus mathematicus (Mathematical course or world). It contains some sixty pages on perspective (Dechales 1674), within which Dechales demonstrated a talent for selecting relevant material. He was acquainted with the works of the masters Guidobaldo and Stevin, and as mentioned in section VI.11, the latter inspired his presentation of the foundation of perspective while Dechales found his inspiration for illustrative examples from the practice of perspective somewhere else. In 1690 Dechales’s Cursus appeared posthumously in a second edition featuring an annotated list of mathematical publications (Dechales 1690, vol. 1, 1–108). He commented positively upon several presentations of perspective, including one written by the above-mentioned Jesuit Andreas Tacquet (1612–1660). Tacquet’s contribution on perspective is found in a section called Optica tribus libris exposita (Optics explained in three books, Tacquet 1669), which is a part of Tacquet’s posthumously published Opera mathematica. Dechales’s praise may have been more a result of his respect for the Jesuit Order than a reflection on the quality of Tacquet’s accomplishments – which compared with other contemporary introductions to perspective is not remarkable. Besides, it is often difficult to follow his arguments because he had not carefully checked that the letters used for points in his text correspond to those used in his figures. The most interesting aspect of Tacquet’s dealing with perspective is his view that a section on the subject should be included in his presentation of optics as an application of geometry.

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Rohault and Ozanam

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he natural philosopher Jacques Rohault (1620–1675), known as a leading advocate of Cartesianism, belonged, like Hérigone, to the group of French private teachers (SchusterS 1975). About 1670 he began planning an edition of his lectures on mathematics, which, however, only appeared posthumously in 1682. Rohault’s oeuvre included twenty five pages on perspective, which, not surprisingly, cover the same theorems and problems as earlier descriptions (Rohault 1682). Nevertheless, rather than being copied from earlier presentations, his survey appears to be a result of an independent analysis of the subject. I, at any rate, am unaware of any previous survey of the theory of perspective that resembles Rohault’s, whereas his own outline was later used by Ozanam. The French mathematician Jacques Ozanam (1640–c. 1717) reportedly liked teaching so much that he did it for free until his financial situation forced him to charge a fee (SchaafS 1974). His lectures resulted in a comprehensive Cours de mathématique (Course on mathematics), which contains a separate treatise covering more than hundred pages and entitled La perspective théorique et pratique (Theoretical and practical perspective, Ozanam 1693). In his preface, Ozanam expressed worries about artists tending to no longer consider perspective a part of their art. He found perspective important, and also believed it was essential to treat the subject in a mathematically correct way. Apparently he hoped to reach practitioners through his work, for his part on the practice of perspective includes examples belonging to their tradition. As indicated, Ozanam based his presentation of the geometrical foundation of perspective upon Rohault, whereas the rest of his work seems to take quite an independent approach to the subject. He treated some of the mathematicians’ favourite topics, such as the question of when the perspective image of a circle is a circle. He also included a few inverse problems of perspective, examples of direct constructions, and examples of throwing shadows into perspective.

Ozanam on Measure Points

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ne of the remarkable aspects of Ozanam’s treatise is his introduction of a concept which later was termed a measure point and which he called a centre diviseur (divider centre). This point is used for constructing the image of a given line segment lying on a horizontal line whose image is known. In section IX.4 we shall see that earlier in the seventeenth century, Aleaume and Migon also introduced a measure point. Ozanam’s introduction of this point was, however, different from theirs. Aleaume and Migon used an angle scale to determine a measure point whereas Ozanam constructed it geometrically. Since measure points came to play an important role in the later development of the theory of perspective, I present Ozanam’s introduction and construction of a measure point.

408 H

IX. France and the Southern Netherlands after 1600 Vl

Ml

P

Z

Ai

G B

R

ll

FIGURE IX.3. Ozanam’s introduction of a measure point.

Let O be the eye point, P the principal vanishing point (figure IX.3), and l a given horizontal line intersecting the ground line in Il and having the vanishing point Vl. To l Ozanam assigned a measure point, say Ml, lying on the horizon on the opposite side of P than Vl and satisfying MlVl = OVl.

(ix.1)

It is worth noticing that Ml is defined uniquely – except if l is an orthogonal. To construct the image Ai of a point A on l, given by the distance AIl, Ozanam cut off a line segment BIl equal to AIl on the ground line (B being to the left of Il if Ml is to the right of Vl, and vice versa), and then determined Ai as the point of intersection of BMl and IlVl (Ozanam 1693, 31). To construct the image of a line segment he applied this procedure twice.2 The construction Ozanam performed to ensure that Ml fulfils the relation (ix.1) is shown in figure IX.4 – whose caption also contain a comparison of various fashions of introducing a measure point.

Op

f 45− 1f 2

H 2

Vl

P

Ml

Z

FIGURE IX.4. An adaptation of Ozanam’s construction of a measure point Ml for a horizontal line l. The principal vanishing point is P, Op is the eye point turned into the picture plane p (and thus OpP is equal to the distance), and Vl is the vanishing point of the line l. Ozanam constructed Ml by making MlVl = OpVl. An easy calculation shows that when ∠POpVl = f (implying that Vl is the vanishing point of the horizontal lines making the angle of j with the normals to p), then ∠MlOpP = (45 – 1/2f). This property was used earlier by Aleaume and Migon (figure IX.16), and later by Lambert.

A more detailed presentation of how to use a measure point is given in the caption of figure IX.16.

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H p

Vl Z

O Ai

A G

l ll R

FIGURE IX.5. The general division theorem.

Although Ozanam did not prove the correctness of his construction, I will, and in so doing introduce a general form of the division theorem. The main theorem and a consideration of the similar triangles (figure IX.5) AiAIl and AiOVl show that the image Ai of A is the point on IlVl for which AiIl: AiVl = AIl : OVl.

(ix.2)

The point Ai that Ozanam constructed in figure IX.3 does indeed fulfil this requirement. Another remarkable aspect of Ozanam’s treatise is that it contains a description of a visual ray construction – as noted in connection with the presentation of ’sGravesande’s version in chapter VII (page 344).

The Encyclopedias

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he material presented in this section shows that in France during the last two-thirds of the seventeenth century, perspective was considered a natural example of applied geometry. The French mathematicians of the next century would not display the same interest for presenting perspective, whereas the topic was taken up by their German and British colleagues – as we shall see in later chapters. The eighteenth century was the age of encyclopaedias, many of which contained articles on perspective. Although these works played a role in spreading the knowledge of perspective projections, they were generally too brief to contribute to any profound understanding of the geometry behind perspective. The contents of these presentations will therefore not be included here.

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IX.3

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The Works of de Caus and Vaulezard

De Caus

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he first seventeenth-century book on perspective in French was printed in London in 1612 (figure IX.6) and written by the maître ingenieur Salomon de Caus (1576–1626). He served European dukes and princes with among other things creating fountains, gardens, and a picture gallery (ZimmermannS 1996). De Caus spent the early 1610s in England, where he came into contact with Henry, the Prince of Wales – who died in 1612 as Prince. De Caus taught perspective to the Prince and his lectures led to the above-mentioned book, which he called La perspective avec la raison des ombres et miroirs (Perspective with the reason for shadows and mirrors). This is neither the first nor the last example in the history of perspective of a book resulting from instruction given to a prince. As mentioned, Stevin’s innovative Van de verschaeuwing belongs to the same category, as does one of Kirby’s books (page 552). Besides La perspective, de Caus published a number of others books, and taken together, they can be seen – in the words of Jurgis Baltrusˇaitis – as “a vast treatise on the wonders of the world” (BaltrusˇaitisS 1977, 37). De Caus’s enthusiasm for the wonders of perspective was greater than his ability to write understandably on the subject (for another presentation of his work, see BessotS 1991). De Caus’s book is actually more in the style of the sixteenth than the seventeenth century, which fits in with the fact that he found much inspiration in Dürer’s work. Figure V.51 shows how de Caus repeated Dürer’s mechanical string method, replacing the lute with a cube. What is more, like Dürer, de Caus presented two geometrical constructions. The first is similar to Dürer’s plan and elevation construction. The second of de Caus’s methods also shows some similarities to Dürer’s second procedure: it is an Alberti construction, but not taken over directly from Dürer. De Caus knew that Dürer’s second construction was problematic, claiming – though without mentioning names – that several authors had made great mistakes in applying this method (De Caus 1612, second chapter). He pointed out that the nature of the mistake was to use a wrong point of intersection to decide how far the images of a transversal should be from the ground line. His own presentation of an Alberti construction is difficult to follow, since he, like Dürer, did not secure a unique correspondence between his description and illustration. His diagram does not display a correct method, whereas his text can be interpreted as an instruction on how to perform a correct construction. It seems that de Caus not only was inspired by Dürer, but also shared with him the familiarity with a plan and elevation construction and a lack of mastering other methods. De Caus followed Dürer in presenting perspectival shadow constructions, based on plans and elevations, but did more than simply copy Dürer’s example. He also applied a plan and elevation construction to neat examples of showing, as actually indicated in his title, how to draw mirror images in perspective (figure IX.7).

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FIGURE IX.6. The frame for title page of de Caus’s book on perspective from 1612. The frame was reused for De CausS 1620.

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B A

B

O

N

Q

A

P

L

DC

D

O

Q

F

C

P

E H

*

E

H

N

F

N

F

G

*

M

FIGURE IX.7. De Caus throwing the mirror image of a cube into perspective. For this purpose he applied the optical law that the mirror image of a point, say E, appears to be the point P lying symmetrical to it with respect to the mirror (cf. relation (x.4)). Thus, to find the perspective image of the configuration with a cube in front of a mirror, de Caus made a plan (at right) and an elevation (at left) of the configuration, including the points M* and N*, which are the plan and elevation of the eye point, as well as the line below, and near to, C and D containing the plan and elevation of the picture plane. To the plan of the cube in front of the mirror he added the plan of its symmetrical cube with respect to the mirror (that is its mirror image), and did the same in the elevation. Finally, he constructed the perspective images of the relevant points. De Caus 1612, near the end of the book (which has no page numbers) with enlarged letters.

De Caus was also aware that the method of throwing a horizontal grid of squares into perspective can be applied for constructing anamorphoses – a procedure to which I will return in section IX.8. He was, however, unable to present this insight clearly. Although de Caus had difficulty describing constructions, he was certainly able to make competent perspective drawings. He strongly advocated the teaching of perspective. To the more traditional arguments about why the discipline is useful, de Caus emphasized that it enables one to make an optical prolongation of rooms – an argument that was later used by Vaulezard as well (Vaulezard 1631, preface).

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Vaulezard on Cylindrical Mirror Anamorphoses

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new and fruitful period for French publications on the practice of perspective began in the early 1630s and lasted for a couple of decades, resulting in a dozen treatises on the subject. The first book from this period is written by Jean Louis Vaulezard, whose career is unclear, though on his title page, reproduced in figure IX.8, he called himself a mathematician. He is known in particular for his French edition from 1630 of François Viète’s famous work on symbolic algebra (VièteS 1591). Before presenting Vaulezard’s contributions to ordinary perspective, I like to describe his work on cylindrical mirror anamorphoses, Perspective cilindrique et conique (Cylindrical and conical perspective, 1630). Like other anamorphoses, a cylindrical mirror anamorphosis is a drawing that looks distorted unless it is viewed in a certain way. The means that makes it seem ‘natural’, is to look at its reflection in a cylindrical mirror, as the name indicates. The drawing itself must be positioned horizontally, and a cylindrical mirror with a given circle as its base must be placed upon it. The art of constructing this kind of anamorphoses seems initially to have been cultivated in China, and most likely passed from there to Europe in the early seventeenth century (BaltrusˇaitisS 1996, 235–242). As far as can be discerned, the Chinese constructions were based on trial and error, whereas in Europe a geometrical rule was introduced, to which I will return. The origin of this rule is unclear, but its existence is documented by the fact that Vaulezard criticized it (Vaulezard 1630, 28). According to his own reports, Vaulezard was asked by some of his students to lecture on cylindrical mirror anamorphoses (Vaulezard 1630, advertisement). He approached the problem in a manner similar to the one often applied for making perspectival anamorphoses and drawings on a vault, namely by introducing the image of a grid of squares and using this image grid as a kind of a coordinate system. In this chapter we will see a procedure for constructing perspectival anamorphoses involving a direct – and not very complicated – construction of a complete grid. We have also seen how Pozzo and other authors working with a curved surface as picture plane generally performed a pointwise construction of a perspective grid (page 390). Their precursor, Vaulezard, did not settle for a pointwise construction of the vertices of the grid, but took up the challenge of determining the lines in the perspective grid. In the case of cylindrical mirror anamorphoses this meant that Vaulezard had to determine a system of curves in a horizontal plane, say g, that would appear to be a vertical grid of squares when reflected in the cylindrical mirror and viewed from the eye point. Therefore his basic problem was this: Determine the two curves in g that, reflected in the mirror, appear respectively to be a given vertical line and as a given horizontal line in the grid (Vaulezard 1630, 23–25).

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FIGURE IX.8. The title page of Vaulezard’s book on cylindrical and conic anamorphoses. It is remarkable that the illustration shows an approximative method (cf. figure IX.38), which Vaulezard criticized, rather than the exact method he developed. The person depicted is probably Louis XIII (BaltrusˇaitisS 1977, 146). Vaulezard 1630.

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Applying the law of refraction, Vaulezard solved these problems very neatly. He deduced that the curve that is reflected as a vertical line is a straight line, whereas the curve that is reflected as a horizontal line is an oval curve (for technical details see AndersenS 1996, 18–23). Vaulezard also considered the situation in which a drawing is reflected on the inner side of a cylindrical mirror. Overall he seems to have been enchanted by all the mathematical problems the theme of mirror anamorphoses offered. However, when he investigated conical anamorphoses, which call for the construction of a grid that is ‘corrected’ by being reflected in the surface of a cone, he refrained from considering the problem in case of an arbitrarily situated eye point, but solved the problem for eye points lying on the axis of the cone above the apex (ibid., 24). Seen from a mathematical point of view Vaulezard’s solutions are interesting, but they did not have any significant impact on how anamorphoses were constructed. His results were, in fact, too complicated to be applied in practice. In his book from 1638 on anamorphoses, Niceron repeated Vaulezard’s construction of the grid for a cylindrical mirror anamorphosis, but advocated the use of a simpler procedure involving the following steps (figure IX.38). The curves that are to appear as vertical lines in the grid are drawn as straight lines passing through one point (in Vaulezard’s exact construction they do not intersect in the same point), and the curves that are to appear as horizontal lines in the grid are drawn as concentric circles with their centres in the point of intersection of the straight lines. This is the method Vaulezard criticized, as noted in the caption of figure IX.8, but despite this it stayed in use.

Vaulezard on Perspective

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n 1631 Vaulezard followed up his Perspective cilindrique with Abrége ou racourcy de la perspective (Summary of or shortcut to perspective). In this work, his aim was to present perspective constructions that did not involve too many steps, as his title indicates. One of his means was to provide a sector with a special scale for perspective – which he called la ligne optique (the optical line). This scale serves in determining the positions of the images of transversals that have a given distance a to the ground line. Hence, Vaulezard marked on the scale the lengths that I have denoted by f (a), and which are given by (cf. (ii.1)) f (a) : h = a : (d + a),

(ix.3)

where, as usual, d is the distance and h the distance between the horizon and the ground line. In an appendix Vaulezard described the construction of his scale without proving its correctness. He let the length of the scale be equal to the distance h between the ground line and the horizon, and the distance d be 60 – both in a given unit. First he showed how to construct f (a) for a = 100, 200, 300, and so forth (in the given unit). Then he subdivided the scale to mark f(a) when a is a multiple of 10, and indicated a further subdivision. In the

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main text of the book he showed how the sector can be used in a traditional way to determine f(a) if other values for h or d are chosen. (This procedure is explained in the caption of figure XII.10.) In 1636 Desargues presented another construction of f(a), and in the 1640s he was accused of having copied Vaulezard’s construction. Therefore in order to be able to point out the differences in the two constructions when dealing with Desargues’s work, I have presented Vaulezard method in the caption of figure IX.9 – and also proved why it is correct.

D

P

S3 S2

T2

S1

G

Q

T1

R

FIGURE IX.9. Diagram to Vaulezard’s construction of foreshortened orthogonal lengths. Let P be the principal vanishing point, GR the ground line, Q the ground point, D the left distance point, PQ = h, PD = d, and QR = b. Vaulezard first constructed f(nb) successively for n = 1, 2, 3, etc. in the following way. He let S1 be the point in which DR meets PQ, T1 the point of intersection of PR and the transversal through S1, S2 the point in which DT1 meets PQ, etc., and then claimed that QSn = f (nb). (1) To prove that this result is correct, I first introduce g(a) = h – f (a) and use relation (6) (page xxxiii) to change (ix.3) into g(a) : h = d : (d + a),

(2)

which is a handier relation to deal with – and which was actually introduced later by Lambert (page 645). Instead of proving (1), I then have to prove that PSn = g(nb), and that I do by looking at similar triangles. First, the pair DPS1 and RQS1 gives PS1 : S1Q = d : b, which combined with (2) (page xxxiii) shows that PS1 : h = d : (d + b).

(3)

A comparison of this result with the relation (2) shows that PS1 is indeed equal to g(b).* *That the line segment S1Q – and thereby PS1 – has the correct length also follows from the fact that it is obtained by means of an Alberti construction.

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FIGURE IX.9. (caption continued ) To prove that PS2 also has the correct length, I first consider the similar triangles PS1T1 and PQR and find that PS1 : h = S1 T1 : b. A combination of the two last relations shows that d : (d + b) = S1T1 : b, and hence according to (9) (page xxxiii), d : S1T1 = (d + b) : b.

(4)

Moreover, a consideration of yet another pair of similar triangles, namely S2S1T1 and S2PD, together with (4), leads to the result that PS2 : S2S1 = d : S1T1 = (d + b) : b. The relation (2) (page xxxiii) applied to this gives PS2 : PS1 = (d + b) : (d + 2b). When I combine this result with (3) as prescribed in (8) (page xxxiii), I get PS2 : h = d : (d + 2b), which according to (2) shows that PS2 = g(2b). The corresponding results for the other Sn’s can be proved similarly. Vaulezard’s next step was to construct f(nb + mc) for c = (1/10)b, n = 0, 1, 2, ..., and m = 1, 2, 3, ..., for which he used a procedure analogous to the one just explained.

As another means of abbreviating perspective constructions Vaulezard suggested the use of a peculiar instrument, which does not resemble any of the other perspective devices I have presented. In the captions of figures IX.10 and IX.11, I have described this instrument and how it was meant to be used. Vaulezard himself gave only a very brief – and rather opaque – presentation of the instrument, promising to return to it in his “universelle perspective” (Vaulezard 1631, 75–76). Apparently, he planned to compose a comprehensive work on perspective, but seems to have abandoned the project. Besides using sectors and a perspective instrument, Vaulezard touched upon a number of other themes in his Abrége ou racourcy de la perspective, such as inverse problems of perspective. He formulated his insights as propositions. Although he did not prove them, he indicated that proofs could be found by applying the geometrical results listed at the beginning of the book as maximes. Altogether, his book is an example of a new approach to perspective, in which some of the results of Guidobaldo’s and Stevin’s theory are applied – though not deduced – and in which new topics are taken up. Vaulezard’s work may have inspired Henry Guenon to write a sort of appendix to an earlier work on using a sector in which he had not treated perspective constructions. In 1640 Guenon published his considerations in a booklet called Pratique nouvelle et universelle de la perspective (New and universal practice of perspective). His goal was to show that it was not necessary to introduce Vaulezard’s special optical line on a sector for making perspective constructions, as they can be performed with the aid of the usual scales.

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FIGURE IX.10. Vaulezard’s perspective instrument. He placed the picture plane ABCD behind the ground plane DCEF. In that respect his instrument is similar to Hayden’s version of Jamnitzer’s instrument (figure V.69), but the two instruments function differently. In Vaulezard’s instrument H is the eye point, and KG a kind of square with a hole at G through which a thread passes – also passing through the hole at H and a hole at the end of the ruler MN. By using this ruler to keep the thread straight, Vaulezard got the point I depicted in the point L. He related this to finding images of points behind the picture plane – as explained in the caption of the next figure. Vaulezard 1631, figure 19, with a letter B changed to an L in accordance with his text.

IX.4

The Work of Aleaume and Migon

The History of the Book by Aleaume and Migon

B

efore Vaulezard published his new approach to perspective and to using a sector, the ingénieur du roy Jacques Aleaume had taken up similar themes. When Aleaume died in 1627, he left what seems to have been an interesting manuscript on perspective. This was bought by the printers and booksellers Pierre Rocolet and Charles Hulpeau, who were granted a royal permission in 1628 to publish Aleaume’s work under the title Introduction à

4. The Work of Aleaume and Migon

419

A

B R D

L

G H C

k h l O

FIGURE IX.11. The geometry behind Vaulezard’s instrument, depicted in figure IX.10. As mentioned in the previous figure caption, Vaulezard’s instrument has the eye point H and maps the point I in the ground plane upon the point L in the picture plane ABCD, working similarly for the other points that together with I form a quadrangle. Vaulezard related this procedure to one in which the image of a quadrangle behind the picture plane is found (Vaulezard 1631, 75–76). To explain his ideas, it is sufficient to look at the point I and its image L. The latter is also the image of the point R in which OI intersects the line HL. Let the given lengths HO (O being the foot) and GI be h and k, and let OR be equal to r. If the point R is given, then the point I having the same image as R can be determined as follows: Since the triangles HOR and GIR are similar, I is the point on the line OR that has the distance x to O, x being determined by x : r = (h − k) : h. When, for instance, a quadrangle with R as a vertex is given, it follows from this relation that a quadrangle through I similar to the one given, and with sides that are in the ratio (h − k):h to the one given, will produce the same image as the given quadrangle.

la perspective, ensemble l’usage du compas optique et perspective (Introduction to perspective, together with the use of the optical and perspectival sector). Some copies of the manuscript, or at least a part of it, were printed before Hulpeau’s death stopped the project.3 Later, in 1641, the mathematician Étienne Migon decided to finish the edition of Aleaume’s work and ended up radically revising it. First he called for copper plates to replace the woodcuts already made, and since he was replacing the figures anyway, he rewrote the text. In 1643 his edition appeared under the title La perspective speculative et pratique (Speculative and practical perspective). Migon did not indicate

3

Aleaume 1643, 156–157; the existence of this print is confirmed in Niceron 1646, 111.

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which things he had changed in Aleaume’s work, but proposed that readers compare his edition with the first one. He had purchased the remaining copies of this and offered them for free to the buyers of the new edition (Aleaume 1643, 154). Migon can no longer be taken up on his offer, nor is any copy of the first print known to exist today. It is therefore difficult to tell exactly what Migon’s changes consisted in. According to the printing permission he received, Migon was responsible for the “speculative” part of the book (Aleaume 1643, 157). Moreover, he left out the part on using a sector announced in Rocolet’s and Hulpeau’s title. Perhaps Migon generally took a more mathematical approach than Aleaume, but on the other hand it is likely that Aleaume himself – who was very interested in mathematics and had learned it under the guidance of Viète4 – had also included mathematical considerations in his work. Being thus unable to decide who was responsible for which ideas in Migon’s edition, I attribute the work jointly to Aleaume and Migon.

Introduction of a Perspective Grid

T

he content of La perspective speculative et pratique is mathematically very interesting, and quite unique for its time. One of the novelties is an elaboration on the idea of using the image of a grid of squares as a coordinate system in the picture plane. Aleaume and Migon did not use the term “coordinates”, but they did refer to a point in the ground plane by writing, for instance, “20,20”. As origin of their system they used the ground point Q in figure IX.12, counting the first coordinate along the ground line – not distinguishing between points to the left and the right of the ground point5 – and the second coordinate away from the ground line along an orthogonal (Aleaume 1643, 91).

H

Z

P

A C

Q

FIGURE IX.12. Aleaume and Migon’s scales for constructing the image of points with given coordinates. Adaptation of Aleaume 1643, 71. 4 5

DictionaireS 1933, vol. 1. Negative coordinates were hardly ever used in the seventeenth century.

4. The Work of Aleaume and Migon

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In order to find the points in the picture plane that are images of the two points m,n in the ground plane, Aleaume and Migon introduced two scales (figure IX.12), the first of which is an equidistant scale on the ground line GR. The second scale is perpendicular to the first and, for a given distance and a given position of the horizon HZ, it is marked with the positions of the images of the transversals in the ground plane with the distance n to the ground line. Aleaume and Migon’s second scale is analogous to the one introduced by Vaulezard for a sector, but it was constructed in a much more transparent way (figure IX.13). Using their scales, Aleaume and Migon made a direct construction of images of points in the ground plane, finding, for instance, the image of the point 5,3 to the right of the eye as the point of intersection of the horizontal line through the point marked 3 and the line connecting the right point on the ground line marked 5 and the principal vanishing point P.

D

P

Bn Bn − 1

Cn Cn + 1 Cn − 1

K

L

B2 B1 G

Q

A1

A2

An − 1 An = R

FIGURE IX.13. Diagram to Aleaume and Migon’s determination of the foreshortened orthogonal lengths f(k), defined in relation (ix.3). Let P be the principal vanishing point, GR the ground line, Q the ground point, and D the left distance point. Aleaume and Migon explained their procedure using an example in which PD = 9 and a line segment, QR, on the ground line is equal to 12 (Aleaume 1643, 70–72), but we can also assume that PD = d and QR = n. Aleaume and Migon first subdivided QR into intervals of length 1, drew lines through the points of division and D, and marked the points of intersection, B1, B2, ..., Bn of these lines and PQ. They thereby obtained the images of orthogonal line segments of lengths 1, 2, ..., n. To obtain the images of orthogonal lengths that are larger than n, they drew an arbitrary line KL nearer to P than QR. Let the points in which KL is intersected by the lines DAn-1 and DAn be Cn-1 and Cn. In determining the points Bi on PQ, intervals of lengths Ci - 1Ci on KL could have been used instead of intervals of length Ai - 1Ai on QR. Hence, by diving the segment CnL into line segments of length Cn-1Cn and joining the points of division with D, Aleaume and Migon determined the foreshortened lengths f(n + 1), f(n + 2), etc., the last number in this series being determined by how many intervals of length Cn-1Cn the segment CnL holds. If the foreshortening of a length greater than this number was required, they repeated the process by drawing a new line closer to P than KL, and so forth.

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Introduction of an Angle Scale

A

leaume and Migon had higher ambitions than making pointwise constructions of points in the ground plane directly in the picture plane. In fact, they wanted to make direct constructions of any horizontal line. As a means of doing this they introduced a third scale, namely an angle scale on the horizon. On this they marked vanishing points (figure IX.14) in such a way that the number f is assigned to the two vanishing points of the two lines in the ground plane that form an angle of f degrees with the normal to the ground line. Their construction of the scale is explained in the caption of figure IX.15. This figure also shows how Aleaume and Migon determined the image of a horizontal line that cuts the ground line in N and makes an angle of 30˚ with the normals to the ground line. In accordance with the main theorem, they joined the point N with the point M marked at 30 on the angle scale. As far as I am aware, Aleaume and Migon were the first to introduce such an angle scale, and thereby the first to develop a practical tool from Guidobaldo’s important insight concerning vanishing points. As noted, after Aleaume and Migon had used angle scales, several other authors did so as well. Building upon the angle scale, Aleaume and Migon treated a number of fascinating problems. These concern the solutions in the picture plane of

H V m p

P Vl

d f f

Z m

O

n G

l

f f Q R

FIGURE IX.14. Explanation of Aleaume and Migon’s angle scale. Let the line HZ in the picture plane p be the horizon, GR the ground line, O the eye point, P the principal vanishing point, d the distance equal to the length OP, n a normal to p, and l and m two horizontal lines at either side of n forming the angle j with n. According to the definition of vanishing points, the vanishing points of l and m, Vl and Vm, are the points on the horizon in which lines through O parallel to l and m, respectively, cuts the horizon. These two points Aleaume and Migon marked with f. These two points can also be characterized as the points having the distance the distance d tanf to P.

4. The Work of Aleaume and Migon

423

FIGURE IX.15. Aleaume and Migon’s construction of their angle scale. The line KL is the horizon, I the principal vanishing point, HI the distance, and the arc IG a quarter of a circle with centre H and radius HI. They divided the arc IG into 90 equal parts and drew lines from H to the points of division intersecting the horizon, marking the points of intersection with the number corresponding to the angle between the line and HI. This construction is in accordance with the explanation in the caption of the previous figure, because if this angle is f, the point of intersection is situated at the distance HI tan f from I. Aleaume 1643, 75.

constructions corresponding to the following usual Euclidean constructions (Aleaume 1643, 105–135). 1. Through a given point, draw a line parallel to a given line. 2. Through a given point on a given line, draw a line that makes a given angle with the given line. 3. Construct a line segment that has a given end point, forms a given angle with the ground line, and has a given length.

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4. Divide a given line segment into a given ratio. 5. Prolong a given line segment with a segment that has a given ratio to the given segment. In Perspectivae libri sex Guidobaldo had treated a few similar problems, as we have seen (page 261), but Aleaume and Migon dealt more systematically with the subject than Guidobaldo had done, and contrary to him they solved all the problems by means of constructions that can be performed directly in the picture plane. When Lambert later worked on direct constructions, he concentrated upon three fundamental problems. Two of his problems are identical to the first two presented by Aleaume and Migon, whereas his third was the perspectival version of constructing a line segment that lies on a given line and has a given end point and a given length. Problem 1 is easy to solve, as Guidobaldo had already shown (page 262). Aleaume and Migon’s solution to the second problem is similar to, though not quite as clear as, the solution Lambert later chose, and which we will see in chapter XII (page 651). To give an impression of their style, I have presented Aleaume and Migon’s solution to the third problem – which involves a measure point – in figure IX.16.

Methods Independent of Vanishing Points

B

y introducing an angle scale and what corresponds to a system of perspective coordinates, Aleaume and Migon had created two methods of performing constructions directly in the picture plane. Wishing to make the virtues of these methods quite clear, Migon chose the following verbose title for his edition of Aleaume’s work: La perspective speculative et pratique ou sont demonstrez les fondemens de cet Art, & de tout ce qui en a esté enseigné jusqu’á present. Ensemble la maniere universelle de la pratiquer non seulement sans plan géometral & sans tiers poinct, dedans ni dehors de la champ du tableau. Mais encore par le moyen de la ligne communément appelée horizontale.6 Besides stressing that the book contains procedures that do not require a ground plane (plan géométral), this title emphasizes that in one of the methods, no “vanishing point (tiers poinct) within or without the picture frame” is applied. In advocating this method Migon seems to have made an attempt to compete with Desargues’s method, which was published seven years earlier under the title Exemple ... touchant la pratique de la perspective sans emploier aucun tiers point ... qui soit hors du champ de l’ouvrage (translation in note 7). Desargues’s method will be presented in section IX.5. There are several other examples

6

Speculative and practical perspective, in which the foundations of this art are proved, and all that has been taught until now. Together with the universal method of practising perspective, not only without a ground plane and without third points – within or without the picture frame – but also by means of the line commonly called the horizon.

4. The Work of Aleaume and Migon H

M

P

V

Z

D

R

425

B

A

G

C

E

FIGURE IX.16. One of Aleaume and Migon’s direct constructions involving a measure point. The horizon HZ and the ground line are given, and are supposed to be equipped with an angle scale and an ordinary length scale, respectively. The principal vanishing point P and the point A are also given. It is required to construct a point B so that the line segment AB is the image of a line segment in the ground plane with a length of 81⁄2 feet, and so that the angle between AB and the transversal through A is the image of an angle of 56˚ in the ground plane (Aleaume 1643, 124). To abbreviate the phrase “being the image of ”, I introduce the symbol =i. Similarly, I use the symbol =p to signify that two magnitudes in the perspective plane are images of equal magnitudes, and I phrase it as they are perspectively equal. Aleaume and Migon solved their problem as follows. They drew the line PA meet1 ing the ground line GR in C, made CD = 8 ⁄2 (scaled) feet, and let PD cut the transversal through A in E. Next, they determined the vanishing point V for horizontal lines, making the angle 56˚ with GR (i.e. the point on the angle scale marked with 34) and the point M on the angle scale on the opposite side of P marked by u defined u + 34 = 180 - 56 , (1) 2 that is u = 28. Finally, they found the required B as the point of intersection of AV and EM. That B is the solution can be seen as follows.* The determinations of V and M imply that ∠BAE =i 56˚, the required angle, and moreover that ∠BEA =i 62˚, and hence ABE is the image of an isosceles triangle which means that AB =p AE. Since ACDE is a perspective rectangle, it is also known that AE =pCD = 81/2, and thus AB has the required length. The point M, introduced by Aleaume and Migon, is the left measure point for horizontal lines that make the angle 34˚ with the normal to the picture plane as the following consideration shows. If we replace the 34 in (1) by v, u is given by u + v = (180 − (90 − v))/2, or equivalently u = 45 − 1⁄2v. Thus, Aleaume and Migon defined the point M by a relation corresponding to the one used for a measure point, as mentioned in the caption of figure IX.4.

*Observation 1 in chapter XII (page 652) is helpful for carrying out these calculations.

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IX. France and the Southern Netherlands after 1600

from mid-seventeenth-century France of earlier titles on perspective being partly copied, as can be seen in the list in appendix four. René Gaultier de Maignannes also published a book with a very long title that stresses the avoidance of vanishing points outside the picture (Gaultier de Maignannes 1648). This work was a manual published in 1648, but according to the printer it had been composed some twenty years earlier. The book could have been quite a useful contribution to French perspective in the 1620s, but by 1648 there was not much to be gained from reading it. Gaultier’s began his title with the words Invention nouvelle (New invention), yet his procedure was not new, since it was based on the technique Alberti had already used: introducing the image of a grid of squares in the picture plane. In constructing the image of a square Gaultier did, however, apply an untraditional procedure, which in principle is a pointwise construction, and is explained as such in the caption of figure IX.17.

Further Issues Treated by Aleaume and Migon

B

esides being concerned with developing new perspective constructions, Aleaume and Migon took up a number of rather academic problems, including some inverse problems of perspective (Aleaume 1643, 101, 113). In treating inverse problems they assumed that the positions of the eye point,

P Ai

G

Q

B

C F

R

FIGURE IX.17. Gaultier’s special procedure for constructing the image of a point A in the ground plane. As usual P is the principal vanishing point and Q the ground point. In addition F is the point on PQ determined by QF = d, d being the distance. Gaultier first constructed the orthogonal projection B of A upon GR, found the point of intersection C of AQ and BF, drew the line through C parallel to PQ, and claimed that the point Ai in which this line meets BP is the image of A. Gaultier did not prove the correctness of his construction, but it can be deduced from the division theorem. According to this theorem, the image of A is the point on BP that divides it in the ratio AB : d = AB : QF. A consideration of the two pairs of similar triangles, BAiC and BPF, and BCA and FCQ, shows that AiB : AiP = BC : CF = AB : QF,

A

so Ai is indeed the image of A.

5. Desargues’s Perspective Method

427

the picture plane p, and the ground plane g are given, and only considered points in p that are images of points in g. This creates a one-to-one correspondence between the points in p and g. Aleaume and Migon used this correspondence to gain information about angles and line segments in original figures from their perspective images. Their method was to reverse the procedures applied for determining the images. In listing the qualities of La perspective speculative et pratique in the full title, quoted above, Migon included its proofs. As mentioned, his contribution to the work presumably consisted in compiling a general theory of perspective in the “speculative” part, to which there are many references throughout the book. His presentation of the theory seems to be inspired by the Guidobaldo school. For instance, he repeated Guidobaldo’s phrase “if the eye sees” (Aleaume 1643, 13). The content of Migon’s theory is equivalent to earlier expositions, but still formulated and argued in his own way with proofs that are sound, although long-winded compared to the elegant proofs by Stevin. It is interesting that in 1605, when Stevin’s book on perspective appeared with a dedication to Prince Maurice, Aleaume was connected to the circle of engineers and mathematicians surrounding the prince (DictionnaireS 1933, vol. 1). We have previously seen that Marolois, too, was a member of this group, and that he wrote a work on perspective without taking Stevin’s achievements into consideration (page 297). Apparently, Aleaume did likewise – at least there is no evidence of any inspiration from Stevin in Migon’s edition of Aleaume’s work. In fact, it is not Aleaume and Migon’s treatment of the theory of perspective that makes their book exceptional, but a number of the themes they took up. My enthusiasm about La perspective speculative et pratique is not based on its role in the historical development. The work was known and frequently referred to, but it seemingly had little influence. The angle scale was, as noted, presented by some of Aleaume and Migon’s successors, as were some of their constructions. However, during the seventeenth century no one took over their entire technique, or made an attempt to generalize their ideas on direct constructions – one possible exception being Bourgoing, whose own influence (if any) was also negligible (page 470). The next important protagonist in the prehistory of perspective geometry was Brook Taylor, who was probably unaware of Aleaume and Migon’s work. One of the innovations in Taylor’s contribution was to deal with direct constructions of the images of three-dimensional figures, a theme left untouched by Aleaume and Migon.

IX.5

N

Desargues’s Perspective Method

ot much is known about the engineer Girard Desargues (1591–1661) before he turned up in Paris in his thirties – apart from his place of birth, Lyon. From about 1630 he joined a group of Parisian mathematicians who gathered around Mersenne to discuss scientific matters. Desargues seems to

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have taken a vivid interest in many subjects, publishing upon several of them. One of his first works to appear was a manual describing a new method for perspective construction. He needed only twelve pages to present this method, but used a very long title to characterize his ideas: Exemple de l’une de manieres universelles touchant la pratique de la perspective sans emploier aucun tiers point, de distance ny d’autre nature, qui soit hors du champs de l’ouvrage7 (1636) – hereafter called La perspective. Desargues’s method received much attention in the mid-seventeenth century. Partly for this reason, and partly because I want to discuss the relationship of this method to Desargues’s own work on projective geometry, I have chosen to present this method in some detail.8 Desargues did not explain the foundations of his method, stating only that: ... to demonstrate the general rules ... only two theorems are needed, which are obvious and familiar to those who are disposed to understand them.9

Actually, I find that he applied more than two theorems, but he may have considered some results so obvious that he did not count them. Desargues’s method basically consists in transferring coordinates from the three-dimensional space to the picture plane. As pointed out earlier, in this respect it resembles the procedures introduced by Vaulezard and by Aleaume and Migon for mapping the ground plane into the picture plane. However, Desargues’s method is more general, because it also covers points that do not lie in the ground plane. He characterized the position of a point in the ground plane by listing its distances to the ground line and a line perpendicular to this passing through a given point on the ground line; and he identified a point outside the ground plane by its orthogonal projection upon this plane and the distance between the point and its projection. Rather unusually, he included the possibility that the point lies below the ground plane. To transfer the coordinates to the picture plane, Desargues constructed scales for foreshortening orthogonal, transversal, and vertical lengths. Like previously treated methods, his construction of the scale for foreshortening orthogonal lengths can be understood by applying a form of the division theorem. As earlier, I let a be the distance between the ground line and a transversal, f(a) the foreshortening of a, h the distance between the eye point and the ground plane, and d the distance. In dealing with Desargues’s

7

Example of one of the universal methods concerning the practice of perspective without using any third point, the distance point or any other kind [of vanishing point], which is outside the field of the picture. 8 The sections on Desargues are an abbreviated and revised version of AndersenS 1991. For other analyses, see Field and GrayS 1987, 25–30; FieldS 19872, 10–17; LaurentS 1994. 9 ... les régles generales ... se demonstrent avec deux seules propositions manifestes & familieres à ceux qui sont disposez à les concevoir. [Desargues 1636, 2–3]

5. Desargues’s Perspective Method

429

method it is convenient, as it was in the caption of figure IX.9, to consider the difference g(a) = h – f(a).

(ix.4)

Inverting the relation (2) in the caption of figure IX.9, I write the division theorem as h : g(a) = (d + a) : d.

(ix.5)

I mention this particular form because it occurs in a section called Autre fondement encore du trait de la perspective . . . (Yet another foundation of the treatment of perspective), which was added, presumably under Desargues’s guidance, to a reprint of La perspective published by Abraham Bosse in 1648 (Bosse 1648, 338 or Desargues Œuvres, 408). We have seen how in their constructions Vaulezard and Aleaume and Migon first chose h and d, and then determined f(a). Desargues’s construction is more general because it works for all d’s and for a’s that are given in relation to d, say a = kd. If this relation is inserted in (ix.5), we get h : g(kd ) = (k + 1) : 1.

(ix.6)

This relation shows, in particular, that g(d ) = 1/2h and f(d ) = h – g(d ) = 1/2h, or in other words, in any perspective picture, the line that is parallel to and situated halfway between the ground line and the horizon is the image of the transversal in the ground plane whose distance to the ground line is d. This observation was later formulated explicitly as a theorem by Dechales in the second edition of his Cursus (Dechales 1690, 512). More generally, the relation (ix.6) implies that a set of transversals whose distances to the ground line are d, 2d, 3d, etc. are depicted in a set of transversals whose distances to the horizon are h/2, h/3, h/4, etc. In his construction Desargues used this result, mentioning explicitly that he obtained foreshortenings firstly by cutting off a half, then by cutting off a third part, then by cutting off a quarter and so on ...10 [Field & GrayS 1987, 152]

His construction of the foreshortenings of the orthogonal lengths d, 2d, 3d, etc. is shown and explained in figure IX.18. He called the scale resulting from this construction échelle des eloignemens (scale of distances), remarking that others called it an optical scale (Desargues 1636, 6). Desargues proceeded by showing how to determine the foreshortenings of lengths that are commensurable with d, that are lengths of the form [n + (p/q)]d where n is a natural number and p/q a positive fraction smaller than 1. His construction is presented in a generalized form in figure IX.19.

10 la ligne ... se trouve retranchée ... premierement en sa moitié, puis en sa troisiéme, puis en sa quatriéme partie, & ainsi de suite ... [Desargues 1636, 7]

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IX. France and the Southern Netherlands after 1600 H

Z

E3 E2 E1

F3

M3

F2

M2 M1

G

F1

R

FIGURE IX.18. Desargues’s construction of the foreshortenings of orthogonal lengths d, 2d, 3d, etc. GHZR is a rectangle with the side GR on the ground line and the height HG = h. Let M1 be the point of intersection of the diagonals HR and ZG, E1 and F1 the points in which the horizontal line through M1 intersects HG and ZR, M2 the point in which ZE1 meets HR, and so forth. Desargues made a claim corresponding to f(nd ) = RFn = GEn. That this is correct can, according to the relations (ix.4) and (ix.6), be seen by proving that HG : HEn = (n + 1) : 1. It follows immediately from the construction of M1 that HG : HE1 = 2 : 1.

(1)

The fact that E1M1M2 and ZM2H, and HE1M1 and HGR, are two pairs of similar triangles and (1) lead to HE2 : E2E1 = HZ : E1M1 = GR : E1M1 = HG : HE1 = 2 : 1. Hence, using result (3) (page xxxiii), we get HE1 : HE2 = 3 : 2.

(2)

A combination of (1) and (2) indeed leads to HG:HE2 = 3 : 1. It can similarly be proved that HE3 : E3E2 = HG : HE2 = 3 : 1, etc.

Desargues illustrated his method with a single example in which he constructed the perspective image of a tower whose plan and elevation are given in the top right-hand diagram in figure IX.20. As the frame of his picture he chose a rectangle with the line segment AB as its base (the largest diagram in figure IX.20), and he let AB represent 12 feet. He assumed that the eye point is 41/2 feet above the ground plane, and hence the horizon, FE, in his picture

5. Desargues’s Perspective Method

431

lies at a distance from AB that is equal to 41⁄2 scaled feet – scaled in the same scale as the unit on AB. He then used the rectangle ACGF to determine the foreshortened orthogonal lengths, producing his échelle des eloignemens. In his example he set the distance to be 24 feet and accordingly ascribed the number 24 to the midpoint T of CG. He similarly ascribed numbers to other points on GC. Desargues’s construction of his échelle des eloignemens is, as noted, independent of the actual length of the distance d, which means that it can be carried out in any rectangle whose height is equal to h. This technique has two advantages. First, it does not put any requirement on the width of the paper used. For instance in his example, d was equal to 2AB, and hence, if the distance had been involved, he would have needed a piece of paper twice as wide as the one defined by the picture frame. Second, once a scale has been constructed, it can be used for different d ’s. In his Maniere universelle (1648), to which I will return, Bosse by and large took over Desargues’s method, but lost the second property of Desargues’s scale. Thus Bosse began by determining f(1) for a fixed d (figure IX.21) in the scale that he called the échelle fuyante (receding scale). So far we have only seen how Desargues constructed foreshortened orthogonal lengths. As for his treatment of transversal and vertical lengths, it is straightforward. He divided AB in figure IX.20 into twelve equal sections each representing 1 foot. By joining the points of division with G, he transferred the scale from the ground line to the images of the transversals. Thus, at any distance from the ground line the transversal line segment between the two lines that connect G with C and the point of division to the right of C represents one foot. By means of this construction Desargues obtained a set of scales for determining the foreshortening of transversal lengths at given distances from the ground line. Since the transition from one of these scales to another can be made with the help of a sector, he conceived of all of them as one scale, which he called the échelle des mesures (scale of measures). He took it for granted that his readers understood that this scale can also be used for vertical lengths. All in all, Desargues had successfully shown how the coordinates of a point in three-dimensional space can the transferred to a picture plane. In Bosse’s later presentation this was illustrated quite neatly, as can be seen in figure IX.22. In principle, disposing over a three-dimensional perspective coordinate system enables a practitioner to throw any object into perspective, and therefore requires no theoretical knowledge. Even so, some knowledge of vanishing points may be useful, as the example of Vredeman de Vries illustrated. He had a kind of three-dimensional coordinate system in his picture, as shown in figure V.74, and yet he did not correctly render the images of all oblique lines, as we saw in figure V.77. I do not doubt that Desargues had understood the theory of vanishing points, and introducing them together with his coordinate system could have been a good idea. However, he seems to have had a special reason for not taking them into consideration, as we shall see in the next section.

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H

B3

E3

A3

E2

B2

E1

A2

C2 B1

X

A1

G

K3 F2 K2 F1 K1

R

FIGURE IX.19. Desargues’s construction of f((p/q)d), f([1+( p/q)]d ), f([2+( p/q)]d ), and so forth. The rectangle GHZR is the same as in the previous figure. On the ground line GR Desargues let A1 be determined by GA1 : GR = GA1 : HZ = p : q;

(1)

and let B1 be the point of intersection of HA1 and GZ. He then drew the horizontal line through B1, cutting ZR in K1, and made a claim corresponding to f ((p/q)d ) = RK1.

(2)

To prove this result, I will return to relation (ix.6), which according to (4) (page xxxiii) is equivalent to

f(kd ) : g(kd ) = (h − g(kd )) : g (kd ) = k : 1.

(3)

To obtain relation (2), I notice that the triangles GB1A1 and ZB1H are similar with heights RK1, and K1Z. This fact and relation (1) yield RK1 : K1Z = GA1 : HZ = p : q,

(4)

which, together with (3), shows that (2) is correct. Desargues proceeded by drawing a line from H to the point A2, in which ZA1 cuts the line E1F1, letting the point of intersection of HA2 and ZG be B2, drawing a horizontal line through B2 cutting ZR in K2, and making a claim that in my notation has the form f ([1 + ( p/q)]d ) = RK2. According to (3) this result is correct if RK2 : K2Z = (1 + ( p/q)) : 1 = (p + q) : q.

(5)

To prove this I introduce C2 as the point of intersection of E1F1 and ZG, X as the point of intersection of B1K1 and A1Z, and first consider the ratio F1K2 : K2Z. From the similar triangles A2B2C2 and HB2Z I deduce F1K2 : K2Z = C2A2 : HZ.

(6)

To determine the right-hand side I include the line segment B1X in the considerations and obtain, from the similar triangles ZC2A2 and ZB1X, C2A2 : B1X = F1Z : K1Z.

(7)

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FIGURE IX.19. (caption continued ) The ratio on the right-hand side of this relation is known, since it has earlier been determined which ratio each of the line segments form with RZ. Thus in the caption of figure IX.18 we saw that E1F1 halves ZR, that is F1Z : RZ = 1 : 2, whereas relation (4), together with (1) (page xxxiii), shows RZ : K1Z = ( p + q) : q. The ‘multiplication’ relation (8) (page xxxiii) applied to the last two relation implies F1Z : K1Z = ( p + q) : (2q), which together with (7) results in CA2 : B1X = ( p + q) : (2q).

(8)

The similarity of the triangles A1B1X and A1HZ and relation (4), combined with (2) (page xxxiii), yield B1X : HZ = p : ( p + q).

(9)

Once again, a ‘multiplication’ of the last two relations leads to C2A2 : HZ = p : (2q). Thus we have finally determined the right-hand side in (6) and obtained F1K2 : K2Z = p : (2q). This can also be written as F1K2 : (F1Z − F1K2) = p : (2q). This, together with relation (7) (page xxxiii), yield RK2 : K2Z = (RF1 + F1K2) : K2Z =(F1Z + F1K2) : (F1Z − F1K2) = (2p + 2 q) : (2q) = ( p + q) : q, which is indeed the requested relation (5). Desargues similarly constructed f([2 + ( p/q)]d ) by drawing a line from H to the point of intersection A3 of ZA1 and E2F2, etc.

Desargues’s Avoidance of Vanishing Points

A

s noted, in the title of his manual Desargues stressed that he did not involve any vanishing points outside the picture frame, and some of his contemporaries actually agreed that precisely this fact made Desargues’s method superior to other procedures. In pursuing a technique of perspective construction that does not involve any points outside the picture frame, Desargues not only disposed of the vanishing points outside the frame, but also of those inside – apart from the principal vanishing point. In this section I show that he avoided vanishing points in his theoretical work as well. An important reason for taking up this issue is that I wish to modify an idea occurring in the literature on the history of mathematics, namely the perception that Desargues to a line added a point at

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FIGURE IX.20. The only illustration in Desargues’s La perspective. The diagram in the upper right-hand corner shows a plan and elevation of a tower, while the diagram at the upper left-hand corner illustrates Desargues’s construction of his échelle des eloignemens. This construction is repeated in the largest diagram, although mirrored. Desargues 1636.

infinity in order to obtain a point on the line that is depicted in its vanishing point (KlineS 1972, 289–290). By looking into some of Desargues’s considerations I argue that this mathematically fascinating idea does not reflect Desargues’s way of treating perspective problems.

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FIGURE IX.21. Bosse’s construction of foreshortened orthogonal lengths. The construction takes place in the left-hand side of the diagram. First, Bosse constructed the point l defined by ZC : El = d : 1, d being the distance. He then let 1 be the point of intersection of Zl and CF. Through 1 he drew a horizontal line, connected the point in which it meets EZ with C obtaining the point 2, and so on. That the scale on the line EZ gives f(1), f(2), f( f(3), etc. can be verified by looking at similar triangles and manipulating proportions until the relation (ix.3) is fulfilled. Bosse 1648, plate 28.

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FIGURE IX.22. Bosse’s perspective coordinate system. Bosse 1648, plate 27.

Theoretical Reflections in La perspective

L

a perspective from 1636 deals mainly with the practice of perspective, but at the very end Desargues devoted about a page to theoretical matters in the form of a number of (unproven) theorems addressed to “contemplators” (Desargues 1636, 11–12). As regards its content, his first theorem bears a striking resemblance to the main theorem, and yet it is formulated quite differently:

When the subject is a point, and when from the subject point and the eye point we draw lines parallel with one another, producing them to meet the picture, the appearance of

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p Im m O

Ai l

A

Il

FIGURE IX.23. Illustration to the first of Desargues’s theoretical theorems. He himself provided no diagram.

the subject lies on the line through the points in which these parallel lines meet the picture ... 11[Translation based on Field and GrayS 1987, 159]

Let A be the subject point (the point to be thrown into perspective) and O the eye point (figure IX.23), and let an arbitrary pair of parallel lines l and m be drawn through A and O so that they meet the picture plane in Il and Im, respectively. In his theorem Desargues concluded that the image Ai of A lies on the line segment IlIm, but neither here nor in any other connection did he introduce Im as the vanishing point of l.

Theoretical Reflections in Aux théoriciens

T

o present two more examples of Desargues’s staying away from vanishing points I must introduce another of his works. According to a statement published by Jean Dubreuil in 1642, Desargues had long been promising to write a book on perspective (Dubreuil 16422, aiijv). If Desargues really had a plan of a more comprehensive work on the subject than La perspective, he never carried it out. He seems, however, to have produced a booklet that Jacques Curabelle referred to as Livret de perspective addressé aux théoriciens (Booklet on perspective addressed to theorists), claiming that it appeared in 1643 (Curabelle 1644, 70). No copies of this booklet are known, but it is highly likely that Bosse incorporated its material in his Maniere universelle from 1648. Bosse had it appear as a section bearing the heading Aux théoriciens, which I use as short title (Desargues 1643). Although this version

11 Quand le sujet est un poinct, & que des poincts de sujet & de l’œil, sont menées iusqu’au tableau des lignes paralelles entre elles, l’aparence du sujet est en la ligne menée par les poincts ausquels ces paralelles rencontrent le tableau ... [Desargues 1636, 11]

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has been edited by Bosse, I consider Desargues to be the author of the text – apart from the introduction, in which Bosse wrote that he wanted to let the section appear to the public like “un enfant perdu” (a lost child) and claimed that its contents were more interesting than useful (Bosse 1648, 311). Aux théoriciens consists of no more than eight pages and eight plates, but it contains much interesting material. From it we learn that Desargues intended to do more with his method on perspective than show pointwise coordinate constructions. Aleaume and Migon’s book presumably inspired him to take up new problems – as has also been suggested by Noël Germinal Poudra (Desargues Œuvres vol. 1, 463). The problem of finding out which forms elementary Euclidean constructions take in the picture plane particularly caught Desargues’s attention. In discussing his attitude to vanishing points, the first problem in Aux théoriciens holds special relevance. Its substance is as follows (figure IX.24). In the picture plane a point M and a line segment BC are given. They are supposed to be images of objects in a ground plane. It is required to construct the line through M that is perspectively parallel to the given BC.12

F

C E

M

B

12

FIGURE IX.24. Desargues’s solution to the problem of determining a perspective parallel. Let the point M and the line segment BC be given in the picture as images of a point and line segment in a horizontal plane; it is required to construct the image of the line that passes through the original of M and is parallel to the original of BC. Through M Desargues drew a transversal meeting BC in the point E, and through C he drew another transversal. Next he used his scales of foreshortening to make the line segment CF perspectively equal to the segment ME, and then claimed that MF is the required line. The correctness of the constructions follows from the facts that the originals of the segments ME and FC are parallel and of equal length, and hence they are opposite sides in a parallelogram. This implies that the originals of MF and BC are also parallel. An adaptation of a section of plate 145 in Bosse 1648.

In Desargues’s formulation:

Par un point d’assiette perspectif d [M] donné de position, mener une droite di [MF], dont la geometrale, soit parallelle a la geometrale d’une droite d’aiette perspective bc [BC], donnée aui de position. [Bosse 1648, text to plate 143]

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Guidobaldo had published a very simple solution to this problem applying the vanishing point of the given line (figure VI.18), and Aleaume and Migon presented the same solution. Desargues did not include a vanishing point, and came up with a less elegant construction in which he obtained the required line by constructing a perspective parallelogram (caption of figure IX.24). Desargues also treated the problem of constructing angles directly in the picture plane and for which purpose he introduced an angle scale. His scale is much more complicated to construct than Aleaume and Migon’s – to prove this claim I discuss Desargues’s construction in detail in the captions of figures IX.25 and IX 26. In addition, his scale is unhandy to use because (as shown in figure IX.26) instead of marking vanishing points it supplies the directions of

U

S

T

Y

n G

V

A

Q

B

R

F

FIGURE IX.25. The configuration for which Desargues constructed his angle scale. The diagram shows a ground plane in which GR is the ground line, F the foot of the eye point, and Q the ground point, hence FQ is equal to the distance d. Desargues wished to construct a scale that defines the images of lines radiating from Q. For this purpose he drew a circle with Q as its centre and radius d 2, then divided its periphery into arcs corresponding to angles of 5˚, and indicated the lines from Q to the points of division. He paid special attention to the two 45˚ points, S and T, drew the transversal ST through them, and marked the points in which FS and FT cut the ground line A and B. If we let U be the midpoint of ST, it follows from the construction that ∠UQT = 45˚, ∠ QUT = 90˚, and QT = d 2, hence UQ = d.

(1)

Redrawing with letters altered and lines added (for later reference) of the upper part of plate 146 in Bosse 1648.

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IX. France and the Southern Netherlands after 1600 H

Z P

S

U

T

X

A

Q

B

FIGURE IX.26. Desargues’s angle scale. The rectangle ABZH is situated in the picture plane, AB being the ground line – which is actually the AB from the previous figure drawn in another scale. The lines AH and BZ are the images of the lines AS and BT from the previous figure (and they are depicted as vertical lines because they pass through the foot F ). The line HZ is the horizon, P the orthogonal projection of the eye point upon the picture, Q the ground point, S the midpoint of AH, and T the midpoint of BZ. The line ST is the image of the transversal situated in the distance d from the ground line (cf. the remark after (ix.6)), meaning it is the image of ST in the previous figure. The numbers marked at the sides AH, HZ, and ZB define the directions of the images of the lines radiating from Q in the previous figure, and thus the image of the left line Q50 in the previous figure is the left line Q50 in this diagram. It is worth noticing that Desargues’s angle scale differs from Aleaume and Migon’s as the latter considered the angles that horizontal lines make with the normal to the picture plane, whereas Desargues measured the angles the lines make with the ground line, that is the complementary angles. As concerns their construction of the points on the angle scale, Desargues and Bosse only revealed that the points from the previous diagram had been “transported as the figures show” (Bosse 1648, text to plate 146). This is only partly understandable, as I am going to explain. In order to reconstruct their procedure I, unlike Desargues, apply vanishing points, and in particular the vanishing points of the lines radiating from Q. For points marked n, the distances between their vanishing points and P are (cf. the caption of figure IX.14, and recalling Desargues worked with complementary angles) d tan(90˚ − n).

(1)

For n’s belonging to the interval [45˚–90˚], these distances are marked on the line ST in the previous figure – QU being equal to d – and can be directly transferred to the present diagram for angles in the interval [65˚–90˚], resulting in the marks on the horizon HZ. Since AS = SH the points of division on SU can be obtained by halving the

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441

FIGURE IX.26. (caption continued ) distances in the preceding figure for angles in the interval [45˚–60˚]. From these points, Desargues and Bosse drew lines through Q to obtain the points on the scale SH. Desargues and Bosse claimed that the points of division on the line AS in the previous figure had been similarly transferred to the line AS in the present diagram. However, since their angle scale is correct and would not have been so if the indicated procedure had been used, they must have used a different procedure. One possibility is that, although a line like SV in figure IX.25 perpendicular to the ground line does not figure in Bosse’s drawings, they still involved it. For an angle n in the interval [0˚–45˚], let the line through Q cut the line SV in Y, then VY = d tan n.

(2)

What is needed in the present diagram is the point X marking the angle n. Let AH = h, then relation (1) and a straightforward calculation show that X is determined by 1 AX = h = h tan n (3) 2 tan (90c - n) 2 With the help of a sector, the line segment VY from (2) can be transferred into the requested line segment in (3). Adaptation of the lower part of plate 146 from Bosse 1648.

the images of horizontal lines that pass through the point Q on the ground line. When working with the direction of any other line l in the ground plane, Desargues had to draw the line through the point Q perspectively parallel to l, which – as just explained – was quite an operation when applying his procedure. A third disadvantage is that Desargues’s angle scale is not only dependent on the distance, as was usual, but also on the distance h between the horizon and the ground line, which meant that choosing an alternative h meant constructing another angle scale. All in all, Desargues’s angle scale seems so unpractical that it was only interesting from a theoretical point of view. While composing Aux théoriciens, Desargues was, as we shall see in section IX.7, in the midst of a polemic about the originality of his method. This might explain why he introduced an angle scale different from the one presented by Aleaume and Migon. He may thus have wanted to demonstrate that his perspective method, in which constructions can be kept inside the picture frame, can be expanded to solve the problems examined by Aleaume and Migon. Even so, I will not rule out the possibility that even if Desargues had felt unconstrained, he would still have chosen the technique he presented.

Conclusion on Desargues and Vanishing Points

T

he examples presented here show that considering vanishing points was not a part of Desargues’s theoretical reasoning. The concept did not occur in his version of the main theorem, nor in connection with his solutions to the problems of drawing perspective parallels and of designing an angle scale. I therefore find it very unlikely that vanishing points should have been his source of inspiration for introducing points at infinity, as claimed by

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Morris Kline. Nevertheless, I do believe that perspective stimulated Desargues to consider points at infinity, and how it might have done so I explain in the following section.

Points at Infinity in Desargues’s Work on Perspective

M

y search for a link between Desargues’s introduction of points at infinity and perspective takes me back to the theoretical section in La perspective. In four of its theorems Desargues discussed the question of how a pencil of lines and a set of parallels are represented in the picture plane (Desargues 1636, 11–12). In both categories of lines he singled out the line that passes through the eye point, referring to it as the “eye line” (ligne de l’œil). He stated that, depending on whether the eye line (i) is parallel to or (ii) cuts the picture plane, 1. the images of a set of parallels are (i) a set of parallels, or (ii) a pencil of lines (figures IX.27 and IX.28). 2. the images of a pencil of lines are, (i) a set of parallels, or (ii) a pencil of lines (figures IX.29 and IX.30). These theorems clearly show that Desargues saw a strong relationship between a pencil of lines and a set of parallels. Further investigation into this matter is likely to have inspired him to treat the two types on a par, the means

p

l li

O

mi

m

L

FIGURE IX.27. Desargues’s first example of representing lines. For a set of parallel lines, represented by l and m, the eye line OL is the line from the set that passes through the eye point O. When OL is parallel to the picture plane p, the parallel lines are depicted in lines that are parallel to OL. In addition, the eye line OL is the line of intersection of all the planes determined by O and each of the original lines. This last result also applies to the situations illustrated in figures IX.28–IX.30.

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p M

l

li O

m

mi

FIGURE IX.28. Desargues’s second example. When the eye line of a set of parallel lines, represented by l and m, cuts the picture plane in a point M, the set is depicted in a pencil of lines whose vertex is M.

being to introduce a point at infinity as the point of intersection of the parallel lines (see also FieldS 19872, 16–17). He would then have found that a pencil of lines is always mapped into a pencil of lines, and that the eye line has the property of connecting the eye point and the vertex of the pencil. Finally, he may have concluded that the point of intersection of the eye line and the picture plane is the vertex of the image of the pencil. In fact, he did more or less state all this at the end of his main work Brouillon project d’une atteinte aux evenemens de rencontres du cone avec un plan (Rough draft of attaining the outcome of intersecting a cone with a plane), where he also

p

mi O

li l m L

FIGURE IX.29. Desargues’s third example. For a pencil of lines, represented by the lines l and m, and with the vertex L, the eye line is OL. When OL is parallel to p, the pencil is depicted in a set of lines that are parallel to OL.

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IX. France and the Southern Netherlands after 1600 L

p

m

M O

mi li

l

FIGURE IX.30. Desargues’s fourth example. When the eye line OL of a pencil of lines, represented by l and m, cuts p in the point M, the pencil is depicted in a pencil of lines whose vertex is M.

introduced points at infinity (DesarguesS 1639/1951, 100). Thus, he put forward a theorem with the following contents (ibid., 180). On a plane the perspective image of a pencil of lines is a pencil of lines; the latter belongs to the same pencil as the line [the eye line] of the (original) pencil that passes through the eye point, and which is the trace of all the planes determined by the eye point and each of the lines in the (original) pencil. Among the wealth of observations in Brouillon project, this is the only one in which Desargues pointed explicitly to perspective images. I see this fact as supporting the hypothesis that the result itself could have originated from the author’s continued study of the perspective images of sets of lines presented in the theoretical section of La perspective. Although I emphasize that a wish to treat a pencil of lines and a set of parallels as one concept seems to have been a strong motivation for Desargues’s introduction of points at infinity, I do not maintain that this wish was his only inspiration. As pointed out by several historians, the idea of a point at infinity emerged in several contexts as a handy tool for avoiding having to distinguish between various positions of lines (FieldS 19872, 24–25; Field and GrayS 1987, 186–187; BkoucheS 1991, 240–243). Returning to the discussion of vanishing points in the previous section, I would like to remark that for a given set of parallels, the point of intersection of the eye line and the picture plane is the vanishing point of the parallels and indeed the image of their point at infinity. Hence, Desargues’s work implicitly contains a correspondence between vanishing points and points at infinity. However, as already stressed, he did not use the concept of vanishing points in his arguments, and I therefore do not think that concept inspired him to introduce points at infinity.

6. Brouillon Project and Perspective

IX.6

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Brouillon project and Perspective

D

esargues’s Brouillon project has, as noted, received much attention among historians of mathematics, and with good reason, for the work contains an approach to geometry that is untraditional – and which in the nineteenth century turned out to be very successful. Besides using the concept of points at infinity, Desargues conceived the ideas of finding properties that are invariant under a central projection and of introducing fundamental concepts similar to some concepts later referred to as “pole” and “polar”. Brouillon project is a theoretical work that is best understood if a large part of it is seen as Desargues’s attempt to revise some of the ideas in the classical Greek work on conics by Apollonius (a point particularly well demonstrated in HogendijkS 1991). Desargues himself, however, seems to have considered Brouillon project useful for practitioners. He harboured ambitions of bridging the gap between the theorists and the mathematical practitioners, and actually possessed some of the qualifications needed for doing so. He was familiar with the practitioners’ approach to geometry as well as with the mathematicians’ way of thinking. This was a good starting position for teaching practitioners useful mathematics, and for getting the mathematicians interested in applied mathematics. However, he did not realize that these were two different processes. In a letter to Desargues dated 19 June 1639, Descartes commented upon Brouillon project and stressed the impossibility of reaching both groups in one book: You may have two designs, which are very good and very praiseworthy, but which do not both require the same course of action. One is to write for the learned, and to instruct them about some new properties of conics with which they are not yet familiar; the other is to write for people who are interested but not learned, and make this subject, which until now has been understood by very few people, but which is nevertheless very useful for Perspective, Architecture, etc., accessible to common people and easily understood by anyone who studies it from your book. If you have the first of these designs ... If you have the second ...13 [Field and GrayS 1987, 176]

Descartes went on to describe what he thought a book had to offer to keep the interest of non-academics alive. It should not contain anything “which seems to them to be less easy of understanding than the description of an enchanted palace in a novel” (Field & GrayS 1987, 176–177). This is a requirement that Brouillon project certainly does not fulfil, but even if it had,

13 Vous pouvez avoir deux desseins, qui sont fort bons et fort louables, mais qui ne requièrent pas tous deux mesme façon de procéder. L’un est d’escrire pour les doctes, et de leur enseigner quelques nouvelles proprietez de ces sections [conics] qui ne leur soient pas encore connues; et l’autre est d’escrire pour les curieux qui ne sont pas doctes, et de faire que cette matiere qui n’a pu jusqu’ici estre entendue que de fort peu de personnes, et qui est néanmoins fort utile pour la Perspective, l’Architecture, &c., devienne vulgaire et facile à tous ceux qui la voudront estudier dans votre livre. Si vous avez le premiere ... Si vous avez le second ... [TatonS 1951, 185]

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one wonders how much of the material buried in it could be of use to practitioners of perspective, for the themes taken up in the book are not especially relevant for perspective constructions. Let me illustrate this point by presenting an example from each of the two parts of the book.

Cross Ratios

I

n the first part of Brouillon project, Desargues mainly dealt with projections from one line to another; he reached mathematically fascinating insights, in particular one about how a special property of three pairs of points – being in involution – is preserved under a central projection (DesarguesS 1639/1951, 127). Desargues’s result implies an important theorem in projective geometry, stating that a cross ratio of four points is preserved under a central projection. In Desargues’s day, however, it was very unlikely that anyone – including Desargues himself – would find his result useful for purposes relating to perspective. In fact, the invariance of cross ratios did not enter the theory of perspective until the nineteenth century. There is one example from the 1710s, as we shall see in chapter X, in which Brook Taylor introduced a ratio similar to the one later called a cross ratio. However, Taylor did not consider the invariance aspect (JonesS 1947, 98; AndersenS 19921, 28–29; see also page 515).

Projection of Conics

I

n the second part of Brouillon project, Desargues focussed upon conic sections. Here, too, he achieved impressive results, though none that are essential for perspective constructions. The problem of throwing conic sections into perspective was on Desargues’s mind when he laconically ended his theoretical section in La perspective with the remark:

The proportion which follows cannot be explained so briefly as the preceding ones. Given to portray a flat section of a cone, draw two lines [in it] whose appearances will become axles of the figure which will represent it.14 [Field and GrayS 1987, 160]

Apparently in 1636 Desargues had the idea of determining the images of conic sections by finding two lines that are projected onto two axes of the image. If this idea played any role in his decision to take up the study that led to Brouillon project, the link to perspective disappeared in the process. This book actually contains no results that are immediately applicable for drawing the perspective images of conic sections. In this book Desargues did not even take up a problem that had baffled so many perspectivists, namely the question of when a circle is depicted as a circle. 14

La proposition qui suit ne se devide pas si briévement que celles qui precedent. Aiant à pourtraire une coupe de cone plate, y mener deux lignes, dont les aparences soient les essieux de la figure qui la representera. [Desargues 1636, 12]

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447

Two Traditions

T

he two examples just presented reflect the general fact that, apart from his introduction of points at infinity, there is not much connection between Desargues’s works on perspective and projective geometry. In my opinion he worked in two traditions, one being the tradition of tracts on perspective, and the other the Apollonian approach to synthetic geometry, essentially developing the latter.15 He dealt with different problems in the two fields – and in a way he dealt with different projections as well. In Brouillon project Desargues treated projections of lines upon lines and of planes upon planes, whereas in La perspective he worked with projections from three-dimensional space onto a plane. By emphasizing the lack of similarities between his work in the two disciplines I do not exclude the possibility that he himself saw an important link between them, but I do claim that in his writings he did not make the relationship clear.16 There is, however, one similarity – perhaps not unifying, but at least characteristic – between Desargues’s work in La perspective and in Brouillon project, namely the almost paradoxical one that the creator of projective geometry approached his topics in a rather ‘unprojective’ manner. His method of perspective was less projective than several other methods, in that he provided a fixed scheme for transferring points from three-dimensional space to the picture plane and did not deal with the properties of a perspective projection. Correspondingly, although he treated projections in Brouillon project, they did not influence his proof technique, which is classical (see also BkoucheS 1991, 258). His approach is, for instance, less projective than the one Blaise Pascal, inspired by Desargues’s ideas, is supposed to have used. Desargues’s proof of the famous theorem, which later came to bear his name and which was first published in Bosse’s Maniere universelle in 1648, is another example of Desargues treating a projective result using traditional methods. Rather ahistorically, let me mention that this theorem can be proved in a fairly straightforward manner if considered as a theorem about a triangle and its perspective image (AndersenS 1991, 82–84).

15 Historians of mathematics are generally of the opinion that Desargues’s work on perspective was a major inspiration for his work on projective geometry – see for instance KlineS 1954, 146; FieldS 19872, 17; Field and GrayS 1987, 26–29; Le GoffS 1994, 157; FregugliaS 1995, 89; and FieldS 1997, 196. 16 The literature from Desargues’s time contains an enigmatic reference to Leçons de ténèbres (Lectures of darkness). Among others Curabelle and Grégoire Huret mentioned them, and according to the latter Desargues should in them have worked on the connection between projective geometry and perspective (Curabelle 1644, 71; Huret 1670, 157–158). However, it is unclear whether the reference is to a book different from Brouillon project – and then lost, to lectures given by Desargues, or something else (for thorough discussions see TatonS 19511, 44–48 and Le GoffS 1994, 182–190).

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Perspectivists at War – and the Work of Dubreuil

D

espite the fact that Desargues had spent less than a dozen pages on presenting his method in La perspective, his work was noticed and praised; actually it was more widely discussed than Brouillon project was at the time. Mersenne had already spoken highly of La perspective two years before it appeared (MersenneS Corr, vol. 4, 333). He sent the booklet to the two most outstanding mathematicians of the day, René Descartes and Pierre Fermat, who both reacted positively to Desargues’s work in letters sent to Mersenne during the spring of 1637 (Descartes Œuvres, vol. 1, 360–361 and 376–377). The following year, Jean François Niceron published a book on perspective in which he surveyed the literature known to him, characterizing Desargues’s method as “generale et fort expeditive” (general and very expedient, Niceron 1638, preface, 3). As noted in section IX.2, Pierre Hérigone also referred to Desargues’s method in the essay on perspective he published in 1642.

Dubreuil

D

esargues’s method was also praised in another book from 1642 called La perspective pratique (Practical perspective) – also known as the Jesuit’s Perspective because the title page revealed only that the author was affiliated to the Jesuit Order. Yet many knew that the person responsible for the work was Jean Dubreuil (1602–1670), who had worked in the book business before taking holy orders. He was an enthusiastic perspectivist who wanted to inform practitioners about the discipline without burdening them with theory. He admitted to having been inspired by many others, referring to “all my private Thefts” (Dubreuil 16421/1743, vii) and mentioning some of the authors who had influenced the field of the practice of perspective: Vredeman de Vries, Serlio, Barbaro, Vignola, Sirigatti, Accolti, and Marolois. In choosing his style Dubreuil took over Vredeman de Vries’s idea of including a large number of diagrams and letting the text comment on them. One example of Dubreuil’s illustrations is shown in figure IX.31. He favoured the distance point construction, but also described other ways of obtaining perspective images, including Desargues’s method, as we shall soon see. In his first book Dubreuil primarily dealt with compositions dominated by the three main directions: orthogonals, transversals and verticals. In 1647 he published a continuation of his perspective work devoting it to objects that are limited by oblique planes, and in particular to perspective images of polyhedra. Finally, in 1649 he added a third part, in which he took up the question of how to make perspective constructions when the picture plane is not vertical or not flat, but for instance a floor, a ceiling, an oblique plane, or a vault. Furthermore, in this third and last volume he also treated all sorts of anamorphoses. The first part of La perspective pratique became very popular. It appeared in many editions in French, and was translated once into German and twice

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FIGURE IX.31. Dubreuil illustrating how to construct perspectival shadows. Dubreuil 16421/1651, figure 137.

into English (Dubreuil 1672, Dubreuil 1710, and Dubreuil 1743). The title page of the second English edition even contains this translated quote from Christian Wolff: “If you would proceed immediately to the Practice of Perspective, without engaging in the Intricasies of the Theory, the Jesuit’s Perspective will answer your Purpose”. This seems to reflect a common opinion, for Dubreuil’s book and Andrea Pozzo’s Perspectivae pictorum became classical textbooks for students who wanted to learn only the practice of perspective. Apparently, Dubreuil himself hoped for some success, because not long after the first volume was published he referred to a future edition (Dubreuil 16422, aiir).

Desargues and Dubreuil

D

ubreuil’s reference to the next edition of his La perspective pratique came up during a polemic with Desargues during which he stated that in revising his work he would omit any favourable mention of Desargues – a promise he actually kept. At first he was very enthusiastic about Desargues’s method. He was, in fact, so positive that in his first edition he presented the method in what he considered to be an improved version, giving full credit to Desargues. Of his changes to Desargues’s procedure Dubreuil wrote:

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Because perhaps not all for whom I work have an adequate background to grasp this practice clearly, I have thought the author would allow me to help them as well as I can, that they may pull out its usefulness.17

Desargues was not pleased with this remark, nor with the discovery that his diagram had been copied by Dubreuil (figure IX.32). He felt he had been

FIGURE IX.32. Dubreuil’s version of Desargues’s tower reproduced in figure IX.20. This illustration occurs as figure 118 in the English 1672 edition and is similar to the original one published in Dubreuil 16421. 17

Comme tous ceux pour qui je travaille, n’ont peut-estre pas assez de fond pour voir clair dans cette Pratique: J’ai creue que l’Autheur me permettroit de leur rendre aisée autant que je pourray, afin qu’ils en tirent de l’utilité. [Dubreuil 16421, 117]

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451

plagiarized and found it important that this became publicly known. He had two sets of posters printed on which he presented his views on Dubreuil’s La perspective pratique. Needless to say, he thereby opened a fierce controversy.

Dubreuil’s Comrades-in-Arms

A

t the beginning of the dispute Dubreuil was Desargues’s main opponent, but in 1644 Dubreuil was forcefully joined by the architect and stonecutter Jacques Curabelle. The latter was mainly upset by two publications on stone cutting written by Desargues and Bosse, respectively (DesarguesS 1640 and BosseS 1643). In his Examen des œuvres du Sieur Desargues (Survey of Master Desargues’s works), Curabelle attacked both works and included a section devoted to a criticism of Desargues’s perspective method (Curabelle 1644). For a social study of the zeal and the scathing arguments some members of the circle of Parisian perspectivists applied to insult each other, the posters, pamphlets, and books printed in connection with the polemic are interesting reading.18 However, seen from the point of view of the development of ideas on perspective, the publications are uninteresting due to their lack of any scientific arguments. In his first reply to Desargues’s attack, Dubreuil wrote that it would have been better for Desargues if his friends had persuaded him not to open the dispute (Dubreuil 16422, aiiv). This seems true, because the episode injured Desargues’s reputation deeply, and caused several of his peers to stop regarding him as the inventor of a new perspective method. This change of attitude is, for instance, reflected in Niceron’s reaction. When a Latin edition of his book on perspective appeared, it was without the earlier mentioned praise of Desargues’s method (Niceron 1646, 111; see also TatonS 19511, 17). Dubreuil seems to have been partially responsible for the shift. In defending himself against Desargues’s accusation of plagiarism, he launched a counterattack in which he reproached Desargues for having copied Vaulezard and Aleaume (Dubreuil 16422, aiir). The fact that he referred to Aleaume’s work in 1642 shows that he knew this work before it appeared in Migon’s edition the following year – but whether Dubreuil was familiar with the original unpublished print or with the then not-yet-published edition I cannot tell. Dubreuil remarked that foreshortening scales did not originate with Desargues. This is correct, for as we have seen, Vaulezard had indeed published such scales in 1631, and Aleaume had presumably also included some in his work. However, it is not necessarily true that Desargues took over the work of others when he introduced his scales. In fact, from the early history of perspective onwards, several authors had been engaged in determining the foreshortening of orthogonal lengths; hence, the making of a scale was a

18

For a survey of the entire polemic see TatonS 1971 and for more details TatonS 19511, 50–55. The latter reference contains a list of the publications related to the polemic, most of which are reprinted in Poudra’s edition of Desargues’s Œuvres.

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natural continuation. Even if Desargues was aware of Aleaume’s and Vaulezard’s work, at least he did not copy their constructions of the scales (captions of figures IX.9, IX.13, and IX.18–19). Finally, Desargues should receive credit for extending the use of scales to three-dimensional figures, because earlier works had only applied scales to plane figures. This section has recounted what could be called the first act in the drama about Desargues’s perspective method. The second act unfolds in section IX.9.

IX.8

D

The Work of Niceron

uring the golden period of French perspective the art of making perspectival anamorphoses got its first proper textbook. It was written by the Minim priest Jean François Niceron (1613–1646), depicted in figure IX.33. He was – to use Michael S. Mahoney’s precise expression – “sympathetic to the natural magic still current at his time ... [and] tended to view optics as the art of illusion rather than the science of light” (MahoneyS 1974, 104). This attitude is reflected in the title of Niceron’s book: La perspective curieuse, ou magie artificiele des effets merveilleux (The curious perspective, or the artificial magic of marvellous effects, 1638) – the marvellous effects being anamorphoses. Niceron’s original intention had been to devote his entire work to anamorphoses (Niceron 1638, preface, 4). However, he also wished to connect the geometric understanding of anamorphoses to the theory of perspective, and therefore engaged in a study of perspective in which he familiarized himself with seventeen books on the subject. Like many other authors on perspective, Niceron was dissatisfied with what he had read, and therefore decided to begin his work on anamorphoses with a section on perspective intended to help the layman. He was especially disappointed at finding no detailed descriptions of how to construct perspective images of the Platonic bodies – objects that since Antiquity have been connected to the wonders of science and creation. Niceron chose to present the distance point method (figure IX.34) and demonstrate a number of applications, dealing in particular with how the regular, and some semiregular, polyhedra are thrown into perspective. Niceron seems to have achieved what he aimed for: an introduction to perspective that offers readers an opportunity to understand the why of the rules without scaring off those unacquainted with geometry. In chapter II we saw how Piero della Francesca constructed anamorphoses by projecting a virtual three-dimensional object upon a horizontal plane situated behind the object. This procedure was not taken up by his successors, who instead dealt with the problem of how a given drawing can be projected so that its image is an anamorphosis. The normal procedure was, as noted, to assume that the drawing was placed in a vertical plane and to equip this plane with a grid of squares. From a given eye point, the grid was projected upon the plane of the anamorphosis – which in Niceron’s examples was a wall

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FIGURE IX.33. Portrait of Niceron. The background shows the Roman church Trinità dei Monti, to which Niceron was attached for some time. Niceron 1638/1652.

perpendicular to the drawing. The essential problem was then to construct the image of the grid. Niceron stressed that this construction procedure is analogous to determining the perspective image of square tiles in a floor (Niceron 1638, 52).

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FIGURE IX.34. Niceron’s presentation of a distance point construction. Niceron 1638/1652, table 3.

Niceron’s Construction of an Anamorphic Grid

T

he principle behind Niceron’s construction is the following (figure IX.35). Let the vertical plane e contain a grid of squares with the diagonal LN – for the sake of simplicity I have included only four squares. Further, let O be the eye point, P its orthogonal projection upon the picture plane p (that is, the wall), and D the point above P on the vertical through P determined by DP = OP. This implies that P is the principal vanishing point, and since the horizontal lines of the grid are orthogonals, they are projected into lines passing through P. The vertical lines of the grid are parallel to the wall, and hence their images are vertical lines whose position can be determined by looking at the diagonal LN. This diagonal has D as its vanishing point and L as its intersection point, and hence it is mapped upon DL. As in any ordinary distance point construction, the points in which the line DL meets the images of the orthogonals determine the positions of the images of the lines parallel to p. By and large, Niceron’s description is similar to the one just given, and his construction is reproduced in figure IX.36, with a more elaborate illustration from 1646 shown in figure IX.37. As mentioned in section IX.3, Niceron also presented a mathematically exact construction of an anamorphic grid for cylindrical mirror anamorphoses, namely the one developed by Vaulezard. Even so, he advocated the use of a simpler grid, reproduced in figure IX.38. Having written a book on perspective and anamorphoses for practitioners, Niceron decided to also attempt to reach a more erudite circle of readers in an expanded Latin version of La perspective curieuse, in which he included more theoretical material – and more illustrations. This second version was published

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455

Ni L

p

Mi

D M N P O

e

FIGURE IX.35. A grid of squares projected upon an anamorphic grid.

under the title Thaumaturgus opticus seu perspectiva curiosa (Thaumaturgic19 optics or the curious perspective) in 1646, the year of Niceron’s death. Actually, he had started to translate the new version into French, but did not finish the project before he died. It was initially carried on by Niceron’s former teacher Marin Mersenne, but in 1648, he too died without completing the translation. Gilles Personne de Roberval then took over the project and saw the second edition of La perspective curieuse through the press. The book finally appeared in 1652 and in a couple of later reprints. One of the new themes Niceron took up in his second edition was the use of instruments. 19

Thaumaturgus means wonder-working.

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FIGURE IX.36. Niceron’s illustration of how to construct an anamorphic grid. Niceron 1638, figure 35.

Niceron not only wrote about anamorphoses, he also had some designed for the Minim convent at Place Royal in Paris (BaltruˇsaitisS 1977, 50, 150). Moreover, he inspired the creation of an anamorphosis at the then Minim convent in Rome, Trinità dei Monti, where he was appointed professor of mathematics in 1639. The anamorphosis was carried out by Emmanuel Maignan (1601–1676), who also worked at the convent in Rome, and who

FIGURE IX.37. An anamorphic projection. The upper layout is Niceron’s own arrangement of the figures, and the lower shows the two diagrams horizontally joined. Niceron 1646, figures 66 and 67.

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FIGURE IX.38. Niceron’s grid for a cylindrical mirror anamorphosis. He drew the lines intended to appear as verticals as lines radiating from the point E; and the lines meant to appear as horizontals he constructed as circles with centre E, and with radii that increase at a ratio of 21:20 (Niceron 1638, 85). The person depicted is Saint Francis of Paola (BaltruˇsaitisS 1977, 147). Niceron 1638, figures 57 and 58.

shared Niceron’s admiration for ‘natural magic’. Maignan took an almost obsessive interest in sundials and wrote a comprehensive volume on their construction, in which he included a small section on anamorphoses, describing the one at Trinità dei Monti (Maignan 1648). This anamorphosis is extremely impressive and can still be admired where it was created – as can one of Maignan’s sundials. Viewed from the eye point the anamorphosis shows a picture of the founder of the Minim Order, Saint Francis of Paola (figure IX.39), whereas seen from other positions it recounts the story of his dangerous crossing of the Strait of Messina (figure IX.40).

IX.9

B

Second Act of the Desargues Drama

efore returning to the further fate of Desargues’s method, I will briefly describe a book that differs from other perspective works published in France during the 1630s and 1640s. Almost all such works appeared in Paris and applied either the classical distance point method or perspective scales. Apparently independently of – though presumably stimulated by – the

FIGURE IX.39. Maignan’s anamorphosis viewed from its eye point. Trinità dei Monti, Rome.

FIGURE IX.40. The previous figure seen from a position that brings out the anamorphic effect and shows a landscape and a seascape with tiny objects.

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Parisian development, the dean Nicolas Baytaz published his Abbreviations des plus difficiles operations de perspective pratique (Abbreviations of the most difficult operations in the practice of perspective) in Annecy in 1644. Addressing his book to “real painters”, he offered them a number of new perspective constructions. It is doubtful that Baytaz’s methods appealed to the painters. They are, however, quite interesting in themselves, because they illustrate that there is no limit to the number of constructions that can be invented, and one gets the impression that Baytaz really took pleasure in finding new constructions. Figure IX.41 presents one of his procedures.20 Baytaz did not argue for the correctness of his constructions, claiming that he did not pretend to “be amused by demonstrations” (Baytaz 1644, 19).

A

P E

B9

C G

R A9 B

FIGURE IX.41. Diagram setting out one of Baytaz’s new techniques. He applied it to throw a pentagon into perspective, but as it is a pointwise construction, I present it as such. Let GR be the ground line, P the principal vanishing point, d the distance, and B a point in the ground plane. On the normal to GR through B (cutting GR in C), Baytaz introduced the auxiliary point B′ defined by CB′ = CB, and on the normal to GR through P two more auxiliary points: A and A′, determined by PA = PA′ = d. He then constructed the image of B as the point of intersection, E, of BA and B′A′. His construction stands alone without any theoretical arguments. Its correctness can, for instance, be demonstrated in the following way. From the division theorem we know that the image Bi of B is the point on PC determined by PBi : BiC = d : BC.

(1)

Hence, we must prove that E lies on PC and that it divides it in the correct ratio. From the construction follows that the triangles AA′E and BB′E are similar, and that the lines EP and EC are medians in them. These facts imply that the points P, E, and C are collinear, and that the triangles PAE and CBE are similar. This last result shows that E divides PC in the ratio AP : BC = d : BC as it should according to (1). Adaptation of figure 47 in Baytaz 1644.

20

For a discussion of some of Baytaz’s other constructions, see PoudraS 1864, 327–330, and PhillipS 1947, 75, 172.

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Desargues’s Supporter Bosse

N

ot much notice was taken of Baytaz’s book in Paris, where the perspectivists were busily engaged in discussing the merits of Desargues’s method. In section IX.7 we saw how several perspectivists turned away from Desargues. Even so, he did not become completely isolated, for he found a true admirer in the well-known engraver Abraham Bosse (1602–1676) – one of whose engravings is reproduced in figure IX.42. Bosse expended much energy on promoting Desargues’s method, and in 1648 he published a book entitled Maniere universelle de Mr Desargues pour pratiquer la perspective (M. Desargues’s universal method for practising perspective). In this work, which I have referred to a couple of times already, Desargues’s own presentation of twelve pages was expanded into about three hundred and fifty pages, and Desargues’s one figure was replaced by more than one hundred and fifty illustrations (including the examples shown in figures II.1, IX.22, IX.43, and IX.44). Bosse was one of the writers who worried about the gap between practitioners of mathematics and mathematicians. He found fault with both groups, reproving the practitioners for not being patient enough in their attempt to understand the mathematicians, and blaming the mathematicians for not trying harder to make themselves understandable. In fact, he believed the mathematicians had limited their commitment to finding the real truth and therefore showed scant concern for enlightening the people who applied perspective (Bosse 1648, 29–33). Notwithstanding this complaint, Bosse himself made no attempt to teach the practitioners the theory of perspective. Maniere universelle is basically a practice of perspective, organised – like the works of Dubreuil and others – so that the text serves as commentary to the illustrations. The book does, however, contain a few theoretical sections that presumably stem directly from Desargues, not all of which are devoted to perspective. Most noteworthy is the previously mentioned result that later came to play an important role in projective geometry, and which is now universally known as Desargues’s theorem (Bosse 1648, 340). Bosse followed his first work on perspective up with a thinner volume on how to make constructions on curved surfaces (Bosse 1653). Bosse’s technique, illustrated in figure IX.45, was the same as one presented by Pozzo (page 390).

Bosse and the Royal Academy of Painting

T

he appearance of Maniere universelle coincided with the founding in Paris of the Académie royale de peinture et sculpture (The Royal Academy of Painting and Sculpture). The academy saw the education of students as one of its most important tasks, and Bosse offered to assist by lecturing on perspective. His proposal was accepted, and he began his lectures in May 1648.

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FIGURE IX.42. Les graveurs, engraving by Abraham Bosse. BlumS 1924, plate 23.

Although the academy did not admit engravers as members, Bosse was appointed an honorary academician in 1651. This was also the year that the editio princeps of selected writings by Leonardo da Vinci appeared in Paris – in both an Italian version and a French translation (Leonardo 16511 and Leonardo 16512). This very edition would soon give rise to a heated debate about Bosse’s lectures. Bosse eagerly told his students that they should create their compositions according to the rules of perspective, and he taught them Desargues’s method. Some academicians, guided by the extremely influential painter Charles Le Brun, found Bosse’s lectures too technical and advocated a more general introduction to the art of painting based on the Leonardo book. Being enthusiastically interested in perspective Bosse could not accept this proposal. This is understandable: if he were to use the Leonardo publication as a textbook, he would not only have to stop presenting Desargues’s method, but he would have to give up teaching the technique of perspective constructions altogether since this technique was not included in the book (page 83). Altogether Bosse was not impressed by the edition of Leonardo texts arguing that the only order in the chapters was their numbering, further stating that in his impression that the book was composed of aphorisms written down at various times (Bosse 1665, 127).

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FIGURE IX.43. Bosse’s example of what he called perspective horizontale, meaning a perspective composition that has a horizontal picture plane. Bosse 1648, figure 108.

This disagreement between Bosse and some members of the academy sparked a second dispute, as bitter as the first, about Desargues’s method.21 Bosse was supported by the painter Laurent de La Hire, whose son Philippe

21

For more details on the dispute see BlumS 1924, 17–36; HeinichS 1983; BaltruˇsaitisS 1977, 71–78; KempS 1987; C. GoldsteinS 1990; KempS 1990, 120–123; and LaurentS 2000 (in which some of the documents are reproduced).

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FIGURE IX.44. An example of Bosse constructing perspectival shadows. Bosse 1648, figure 112.

de la Hire was one of the few who followed up Desargues’s creative ideas on geometry – Blaise Pascal being another, as noted. Following the death of Laurent de la Hire in 1656, Bosse’s situation at the academy became more difficult. A bit later he once again turned down a suggestion of using a new textbook. This time the book in question was Traité de perspective (Treatise on perspective), composed by academy member Jacques le Bicheur and dedicated to Le Brun and appearing in 1660 with handwritten text (DescarguesS

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FIGURE IX.45. Bosse projecting a grid of squares upon a curved surface. Bosse 1653, part of plate 2.

1976, 108). Copies of le Bicheur’s book are apparently very rare, at least I have found none and am therefore unable to describe Bicheur’s approach to perspective. Ultimately in 1661, the year of Desargues’s death, Bosse was dismissed from the academy. Even after leaving the academy Bosse kept fighting for Desargues’s method, and in 1665 he published the lectures he had given at the academy in a volume entitled Traité des pratiques géométrales et perspectives enseignées dans l’Académie royale de la peinture et sculpture (Treatise on the practices of geometrical planes and perspectives taught at the Royal Academy of Painting and Sculpture). He had previously attempted, in vain, to get the academy’s approval of this publication, which also met opposition, as we shall see shortly. If Bosse had indeed followed his textbook when lecturing, then an introduction to elementary practical geometry rather than perspective would have been the major part of his course, and he would indeed have been rather

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465

technical, though not exceedingly so in connection with presenting perspective constructions. Looking back on how Desargues’s method was received in France, it is worth noting that the debate it raised more concerned emotions than rational arguments. After Desargues’s death, his method came to be appreciated outside France. Bosse’s Maniere universelle was translated into Dutch in 1664, as we saw in section VII.8, and Desargues’s method was applied by the Dutch engravers Dirk Bosboom and Caspar Jacobszoon Philips (Bosboom 1703 and Philips 1765). Either directly or through the Dutch translation, Bosse’s presentation of Desargues’s method also became known in Germany, where it was taken up by Johann Leonhard Rost and Johann Jacob Schübler (Schübler & Rost 1724). Finally, Bosse’s book appeared in Japanese (TatonS 19511, 57), but I cannot say whether this edition of the work had any influence.

IX.10

The 1660s and 1670s

Huret

T

he draughtsman and engraver Grégoire Huret (1610–1670) – an academician who was also close to the king – took over Bosse’s lectures at the Royal Academy of Painting and Sculpture (BaltruˇsaitisS 1977, 75). In 1670, Huret published Optique de portraiture et peinture (Optics of portraying and painting) in which he expressed great concern about the state of the art, particularly criticizing Bosse’s book from 1665 and previous publications on Desargues’s method. Huret claimed that his book should be seen as a contribution to the education of the young, especially to teaching them useful rules for drawing in perspective. Following the general layout used by Aleaume and Migon, Huret divided his book into two parts entitled la perspective pratique (practical perspective) and la perspective speculative (speculative perspective). Yet Optique is far from being a textbook that gradually builds up a system of perspective constructions. Instead it consists of a curious mixture of observations, where the author offered useful constructions in some, but in others presented incorrect conclusions (PoudraS 1864, 430–463). Most of his book is devoted to comments – largely unkind ones – on earlier publications on perspective, and particularly to negative comments on Desargues’s method. As pointed out by Poudra, a few of Huret’s observations on mistakes made by Bosse and others are justified, but most of them are not (PoudraS 1864, 441). Huret was well read and discussed books on other subjects than perspective. Naturally enough these included general works on painting, but also Desargues’s and Pascal’s works on projective geometry (Huret 1670,

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157–158), despite only knowing Desargues’s Brouillon project from a copy made by Philippe de La Hire. Huret even wrote a few favourable words about Brouillon project, although it received more negative remarks than positive. Among the more productive passages in Huret’s book one can find a presentation of the construction and use of an angle scale (Huret 1670, 116). He also dealt with a point most authors avoided, namely the use of scaled lengths in perspective constructions. In this connection he presented the technique for applying a reduced distance in a distance point construction – without reducing the actual distance – that ’sGravesande applied later (page 336). Determining whether Optique de portraiture et peinture has some kind of underlying theme would require a much more detailed analysis than I have carried out. Among other things, Huret may have been searching for a closer link between perspective and projective geometry. It also seems he had some doubts concerning perspective that were similar to those expressed by Leonardo da Vinci.

Le Clerc

H

uret’s preoccupation with perspective seems to have given rise to lively discussions within a circle connected to the Académie royale de peinture et sculpture. References to debates on perspective – in which Huret

FIGURE IX.46. Sébastien Leclerc’s engraving portraying the activities of the Parisian academies of the sciences and arts, 1698.

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participated – are made by the engraver Sébastien Leclerc (1637–1714) in a book devoted to a discussion of the mathematical model for perspective drawing (Leclerc 1679). Leclerc himself practised and taught perspective. In 1672 he was admitted to the academy and became professor of geometry and perspective there, keeping this position for thirty years (WatsonS 1939, 558). In 1721 Leclerc’s son, also called Sébastien Leclerc, took over his father’s teaching position as well as his lecture notes, which he enlarged over the years (Jeaurat 1750, vi). The entire material was inherited by the older Sébastien Leclerc’s grandson Edme Sébastien Jeaurat, who edited it and had it published in 1750 – in section IX.11 I will return to his edition. Leclerc, the older, made many engravings for the Académie des sciences (WatsonS 1939), of which the most famous is probably his engraving from 1698 (figure IX.46) showing the activities of the academies of the sciences and arts, including some people occupied with a perspective instrument (figure IX.47).

Bourgoing

I

n this chapter we have so far seen members of the Minim and Jesuit Orders as authors on perspective, but the Augustine Order was also represented, in the person of Charles Bourgoing. He wrote the truly remarkable book La perspective affranchie ... sans tracer ny supposer le plan geometral ordinarie (The perspective freed ... without drawing nor assuming the ordinary geometrical plane), which was published in 1661. Bourgoing’s work is vaguely connected to the dispute about Desargues’s

FIGURE IX.47. Detail from figure IX.46 showing a perspective instrument.

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method at the Académie royale de peinture et sculpture. During the controversy – in July 1657, to be more precise – Desargues offered a prize of thousand francs to any Frenchman who could present a perspective method that was more universal, more easily understandable, and more quickly performed than his own (Bourgoing 1661, Preface 1r; figure IX.48). Bourgoing professed himself a candidate for Desargues’s prize, and in his own estimation his book embodied all the qualities Desargues had requested, and quite a few more. Let me state at once that clarity is not among the book’s virtues. It is often difficult to understand precisely what Bourgoing had in mind, and even previous knowledge of the subject is not always enough, since the author worked with rather unorthodox concepts and applied unusual terminology. In addition, it is sometimes difficult to decipher his text at all, for it is a stencilled version of a handwritten manuscript (figure IX. 48). Nevertheless, it does pay to struggle with Bourgoing’s text, because hidden within it are several original ideas resembling thoughts presented later by Taylor and Lambert. In fact, the French title of Lambert’s book, La perspective affranchie de l’embaras du plan géométral (The perspective freed from the nuisance of the geometrical

FIGURE IX.48. A passage in which Bourgoing presented Desargues’s prize. The text reads: ... I thought I could not choose a better means than to claim the prize that the very learned and generous man, Monsieur Desargues has proposed to he among us Frenchmen who could find a perspective method that is better and more perfect than his, doing so in a letter he wrote to Monsieur Bosse which was printed in Paris on July the 25th in the year 1657; a letter read to the Royal Academy on the following 29th by the aforementioned Monsieur Bosse and then distributed to all those interested in perspective.*

*... j’ai creu ne pouvoir choisir un moien plus favorable que de pretendre aux prix que Mr Desargues, home tres scavant et genereux, a proposé a celui de nos françois, qui trouveroit une maniere de perspective meilleure et plus parfaite que la siene, par une lettre qu’il a ecrivoit à Mr Bosse, qui a este imprimée a Paris le 25 Juillet de l’année 1657: et qui a este leüe a l’academie Roiale par led. Mr Bosse le 29. suivant, et distribuée ensuite à tous les curieux de la perspective. [Bourgoing 1661, Preface, 1r]

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469

FIGURE IX.49. Bourgoing’s construction of a perspective reflection. Bourgoing 1661, 140.

plane) resembles that of Bourgoing’s, but as Bourgoing’s work received little attention, I find it quite unlikely that Lambert became aware of the book almost a hundred years after its publication. As Bourgoing stressed in his title, he aimed at performing perspective constructions without using plans. Although he did not refer to earlier writers, he might have been inspired by Aleaume and Migon. He went further than they had, however, by also dealing with three-dimensional objects. First he considered a line with a given direction, but not parallel to the picture plane, and assigned to this line two points de veüe, un reelle & un feint (two viewpoints, one real and one feigned, Bourgoing 1661, 3–4). Contrary to what one might expect, the real one is the point at infinity defined by the direction of the line, while the feigned one is the line’s vanishing point in a

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given picture plane. Bourgoing mentioned that to an eye, the two points appear to be on a line (ibid., 5). He then generalized these concepts, similarly ascribing a real and a feigned line to a plane that is not parallel to the picture plane. He thereby became one of the first to work with the general concept of a vanishing line and to pair it with a concept corresponding to a line at infinity. Bourgoing presented a couple of methods for constructing the vanishing point of a horizontal line, one of which is similar to Aleaume and Migon’s and is based on the use of an angle scale. Bourgoing applied his theory to design a method of making perspective constructions that he illustrated by throwing polygons and polyhedra into perspective. His method is remarkable in that it avoids the problem of reversing (section VII.7) – a point upon which Bourgoing himself did not comment. He also demonstrated how his theory can be used for constructing perspective shadows and reflections, the latter being illustrated in figure IX.49. Since Bourgoing’s work completely lacks clarity and leaves many details to the reader, it is highly unlikely that it was much noticed by practitioners of perspective. Bosse was aware of its existence, but did not consider it useful (Bosse 1665, 127). It is also doubtful that Bourgoing’s La perspective had any influence on the theoretical development, yet despite the book’s shortcomings, it is interesting to notice that its author attempted a new approach to perspective that later turned out to be mathematically fruitful.

IX.11

Perspective and the Educated Mathematician

I

n chapter VII we saw how in the Netherlands the promising young scientist Christiaan Huygens was advised to study perspective – and how he found the subject rather trivial. In France, several of the leading mathematicians demonstrated a knowledge of perspective while at the same time taking a less arrogant attitude towards the topic than Huygens. Fermat and Descartes, for example, had such an extensive familiarity with perspective that they found it natural to express their appreciation of Desargues’s perspective procedure (page 448). Likewise, Roberval, whose main interest was pure science and who is known for his contributions to infinitesimal calculus and optics, agreed in 1648 to edit the last edition of Niceron’s work on perspective (page 455), which must mean he had studied the topic. There is even evidence that another of the famous French mathematicians, Blaise Pascal, made his own contribution to perspective. In an address he sent to “the Parisian Academy”,22 he announced that he was preparing a

22

Presumably denoting the informal group of scientists also known as l’Académie Le Pailleur (MesnardS 1964, 10)

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very brief, attractive, and quick method for perspective construction (PascalS 1654). In this section I would also like to mention the famous philosopher and scientist Gottfried Wilhelm Leibniz, for although he was a German, his principal introduction to mathematics took place during his stay in Paris in the period 1672–1676. He took an interest in Desargues’s and Pascal’s approach to geometry, which seems to have led him to explore perspective. He read and made marginal notes in the books written by Aleaume, Dubreuil, and Bosse (EcheverríaS 1994, 283). However, although he became acquainted with perspective in this way, he was presumably more intent on looking for ideas associated with projective geometry than on studying perspective construction techniques.

IX.12 French Eighteenth-Century Literature on Perspective

A

t the end of the seventeenth century, the interest in perspective declined in France. The remarkable contributions from the middle of the century were not followed up, and not a single book devoted solely to perspective was published in France during the last three decades of the seventeenth century (cf. the survey of French literature in appendix four). The eighteenth century did not bring a revival of French writings on perspective: apart from short articles, I am aware of only a dozen publications appearing in the eighteenth century. The authors were almost evenly divided between theoreticians and practitioners. Both groups focussed upon the art of perspective construction, keeping to the most popular methods. Several authors maintained that although the literature on perspective was considerable, there was a need for a clear presentation of constructions. Authors with a mathematical background included some geometry in their writings, but a less abstract theory of perspective than the one presented in the previous century. Rather than basing their arguments on the main theorem, they relied on the division theorem and a widespread use of similar triangles. In presenting the French eighteenth-century publications on perspective in greater detail, I by and large follow their chronological order.

Lamy

T

he first book discussed here appeared at the very beginning of the century, in 1701, bearing the traditional title Traité de perspective (Treatise on perspective). Its author was the Oratorian priest Bernard Lamy (c. 1640–1715), who had a wide variety of interests and published on several scientific and theological topics. He was very fond of mathematics and published a book on geometry in 1685, which he had initially planned to

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extend to a cours de mathématique. He became distracted by other projects, however, one of which concerned the Temple of Jerusalem. This project inspired him to refresh his knowledge of perspective (Lamy 1701, preface). His studies resulted in a charming volume that illustrates the above-mentioned eighteenth-century attitude among the French authors of avoiding excessive abstraction in presenting the theory of perspective. Perhaps inspired by Leonardo da Vinci (page 85), Lamy distinguished between perspective aërienne (aerial) and perspective lineale (linear), which was not very common in his day. He mainly concerned himself with linear perspective. In his practice of perspective Lamy concentrated upon the construction of the image of a grid of squares by means of a distance point method. He then used the perspective grid for various procedures, such as determining the perspective image of a plane configuration (figure IX.50), constructing images on a vault (figure IX.51), and creating an anamorphosis (figure IX.52). Lamy also touched upon a few mathematically complicated problems, such as how to work with an oblique picture plane (Lamy 1701, 166ff ). In his solutions he applied the rotation introduced by Stevin (page 271 and caption of figure VI.24). Lamy’s geometrical explanations are often interrupted by long sections in which he discussed various matters concerning painting, such as the choice of parameters in a picture. He also voiced his opinion on the themes suitable for a painting, and commented upon the works of some of the Italian masters. He stressed several times that geometry does not make a painting, but supports it, asserting, among other things:

FIGURE IX.50. Lamy’s use of a perspective grid of squares for constructing the perspective image of a fortification. This illustration is very similar to one Stevin published about hundred years earlier (figure VI.36). Lamy 1710, plate 18.

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FIGURE IX.51. An optical projection of the image of a square grid to be used for drawing a perspective composition on a vault. As pointed out by Pozzo, it is doubtful whether this method would work in practice (page 390). Lamy 1710, plate 27.

Mathematicians draw only Lines, they cannot finish a Picture; And on the other hand, Painters cannot begin it without a regard to the Rules taught by the Mathematicians.23 [Lamy 1710, 6] 23 Les Mathematiciens ne tirent que des lignes, ils ne peuvent donc pas achever un tableau; mais aussi les Peintres ne le peuvent commencer, s’ils ne se fondent sur les régles que les Mathematiques enseignent. [Lamy 1701, 6]

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FIGURE IX.52. Lamy’s grid for constructing an anamorphosis on a wall. Lamy 1710, plate 25.

To justify his treating the art of painting as a non-expert Lamy ended his book by quoting Socrates’s discussion with the painter Parrhasius and the sculptor Kleiton24 as created by Xenophon in his Memorabilia. Lamy does not seem to have influenced the French literature on perspective, but his book, which was translated into English in 1702, is likely to have given Taylor some ideas – a point to which I will return in chapter X.

Bretez, Courtonne, Deidier, and Roy

I

n contrast to Bosse, the engraver and dessinateur d’architectures (architectural draughtsman) Louis Bretez was accepted as a member of the Académie royale de peinture et sculpture, where he also taught perspective (SørensenS 1996). In 1701 Bretez published La perspective practique de l’architecture (The practical perspective of architecture) with the primary aim, it seems, of teaching his students the subject through a study of instructive perspective compositions, which make up the staple of his book (figure IX.53). His sparse commentary, printed in hand-written lettering, included a presentation of a distance point construction.

24

Kleiton is not known in other connections, thus he may be Xenophon’s own creation.

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FIGURE IX.53. Bretez’s example of throwing a staircase into perspective. Bretez 1706, figure 37.

A distance point method was similarly basic in another manual on drawing architectural elements in perspective: Traité de la perspective pratique avec des remarques sur l’architecture (Treatise on the practice of perspective with remarks on architecture), published in 1725 by the architect Jean Courtonne (1671–1739). He became a member of the Académie royale d’architecture in

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1728 and began lecturing there in 1730 (NeumanS 1996, 64). In his book Courtonne acknowledged having been inspired by Niceron and others, but noted that he was not content with their publications and found that they lacked clarity: I admit that had I had some methodic or sufficiently understandable treatises on this art in my hands ... I would perhaps not have thought of this project. However, in the majority of the books on perspective I have seen thus far, I have found so many obscurities and so little order concerning the method as well as the material that I have decided to gather, from all of these authors, some detached bits and pieces.25

Courtonne had apparently struggled to obtain an understanding of perspective from the existing literature, and having finally managed to do this, he wished to share his insights with others. A similar story can be told for a large part of the literature on perspective, from all the areas considered. We have already come across several examples of this, and we will see more.26 As a matter of fact, it almost became a tradition to introduce a book by complaining about previous works. Some authors were dissatisfied with the way the practice of perspective was treated, and others were unimpressed with the presentations of the theory. Another French teacher, the Jesuit Abbé Deidier (1696–1746), also chose to base his presentation of perspective on the distant point construction occurring in Traité de perspective théorique et pratique (Treatise on the theory and practice of perspective, 1744). The following year he republished this work in a general textbook on mathematics that was compiled of lectures he gave as a professor of mathematics at the military school at Fère (Deidier 1745). This book is the only instance I have found of perspective being included in a general book on mathematics published in France in the eighteenth century. Despite the fact that Deidier included the word “theoretical” in the title of his work, he was not overly concerned with the geometric theory behind perspective, instead offering his students a substantial text spanning hundred pages and accompanied by one hundred and three figures. Distance point methods also met with criticism, not least from the engraver and glass painter Claude de Roy in his Essai sur la perspective pratique (Essay on the practice of perspective, 1756). He believed that perspective constructions generally contained too many auxiliary lines, and that these lines were confusing when the final image was to be drawn. He found this particularly

25

J’avoue que s’il m’étoit tombé entre les mains quelques Ecrits méthodiques de cet Art, ou assez intelligibles ... je n’aurois peut-être jamais pensé à ce Projet; mais je trouvois dans la plûpart des Livres de Perspective que j’avois vû jusqu’alors tant d’obscuritez & si peu d’ordre, aussi-bien dans la méthode que dans la matiere, que je me déterminai à ramasser de tous ces Auteurs quelques morceaux détachez. [Courtonne 1725, Preface] 26 For the situation in Germany around 1800, see SiebelS 1999, 161.

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true of distance point constructions and constructions performed with the aid of perspective scales. Even though he opposed both methods, he still suggested a new method of constructing scales (Roy 1756, 13–17). The way he avoided the many lines was by means of arithmetic and proportions. Focussing on the foreshortening of orthogonal lengths, he deduced the relation we already saw in Piero della Francesca’s work (cf. relation (ii.1)). Roy worked on a form that states, in my notation (d + a) : h = a : f(a). Instead of a geometrical construction, Roy suggested calculating f(a) from this relation, and he made a small table for various values of d, h, and a.

Petitot and Curel

A

lthough distance point constructions were popular, they did not completely dominate the textbooks. During the second half of the eighteenth century, some French perspectivists returned to the very early perspective procedure of applying plan and elevation constructions. The French architect Ennemond Alexandre Petitot (1727–1801), who worked in Italy, settled for this kind of method in a booklet appearing in 1758 (figure IX.54) with parallel French and Italian text. The chevalier and engineer Nicolas François de Curel also advocated the use of the plan and elevation construction in his Essai sur la perspective linéaire (Essay on linear perspective), published in 1766. This title is in itself remarkable for being the earliest French title containing the expression “linear perspective. Whether Curel got his inspiration directly from Leonardo da Vinci, from his French predecessors, or from Brook Taylor I cannot say. Curel’s book also deviates from the norm with respect to its contents. Inspired by optics, Curel divided perspective projections into three types: depending on whether they provided representations of objects 1. that are perceived directly 2. that have been reflected in a mirror 3. that are seen after a refraction has taken place. He believed that for artists the third type is “more curious than useful”, the second not really needed, but the first very important (Curel 1766, v). Curel accused earlier writers of making the mathematical theory of perspective unnecessarily complicated. According to him, “the entire theory of perspective is nothing but a simple application of similar triangles” (ibid., iii). Curel’s treatment of perspective provides fascinating problems in descriptive geometry – even before this discipline was created – for he solved most of his problems by looking at plans and elevations. Among other things, Curel used a plan and an elevation to prove how parallel lines that are not parallel to the picture plane “entirely lose their property of being parallel in the picture plane” (ibid., 21). His treatment of perspective reflections and perspectival shadows

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FIGURE IX.54. The French title page of Petitot’s book.

was similarly based on plans and elevations. He formulated a number of rather intricate problems, one being the following: Determine the shadow cast on a cylinder by a sphere when the source of light is the sun (ibid., 50). However, he only treated his problems in their simplest form. Although Curel was inspired by optics in dividing perspective into three categories, he was strongly against treating perspective together with optics.

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He claimed that doing so would detract from perspective, and that optics would gain nothing (Curel 1766, ii).

Lacaille

C

urel recommendation of keeping perspective and optics apart did not mean a new approach, for this had actually already been customary for quite some time. Ten years earlier, however, Nicolas Louis de Lacaille (1713–1762) had continued the earlier tradition and included perspective in his Leçons élémentaires d’optiques (Elementary lectures on optics, Lacaille 1756). Lacaille is known first and foremost for his astronomical and geodetic measurements. He took part in the project of settling one of the major points of controversy between Cartesians and Newtonians, namely the shape of the earth. Measurements carried out by Lacaille and others confirmed that Newton had been right in assuming that the earth is flattened at the poles (GingerichS 1973, 542). Lacaille’s excellent reputation won him a chair in mathematical sciences at Colle`ge Mazarin in Paris in 1739, and led to membership of L’Académie royale des sciences in 1741. Explaining why he had chosen to write yet another presentation of perspective, Lacaille put forward an argument very reminiscent of those described earlier in this section: On this part of optics we have a great number of books ..., but to my knowledge none that includes the principles in a sufficiently general manner. Commonly one finds only vague practices, presented obscurely, without order and without proofs.27

His treatment of perspective is not merely a repetition of old material, however, but shows quite a bit of original thinking – partially inspired by optics and astronomy. Lacaille’s introduction of vanishing points, for instance, is very untraditional. He began with the optical result that parallel lines appear to converge, introduced a virtual convergence point he called “the apparent common point”, and proceeded as follows (Lacaille 1756, §§82 and 343 – the notation is mine): Together with a set of parallel lines that are not parallel to the picture plane p, Lacaille considered the line connecting the eye point O and the apparent common point of the parallel lines. This line cuts p in a point, say V, that has the property of being a convergence point for the images of the parallel lines. Finally claiming that the apparent common point of the parallel lines is a point at infinity, Lacaille concluded that V is the point of intersection of p and the line through O parallel to the parallel lines.

27 Nous avons sur cette partie de l’Optique un bien grand nombre de livres ..., mais aucun que je sache n’en renferme les principes d’une maniere assez générale. On ne trouve communément que des pratiques vagues, obscurément énoncées, sans ordre & sans démonstrations. [Lacaille 1756, iii–iv]

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To describe the determination of perspective images, Lacaille introduced two planes of reference, which he called the horizontal plane and the vertical plane, respectively (which concepts I have actually applied throughout this book). Lacaille let a point D be given with respect to these two planes – that is, he let D’s orthogonal projections on these planes be given – and he then presented two general methods for finding “la perspective d ” of D (ibid., §350). His first procedure is based on proportions and corresponds to a determination of the coordinates of d (ibid., §351). His second method is a distance point construction combined with a procedure for throwing vertical line segments into perspective (ibid., §367). After describing various traditional constructions, Lacaille introduced what he called a chassis perspectief. This consists of three scales: an ordinary length scale, a scale for foreshortening orthogonal lengths, and an angle scale on the horizon, all similar to the ones used by Aleaume and Migon. While Lacaille may have been inspired by these authors, he constructed his scales differently than they had. To prove that his construction was correct, Lacaille applied the ultimate tool of choice among French perspectivists at the time, namely the theory of similar triangles – and there really are a lot of similar triangles in Lacaille’s work. As noted, Aleaume and Migon only considered vanishing points in connection with horizontal lines, whereas Lacaille went a step further and constructed a vanishing point V for any line that was not parallel to the picture plane p. He let a line, l, or rather its direction, be given by two angles, u and v. The first, u, is the angle that l forms with its orthogonal projection, say m, upon a horizontal plane, and v is the angle that the vertical plane, say a, through l forms with the vertical plane. Lacaille presented a geometrical construction of the vanishing point V (ibid., §§410–411) that corresponds to the following trigonometric determination (figure IX.55). Let d be the distance, P the principal vanishing point, and A the point on the horizon defined by28 PA = d tan v, V is then the point on the vertical line through A given by d tan u, AV = cos v

(ix.7)

It is easy to deduce these two relations by looking at two triangles, as I have explained in the caption of figure IX.55. In proving the correctness of his construction, Lacaille found the position of the point A in a similar fashion, but he obtained the distance AV in (ix.7) in a very complicated manner. This is interesting because it reflects his habit of thinking as an astronomer, and I have therefore outlined his argument in the caption of figure IX.55.

28

It is assumed that the point A lies to the same side of P as the plane a of the vertical plane, and that V lies to same the side of A as the line l of the line m.

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p

H

V P

O

v u

A

Z

FIGURE IX.55. Lacaille’s method for determining the vanishing point of oblique lines. Let p be the picture plane, O the eye point, P the principal vanishing point, d the distance, and V the vanishing point of lines that make the angle u with the horizontal plane and the angle v with the vertical plane. Since OV is parallel to these lines, the line OV is also determined by these two angles. Finally, let OA be the projection of OV upon the horizontal plane through O. From triangle OPA it can be deduced that d . PA = d tan v and OA = cos v Similarly, from triangle OVA it can be concluded that d tan u . AV = cos v Lacaille’s own determination of AV runs as follows (Lacaille 1756, §§412–420). He first introduced a sphere with the eye point O as its centre, and on the spherical surface he considered the circle, which is parallel to the sphere’s horizontal great circle and characterized by the angle u. It is evident that the lines connecting O and the vanishing points of the lines whose direction has u as the first angle cut this circle. Taking O as the centre of projection, Lacaille projected the circle upon the picture plane, which corresponds to determining the curve defined by the vanishing points of all directions whose first angle is u – the curve being a segment of a hyperbola with its vertex on the vertical line through P. After quite a number of calculations and considerations of similar triangles, Lacaille found what corresponds to the equation of the hyperbola. Finally, he determined the distance AV by intersecting the hyperbola with the vertical line through A. Lacaille also used his sphere for presenting a construction of celestial maps that he ascribed to the French Jesuit physicist Ignace Gaston Pardies (ibid., §422).

Lacaille limited the application of his theory to one example in which he carefully described how to construct the image of a column (ibid., §§431–450). He concluded his treatise with a thorough presentation of the theory behind perspectival shadows.

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Jeaurat

T

he career of Edme Sébastien Jeaurat (1724–1803) resembles that of Lacaille in several ways. He was active in the fields of astronomy and geodesy, published on perspective, held a position as professor of mathematics at the military school at Vincennes from 1755, and became a member of the Académie royale des sciences in 1763. In his youth Jeaurat found it difficult to choose between drawing and mathematics. His interest in the first subject was inherited, so to speak, for Sébastien Leclerc was his maternal grandfather and Étienne Jeaurat, the court painter, his uncle. Before settling on the mathematical sciences, Jeaurat was awarded a medal from the Académie de peinture et sculpture in 1746. His combined interest in drawing and mathematics led him to study the notes his grandfather and another uncle, Sébastien Leclerc, had used for their lectures on perspective (page 467). He edited these lectures, as noted previously in this chapter, publishing them in 1750 under the title Traité de perspective à l’usage des artistes (Treatise on perspective for the use of artists) and illustrating them richly with perspective diagrams and vignettes (figure IX.56). Jeaurat included a bit on the theory of perspective in the first part of the book, mainly based on similar triangles. He paid considerable attention to the question of how to decide on the parameters in a picture, and one of his rules of thumb has already been presented in section VII.9 (page 367). The second part of his book treats the practice of perspective and contains a number of unorthodox constructions (PoudraS 1864, 503–505).

Taylor’s Theory Introduced in France

I

n 1757 the Jesuit writer Antoine Rivoire (1709–c. 1790) published a French translation of Brook Taylor’s New Principles of Linear Perspective from

FIGURE IX.56. Vignette introducing Jeaurat 1750.

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1719 under the faithful title Noveaux principes de la perspective linéaire. In the preface Rivoire expressed much concern of the state of practitioners’ – not least architects’ – education in perspective. In order to remedy the situation, which he found deplorable, Rivoire had decided to translate Taylor’s work, expressing his dissatisfaction with other books on perspective in a rather untraditional way: We do not lack books on this subject; although this part of mathematics has not been pursued as far as the others, we have some good books on perspective. However, some of them belong to an entire course, which an artist, needing only a small part, would hesitate to buy. Those that are printed separately can only be acquired at an immense price, either because they are voluminous, or because they have been overloaded with a bulk of plates. In the book whose translation I bring here, this double inconvenience has been avoided without falling into the usual trap of becoming obscure and unintelligible when insisting on being brief. In this book the author [Taylor] establishes the most general and comprehensive principles, and develops them with clarity and exactness. And even though he treats his subject as a great geometer, he knows how to make it accessible to those who have only a very slight knowledge of geometry.29

Thus, to the various reasons given earlier for publishing a book on perspective Rivoire added one relating to costs. It is true that Taylor’s New Principles offered the most developed theory in a small number of pages; but if Rivoire really thought this book could be understood by people who were virtually unacquainted with geometry, his expectations were unrealistic, as we shall see in chapter X. Perhaps Rivoire’s arguments should not be taken at face value, for he did not quite stick to his idea of publishing a brief book on perspective. He not only add an unremarkable introduction to perspective by the French astronomer Guillaume Saint-Jacques de Silvabelle (1722–1801), but also included a translation of the part of the English mathematician Patrick Murdoch’s Newtoni genesis curvarum per umbras (Newton’s generation of curves by shadows) that deals with perspective (Murdoch 1746). I will return to the latter in chapter X, where I argue that it is difficult to understand why Rivoire decided to publish Murdoch’s work (page 594).

29 Ce n’est pas que nous manquions de Livres sur cette matiere; & quoique cette partie des Mathématiques n’ait pas été poussée aussi loin que les autres, nous avons pourtant quelques bons ouvrages sur la Perspective: mais quelques-uns sont à la suite d’un cours entier, qu’un Artiste ne se détermineroit à acheter qu’avec peine, pour acquerir cette seule partie dont il a besoin. Ceux qui ont été imprimés séparément ne laissent pas d’être d’un prix excessif, soit par la grosseur du volume, soit par la quantité des planches dont ils sont surchargés. On a évité ce double inconvénient dans le Livre dont je donne ici la traduction, sans tomber dans celui où l’on tombe communément, quand à force de vouloir être court, on se rend obscur & inintelligible. L’Auteur [Taylor] y établit les principes les plus généraux & les plus étendus, il les y develope avec clarté & précision, & lors même qu’il traite son sujet en grand Géométre, il sçait le mettre à la portée de ceux qui n’auroient que les plus foibles connaissances de la Géométrie. [Taylor 1757/1759, v–vi]

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Rivoire himself explained that he had included Silvabelle’s section because he found it provided a background for an appendix in which Taylor touched upon the problem of constructing perspective compositions upon curved surfaces. In fact, Silvabelle outlined a method for using the intersections of the vertical plane and the ground plane and picture plane, respectively, as axes for a perspective construction that would result in a sort of a plan and elevation construction. Since Silvabelle did not work out this method, and especially since it would not be of much help for working with curved surfaces as picture planes, it completely escapes me why Rivoire would want to print Silvabelle’s composition – unless for some personal reason that has nothing to do with Taylor’s book. Taylor’s work had an enormous impact on perspective in England in the second half of the eighteenth century, as will become evident in chapter X. The French translation of his work had no comparable influence in France, which is similar to what happened in Italy with the translation into Italian (page 399). Rivoire’s edition did, however, presumably have the effect that the expression ‘linear perspective’ became more widely used in France.

Michel

O

ne French author, namely the mathematician S.N. Michel, took over much more than the word ‘linear’ from Taylor. In 1771 Michel published a slim manual on how to perform perspective constructions, entitled Traité de perspective linéaire (Treatise on linear perspective), in which he claimed that “the entire art of linear perspective was invented by the celebrated Englishman Doctor Brook Taylor” (Michel 1771, 6). What Michel was probably referring to was Taylor’s extensive and elegant use of vanishing points and lines, because it was this aspect Michel took over in his constructions. Contrary to Taylor himself, Michel stressed that Taylor’s method had the advantage that it did not require a plan and an elevation of the object meant to be thrown into perspective. It is likely that the French seventeenth-century publications whose titles mention the avoidance of le plan géométral stimulated Michel to focus upon precisely this aspect of Taylor’s work, but his inspiration could also have come from the title of Lambert’s book (Lambert 1759). Michel did not cite any of these books, whereas he did refer to Lacaille (Michel 1771, 2) – though without using his material. Taylor was really the main source for Michel’s treatment of perspective construction, but Michel also introduced a non-Taylorian concept, namely that of the distance circle (ibid., 4). This circle consists of all vanishing points whose distance to the principal vanishing point equals the distance. As we saw in section VII.9, such a circle also occurs in Jeaurat’s work – as the limit for the size of the picture – but without a name.

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485

Valenciennes

T

hus far, we have not seen any painters among the French eighteenth-century writers on perspective. In the very last year of the century, 1800 or rather An VIII, however, we do come across one painter – and a fairly wellknown one at that: Pierre Henri Valenciennes (1750–1819). He was received into the Académie de peinture et sculpture in 1787 and became rather influential as a teacher, being appointed as a professor of perspective at the École impériale des beaux-arts (The imperial school of fine arts) in 1812. At that point he had already published his Elémens de perspective pratique à l’usage des artistes (Elements of the practice of perspective for the use of artists, 1800). Valenciennes joined the ranks of those claiming that a large number of authors had already written on the subject, but not satisfactorily. His particular complaint was that the authors were mathematicians and architects who did not understand what young painters needed. Valenciennes seems to have held the opinion that what students of painting had been missing was a wordy manual. He thus spent some three hundred pages on teaching them the art of perspective and instructing them in how to apply a distance point method, but without giving any mathematical explanations of why the method worked. He also included a brief section presenting a plan and elevation method. His one hundred and thirty or so illustrations were rather conventional and not very elaborate, depicting geometrical figures, staircases, arches, and columns in perspective. Two of his more pedagogical drawings are shown in figures IX.57 and IX.58.

IX.13

L

The French Development

ooking back upon the literature on perspective in France during the seventeenth and eighteenth centuries, we can conclude that the field was dominated by mathematicians and mathematical practitioners. Moreover, almost all the innovations took place in the relatively brief period from the late 1620s to the mid-1640s. During these decades several authors introduced perspective scales and worked with direct constructions in the picture plane. The new approach was not followed up, however, instead authors on perspective returned to distance point constructions, which had already become popular in the sixteenth century. In the second half of the eighteenth century, a couple of authors took up plan and elevation constructions, but their attempts to reintroduce this method never seriously challenged the distance point construction. As concerns the mathematical understanding of perspective procedures, we have not encountered any progress in this chapter. In fact, there was a development away from Guidobaldo’s and Stevin’s theories and towards the use of the division theorem, combined with the theory of similar triangles,

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IX. France and the Southern Netherlands after 1600

FIGURE IX.57. Valenciennes’s introduction to a plan and elevation method. The diagram shows the plan and elevation of a cylinder, the picture plane, and the eye point. Valenciennes 1800, plate 26, figures 1 and 2.

and there was no reaction to the innovative works by Taylor and Lambert, both of which were available in French. Only a few architects and even fewer painters contributed to the French literature on perspective published between 1600 and 1800. I am not familiar enough with the French tradition of architectural drawing to tell whether there may be some explanation as to why so few contributed to the literature on architecture in perspective, or whether this was coincidental. My knowledge of French painting is also very limited, but I am nevertheless tempted to say I find it understandable that painters in general were not very interested in perspective, and hence did not write on the subject. Most of the mainstream French paintings do not contain the kind of architectural or geometrical elements that would require a careful perspective construction. To the degree that painters wished to organize their motifs according to the rules of perspective, a basic grid of perspective squares – like the one Leonardo da Vinci applied in The Adoration of the Magi (figure III.28) – would suffice.

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FIGURE IX.58. Valenciennes’s illustration of how to decide on perspective heights.

Rather than seeing perspective in paintings, we find perspective compositions in book illustrations and prints. This very appropriately reflects the considerable interest in perspective within the French world of drawing, engraving, and printing during the seventeenth and eighteenth centuries.

Chapter X Britain

X.1

Starting Late

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he different waves that had awakened an interest in perspective in various circles on the Continent did not reach the British Isles until the end of the seventeenth century. Paintings were unwanted in the Protestant churches, and in Britain they were not demanded in great quantities in private homes, as had been the case in the Netherlands. The few British painters who could make a living in the early modern period mainly painted portraits with backgrounds that did not require any knowledge of perspective. In addition, the group of British mathematical practitioners and mathematicians did not take up the topic of perspective until the final decades of the seventeenth century. From then on and until around 1750, British tracts on perspective appeared at regular intervals. While all of Continental Europe, except the German states, saw the interest in writing about the subject declined during the second half of the eighteenth century, Britain experienced a boom with almost thirty new publications. In the history of British works on perspective, the mathematician Brook Taylor is an absolutely central figure, and I therefore devote a large part of this chapter to his work. I begin with a brief presentation of the pre-Taylor period and end with four sections on Taylor’s successors, as well as a few pages on the rest of the British story, and my style changes as the chapter progresses. Taylor’s contributions were seminal, not only from a British point of view, but to the entire history of the mathematical theory of perspective, and they therefore justify thorough treatment. Since his British successors were mainly repeating the ideas conceived by Taylor and others, I do not describe their works in detail.

X.2

British Literature on Perspective Before Taylor

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he first publication on perspective to appear in Dutch was a translation from 1553 of Serlio’s book on the subject. Strangely enough, the first British book on perspective was an English translation by Robert Peake of this Dutch edition. In his dedication to the Prince of Wales (Prince Henry), Peake declared that he had translated Serlio because there had long been “an 489

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ignorance and want” of perspective “in most parts of this Kingdome” (Serlio 1611, dedication). His translation appeared in 1611 and remained the only English-language publication on perspective for almost six decades. As argued earlier, Serlio’s work was not a very satisfactory introduction to the subject (page 116). Some Britons learned perspective from other sources, and from around 1670 publications document a nascent British interest in the field.

Wren, Moxon, and Salmon

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he famous architect Christopher Wren (1632–1723), who is also well known in the history of mathematics for being the first to rectify the cycloid, took a particular interest in perspective drawing instruments. In the first half of the 1650s he designed his own version (figure X.1) that was described and illustrated in the Philosophical Transactions of the Royal Society, most likely by the society’s secretary, Henry Oldenburg1 (BennettS 1982, 75). According to historian Jim A. Bennett, this instrument was actually constructed (ibid.). About a hundred years later, Wren’s ideas were followed up by some of his countrymen, like James Watt and George Adams, who experimented with improving earlier perspective apparatuses (cf. page 596; DawesS 1988, 4; KempS 1990, 182). In 1670 the royal hydrographer Joseph Moxon (1627–1700) published a book on perspective constructions entitled Practical Perspective, or, Perspective Made Easie. Moxon taught mathematics and wrote several books on the mathematical sciences, becoming a Fellow to the Royal Society in 1678. In his Practical Perspective he maintained that the British needed a better book than Serlio’s. Even so, Moxon had so much respect for Serlio’s fame as an architect that rather than ascribing the incomprehensibility of Serlio’s text to the author himself, Moxon saw it as a consequence of the book having gone through the hands of two translators (Moxon 1670, preface). In his own book Moxon collected what he considered to be the most instructive illustrations of the art of perspective available from the Continental literature. He chose his drawings from the works of Dürer, Cousin, Hondius, Marolois, and Niceron, and from Bosse’s presentation of Desargues’s method, adding a few diagrams of his own. Altogether his work consists of 60 illustrations and descriptions of how they were produced – his basic construction being a distance point method. Moxon’s book seems to have received little attention, so its greatest significance lies in being the first book on perspective ever published by a Briton. It also happened to be the first of a series of books whose titles claimed to make perspective easy.2 1

The publication is listed as Oldenburg 1669. Five years before the instrument was presented in Philosophical Transactions, Oldenburg described it in a letter to Balthazar de Moncony (OldenburgS Corr, vol. 2, 285–288). 2 The others being Lamy 1710, Halfpenny 1731, Kirby 1754, Bardwell 1756, and Ferguson 1775.

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FIGURE X.1. Wren’s instrument. “The manner of Using it is this: Set the Instrument on a Table, and fix the Sight A, at what height above the Table, and what distance from the Frame SSSS, you please. Then, looking through the Sight A, and holding the Pen I in your hand, move the Head of the Pin P up and down the Out-lines of the Object, and the point I will describe on the Paper OOOO, the Shape of the Object so trac’d” (Oldenburg 1669, 899). Like with the pantograph (figure VIII.10), the outcome will be a drawing that has to be moved if the point A is kept as eye point. In the case of Wren’s instrument, the picture should be moved the distance IP to the left. It can also be noticed that the distance cannot be more than an arm’s length. Oldenburg 1669, figure 1.

In 1672 a better introduction to perspective than Moxon’s appeared in Britain, namely an English translation of Dubreuil’s first book on perspective, edited by Robert Pricke. That same year, the empirical physician and plant collector William Salmon presented perspective constructions in a general introduction to the art of drawing (Salmon 1672). It is interesting that he included a section on perspective in his book, but his description only makes sense to readers who are already familiar with the topic. The publishing activity in the early 1670s came to a standstill for more than three decades. The next perspective book to appear was also a translation, but unlike earlier works this was a translation of a recent publication, namely Lamy’s French treatise from 1701 (page 471), which already appeared in English in 1702. Lamy’s book produced a reaction from the astronomer John Shuttleworth (1671–1750), who after studying Lamy drew a perspective composition he

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found distorted at the edges. This moved him to investigate visual perception. Like many of his Continental predecessors, he came to the conclusion that there was a contradiction between the theories of vision and perspective, and he later published on this theme (ShuttleworthS 1709).

Ditton

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he English edition of Lamy’s work was the first manifestation of an interest in the mathematical theory of perspective awakening in Britain. The next signs were provided by the two mathematicians Ditton and Taylor, who both published on the subject in the first half of 1710s. The two scholars also shared a great esteem for Isaac Newton’s work, and they both wrote about his method of fluxions. In 1705, Humphry Ditton (1675–1715) published a description of the ideas presented in Newton’s Principia, and the following year, presumably thanks to Newton’s influence, he was appointed a master at the New Mathematical School at Christ’s Hospital. The curriculum there included perspective, a subject for which Ditton created his own course. Based on his lecture notes he published the textbook A Treatise on Perspective Demonstrative and Practical in 1712. In his preface Ditton referred to “Writers on this Science”, but did not mention any names. His text reveals, however, that in preparing its theoretical part he was greatly influenced by Aguilon. Ditton for instance followed Aguilon in considering other central projections than the one applied for perspective – which was not common in tracts on perspective. Like Aguilon, Ditton considered a parallel projection as a central projection with its centre in a point at infinity, and he also treated stereographic and gnomonical projections (the latter have their centres at the centre of a sphere and map the sphere upon one of its tangent planes). Aguilon used the term linea directa for an orthogonal, and Ditton took over this term, rendering it as a direct line. Aguilon’s rather unusual theorems concerning pencils of lines that are depicted in sets of parallel lines also found their way into Ditton’s book (Aguilon 1613, propositions 133 & 134; Ditton 1712, proposition 7). Likewise, Ditton kept fairly close to Aguilon in treating inverse problems of perspective. Both authors assumed that the positions of the eye point and the plane of an original figure are given in relation to the picture plane. Hence, to them inverse problems became a question of projecting images back upon their original planes. Ditton introduced the concepts of direct and inverse perspective and interestingly compared them with direct and inverse methods of fluxions, and to analysis and synthesis (Ditton 1712, 72). In spite of these similarities, Ditton’s book is more than a paraphrase of Aguilon’s results. For one thing, he included other results from the literature of the Guidobaldo–Stevin tradition, and he also added reflections of his own. In one example of the latter, he emphasized the importance of the vanishing

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point theorem, as mentioned earlier in connection with Guidobaldo’s treatment of this theorem (page 245). Another of Ditton’s own contributions concerns the theory of apparent sizes. He followed the old tradition of letting the theory of perspective be preceded by a section on visual perception. Referring to personal experience, he expressed doubts about the angle axiom (ibid., definition 22). In addition, he stressed that the theory of apparent sizes has shortcomings in relation to perspective. In fact, he showed an example in which the theory of visual perception contradicts the rules of perspective (ibid., 17), a point I have touched upon in appendix one (page 726). Ditton’s general approach to perspective was very theoretical, one example being his determination of the axes of an ellipse that is the perspective image of a circle (ibid., problem 12), and another his comparison of reflection and central projection (figure X.2). In the practical part of his book, Ditton mainly, but not exclusively, applied a distance point construction. Some of his other constructions were

FIGURE X.2. Ditton’s comparison of a reflection and a central projection. He considered the situation in which are given a light at E, a horizontal ceiling LONT, a rectangular mirror ABDC with the sides AB and CD parallel to the ceiling and lying in a vertical plane that cuts the horizontal plane through E in the line ST, and finally the point F lying symmetrically to E in relation to the vertical plane through ST. Ditton determined the reflection abdc of the mirror upon the ceiling and showed that it is identical to the central projection abek of ABDC from F upon the ceiling. Ditton 1712, figure 32.

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based on the division theorem (ibid., problem 15). He also explained how to carry out perspective constructions for oblique picture planes (ibid., 129), and criticized Lamy for reducing such a construction to a problem of constructing the image in a vertical plane – seen from another eye point – an idea Lamy actually had taken over from Stevin (page 472). Finally with a reference to Pozzo, Ditton briefly touched upon stage designs (ibid., 155). Ditton’s first British treatment of the theory of perspective was of better quality than several other introductions to the subject. Nevertheless, it did not have much direct influence, for Taylor’s contributions came to dominate the further development in Britain. Indirectly, however, a few of Ditton’s ideas may have lived on, thus I think that even though Taylor basically worked independently of Ditton, he picked a few insights from him.

X.3

Taylor and His Work on Perspective

Taylor’s Background

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n mathematics the name of Taylor is immortalized because of a very central tool in the calculus called “Taylor series”. Taylor introduced these series in his Methodus incrementorum directa et inversa (The direct and inverse method of increments, 1715), but he was not the only person to develop this concept, nor was he the first. Taylor series occur in the works of all the talented mathematicians who occupied themselves with the calculus in Taylor’s day (FeigenbaumS 1985, 258). Hence, it is accidental that the series were named after him. Methodus is an important contribution to calculus, but – contrary to Taylor’s contributions to perspective – it is not an outstanding work that revolutionized a mathematical discipline. Nonetheless, calculus was and still is considered so much more important than the theory of perspective that Taylor is generally remembered for his contribution to the former discipline rather than the latter, not just in the history of mathematics, but even in the history of art.3 In 1715 Brook Taylor (1685–1731) not only published Methodus, but also a work entitled Linear Perspective. At the time he was in a very creative phase of his life, having become Fellow to the Royal Society three years earlier, graduated in law from Cambridge, and been appointed secretary to the Royal Society in 1714. From 1712 until 1724 Taylor was a frequent contributor to Philosophical Transactions, submitting papers on mathematics and natural philosophy. In addition, he found time to practise music and to paint. According to his first biographer, his grandson William Young, Taylor was a accomplished landscape painter: 3

The New Grove Dictionary of Music and Musicians naturally describes Taylor’s contributions to music, and it also mentions his work in calculus, but says nothing of his achievements in perspective (Jeans & GoukS 2001).

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His [Taylor’s] drawings and paintings preserved in our family require not those allowances for error and imperfection with which we scan performances of even the superior dilettanti: – they will bear the test of scrutiny and criticism from artists themselves, and those of the first genius and professional abilities. [TaylorS 1793, 15]

The nearest I have come to tracing a painting by Taylor is to assume he did the landscape painting included in his portrait, shown in figure X.3.

FIGURE X.3. Taylor displaying his manuscript of a book on perspective. The figure occurring in the manuscript is included in a mirrored version in his New Principles of Linear Perspective from 1719 (cf. figure X.33). The painter is ostensibly Joseph Goupy, who was a friend of Newton (JefreeS 1996, 229). National Portrait Gallery, London.

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Taylor’s Inspiration

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n Taylor’s time, synthetic geometry was not en vogue on the Continent, whereas in Britain Newton extolled its virtues. It was presumably under Newton’s influence that Taylor defended the discipline, stating that

... it is my opinion that the prevailing humour of treating Geometry so much in an Algebraic way has prevented many noble discoveries, that might otherwise have been made, by following the Methods of the Ancient geometricians. [TaylorS 1711, quoted from FeigenbaumS 1986, 53]

It is very likely that Taylor’s interest in classical geometry, combined with his painting activities, led him to study perspective. Like so many of his predecessors, he was not satisfied with the presentations he had read, and believed he could write a better one himself. He did not reveal which authors he had studied, but gave the impression that he had not seen any book on perspective that was written by an able mathematician (Taylor 1719, iv). I find his statement unfair to ’sGravesande, for it is beyond doubt, as we shall see, that Taylor benefited from reading ’sGravesande’s Essai de perspective. There are also a number of indications, to which I will return later in this chapter, that Taylor found some inspiration in reading Lamy. Finally, as noted earlier, I suspect but cannot prove that a few of Taylor’s ideas came from Ditton. Despite the impulses he received from other mathematicians, Taylor’s contribution to the theory of perspective remains highly original. He plucked from the works of others the ideas that were mathematically the most fruitful, and taking these as a starting point, he built up a theory that was far more general, comprehensive, and systematic than anything seen before.

Taylor’s Aim

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lthough extremely fascinating, Linear Perspective is difficult reading because it is very concise and leaves many arguments – and not necessarily trivial ones – to the reader. Warren Mild has aptly characterized Taylor as being “given more to uncovering difficulties for himself than to explaining their solutions fully and clearly to others” (MildS 1990, 58). It is a style that may appeal to some mathematicians. However, Taylor did not offer this work to his fellow mathematicians, but to painters, architects, and other practitioners, even going as far as to claim on the title page that his book was necessary reading for them (figure X.4). As in the case of ’sGravesande (page 360), I wonder why Taylor did not admit he had written a book for mathematicians. Did he merely follow his predecessors, who had dedicated books on perspective to practitioners? Or was he really so naive that he thought they would benefit from his work? The answer is presumably a conditional ‘yes’ to both questions. He was part of a tradition in which the only legitimate addressees for books on perspective seem to have been practitioners of perspective. Moreover, like several of his fellow mathematicians, he had great expectations

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when it came to practitioners’ inclination and ability to read mathematics. This is clear from the preface to the second edition of his book, where he described some of his thoughts concerning the importance of perspective, and of the subject being taught. In his view, the demand that a drawing respect the laws of perspective was extremely fundamental: A Figure in a Picture, which is not drawn according to the Rules of Perspective, does not represent what is intended, but something else. So that it seems to me, that a Picture which is faulty in this particular, is as blameable, or more so, than any Composition in Writing, which is faulty in point of Orthography, or Grammar. [Taylor 1719, ix]

FIGURE X.4. The title page of Linear Perspective.

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The attitude that correct use of perspective is a sine qua non for a painting made him suggest the following education for a painter: I would first have him learn the most common Effections of Practical Geometry, and the first Elements of Plain Geometry, and common Arithmetic. When he is sufficiently perfect in these, I would have him learn Perspective. [ibid., xii]

We see some similarities here with Alberti’s and Dürer’s requests for the education of painters (pages 18 and 189). Taylor’s idea was that not only should a painter learn to perform some basic constructions, but he should learn the subject so well that he really understood it, thereby saving time: Nothing ought to be more familiar to a Painter than Perspective; for it is the only thing that can make the Judgment correct, and will help the Fancy to invent with ten times the ease that it could do without it. [ibid., xiii]

Still, it remains a question whether Taylor really thought his work would be the one appropriate for educating painters. My impression is that in his most optimistic moods, Taylor hoped practitioners would be able to follow his arguments, but that he was also aware that he was describing an ideal situation somewhat removed from reality. The most he could aspire to was that some enthusiastic practitioners would come to understand his work and use their knowledge in teaching other practitioners. And to some extent, this actually happened in the middle of the eighteenth century, as we shall see later in this chapter.

Taylor’s Two Books on Perspective

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nly four years after the appearance of Linear Perspective Taylor published a thoroughly revised version, New Principles of Linear Perspective (figure X.5), which I give the short title New Principles. Taylor’s decision to rewrite his work was seemingly motivated by criticism levelled at Linear Perspective for being too abstract (Taylor 1719, vi–vii). In the new version, Taylor expanded the number of pages from 42 to 70, but reduced the number of figures from 43 to 25. The contents of the two books are basically the same, the main difference being that in New Principles Taylor attempted to make his material more accessible. Consequently some of the most abstract material from Linear Perspective is left out in New Principles, which also has the mathematical proofs worked out in greater detail. Moreover, Taylor split some of his general results up into several theorems and problems, and listed some fundamental geometrical results – for instance that two intersecting or parallel lines determine a plane – as axioms (see also BkoucheS 1991, 268). Taylor may have got this idea from Lamy, who had done the same thing. Taylor also changed his terminology slightly, one example being that in Linear Perspective he called a perspective image a “representation”, a term Ditton also used, whereas in New Principles he called it a “projection”. Finally, Taylor was a bit more generous with comments to his theorems and problems in New Principles.

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FIGURE X.5. The title page of New Principles.

The rewriting process also resulted in a permutation that was actually unfortunate: it is more difficult to find the red thread running through New Principles than to do so in Linear Perspective. In presenting Taylor’s results I keep, by and large, to the order of the latter. Taylor took up some of Leonardo da Vinci’s expressions, like him distinguishing between linear perspective and other drawing techniques. Thus Taylor mentioned the use of colouring to create the impression of depth – often called ‘aerial perspective’ – and the use of light and shadow to emphasize the spatial illusion. Taylor called the latter chiaroscuro, a word Leonardo

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FIGURE X.6. Taylor’s illustration of a perspective projection. He drew the image on the opposite side of the picture plane than the one seen from the eye point O, producing a reversed image. Taylor 1719, figure 1.

da Vinci also had used (Taylor 1719, 2; Leonardo 1989, 16). As noted earlier, my impression is that Taylor’s use of the term ‘linear perspective’ is what caused it to become common. Taylor’s Linear Perspective consists of five sections, the first of which contains fundamental definitions and theorems. The second section treats a number of problems related to determining perspective images. The third and fourth sections take up problems of shadows and reflections in the picture plane, and finally, the fifth section presents some inverse problems of perspective. In New Principles Taylor added two more topics, which appear in appendices. In the first appendix, Taylor dealt with the problem of finding the image of a line upon an irregular surface as a shadow (Taylor 1719, 59–61; AndersenS 19921, 48). He had previously outlined the shadow procedure in a description of Linear Perspective he wrote for Philosophical Transactions (Taylor 17152) – an idea he might have picked up from Lamy (figure IX.51). We have seen that two decades earlier Pozzo had touched upon the theoretical possibility of applying this method, and claimed that it did not work in practice (page 390). As light sources had not changed much in the mean time it is doubtful that Taylor’s shadow procedure would work. Taylor himself admitted he had not had “an opportunity of putting it in practice” (Taylor 1719, 61).

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FIGURE X.7. Lamy introducing a generalized visual pyramid (cf. figure II.1). Lamy 1701, figure on page 34.

Taylor’s second appendix bears the heading “A New Theory for mixing colours taken from Sir Isaac Newton’s Opticks”. Here Taylor adapted Newton’s model for explaining the mixing of light rays – later known as Newton’s colour circle – to the mixing of colour pigments (AndersenS 19921, 48–51). Taylor’s inclusion of this appendix manifests his wish to have the art of painting based on scientific methods. For drawing lines he presented a very advanced theory, partly of his own devising, and for mixing colours he applied the newest theory of light and colours.4 In describing Taylor’s treatment of perspective I concentrate on the theoretical aspect of his work. Thus, besides looking at his own presentation of the theory of perspective, I examine the foundations upon which his applications are built, generally treating the examples that are theoretically most demanding.

4

For reactions from the art community to the use of Newton’s theory, see KempS 1990, 292–306. For similarities and differences between the theory of mixing light colours and the theory of mixing colours for paint, and for the interplay between the developments of the two theories, see ShapiroS 1994.

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FIGURE X.8. Taylor’s basic configuration involving an oblique picture plane. Taylor 17151, figure 2.

X.4

Taylor’s Fundamental Concepts and Results

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n Linear Perspective Taylor concentrated the basic theory of perspective in ten definitions and four theorems. In New Principles he included another nine definitions. As indicated, he also added four axioms concerning fundamental geometrical results that can be found as theorems in Euclid’s Elements, and doubled the number of theorems. One of his new theorems states that a perspective representation is the same as a central projection – which he called a ‘scenographic projection’5 (Taylor 1719, 12). His diagram of a perspective projection (figure X.6) may have been inspired by Lamy’s illustration (figure X.7) of the step previous to introducing the perspective image, namely the one of an observer looking at an object. From the outset Taylor worked with a general situation in which the picture plane is oblique, as illustrated in figure X.8. In this set-up, O is the eye point, QSD the picture plane, and ABQ a horizontal plane of reference. In relation to this diagram Taylor introduced a set of new concepts and a new terminology, which I present in my own notation.

5

The enunciation of the theorem contains the term “the Ichnographic Projection”, but it is clear from the context that this is either a slip of the pen or a misprint, and that Taylor meant “Schenographic Projection”.

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Vanishing Points and Lines o a line l, not parallel to the picture plane π, Taylor assigned the two points Il and Vl in p (figure X.9), in which π is intersected by l and the line through the eye point O is parallel to l. He gave these points the names I have used throughout this book: the intersection and the vanishing point of the line l. In New Principles he explained why he had chosen the term “vanishing point”, actually giving two reasons. The first (figure X.9), which is rather less convincing than the second (Taylor 1719, 15), is that a line m that passes through its vanishing point Vm is depicted only in that one point “and may be said to vanish”. The second reason concerns images of equal line segments on a line l (figure X.10). Taylor noticed that the further away a line segment on l of a given length is from p, the smaller and the nearer to the vanishing Vl its image will be, and when the image “comes into this [vanishing] Point, its Magnitude vanishes because the Original Object is at an infinite Distance”. The first theorem in Linear Perspective is the main theorem, which makes the importance of the concepts of intersection and vanishing point evident. As we have seen, after the publication of Guidobaldo’s work the main theorem had been extremely central in the theory of perspective without this fact being stressed. Taylor did not do so in Linear Perspective either, but in New Principles he wrote as follows.

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This Theorem being the principal Foundation of all the Practice of Perspective, the Reader would do well to make it very familiar to him. [Taylor 1719, 14] m

Vm π

Vl

l

O

Il

FIGURE X.9. Taylor’s introduction of the intersection (Il) and the vanishing point (Vl) of a line.

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p

O

Vl

D

Di Ci

C

Bi Ai

B A

Il

FIGURE X.10. Taylor’s second argument for using the term “vanishing point”. The line l is depicted in IlVl. The line segments AB, BC, and CD on l are equal, but the lengths of their images decreases in the direction of Vl.

For a line segment parallel to p, Taylor deduced the traditional result (page 242), that its image is parallel to it (Taylor 17151, 8; Taylor 1719, 15–17). When the position of eye point O in relation to the picture plane p is given, then a given line lp in p is the image of all the lines situated in the plane determined by O and lp. However, if a point of intersection and a vanishing point are given on lp, then the original line of lp is determined uniquely. Taylor did not mention this uniqueness explicitly, but it seems to have played a role for his generalization of the inherited theory of perspective and his investigation of how elements in space can be characterized by their images in p. Generalizing the concepts of intersection and vanishing point, Taylor used two lines to characterize a plane a (not parallel to p) in p. These two lines, which are parallel (figure X.11), are the trace ia of a and p – which Taylor called a’s intersection – and the intersection, va, of p and a plane through O parallel to a – which he called a’s vanishing line. Vanishing lines had occurred more or less explicitly several times since Guidobaldo, but Taylor was the first to focus on the concept and give it a central role in the theory of perspective coining the term “vanishing line” in the process. One of Taylor’s aims with this set-up was to perform constructions directly in the picture plane p – an idea that was later fully developed by Lambert when he created the notion of perspective geometry. For Taylor’s purpose, vanishing lines were very useful since they enabled him to work with threedimensional problems in p. Among other things, they provided him with results that correspond to fundamental results in the three-dimensional space, such as that two intersecting or parallel lines determine a plane, and that two non-parallel planes determine a line. The results in the picture plane for lines and planes that are not parallel to p are easily deduced from the definitions of vanishing lines and intersections. For later use, I nevertheless

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Pa a da O

va

P

d

ia

FIGURE X.11. Some of Taylor’s new concepts. To a plane a that is not parallel to the picture plane p he assigned the intersection (ia), the vanishing line (va), the centre of the vanishing line (Pa), and the distance of the vanishing line (da).

formulate them explicitly – as was also done by Taylor’s successor John Lodge Cowley (Cowley 1765, 32 and 44). Observation1. Given are two lines l and m that lie in a plane a and are not parallel to the picture plane p. The vanishing line va of a is determined by the vanishing points Vl and Vm, and analogously the intersection ia is determined by the intersections Il and Im. In general, va is the line joining the two vanishing points Vl and Vm, and similarly ia is the line joining Il and Im. If the two lines l and m happen to be parallel, the vanishing line va can be constructed as the line that passes through the common vanishing point of l and m and is parallel to ia. If Il and Im coincide, the intersection ia can be constructed analogously. The second theorem to be mentioned is the dual of the first: Observation 2. When the planes a and b intersect in a line l that is not parallel to or situated in the picture plane p, the vanishing point Vl of l is the point of intersection of the two vanishing lines va and vb, and similarly l’s intersection Il is the point of intersection of the intersections ia and ib. In a few cases Taylor took points at infinity into consideration, but not explicitly as vanishing points of lines parallel to the picture plane. In Linear Perspective Taylor applied the observations 1 and 2 frequently without further comment, whereas in New Principles he stated them as corollaries to the following two theorems: The Vanishing Points of all Lines in any Original Plane, are in the Vanishing line of that Plane. [Taylor 1719, 18] The Intersections of all Lines in the same Original Plane, are in the Intersection of that Plane. [ibid., 19]

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For a traditional presentation of perspective in which the picture plane is vertical and the representation of horizontal lines is the main issue, the principal vanishing point and the distance are essential. Taylor also introduced these concepts, calling the principal vanishing point the centre of the picture. However, for his general approach he needed broader concepts. Thus, to a given vanishing line he assigned the centre of the vanishing line and the distance of the vanishing line (figure X.11). The first is the point Pa on the vanishing line va, in which the normal from the eye point O to va cuts it, and the latter is the distance OPa – which I often denote da. In dealing with these new concepts, it is useful to be aware of the following. Result 1. The line joining the principal vanishing point P and the centre Pa of a vanishing line va is perpendicular to va (Taylor 1719, 11) Taylor stated this result as the first theorem in New Principles and referred to a proposition in book XI of Euclid’s Elements for a proof. The result can also be obtained by observing that since OP and OPa are both perpendicular to the vanishing line va, the latter is a normal to the plane OPPa, and hence in particular perpendicular to PPa.

The Directing Plane

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he Taylor drawing reproduced in figure X.8 includes a plane through the eye point O parallel to the picture plane. Taylor called this the directing plane – which later became commonly known as the vanishing plane. Connected to this plane, shown as d in figure X.12, Taylor introduced the point Dl, in which a line l (not parallel to the picture plane p) meets d as a directing point. In New Principles he generalized this concept (Taylor 1719, 8 and 18) and defined the directing line of a plane a (not parallel to p) as the trace between

p

Vl

a va

d

O

l

Il

I

ia

Dl J

FIGURE X.12. A configuration involving the directing plane d.

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a and d (IJ in figure X.12) and noticed that this line is parallel to ia and va. For a vertical picture plane, Taylor’s directing line is the same as ’sGravesande’s geometrical line. In both his books Taylor formulated the following theorem. Result 2. The image of a line l that is not parallel to the picture plane is parallel to the line that passes through the eye point O and the directing point Dl (Taylor 17151, 9; Taylor 1719, 17). In the last of the introductory propositions in Linear Perspective, theorem 4, Taylor dealt with line segments defined in relation to the directing plane, but he never applied this theorem later in the book. While revising his work, Taylor apparently realized that the theorem was superfluous, because he left it out of New Principles. Altogether, Taylor made very little use of the concepts connected to the directing plane in both of his books, so I wonder what their function was. A look at one of the few constructions in which Taylor applied these concepts may provide a clue. The procedure – presented in the caption of figure X.13 – greatly resembles one of ’sGravesande’s constructions (figure VII.56). O

I

Dm

Dl

J va

Ai

Il

Im

ia

l

m

A

FIGURE X.13. One of Taylor’s constructions applying directing points. The intersection ia and the vanishing line va of a plane a are given, and it is required to throw a given point A in a into perspective. To work with a plane diagram Taylor used a rabatment that can be obtained using the following procedure. First turn the plane d (figure X.12) into a around their line of intersection IJ, and then turn a into the picture plane p around ia. In his construction, Taylor drew two lines l and m through A, cutting ia in the intersections Il and Im and IJ in the points Dl and Dm. According to result 2, the images of l and m are parallel to ODl and ODm.Taylor applied this to construct Ai as the point of intersection of the line through Il parallel ODl and the line through Im parallel to ODm (Taylor 17151, 22–23; Taylor 1719, 27).

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Indeed, the steps carried out in the two constructions are identical. Taylor’s construction, however, is more general than ’sGravesande’s in that Taylor’s procedure applies to any point in any plane that is not parallel to p, whereas ’sGravesande’s procedure was only designed for a point in a ground plane. It is possible Taylor was led to introduce the directing plane while searching for an elegant way of generalizing ’sGravesande’s procedure. The fact that he included the theory related to this plane in Linear Perspective may be a sign that at some stage he had made more use of it than he did in the final presentation. Whatever the case, he seems to have been so attached to the directing plane that he did not want to let it go completely, neither in Linear Perspective nor in New Principles. When commenting upon Taylor’s introduction of the directing plane and directing lines, Thomas Malton wrote they “are most essential, in Theory, though but of little use, in Practice” (T. Malton 1775, 109).

X.5

Taylor’s Basic Constructions

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nlike a number of French seventeenth-century perspectivists, and unlike Lambert, Taylor did not explicitly point to the difference between constructions involving a plan and constructions that can be performed directly in the picture plane – which I call direct constructions or, following Lambert, free constructions. In Linear Perspective Taylor suggested the general strategy of applying free constructions as soon as the image of a “Remarkable Line [Segment] in the Design” has been found (Taylor 17151, 21). In New Principles he added another method that is quite fascinating from a theoretical point of view, for it combines the traditional use of a plan and elevation with the method of constructing directly in the picture plane, as explained in greater detail in section X.8.

Pointwise Constructions

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or the first step in his general procedure – throwing a line segment into perspective – Taylor introduced a pointwise construction in order to determine the images of the two endpoints of the segment. In both his books he presented two methods of constructing the image of a point: a direct construction based on the division theorem, and a construction that involves a plan and is based on the main theorem (Taylor 17151, 10–11; Taylor 1719, 20–21). In addition to these procedures, Taylor also applied one that he did not present in connection with his introduction of point constructions, namely a method involving “visual rays” (cf. page 347).6 As mentioned earlier, I find it very likely that Taylor had noticed ’sGravesande’s use of a visual ray construction and realized how elegant it

6

For instance in Taylor 17151, figures 16 and 20; and Taylor 1719, figures 10 and 13.

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was. I am sure he had also convinced himself of the correctness of the construction, but he must somehow have found it too cumbersome to give a thorough introduction to it – or forgotten to include one. The foundation of a visual ray construction is the following, as already remarked in observation 1 in chapter VII (ibid.). Observation 3 (figure X.14). Let O be the eye point, p the picture plane, and A a given point in the plane a whose intersection, vanishing line, and centre are given as ia, va, and Pa, and let Ai be the image of the point A. If a is turned around ia into p, and OPa is similarly rotated into p around va, so O and A fall in Op and Ap, respectively, then the points Ai, Ap, and Op are collinear. In Linear Perspective Taylor took this result for granted, whereas in New Principles it occurs between the lines as part of a very concise argument based on the division theorem (Taylor 1719, 22; see also AndersenS 19921, 16–18). The proof based on ’sGravesande’s idea, presented in chapter VII (page 346), also applies for Taylor’s more general situation. A third way of obtaining the same result is presented in the caption of figure X.14, and a fourth method can be found in JonesS 1947, 95. Observation 3 shows that the line OpAp can be used as one of the lines defining the image Ai of A. Taylor applied this result a few times, for instance in the Op

va

Ai

FIGURE X.14. The visual ray result described in observation 3. To prove this result, I look at the lines AAπ and OOp. A straightforward consideration shows that these two lines are parallel, and hence Op is the vanishing point of the line AAp, whose intersection is the point Ap. According to the main theorem, this means that AAp is mapped upon Ap, implying that Ai lies on this line.

p

Pa

O

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A ia

Ap

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construction reproduced as figure X.15. Two other examples are presented in figure X.25, showing different choices for the second line passing through A. Taylor’s procedure for determining the image of a line segment can easily be repeated to find the image of a plane figure. In a few examples he demonstrated how this is done – as shown in figure X.15. However, as noted above, Taylor found it mathematically more satisfying to proceed with a direct construction.

Taylor’s Inspiration from ’sGravesande

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efore finishing this section, allow me a digression to discuss ’sGravesande’s influence on Taylor’s thinking. I have claimed that ’sGravesande seems to be behind Taylor’s use of directing points, as well as his introduction of the directing plane (page 506). I have further suggested that ’sGravesande could also have provided Taylor with the idea of using “visual rays” in his constructions, and hence also of working with a turned-in eye point above the horizon – or any other vanishing line. What is more, as Phillip S. Jones has also pointed out, a few of Taylor’s other constructions resemble some of ’sGravesande’s7 (JonesS 1947, 115–118; AndersenS 19921, 20–21).

FIGURE X.15. Two of Taylor’s constructions of perspective images of plane figures. In these examples he involved the figures themselves. To find the image of the point B, he applied a visual ray construction. Taylor 17151, figure 16.

7

Some of the similarities can be observed by comparing the procedures presented in figures VII.56 and X.13, and the constructions applied in figure VII.60 and the righthand diagram of figure X.25.

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Finally, there is another remarkable similarity between the works of ’sGravesande and Taylor. As we saw in chapter VII, ’sGravesande was one of the few who commented upon the problem of reversing stating among other things that a procedure like the one used in a visual ray construction has “the same effect as looking through the back-side of a paper, at a picture drawn thereon” (page 331). Taylor similarly remarked that the image is “seen on the back-side” (Taylor 17151, 22), or “seen on the Reverse as Objects appear in a Looking Glass” (Taylor 1719, 26). Like Jones, I find that these resemblances are too striking to be accidental, concluding that Taylor had read and was inspired by ’sGravesande.

X.6 Taylor’s Contributions to Plane Perspective Geometry

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et me recall some of the specific vocabulary and ad hoc notation I have introduced to describe properties and relations in the picture plane p. The expression “perspectively parallel”, or parallel in p, means that two lines are images of parallel lines. And I similarly say that two line segments in p that are images of equal line segments are “perspectively equal”, or equal in p, and use the symbol =p. Finally I find it convenient to use the symbol =i as a short form of “being the image of ”. Contrary to Taylor himself, I split his direct constructions into two groups that concern plane and solid figures, respectively, and in this section I deal with the first. In the ordinary plane, three of the most fundamental operations are to construct a given angle, a given line segment, and a line parallel to a given line through a given point. As we have seen, Aleaume and Migon considered problems that cover the perspective versions of these operations (pages 423–424), and Lambert later took up precisely these three constructions in p (pages 651–654). When Taylor applied the procedure of completing a perspective figure from a given line segment in p, he needed similar constructions. He did not treat the determination of a perspective parallel as a separate problem, but he certainly knew how to perform this construction. Neither did he formulate the general problem of cutting off a line segment. In fact, because he assumed that one line segment in p is given, he was able to make due with less and included in both Linear Perspective and New Principles two basic constructions that correspond to the following (Taylor 17151, 17–18; Taylor 1719, 32, 23).

Problem 1. Given are an angle j, and in p the vanishing line va of a plane a together with its centre Pa and its distance da, as well as the image l of a line in a and a point A on l. Construct in p a line m through A so that ∠ (l,m) =i j. Problem 2. Given are a ratio r : s, and in p a line segment AB and its vanishing point V. Construct the point C on AB so that AC : BC =i r : s.

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In Linear Perspective Taylor also included the problem of constructing a perspectival fourth proportional in p (Taylor 17151, 19). Additionally, he presented a construction that enabled him to transfer a line segment from one line in p to another (ibid.): Problem 3. Given are a ratio r : s, and in p the vanishing line va of a plane a together with its centre Pa, and its distance da, the images l and m of two lines in a, a line segment AB on l, and a point D on m. Construct a point C on m so that AB : DC =i r : s. With the help of his solutions to problems 1–3, Taylor constructed the image of a given polygon directly in the picture plane p when the image of one of its sides was given in p.

Taylor’s Solution to Problem 1

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o solve the first problem for lines in a horizontal plane, some of Taylor’s French predecessors had introduced an angle scale. This procedure can be generalized to other planes – a step Lambert later took, as we shall see (page 657). I do not know whether Taylor was familiar with angle scales, but if he was he would presumably have found it too cumbersome to construct a full scale of vanishing points if his construction only called for a few. At any rate, he used ’sGravesande’s rabatment of the eye point into p to construct the vanishing points he needed, thereby solving problem 1 straightforwardly and elegantly, as I have explained in the caption of figure X.16. Mathematically there is no difference between applying an angle scale and Taylor’s construction. In practical performance Taylor’s procedure seems to be the most handy, unless one is solving a problem that involves constructing a large number of perspectival angles for the same distance. It is worth noticing that in formulating problem 1, Taylor did not suppose that an eye point and an original plane were given. This reflects a general attitude he had of assuming as little as possible to be known. Thus, his construction for solving problem 1 applies to any pair consisting of an eye point O and an original plane b that fulfil the following requirements: the plane b has va as its vanishing line, and O lies on a circle that is perpendicular to va, has Pa as its centre, and da as its radius.

Taylor’s Solutions to Problems 2 and 3

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aylor’s two last basic problems deal with the construction of line segments in p. Several authors involved a measure point for this kind of constructions, but Taylor was not one of them. A measure point does, however, occur implicitly in one of his problems – which deals with the determination of the original length of a line segment given in p (Taylor 1719, 31). Taylor’s solution to problem 3 is rather complicated (Taylor 17151, 19–20). It seems he also thought so himself, for he did not include this problem in

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Op

Vm

Vl

Va

A

l m

FIGURE X.16. Taylor’s construction of an angle in p. Based on the givens in problem 1, Taylor constructed the turned-in eye point Op and the vanishing point Vl of l. He then made the angle VlOpVm equal to the given angle j, defining Vm as the point of intersection of the second leg of the angle and va, concluding that the line VmA is the required line m. Taylor 1719, adaptation of a section of figure 10.

FIGURE X.17. Taylor’s method for constructing the perspective image of the plan of an icosahedron. He assumed the image ab of the side AB to be given, and then used his results for direct constructions to complete the perspective plan. Taylor 1719, figure 15.

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New Principles. Instead he based his constructions on triangulations and applied the solution to problem 1 twice to construct a triangle directly in p when one of its sides is given (Taylor 1719, 33–36). One of Taylor’s examples for using this technique is shown in figure X.17. Problem 2, that of dividing a line segment perspectively in a given ratio, had caught the attention of several perspectivists, and ’sGravesande gave a remarkably simple solution based on a visual ray construction (figure VII.54). Taylor’s solution was very easy to perform, but less transparent, and I therefore invite the reader to begin by looking at the three-dimensional configuration in figure X.18. Let us assume that the given line segment AB in p has been obtained as the image of a line segment AoBo on a line l, having the given vanishing point V, under a central projection from a point O. Let it be required to construct the point C so that it is the image of the point Co on l determined by AoCo : CoBo = r : s.

(x.1)

If we turn the plane, defined by O and l, into p around the line AB, then C can be constructed in the following way. Determine the point Co on l, which is defined by (x.1), and construct C as the point of intersection of OCo and the line AB. This construction is neither dependent on O’s distance to V, nor on the positions of OV and l, the only requirement being that these two lines are parallel. Taylor applied this result by performing the following construction (figure X.19). He drew an arbitrary line segment OV in p, and at an arbitrary distance drew a line parallel to it cutting the lines OA and OB in the points Ao and Bo. He then constructed Co so that it divides AoBo in the given ratio r : s, and found the desired point C as the point of intersection of AB and OCo. In New Principles Taylor added the interesting remark that the “Mathematical Reader will easily find” that (AC . BV ) : (BC . AV ) = AoCo : CoBo = r : s

(x.2)

(Taylor 1719, 23). Indeed, the mathematically inclined reader can confirm this relation by considering similar triangles, as I have shown in the caption of figure X.19. Such a reader will also see that the relation (x.2) confirms the

V O

Bo

B C Co

A Ao

FIGURE X.18. The idea behind Taylor’s solution to problem 3.

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V

O

B E

F

C A Ao

Co

Bo

FIGURE X.19. Taylor’s solution to problem 3. To prove relation (x.2), that is (AC . BV ) : (BC . AV ) = r : s,

(1)

I draw the line through C parallel to AoBo and let it meet OAo in E, and meet OBo in F. From the two pairs of similar triangles ACE, AVO and BVO, BCF I find AC : AV = EC : OV and BV : BC = OV : CF, and hence, by the ‘multiplication’ relation (8) (page xxxiii), (AC . BV ) : (BC . AV ) = EC : CF.

(2)

Because EF is parallel to AoBo, I also know that EC : CF = AoCo : BoCo = r : s.

(3)

A combination of (2) and (3) implies the required result (1).

above-mentioned independence of the placement of the line OV and the line parallel to it. The ratio (AC.BV ) : (BC.AV ) later became a key concept in projective geometry and became known as the ‘cross ratio’ of the four points A, B, C, and V. Commenting upon the relation (x.2), Phillip Jones observed that it “amounts to a use of the invariance of the cross ratio of four points under a projection in which one point goes to infinity” (JonesS 1947, 98). One could even go a step further and say that Taylor’s relation (x.2) implies a basic result in projective geometry, namely that the cross ratio of four points is invariant under any central projection, as I have shown elsewhere (AndersenS 19921, 29). Although there is no indication whatsoever that Taylor looked upon his result in this manner, it is still interesting that he was led to consider the ratio (AC.BV ) : (BC.AV ).

X.7 Taylor’s Contributions to Solid Perspective Geometry n order to also construct polyhedra in p, Taylor considered a number of problems that by and large correspond to the following problems in the usual three-dimensional space.

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Problem 4′. Determine the direction of the normals to a given plane. Problem 5′. Determine the direction of the planes perpendicular to a given line. Problem 6′. Given a plane a and a line l that neither lies in it nor is perpendicular to it, determine the plane h perpendicular to a that contains l. Problem 7′. Given a plane a and a line l in it, determine the plane b that contains l and forms a given angle j with a. In the rest of this section I describe how Taylor solved the perspective versions of these problems, presenting his first solution in detail and outlining his solutions to the other problems. Taylor’s perspective version of problem 4′ – in my notation – was as follows (figure X.20). Problem 4. Given the principal vanishing point P, the distance, and the vanishing line va of the plane a, determine the vanishing point Na of lines perpendicular to a (Taylor 17151, 13; Taylor 1719, 37). Albeit very simple, Taylor’s construction requires some explanation. To provide this, let me analyse the problem before presenting his solution. Since P and the distance are given, the position of the eye point O is known. Let Pa be the centre of va; according to result 1, this point can be constructed as the point of intersection of the line va and the normal to va through P (page 506). Let b be the plane through O parallel to a; by definition b cuts p in va. Also by definition, the required Na is the point of intersection of the picture plane p and the line through O perpendicular to a. Let e be the plane determined by the points O, Pa, and P. I want to show that Na lies in e and is the point of intersection of PaP and the perpendicular n through O in e to OPa. I will have done this if I can prove that n is a normal to a, and that Na lies on PaP. First I notice that since va is perpendicular

p Pa b va P

O

e Na

FIGURE X.20. Taylor’s solution to problem 4.

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to the two lines OPa and OP, it is perpendicular to all lines in the plane e, including n. This implies that line n is a normal to b and hence to a – the two planes being parallel, because it is orthogonal to va and to OPa (per definition). That Na lies on Pa P follows from the fact that this line is the trace of e and p. It was precisely this characterization of Na that Taylor applied for his construction (figure X.21), in which he turned the plane e into p. In a note he remarked that the point Na is determined on the line PPa by the relation Pα P : Pα O = Pα O : PαNα,

(x.3)

a result easily obtained by looking at similar triangles. The considerations presented thus far do not apply when the principal vanishing point P lies on the given vanishing line va. In New Principles, Taylor overcame this problem elegantly by involving a point at infinity, noticing that in the described situation the point Na “will be infinitely distant” (the lines ONa and PNa being parallel). This implies that the images of the normals to the plane a are the lines in p perpendicular to na. Taylor remarked that this result is in accordance with the fact that when P lies on na, the plane b is perpendicular to the picture plane p, and hence its normals are parallel to p and their images parallel to the originals (Taylor 1719, 38). In the case where the original plane a is parallel to p, Taylor also used infinite distances to conclude the not surprising result that Na then coincides with the principal vanishing point P (ibid., 38–39). Taylor’s construction of problem 4 led him to straightforwardly solve the perspective version of problem 5′ (Taylor 17151, 14; Taylor 1719, 39). He assumed (figure X.21) that the principal vanishing point P, the distance OP,

Pa

O

va

P

Vl

FIGURE X.21. Diagram to Taylor’s solutions to problem 4 and the perspectival version of problems 5′ and 6′. Taylor 1719, adaptation of a section of figure 18.

Na

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and the vanishing point Na are given, and asked for the vanishing line na of the planes perpendicular to the lines whose vanishing point is Na. His solution was to determine Pa on PNa by the relation (x.3) – for which there is an easy geometrical construction – and to construct na as the line that passes through Pa and is perpendicular to PNa. Similarly, with the solution to problem 4 at his disposal, Taylor smoothly solved the perspective version of problem 6′ (Taylor 17151, 15; Taylor 1719, 40). This time he assumed (figure X.21) that the principal vanishing point P, the distance OP, the vanishing line na, and a vanishing point Vl are given. He searched the vanishing line of the planes that are perpendicular to the planes having na as their vanishing line and that contain a line with vanishing point Vl. Taylor constructed this line as the one determined by Na and Vl. To solve the perspectival version of problem 7′, Taylor performed the following operations directly in p (Taylor 17151, 16; Taylor 1719, 41). He applied the solution to the perspective version of problem 5′ to construct the normal plane h to l (figure X.22), used observation 2 to find the intersection m of a and h, and then, with the aid of the solution to problem 1, constructed the line n in h that forms the given angle j with m, finally concluding that b is the plane defined by the lines l and n (for a detailed presentation of Taylor’s construction, see AndersenS 19921, 37–38). The material described thus far shows that Taylor’s theory of perspective contains many insights into direct constructions. However, he did not pursue this approach systematically and, for instance, did not explicitly address the problem of how to mark off a given length on a line in p. Thus, despite his fascination with direct constructions, Taylor left to Lambert the final decisive step of providing the picture plane with its own geometry.

h

n

a l

f

m

FIGURE X.22. Taylor’s solution to problem 7.

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X.8 Taylor’s Examples of Drawing Figures in Perspective

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n Linear Perspective Taylor did not show much concern for teaching his readers how to apply his theory. In fact, he provided only eight examples, including a couple of illustrations of how to construct the perspective images of polygons (among them figure X.15). Most of his other examples are more advanced, and directed more towards mathematicians than towards practitioners – his most practical example, reproduced in figure X.23, being one of throwing a staircase into perspective. Three of his examples deal with circles (Taylor 17151, 24–25). In two of them Taylor showed how to construct further points on a perspective circle when its centre and one of its radii are given, or when three points upon it are given. The third example is devoted to showing how the Euclidean problem of constructing tangents from a given point to a given circle can be dealt with in the picture plane (figure X.24). Taylor remarked that this problem was of relevance for determining the visible part of a perspective cone or cylinder, but he did not work this out any further. When he revised his material, he skipped the above-mentioned circle examples altogether. Instead he constructed the image of a circle from scratch using a neat pointwise construction (figure X.25). The remaining examples in Linear Perspective concern the construction of the image of the visible part of a sphere, and the direct construction of a perspective tetrahedron, one side of which is given in p. The latter is the only illustrative example in Linear Perspective of how to apply Taylor’s elaborated

FIGURE X.23. Taylor’s example of how to draw a staircase in perspective. Taylor 17151, figure 24.

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va T1

B

E C T2

t1

A b

c

a

t2

FIGURE X.24. Taylor’s perspective version of constructing tangents to a given circle through a given point (Taylor 17151, 25). The line va is the vanishing line of the plane of a circle, and Oπ is the eye point rotated into the picture plane p, as shown in figure X.14. A circle is given in p by the images C and CE of its centre and a radius, it is required to construct the tangents to the perspective circle through the given point B. Taylor prescribed a procedure for which I argue during the presentation below. Taylor introduced the vanishing point V of the line BC (the point of intersection of BC and va), and with the aid of his solution to a previous problem he constructed one of the points, say A, that lies on BC and the perspective circle. The points A, B, and C, then, are images of three collinear points a, b, and c, and from relation (x.2) he knew that ac : cb = (AC.BV ) : (BC.AV ). He chose three points a, b, and c, for which this relation applies, drew the circle with radius ca, and determined the points t1 and t2, in which the tangents through b meet this circle. Finally, he constructed the two points of tangency T1 and T2 by applying his technique of completing a figure in p when the image of one side is given. Taylor’s construction is based on the assumption that a perspective projection preserves tangency – which all mathematicians at his time took for granted. Adaptation of figure 23 in Taylor 17151.

theory contained in the perspective versions of problems 4′–7′. Indirectly, Taylor even indicated that rather than performing a direct construction in p, when dealing with a solid figure in practice it is often easier to first throw its plan into perspective and then, based on an elevation, add the perspective heights. Thus, without saying why, Taylor included a description of how to construct the plans and elevations of the regular octahedron, dodecahedron, and icosahedron (Taylor 17151, 26–28). Responding to the criticism that his examples were too scarce in Linear Perspective, Taylor added a few more in New Principles and even made some of these examples, as he himself remarked, more “ornamental” (Taylor 1719,

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FIGURE X.25. Two of Taylor’s examples of throwing a circle into perspective. In the right-hand figure, FV is the vanishing line of the plane of the original circle, O the turned-in eye point, and V the vanishing point of lines of a chosen direction. Taylor found the image a of a point A as the point of intersection of “the visual ray” OA and the image of the line through A with the chosen direction. Like Guidobaldo and ’sGravesande, Taylor considered a number of points on the circle lying on chords with the chosen direction, and he found the images of these in the same way as he found a. Through the image points he, like other authors, drew a smooth curve. In the left-hand figure, Taylor, a bit confusingly, let FV be the directing line of the plane of the original circle and V the directing point of the line AD; O is still the turned-in eye point. Again, Taylor involved the “visual ray” OA and found the image of A as the point of intersection of OA and the line through D parallel to OV, applying the result that the latter line is the image of AD (result 2). In this construction he chose the other points on the circle so that they lie on lines passing through V. Taylor 1719, figure 13.

viii). One of the new examples is shown in figure X.26. He had also been criticized for being too concise in describing his constructions, but this point he did not take, wanting his readers to learn from their own thinking: It would have been easy to have multiplied Examples, and to have enlarged upon several things that I have only given Hints of, which may easily be pursued by those who have made themselves Masters of these Principles. Perhaps some People would have been better pleased with my Book, if I had done this: but I must take the freedom to tell them, that tho’ it might have amused their Fancy something more by this means, it would not have been more instructive to them. [Taylor 1719, vii]

The function of his additional examples seems to have been to demonstrate the power of his theory rather than to illustrate simple applications of it. Building upon the problems 1–3 and the perspective equivalents of problems 4′–7′ and leaving all the technical details to the readers, Taylor managed to produce impressive examples indeed, some of which we will see later.

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FIGURE X.26. One of Taylor’s “ornamental” illustrations. Taylor 1719, figure 20.

FIGURE X.27. Taylor’s construction of a dodecahedron in perspective (Taylor 1719, 33–45). He assumed that the following are given: the principal vanishing point H, the distance OH, the vanishing line Fs (with centre F ) of the original of the face ABCDE, and the perspective image AB of one of the sides of the dodecahedron. He also supposed that Fs and AB are parallel. Taylor first constructed the perspective image of a plan of the dodecahedron in a plane, say a, perspectively parallel to ABCDE. In p this meant to project the dodecahedron orthogonally upon a, which is to perform a central projection from the vanishing point I of the perpendiculars to a. He applied the solution to problem 4 to determine I on the line through the principal vanishing point H perpendicular to Fs. He then looked first for the projection of AB: since this line segment is parallel to the vanishing line Fs of a, its projection upon a in p is also parallel to Fs. He chose this projection, ab, as a line segments that has its end points on IA and IB and is parallel to AB. Having decided upon the position and length of ab, Taylor told his readers to complete the perspective plan of the dodecahedron, implying to construct a polygon directly when the image of one of its sides is known. For the perspective elevation Taylor chose a plane say b, which has FI as its vanishing line. Since the principal vanishing point H lies on this vanishing line, the images of lines perpendicular to b are lines perpendicular to FI. This means that in p the orthogonal projection of the dodecahedron upon b is a parallel projection with the direction Fs. To construct the elevation, Taylor needed to know one of its sides, and for this AB

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FIGURE X.27. (caption continued ) was of no use since its projection upon b is a point a. To obtain another point e in the elevation, he first constructed the vertex E of the dodecahedron directly based on the following considerations. Because the face ABCDE and the plane of the plan are perspectively parallel, the lines ae and AE are also perspectively parallel, hence the vanishing point, G, of ae is also the vanishing point of AE. This implies that the point E is the point of intersection of AG and Ie. Since the point A was given, Taylor now had AE at his disposal, and he constructed its projection ae upon the elevation by applying the facts that Aa and Ee are parallel to Fs and that ae has the vanishing point F (since ae lies in the plane ABCDE with vanishing line Fs, as well as in a plane with vanishing line FI). His further procedure was to complete the perspective elevation from ae, and finally to compose the perspective plan and elevation to one image by determining the points where the lines from I to the vertices of the plan meet the lines parallel to Fs through the vertices of the elevation. Taylor 1719, figure 19, with the letter H coloured white.

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Constructions as an Intellectual Experiment

S

een from a mathematical point of view, Taylor’s programme for performing perspective constructions is exquisite, but rather intricate to apply in practice. Several of Taylor’s constructions share their complication with many of Euclid’s constructions, in the sense that quite often a construction procedure is more a thought process than a practical means of obtaining a figure with given properties. The reason is that the solution to a given problem involves one construction that builds upon another, which in turn presupposes that a third has been carried out, and so forth. In most cases the idea is not to perform an entire construction of a problem, but to describe standard procedures by which it can be solved. Accordingly, in solving his problems Taylor did not always start from the beginning, but simply mentioned that a certain point could be constructed as shown in a previously presented procedure. Another complication in Taylor’s programme is that there are many vanishing points and lines to keep track of – which is partly due to Taylor’s insistence on presenting the most general cases. It seems indeed that the practical aspect of performing constructions did not really interest Taylor.

A Direct Plan and Elevation Construction

T

he impression that Taylor was more interested in the theoretical potential of his approach to perspective than in its applicability is confirmed by Taylor’s presentation in New Principles of a technique, mentioned in section X.5, that combines the use of a plan and elevation with direct constructions. Taylor’s idea was first to throw a plan and an elevation of an object into perspective, and then obtain the final image by a composition. To illustrate the theoretical insight this procedure requires I have paraphrased Taylor’s description of how to construct the perspective image of a regular dodecahedron – in fact his only example of how to apply the method – in the caption of figure X.27. His method of composing a plan and an elevation in p is intellectually attractive, but performing all its steps in practice is extremely cumbersome. It is particularly complicated because in order to display the generality of his method, Taylor chose a picture plane that is neither perpendicular nor parallel to any of the faces of the dodecahedron.

X.9

T

Taylor’s Treatment of Shadows

aylor’s theory of vanishing points and lines was ideal for handling the question of how to construct shadows and reflections in a perspective composition. In Linear Perspective he demonstrated this by presenting

9. Taylor’s Treatment of Shadows

525

some general results – though no concrete examples. In New Principles he illustrated the strength of his theory by including some examples, but here, on the other hand, he skipped the general considerations. In dealing with shadows, Taylor, like his predecessors, assumed that the light rays either come from one point or are a set of parallel lines. In the latter case he also considered the source of light to be a point – but one located at an infinite distance (Taylor 17151, 32). Concerning the treatment of the light source as a point he wrote: it being sufficient for Practice to regard only the Center of the Luminous Body; and having found the Contour of the Shadow in this Case, the Penumbra may be drawn by a good Judgment founded on much Observation; it being difficult to bring everything to exact mathematical Constructions, at least so as to be most convenient for Practice. [Taylor 17151, 30–31]

The quote shows that – like ’sGravesande had done (page 358) – Taylor acknowledged that some problems concerning perspective constructions can become so complicated that it is better to rely on visual impression than to attempt to solve them mathematically. Again like ’sGravesande, Taylor saw the central problem of constructing shadows in p as a problem of constructing the shadow a given point A casts upon a given plane a. As we have seen earlier, this involves constructing in p the point of intersection of a light ray through A and its orthogonal projections upon a. In general Taylor supposed that the following were given in p: The light source (if it is at an infinite distance it means the vanishing point of the light rays), the considered point, and the orthogonal projections of these two points upon a.8 Thus, his key problem was the following. Problem 8. Given the distance, and in p the principal vanishing point, the vanishing line of a plane a, a point A, a source of light L, and the orthogonal projections of A and L upon a, construct in p the shadow of the point A upon a (Taylor 17151, 31). He solved this straightforwardly, and in addition he determined the vanishing point of the light ray LA when L is a point at a finite distance. Taylor proceeded by assuming that in p are given a line l, its vanishing point, a point A on l, the orthogonal projection of A upon a plane a, and the vanishing line of a (Taylor 17151, 32). He then showed how the orthogonal projection of l upon a could be constructed. This result, combined with the solution to problem 8, allowed him to construct the shadow cast upon a by

8 When the vanishing point S of a set of sunrays is given, the vanishing point of their projections upon a can be constructed by finding the point of intersection of the vanishing line va of a and the line joining S with the vanishing point Na of the normals to a , Na being constructed according to the solution to problem 4.

526

X. Britain Vq

nb

Vn

k Vk

Vm

B

na

Vi A l m

Na T

na

Nb

n nb

T

q

FIGURE X.28. One of Taylor’s perspectival projection problems. Taylor first described his construction and then proved its correctness (Taylor 17151, 33–34). In presenting his solution I combine the two processes. Given are the distance, and in p the principal vanishing point (not marked in the drawing cf. note *), the vanishing lines va and vb of two planes a and b, the intersection k of these two planes, and a line l and its orthogonal projection m upon a. The elements are supposed to be given in such a way that the intersections considered in the following exist (I use the symbol ∩ to designate an intersection). It is required to construct the orthogonal projection n of l upon b. Let a ⊥ and b ⊥ be the planes through l perpendicular to a and b, respectively, which obviously means that l = a ⊥ ∩ b ⊥ and m = a ∩ a ⊥, and that the required line n is b ∩ b ⊥. The relations considered thus far are: k = a ∩ b,

(1)

l = a ⊥ ∩ b ⊥,

(2)

m = a ∩ a ⊥,

(3)

n = b ∩ b ⊥.

(4)

Taylor’s idea was to construct n by determining a point upon it and its vanishing point, that is: Determine a point A on n.

(5)

Determine Vn.

(6)

There is no immediate solution to (5), so Taylor involved yet another line, q, defined by q = b ∩ a ⊥.

(7)

His idea was then to determine A as l ∩ q (= a ⊥ ∩ b ⊥ ∩ b ∩ a ⊥ = a ⊥ ∩ n).

(8)

First, however, he had to construct q. Once more he used the strategy of determining a line in p by a point on it and its vanishing point, that is:

9. Taylor’s Treatment of Shadows

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FIGURE X.28. (caption continued ) Determine a point B on q.

(9)

Determine Vq.

(10)

He solved problem (9) easily by setting B = k ∩ m.

(11)

That B lies on q follows immediately from (1), (3) (k ∩ m = a ∩ b ∩ a ∩ a ⊥), and (7). To construct the vanishing point Vq of the line q, Taylor had to involve a number of points and lines. He began by constructing the vanishing point Vm of the line m as the point of intersection of m and va , and with the aid of the solution to problem 4 he constructed the vanishing points Na and Nb of the normals to a and b.* From the definitions of m and a ⊥ it follows that va ⊥ = NaVm.

(12)

Since vb is given, Taylor then solved (10) using the following relation: Vq = vb ∩ va ⊥ = vb ∩ Na Vm.

(13)

Thus, he had finally found his line q as q = BVq.

(14)

Since l lies in a ⊥, the relation (12) implies that Vl = NaVm ∩ l.

(15)

Analogous to (12) it can be seen that

vb ⊥ = NbVl.

(16)

Based on (4) Taylor then concluded that Vn = vb ∩ vb ⊥ = vb ∩ Nb Vl.

(17)

Having eventually found the point Vn, and seeing straightforwardly that A lies on n (relation (8)), Taylor offered the solution n = AVn.

(18)

Adaptation of figure 35 in Taylor 17151.

a polyhedron given in p. However, as noted, he provided no examples of this technique in Linear Perspective. Taylor’s third and last shadow problem in Linear Perspective deals with how to determine in p the orthogonal projection of a line upon a given plane when its projection upon another plane is known. His idea was that the solution of this problem should be applied in situations where the shadow of a line cast upon one plane is known in p and its shadow cast upon another plane has to be determined. The solution itself required many steps, because

*In his diagram Taylor did not show how he had constructed Na and Nb, nor did he indicate the given principal vanishing point and the distance in the diagram. He apparently trusted his readers to recall that Na and Nb can be constructed (cf. page 524).

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FIGURE X.29. Shadows in a room with a light source assumed to be concentrated in one point. Taylor 1719, figure 21.

determining orthogonal projections in p is not as straightforward as it sounds – the complication being to determine in p the point of intersection of a line and a plane. In the caption of figure X.28 I have presented Taylor’s solution. It clearly demonstrates how elegant Taylor’s theory is from an academic point of view, and how in practice it can become an elaborate juggling act with vanishing points and lines. Taylor concluded his section on shadows in Linear Perspective with the remark that shadows in the picture plane can be constructed by “putting the

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... Rules of Perspective into Perspective” (Taylor 17151, 34), or in other words by performing a central projection directly in p. In New Principles Taylor, as noted earlier, omitted the theoretical background for constructing perspectival shadows and just briefly described how he had obtained some of the shadows occurring in his illustrations, for instance those cast by the dodecahedron in figure X.27, and the shadows occurring in figures X.26 and X.29.

X.10

Taylor on Reflections

B

efore treating perspectival reflections, Taylor introduced the concept of an apparent place of a given point A in relation to a reflecting plane or a mirror m (Taylor 17151, 35). This point, which I call the mirror point or the mirror image of A and denote Am, is the point that lies symmetrically to A with respect to m (figure X.30). Hence when A′ is the orthogonal projection of A upon m, Am is defined as being the point on AA′ for which AmA′ =p A′A.

(x.4)

Let O be an eye point and let Ar be the reflection point of A, meaning the point in m in which the light path from A to O is reflected. According to the theory of reflection, for any eye point O, the point Am has the property that the ray ArO passes through it. For Taylor the three main problems associated with reflection were to determine the perspectival mirror images of a point, of a line, and of a plane with respect to a mirror given in p. More precisely, he aimed to solve the following problems. Problem 9. Given are the distance, and in p the principal vanishing point, the vanishing line nm of a reflecting plane m, and a point A and its orthogonal projection A′ upon m. Construct the perspective mirror point Am of A (Taylor 17151, 36). Problem 10. Given are the distance, and in p the principal vanishing point, the vanishing line nm of a reflecting plane m, a line l, its vanishing point Vl, and the orthogonal projection r of l upon m. Construct the mirror image, say s, of l and its vanishing point Vs (ibid.). Problem 11. Given are the distance, and in p the principal vanishing point, the vanishing line nm of a reflecting plane m, and the vanishing line na of a plane a. Construct the vanishing line of the mirror image of the plane a (ibid., 37). Taylor’s solution to problem 9 consisted of transferring the relation (x.4) to p. Thus, in p he constructed the vanishing point Nm of the normals to m (problem 4), and then by using problem 2 he constructed the point Am on the normal to m through A′ determined by Am A′ =p A′ A.

(x.5)

530

X. Britain m

O

Ar

Am A9 A

FIGURE X.30. A mirror point and a reflection point.

In solving problem 10 Taylor made one assumption more than those explicitly stated in the formulation of the problem, namely that the lines l and r meet in the point I (figure X.31). Since I lies on the line r situated in m, it is its own mirror point, which implies that the required line s passes through I. Taylor needed another point to determine s and chose its vanishing point Vs which he obtained in the following way. He noticed that the point of intersection of r Vl

A Vr

nm A9 Vs

I Am

r I

Nm

FIGURE X.31. Taylor’s solution to problem 10. One way of interpreting his claim that Vs is the point on the line VlVr defined by VsVr =p VrVl is the following. Points A, A′, and Am on l, r, and s fulfil the relation (x.5), that is, AmA′ =p A′A, which in other words means that A′ divides AAm perspectively in the ratio 1 : 1. According to (x.2) this implies that (1) (AA′ . A N ) : (A A′ . AN ) = 1 : 1. m

m

m

m

Taylor’s claim is equivalent to saying that this relationship also applies for the points Vl, Vr, and Vs. In modern terms this argument can be formulated as the result that the cross ratio on the left-hand side in (1) is invariant under a central projection from the point I – which is true. Adaptation of figure 39 in Taylor 17151.

10. Taylor on Reflections

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and nm is the vanishing point Vr of r, and that the line VlVr is the vanishing line of the plane determined by the lines l and r. Since the line s lies in this plane, Vs lies on the line VlVr, and Taylor claimed that Vs was the point on this line determined by Vs Vr =p Vr Vl.

(x.6)

To argue for the validity of (x.6), Taylor first observed that any set of points A, A′, and Am that lies on a perspective normal to m and on the lines l, r, and s, respectively, satisfies (x.5), and he then stated that this relation also applies if instead of considering points on the lines we consider their vanishing points (Taylor 17151, 37). In the caption of figure X.31, I have given a modern interpretation of Taylor’s claim. na

V

U

V'

nm

V"

Nm

FIGURE X.32. Taylor’s solution to problem 11. In p are given the vanishing line vm of a reflecting plane m, and the vanishing line va of a plane a. Although not figuring in the diagram, the distance and the principal vanishing point are also known (cf. note * in the caption of figure X.28). It is required to construct the vanishing line of the mirror image of the plane a. Since this problem only concerns vanishing elements, the correct thing to do would be to consider the set of planes or lines belonging to the given vanishing lines and points. For reasons of brevity, however, I follow Taylor and only mention one item from each set. Taylor assumed that the given vanishing lines va and vm meet in a point U. This point, then, is the vanishing point of the intersection of a and m, that is, of a line in a that is its own mirror image. The point U therefore lies on the required vanishing line. To find a second point on this line, Taylor chose a vanishing point V on va, drew the line through this point and Nm (the vanishing point of normals to m), and marked as V¢ the point in which VNm intersects vm. Then, letting V¢ and V¢¢ take the roles of Vr and Vs in relation (x.6), he constructed V¢¢ as the second point on the requested vanishing line. Redrawing, with altered letters, of figure 39 in Taylor 17151.

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It is actually possible to solve problem 10 without involving the relation (x.6), for instance as follows. One can choose a point A on l, find the corresponding point A’ as the point of intersection of r and ANm, construct Am on s with help of (x.5), and let s be the line IAm. Taylor, however, needed relation (x.6) for his solution of problem 11 – which is presented in the caption of figure X.32. At the end of the section on reflection in Linear Perspective, Taylor stated that with the aid of the solutions to problems 9–11, one can construct perspectival reflections just like one constructs other figures in perspective (Taylor 17151, 38). In theory this is true, but in practice it is difficult to keep track of all the lines involved. There is also another complication, namely that Taylor had assumed that the orthogonal projections of the considered elements upon the mirror were given in p. As demonstrated in the caption of figure X.28, it can be quite laborious to determine these projections. The same becomes clear from an example of determining reflections that Taylor included in New Principles (figure X.33). To the right in the picture is an oblique easel, and to the left an oblique mirror. Taylor set himself the task of constructing the reflection of the picture on the easel, but in fact he only showed how the reflection of the point G is constructed, and that in itself is a very long construction (for a presentation of Taylor’s procedure, see AndersenS 19921, 43–44, note 14). In the French edition of New Principles, Taylor’s illustration was changed slightly (figure X.34), but without making the example any easier. Taylor gave another example of reflection (figure X.35) that is more straightforward, but which includes complicated shadows, such as the one cast by the cylinder on the cone.

FIGURE X.33. One of Taylor’s most intricate demonstrations of a direct construction. Taylor 1719, section of figure 23.

10. Taylor on Reflections

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FIGURE X.34. The previous illustration adapted for the French edition of Taylor’s New Principles. Section of figure 23 in Taylor 1757/1759.

FIGURE X.35. Mirror images and shadows. Taylor 1719, figure 22.

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X.11

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Taylor on Inverse Problems of Perspective

I

n his description of Linear Perspective, published in Philosophical Transactions, Taylor introduced a natural distinction between two kinds of inverse problems of perspective – a distinction I took over in chapter VI. Thus, he claimed that inverse problems required one

... to find out what Point the Picture is to be seen from, or having that given to find what the Figures are which are described on the Picture. [Taylor 17152, 303]

Problems about Determining the Eye Point

I

n presenting the problems on inverse perspective occurring in Taylor’s two books, I begin with his examples concerning an unknown eye point. In all but one example, he dealt with perspective polygons and kept to his usual generality, which means that he determined a vanishing line na, its centre Pa, and its distance da. The eye points belonging to the composition were then situated on a circle with centre Pa and radius da in the vertical plane through Pa perpendicular to na. In other words, there is complete freedom of choice in selecting the plane a of the original figure among those with na as their vanishing line, and not until a has been chosen is the eye point determined. The most general of Taylor’s examples in this category concerns a perspective quadrangle whose original shape – that is, the angles and the ratios between the sides – is known: Problem 12. Let a quadrangle in p be given. Determine the vanishing line of the quadrangles of a given shape that are mapped into the given image, and determine the centre and the distance of the vanishing line (Taylor 17151, 40; Taylor 1719, 57). Taylor solved this problem with the aid of his own solutions to two of his previous problems. I first present these and another result he applied, and then will return to problem 12. Problem 13. Given the three points A, B, and C on a line l in p, and given that AC : CB =i r : s, construct the vanishing point V of l (Taylor 17151, 38; Taylor 1719, 55). Taylor presented a geometric construction of V and gave an algebraic expression for the length BV – which he obtained using relation (x.2): BV = r $ AB $ CB . s $ AC - r $ CB To solve his next problem Taylor applied the following result, which is a direct consequence of his definition of vanishing lines, and which he considered to be so obvious that he did not mention it. Observation 4 (figure X.36). Given in p are two perspectively coplanar lines l and m, as well as their vanishing points Vl and Vm. Let it also be given that ∠(l,m) =i φ. When the eye points for this composition are rotated around the

11. Taylor on Inverse Problems of Perspective

Vl

Vm

nm

l

m

FIGURE X.36. Diagram to observation 4.

535

vanishing line VlVm into p, they lie on the circular arc from which the segment VlVm is seen within the angle φ.9 Using this result, Taylor straightforwardly solved the following problem, as explained in the caption of figure X.37.

Op

Vl

Vm

Pa

Vn

na

B C A

FIGURE X.37. Taylor’s problem 14. The triangle ABC and the vanishing line v a are given, along with the original shape of ABC. To determine the centre and the distance of va, Taylor first prolonged the sides of the triangle ABC to meet va in the vanishing points Vl, Vm, and Vn. From observation 4 he knew that the turned-in eye point Op lies on the circular arc from which VlVm is seen within the original of angle ⬔BAC, and similarly on the circular arc from which VmVn is seen within the original of ⬔CBA, and he thus found Op as the point of intersection of these two arcs. The orthogonal projection Pa upon va is therefore the required centre, and PaOp the distance of va. Based on figure 40 in Taylor 17151. 9

How this arc is constructed is explained in Euclid’s Elements, book III, proposition 33.

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FIGURE X.38. Taylor’s solution to problem 12. Taylor 17151, figure 42.

Problem 14. Let the triangle ABC and its vanishing line va be given in p, and let it be known that ABC is the image of triangles with a given shape. Determine the centre of the vanishing line va and its distance (Taylor 17151, 39; Taylor 1719, 56). Based on the solutions to problems 13 and 14, Taylor solved problem 12 as follows. Let ABCD (figure X.38) be the given quadrangle. By drawing the diagonals Taylor obtained two line segments that are perspectively divided in given ratios (because the shape of a quadrangle that is mapped into ABCD is given). By means of the solution to problem 13 he found the vanishing points F and G of the diagonals, and hence the vanishing line of the quadrangle, and using problem 14 he found the centre and the distance of the vanishing line. As we saw in chapter VI (page 280), Stevin also treated the problem of determining the eye point for a perspective quadrangle that is the image of a figure of a given shape. Through a number of steps he arrived at his most general case. This was not nearly as general as Taylor’s problem concerning a quadrangle. Indeed, in the hundred years that had passed since Stevin worked on his problem, the theory of perspective had developed considerably, not least thanks to Taylor’s own contributions.

Problems Concerning the Shape of an Original Figure

I

n his description of Linear Perspective, Taylor mentioned problems associated with determining the original shapes of figures when a perspective figure and its eye point are given, but without presenting any typical examples in the book. In New Principles he presented a couple of problems of this kind (Taylor 1719, 54–56). He based his solutions upon an observation he did not state explicitly, but which concerned the correspondence between figures in the picture plane p and figures in three-dimensional space: Observation 5. When for a plane a the intersection ia, the vanishing line va, its centre Pa, and its distance da are given along with a polygon in p, then there

537

11. Taylor on Inverse Problems of Perspective Op

da V v a

Pa

FIGURE X.39. Illustration to observation 5. It is sufficient to prove the result for the image of one line segment, say AB. Let then for the plane a the following quantities be given in p: the intersection ia, the vanishing line va along with its centre Pa and its distance da (which implies that the turned-in eye point Op is known as well). To determine the original line segment AoBo Taylor constructed the points of intersection, I and V, of the line AB and the lines ia and va, respectively, drew the line l through I parallel to OpV, found Ao as the point of intersection of OpA and l, and Bo as the point of intersection of OpB and l.

B A

ia

I

Ao

Bo

l

is a unique polygon in a that has the given polygon in p as its perspective image. (If the angle between a and p is also given, then not only the form of the original polygon but also its position in space is known.) This result can be obtained by inverting Taylor’s visual ray construction, as I have described in the caption of figure X.39.

Determining the Eye Point as Well as the Shape

I

n both his books on perspective Taylor included a problem concerning a certain polyhedron, namely:

Problem 15. Given in p is the image of a right-angled parallelepiped (figure X.40). Determine the eye point as well as the ratios between the original sides (Taylor 17151, 41; Taylor 1719, 57). From the givens, Taylor straightforwardly determined the principal vanishing point S and the distance OS. In New Principles he used observation 5 to determine the ratios between the sides of the parallelepiped, whereas in Linear Perspective he had to work a bit harder to complete the same task. Taylor claimed the example was relevant for stage design (Taylor 17151, 42; Taylor 1719, 58), pointing out that if the edges AB, CG, and ED (figure X.41) are parallel, it can only be concluded that the eye point lies on a horizontal circle with a known diameter (KI). What he presumably had in mind was

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FIGURE X.40. One of Taylor’s illustrations to problem 16. Taylor 1719, figure 24.

that – even though the apparent ratios between the sides of the parallelepiped will vary with the position of the eye – if the eyes of a spectator in a theatre are located near the mentioned arc, he will experience the drawn image as a right-angled parallelepiped.

X.12

I

The Immediate Response to Taylor’s Work

t would be interesting to know how Taylor’s contemporaries reacted to the relatively advanced and fascinating theory of perspective he had created. He himself stated, as we have already seen (page 498), that he had written New Principles in reaction to the criticism to which Linear Perspective had given rise. The responses he received may have been oral, or they may have been published in journals that are not obvious sources to search. At any rate, I have not been able to find any contemporary reviews of Linear Perspective. Taylor’s grandson Young claimed that Johann Bernoulli had characterized Linear Perspective as being “abstruse to all and unintelligible to artists for

12. The Immediate Response to Taylor’s Work

539

FIGURE X.41. A special case of problem 16. Taylor 17151, figure 43.2.

whom it was more especially written” (Young in TaylorS 1793, 29). However, this quote has not been verified (AndersenS 19921, 53, note 20). We saw earlier that Bernoulli was so interested in perspective that he read, and became very enthusiastic about, ’sGravesande’s Essay de perspective (page 359). If Bernoulli actually read Taylor’s work on perspective, he would certainly have been able to see its qualities, but it is very unlikely he would have admitted this, for Bernoulli and Taylor had been opponents in a bitter cross-Channel fight about who had priority to the invention of the calculus. I am not aware of any reviews of New Principles either, but those who according to Taylor criticised Linear Perspective would presumably have been no more pleased with his second book, since, as noted, it is no clearer than the first. It actually seems that Taylor made a third attempt at being understood, but died before he was ready to publish – or at least that is what Cowley reported (Cowley 1765, vii). As will become clear, Taylor’s work came to have an enormous influence on perspective literature in Great Britain, but not until after he had died. In fact, the first sign of Taylor being read in Britain appeared in 1738 – seven years after his death – whereas he only began to have a major impact in the mid-1750s. In sections VIII.7 and IX.12 we saw that Taylor’s work on perspective was noticed abroad, but this, too, was after his death. Apparently his work was too advanced for his time, though it should be said that had they taken an interest in it, many mathematicians would have understood his thinking.

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Taylor’s Work in History

T

he 1750s brought a renaissance for Taylor’s ideas in Britain as well as on the Continent, but his work was treated in two very different ways. In his own country the writers on perspective attempted to explain Taylor’s insights to an audience of non-mathematicians, whereas in Italy and France Taylor’s own text became available in translation, without this giving rise to a second generation of books based on his theory – one pre-1800 exception being Michel’s Traité de perspective lineaire from 1771 (page 484). On neither side of the Channel did any further development take place in the eighteenth century based directly on Taylor’s ideas. Despite the fact that Taylor’s theory was presented in only a handful of continental books, a knowledge of and admiration for it were kept alive on the Continent. In his Histoire des mathématiques, Jean Étienne Montucla emphasized that Taylor had treated perspective in a new manner (MontuclaS 1758/1799, 638/711). Referring to Montucla, Abbé de la Chapelle characterized ’sGravesande’s Essai de perspective and one of Taylor’s books, presumably New Principles, as “the two best works that we have on this subject” (ChapelleS 1780, 454). Michel Chasles repeated this opinion, claiming that “S’Gravezande and Taylor are often mentioned, and rightly so, for having treated perspective in a new and scientific manner” (cf. note 16 in chapter VI). As late as 1865, the Italian mathematician Luigi Cremona – using the anagram Marco Uglieni as a pseudonym – published a paper on Taylor’s theory. Cremona presented nine of Taylor’s problems from New Principles, keeping fairly close to Taylor’s own formulation. He also paraphrased Taylor’s constructions of the problems, but gave new proofs of their correctness. Cremona titled his paper I principii della prospettiva lineare secondo Taylor (The principles of linear perspective according to Taylor), which indicates that he conceived of Taylor’s principles as fundamental construction problems.10 What they were to Taylor’s British successors will be discussed later in this chapter. In 1908 another Italian mathematician, Gino Loria, wrote on the history of perspective seen from a mathematical point of view. He was quite enthusiastic about Taylor’s New Principles, as the following shows. The reading of this short but excellent work surprises the modern reader most pleasantly, because in it, in a word, one finds all the fundamental concepts (except perhaps the “circle of distance”) and all the methods of central projection, as one finds them presented in for instance the classical textbook by W. Fiedler.11

10

The nine problems Cremona chose have the following numbers in New Principles: 1, 2, 3, 5, 7, 11, 14, 16, and 17. 11 Die Lektüre dieses kürzen, aber vortrefflichen Werkes bereitet dem modernen Leser eine der angenehmsten Überraschungen, da man, um es ganz kurz zu sagen, darin alle Grundbegriffe (nur etwa den “Distanzkreis” ausgenommen) und alle Methoden der Zentralprojektion vorfindet, wie man sie z. B. in dem klassischen Lehrbuch von W. Fiedler ausgeführt findet. [LoriaS 1908, 597–598] The book by Fiedler to which Loria referred to was presumably FiedlerS 1871.

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541

Notwithstanding these positive assessments of Taylor’s work, as mentioned in section X.3, it is not for his work on perspective that Taylor is remembered, but for his work in the calculus.

X.14

Hamilton’s Comprehensive Work on Perspective

Hamilton’s Background

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he first work showing Taylor’s influence is a substantial book from 1738 spanning four hundreds pages and entitled Stereography or a Compleat Body of Perspective written by a perspective enthusiast named John Hamilton. He opened his work by thanking Joseph Jekyll for placing him “in a more easy Station of Life” (Hamilton 1738, dedication). This step had allowed Hamilton to finish his book, which “was begun many years since and carried on at short and distant Intervals of Recess from a Hurry of Business” (ibid.). He also mentioned that “the Subject, indeed is foreign to the Profession in which I have been bred” (ibid.). On the title page of Stereography we learn that Hamilton called himself “esquire” and was a Fellow of the Royal Society, and that is all the information I have found about him. However, the fact that he mentioned working under Joseph Jekyll gives a clue to Hamilton’s possible profession, namely law. Jekyll became serjeant-at-law in 1700, and seventeen years later master of the rolls – a position he held until his death (HutchinsonS 1902). It was presumably during the latter period that Hamilton worked with Jekyll. At the end of the eighteenth century Thomas Malton also came to the conclusion that Hamilton “practised the Law” (T. Malton 1783, 98). Hamilton’s interest in perspective seems to have been deep and to have involved studying a large part of the literature which he – like so many other writers on the subject – found unimpressive. Thus, in his preface he wondered why a subject that had been treated over such a long period of time had not been better shaped, and he suggested the following answer. The Reason of this seems to be, that the first Writers having set out upon very narrow Principles, and prescribed difficult and inconvenient Operations, those who followed, rather applied themselves to facilitate the Practice, than to enlarge the Foundation: This might induce the Mathematicians amongst them to imagine the Subject incapable of any great Advancements in the Theory, and so not worthy of their closer Application;12 and the Artists, to substitute Drawing and Designing in its Stead, to supply the Imperfection and the Deficiency of the Rules ... [Hamilton 1738, a2–a3]

Hamilton’s remark about mathematicians’ lack of expectations to the theoretical aspect of perspective is very much to the point. The mathematically most advanced books on the subject that existed when Hamilton published his work in 1738 were the works by ’sGravesande and Taylor, which, as noted, 12

This is close to what ’sGravesande had written earlier (’sGravesande 1724, iv, quoted page 360).

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were addressed to the practitioners of perspective rather than to mathematicians. Lambert would later follow the same path. And while these three mathematicians toned down the role geometry played in their works, the non-mathematician Hamilton dared to acknowledge his mathematical approach.

Perspective and Conic Sections

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amilton combined his presentation of perspective with a study of projections of conic sections and harmonic division.13 By including these subjects he produced a work that more than any other pre-1800 book on perspective belongs to the prehistory of projective geometry. Hamilton praised Philippe de La Hire for his approach to conic sections, which Hamilton preferred to Guillaume François Antoine L’Hospital’s analytic treatment (ibid., a2).14 Hamilton included a small section on “geometrical” (parallel) projections, but his main concern was “stereographic” (central) projections. His treatment of the latter subject falls into two rather detached sections, one on projections of conics, and one on perspective. He dealt with conics in a way that, generally speaking, is too academic to be of use in perspective (Hamilton 1738, 80–150). When looking at projections of circles, for instance, he distinguished between three cases (figures X.42–X.44), of which only the first normally is relevant in connection with perspective constructions. He also expended quite a bit of energy on determining the axes or a set of conjugate diameters of the conics upon which the circles are projected (ibid., 104 ff.), just as ’sGravesande and Ditton had done when the images were ellipses. This is far too complicated a method to use for constructing the perspective images of circles – the ordinary method being to construct a number of points on a perspective circle, as we have seen. Hamilton was also intrigued by the subject of projecting the various conic sections upon each other, and in this connection looked at harmonic division.

Hamilton on Linear Perspective

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s for his treatment of perspective, Hamilton pointed to Taylor as his main source of inspiration. Hamilton claimed that Taylor had “in a few Pages, made more Advances towards perfecting the Science than all Writers who went before him” (ibid., a4). In presenting the subject, Hamilton dealt with general situations, but contrary to Taylor, he approached them stepwise, starting with simple cases.

13

In Hamilton’s day a line was said to be divided harmonically by four points A, B, C, and D when (AC . BD) : (AD . BC) = 1; later when mathematicians oriented line segments, the definition became (AC . BD) : (AD . BC) = –1. 14 Hamilton’s second reference was to L’HospitalS 1720, but it is not quite clear whether he had in mind La HireS 1673 or La HireS 1685 – or both. In the first La Hire applied synthetic methods, whereas in the second he made the concept of harmonic division a central concept.

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FIGURE X.42. Hamilton’s illustration – drawn in perspective – of his first example of projecting a circle. A cone with a circular base in a horizontal ground plane is given, along with the picture plane GH and the apex I, which is also the eye or projection point. Through I Hamilton drew the directing plane parallel to the plane GH, introducing the line LM in which the directing plane cuts the plane of the base, calling it, as Taylor had done, the directing line. He proved that when this line lies outside the base circle, the latter is projected into an ellipse (or a circle). Hamilton 1738, figure 54.3.

FIGURE X.43. Hamilton’s second example. Here, the directing line LM (cf. the caption of the previous figure) is a tangent to the base circle, and Hamilton proved that the circle is projected upon a parabola. Hamilton 1738, figure 54.4.

Hamilton devoted a section to the foreshortening of orthogonal line segments and, as Lambert would later do, he introduced what could be called the curve of foreshortening. When the distance between a point in the ground plane and the ground line is x and the distance between the point’s perspective image and the ground line is y, Hamilton showed that the points (x,y) lie on a hyperbola (Hamilton 1738, 34; for more technical details, see page 644).

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FIGURE X.44. Hamilton’s third example. Here, the directing line LM (cf. the caption of figure X.42) cuts the base circle, and Hamilton proved that the circle is projected into two branches of a hyperbola. Hamilton 1738, figure 54.5.

While perspectivists since Piero della Francesca had been looking at foreshortening of orthogonal lengths (cf. (ii.1)), Hamilton also investigated the relation between the lengths of line segments and their perspective images, which fits in with his interest in finding out what happens with ratios of line segments under a central projection. Thus, when O is the eye point (figure X.45), F its foot, P the principal vanishing point, Q the ground point, and A, B, and C three points on the line FQ, Hamilton proved – by considering similar triangles – that (ibid., 40) Ai Bi $ C i P = AB . Bi C i $ Ai P BC This result is actually similar to Taylor’s slightly more general result (x.2) page 514. In a corollary, Hamilton noticed that when AB and BC are equal, then AiP is divided harmonically. In presenting the Taylorian theory of perspective, Hamilton worked much more with concepts belonging to the directing plane (figure X.46) than the master himself had done. Otherwise Hamilton kept fairly close to Taylor’s ideas, but spent many more words than Taylor had on explaining the theory

O

P Ci Bi Ai

F

Q

A

B

C

FIGURE X.45. Illustration for one of Hamilton’s results on foreshortening. Adaptation of figure 21 in Hamilton 1738.

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FIGURE X.46. Hamilton’s illustration of, among other things, concepts belonging to the directing plane. He let the eye point be I and the principal vanishing point be O. The vertical planes through I and O are the directing plane and the picture plane, respectively. Like Taylor had done in the diagram reproduced as figure X.6, Hamilton considered the opposite side of the picture plane rather than that seen from the eye point. Hamilton 1738, figure 12.

and presenting examples. All of Hamilton’s examples concern geometrical figures, more specifically polygons, circles, the conics, the Platonic solids (figure X.47), cones, cylinders, spheres, and annuli. Taylor had been rather brief in treating the construction of perspectival shadows and reflections. Hamilton presented these themes thoroughly based on Taylor’s theory; that is, he used vanishing points and lines. Particularly in connection with reflections he went rather far in treating problems such as the following. The Center and Distance of the Picture, and the Vanishing Line of a Reflecting Plane, being given, together with the Vanishing Line of an Original Plane, and its Reflection; thence to find the Reflection of any other Vanishing Line proposed. [Hamilton 1738, Problem 34]

Unlike Taylor, Hamilton did not devote a section to inverse problems of perspective, but he included some problems in which he searched for the simplest of the figures having a given figure in the picture plane as its image. One of his examples features a given quadrangle (figure X.48), the problem being to find the position of a vanishing line with respect to which the given quadrangle represents a parallelogram (ibid., 160). In the very last part of Stereography, Hamilton addressed eleven themes related to the practice of perspective. These include choosing the parameters of a painting and topics with headings like “Of the Consequence of viewing a Picture from any other Point than the true Point of Sight”, “Aerial Perspective”, “Scenography”,15 and “Anamorphoses”. His commentary is 15 By this he meant the art of designing theatre stages in perspective. For more on the term scenography, see note 3 in the introduction.

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FIGURE X.47. Hamilton’s construction of the perspective image of a dodecahedron. Like Taylor (figure X.27), Hamilton composed a perspective plan and a perspective elevation. Hamilton 1738, figure 136.

generally competent, and his presentation of anamorphoses, for instance, is very clear and illustrative.

Hamilton’s Influence

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here is no doubt that Hamilton put a lot of work into composing Stereography, and that it was a good book for mathematically inclined readers. However, there are not many traces of him having a possible influence on posterity. We can be sure, as we shall soon see, that Kirby studied Stereography carefully, and benefited from it. Thomas Malton also read Hamilton’s work, but if he found any inspiration in it, he did not admit this, claiming: “I fear that his Reward was by no means proportioned to the Labour attending the execution of it” (T. Malton 1783, 98). Malton found that “to go through the drudgery of it, must render it extremely irksome” (ibid.).

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FIGURE X.48. One of Hamilton’s inverse problems of perspective. Hamilton 1738, figure 96.

Some of Hamilton’s ideas can also be found in Lambert’s work, for instance the foreshortening hyperbola and the use of the phrase ‘perspective geometry’, which Hamilton applied in his introduction (Hamilton 1738, b4) and in the title “Rules of Perspective Geometry” that he gave to the second part of his book. However, he did not make it a central concept, like Lambert would later do. All in all, I do not find the similarities between Lambert’s and Hamilton’s approaches so strong that I would conclude Lambert was familiar with Hamilton’s Stereography.

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Kirby and Highmore

s we saw, Taylor had wished to reach the practitioners of perspective, but his style made this all but impossible. A bridge was built by two nonmathematicians, Kirby and Highmore, who had come to appreciate Taylor’s work so much that they made a great effort into understanding and transmitting his thoughts to a wider audience. According to Warren Mild, Joseph Highmore (1692–1780) was the first of the two to take up perspective à la Taylor, and he may even have played a role in motivating John Joshua Kirby (1716–1774) to do likewise. Mild described the link as follows (MildS 1990, 350–351). Highmore’s enthusiasm for Taylor’s approach to perspective convinced the painter and engraver Hubert Gravelot that it was the best, and Gravelot began to teach this approach at his drawing school. Here Thomas Gainsborough, a pupil of Gravelot’s who would go on to become one of Britain’s most famous painters, became so thrilled about Taylor’s perspective that he introduced it to his neighbour in Ipswich, John Kirby.

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Kirby published on Taylor’s methods before Highmore did. He also had the clearest style of presentation and therefore became the most influential of the two men in making Taylor’s ideas known. I consequently pay more attention to Kirby’s work than to Highmore’s.

Kirby’s Publications on Perspective

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irby (figure X.49) began his career as a coach and house painter, but inspired by Gainsborough, he later took up landscape painting and drawing. He also taught drawing, and according to his own testimony it was his friend William Hogarth who encouraged him to publish on perspective (Kirby 1754, dedication). This was likely connected with Hogarth having asked Kirby to teach perspective at the Academy of Painting, Sculpture, and Architecture in London, the so-called St. Martin’s Lane Academy.16 Kirby’s activities resulted in Dr. Brook Taylor’s Method of Perspective Made Easy, Both in Theory and Practice (hereafter called Taylor’s Method Made Easy), for which he announced for subscribers in May 1751 (MildS 1990, 351). The book appeared in 1754, with a charming frontispiece by Hogarth (figure X.50) to whom Kirby had dedicated the first part of his book. Hogarth’s creation shows what can happen when an artist fails to properly learn perspective, and it has found its way into virtually every book on the history of perspective.

FIGURE X.49. A portrait of Kirby. CromwellS 1820.

16

Hogarth had established this academy in 1734 as a kind of continuation of an academy the painter Godfrey Kneller had opened in 1711.

FIGURE X.50. Hogarth’s frontispiece to Kirby 1754.

FIGURE X.51. David Hockney, Kerby (After Hogarth) Useful Knowledge, 1975. Museum of Modern Art, New York.

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Hogarth’s piece is also sometimes seen as a forerunner for the Dutch artist Maurits C. Escher’s drawings of impossible situations, and in the twentieth century it inspired the British painter David Hockney (figure X.51). Apparently Kirby’s book achieved greater success than expected, for it was already reissued, without many changes, the following year. In 1765, a third essentially revised edition was published, both in an impressive folio volume – in which several of the figures can be folded out to become three-dimensional – and in quarto. Both these editions were reissued in 1768. In revising his work Kirby had, in particular, added new mathematical arguments. Kirby published two more works on perspective, one of which was provoked by a remark he read in the Public Advertiser from 14 March 1755. According to his own description, it read as follows: The best author that ever treated on Perspective is now translated from the Italian Language into English , which Work when completed will undoubtedly be the most useful of its kind. [Kirby, s.a., preface]

The work in question was Sirigatti’s Practice of Perspective, which was published in an English translation in 1756 – more than one and a half centuries after its composition. Kirby’s reaction was to write a long review entitled Dr. Brook Taylor’s Method of Perspective, compared with Examples lately publish’d on this Subject as Sirigatti’s by Isaac Ware. This appeared without a date, but presumably in 1757. Wanting to promote Taylor, Kirby was far from kind to Sirigatti and Ware. Kirby’s third work on perspective was a de luxe edition (figure X.52) entitled The Perspective of Architecture ... Deduced from the Principles of Dr. Brook Taylor. The first part of this book contains a description of how to use an instrument for drawing plans and elevations. This instrument, which Kirby called the “Architectonic Sector”, is a sector with special scales. In the second part of the book Kirby taught his readers how to throw geometrical figures and architectural elements into perspective (figures X.53 and X.54). He may have found help in Taylor’s theory while composing this part,

FIGURE X.52. Vignette by Samuel Wale to Kirby’s The Perspective of Architecture. Kirby 1761, part two, 1.

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FIGURE X.53. An architectural element in perspective. Kirby 1761, plate 49.

FIGURE X.54. The previous drawing completed. Kirby 1761, plate 50.

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but there is not much of the Taylorian spirit in it. In fact, it contains no geometrical explanations at all, but only presentations of various constructions. In one of these Kirby applied a measure point for diagonals, though without giving it a name, to construct line segments that are perspective images of given lengths on a diagonal (Kirby 1761, part two, 5). The Perspective of Architecture was dedicated to, and presumably funded by, King George III, who – as Prince of Wales – had encouraged Kirby to write the book after Kirby had taught him perspective. Earlier in this history we have seen other examples of perspectivists who taught princes, namely Stevin and de Caus. Kirby’s connection to George III gained him a position as surveyor for the buildings at Kew Palace.

Kirby’s Main Work on Perspective

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irby opened his Taylor’s Method Made Easy by joining the ranks of those complaining about the earlier literature:

For certain it is, that no Subject hath been treated in a worse Manner than this [perspective], notwithstanding the many Volumes which have been wrote upon it; some purely Mathematical, and therefore unfit for the Generality of Persons who are concerned in the Arts of Design; others wholly Mechanical, made up of incoherent Schemes, unapplicable Examples, and such a Confusion of unnecessary Lines, as tend only to puzzle and discourage the Learner. [Kirby 1754, first book, i]

Like some of his predecessors, Kirby wanted to bridge the gap between the theory and practice of perspective. Cowley described the process Kirby went through as follows. This ingenious author, by attentively examining and applying Dr. Taylor’s new principles of Perspective to practice, was gradually led to a discovery of their generality and facility in operation, saw how preferable and excellent they were in practical applications, how simple and extensive their constructions ..., and how beneficial they would be if generally known to artists concerned in works of design; possessed with these and such like considerations, he employed himself zealously to retrieve them from the state of darkness in which their author’s brevity of expression and manner of writing had concealed them, and became the first among artists, who appeared in publick, to explain their true nature and use in adapting them suitably to the arts of design. [Cowley 1765, viii–ix]

To some extent I share Cowley’s admiration of Kirby, thus finding him more successful than most in presenting the mathematics behind perspective to practitioners of the field. He was also good at making illustrative drawings (figure X.55) that helped the reader visualize his explanations – and he became even better in his later editions. My positive attitude towards Kirby as a teacher presumably reflects my own mathematical background. Looking at the matter from the art historian’s point of view, Martin Kemp finds that many of Kirby’s problems “are of an uncompromisingly geometrical nature” (KempS 1990, 154).

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FIGURE X.55. A rabatment in which the eye point is turned into the picture plane above the horizon. Kirby 1765, book one, plate 5, figure 1.

The structure of Taylor’s Method Made Easy is quite unusual. As we have seen, most authors who treated both the mathematical theory of perspective and its practical application started out with the theory, then deduced some constructions from it, and finally presented a number of more or less elaborate examples of perspective constructions. Books in this genre were often divided into two parts that were entitled the “theory of perspective” and the “practice of perspective” or something similar. Kirby’s work also has two parts, called the “first book” and “second book” and separately paginated, but the theory does not occur in isolation. His first book offers the theory as well as its application, the text being addressed to readers “who do not like to

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take Things for granted, but choose to be convinced by Demonstration, and to have the Reason of Things explained upon certain Principles” (Kirby 1754, first book, iii). His second book is dedicated to the Academy of Painting, Sculpture, and Architecture in London and is written for people “who either want Time or Capacity to go regularly through the Theoretical Part” (ibid.). The second book only contains the practice without making any references to the first book, being, in fact, independent of it. Kirby could just as well have published the two books as separate items, but in that case he would probably have sold fewer copies in total.

Kirby’s Inspiration

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y mentioning Taylor in the title of his book, Kirby very explicitly admitted that he had been inspired by Taylor’s work. Taylor was not Kirby’s only source for the mathematical theory of perspective, however. He also found Hamilton’s book extremely useful, as can be seen from the following:

... I must frankly acknowledge, that I think it [Hamilton’s Stereography] the best System of Perspective Projection, ... that ever was, or, perhaps, ever will be made publick; and I should be very ungenerous in not confessing that it has been of great Service to me in several Parts of my Work; and that I am indebted to it for some Things which I should never have thought of, had not that ingenious Gentleman pointed them out to me ... [Kirby 1754, first book, ii]

Hamilton’s Stereography presumably helped Kirby in his struggle to understand Taylor. Besides this, Kirby benefited from the work when treating stage design and how colouring influences perspective (ibid., second book, 76). Kirby did not similarly point to any inspiration from books on the practice of perspective. It is my impression that in choosing examples of perspective constructions he consulted earlier works, but was not dependent on any specific author. In fact, he included a chapter called “An Abstract of several Methods of Perspective; transcribed from the most eminent Authors” (ibid., 81–84), in which he presented methods applied by Vignola, Vredeman de Vries, Marolois, “the Jesuit” (Dubreuil), and Pozzo. He also mentioned Kircher and Lamy. Giving full reference, he copied some drawings from these authors, such as Pozzo’s construction of the image of a cupola (ibid., figures 81 and 82, cf. figures VIII.18 and VIII.19).

Kirby on the Theory of Perspective

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n treating the theory of perspective Kirby started from scratch by introducing all the geometrical terms involved. Contrary to Taylor, Kirby also included a bit of optical theory, particularly the theory of apparent sizes. As for the rest of the subject matter he was less ambitious than Taylor, his main aim being to make the concepts of vanishing points and lines familiar to his readers. In fact, in his praise of Taylor, Kirby particularly mentioned that Taylor

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FIGURE X.56. One of Kirby’s fairly straightforward examples. Kirby 1765, book two, plate 14, figure 6.

had furnished him “with Principles to build upon” (Kirby 1754, first book, i), by which he presumably meant the theory of vanishing points and lines – a point to which I will return in the section on Highmore. Kirby applied this theory in his mathematical arguments, but unlike Taylor he presented it stepwise rather than starting with the most general situation. For instance, Kirby passed lightly over how to perform perspective constructions in oblique planes, claiming that this problem is “rather more curious than useful” (ibid., iv).

Kirby on the Practice of Perspective

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ach of the two books of Taylor’s Method Made Easy contains several methods for determining the perspective image of a point. Kirby’s favourite constructions were a visual ray construction, and some procedures – including a distance point construction – that involve two vanishing points. In most cases Kirby used a rabatment similar to Taylor’s, that is, one in which the eye point is turned into the picture plane (figure X.55). In his first book, Kirby’s examples of perspective constructions mainly concerned squares, triangles, and circles – his reason being that these are “the simple Materials of Shapes in general” and of buildings in particular (ibid., v). In the second book Kirby taught his readers how to throw some geometrical figures and a few other – not very elaborate – objects into perspective (figure X.56), but he also included a few illustrations that are quite spectacular (figure X.57). Like Hamilton, Kirby presented the construction of perspectival shadows much more thoroughly than Taylor had (ibid., first book, 56–64; second book,

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FIGURE X.57. An elaborate composition. Kirby 1754, book two, figure 70.

66–73). He also provided many illustrations of perspectival shadows, including no less than ten in his first book and eleven in his second – one of which is reproduced in figure X.58. Perspectival reflections, on the other hand, did not appeal to Kirby. He presumably found it too difficult a subject to present, and his treatment is limited to two pages (ibid., second book, 73–75). Kirby also took up a number of practical aspects. For one thing he devoted a full chapter to the choice of parameters of a perspective composition, presenting some rules of thumb, and referring to Pozzo in his later editions (Kirby 1765, first book, 92). Besides this, Kirby taught his readers how to construct perspective images on domes, vaulted roofs, and other curved surfaces. In contrast to most other perspectivists, such as Bosse and Pozzo

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FIGURE X.58. Perspectival shadows. Kirby 1754, book two, figure 91.

(figure IX.45 and VIII. 22), Kirby did not project a horizontal grid of squares upon a cylindrical vault, but chose a solution that seems more theoretically interesting than practical: He searched for a horizontal grid that is projected into a system of half-circles and straight lines upon the vault (figures X.59 and X.60). This procedure has the advantage that the cylindrical picture plane can be furnished with a coordinate system, but the disadvantage that the ellipses that are projected in the wanted circles are not so easy to determine. For reasons soon to be explained, Kirby already published an appendix to Taylor’s Method Made Easy in 1754 – which was issued together with the 1755 edition and incorporated in later editions. In this appendix he devoted a section to perspective instruments (Kirby 1754/1755, 12–15), among them a “pocket” camera obscura (figure X.61).

Kirby and the Column Problem

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t one point Kirby directly opposed the mathematical approach to perspective, namely in connection with the column problem (cf. section II.15). Taylor had not treated this problem, but given his attitude that a perspective representation of an object should appear so that a viewer could not distinguish between the representation and the real original object (Taylor 1719, 1), there can be no doubt that Taylor would have chosen the solution dictated by the laws of perspective. Kirby, on the other hand, joined the

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FIGURE X.59. Kirby’s grid of lines and half-circles on a cylindrical vault. Under the vault is a horizontal rectangle HIGK divided by the lines ML and the line perpendicular to ML through the point C. As Kirby has drawn his illustration in perspective with E as principal vanishing point, reading it can be confusing. Kirby 1765, plate 11, figure 1.

FIGURE X.60. The grid from the previous figure projected upon the plane GHIK from the eye point E. Kirby 1765, book one, plate 11, figure 4.

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FIGURE X.61. Kirby’s illustration of a “pocket” camera obscura. Kirby 1754/1755, appendix, figure 14.

group that called for correspondence between the perspective image of an object and the object’s visual impression. In fact, he claimed that the aim of perspective is “to draw the Representations of Objects as they appear to the Eye” and not to draw them “by the Rules of Geometry” (Kirby 1754, first book, 70). Based on a mathematically incorrect argument, Kirby found a solution to the column problem that is different from any other of which I am aware. He would not follow the “strict Rules of Perspective” for “round Objects”, but would do so for “angular ones” (ibid.,72). He reached this standpoint by looking at two sets of columns (figure X. 62) with horizontal sections that are circles and squares, respectively, and studying them as follows. In a cylindrical column he considered the circle defined by the horizontal section through the eye point E and drew in it the diameter perpendicular to the line joining E and its centre. He let the angle at E defined by this diameter determine the visible part of the column.17 This is a determination similar to the one Piero della Francesca presumably used (page 58). Like several of his predecessors, though not Piero, Kirby argued that since the visible parts of the columns appear to be smaller the further they are away from the eye point E, they should also be represented as smaller and smaller. That made him reject representations obtained by means of a perspective projection, as in these the columns become larger and larger as their distance to the eye point increases. For the quadratic columns, Kirby similarly considered the squares defined by a horizontal plane through the eye point. He let the visible part of the square nearest to the eye be determined by its side parallel to the horizon and the visible part of the next square by one of its diagonals. 17

On the determination of the visible part of a cylinder, see page 56.

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FIGURE X.62. A horizontal section of two rows of columns. Kirby 1754, book one, figure 108.

From this introduction of visible parts he drew the faulty conclusion that the visible parts of square columns appear to increase with increasing distances to the eye,18 and that made Kirby accept to follow the rules of perspective for “angular” columns. In 1754 Highmore published a pamphlet in which he included some comments about Kirby’s Taylor’s Method Made Easy. In general, Highmore wrote positively about Kirby’s work, but found that his treatment of the column problem betrayed Taylor. In one of his disapproving remarks Highmore mentioned that Kirby accepted the practice of representing a long rectangular wall parallel to the picture plane in such away that its height at the end is the same as its height directly opposite the eye, “notwithstanding that it appears of less Height, the farther it is extended” (Highmore 1754, vi). 18 Kirby’s two squares are actually positioned so that the side of the square nearest to the eye is seen within a smaller angle than the diagonal in the next square. However, if the second square were positioned differently, it is possible that the second angle becomes smaller than the first. Moreover, and perhaps more importantly, if Kirby had involved a third square equidistant to the first two, its diagonal would always be seen within an angle that is smaller than the angle within which the diagonal of the second square is seen. This is similar to what happened with the diameters in the circles he considered. Thus, if he did not want to accept the rule of perspective for circular columns, he should also have refused to follow the rule for square columns – had his mathematical reason not misled him.

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Kirby reacted to Highmore’s criticism by writing the appendix to Taylor’s Method Made Easy, mentioned earlier. Kirby did not take Highmore’s point and refusing to give in, he claimed, among other things: Now I should very readily have retracted what I said upon this controverted Affair, and without any Difficulty should have acknowledged myself in an Error, if Mr Highmore had proved that I was mistaken; but since what he hath given us, does not, upon the most candid Examination, appear to me clear and satisfactory, I must therefore beg to retain my former Opinion; ... [Kirby 1754/1755, appendix]

In defending the argument for not always applying “the strict Rules of mathematical Perspective”, Kirby explained that if a family picture with many people had to be drawn and the rules were followed, then those near the edges would appear as though they were in an anamorphosis. Instead they should by drawn “under the most pleasing and agreeable Shapes” (ibid.). Kirby’s solution was later criticized by Edward Noble as well.19 He remarked that “the appearance of an object” is often confounded with its perspective representation (Noble 1771, 78). He took up the issue because the reputation of this gentleman (which excepting in this instance is not wholly undeserved) may give sanction to errors tending to lessen the utility of perspective. [ibid., 147]

Noble did not mention “this gentleman” by name, but insiders are able to identify him, for Noble quoted three pages of Kirby’s treatment of the column problem (namely those occurring in Kirby 1754, first book, 69–72). To this quote Noble added his own comments as well as some quotations from Taylor (Noble 1771, 148–164).

Kirby’s Service to Taylor

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bout hundred years after Kirby’s presentation of Taylor’s theory, the English mathematician Augustus De Morgan claimed that “Taylor would not have thanked Kirby” for his service (De MorganS 1861, 728). De Morgan could be right in so far as Taylor would most likely have been disappointed that Kirby only applied a small part of his powerful theory, and would have disapproved of Kirby’s solution to the column problem. However, Taylor might nevertheless have been thankful to Kirby for the great effort he put into making Taylor’s theory known – and for explicitly mentioning Taylor’s name in the titles of all his publications on perspective. Whether Taylor would also have appreciated the homage Kirby paid to him by designing the engraving reproduced in figure X.63, I cannot say.

19 For more on the discussion about the column problem among British perspectivists in periodicals and pamphlets, see MildS 1990, 352–357.

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FIGURE X.63. Kirby’s engraving honouring Taylor. Under it Kirby wrote: “To Mrs Younge, Daughter of Dr Brook Taylor; This Plate as a Tribute Due to her Father’s Merit, is Dedicated by Her unknown but most Respectful humble Servant Joshua Kirby”. Kirby 1761, plate 55.

Highmore

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rom his childhood onwards Joseph Highmore (1692–1780) showed a keen interest in drawing and painting, and his greatest wish was to become an apprentice in the atelier of his uncle Thomas Highmore, who was a serjeant painter at the court. However, his uncle already had two apprentices, and did not want any more (MildS 1990, 51). Instead Highmore became clerk to an attorney in 1707, spending his leisure time attending anatomical lectures and studying geometry and perspective. In 1714 he enrolled at a painting academy. Eight years later he illustrated a book written by his anatomy professor, and quickly earned himself a name as a skilled portraitist (figure X.64). He also completed a series of paintings based on Samuel Richardson’s popular novel Pamela or Virtue Rewarded – one of which is reproduced in figure X.65.20 In 1754 Highmore published A Critical Examination of those two Paintings on the Ceiling of the Banqueting-House at Whitehall, in which Architecture is introduced so far as relates to the Perspective. His introduction stated that he had considered letting the pamphlet appear as an appendix to a book on perspective that he had been working on for quite a while, and which he had based on Taylor’s ideas. According to Warren Mild, Highmore had started studying Taylor before the master had died, and had contacted him with a

20

For a very detailed biography of Highmore, see MildS 1990.

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FIGURE X.64. Joseph Highmore, self-portrait, c. 1730. National Gallery of Victoria, Melbourne.

view to discussing perspective. He had also painted portraits of Taylor and one of his wives – but unfortunately, contrary to his usual style, Mild gave no references to document that Taylor and Highmore met (MildS 1990, 59 and 350). In any case, there is no doubt that Highmore was strongly interested in Taylor’s work on perspective, therefore it was presumably with regret that Highmore saw his book project preceded by Kirby’s Taylor’s Method Made Easy, and he used A Critical Examination to write some comments on Kirby’s book, as previously noted. The better part of A Critical Examination is devoted to an analysis of how the perspective in the two paintings mentioned in the title had been constructed – strangely enough, Highmore did not mention that they were works by Rubens. One part of his analysis is actually slightly related to his criticism

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FIGURE X.65. Painting by Highmore, based upon which an illustration for Samuel Richardson’s Pamela or Virtue Rewarded was made. Highmore, Pamela and Mr B in the Summerhouse, 1744, Fitzwilliam Museum, University of Cambridge.

of Kirby’s treatment of the column problem, because Highmore found mathematical errors in the way Rubens had painted vertical columns on the horizontal ceiling. Using beautiful illustrations and a thorough text, Highmore explained the correct geometrical solution. Nine years after announcing his book, Highmore finally published it, even stressing its delayed appearance in the title, The Practice of Perspective on the Principles of Dr. Brook Taylor, ... Written Many Years Since, but now First Published – which I abbreviate to the Practice of Perspective. Like Kirby, Highmore admired Taylor’s work, even while finding it hard to read. He wrote that Taylor has invented, and, in a very short compass, exhibited an universal theory; the truth, and excellence of which is acknowledged by all who have read, and considered it, at the same time that they complain of its obscurity. The attention and application which the reading, and understanding this little book [Linear Perspective] require ... has discouraged the generality of those, for whose service it was chiefly designed, from the attempt; so that very few have profited by the best treatise that has been published on the subject ... But though this author has been studied by few, yet with these he is in highest esteem, as the inventor of the true universal system. [Highmore 1763, v]

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Based on Taylor’s ideas, Highmore aspired to give the surest, and shortest rules for representing all sorts of objects, and this in a popular, familiar manner, without constant strict mathematical demonstrations; although illustrations, and even demonstrations, are not omitted, where they have been thought necessary. [ibid., vi]

It was, in fact, not often that Highmore thought demonstrations necessary, and the ones he included more take the form of persuasions than mathematical proofs. Like in many other books on the practice of perspective, Highmore’s text to a large extent consists of explanations of how his figures were produced, rather than of why he had performed the constructions the way he had. Highmore divided his book of some hundred and thirty pages into five parts. In the first he introduced a couple of constructions for determining the image of a point given in a ground plane, one of which was a visual ray construction. The second part Highmore called “a comparative perspective” and devoted it to showing that Taylor’s approach to constructions was superior to earlier methods. He claimed that pre-Taylorian procedures correspond to writing down the number 278 three hundred times and using addition to find the sum, when one could simply multiply 278 by 300, because the common methods are attended with such tedious operations, such a multitude of unnecessary lines, and, in some situations, with such perplexed and intricate schemes, as require more than human patience to execute, and, after all, render mistakes almost unavoidable, of which any one will be convinced who shall examine the plates of Pozzo, ... [ibid., 11]

In several examples Highmore compared a traditional construction with one based on Taylor’s theory, as shown for a perspectival cube in figures X.66 and X.67. Indirectly it becomes clear that what Highmore found so supreme in Taylor’s method was his extensive use of vanishing points. Although he applied them, Highmore put less stress on vanishing lines, presumably finding this part of Taylor’s theory generally too complicated for practical purposes. In the title of his book, Highmore claimed that it was based on the “Principles of Dr. Brook Taylor”, by which he apparently meant the use of vanishing points and vanishing lines. My understanding is that this was also how other British authors conceived of Taylor’s principles – and we shall actually see an example of this in the section on Thomas Malton.21 In general, the British perspectivists seem to have been of the opinion that these concepts were Taylor’s invention – Malton being an exception (page 580). Although this was not the case, it is true that Taylor was the first to emphasize so strongly the usefulness of vanishing points and vanishing lines, and to actually applying the latter. By the time Highmore’s book appeared, Lambert

21 For a more detailed discussion of what was meant by “Taylor’s principles”, see AndersenS 19921, 62–65.

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FIGURE X.66. Highmore's illustration of "the old method" for throwing a cube into perspective. Highmore 1763, figure 21.

had already published his main work on perspective (1759), in which vanishing elements were used even more than in Taylor's books. It is, however, highly unlikely that Lambert's work was noticed in Britain, at least there are no traces of it being read by British perspectivists. In his "comparative perspective" Highmore also reproduced and commented upon examples by earlier authors in which he had found mistakes. He

FIGURE X.67. Highmore's illustration of "the new method" for drawing a cube in perspective. Highmore 1763, figure 22.

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was particularly concerned about obscurities in Bosse’s presentation of Desargues’s method, stating that this method “is still esteemed by some moderns” (ibid., 14) – a statement I find interesting and somewhat surprising, since I am not aware of any publication in Britain or any other country from the decades before Highmore’s book appeared (in 1763) that show an influence from Desargues’ ideas.22 In the third part of his Practice of Perspective Highmore concentrated on how to throw three-dimensional objects into perspective. For some of these he first constructed the image of a plan and then added the perspective heights, and for others he performed direct constructions involving vanishing lines. He chose the Platonic solids as some of his examples, explaining that they furnish “occasion for almost every case that has any difficulty in perspective” (ibid., vii). The fourth part of Highmore’s book deals with perspective division and also contains some examples in which he made short cuts in his general construction procedures. In the fifth and final part, he took up perspectival shadows (figure X.68) and reflections. For the latter he provided an example, reproduced in figure X.69, that displays the same degree of knottiness as

FIGURE X.68. A perspectival shadow. Highmore 1763, figure 72.3. 22 Two years after Highmore’s book appeared, Philips presented Desargues’s ideas, and ten years later Karsten did the same (Philips 1765 and Karsten 1775), so Desargues’s approach may have been discussed in some perspectivist circles without appearing in any recent books.

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FIGURE X.69. Perspective composition with mirrors. Noticeably, the person at B is reflected in two mirrors. Highmore 1763, figure 74.1.

Taylor’s illustration of the theme (figure X.33). Taylor, however, had provided a more detailed explanation than Highmore of how to construct the image of a reflected point. Highmore’s enthusiasm for perspective does not seem to have had much influence on his performance as an artist; at least none of the Highmore paintings I have seen (JohnstonS 1963, MildS 1990) contain striking perspectives.

X.16

The Taylor Tradition Continued

I

n this section I deal with about a dozen publications in the Taylor tradition, appearing in the period from 1761 to 1800 and solely devoted to perspective. While their authors have quite different backgrounds, the books were all addressed to practitioners of perspective. Generally speaking, the books do not contain any novelties, and hence I describe them rather briefly, but quote them frequently and at length to illustrate the authors’ attitudes to their subject matter. Let me begin with an author who treated perspective in a book on painting, and who was sceptical of Taylorian thinking.

Bardwell Protesting

I

n 1756 the painter Thomas Bardwell (1704–1767) published The Practice of Painting and Perspective Made Easy, in which he devoted some twenty pages to perspective, introduced by the following opinion.

We are much obliged to the Learned in Mathematicks, who in the beginning of this Century made such great Improvements in the Principles of Perspective. ... But for

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want of understanding the Art of Painting and the Practice of Designing, they are intelligible only to those Readers who have a sufficient fund of Geometry. ... Were we to follow the Rules . . . we should find those useless Operations too tedious and difficult for Human Patience. And how we are to represent all Sorts of Objects according to the Rules of Perspective, is to me as surprising as impossible. [Bardwell 1756, 42–43]

Despite his dissatisfaction with Taylor’s approach, Bardwell was clearly inspired by it. To throw a point in the ground plane into perspective, he chose – besides a distance point construction – a visual ray construction. He also went further than merely using the horizon, as can be seen in figure X.70, but made no attempt to explain the geometry behind his constructions. So what Bardwell offered was, in fact, a manual in perspective constructions in a Taylorian tradition. Some fifty years later, the painter Edward Edwards characterized Bardwell’s book as generally being “the best that have hitherto been published”, but he found that “the perspective of the work does not deserve equal praise” (EdwardsS 1808, 7) At the end of his section on perspective, Bardwell reacted to Highmore’s Critical Examination, finding some of the criticism unfair and claiming that the “noble Design” functioned well from the centre of the room containing Ruben’s two paintings under discussion (Bardwell 1756, 63).

FIGURE X.70. Bardwell throwing a “large Block of Stone” into perspective. Bardwell 1756, figure 12.

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Fournier and Cowley Addressing Students at Military Academies

D

aniel Fournier published, in 1761, A Treatise of the Theory and Practice of Perspective. Wherein the Principles ... by Dr. B. Taylor are fully and clearly explained. On the title page he described himself as a “drawing master and teacher of perspective”, and wrote that the book was to be used for “noblemen’s and gentlemen’s military education”. He offered these gentlemen two sections, the first of which was a paraphrase – sometimes even a verbatim copy of – Taylor’s presentation of the theory of perspective in New Principles. Figure X.71 shows how Fournier stayed close to Taylor in his illustrations as well. Fournier’s second part similarly keeps very close to Kirby’s presentation of the practice of perspective, or, as Cowley phrased it, Fournier found the 1755 edition of Kirby “worthy for him to make free use of ” (Cowley 1765, viii). John Lodge Cowley himself (1719–1797) was the author of a textbook entitled The Theory of Perspective, Demonstrated in a Method Entirely New (1765), presumably also intended for use in military academies. He was professor of mathematics at the Royal Military Academy at Woolwich and had previously published a commentary on Euclid’s Elements. Cowley began his book on perspective by presenting the history of the literature on the subject, and in this connection he praised Kirby fulsomely. Kirby, on his part, spoke of “my ingenious and worthy friend Mr Cowley” (Kirby 1754/1765, advertisement). Incidentally, it is interesting to notice that Cowley ascribed a

FIGURE X.71. Fournier’s illustration of a perspective projection. His drawing was clearly inspired by the one by Taylor presented in figure X.6, which in turn was influenced by Lamy’s creation reproduced as figure X.7. Fournier 1761, plate 2, figure 1.

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distance point construction to Peruzzi, and claimed that Vignola had copied Peruzzi (Cowley 1765, iv). In Cowley’s opinion, the two most important perspectivists before Taylor were Peruzzi and Guidobaldo (ibid., vi). Cowley’s work is unique in the literature on perspective because his main concern is not to present perspective constructions, but to make his readers familiar with three-dimensional geometry. To this end he designed pop-up diagrams that fold out from the page and can be shaped in a given threedimensional form. This is presumably the technique to which he referred in his title when claiming that the theory is demonstrated in an “entirely new” way, because his proving style is the same as Euclid’s. Cowley’s figures, thirteen in total, are well meant, but folding them into the right shapes tests not only one’s intelligence, but one’s patience and dexterity as well. In the first part of his book Cowley taught three-dimensional geometry. In the second he introduced all the concepts belonging to perspective, including those relating to a directing plane. In accordance with the British tradition, he thought that most of these concepts originated with Taylor. The largest part of his theory consists of theorems about relations between the fundamental concepts. He carefully included all details, such as the result, mentioned earlier, that the point of intersection of two lines is mapped upon the point of intersection of the images of two lines (page 254). In the third and last part of his book, Cowley presented a few examples of constructing the perspective images of plane figures. In this connection he also discussed inverse perspective – once again more from a theoretical point of view than with the purpose of solving problems. Finally, he treated stereographic and parallel projections.

Emerson, the Textbook Writer

W

illiam Emerson (1701–1782) was an autodidact, who initially tried to follow in the steps of his father as a school teacher but found he did not have the right temperament for it. Managing to live on a small inheritance from his parents, he devoted most of his time to studying and writing textbooks. This resulted in an impressive list of publications on mathematics and mathematical sciences that caught the attention of the Royal Society. Emerson, however, declined the society’s offer of a fellowship, remarking that he did not want to pay so much every year merely for the honour of writing the letters FRS after his name (VianS 1889, 351). Emerson first published on perspective in volume six of his Cyclomathesis (Emerson 1765). This volume is so rare that I have not been able to trace a copy I could examine. I imagine, but cannot be sure, that his treatment of perspective in that book is similar to the contents of his Perspective: Or the Art of Drawing the Representations of all Objects upon a Plane (1768). Emerson opened this work by claiming that perspective is a part of optics, since it is based on the assumption that light rays are straight lines (ibid., iii), and he saw it as an enjoyable subdiscipline:

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I took notice in the Optics how delightful a study that is; but the greatest share of that pleasure belongs to Perspective. [ibid., iv]

Otherwise, Emerson did not link perspective to optics. Like so many of his predecessors, he claimed that a knowledge of perspective was absolutely necessary for painters, architects, and others – and in some cases he even recycled, without reference, some of his predecessors’ statements, such as Taylor’s paragraph, quoted on page 498, beginning “Nothing ought to be more familiar to a painter than perspective” (ibid.). Emerson’s Perspective is, as was customary, divided into a theoretical and a practical part. His sections on the theory of perspective contain twenty three theorems and cover all the material that most British perspectivists dealt with, but he included much more on shadows than was usual, while treating less than usual of the theory necessary for constructing perspective solids directly. In his practical part, Emerson presented twenty eight problems that also resemble the ones his predecessors had dealt with. Emerson, however, took up two not-so-common issues, the first being how to construct images obtained by parallel projections. The second was a discussion of which rabatment to use in perspective constructions. He advised his readers to mirror a plane figure before starting to construct its perspective image – as shown in figure X.72. In commenting on this procedure he only mentioned that placing the figure above the ground line (which involves applying a rabatment like Stevin’s, cf. page 276) is impractical, for it causes the figure and its image to overlap (ibid., 34–35). As we have seen, when moving the figure to the other side of the ground line, most authors ran into what I have called the ‘problem of reversing’ (page 330). Emerson’s solution avoids this problem, and even though this is most likely the reason he introduced the mirroring technique, he did not reveal his motives to his readers. All in all, Emerson had a thorough understanding of perspective, but apparently this did not help him much when he created his own perspective compositions – a claim to which figure X.73 bears witness.

FIGURE X.72. Emerson’s procedure for mirroring a polygon. Adaptation of figure 22 in Emerson 1768.

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FIGURE X.73. A perspective composition. Emerson 1768, figure 68.

The Scientist Priestley Entering the Field

J

oseph Priestley (1733–1804) was a man of many interests and is now best known in the history of chemistry, as the discoverer of what is now known as oxygen. He made his living as a teacher and a preacher, besides which he published on various scientific subjects as well as theology, politics, and a number of other fields (SchofieldS 1975, 139). According to his own account, Priestley came to take an interest in perspective while he was preparing his History and Present State of Electricity: Though, for my own part, I got a general idea of the theory of perspective pretty early, at the time I attended to other branches of mathematical science, I was not capable of making a draught of any thing, till I was under a necessity of having original drawings of electrical machines and apparatus, and was in a situation where I could not find any person to make them for me. [Priestley 1770, viii]

He found that the existing books did not help him much, because he had to rethink it all for himself (ibid.). Thus, Priestley joined the large group of authors on perspective who were dissatisfied with what they read and thought they could make a better job than their predecessors of presenting the subject in an understandable way. In fact, he claimed that perspective could be learned from his book in a very short time (ibid., ix), and like so many other authors he deeply regretted that so few mastered perspective (ibid., vi–vii). One of the results of Priestley drawing exercises, published in his History of Electricity in 1767, can be seen in figure X.74.

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FIGURE X.74. Priestley’s drawing of “an electrical battery”. First published in PriestleyS 1767 and reproduced in Priestley 1770.

Although Priestley was discontented with earlier books, he was quite pleased with the state of the art, and especially with the contributions by his countryman Taylor, as the following passage shows. As in all other branches of mathematical knowledge the progress of this art has been slow, but sure: and the English writers (particularly Dr. Brooke Taylor) seem to have carried it to a degree of perfection we can hardly conceive it possible to be exceeded. [ibid. v–vi].

Priestley published A Familiar Introduction to the Theory and Practice of Perspective in 1770 and dedicated it to Joshua Reynolds. Even though Priestley gave his book a conventional title, he did not follow the tradition of first presenting the theory of perspective and then deducing the practice from it. Instead he began with the practice, presenting it as a kind of manual without any theory. The latter then occurs as a very brief section at the end, and as endnotes giving the theoretical background for some of Priestley’s constructions. For assistance with some of these constructions he thanked his namesake “Joseph Priestley of Halifax” (ibid., xiii). Priestley’s predominant approach was to focus upon directions of lines and show how the images of lines with given directions can be constructed. For this he applied vanishing points, though initially without referring to them as such. He considered six cases in the following order: transversals, orthogonals, arbitrary horizontal lines, lines perpendicular to a ground plane g, lines whose projections upon g are orthogonals, and finally arbitrary oblique lines. Priestley presented a single example of throwing a horizontal polygon into perspective (figure X.75), for which he applied a visual ray construction

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X.75. A visual ray construction. Priestley 1770, figure 23.

(ibid., 52-53). But he found it bothersome to first draw a plan and elevation - in some scale - of an object before drawing its perspective image. Therefore he was rather enthusiastic about a method that did not require lengths, but only angles, obtained for instance with the help of a theodolite (ibid., 57-59) - as in the example presented in figure X.76. This perspective construction is one I have not seen presented as a basic method by any other author. Lambert applied a similar construction to determine the vanishing point of sun rays (figure XII.23), but not to obtain images of points in general. Priestley's wish to avoid the use of plans and elevations resembles Lambert's approach to perspective. In principle, Priestley could have been familiar with Lambert's book, published in French and German in 1759. However, as previously noted, I find it unlikely that Lambert's work on perspective was known in Britain in Priestley's day. Perhaps inspired by Emerson, whom he quoted a couple of times (ibid., 95-97), Priestley also treated the theme of parallel projections, referring to it as orthographical perspective (ibid., 63-72). In his History and Present State of Discoveries Relating to Vision, Light and Colours, Priestley included five pages on the history of perspective (Priestley 1772). His reason was that although perspective "is more a business of geometry than opticks" (ibid., 91),

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FIGURE X.76. Priestley’s method of constructing the perspective image of a point whose declination and elevation are known. He imagined that a point, say xo, which is mapped in the point x, is determined by two angles – which he called “declination and elevation” – that have their vertex in the eye point. The declination is the angle between the vertical plane and the vertical plane, say a, through xo and the eye point. The elevation is the angle in a between the line through the eye point and xo and its projection upon a horizontal plane. In his example, Priestley let the two angles be 10˚ and 20˚, respectively, introduced E as the eye point turned into p, and D as the principal vanishing point. To construct the image x, he made the angle DEa equal to 10˚ with a positioned on the horizon, concluding that x has to be situated on the vertical line through a (as the vertical through x is mapped upon this line). He used the elevation to decide where on the vertical line x should lie by making ab = Ea and constructing a 20˚ angle with its right leg on the horizon and vertex in b, he found x as the point of intersection of the left leg of the angle and the vertical through a. Priestley 1770, figure 26.

... it is derived from optical principles, and as the use of it is to give pleasure to the eye, by a just representation of natural objects, I should do wrong not to give a short account of its rise and progress. [ibid.]

Just as Lambert would do a few years later, Priestley mainly based his presentation of his predecessors’ works on accounts given by Montucla and Savérien (MontuclaS 1758; SavérienS 1766). Among other things, Priestley repeated a statement by Savérien claiming that Stevin was the first to write on anamorphoses (Savérien 1766, 256; Priestley 1772, 93). The fact of matter was, Stevin touched only briefly upon the subject, namely in the text quoted (page 279), whereas Barbaro had written rather more about the subject, though not very understandably (page 155).

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Noble Attempting to Bridge the Gap Between Theory and Practice

E

dward Noble, who was mentioned in connection with Kirby’s treatment of the column problem, had the ambitious programme of writing a textbook that should enable people “without any other previous Mathematical knowledge” to perform all sorts of perspective constructions and to understand the underlying geometry. Most of this he stated in an incredibly long title which I abbreviate to The Elements of Linear Perspective (Noble 1771). Noble dedicated his book to the President of the Royal Academy of Art – and as this was Joshua Reynolds, Noble made the same choice as Priestley. Perhaps both men had hoped their books would be used at the academy. Noble, whose biographical details I have not been able to find, may even have hoped for a teaching position at the academy, which had been founded only three years earlier. It had as its first professor of perspective the painter Samuel Wale,23 who was succeeded in 1786 by Edward Edwards (page 598). Noble divided the writers on perspective into two categories (Noble 1771, iv). In the first he put those “investigating its rules with geometrical rigour, and applying them to practice with that universality, which distinguishes the works of mathematicians”. These writers he found “unintelligible where they are most wanted”. He let the second class comprise

those who endeavour to demonstrate the principles of perspective without having recourse to mathematical speculations. These gentlemen may be compared to an architect who would support a building whilst the foundation is removed. [ibid.]

Noble aspired to combine the best from both traditions. Like many of his British predecessors, Noble pointed to Taylor, claiming that he was “the most excellent, as well as the most concise writer on this subject”, and that “his fate has been to be more admired than understood” (ibid., vi). Though describing two categories of authors, Noble only named a very few, including Benjamin Martin – to whose work I will return – and ’sGravesande. In addition, he mentioned that Thomas Malton was a good lecturer on perspective. We meet him as an author in the next section. After spending about hundred pages on treating elementary geometry, Noble began the perspective part of his book with two chapters on the theory of perspective. In his selection of topics Noble followed Taylor, but he provided no proofs, while on the other hand he did add some more practical aspects, like that of choosing the parameters of a picture. Noble proceeded by teaching his readers how to throw plane figures into perspective and how to construct images of three-dimensional figures whose plan and elevation were given. To this end he introduced a scale to be used for foreshortening 23 The persons responsible for teaching perspective at the academy were titular professors when they were also academicians, and otherwise teachers (HutchisonS 1986, 270).

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vertical line segments. Only later in the book did he take up the theory of vanishing lines and show how it can be applied to construct the images of three-dimensional figures (figure X.77). Noble chose to use constructions of houses as examples, claiming that “the Platonic Bodies, which are so much paraded with in most books on the subject ... are ... void of utility to the Artist” (ibid., x). It is interesting that Noble felt a need to protest against the custom of throwing the regular polyhedra into perspective. They had indeed figured prominently as examples in textbooks on perspective, entering the literature during a period when new Platonic ideas were influential and remaining there long after their symbolism was gone. The Elements of Linear Perspective also contains chapters on perspectival shadows, reflections, and inverse problems of perspective – all very typical for the Taylorian line of presenting perspective. Noble additionally devoted a chapter to anamorphoses, applying the traditional methods of constructing a perspectival grid that looks distorted when not seen from its eye point, as illustrated in figure IX.36. Finally, Noble included a highly interesting chapter on direct constructions, during the writing of which he seems to have forgotten his aim of teaching practitioners of perspective useful matters. In this chapter, his ninth, Noble not only dealt with how some basic constructions can be performed in the picture plane, but chose no less than fourteen construction problems from Euclid’s Elements and demonstrated how the perspective versions of these problems can be solved, one example being the problem of constructing the

FIGURE X.77. Houses in perspective. Noble 1771, figure 72*.

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centre of a perspective circle. He went much farther on this issue than the master Taylor, apparently guided by a fascination with the theme of direct constructions. With his Elements of Linear Perspective Noble showed that he had understood Taylor and was better at explaining his ideas than Taylor himself had been. Readers who had no mathematical background – and who were not put off by mathematics – would be able to understand parts of Noble’s work, but it is more than doubtful that they would have understood all of it, as Noble had intended.

Malton and Son

T

homas Malton (1726–1801) seems to originally have run an upholstery shop on the Strand in London. He later made a living from privately teaching mathematics and perspective, and from selling architectural drawings, some of which he exhibited at the Royal Academy of Art. In 1785, finding it difficult to make ends meet in London, he moved to Dublin, where he taught perspective. While in England, Malton published The Royal Road to Geometry (1774), as well as A Compleat Treatise on Perspective in Theory and Practice on the True Principles of Dr. Brook Taylor, Made Clear (1775), which was reissued three times over a period of four years. He followed up the latter work in 1783 with An Appendix or Second Part to the Compleat Treatise on Perspective. Like Noble and Priestley, Malton dedicated his book on perspective to the President of the Royal Academy of Art. He also shared Noble’s aim, namely to write a book containing the theory of perspective, based on Taylor’s ideas, that could be understood by practitioners of perspective. His reason for writing yet another book with a Taylorian approach seems to have become almost traditional:

... it is a certain truth; there are indeed a sufficient number of Authors on the Subject [perspective]; and yet perhaps, no subject has been worse handled, in general ... I do not pretend to have found out new Principles, nor do I think, there can or need be any other; those given by Brook Taylor, being sufficient for any purpose, whatever; and that, the Principles, on which he has founded his System, are the most simple and perfect that can possibly be conceived. ... Now, although I have not the least pretence to the invention of New Principles, yet I am firmly persuaded, that I have made use of those we have to the best advantage; that, from the irregular and imperfect Order, as they are given by Dr. Brook Taylor, I have digested it into an useful and practical System; not involved in a labyrinth of mathematical Demonstration ... [T. Malton 1775, i–ii]

Malton was of the opinion that although perspective does not belong to “the Education of a Gentleman yet it must be looked on as a genteel and polite accomplishment” (ibid.). With this attitude he composed an independent adaptation of Taylor’s ideas. He succeeded quite well in explaining the theoretical part, but, like Noble, he was presumably too optimistic as to how much a

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reader without a solid background in geometry would be able to absorb. He took up a few themes that had not been treated by Taylor, but by some of Taylor’s other followers, such as the column problem. He advocated keeping to the rules of perspective based on various arguments, including the following. If a Person, not knowing how to choose a proper Distance, take, into the Picture, more than the Eye is capable of taking in at one View ... is the fault in Perspective, or in his Judgement? In Perspective there is not, nor can be, on the Principles here laid down, any, the least error, if the Elements of Euclid are to be depended on, upon which the whole Fabric is erected: if one falls the other falls with it. [T. Malton 1775, 98]

In showing how to apply the theory of perspective constructions Malton chose some practical examples (figure X.78), including drawings of furniture – which had actually been done before, as we shall see in a moment. Like Kirby, Malton also designed a picture that paid homage to Taylor (figure X.79). In his 1783 Appendix, Malton added a few themes not treated in his book, for instance perspective applied to stage design and perspective representations of ships. He also supplemented some of his earlier presentations, such as how to throw round objects into perspective (figure X.80), and he described some new considerations, for example the effect of ascent and descent (figure X.81). In addition, he used the occasion to air some of his opinions, like how he failed to see the point of applying parallel rather than perspective projections for depicting fortifications (T. Malton, 1783, 28). Most of Malton’s Appendix is devoted to an overview of important continental books and most of the British literature on perspective (ibid., 4–25, 80–154). Malton’s reviews of his countrymen’s contributions to the field, which are fairly detailed and often very critical, make quite an interesting read. More exciting, however, is his elevation of Taylor’s role in the development of the theory of perspective. When writing A Compleat Treatise on Perspective, Malton does not seem to have doubted that Taylor had played a revolutionary part in the development of the theory of the discipline. However, Malton changed this view in his Appendix. In commenting upon Guidobaldo del Monte’s solution to a problem in which he applied the main theorem, Malton wrote: “In this Example may be clearly seen the true Principles of Perspective” (ibid., 81, my emphasis). In other parts of Guidobaldo’s book Malton also found ideas similar to Taylor’s, but he could still maintain that Taylor had extended Guidobaldo’s theory. Praising Taylor as an inventor of special principles became impossible for Malton, however, when he compared ’sGravesande’s and Taylor’s works. He was then led to a conclusion he seemed none too happy to draw: It is far from my Intention, nor have I a wish to lessen the Merits of my own countryman, Dr. Taylor; I had much rather attributed to him the sole invention of the new Principles, could it be done with candour; but ‘tis my determination to give the praise due to the Author, wherever I find occasion. In this work [Essai de perspective by ’sGravesande] it is manifest, that a foundation is laid for those universal Principles ...[ibid., 85]

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FIGURE X.78. Two examples of perspective constructions by Thomas Malton. He was so economical with space that the two illustrations overlap a bit. T. Malton 1775, plate 30, figures 1 and 2.

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FIGURE X.79. Thomas Malton’s homage to Taylor. The text in the picture reads “To the Memory of Dr. Brook Taylor in Gratitude for his sublime Principles on perspective”. T. Malton 1775.

Malton’s understanding of the history of “Taylor’s principles” did not seem to have much influence, and thus the idea that Taylor furnished the theory of perspective with a special foundation survived long after Malton. Actually, as late the 1880s, in a textbook on perspective the headmaster of the School of Art in Manchester, George O. Blacker, stated that Taylor was “the father of all modern Perspective” (Blacker 1885, preface). Malton had two sons, Thomas and James, who also worked as architectural draughtsmen. James Malton additionally followed his father in publishing on

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FIGURE X.80. The result of a complicated construction of images of round objects. T. Malton 1783, figure 27.

perspective. He did so with The Young Painter’s Maulstick: being a Practical Treatise on Perspective founded on the process of Vignola and Sirigatti, ... united with the theoretic principles of ... B. Taylor (1800). In an introductory “apology” James Malton stated that he had given courses in perspective and

FIGURE X.81. Thomas Malton’s illustration of ascent and descent. This is similar to the Noble drawing reproduced in figure X.77. Malton’s diagrams accompany some thoughts he had about how it can sometimes be difficult to determine the level of the horizon in a perspective composition. T. Malton 1783, figures 21 and 22.

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found it impossible to teach according to “Taylor’s principles” and “making it engaging at the same time” (J. Malton, 1800, i). This remark shows that in some circles Taylor’s approach had become so much of a paradigm that authors made an excuse for not using it. The path James Malton decided to follow was a modified plan and elevation method – the modification being that he allowed the use of vanishing points. As his long title indicates, he saw this approach as a synthesis of a method used by Vignola and Sirigatti and Taylor’s ideas. Actually, Vignola himself sometimes made use of the principal vanishing point when performing a plan and elevation construction (cf. figure IV. 16), so the element Malton got from Taylor was to involve other vanishing points than the principal one – although he went no further than considering those situated on the horizon. An example of this is shown in figure X.82.

Clarke Presenting Perspective for Young Gentlemen

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he year after Thomas Malton’s first book on perspective appeared, a similar textbook was published called Practical Perspective being a Course or Lessons Exhibiting Easy and Concise Rules for Drawing Justly all Sorts of Objects (1776). The author, Henry Clarke (1745–1818), first made his living running a “commercial and mathematical school” near Manchester (Clarke 1776, preface). The students at the school could board and were instructed in topics “which qualify them either for the Army, Navy, Counting-House, or any Artificer’s Business” (ibid., end). Clarke later worked as a surveyor, and then became a professor at a military college. As he began by stating in his book:

It may, perhaps, have Appearance of Vanity to attempt to write upon a subject which seems already to have been quite exhausted. [Clarke 1776, v]

His reason for doing so, however, was that he wished to address a young audience. He surveyed a part of his predecessors’ work and was rather critical about two of the classics in the practice of perspective, namely the works by Dubreuil and Pozzo (ibid., xi–xii). Clarke’s own work contains just over a hundred pages and does not differ remarkably from the other Taylorian books from the period. In particular, it is not immediately clear why he thought his book would suit young readers better than the other books. He treated the usual themes, including perspectival reflections – which he called “catoptric perspective” (ibid., 83) – but skipped inverse problems of perspective. He also had a section on theatrical perspective (figure X.83). Compared with Noble and Thomas Malton, Clarke paid less attention to theory and put more emphasis on examples. Clarke announced that a second volume on the practice of perspective with more spectacular applications would be published “as soon as ever the plates can be got ready” (ibid., advertisement at the end of the book). Apparently this never happened.

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FIGURE X.82. James Malton’s method. In Fig. 1 and Fig. 2 Malton showed the plan and a profile of the church he wanted to throw into perspective. The point S is the plan of the eye point, and it is used to construct an image of the plan of the building (Fig. 3). To complete the drawing, Malton also applied two vanishing points on the horizon. J. Malton 1800, plate 16.

Wood Writing for Painters

I

n 1799 John Wood (1775–1822) published Elements of Perspective, Containing the Nature of Light and Colours and the Theory and Practice of Perspective.24 On the title page he described himself as “Master of the Drawing

24

For a later work by Wood on perspective, see KempS 1990, 185–186.

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FIGURE X.83. Clarke on perspective in a theatre. His drawing resembles an illustration by Martin published five years earlier (figure X.86). Clarke 1776, figure 49.

Academy in Edinburgh”. Evidently, like several other eighteenth-century British books on perspective, Wood’s work was a result of teaching the subject. Wood began his book with a rather comprehensive history of the field, claiming that the first book on perspective was published in 1440 (sic) by Bartolomeo Bramantino (Wood 1799, vi). After mentioning many names, he came to Taylor and found that his “principles are far more general than those of any of his predecessors” (ibid., ix). A bit surprisingly, he also claimed that “Besides, his system is the only one calculated for artists and practitioners in the art of design”. Among the authors after Taylor, Wood particularly favoured Hamilton, Kirby, Cowley, and Emerson (ibid.). Commenting upon the target groups of the various works, Wood noticed that most of them were not addressed to mathematicians, but to artists, continuing as follows. ... indeed the artists themselves have in general an unfortunate idea, that a knowledge of the mathematics has only a tendency to cramp the freedom of the imagination, and that geometry furnishes rules that are only so many fetters to men of real genius. They assert also, that those persons who distinguish themselves by a strong predilection for mathematical sciences, however important they may be deemed, in fact terminate in speculation. [ibid., xi]

It seems that Wood had some difficulty convincing his students they should try to understand the geometry behind perspective. Wood divided his book in three parts. In the first he dealt with optics (ibid., 1–43), although he agreed with Emerson on the relationship between perspective and optics:

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The Art of Perspective has been regarded by some as a branch of optics, by others as a problem in geometry; its first principle, which is purely optical being once admitted, all its operations are afterward geometrical ... [ibid., v]

In the second part he presented the theory of perspective (ibid., 44–107), and in the third and last he treated the practice of perspective (ibid., 108–125). In this connection he described a perspective instrument and discussed how the use of colours influence perspective representations.

Taylor’s Influence on the Drawing of Chairs

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he search for traces of Taylorian influence in the literature leads one to the title A New Book of Chinese, Gothic, and Modern Chairs, or rather to its subtitle “with the Manner of putting them in Perspective according to Brook Taylor”. Published in 1751 by the well-known engraver Matthew Darly, this small pamphlet, consisting of just seven plates, seems to have been influential in making it common in Britain to draw chairs in perspective (RococoS 1984, 167–168).25 Unfortunately the only part of the publication I have seen is the plate reproduced in figure X.84, but as far as I have understood Darly included no explanations of how his drawings were produced. As they could have been obtained by any perspective method, it is interesting that Darly insisted his construction was based on Taylor’s ideas. What is particularly remarkable is that he did so as early as 1751, for at that time Taylor’s approach to perspective was still little known. At the end of the eighteenth century Thomas Sheraton followed up on the theme of drawing furniture in perspective in his work The Cabinet-maker and

FIGURE X.84. Darly’s drawing of chairs in perspective. Darly 1751.

25

I am thankful to June Barrow-Green for having made me aware of this publication.

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FIGURE X.85. A desk in perspective. Sheraton 1793, figure 41.

Upholsterer’s Drawing Book (figure X.85), in which he devoted almost two hundred pages to perspective (Sheraton 1793). It cannot have been encouraging for his readers to learn that he was of the following opinion: “... I will venture to say that perspective can never be clearly comprehended” (ibid., 211). As for his methods, Sheraton claimed to be using “Dr. Brook Taylor’s system” (ibid., 224). In his examples he mainly applied a distance point method. Despite his humble attitude towards perspective, Sheraton had understood his subject quite well.

X.17

H

Perspective in Textbooks on Mathematics

amilton claimed that “... of late Years, no general Courses of Mathematicks have been esteemed compleat, without a particular Treatise on that Subject [Perspective]” (Hamilton 1738, a2). I do not know to

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what extent he was thinking of textbooks in English, but if he were, they are not among the publications I have found. Thus, as far as books that can be seen as part of a general course in mathematics are concerned, the only English book in this genre from before 1738 that I am aware of is Ditton’s work on perspective. Later, when English publications on perspective reached a quantitative peak in the 1760s and 1770s, four presentations of the subject appeared in books on mathematics. One of these – volume six of Emerson’s Cyclomathesis – I have been unable to study, as previously noted.

Martin

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he book A New and Comprehensive System of Mathematical Institutions (1764) included eighty pages on perspective. Its author, Benjamin Martin (1704–1782), was a popular travelling lecturer who demonstrated physical experiments, as well as an industrious writer. Among other things, he published descriptions of all sorts of scientific instruments, textbooks on astronomy, geography, optics, mathematics, and Newton’s philosophy. He settled in London, where, besides lecturing, he sold scientific instruments, some of which he designed or improved himself (TurnerS 1974, 141). Apparently the field of optics was particularly close to his heart and may have brought him to perspective, which he considered to include the theory of apparent sizes. In his enthusiasm for perspective he even went so far as to call it “the most delightful and necessary of all the mathematical sciences” (Martin 1764, 148). Martin gave a detailed introduction to perspective, and to the themes of anamorphosis and theatre stage design (figure X.86). He based his constructions on a distance point method (figure X.87) and proved that his constructions were correct. He claimed, surprisingly, that this was an endeavour that has “the Face of Novelty” (ibid., 157). It seems he had not read many mathematical books on perspective – and apparently none in the Taylorian tradition. He also published a book exclusively on perspective that is undated, but which presumably appeared later than his Mathematical

FIGURE X.86. Martin illustrating perspective in connection with stage design. Martin 1771, plate 4, figure 4.

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FIGURE X.87. A distance point construction. Martin 1764, plate 3, figure 1.

Institutions. Here, too, he demonstrated his ignorance of earlier works on the theory of perspective: No mathematical Science requires a Theory more than PERSPECTIVE, and perhaps no one has been less attended with it in Publications of this Kind. [Martin, s.a.]

Martin’s two treatments of perspective are different, although some of the figures are recycled. In his book he defined a perspective image with the aid of Benedetti’s prism, and he also used this and a movement to introduce the principal vanishing point (figure X.88). The idea of explaining the property of vanishing points by letting a point move was also used by Lambert (figure XII.11).

N

P p O

Ti T

FIGURE X.88. Martin’s introduction of the principal vanishing point. He considered a prism in which O is the eye point, P its orthogonal projection upon the picture plane p, T a point, Ti its image, and N the point of intersection of OP and the vertical plane through T. He introduced the orthogonal through T, imagined that T moves on this line away from p, and concluded that “the Angle [TON] constantly decreases; till ... the Point gets at an indefinite Distance ... then the Angle vanishes and the Ray [OT] coincides with the Line [ON]” (Martin s.a., 3). From this he concluded that the image of the orthogonal through T prolonged passes through P. In a similar manner, Martin introduced the distance point.

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It is, however, very unlikely that Martin was aware of this, as there is no evidence, as noted several times, that Lambert’s work was known in Britain. In an appendix on the column problem, Martin argued in favour of following the rules of perspective. The first eight of Martin’s nine chapters were quite literally copied – apart from very minor changes in a few places – by Robert Bradberry. To Martin’s text Bradberry added a section including two figures on a perspective instrument, which he had taken from William Hooper, who will be presented shortly. In 1771 Martin published a pamphlet of nine pages presenting an instrument that was to be used for perspective drawing, which he called a graphical perspective (Martin 1771). He considered this publication an addition to the section on instruments in his previous book on perspective. Martin described his new device as an optical instrument, but provided no illustration of it. As I understand him, his idea was to provide the instrument with a wire netting so what was seen through this could be copied onto squared paper. Perhaps it was similar to the just-mentioned instrument Hooper later illustrated (figure X.90). Martin found the instrument useful because: “There appears an universal Desire ... in Mankind, of both Sexes, to imitate the visible, or pituresque Appearances of natural Objects” and because such imitations – which he considered to be “among the most innocent, ingenious, and delightful Recreations of Life” – only work by keeping to “the true Perspective” (Martin 1771, 1–2).

Muller

O

n the title page of the third edition of his Elements of Mathematics the professor of artillery and fortification John Muller announced that now he had also treated perspective (Muller 1765). His section on the subject only covers nine pages, but it nevertheless gives a reasonable introduction that was inspired by Taylor and mainly focussed on a visual ray construction.

Wright

I

n his textbook Elements of Trigonometry, John Wright (†1813) gave an unusual reason for writing on perspective. He announced that before presenting spherical trigonometry, he wanted to teach his readers how to illustrate the subject and “because projection of the sphere is but a particular case of perspective”, he “found it of great advantage to make the learner first acquainted with perspective” (Wright 1772, xi). In accordance with this idea, Wright devoted some thirty five pages on introducing the subject and making his readers familiar with a distant point construction. Rather than involving vanishing points, he proved that this construction was correct using similar triangles. Wright’s preoccupation with mathematics was a step in a career where he began as a humble shoemaker and made it to become a lecturer on law. He first managed to learn Latin, then entered Glasgow University and later moved on to the university of Edinburgh to study law, earning his living teaching mathematics and law (PatersonS 1885).

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British Individualists

I

n this section I briefly survey the part of the British literature that does not belong within any of the categories considered thus far.

Halfpenny

I

n the rather stagnant period for the literature on perspective that followed after the appearance of Taylor’s books, the architect and carpenter William Halfpenny, alias Michael Hoare (†1755), published his Perspective Made Easy: Or a New Method for Practical Perspective (1731). It begins with a six-page description of an instrument recommended to Halfpenny by the instrument maker Thomas Heath and called a scenographical protractor. In the remaining twenty eight pages of the booklet, Halfpenny gave a description – without mathematical arguments – of how to apply a distance point construction, for which he used a rather unconventional rabatment. He included a large number of drawings, especially of architectural elements, some of which are quite elaborate.

Hodgson

W

hen plans were under way to make an English translation of Dubreuil’s popular book on perspective, a decision was also made to incorporate an English introduction on perspective theory. This was written by James Hodgson (1672–1755), a Fellow of the Royal Society who had published on several themes from pure and mixed mathematics. He managed to cover quite a lot of perspective issues on sixteen pages (Hodgson 1743). He was presumably inspired by Taylor in letting the requested knowledge from Euclid occur as axioms, but otherwise Hodgson took his own approach.

Murdoch

T

he work presented in this section has a prehistory that is connected to the classification of algebraic curves of the third order, also known as cubic curves. In 1704 Newton published a treatise (NewtonS 1704) on cubic curves – a topic that had actually engaged him since the late 1660s. He distinguished between no less than 72 different types of cubics (with later authors adding another 6). Presumably inspired by the facts that the class of second-order curves consists of the conic sections, and that all of them can be obtained from a circle by means of a central projection upon a plane, Newton approached the cubic curves projectively and found that they can be obtained from five particular types – which he referred to as diverging parabolas. Newton’s work on cubics – in which many statements were left unproved – inspired several mathematicians to continue the research programme Newton had initiated. Among them was a professor of mathematics in Edinburgh, Colin MacLaurin, who in turn stimulated his pupil Patrick Murdoch (†1774)

18. British Individualists p

593

P

d O

A'i

A'

Ai

Dx A x B

F

FIGURE X.89. Murdoch’s introduction of velocities (Murdoch 1746, 12). In paraphrasing his ideas, I have changed his symbols. Let O be the eye point, P its orthogonal projection upon the picture plane p, A a point in the ground plane, and B its orthogonal projection upon the ground line. Deducing a result equivalent to the division theorem, Murdoch found that for OP = d and BA = x BAi = BP # x . (1) d+x He then imagined that the point A moves uniformly on BA to the point A′, the distance AA′ being Δx. From (1) he deduced that BP # (x + Dx) . BA′i = (2) d + x + Dx Subtracting (1) from (2), Murdoch found that Ai has moved the distance BP # d # Dx . (d + x) (d + x + Dx) As A moves uniformly, Δx is proportional to the increase in time, from which Murdoch concluded that the velocity of Ai is proportional to the limit of Ai Ali , Dx and hence inversely proportional to the distance (d + x)2. AiA′i =

to study the cubic curves projectively. As for so many early modern scholars, mathematics was a hobby for Murdoch, who graduated in theology and worked as a priest. He became a Fellow of the Royal Society in 1747. According to his own account, Murdoch finished his work on cubic curves in 1732 (Murdoch 1746, vii), but it did not appeared until fourteen years later under the title Neutoni genesis curvarum per umbras seu perspectivae universalis elementa (Newton’s generation of curves by shadows or elements of general perspective). Before presenting his work on the projection of curves, Murdoch devoted a chapter, called “the principles of linear perspective”, to methods for determining the images of points and straight lines under a central or perspective projection (Murdoch 1746). Murdoch’s chapter is remarkable by apparently being independent of any earlier presentation of perspective. Like other mathematicians, he considered the basic problem of perspective to be one of determining the

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FIGURE X.90. A perspective instrument described by Hooper. Hooper 1774, figure 3.

image of an arbitrary point when the picture plane and the eye point are given – referring to the latter as the “pole” (Murdoch 1746, 5); but his solution is unique. He first gave a perfectly sound proof of how the image of the point can be determined using the division theorem. From this he deduced a rather cumbersome construction – which Wieleitner characterized as tedious (WieleitnerS 1914, 324). Based on his main result, Murdoch then derived some of the standard results of perspective in 13 corollaries (a more detailed description can be found in ibid., 322–327). One of his corollaries is, however, highly unorthodox, as it involves movement of a point and its perspective image and the ratio between their velocities (figure X.89). When Rivoire translated Taylor’s New Principles of Linear Perspective into French, he added to it, as noted, a translation of Murdoch’s chapter on linear perspective (page 483). It is difficult to understand why among all the possibilities, Rivoire chose specifically Murdoch’s introduction to perspective, for Murdoch did not apply the theory of vanishing points, and hence did not help to promote an understanding of Taylor’s theory.

Hooper

I

n his Rational Recreations William Hooper included a section on “Perspective Recreations” (Hooper 1774). He devoted this section to anamorphoses, a theme with which he seems to have been well acquainted, reporting among other things:

There are at the convent of Minims in Place Royale at Paris, several subjects of this kind, painted on the walls of the cloister by P. Niceron, who has published an excellent treatise on this art. Among others, the figure of a Magdalene daily excites the curiosity of a number of connoisseurs. [Hooper 1774, 174]

Hooper regretted that these anamorphoses were not well maintained.

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FIGURE X.91. Ferguson’s erroneous distance point construction. The line HZ is the horizon with the principal vanishing point P, GR is the ground line with a given line segment AB upon which Ferguson wanted to construct the image of a square, and under the ground is given the foot F of the eye point. Ferguson proceeded by letting W on the HZ be determined by PW = PF, and using W as a distance point he obtained his perspective square ABCiEi. The correct procedure would have been to use the distance between F and AB as the distance, but instead Ferguson determined W by letting it have a distance to P that is equal to the sum of the distance and the height of the eye point. Adaptation of figure 6 in Ferguson 1775.

Despite his admiration for Niceron, Hooper did not base his own presentation on Niceron’s work, claiming that an exposition by Guyot was even better than Niceron’s and Ozanam’s (ibid., 172). I assume that this “Guyot” was Guillaume Germain Guyot, and that he dealt with anamorphoses in his book on physical and mathematical recreations (Guyot 1769). Although, regrettably, I have not seen Guyot’s treatment of anamorphoses, I think that Hooper referred to Guyot rather than to Niceron because Guyot’s publication was of the same type as Hooper’s own work, and not because Guyot made anamorphoses more understandable than Niceron. For perspective drawing Hooper presented the old technique of applying a grid of squares, which was already presented by Alberti, and which we have seen in Dürer’s illustration (figure II. 3). However, as shown in figure X.90, Hooper’s instrument was more elaborate than the earlier designs.

Ferguson

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n his survey of the British literature on perspective, Thomas Malton included The Art of Drawing in Perspective Made Easy published in 1775. Malton remarked that its author was so well known that his very name – Ferguson – ought to be a recommendation (T. Malton 1783, 145). James Ferguson (1710–1776), who became Fellow of the Royal Society in 1763, was a highly skilled instrument designer and a popularizer of science, including Newton’s ideas. His favourite topic was astronomy, but he also worked on other subjects and wrote a variety of textbooks, some of which were used up until the 1840s (LaudanS 1971, 565). Despite his interest in mathematical sciences, Ferguson seems to have lacked a deeper understanding of mathematics (ibid.).

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FIGURE X.92. Two perspective instruments described by Adams. Figures 1 and 2 on plate 32 in Adams 1791.

In his preface to The Art of Drawing, Ferguson stated that due to failing health “that is very apt to effect the mental faculties” he had decided not to write any more books. However, as he did not like idleness and had recently been amused by perspective, he had changed his mind and composed a book on the subject. As regards the quality of the result, Malton wrote: “As it is almost the last published on Perspective, so I affirm it to be the very last in merit, that has fallen into my hands” (T. Malton 1783, 145). Malton then spent five pages pointing out some of Ferguson’s mistakes. I shall limit my comments to saying that Ferguson presented an erroneous construction of the perspective image of a square, his mistake being to use a wrong distance, as I have explained in the caption of figure X.91. Although mathematics was not his forte, it is remarkable that Ferguson managed to get the distance wrong – an error not seen in print since Serlio’s work appeared in 1545.

Adams

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n 1791 a maker of mathematical instruments named George Adams (1750–1795) published a large book on instruments entitled Geometrical and Graphical Essays. Among his instruments are two perspectographs (figure X.92). His description of these devices is very brief, but he did begin with a twenty-two-page introduction on perspective in which he included some mathematical considerations (Adams 1791). With the argument that “so great is the difficulty, and so tedious the operation of putting objects in true perspective” (ibid.,482–483), Adams recommended the use of an instrument, and he reported that of all those known to him he preferred the two he had depicted – and of those two, the one on the left (ibid., 484).

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FIGURE X.93. Newton’s sketch of his famous light experiment. He first let light from the sun be refracted through a prism, and from the resulting spectrum he then let one of the colours be refracted through a second prism and saw that no further colour change took place. New College, Oxford.

X.19

B

British Mathematicians and Perspective

efore closing this chapter I want to mention that perspective was presumably quite well known, though not actively cultivated, in a wider circle of British mathematicians than those discussed in this chapter. In earlier chapters we saw how some Continental mathematicians who did not write on the subject, among them Gottfried Wilhelm Leibniz, were familiar with perspective. His British counterpart, Isaac Newton, was probably also acquainted with the subject. It is worth noting (figure X.93) that when illustrating his so-called experimentum crucis supporting his theory that white light from the Sun is composed of the colours of the rainbow, he made a perspective sketch of the room. In Newton’s correspondence we find an interesting remark implying that the knowledge of perspective signals an interest in science. The Swiss mathematician Nicolas Fatio de Duillier, who was a close friend of Newton’s for a while, wanted to establish contact between his brother and Newton in 1692. In arguing that Newton might find his brother interesting, Duillier listed subjects from pure and mixed mathematics that his brother understood, including perspective (NewtonS Corr, vol. 3, 232). As a curiosity I would like to mention that one of Britain’s well-known female scientists, Mary Fairfax Somerville (1780–1872), initially came to mathematics through perspective. She received a traditional education for girls, which did not involve any mathematics or science, but did include drawing lessons. During one such lesson the teacher was discussing perspective with two

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students, and Mary Somerville overheard him advise them to read Euclid’s Elements, claiming that this work was “the foundation not only of perspective, but of astronomy and all mechanical science” (SomervilleS 1873, 52). Her comment to this episode was: “Here, in the most unexpected manner, I got the information, I wanted (ibid.).” She soon convinced a teacher of one of her brothers to buy her a copy of Euclid.

X.20

The British Chapter

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he two facts that there were so few British publications on perspective before Taylor, and that he came to have an enormous influence on the British literature during the rest of the eighteenth century make the history of perspective more homogeneous in Britain than on the Continent. The British did not reinvent perspective: Ditton was influenced by Aguilon and Taylor by ’sGravesande. However, Taylor distilled the inherited material and made it clear how important intersections, vanishing points, and vanishing lines are. Besides this, he formulated most of the theorems necessary for working in the picture plane without taking recourse to a plan and an elevation of an object. From the 1750s, a part of Taylor’s theory based on vanishing points and lines was used by British writers on the practice of perspective. At the same time, generally on the Continent only vanishing points were applied. Although the British literature was dominated by Taylor, Britain was not isolated. In fact, many of the British authors described the Continental literature on perspective – though without including Lambert’s work. If we look at the authors, it is notable that there are many teachers of perspective among them – which is not so surprising. I have also observed that among those who wrote more than a few pages on perspective and were theorists or practitioners of science and mathematics, well over half were Fellows of the Royal Society,26 but this fact does not lead to any real conclusions. In connection with Britain, my project of only surveying the literature published up to 1800 is somewhat unfortunate, for the Taylorian approach lived on after that date. For instance, Edward Edwards, painter and teacher of perspective at the Royal Academy of Art, published A Practical Treatise of Perspective, on the Principles of Dr. B. Taylor in 1803. What is more, Taylor’s own New Principles were reissued twice during the first third of the nineteenth century (Taylor 1719/1811 and Taylor 1719/1835). Although I have not traced the influence of Taylor to the end, I dare say that despite the many who took an interest in his work, no one in Britain ever seriously attempted to develop his innovative theoretical ideas further.

26

These were, in chronological order: Moxon, Taylor, Hamilton, Hodgson, Murdoch, Cowley, Priestley, and Ferguson. Emerson was also offered a fellowship, but declined.

Chapter XI The German-Speaking Areas after 1600

XLI

Categorization of the German Literature

T

his chapter deals with the literature on perspective published in the seventeenth and eighteenth centuries in the German states, Austria and Switzerland. Although written in German, the contributions by the Alsatian scientist Johann Heinrich Lambert are not discussed in this chapter, because they are so outstanding in the history of perspective that they deserve a separate chapter. Before 1600, German perspectivists had no influence on the development of perspective techniques, or on how these techniques were understood. By and large this continued to be the case, but the Germans' interest in the subject grew during the seventeenth century. At the beginning of the eighteenth century, several authors also took up the theory of perspective. Most of the material in the German publications on perspective was derivative. I shall therefore be rather brief in dealing with the technical contents of the publications, focussing instead upon the identity of the authors and their relationships to perspective. To bring some structure to my survey of about 40 titles, I have grouped them as follows. I begin with publications dealing with perspective instruments and anamorphoses. Next, I present the contributions by practitioners not already mentioned, then turn to the contributions by mathematicians, and finally describe traces of Lambert's work. While all the sixteenth-century German authors were practitioners, academics took over the field in the following centuries, and particularly during the eighteenth century mathematicians came to dominate the scene.

XL2 Perspective Instruments

G

uidobaldo's creation of the theory of perspective, published in 1600, did not change the German approach to the discipline during the seventeenth century. The German writers on perspective continued along the lines laid out by their predecessors. This was not because Guidobaldo's work was 599

600

XI. The German-Speaking Areas after 1600

completely unknown in Germany - it was, for instance, described in 1615 by Lucas Brunn as consisting of sechs herrliche Bucher (six magnificent books, Brunn 1615, 1)1 - but simply because the Germans dealing with perspective were not attracted by mathematical theory. Their interest was captured by the wonders perspective could create, and by the marvels, or machines, that could produce the wonderful phenomenon of perspective. The German literature from the early seventeenth century is indeed dominated by descriptions of perspective instruments, and not written primarily by artisans who used the devices in their work, but by authors who were fascinated by the potential uses of perspective instruments (a similar impression is expressed in Kemps 1990, 183).

Faulhaber and Bramer

T

he Rechenmeister (reckoning master) Johann Faulhaber (1580-1635), depicted in figure XLI, was one of the authors who was captivated by natural magic (Schneiders 1993). In 1610 he published the booklet Newe geometrische und perspectivische Inventiones (New geometrical and perspective inventions). He documented an interest in perspective instruments outside the circle of practitioners by recounting that the very learned medicus Johann Terrentius "who had visited 40 universities in 10 years" had introduced him to a perspective machine. After extensive investigations Faulhaber came to the conclusion that this instrument had been constructed by Jamnitzer. Faulhaber could have found information on Jamnitzer's instrument in Pfinzing's book on geometry and perspective from 1599, but he was presumably unaware of this privately printed work (page 230). Besides accounting for the history of Jarnnitzer's instrument, Faulhaber described its mode of operation and showed what it looks like (figures XL2 and XU). He returned to the instrument in his textbook Ingenieurs Schul (Engineer's school), which appeared in 1633. Faulhaber's name was also found in the title of another book on a perspective instrument published twenty years after his Inventiones (Bramer 1630). Its author, Benjamin Bramer (c. 1588-1652), was a mathematical practitioner partly educated by his brother-in-law, the instrument maker and calculator Joost Burgi, who is best remembered in the history of mathematics for his creation of a table of logarithms. From his early childhood Bramer lived in the home of his sister and Burgi, and for some years he followed Burgi's work closely (KirchvogelS 1970). Bramer cultivated various fields of applied mathematics, and he became particularly esteemed as a designer of fortifications. As early as 1604 - when Bramer was about 16 - Burgi may already have shown him a perspective instrument of own design. In 1617

lLater in the century Guidobaldo was also mentioned by, among others, Haesell and Hartnack.

2. Perspective Instruments

FIGURE XI.!. A portrait of Faulhaber. Frontispiece from Faulhaber 1610.

FIGURE XI.2. A perspective instrument. Faulhaber 1610, 32.

601

602

XI The German-Speaking Areas after 1600

FIGURE XL3. Another perspective instrument. Faulhaber seems to have been more interested in making a nice illustration than in getting the details correct. Thus, the eye point of the instrument (on the wall) is not the eye point of the drawn perspective image of the plan shown on the table. Faulhaber 1610,37.

Bramer described a perspective machine in a publication on trigonometry (ibid.), and in his above-mentioned publication from 1630 he explained how he had improved the principle of this instrument and how it compared to the instrument presented by Faulhaber.

Brunn and Scheiner

A

nother example of the general attraction of perspective instruments is provided by Lucas Brunn (tI640). He had a university education and ended his career as Electoral mathematicus and inspector for the Kunstkammer in Dresden. Presumably inspired by Lencker - to whom he referred - Brunn published a book that contains elaborate perspective images of the letters of the alphabet (Brunn 1615). He claimed that he had constructed these images using an instrument (figure XI. 4) he ascribed to professor Johann Praetorius of Altdorf. The Jesuit scientist Christoph Scheiner (1573-1650) is known, among other things, for his observations of the sun, for his discussions with Galileo concerning the interpretation of the observations, and for his invention of the pantograph. As shown in chapter VIII, the main function of this instrument

2. Perspective Instruments

FIGURE

603

XI.4. Brunn's drawing of a perspective instrument. Brunn 1615.

was to redraw a contour in another scale. Scheiner additionally suggested using the pantograph for perspective drawing (figure VULlO).

Halt

A

mong the early seventeenth-century German writers on instruments was one who presumably performed perspective constructions in connection with his work, namely the stone cutter and draughtsman Peter Halt. In 1625 Halt published Perspectivische Reisskunst (The art of perspective drawing), which contains a laudatory introduction in verse by Faulhaber and which, with its two hundred and twenty nine pages and one hundred and seventy five drawings, is the largest seventeenth-century German book on perspective. The book itself clearly shows signs of inspiration from Durer and Jamnitzer, praising the latter for having stressed the importance of the regular polyhedra. In fact, according to Halt, the Platonic solids strengthen art like vowels strengthen language (Halt 1625, preface) - one wonders whether it played a role for this comparison that there are five regular polyhedra and five vowels (in Latin). Halt based his perspective constructions on a plan and elevation method, which he presented together with an instrument he claimed was his own invention. His description of how to use the instrument is not too clear, but if I have understood him correctly, he applied it not only for making perspective compositions, but also for drawing plans. Already a year after the appearance of Perspectivische Reisskunst, Halt published a second book on perspective and perspective instruments (Halt 1626), which I have been unable to examine.

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XL The German-Speaking Areas after 1600

Hartnack y the mid-1630s, the interest in perspective instruments seems to have declined in the German states - or at least the enthusiasm for writing B about them. Fifty years later the subject was taken up again, this time by Daniel Hartnack (1642-1708). According to some biographies, for instance Rotermunds 1818, Hartnack was one of the more colourful perspectivists. He was often on the run, first because as a young teenager he got a woman pregnant, then from prison, to which he had been sentenced because of an unpaid debt, then from his position because he was found to lack the education he had claimed, and so forth. Despite his tumultuous life, Hartnack managed to work as a teacher and headmaster for most of his life, and to get at least seventy titles published. One was the booklet Perspectiva mechanica und eigentliche Beschreibung derer vornehmsten Instrumenten ... zum perspectivischen Reissen bissher erfunden worden (Mechanical perspective and a proper description of the most refined instruments that until now have been invented for perspective drawings ...), published in 1683. 2 This booklet described some seven instruments, though without bringing anything new. Hartnack began with a list of authors on perspective and included five nonGermans. Although there is no guarantee that Hartnack's selection of foreign literature reflects what German perspectivists were generally acquainted with in his day, I still find it interesting to notice that he pointed to the two very popular authors Serlio and Dubreuil, and to the two frequently quoted authors Marolois and Niceron, finally also mentioning Guidobaldo (Hartnack 1683, preface).

Meister and Hoffmann n the 1750s we find a renewed interest in perspective instruments docuby Albrecht Ludwig Friedrich Meister (1724-1788). As the topic Iformented dissertation he chose a description of a perspective machine, his

magister and his introduction was a survey of earlier instruments (Meister 1753). He referred, among others, to Biirgi, Bramer, and Christopher Wren. Meister pursued an academic career and became a professor of philosophy in Gottingen, in which capacity he taught mathematics and special courses on perspective (PiitterS 1765). As late as 1780, the painter Johann Leonhard Hoffmann (174Q-c.1812) published more than fifty pages on a perspective instrument whose main virtue apparently was that it did not involve lenses (Hoffmann 1780). He found that good lenses were too expensive, and criticized the camera obscura for requiring too much light (ibid., 7-8). As far as I understand his description, Hoffmann had designed an instrument that was not much different

21m am very thankful to Herbert Breger for providing me with a copy of this book.

3. Anamorphoses

605

from the one Stevin had introduced in the beginning of the seventeenth century (figure VI.38). Before presenting his own instrument, Hoffmann gave a cursory introduction to perspective. He also published books on how to draw and paint and on colours. In 1799 he was appointed Zeichenmeister (master of drawing) in Erlangen.

Bischoff and Biirja

T

he three eighteenth-century mathematicians Bischoff, Lambert, and Biirja each invented and described a perspective instrument (Bischoff 1741, Lambert 1752, Biirja 1795). Biirja's presentation occurred in a book on perspective, to which I come back. On perspective machines in general, he had the following comment. I have noticed various troubles and inconveniences in the use of those perspective instruments with which I have become acquainted. 3

This statement confirms my impression, indicated earlier, that perspective instruments were better as creations of the mind than as practical tools. Biirja's own instrument resembles Scheiner's pantograph, and it is difficult to say whether it really was convenient to use. I return in chapter XII to Lambert's instrument, but find it appropriate to mention here that it is also doubtful whether Lambert's machine would have worked smoothly (page 643). Johann Christoph Bischoff (t 1774) described his instrument in KurtzgeJasste Einleitung zur Perspectiv, darinnen nebst dem wahren Fundamente derselben ...

(Brief introduction to perspective containing its true foundation together with ...,1741). Unfortunately I can say nothing about his instrument, because I have only been able to examine a copy of Bischoff's book in which the plates were missing. In treating perspective, Bischoff based his presentation on a distance point construction that he introduced without including any theory despite the title of his book. In his preface, he wrote that his reason for writing yet another book on perspective was that many of the existing works could only be accessed with difficulty, if at all.

XI. 3 Anamorphoses Albrecht

T

he diagram reproduced in figure XI. 5 shows that a kind of distance point method for constructing anamorphoses was already applied in Germany in the mid-sixteenth century. As we saw in chapter IX, de Caus

3Bei denjeningen perspektivischen Instrumenten, die zu meiner Kenntniss gekommen sind, habe ich allemal in der Anwendung verschiedene Schwierigkeiten und Unbequemlichkeiten bemerket. [Biirja 1795, 225]

606

XI. The German-Speaking Areas after 1600

FIGURE XI.5. A diagonal method applied for an anamorphic drawing. The author is known as Master H.R. from Nuremberg, c. 1540. Universitatsbibliothek ErlangenNiirnberg drawing B 448.

described such a method in 1612, though not very understandably - the first lucid exposition in French being Niceron's presentation from 1638. Between the appearances of these two works, Albrecht published the first German description of how to construct a perspectival anamorphosis, entitled Zwey Bucher. Das erste von der ohne und durch die Arithmetica gefundenen Perspectiva. Das andere von dem darzu gehorigen Schatten (Two books. The first on perspective found without and by means of arithmetic. The other about the shadow belonging to it), published in 1623 (figure XI.6). Before telling any more about Albrecht's anamorphoses, let me present the author and his book shedding light on its rather enigmatic title. Andreas Albrecht (t1628) was as a mathematical practitioner earning his daily bread as a calculator and engineer. He evidently did not tire easily of arithmetical operations, because he devoted a large part of his work on perspective to calculations. Like Durer, Albrecht centred his work around a plan and elevation construction of the perspective image of a cube with two sides parallel to the picture plane n. Unlike Durer, however, Albrecht assumed that the eye point is situated centrally with respect to the cube - or more precisely, that the eye point lies in the vertical plane orthogonal to nthat bisects the cube. Having presented a geometrical construction of the image of the cube, Albrecht proceeded with calculations. He chose a fixed distance and made some tables listing the coordinates (though he did not use this term) of the perspective image of a vertex in the cube for various values of the height of the eye point above the ground plane, the distance between the cube and the picture plane, and the side of the cube (figure XI. 7). He stressed that in choosing the various variables, it was important to be aware that the resulting visual angle should have a reasonable size under 90° (Albrecht 1623, chapter one). Thus far we have not met an explicitly arithmetical approach to perspective, by which I mean an approach in which an author recommends his reader to calculate what corresponds to the coordinates of a perspective image. This is understandable, since before the age of modern computers it was, as noted previously, much quicker to find the image using a geometrical construction than it was to calculate its coordinates. However, we have seen methods that implicitly involve arithmetic, namely the procedures that

3. Anamorphoses

FIGURE

607

XI.6. The title page of Albrecht's book.

were based on the use of special scales - often engraved on a sector. This solution was one that was preferred, most notably, by a number of French perspectivists from around 1630 onwards (while others constructed foreshortening scales geometrically). Albrecht's procedure of producing tables rather than scales is quite exceptional; one of the few others to use this approach was Claude de Roy (page 477), who presumably did this completely independently of Albrecht. As indicated in the title of Albrecht's book, he also treated shadows. He included a few more examples than Durer, but basically followed Durer and like him, drew a sun as the source of light, but still treated the source as a point. In extension of his treatment of shadows, Albrecht turned to an example of an anamorphosis, which he called an Ongestalt. 4 He applied a plan and eleva4In modern German: Ungeslall (deformed object).

608

Xl The German-Speaking Areas after 1600

c y

d

e

FIGURE XL7. Diagram to Albrecht's tables. The upper drawing shows the elevations 0e' A e, and Ce of the eye point 0, a point A in the ground plane, and a point C vertically above A - the picture plane tr being represented by PeQe. Similarly, in the lower drawing Op and A are the projections of 0 and A into the plan. Moreover, d is the distance, h the p height of 0 above the plan, a the distance between A and the picture plane, b the distance between A and the vertical plane, and c the distance between C and the plan. Finally x, y, and z are the foreshortened b, a, and c respectively. Albrecht's arithmetical approach to perspective consisted in calculating tables, in which x, y, and z could be looked up. From similar triangles we see that the required quantities are given by the following relations, x = a

bd

+ d' Y

= a

ah

d

+ d' an z

= a

cd

+d

Albrecht set d equal to 10 and let a, c, and h be integers in the interval [1,16] and b be integers in the interval [1,8]. He first used the rule of three to find x, y, and z, but later he also introduced some angles and their tangents, and obtained some of his results in a rather cumbersome way. With his method of calculating x, y, and z Albrecht seems to have confused himself, for he did not notice that z is independent of h.

tion technique in a correct and straightforward manner to construct the anamorphosis. The caption of figure XI.8 explains the principle of Albrecht's construction, and his own illustration is reproduced in figure XI. 9. Besides his very detailed explanation of how a perspective cube can be obtained in various ways, Albrecht included a number of illustrations of

3. Anamorphoses FIGURE XI.8. The principle behind Albrecht's construction of anamorphoses. To coordinate this diagram with the one by Albrecht, reproduced in the next figure, I have used his letters - and added a few of my own. The square abed is to be projected upon the wall Jr from the eye point 0. Albrecth did this with the help of the square's plan and elevation, where, as plan he used the ground plane r and as elevation Jr itself, EF being the line of intersection of these two planes, and G and N the plan and elevation of 0. In the diagram I have showed how he constructed the image of the point A (5,5): The plan and the elevation of the line 0A are the lines G5p and N5. Using a traditional plan and elevation construction, Albrecht found the point of intersection m of EF and G5p' and finally determined the image Ai as the point of intersection of N5 and the vertical line throughm.

609

c Ai

F N

E

G

elaborate perspective compositions, among them a spiral staircase, which became a much-used motif among Durer's followers. As the first seventeenth-century German work completely devoted to perspective, Albrecht's contribution seems to have become fairly well known. It was translated into Latin, 5 and was often referred to by later writers. However, it was more the existence of Albrecht's work than its content that was mentioned. His treatment of anamorphoses seems to have been particularly overlooked: the subject was taken up by several other German writers, but they used Niceron's method.

Kircher and Schott chapter IX we saw how Niceron's interest in anamorphoses was linked IA tonsimilar a general enthusiasm for the aspects of science that can seem magical. attitude towards science was displayed by the Jesuit polymath Athanasius Kircher (c. 1602-1680) who had contacts with the Minims - the Order to which Niceron belonged. Kircher was attracted by virtually all the 5The Latin edition I have seen was partially in German, but whether this applies to all copies, I cannot say.

610

XI. The German-Speaking Areas after 1600

----- ....

,~

.....

N. ,.-.......•...............,. •• -..J••••••••••••••

....

....

.,Jl~~~~.

......

i

:

f

m·

........ ~..•.

.....

.-

..."

....

....

. .., .-" .. ....... .....

....

.....

f

o,l.,;)'

.....

FIGURE XI.9. Albrecht's construction of an anamorphosis of a Nuremberg eagle. Albrecht 1623, figures 129 -131.

theoretical disciplines of his time within science and the humanities - including hieroglyph deciphering (Kangro s 1973). Moreover, he was very eager to transmit his knowledge and wrote more than forty books. Kircher not only studied curiosities, but also collected them. During a prolonged stay in Rome he collected a number of strange artifacts to a museum, which, although not created by him, came to be known as Kircher's museum. The museum is likely to have contained some anamorphoses, for Kircher took a vivid interest in them (for more on Kircher and anamorphoses - and other curious pictures see Baltrusaitis S 1977, 79-86).

3. Anamorphoses

611

In 1646 Kircher published Ars magna lucis et umbrae (The great art of light and shadow), which dealt with various disciplines involving light rays and included perspective and anamorphoses (Kircher 1646). He used a number of uncommon terms, for instance planum mesopticum for the picture plane and rather confusingly, punctum oculi (eye point) for a distance point. He presented a few mathematical results such as the convergence rule - but did not introduce the general concept of a vanishing point. As his method of construction he chose a distance point construction, which he also applied for perspectival anamorphoses. His treatment of various sorts of anamorphoses seems to be inspired by Niceron's work. For cylindrical mirror anamorphoses, Kircher presented the same construction as Niceron (figure IX.38), and like Niceron, he suggested that the radii of two consecutive circles should have the ratio 21:20. Kircher exerted a great influence on his pupil Gaspar Schott (1608-1666), who also entered the Jesuit Order. Like his master, Schott was much fascinated by curious phenomena - natural as well as supernatural. Today he is best known for his role as a correspondent who kept himself and others abreast of various experiments, in particular some concerning vacuum (KellerS 1975, 210-211). Schott wrote about his observations in Magia universalis naturae et artis (Universal magic of nature and art). In this book he included optics and devoted seventy pages to the "magic of anamorphoses" (Schott 1657). He followed Kircher in presenting perspective in connection with anamorphoses (for more on Schott and anamorphoses, see BaltrusaitisS 1977, 85-90). As we shall see, Schott also took up perspective in a textbook on mathematics that appeared posthumously.

Leupold

A

fter the l650s anamorphoses apparently lost some of their attractionand presumably went out of fashion. They continued to be treated in books on perspective, but did not receive much attention. The golden age of anamorphoses partially overlaps with the period when perspective instruments were greatly in focus. One could imagine that during this time some people got the idea of creating machines that could, at least in principle, construct anamorphoses. I do not know whether this really happened at the time, but such instruments were described later, by the instrument maker Jakob Leupold. In 1713 Leupold published Anamorphosis mechanica nova, oder Beschreibung dreyer neuen Maschinen (New machines for anamorphoses, or description of three new machines). It is a kind of catalogue consisting of six pages that describe, indistinctly, three rather complicated machines, all meant for constructing mirror anamorphoses: one for plane mirrors, one for conical mirrors, and one for cylindrical mirrors (figure XLlO). For further reading Leupold recommended, among others, Niceron, Kircher, Schott, and Sturm.

612

XI. The German-Speaking Areas after 1600

CJ(;I

FIGURE XLI O. Leupold's machine for creating cylindrical anamorphoses. Contrary to my general policy, here 1 abstain from describing the machine's mode of operation, but 1 can tell as much as the drawing fixed to the cylinder infig. 3 is redrawn as an anamorphosis on the horizontal board under the cylinder. Leupold 1713, being a reproduction of a figure published a year earlier in Acta eruditorum.

Mathematischer Lust und Nutzgarten

W

ith the only indication that it was written by "S.R.", a book appeared in 1724 under the title Mathematischer Lust und Nutzgarten (The mathematical pleasure and vegetable garden). Among many other things, this work treated anamorphoses and carried a brief introduction to perspective. The search to identify S.R. has been a puzzle, but I am now convinced that two authors were involved, namely the artist Johann Jacob Schiibler (1689-1741) - to whom I return - and the astronomer Johann Leonhard Rost (1688-1727). To support my conclusion I first relied on the following two arguments. First, one of the older biographies of Schiibler mentioned his being the author of a Mathematischer Lustgarten (WillS 1757). Second, in 1745, after both men had died, another Mathematischer Lust und Nutzgarten was issued with Rost's name on the title page and with the rest of the title similar to that of the S. R. edition (Rost 1745).6 Later I found an irresistible

61 cannot tell whether Schiibler's name also figures in 1745, nor compare the contents of S.R. 1724 and Rost 1745, for 1 have not managed to see a copy of the latter.

3. Anamorphoses

613

proof: In 1786 Johann Tobias Mayer - to whom I also return - published a book stating that it was a revised version of the "S.R." book from 1724 and that the authors were Schiibler and Rost (Mayer 1786, title page and iv). The section on perspective in Mathematischer Lust und Nutzgarten accommodates a nicely illustrated presentation of Desargues's method as described by Bosse (figure XLII). It also contains references to Niceron and Laraisse.

c '

......

\.

.

~...

"

I., \

~.

~

':'

........ : J

...

:• . • t.••

...

:.....~

...:

.,

:

"

FIGURE XI.ll. A perspectival coordinate system. Figure 208 in Schiibler & Rost 1724.

614

XI The German-Speaking Areas after 1600

XI.4 Perspective Presented for Practitioners n this section I present the part of the German literature not yet described, which was meant for other readers than students of mathematics. I and

The Unknown Filllisch

j

ohann Jacob Fiillisch - about whom I have been unable to find any biographical information - included perspective in his Compendium artis ...

Das ist kurzer ... Unterricht von der geometrisch-ignographischen Zeugnungs und Baukunst (Compendium on art ... That is a brief ... instruction in the geo-

metrico-ichnographical arts of drawing and building). Even though the book appeared in 1680, its style is reminiscent of books that are a hundred years older. Fiillisch made rather primitive drawings of what looks like an Alberti construction and a distance point construction without explaining his diagrams.

The Philomath Haesell

T

heodosius Haesell shared with some of the previously mentioned authors an approach to perspective that saw it as part of a mystical and divine cosmos. In 1652 he published Geistliche perspectiva (Spiritual perspective), whose title page describes Haesell as a philomath who worked as a secretary for the Elector of Sachsen. His book contains frequent references to the bible, a drawing of Kepler's famous model of the orbits of the planets (figure XI.l2), and an elaborate diagram showing various holy names thrown into perspective (figure XI.l3). Haesell included a list of fifteen earlier writers on perspective, which he presumably copied from an older book, since he solely referred to works that appeared before 1632.

The Painters Sandrart and Heinecke

O

ne of the few painters we meet in this chapter, Joachim Sandrart (1606-1688), was also an influential teacher of art In 1675 he published a general textbook on architecture, sculpture, and painting called Teutsche Academie (The German academy), in which he devoted ten pages to perspective. He presented a distance point method and included references to the rather old works of Serlio and Danti, but he also showed an awareness of newer literature, and of another perspective method, by referring to Desargues and Bosse. Another example of an eighteenth-century author from the world of painters, or at least from the world of miniature painters and enamellers, is Paul Heinecke. In 1727, Heinecke published a booklet on perspective with more than hundred illustrations, several of which are quite impressive. He introduced a distance point construction as well as a plan and elevation

4. Perspective Presented for Practitioners

615

FIGURE XI.12. Haesell's reproduction of Johann Kepler's planetary model in Geistliche perspectiva. Haesell1652.

construction, and referred to a writer who had done the same in the sixteenth century, namely Vignola.

The Architect and Drawer Schiibler

S

andrart may have inspired a pupil of his, the architect and drawer Johann Jacob Schiibler (1689-1741), to take up perspective. Schiibler manifested his interest in the subject in Perspectiva pes picturae. Das ist kurze und leichte Verfassung der practicabelsten Regul zur perspectivischen Zeichungskunst (Perspective, the basis of painting. That is a brief and easy composition of the most practical rules of the art of perspective drawing). This work contains a series of splendid illustrations and an extensive survey of the works of earlier perspectivists (Schiibler 1719-1720). Schiibler was most impressed by

616

Xl. The German-Speaking Areas after 1600

FIGURE XI.l3. A perspective composition by Haesell. It is worth noticing that he followed a tradition of including the Platonic solids, which, presumably inspired by Kepler, he decorated to illustrate the elements they represent. Haeselll652.

4. Perspective Presentedfor Practitioners

617

the works of Sirigatti and Dubreuil, whereas he found little use for the books that only presented a perspective method without giving illustrative examples. As mentioned, Schubler returned to perspective as one of the authors with the pen name "S.R." (page 612). Besides architecture and perspective, Schubler also engaged in other fields of applied mathematics and became a member of the Royal Prussian Academy of Sciences in 1735.

The Engraver Werner

P

lan and elevation methods, which fell into disuse in other countries, continued to be used by German practitioners - or was at any rate described in their textbooks. The engraver and medallist Georg Heinrich Werner (c. 1722-1788), who became a member of several Academies, also took up such a method. He presented his version in Erlernung der Zeichenkunst durch Geometrie und Perspektiv (The learning of the art of drawing through geometry and perspective, 1763), which appeared in a revised and posthumous edition under a new title thirty three years later (Werner 1796). The first edition is divided into two separately paginated parts in which the title page of the part on perspective only states the author's name as G.H.W, whereas the second part dealing with geometry, gives his full name. In his Erlernung Werner included an unusually long list of predecessors' works on perspective, beginning with classical Greek titles on optics (Werner 1763, 7-11). The first 'real' perspectivist he mentioned was Durer, whose name is followed by a large part of the sixteenth-century authors on the topic. Werner claimed that perspective was advanced considerably in 1590 by a certain Johann Heyder - whose life and work remains a mystery for me, since I have been unable to find any information on him. The non-Germans from the sixteenth century, and indeed all authors after that time, seem to have been chosen rather at random in Werner's list. It is interesting, though, to notice that John Hamilton's book from 1738 is included and characterized as a "very nice and excellent work" (ibid., 10). Werner did not compose a treatise in the style of Hamilton, however, but a fairly elementary manual without any mathematical arguments. As noted, he decided on a plan and elevation construction, claiming that he was following the rule of Sirigatti (ibid., 11). He did not expect much independent thinking of his readers. Thus, he first showed how an equilateral triangle, then an isosceles triangle, and finally a general triangle is thrown into perspective. He then repeated the procedure of going from special to general figures for quadrangles and pentagons as well.

The Master Carpenter Rodel

A

plan and elevation method is also central in another book on perspective, published in 1784 by the Hof und Rathzimmermeister (court and council master carpenter) Johann Michael ROdel (1734-1784). Mter presenting this

618

Xl The German-Speaking Areas after 1600

method, Rodel proceeded to introduce the concept of a vanishing point, for which he used the translation zufiilliger Punkt (accidental point), from the Latin term punctum accidentale, introduced into Germany by Wolff, as we shall see (page 622). ROdel stressed his use of vanishing points by calling his book Abhandlung von den zufiilligen Punkten in der Perspektivkunst fiir Werkmeister

(Treatise on accidental points in the art of perspective for foremen). Rodel's book is not particularly remarkable, but the mathematician Kastner nevertheless graced the work with a lengthy introduction. This is a bit ironic, given that Kastner was not particularly interested in using vanishing points as the foundation of perspective theory. In fact, Kastner had his own analytical approach, as we shall see shortly, and he spent part of the introduction advocating his own method. Acknowledging that geometrical constructions can be of use, Kastner pointed to Lambert's excellent treatment of perspective while letting the virtues of ROdel's book remain unclear. From a historical point of view the existence of Rodel's book is more important than its quality, for it shows that at least one German outside the circle of mathematicians was aware of vanishing points. Strangely, the German mathematicians generally only let vanishing points playa minor role in their presentations of perspective.

Gericke and Weidemann, Professors at the Academy of Art here is an activity in perspective whose history still calls for some investigation, namely the teaching of the subject at art academies. In chapter IX T we saw how the contents of a perspective course were disputed at the Paris Academy of Art (page 461). Irrespective of a disagreement, neither of the parties seems to have doubted that the discipline should be taught. The idea of teaching perspective was also supported at the Akademie der Kiinste (Academy of arts), founded in Berlin in 1696, which soon established a professorship of perspective. In 1699 the academy's first perspective professor, the court painter Samuel Theodor Gericke (1665-1730) - who was also rector of the academy - announced his first course (Gerickes 1699). Teaching took place two hours each Saturday morning for four months, which, according to Gericke, was sorely needed, because a similar course was given in Paris and because so many famous painters had written about the necessity of knowing perspective. Three decades later another professor of perspective at the academy, Friedrich Wilhelm Weidemann (1668-1750), published his lecture notes under the title Kurtze Einleitung zu der optischen Perspectiv nebst deren ersten Grund und Lehrsiitzen (Brief introduction to the optical perspective together with its foundation and theorems, 1733), of which I have only seen the second edition from 1746. Weidemann's idea was that his students should learn not only the rules of perspective drawing, but also the reasons for these rules. Thus, he chose a mathematical approach based on theorems that he supplied with proofs. The style of his book resembles that used by the German mathematicians - soon to be presented.

5. Mathematical Works on Perspective

619

The Theologian Horstig

T

he last book to be mentioned in this section may actually be slightly outside the category, since it was not addressed to practitioners, but to a liebe Freundin (dear - female - friend). Who else was meant to read it is difficult to say. The book in question is Brieft iiber die mahlerische Perspektive (Letters about perspective for painting, 1797) written by the theologian and teacher Karl Gottlieb Horstig (1763-1835). In a total of one hundred and thirty one letters he sought to introduce his friend gently to the ideas behind perspective. He did actually not reveal to her how a perspective construction can be made, but only why perspective representation is used. This led him to the theory of vision, and it turns out, Horstig belonged to the group who thought that in principle, the visual impression is what should be drawn. Thus he wrote: From my last letter you will have understood as well, that straight lines in perspective should not at all have the appearance of straight lines. 7

According to Horstig, no painter dared to draw two straight parallel lines in their true perspective foreshortening (Horstig 1797, 53). However, he did not find this too bad, stating that the eye has become used to the fashion of drawing and knows how to compensate for perspectival representations. According to Horstig, although a house ought be drawn like the one at left in figure XI.l4, the eye still perceives the house at right correctly (ibid.).

XI.5 Mathematical Works on Perspective

F

rom the end of the seventeenth century, perspective became a standard element in German textbooks on mathematics. The space German mathematicians devoted to the subject varies considerably, the shortest presentation

OODODDOOO~ . DOD

~! DODO

00000000000 DO 00 =[1 0000

FIGURE XI.l4. At left is Horstig's idea of how a house should be drawn in perspective, and at right how it is customarily done. Horstig 1797, figures 4 and 5.

7Aus meinen vorigen Brief werden Sie zugleich begriffen haben, dass grade Linien in der Perspective nicht durchaus die Gestalt von graden Linien haben durfen. [Horstig 1797, 52-53]

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XI The German-Speaking Areas after 1600

taking up less than half a page and the longest spanning more than three hundred pages. The unofficial record for the shortest presentation is held by Schott, who, as previously indicated, included perspective in his work Cursus mathematicus (Schott 1677), which appeared twenty years after his Magia universalis. In my observation, Schott's Cursus is the first general German textbook on mathematics that includes perspective, and the next to do so is a textbook for adolescents called Mathesis juvenilis (Mathematics for the young, Sturm 1699/1701). Its author was Johann Christoph Sturm (1635-1703) who had been a professor of mathematics at the University of Altdorf since 1669. In his work, which appeared in Latin as well as German editions, Sturm spent ten pages presenting a distance point construction and anamorphoses, making references to Niceron, Schott, and Pozzo. Before turning to the other mathematical textbooks I would like to mention a very special treatment of perspective that is different from all others presented in this book. It occurs in Beytrage zur Mathematik (Contributions to mathematics), which was published in 1781 by Karl Scherffer, a Jesuit professor of mathematics at the university of Vienna. Scherffer's aim was not to give an introduction of perspective, but to create complicated problems involving the trigonometric functions and their logarithms (Scherffer 1781). Referring to Lacaille, the author assumed his readers were familiar with using an angle scale - or, as he called it, a tangent scale - to construct vanishing points in the picture plane. Based on this scale Scherffer formulated twelve trigonometrical problems that would never occur in the practical performance of perspective drawings.

The Wolf/ian Tradition or almost two hundred years the German literature on perspective had avoided proofs. The much-applied plan and elevation constructions were, P as stressed earlier, so transparent that a proof of their correctness was not really necessary. However, it is difficult to see why a distance point construction leads to a correct result. In Italy, the question of the correctness of perspective constructions was taken up early, not only by Piero, who was greatly interested in mathematics, but also by the less mathematically inclined Vignola. In the German-language states there was seemingly no similar urge to know why such a construction works. It was deemed sufficient to know how to perform it - which could in itself be difficult enough to learn from presentations written in German. Indeed, in the German literature, treating the question of a distance point construction's correctness was left to the mathematicians of the eighteenth century. To my knowledge Wolff was the first to publish a proof - or rather, to publish an attempt at a proof. Christian Wolff (1679-1754) was an important figure in German intellectuallife during the first half of the eighteenth century. He was not highly original, being rather dependent on Leibniz and others, but he was enthusiastic in

5. Mathematical Works on Perspective

621

his discussions of fundamental philosophical questions. Like Descartes, Pascal, and Leibniz, Wolff combined an interest in philosophy with an interest in mathematics. Even though his mathematical talent could not be compared with that of these three famous philosophers, Wolff was far more active than they were in spreading the knowledge of mathematics. For most of his adult life he held a professorship in mathematics and natural philosophy in Halle, and from 1710 onwards he published various general textbooks on mathematics in German and Latin, which also were translated into various other languages. Wolff's exposition of the theory of perspective appeared in his Elementa matheseos universae (Elements of universal mathematics, 1713-1715). This work contains a section of about thirty pages, Elementa perspectivae (Elements of perspective, Wolff 1715), which begins with eighteen definitions and introduces, among other things, the concept of a visual pyramid (figure XI.I5). In addition, the Elementa perspectivae has two introductory propositions, of which the first states that the perspective image of a straight line is a straight line (ibid., §28). In the second proposition, Wolff determined a proportion that gives the height above the ground line of the image of a point in the ground plane whose distance to the ground line is known (ibid., §32). His result is equivalent to the second equation in the caption of figure XI.7. Wolff applied this result to argue the correctness of his distance point construction (ibid., §33). However, he tacitly assumed that the images of orthogonals pass through the principal vanishing point. In the Renaissance this result was also taken for granted, but in early modern times mathematicians - at least those who wanted to prove their theorems - would give a proof of this result. Wolff seems to have based his presentation of the theory of perspective upon Tacquet, to whom he referred. Tacquet had centred his exposition around proportionality, gradually building up the theory by means of six

FIGURE XI. 15. Wolff's visual pyramid. Wolff 1715/1735, figure I - the top of the viewer's head is also missing in Wolff's book.

622

Xl The German-Speaking Areas after 1600

theorems (Tacquet 1669, 159-163). By including only two of these theorems, Wolff opted for a mathematically insufficient solution. Wolff's approach to perspective was criticized by Karsten, whose own work on the subject we will meet later. Karsten showed quite a degree of respect for Wolff's contributions to the teaching of mathematics, but he was not happy with Wolff's treatment of perspective, regretting, among other things, the scant attention Wolff had paid to the vanishing point theorem (Karsten 1775, §45). I emphasize the incompleteness of Wolff's proof because I want to point out that even for an eighteenth-century mathematician the geometrical theory of perspective could cause problems. In fact, Wolff was not alone in his inaccuracy, but was followed by other German mathematicians who copied Wolff's proof without completing it. Irrespective of whether their proofs were incomplete or not, most German mathematicians followed Wolff in building the theory of perspective on proportionality rather than on the general concept of vanishing points - a tendency we also met in eighteenth-century France. In addition to presenting a distance point construction, Wolff treated thirty five problems in which he explained how to use a frame equipped with a grid of squares for drawing perspective compositions (figure XI.16), gave some examples of how to use his distance point construction, and described how vertical lengths can be foreshortened. Wolff did introduce the concept of a vanishing point, but only in a scholium and only in connection with horizontal lines (Wolff 1715, §79). As mentioned, Wolff called a vanishing point punctum accidentale - an expression we met in connection with Cousin (page 178). Wolff concluded his treatment of perspective with a small section on anamorphoses (ibid., §§104-134). For readers who did not know Latin, Wolff compiled Anfangsgriinde aller mathematischen Wissenschaften (Elements of all mathematical sciences, Wolff 1737). Here, too, he treated perspective, but more briefly than in the Elementa and without attempting to give proofs.

FIGURE XI.16. A frame with a grid of squares for making perspective drawings. Wolff 1735, figure 2.

5. Mathematical Works on Perspective

623

Weidler

A

n influence from Wolff is noticeable in Institutiones matheseos (Instructions in mathematics). The work was first published in 1718 by Johann Friedrich Weidler (1692-1755), who became a professor of mathematics in Wittenberg the following year. Institutiones was subsequently reissued at least five times. The oldest edition I have seen, from 1736, contains ten pages on perspective (Weidler 1718/1736), beginning with a theoretical part that is very similar to Wolff's. In those few pages Weidler moreover managed to deal with a plan and elevation construction, shadows, and anamorphoses.

lena Scholars n Jena the Wolffian tradition was perpetuated by three textbook writers. First the mathematician Johann Wenceslaus Kaschube devoted some ten Ipages and sixteen (tiny) figures to presenting a distance point construction, the foreshortening of vertical lengths, shadows, and anamorphoses in the section entitled Die Perspectiv-Kunst (The art of perspective, Kaschube 1717) in a general textbook on mathematics. Eight years later, the professor of mathematics Johann Bernhard Wiedeburg (born 1687) likewise included perspective in his Einleitung zu denen mathematischen Wissenschaften (Introduction to the mathematical sciences, Wiedeburg 1725/1735). His presentation was similar to Kaschube's. Finally the philosopher and jurist Joachim Georg Darjes (c. 1715-1791) spent some ten pages describing the Elementa perspectivae (Elements of perspective) in an introductory work to mathematics (Darjes 1747) that appeared in several editions. Besides describing a distance point construction, he presented some problems and their solutions. Jena also provided the literature with two dissertations on perspective. The first of these was published in 1717 by Anton Bernhard Lauterbach (c. 1688-1713) and defended by Matthaeus Honoldus. Its title, CIa vis perspectivae verticalis geometrica (A geometrical key to vertical perspective), is puzzling. As I have been unable to see the dissertation, I cannot say what its author meant by vertical perspective - perhaps simply that the picture plane should be vertical. The second dissertation is entitled Dissertatio mathematica sistens leges perspectivae ad situm plani transparentis mutatum adplicatas (Mathematical dissertation presenting the rules of perspective connected to a changed position of the transparent plane). It was written by a professor of medicine at Jena university, Georg Erhard Hamberger (1697-1755), and defended in 1719 by Johann Christoph Schilling. It remains unclear to me whether the dissertation was published at the time and then republished in 1747, or first published in 1747. At any rate, the copy I have seen is from 1747. Its theme is how to determine the placement of the picture plane when the positions of the eye point and the objects are given. The criteria

624

XI. The German-Speaking Areas after 1600

according to which the determination is made are not presented very clearly, but the dissertation contains an impressive number of references to Euclid and Ozanam. Hamberger, who is listed in some library catalogues as Georg Erhard, wrote dissertations on many other subjects besides perspective, one of which was defended by Johann Andreas Segner, whom we will meet shortly.

H ennert and Lorenz

A

s Lacaille had done in the second edition of his book on optics from 1756, German Johann Friedrich Hennert (1733-1813), a professor of mathematics, astronomy, and philosophy at Utrecht University, included a section on perspective in his Elementa optices (Elements of optics), a work published in 1770. And like Lacaille, Hennert based his theory of perspective on the theory of proportion - but there the similarities stop. Lacaille had given a thorough presentation of the subject, whereas Hennert was superficial. Hennert's outline was presumably influenced by Wolff, for like him, Hennert only gave half of the argument for proving that his distance point construction is correct (Hennert 1770, §101) - and his visual pyramid (figure XU7) bears a strong resemblance to Wolff's (figure XI.l5). The fact that he worked in the Netherlands does not seem to have inspired Hennert to look at how scholars there had treated perspective. When the mathematician and popular textbook writer Johann Friedrich Lorenz (1738-1807) decided to include a short section on perspective in a general book on mathematics (Lorenz 1799), he apparently consulted Wolff's and Kastner's presentations. He took a bit from each, but since he did not follow Kastner's general approach - which will be presented shortly Lorenz's result came to be fairly similar to Wolff's text.

~----.B

FIGURE

XU? Hennert's visual pyramid. Hennert 1770, figure 28.

5. Mathematical Works on Perspective

625

Segner and Biirja

P

resumably I have not traced all the presentations of perspective occurring in German-language textbooks on mathematics; but the number I have found clearly shows that perspective became a part of mathematics in Germany in the eighteenth century, and that the various textbook authors treated the subject in the same way. The two German mathematicians Segner and Biirja both became so interested in perspective that they devoted entire books to the subject. Johann Andreas von Segner (1704-1777) was a professor of mathematics and physics, first in Gottingen and later in Halle, who became a member of the Academies of Sciences in Berlin and St. Petersburg, and a Fellow of the Royal Society in London. In 1779, Segner's son followed the will of his father and published Griinde der Perspectiv (Foundations of perspective) after a manuscript he had left behind. This resulted in a book of slightly less than hundred pages, in which Segner aimed to give practitioners of the discipline a feeling for the mathematical foundation of perspective, but without presenting it as a traditional piece of theoretical geometry, explaining the contents of the main theorem without proving it (Segner 1779, 9-10). In the places Segner did add proofs to support his claims, he adhered to the general German approach of using the theory of proportion. He chose his own vocabulary, calling an image a Vorstellung (imagination or representation), and a vanishing point a Griinzpunkt (limit point). Abel Biirja (1752-1816), whose perspective instrument I touched upon earlier, was also a professor of mathematics and a member of the academies in Berlin and St. Petersburg. He published a considerable number of textbooks on mathematics and its applications. The latter area included perspective, to which he devoted an entire book entitled Der mathematische Maler (The mathematical painter, 1795). Like so many of his countrymen, Biirja based many of his geometrical arguments on the theory of proportions, but he went further with the theory of perspective than most other German authors. He introduced a tangent scale or angle scale for vanishing points with reference to Lacaille and Lambert, constructed ein perspektivisches Netz (a perspective grid) that resembles the grid introduced by Desargues (figure IX.22), and explained how a sector with perspective scales can be applied for constructions (Biirja 1795,51,101, and 153). Moreover, he included several considerations relevant for painters choosing parameters for a painting; in fact, he recommended specific choices for small paintings, for larger paintings, and for set pieces to be used in theatres. In his opinion, a painter ought to compose his picture in such a way that its horizon was easily discernible, because otherwise the viewers would not be able to fully appreciate it (ibid., 39). Perspectival shadows also received considerable attention in Der mathematische Maler - much more, in fact, than was usual in the pre-1800 literature.

626

Xl The German-Speaking Areas after 1600

It is lastly interesting to notice that Biirja, like Lambert, treated "military perspective", meaning a parallel projection, as a special case of perspective, namely one in which the eye point is a point at infinity (ibid., 88).

Kastner's Analytical Approach

A

braham Gotthelf Kastner (1719-1800) succeeded Segner as a professor of mathematics and physics in Gottingen in 1756, after working first as a Privatdozent and later an extraordinary professor in Leipzig. Like Wolff, Kastner is primarily remembered for his indefatigable efforts to make the world of mathematics and its applications known. This resulted in an impressive list of publications. Many of his ideas are collected in Mathematische Anfangsgrunde (Mathematical elements), which appeared for the first time in the 1750s and was reissued often. In the third revised edition of the volume, called Anfangsgrunde der Arithmetik, Geometrie... und Perspectiv (Elements of arithmetic, geometry ... and perspective), Kastner devoted some twenty pages to perspective (Kastner 1774). This presentation of the subject is a simplification of ideas contained in the booklet Perspectivae et projectionum theoria generalis ana/ytica (General analytical theory of perspective and projections, abbreviated as Perspevtivae theoria ana/ytica) from 1752. During his last years Kastner composed a Geschichte der Mathematik (History of mathematics, 1796-1800), in which he included the history of perspective until 1602 (Kastners 1797). We have seen that in the early seventeenth century Albrecht published an arithmetical approach to perspective, in which he provided tables for determining the length of line segments as an alternative to constructing them geometrically or obtaining them by use of scales on a sector. Kastner went a step further and treated perspective analytically. Although Perspectivae theoria ana/ytica is Kastner's oldest work, I first present his treatment of perspective in Anfangsgrunde, since a familiarity with the latter makes it easier to understand the contents of the former. Rodel interpreted Kastner as having the opinion that the best approach to perspective is to calculate what corresponds to the coordinates of the image point, as its geometrical construction involves a number of lines that have no function in the final composition (ROdel 1796, preface) - which is actually the same argument as Claude de Roy used (page 477). My impression is that Kastner was after something else. Thus, I do not think he was particularly interested in the actual performance of perspective constructions, but in demonstrating that some of the key results in perspective could be obtained by algebraic calculations - of which I will give some examples shortly. In Anfangsgrunde Kastner dealt with the usual situation of a horizontal reference plane and a vertical picture plane. His own diagram is shown in figure XI.18, and my redrawing of it in figure XI.l9. As the origin of his coordinate system (a phrase he did not use himself) he chose the ground point which is the point Q in figure XI.l9. Let A be a point in the ground plane and

5. Mathematical Works on Perspective

627

XI.18. Kastner's representation of the threedimensional configuration related to determining the image of a point. Kastner 1752/1774, table 11, figure 1. FIGURE

C a point vertically above it, and let the orthogonal projection of A upon the ground line be B. The points A and C are given by the lengths AB

=a,

BQ

=b,

AC

=c.

Furthermore, let A; be the image of A, S the orthogonal projection of A; upon the ground line, and C; the image of C. The positions of A; and C; are given by the lengths x QS, Y SA;, and z A;C; - the last of which Kastner called the perspective height. Looking at similar triangles, he found results corresponding to the ones Albrecht had used earlier for his arithmetical calculations (detailed in the caption of figure XI.7), namely

=

=

=

bd x = a + d'

y = a

and

(xi.l)

ah

+ d'

(xi.2)

cd

(xU)

z= a+d'

c c

p

z

OF-I-tT--fT----+--_~~A I Ai a

FIGURE

XI.19. Diagram illustrating Kastner's calculations.

628

XL The German-Speaking Areas after 1600

Based on the first two of these relations Kastner proved analytically that the points B, Ai' and P are collinear. This can be interpreted as the result that the orthogonal through B is depicted in the line BP, or in other words the convergence rule. Kastner continued by deducing a distance point construction analytically, and similarly, now involving (xi.3), by arguing for a geometrical construction of Cj' He also included a few examples in which he ascribed numbers to the variables and obtained results similar to Albrecht's arithmetical examples.

Kastner's General Theory n Perspectivae theoria analytica Kastner considered a general situation in I(xi.3)which the picture plane is not vertical, so instead of the formulae (xi. 1) he obtained equations that included a greater number of terms (Kastner 1752, IV). It is striking that in deducing the fundamental formulae for the coordinates, Kastner kept to classical geometrical means without involving, for instance, trigonometric expressions - Karsten later did. Thus, to deduce the generalization of (xi.2), Kastner considered no less than five pairs of similar triangles obtaining a number of proportions from which he then deduced his final relation by applying Euclid's theory of proportions as set out in book V of the Elements. In the remaining part of his booklet Kastner dealt with the problem of determining images of curves. He first considered curves in the ground plane. From the formulae (xi.l) and (xi.2) he deduced that yd xh a=-h-y and b=-h-' -y

(xiA)

Since a point in the ground plane is given by a and b, a curve in the ground plane is given by an equation between a and b. Kastner's idea was to use the curve equation together with the relations in (xi.4) to obtain an equation in x and y that would describe the image of the curve (Kastner 1752, V-VI). He dealt especially with circles, and came to the - unsurprising - conclusion that their images are conic sections. His formula enabled him to state the conditions under which the various conic sections are obtained. Pointing to a mistake by the "anonymous Frenchman", that is Dubreuil, he remarked that some earlier writers had overlooked the possibilities that the perspective image of a circle may be a hyperbola or parabola (ibid., VI).8 Second, Kastner assumed that the curve to be drawn in perspective lies in a plane that makes a given angle with the horizontal plane, and he similarly showed how the equation of the curve leads to an equation of its image. As an example he considered the images of great circles on a sphere under a 8This requires the picture plane to intersect the circle, cf. figures X,43-X,44.

5. Mathematical Works on Perspective

629

stereographic projection. Thus, as Commandino had done two hundred years before, Kastner linked the perspective and the stereographic projections. As to results, Kastner's booklet did not offer anything new, for many publications on general central projections already existed, and conic sections had previously been treated analytically by many others and projectively by writers such as Desargues, Philippe de La Hire, and John Hamilton (Desargues s 1639; La Hires 1673 and 1685; Hamilton 1738). Still, for a publication on perspective Kastner's approach was new, and it is quite impressive how much material he managed to cover in a mere twelve pages. The predecessors he referred to were Apollonius and Newton, and he claimed that a further elaboration of his method would yield many of the results contained in Apollonius's Conics and Newton's work on cubics (NewtonS 1704, cf. page 592). In the same year as Kastner's Perspectivae theoria analytica appeared, which was 1752, Lambert presumably wrote a manuscript on perspective. The contents of this manuscript will be presented in chapter XII (page 642), but let me already point out here that in this manuscript Lambert also took an analytical approach. Like Kastner, Lambert deduced formulae for the coordinates of the image of a point and used them to prove the correctness of a particular perspective construction. Though there are several similarities between Kastner's and Lambert's treatments of perspective, there are also differences, so it is extremely difficult to decide whether Lambert was acquainted with Kastner's booklet when he wrote his own manuscript. Using the theory of proportions to determine foreshortenings of line segments had long been a characteristic feature in German perspective literature, and it is therefore quite possible that Kastner and Lambert simultaneously but independently took an analytical approach to perspective. By the time Freye Perspektive, Lambert's chef-d'(£uvre, appeared in 1759, he had left the analytical approach, as we shall see in chapter XII.

Karsten's Mastodon

W

ith reference to Kastner, another German mathematician, Wenceslaus Johann Gustav Karsten (1732-1787) advocated treating perspective analytically (Karsten 1775, preface). In October 1773 Karsten wrote to Lambert that he was seeing his own treatment of perspective through the press to appear as a volume of his Lehrbegrif der gesamten Mathematik (A system9 of the entire mathematics). Having read that a second edition of Lambert's Freye Perspektive was going to appear, Karsten was interested in information on whether Lambert planned to make any essential changes. In particular, Karsten was keen to know what Lambert thought about his idea of applying analysis to perspective (LambertS Briefe, vol. 4, 322-323). In his reply Lambert expressed some reservations concerning Karsten's ideas. 91 am grateful to Christoph Scriba for suggesting possible translations of the word Lehrbegriff.

630

XI. The German-Speaking Areas after 1600

Karsten seems to have found Lambert's reaction disquieting, and he added some remarks to his book in defence of what he called "analytical perspective", asserting that this discipline was especially useful for astronomers and geographers (Karsten 1775, preface). When his treatment of perspective appeared, published as Die Perspectiv in the seventh volume of Lehrbegrif, Karsten was a professor of mathematics in Butzow and member of the learned societies in Munich, Haarlem, and Copenhagen. His membership of the latter was actually so new that he dedicated Die Perspectiv to the Royal Danish Academy of Sciences and Letters. The Lehrbegrif itself is an ambitious presentation of mathematics that appeared in eight volumes during the period 1767-1777. The year after the last volume was published Karsten became a professor of mathematics and physics in Halle. Karsten's Die Perspectiv contains more than eight hundred pages, but only some three hundred deal with perspective in the sense of the term as used in this book - the rest being about other forms of central projections and other types of projection. 1O In fact, Karsten's work contains a lot about conic sections, projections of such sections, and projections of spheres. To some extent the book is similar to Hamilton's Stereography, or a Compleat Body oj Perspective, but I do not think Karsten was familiar with this work. In a historical section of Die Perspectiv Karsten praised Guidobaldo highly (Karsten 1775, §45), and in the above-mentioned letter to Lambert he claimed that no real progress in the theory of perspective had been made between Guidobaldo and Lambert (LambertS BrieJe, vol. 4, 322). Although this was presumably written to please Lambert, it may also be a sign that Karsten was not familiar with the contents of 'sGravesande's and Taylor's books, despite the fact that he did refer to their titles in another, later, connection (Karsten 1775, §44). In addition to expressing his esteem for Guidobaldo and Lambert, Karsten was also quite enthusiastic about Desargues's work as presented by Bosse (ibid). Karsten's own work was inspired by precisely the perspectivists he praised, and to their ideas he added one of his own, namely to prove a substantial part of his results analytically. To some extent Kastner had done the same, but he had only dealt with fairly elementary problems, whereas Karsten treated many complicated problems. Die Perspectiv is actually full of formulae containing numerous trigonometric expressions that Karsten found "easy to take in" (LambertS BrieJe, vol. 4, 323). Despite Karsten's many calculations, his approach to perspective was highly influenced by Lambert and to a lesser degree by Guidobaldo and Desargues. For instance, Karsten included a section on direct or free constructions in the picture plane. In this section he combined the use of an angle scale - which he introduced trigonometrically - with Desargues's idea of using scales for foreshortening orthogonal and transversal lengths (Karsten 1775, §§28-36). lOPor another presentation of Karsten's work on perspective, see Lorias 1908, 614-616.

6. Traces of Lambert

631

Karsten also devoted considerable space to explaining how perspective constructions can be performed with the aid of perspective scales on a sector (ibid., §§37--42). What is more, he obviously found inspiration in Lambert's figures, as can be seen by comparing figures XI.20 and XII. 50. After about a hundred pages of rather traditional material, Karsten addressed special problems. Among many other topics, he took up the problem of what happens when an object drawn in perspective is seen from a point other than its eye point (ibid., §§136-l48). This theme, too, gave Karsten the opportunity to perform an abundance of calculations. In one case dealing with perspectival shadows he refrained from calculating, however, and followed the custom of performing geometrical constructions. He treated this subject in much greater detail than his German predecessors (ibid., §§149-175), and contrary to all the traditional presentations, Karsten also discussed setups involving more than one source of light (§160).

XI. 6 Traces of Lambert

A

lthough Lambert's contributions to the theory of perspective will not be treated until the next chapter, J find it relevant in this chapter to point out the eighteenth-century German-speaking authors who were inspired by Lambert's ideas. As we have seen, Karsten was very impressed by Lambert

.'

XI.20. Karsten partly recycling a figure by Lambert (figure XII.50). Karsten 1775, figure 68.

FIGURE

632

XI The German-Speaking Areas after 1600

and took up some of his ideas. Nevertheless, Karsten was no total devotee of Lambert, for he wanted to launch his own analytical perspective. The mathematician Biirja also acknowledged Lambert's ideas and incorporated some of them in Der mathematische Maler. In general, however, Biirja chose an approach that was different from Lambert's. The reason could be that Biirja wished to reach a larger audience and found that Lambert's theory of perspective geometry was too abstract for his potential readers. On the initiative of the publishing house Weigel and Schneider, the professor of mathematics and physics Johann Tobias Mayer (1752-1814) revised Schiibler and Rost's Mathematischer Lust und Nutzgarten from 1724 (Mayer 1786, iv). According to his own account, Mayer made particularly many changes in the sections on perspective, having new copper plates engraved and wanting to teach his readers the use of perspective scales (ibid., viii). Mayer found inspiration for many of his additions in Lambert's Freye Perspektive, in fact directly copying some of Lambert's figures, such as the ones concerning shadows reproduced as figures XII.23 and XII.26 - and as far as I have noticed without making any references. Outside the world of mathematicians, Lambert's work was appreciated by the aesthete Johann Georg Sulzer (1720-1799). Sulzer was born in Switzerland, studied a wide variety of subjects, and went to Berlin in the 1740s, working there for some time as a professor of mathematics at a Gymnasium (high school). In 1750 he was elected a member of the Royal Prussian Academy of Sciences and became director of its class of philosophy in 1775. He probably met Lambert often at the academy. In addition to a common passion for philosophy, the two scholars shared an interest in perspective. When Sulzer published his Allgemeine Theorie der schOnen Kiinste (General theory of the fine arts), he included a section on perspective based on Lambert's ideas (Sulzer 1787). Lambert had actually explained his ideas in a tract composed especially for Sulzer in 1771, of which Sulzer used a large part, and which was published posthumously (Lambert 1799, cf. page 640). Sulzer also treated the history of perspective in his book, listing some sixty works on the subject (Sulzer 1787, 566-568). Some of Lambert's ideas were also taken up by another member of the Berlin Academy of Sciences, Bernhard Friedrich Monnich (1741-1800). Like Lambert, Monnich became koniglicher Oberbaurat (- royal chief building surveyor) - after having been a professor of mathematics and physics. Monnich aimed to make the use of a perspective sector, an instrument designed by Lambert, understandable to practitioners of perspective. He made his initial attempt in a series of lectures held at the Akademie der Kiinste (Monnich 179411801, fol. A4r--4v). Apparently this encouraged him to publish Versuch die mathematischen Regeln der Perspektive filr den Kilnstler ohne Theorie anwendbar zu machen (Attempt to make the mathematical rules of perspective applicable for artists without involving theory, 1794). Showing great respect for Lambert, Monnich cautiously indicated that Lambert had

7. Perspective in the German Countries

633

FIGURE XI.21. Monnich illustrating the problem of scaling. Monnich 1794/1801, figure 1.

been too concise in demonstrating how the scales of his sector relate to the concrete problem of drawing a perspective picture (ibid., A3 v-4 r). Monnich, who claimed he had not engaged in profound science for a very long time, guided his readers carefully. He began by presenting the problem of scaling, which, as noted earlier, was generally ignored (page 329). Monnich chose a solution in which the entire configuration, including the distance, is downscaled (figure XI. 21). The rest of Monnich's book contains a thorough introduction to making direct constructions with the aid of the scales on Lambert's sector. And as Monnich had promised in the title of his book, he did not include any mathematical arguments.

XI.7 Perspective in the German Countries

A

s to the number of publications containing perspective, the Germanspeaking countries were in the lead until 1800. 11 This presumably reflects a genuine interest in perspective over a prolonged period. It took a long time, however, before a mathematical understanding of perspective was reflected in the German literature. In fact, the geometrical interpretation, introduced by Guidobaldo and followed up by Stevin in the beginning of the seventeenth century, did not make itself felt in the German literature until about one and llIn round figures the distribution of the publications included in my first bibliography is as follows: Germany 60, Italy 50, France 50, and Britain 40.

634

Xl The German-Speaking Areas after 1600

a half centuries later. This is, however, not that surprising if the general situation of mathematics in Germany is taken into consideration. Probably due in part to all the suffering the Thirty Years War caused throughout the German-language region, these countries did not participate in the growing innovation of and interest in pure mathematics that took place in Europe after 1630. Readers having Leibniz in mind may wonder how I can write like this. I can, because Leibniz's mathematical education did not take place in Germany, but in Paris, and after his return to Germany he did not influence the mathematical climate there. During the second half of the eighteenth century the German publications on perspective were, in general, on a level comparable to the literature in the other countries considered in this book. The fact that Lambert's ideas did not give rise to a new approach to perspective in Germany proved to be no different than what had happened elsewhere: in the British parallel, Taylor's innovations likewise went undeveloped in his own country and in other countries where his work was known.

Chapter XII Lambert

XII. 1 Lambert's Special Position

I

n the history of the mathematical theory of perspective, Lambert's work constitutes a chapter in itself, partly because of its excellent quality and partly because it is not an obvious part of any continuous development. Lambert put the final touches on thoughts that had been under way for one and a half centuries. Still, it is impossible to say where he got his inspiration. On the other hand, there is no evidence that his way of dealing with perspective had any influence on mathematicians who took up ideas comparable to his. Lambert's approach, as previously mentioned, involved looking directly at the geometry in the picture plane. Later in the eighteenth century, Gaspard Monge created descriptive geometry by applying a similar idea to a plane containing a plan and an elevation, but it is not very likely that Monge was inspired by Lambert. Similarly, some of Lambert's ideas pointed towards treating geometry projectively, but once again there are no indications that Lambert's work influenced the development that led to projective geometry. Let me remark that although I see similarities between Lambert's and Monge's ways of thinking, I do not suggest that Lambert's perspective geometry should be seen as a forerunner of Monge's descriptive geometry. Nor do I find that Lambert's perspective geometry, although later considered to be a part of projective geometry, should generally be interpreted as projective geometry. The instances of Lambert treating problems projectively occur in connection with his speculations about what he called Linealgeometrie (ruler geometry). This theory is presented at the end of the chapter, following sections that deal with Lambert's contributions to perspective proper. Chapter XIII contains a presentation of the basic idea behind Monge's descriptive geometry and a discussion of the similarities between this and Lambert's perspective geometry.

XII.2

T

Life and Work on Perspective

he Alsatian scientist and philosopher Johann Heinrich Lambert (1728-1777) received no more formal education than elementary school 635

636

XII. Lambert

instruction and a brief introduction to Latin and French, German being his mother tongue. He would nevertheless rise to one of the most attractive academic positions of his time: a salaried membership of the Royal Prussian Academy of Sciences in Berlin. He obtained this position in 1765 - after it had taken a year to convince Frederick the Great that this man of simple manners was a prominent scientist, who could also be consulted on practical matters (Scribas 1973, 597). In 1770, Lambert received the title of Koniglicher Oberbaurat (~ royal chief building surveyor). From his childhood, Lambert was devoted to reading. His first biographer perhaps Georg Christoph Lichtenberg l - told a moving story of how Lambert devised a way to buy candles, enabling him to read after sunset: While calming his little sister by rocking her cradle with his feet, he made sketches that he sold (Lambert 1943, 12-13). His later interest in perspective may have been awakened by his drawing activity - which he kept up (figure XII. I). He himself refused to have his portrait done (JaquelS 1979,44). Lambert's path to the academy in Berlin went via jobs as a clerk and as a tutor for the family von Salis for ten years - the last two of which he spent on a Bildungsreise with two of his students. Upon returning from this journey in 1758, he searched for a permanent academic position while taking part in various projects for brief periods of time in Germany and Switzerland. Lambert had a broad spectrum of interests, but devoted most of his study time to astronomy, mathematics, philosophy, and physics, which were also the main subjects of his publications (Scriba s 1973, 597-600). Although seemingly diverse, his interests are to some extent unified by his wish to use mathematics and measurement to understand a major part of human activities and all aspects of nature, which he saw as God's creation (Schreibers 1978, 1). An industrious writer, Lambert composed more than 150 manuscripts - some

XII. I. Drawing by Lambert. He might have planned to let this illustrate aerial perspective in his Freye Perspektive. Bibliothek der Universitat Basel L.r. a 736/Nr.7, 195.

FIGURE

lAccording to Max Steck in Lambert 1943,7.

2. Life and Work on Perspective

637

of which are substantial tomes. He combined his interests in theory with a concern for practice and was particularly fascinated with scientific instruments, constructing several himself. Before he died, he had collected no less than one hundred and six instruments (Steck in Lambert 1943,5). In the history of mathematics, Lambert is best remembered for making clever observations without creating a theory, examples being his proof that the number n is irrational, and his work on Euclid's fifth postulate, the so-called parallel postulate. In the latter he came close to the idea of a nonEuclidean geometry, but did not develop it. His contribution to perspective is thus different from the rest of his mathematical work, because as far as perspective is concerned, he provided a complete theory. A description of Lambert as a private teacher in the von Salis household reports that while spotting an error in one of his students' attempt to solve an algebraic problem, Lambert suddenly claimed: "This mistake leads me to a discovery"2 - this discovery was supposed to be related to a perspective instrument. The unknown author 3 of the report thought the incident was accidental. It is, however, unlikely that an algebraic problem would lead to the invention of a perspective instrument out of the blue, so I take the episode as an indication that at the time it occurred, Lambert was already taking an algebraic approach to perspective. He actually left a manuscript, called Anlage zur Perspektive (Essay on perspective), in which he treated perspective algebraically and also described an instrument for perspective constructions. Anlage zur Perspektive is usually dated to 1752 for the following reason. 4 Lambert kept a diary in which he entered brief remarks on what he had been working on from month to month, and which is therefore known as the Monatsbuch (Month book, LambertS 1916). The entry for August 1752 contains the term "Ichnogr." - ichnography - which Lambert seems to have used for perspective. 5 Since no other early Lambertian manuscripts on perspective are known, it seems safe to assume that Lambert was referring to Anlage zur Perspektive in August 1752. This was the same year the German mathematician Kastner published his booklet on perspective, which also contained algebraic formulae used to determine the images of line segments (page 629). Whether this publication had any influence on Lambert's work is, as mentioned, difficult to decide. I am inclined, though, to think that it did not. Lambert's original intentions with Anlage zur Perspektive are unclear. It is possible that he initially planned to publish it, but gave up the idea when he became so well-established that he could get his manuscripts printed - the

2Dieser [rrthum bringt mich au/ eine Entdeckung. [LambertS Brie/e, vol. 2, 13] 3The editor of Lambert's Brie/e, Johann (III) Bernoulli, assumed that the author was Andreas von Salis (LambertS Brie/e, vol. 2, 11). 4K.arl Bopp in LambertS 1916,34; Steck in Lambert 1943,42. 5Vitruvius, as noted in chapter II, used the word ichnographia for a plan (Vitruviuss Arch, book I, chapter 2, §2).

638

XII Lambert

reason being that in the meantime he had changed his ideas on how best to present perspective. The manuscript remained unpublished until 1943, when Max Steck included it in his comprehensive book on Lambert's work on perspective (Lambert 1943, 161-186). When Lambert the tutor and his students set off on their formative journey, they first went to Gottingen where Lambert met Kastner. Since both men had worked on perspective, it is likely that they also discussed the subject, and that Lambert was thereby inspired to return to it. In any case, while in Gottingen, Lambert mentioned perspective in his Monatsbuch, writing under November 1756: "Artis perspectivae problemata inversa" (inverse problems of the art of perspective). This must mean that he took up inverse problems of perspective. In February 1757, Lambert was once again so preoccupied with perspective that he made a note of it in his Monatsbuch. An important entry follows in September 1758, when he wrote "Massilae perspectivae fundamenta conieci" (in Marseille I put down the foundations of perspective). From that point up until the publication in 1759 of his main work on perspective, Lambert's Monatsbuch contains several remarks on the subject. In March 1759 Lambert stated "perspectivam germanice a gallico typis mandavi" (I put the [translation of my] perspective from French into German in the hands of the printer). It thus seems Lambert had a French manuscript that he wanted to publish in German. His wish to have the book printed in German is confirmed by a letter from November 1758 that he sent to Johann Gessner, a professor of mathematics in Zurich (LambertS Briefe, vol. 2, 175). Ultimately, however, a decision was made to print both a French and a German edition. It is unclear who was responsible for the German translation, but the fact that it was not literal6 indicates that it was someone who understood the text well- perhaps even Lambert himself. The two books 6The relative freedom of the German translation can be illustrated by quoting §139 in the two languages: L'ombre des corps, contribue beaucoup, a donner du relief au parties du tableau, ales distinguer d'une simple figure geometrique, & a faire paroitre les corps comme tels. C'est un art du peintre, que de savoir la distribuer a propos, & de lui donner les degres de force, qu'elle doit avoir. La Perspective ne se mele que de sa grandeur, & de sa direction, qu'elle enseigne a determiner. Les regles, qu'elle donne pour cet effet, n'ont point de difficulte, & pour les pratiquer if ne faut, que sravoir, de quelle part vient la lumiere. (The shadow

of bodies contributes much to give relief to the parts of the picture, to distinguish them from a simple geometrical figure, and to let the bodies appear as they are. This is a painter's art, to know how to distribute the shadow on purpose and to give it the degree of force it ought to have. Perspective is only involved in teaching how to determine the shadow's size and direction. The rules which perspective gives for that purpose are not difficult at all, and to apply them one need only know from which side the light comes.) Der Schatten die Korper von sich werfen, mufJ in den perspektivischen Aufrissen ebenfalls gezeichnet werden. Es giebt aber dabey keine Schwiirigkeit, so bald das Licht gezeichnet ist, von dem der Schatten herkomm. (The shadow cast by bodies must also be

delineated in the perspective drawings. As soon as the light, from whence the shadow originates, is drawn, there is, however, no difficulty involved.)

2. Life and Work on Perspective

639

were given the titles La perspective affranchie de l'embaras du plan geometral (Perspective freed from the nuisance of a geometrical plane, figure XII.2) and Die freye Perspektive, oder Anweisungjeden perspektivischen Aufriss von freyen Stiicken und ohne Grundriss zu verfertigen (Free perspective, or instruction in making any perspective drawing of one's own choice without a plan). I use Freye Perspektive as the short title. While his books were being typeset, Lambert stayed in close contact with the printers in Zurich, lamenting to Daniel Bernoulli in a letter from April 8, 1759 that for this reason he could not leave the city in the coming weeks (LambertS 1979,59). As an autodidact, Lambert, as he himself wrote in 1750, knew the difficulty of learning from books when there was no teacher to help answer the questions to which the books gave rise (LambertS Briefe, vol. 2, 10). Nevertheless, the author's personal experience did not induce him to write very pedagogical texts that explained all the details. On the contrary, he expected quite a lot from his readers. This is already evident in Freye Perspektive, and no less so in the text connected to his next project on perspective. The latter concerns an idea, similar to one occurring in seventeenth-century France, namely the idea of having special scales engraved on a sector, resulting in what Lambert called a perspective sector (perspektivischer Proportionalzirkel). He discussed his plan with Georg Friedrich Brander, a well-known instrument maker with whom he had collaborated since 1759. Brander was positive about making the instrument. He also recognized the need for a manual and asked Lambert to write one, stressing that it should contain no theoretical material, but many examples of applications that could be understood by painters and draughtsmen (LambertS Brieft, vol. 3, 22). Lambert sent his manuscript for a manual to Brander in May 1767, believing it would serve its function - provided the readers were not ganz dumm (completely stupid). Receiving a text quite different from what he had asked for, Brander had some practitioners tryout the draft manual and the scales, and he described his experience four months later in a letter to Lambert: ... the tangents and secants occurring in the treatise were all Greek to them, but after I have shown one and another an example of the use of the sector they now look upon the matter with quite different eyes acknowledging the advantage gained by it.?

The instrument was produced in 1768 (Lambert 1774, 104), and although Brander seems to have proved that Lambert's manual was difficult to understand without oral instruction (which a reading of the text confirms), he nevertheless arranged for its publication the same year. When the time came to republish Freye Perspektive, Brander sought to persuade Lambert to add a thorough description of the perspective sector and to illustrate its use with numerous examples (LambertS Briefe, vol. 3, 345), but Lambert did not comply.

7... die in der Abhandlung vorkommenden Tangenten und Secanten waren ihnen schon bOhmische Darfer; nachdem ich ein und dem anderen aber einen Casum mit dem Proportional Circul vorgezeigt, so sehen sie es nun mit ganz anderen Augen an und erkennen nunmehr die Vortheile die man hierdurch gewinnet. [LambertS Briefe, vol. 3, 38]

640

XII. Lambert

LA

PERSPECTIVE affranchic: de l'embaras du

PI an g eometral. Par

J. H. LAM BE RT.

Z U R Ie, CHEZ HEIDEGGUER M Dec L 1 X,

• ET

COMP

FIGURE XII.2. The title page of the French edition of Lambert's main work on perspective.

As mentioned in chapter XI, in 1771 Sulzer inspired Lambert to compose a special treatise on perspective (page 632). About three decades later, Johann (III) Bernoulli published this (figure XII.3) work posthumously (Lambert 1799). Meanwhile, in 1772 Lambert once again returned to perspective, not on his own initiative at first, but because of an incident he described in a letter written in May to an unidentified recipient: In the current fair catalogue I see that the gentlemen Orell, GeJ3ner and Compo will publish a new and even improved version of my freye Perspective. They have not informed me with one word about this. But if the edition is to be called improved I should at least be consulted, particularly because since that time I have collected much additional material. 8 8Aus dem dermalingen Meflcatalogo sehe ich, dafl die Herren Orell, Geflner and Camp. meine freye Perspective neu and zwar verbessert wollen auflegen lassen. Indessen haben sie mir kein Wort davon berichtet. Wennja die Auflage verbessert heissen solite, so sollte ich weningstens darum befragt werden zumal da ich seit dem noch vielen Stoff dazu gesammelt habe. [Lambert S Briefe, vol. 2, 53]

2. Life and Work on Perspective

r

641

,.,.,.

FIGURE XII.3. Illustration from Lambert's treatise written for Sulzer. Lambert 1943, opposite page 384.

Two days later Lambert wrote to the publishers that he did not see the need for an improved version of Freye Perspektive, but that he would like to include a number of additions, which, in his opinion, would be most appropriately placed at the end of the book. Out of consideration for those who already had the first edition, he asked for the additions to appear also in a separate volume (LambertS Briefe, vol. 2, 54). In November 1773 he sent a printed copy of his additions to Karsten (ibid., vol. 4, 324), and in 1774 the new editions appeared in the forms Lambert had requested (figure XII.4). In referring to the supplementary information Lambert provided I use his German term and call them Zusatze (additions). Besides adding new observations concerning perspective in Zusatze, Lambert also included a section on the history of the discipline. Based on Jean Etienne Montucla's Histoire des mathematiques (Montuclas 1758), Alexandre Saverien's Histoire des progres de l'esprit humain (Saverien s 1766) and Egnazio Danti's and Jean Fran90is Niceron's remarks on earlier perspectivists, Lambert listed a couple of dozen publications on perspective, but wrote very little about their content (Lambert 1774,15-31). He did, though, introduced his historical survey with a fairly thorough and interesting discussion of whether the ancient Greek sciences included perspective (ibid., 5-14; see also Andersen s 1987 2 , 86). Lambert himself does not seem to have collected works on perspective, for his Nachlass on this subject only contained his own publications and Brook Taylor's main work in the French translation from 1757 (Lambert 1943,48). In addition, Lambert also possessed a German edition from 1747 of the first printing of some of Leonardo da Vinci's ideas on painting (Leonardo 1724).

642

XII Lambert

~. ~.

~ Ct

2 amb er t 8

fp t cti )J t, ober

9nfVfifung, jeben

"trfj)ertl~lr
etUcttD un~

91llfrt8

obne e}runbti8

\lOll

frt\Jfn

IU tlnfcrti!lttl.

~uttd), ~(9

DrcU, ~cgnfr I ~ucf.fi\l «nb qom"n'lnft.

XII.3

I

I "....

FIGURE XIJ.4. The title page of the second part of the enlarged 1774 edition of Freye Perspektive.

Early Approach to Perspective

n Anlage zur Perspektive, which, as noted, most likely dates from 1752, Lambert began by taking a rather conventional approach to perspective, which involved basing the construction of a figure on its plan. Less conventionally, he decided to apply algebraic formulae in his proofs. To explain his idea it is sufficient to consider one point in the ground plane. Let this be the point A in figure XII.5, in which most of the other letters have their usual significations, meaning 0 is the eye point, F the foot, P the principal vanishing point, and Q the ground point; furthermore S is the point of intersection of FA and GR. Like Kastner in his 1752 treatise, Lambert - despite applying a different terminology - aimed to determine the coordinates x = QS and y = SA; of the image Ai of A. By considering similar triangles he found algebraic expressions for x and y in terms of quantities that are given when the positions of the

3. Early Approach to Perspective

643

FIGURE XII.5. Lambert determining the image of a point. Adaptation of Lambert's drawing reproduced in Lambert 1943, opposite page 161.

point A and the eye point are given. He then used the formulae to argue the correctness of a particular construction (Lambert 1752, §§10--16). This was actually Guidobaldo's twenty-first method, a construction both Guidobaldo himself and Marolois counted among their favourites (pages 256 and 301). It is not possible to decide whether Lambert had seen this construction in Guidobaldo's or Marolois's work or whether he reached it independently. In any case, it seems he was searching for a construction that could form the foundation of a perspective instrument. Before presenting this instrument, let me repeat the description of Guidobaldo's twenty-first metnod (figure XII.6). Let A be a point in the ground plane, F the foot, O· the point on the line through F parallel to GR determined by FO' =FO =h, and T the point of intersection of AO' and GR. The image A; of A is determined by making SA; equal to ST on the orthogonal to GR through S (for a proof of this, see the caption of figure VI.I3). Probably having a distance point construction in mind, Lambert remarked that this configuration is unusual in the sense that it works with a point F whose distance to the ground line is not h but the distance, d, and similarly that FO' is not d, but h (Lambert 1752, §38). For a given hand d, the points F and O· are fixed in relation to the line GR; the problem is then to construct the points Sand T for various As and to transfer the length ST to a perpendicular through S. Lambert designed his instrument to solve this problem (figure XII.7) and gave quite a detailed description of how it should be built. The theory of the instrument is beautiful, but I doubt it would have worked well in practice, as the mechanical parts of Lambert's devices were not fit for a smooth movement. 9 Lambert 9This information I have from Peter Schreiber, who has reconstructed Lambert's instrument.

644

XII Lambert

..-

---, o·

R

A

FIGURE XII.6. Lambert's procedure for constructing the image of a point. This figure is similar to figure VI.13, but rotated 180· in accordance with Lambert's own drawings.

remained interested in the construction for which the instrument had been built, returning to it in his Zusiitze (Lambert 1774, 37-38). In Anlage zur Perspektive, Lambert proceeded with other algebraic considerations - and I deliberately use the term algebraic rather than analytical because Lambert himself preferred the former expression, or at least he had come to prefer it in 1773 when he exchanged letters with Karsten about their work on perspective. Karsten, as mentioned in chapter XI, was particularly interested in learning what Lambert thought of treating perspective analytically. In his reply dated November 6, Lambert stated that geometrical constructions were much more convenient for practising perspective than algebraic formulae, continuing: Besides, a proper analytical part of perspective can very well be imagined. This consists, however, of something quite different from algebraic expressions for the synthetic basic rules. 10

In section XII.l2 I return to what was on Lambert's mind. Lambert's basic calculations in Anlage zur Perspektive concerned the problem of determining the foreshortened length of a given orthogonal line segment that has one end point on the ground line - a problem that had been dealt with many times in the literature on perspective, from Piero della Francesca's De prospectiva pingendi onwards. In 1738 John Hamilton had introduced a hyperbola in connection with this problem (page 543). Lambert got the same idea - independently of Hamilton, I assume. To be more explicit, let me recapitulate some earlier considerations, referencing the elevation shown in figure XII.8. Here QA is an orthogonal line segment situated 10Es liisst sich ilbrigens auch ganz wahl ein eigentlich analytischer Theil der Perspective gedenken. Dieser bestehet aber in ganz was anderm, als in algebraischen Ausdrilcken der synthetischen Grundregeln. [LambertS Briefe, vol. 4, 325]

3. Early Approach to Perspective

a~---

z

G

~:;:;:::,..-;;;;:;-.

_~~_

645

®J

d~

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,.

,

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L

------.-J

.7L:.=================~~ FIGURE XII.? Lambert's drawing of his perspective instrument with some letters changed. The instrument has two rulers that can rotate around the points F and 0', respectively, and which are turned until they meet at the point A, the point to be thrown into perspective. These rulers cut the ground line at the points Sand T, respectively. The distance ST is transformed into an orthogonal distance with the help of a device consisting of the three moving rulers b, P, and x, which are organized so that P and x are perpendicular to each other and b makes an angle of 4Y with them. Lambert 1943, opposite page 176 with some letters altered and others enlarged.

in the vertical plane; the end point Q lies in Jr, and QA has the length a. The problem is to determine the foreshortening of a,f(a), which is the length of QA i, Ai being the image of A, as usual. Lambert found it more convenient to consider another length, however, namely the one I have previously introduced as g(a). This is the length of PA i which for OF = h fulfils the relation (xii. 1)

g(a) = h - I(a). O~_---=d=-- _ _----,P

z = gray h

FIGURE

z = g(a).

Ai

XII.S. Elevation showing F

Q

a

A

646

XII Lambert

Lambert denoted this length by z and called it the Erniedrung (descent, Lambert 1752, §40) - obviously thinking of how far below P the point Ai lies. A consideration of similar triangles led him to the result (cf. (ix.4) page 429)

= d dh + a'

(xii.2) in which d is the distance. 11 The points (x, y) = (a, g(a)), which for the given d and h satisfy equation (xii.2) in a rectangular coordinate system, lie on a hyperbola. Lambert first described a procedure for constructing a number of points on this hyperbola. Subsequently, assuming the complete hyperbola to be known, he showed how it can be used to solve various perspective problems. One of his examples is the following (figure XII.9): Given are the ground line GR, the principal vanishing point P, the distance d, and the image of a transversal passing through the point Ml' Construct the points M 2 , M 3 , etc. so that the horizontal lines through the points M n are images of equidistant transversals. His solution is extremely elegant, allowing the hyperbola - drawn for a particular choice of hand d - to also be used for any other choice of hand d (Lambert 1752, §46). In his Freye Perspektive, Lambert briefly returned to the hyperbola (Lambert 1759, §115). As an alternative to determining g(a) defined by (xii.2) with the aid of a hyperbola, Lambert offered a solution involving a sector with two identical reciprocal scales. In Freye Perspektive he elaborated this idea of using a sector. Here he suggested having five scales listing g(a)

d

~a

for d

=2,4, ..., 10 and a suitable number of as,

p

FIGURE

G

Q

XII.9. Lambert's problem concern-

Ring images of equidistant transversals.

liThe advantage of using g(a) can be seen by comparing formula (xii.2) with formula (ii.l), which determines f(a)=QAj' In the former the variable a only occurs once, whereas it occurs twice in the latter.

4. The Contents of Freye Perspektive

-

~"'_"'_-

i»

647 F

"".'

~

.,

••

•

..

.•

t

FIGURE XlI.lO. Lambert's "perspective lines". He constructed these lines as reciprocal scales that function as follows. He marked the outer lines with a 10 at the ends and meant for these lines to be used when the distance, d, is 10 - in the unit where the length OF corresponds to the height h of the eye. Similarly, he meant for the other scales to be used when d is equal to 8, 6,4, or 2. On a pair of corresponding scales, Lambert put the number n at the point A, whose distance from the origin 0 is determined by OA : OF d: n. For h OF and a defined by d + a = n, relation (xii.2) shows that

=

OA : h

= OA

: OF = d: (d + a)

=

= g(a) : h.

In agreement with this, Lambert concluded that the line segment OA is equal to g(a). Thus, for d = 10 and a = 5 the required g(a) is found as the length on the outer scale from 0 to the point marked 15. If d is kept constant, but h changed to h', Lambert applied the sector in the usual way for constructing a fourth proportional to three given line segments. When the g(a),s corresponding to hand h' are denoted by g,,(a) and gia), respectively, it follows from relation (xii.2) that g",(a) : h' = gila) : h. Since OF = h, Lambert's idea was to open the sector so the distance between the end points of the pair of scales, used for d, is h', and finally to determine g",(a) as the distance between the points on the scales that gave gla). Lambert 1768, half of figure 1 with letters added (the other half is reproduced in figure XII.47).

engraved on the instrument. The sector could then be used to determine the length g(a) (Lambert 1759, §§116-122). These scales were actually engraved on one side of Lambert's perspective sector fabricated, as noted, by Brander (figure XII. 10). The scales on the other side of the instrument relate to some of the ideas Lambert presented in Freye Perspektive - to which I return in section XII.ll.

XIIA

L

The Contents of Freye Perspektive

ambert divided his Freye Perspektive into eight chapters, subdivided into three hundred and fifteen numbered paragraphs. In a free translation (based on the French text), the chapter titles run as follows: I. On the foundation of perspective and the universal laws for throwing plane figures and solids standing upon them into perspective 2. On the convenient choices of the position of the eye point and its distance to the picture plane

648

XII. Lambert

3. 4. 5. 6.

On various instruments that can shorten the practice of perspective The rules applied to examples with many details On perspective projections of oblique planes and objects in them Comments upon the picture plane and examples to illustrate what was taught about oblique planes 7. On orthographic projections using a point at infinity 8. Inverse problems of perspective

Lambert did not tell for whom he had written his work apart from mentioning that he did not expect his readers to know much about perspective (Lambert 1759" preface, 4f ; 17592, Vorrede, 5V ). One gets the impression, however, that he hoped to reach people who were, or were going to be, actively performing perspective constructions, and that he therefore chose a style different from the one normally used by mathematicians. Thus, Freye Perspektive does not contain traditional theorems, but statements, which rather than being introduced as general results, are related to particular examples. (Following my usual strategy of presentation, I generalize the Lambertian observations.) More often than not Lambert argued to substantiate his results, but avoided formal mathematical proofs - as exemplified in the caption of figure XII. 11. He also refrained from long arguments and left statements that would require such arguments unproved. While writing his Zusiitze, his passion for mathematics took over, so in this work he did actually add several of the missing proofs. In Freye Perspektive, twenty six of Lambert's results are solutions to construction problems. Although Lambert covered a complex material, he did not find it necessary to use many diagrams. The entire book contains only thirty two figures, of which as few as nine are illustrations of perspective compositions. In presenting Lambert's Freye Perspektive - including his Zusiitze - I first concentrate on the author's results concerning perspective geometry. I then show how he linked parallel projections to perspective, and how he treated inverse problems of perspective before finally discussing his examples. An appreciation of Lambert's treatment of perspective geometry requires some familiarity with his insights, and the following sections therefore involve quite a number of technicalities. Before moving on to these, however, I will briefly discuss his sources of inspiration.

Lambert's Possible Sources Lambert described his achievement in Freye Perspektive in the following way. I have no desire to take credit for having discovered theorems to which others have a more just claim. Yet even if all the propositions this book contains happen to be known, they are only to be found dispersed in manifold tracts ... 12 12Je n'ambitionnerai pas l'honneur d'avoir decouvert des propositions, que d'autres pourront s'attribuer a plus juste titre. Encore que toutes celles, que cet ouvrage renferme, eussent ete connues, on ne les trouveroit que dispersees en plusieurs traites ... [Lambert

17591' preface, 5-6]

4. The Contents of Freye Perspektive

649

°r:::::::::::::::-t---+-_-=::::::::::::,..v G

F

FIGURE XII.11. Lambert's argument for the main theorem (Lambert 1759, §§15-17). Let rbe a ground plane, ,. a vertical picture plane, GR the ground line, 0 the eye point, F its foot, P the principal vanishing point, and Q the ground point. Lambert's main argument concerns a line 1that lies in rand passes through Q. First he drew the lines FB and OV parallel to 1 and cutting,. in the points B and V, respectively. To determine the image of I, he focussed upon a point A on 1 and on its image Ai" He let the line FA cut GR at C and first argued that Ai lies on the vertical line through C (the line FC is the orthogonal projection of the OA upon 11. Then he imagined that A moves away from Q and noticed that the further away A is, the greater the angle AFQ becomes, the nearer Cmoves to B, the greater the angle AOFbecomes, and the higher Ai is over GR. He claimed that this process continues until A is infinitely far away from Q, and that in this situation FC is parallel to 1 (hence coinciding with FB) and OA is horizontal (coinciding with OV). Based on this he concluded that the perspective projection of the point A at infinity is the point V, and that the image of 1 is QV. From this special version of the main theorem he deduced the theorem for all lines parallel to 1 and hence for all lines that are not parallel to ,. - as the direction of 1was arbitrary. Adaptation of a section of figure I in Lambert 1759.

He was right, but also far too modest. As previously explained, the germs of Lambert's approach and many of his results can be found in the literature preceding Freye Per,spective. Still, Lambert did more than collect material from various sources; he unified earlier results and was the first to have the clearly stated aim of looking directly at constructions in the picture plane Jr, and to conceive of the latter as having its own geometry - which he termed perspective geometry (Lambert 1759, §30).13 He adapted his language to this view, for instance calling two lines in Jr parallel when they are images of parallellines - and in cases where confusion could arise, he used the expression

13The term had previously been used by Hamilton. The works of the two perspectivists also resemble each other in the introduction of a foreshortening hyperbola, but I ascribe these two circumstances more to the authors getting similar ideas than to Lambert being familiar with Hamilton's ideas (page 547).

650

XII Lambert

perspectively parallel. This vocabulary I have taken over. Similarly, Lambert

considered the properties of being perpendicular, equal to, and so forth in n. As I have done earlier, I sometimes use the notation = to signify perspectively equal, and the symbol a =j b to express that a length o~ an angle a is the image of a length or an angle b. In placing Lambert's achievements within the history of perspective, it would be particularly relevant to know whether he was inspired by Taylor. It is certain that at some point in his life Lambert became acquainted with Taylor's New Principles. As mentioned previously, Lambert actually had a copy of the French edition from 1757 in his personal library (page 641). However, we do not and probably never will know whether he was already familiar with the contents of Taylor's work on perspective when he composed Freye Perspective. In Zusiitze (1774), Lambert mentioned Taylor's work - but not exactly favourably: Taylor treats the theory [of perspective] very generally, because from the beginning he assumes tnat the picture plane is oblique. In addition, he uses mainly new and unnecessarily numerous terms, which - although they give him more theorems - make the theory unnecessarily long-winded. 14

Notwithstanding this reservation, which Lambert presumably wrote partly with Taylor's theory concerning the directing plane in mind, the possibility exists that he had noticed the interesting approach of making direct constructions in Taylor's "long-winded" theory before planning his own book in September 1758. However, it is just as likely, or - I think, though without having conclusive arguments - even more likely, that Lambert first saw Taylor's work after he had created perspective geometry. I later present some observations supporting this point of view.

XII.5

Constructing Polygons in the Picture Plane

I

n the first part of Freye Perspektive, Lambert assumed that the picture plane n is vertical and addressed the problem of how to construct images of horizontal polygons. His programme was to copy the procedure applied in the Euclidean plane for constructing a polygon in which the sides and angles (or rather a sufficient number of them) are given (Lambert 1759, §31). For this programme he needed a number of standard constructions in n. These he presented after first providing n with an angle scale - which he called Ie transporteur perspectif in French and der Winkelmesser in German (Lambert 1759, §32). 14Taylor handelt die Theorie sehr allgemein abo weil er die Tafel gleich anfangs als schiefliegend annimmt. Ueber dieses hat er meistens neue und iiberfliissig viele Benennungen, die ihm zwar mehrere Lehrsiitze geben, dabey aber die Theorie ohne Nothwendigkeit weitliiuftiger machen. [Lambert 1774, 29]

5. Constructing Polygons in the Picture Plane

651

Lambert's angle scale is similar to the one introduced by Aleaume and Migon (page 422) and later described by several other French authors. Thus, on the horizon Lambert marked a point with the number (/) when this point is the vanishing point of horizontal lines that form an angle of (/) with the normals to 1r. According to Lambert himself, he had conceived the idea leading to the angle scale in the summer of 1758 (Lambert 1774, 29). However, later he realized that he had not been the first to introduce this concept, but had been preceded by Lacaille, who had published an angle scale in 1756. Apparently Lambert was not aware of the fact that an angle scale had been presented not only two years, but more than a hundred years before he created his own version. This is in accordance with his survey of perspective literature, which does not include his French seventeenth-century predecessors. Several of Lambert's examples of how to perform constructions directly in the picture plane 1r are similar to problems treated by earlier perspectivists. Lambert, like them, took up two fundamental constructions for dealing with polygons, namely the construction of line segments and that of angles. More precisely, he explained how the following problems should be solved in 1r (I have renumbered his problems).

Problem 1. Through a given point on a given line, draw a line making a given angle with the given line (Lambert 1759, §33). Problem 2. Cut off a line segment of a given length on a given line from a given point (ibid., §52). Because these constructions are so crucial in Freye Perspektive, I have chosen to present Lambert's solutions in some detail. In problem 1 (figure XII. 12), let A be the given point on the given line I cutting the horizon HZ in the point B at which the angle scale is marked with a (/). Let it be required to draw a line m through A so that it perspectively forms the angle (J with I. There are various cases to take into consideration: B can be situated to the right or the left of the principal vanishing point P (or coincide with it); the angle (Jcan be smaller or larger than (/) (or equal to it); and m can be required to be a left or a right leg

H

FIGURE XII.12. Lambert's construction of angles in Jr.

G

p

z

R

652

XII. Lambert

of the angle. In his example Lambert assumed that 0 < if> and that the line m should be a left leg, and for this case his instruction was the following. Find the point C on HZ reached by counting 0 degrees to the left of Band draw the line A C which is the required line. In the situation illustrated in figure XII. 12, the point C lies at the same side of P as B and is marked if> - o. Lambert's instruction implies that if 0 > if>, we should search for the point to the left of P that is marked 0 - if>. Similarly, if B is situated on the opposite side of P, we should use the point 0 + if> when 0+ if> < 90' or otherwise construct m as the right leg of an angle forming the supplementary angle 180' - 0 with I (Lambert 1759, §34). Finally when m has to be the right leg of the angle the counting should be to the right in a way analogous to the procedure explained for counting to the left. The construction just presented was among those for which Lambert gave no proof. It is a consequence of the definition of the angle scale and can, for instance, be obtained by means of the following fairly straightforward result.

Observation 1 (figure XII. B). Let if> be a mark on the angle scale to the right of the principal vanishing point P, and VI a mark to the left. For any line I that has the point marked if> as its vanishing point and intersects the ground line GR at 1[, the angle I makes with the half-line I[R is 90' -if>. Analogously, for a line m having the point marked VIas its vanishing point and intersecting GR at 1m' the angle between m and ImR is 90'+VI. If we apply this observation to the situation drawn in figure XII.l2 and (as shown in figure XII.l4) prolong AB and AC to cut GR at E and D, we find that LADR =j 90' -if> + 0 and LAED =j 90' -if>; hence LBAC = pLEAD =j O.

F FIGURE

XII.B. Illustration for observation 1.

5. Constructing Polygons in the Picture Plane

653

z

R FIGURE

XII.l4. Calculating the angles in the triangle EAD.

For all the other possible cases, mentioned above, similar calculations show the correctness of Lambert's construction. Problem 2 was taken up by Lambert several times, and he gave various solutions, most of which involved a measure point. 15 The first time he applied this point he gave it no special name (Lambert 1759, §52), but later in his main work he often referred to it as Ie point de division in French and Teilungspunkt in German (for instance ibid., §138). His initial introduction of a measure point was based on the angle scale on the horizon HZ (figure XILI5) Thus, to lines having direction cf> - that is, lines with the point V marked cf> as their vanishing point - Lambert assigned as measure point the point M, which lies on HZ on the opposite side of P than V does, and which is marked 45° - 112cf>. Lambert applied the measure point to determine lengths on images of horizontallines. Let AVbe a given line, and let it be required to construct the point B on A V so that AB is the image of a given length a. Lambert solved this problem as follows. He found the measure point M belonging to AV, drew MA cutting the ground line GR at S, marked the point T on GR determined by ST = a, and finally constructed the required point B as the point of intersection of MT and A V. In Zusiitze Lambert gave a proof for the correctness of this construction (Lambert 1774, 64). I follow the idea of his proof, but have changed it slightly using observation 1 presented above. When the line VA cuts the ground line GR at the point C, it follows from the observation that LACS = i 90° - cf> and LTSA =j 90° + 45° - 112cf> implying that LASC =j 45" + 112cf>.

15Among the places where Lambert treated the problem of constructing lengths are Freye Perspektive §§52, 110, 132, 135, 138, and 299 and Zusiitze 64, 103, and 106.

654

XII Lambert

v

HM

T

G FIGURE

z

R

XII.15. Lambert's introduction of a measure point.

Hence also LCAS =i 45° + 1/21/>, which means that the triangle SCA is perspectively isosceles and CA = CS. Similarly CB = CT, and hence AB =pST =a. The procedure of co~mting on the angle scale to construct angles and lengths brought a sort of unity to Lambert's theory. Often, however, he preferred to construct a measure point without using an angle scale and applied a construction similar to Ozanam's. As previously outlined in connection with relation (ix.I), this construction provides the measure point M for horizontal lines with the vanishing point Vas the point that lies to the opposite side of P than V and fulfils MV=

av,

(xii.3)

o being the eye point. Lambert gave both a geometrical and a trigonometri-

cal argument to show that this construction yields the same measure point as the earlier one (Lambert 1759, §135). Lambert's basic constructions also included that of drawing, through a given point, a perspective parallel to a given line (ibid., §29) - which several of his predecessors also considered (pages 261, 423, and 511). At the end of the nineteenth century the German mathematician David Hilbert investigated the foundation of geometry and the fundamental construction problems in plane geometry. His thorough investigation resulted in a number of constructions that include the three taken up by Lambert, meaning the one just mentioned and the two presented above as problems 1 and 2 (HilbertS 1899/1909, §36 problems 2-4). Having got the basic constructions in Tr under control, Lambert demonstrated how they can be applied to a number of problems concerning direct constructions of polygons in Tr. Like several of his predecessors he restricted his investigation of curves to circles and addressed the problem of constructing

6. Oblique Figures

655

z

H

E

A G

R

XII.16. Lambert's construction of points on a perspective circle. It is given that the line segment AB in tr is a chord subtending - under it - the angle 24> in a perspective circle. It is required to construct additional points on the circle (Lambert 1759, §40). Lambert applied the fact that for any point C situated on the circle above AB we have LACB =j 4>. This gave rise to his procedure of drawing an arbitrary line AD, using the angle scale to find the perspective size, say lfI, of LDAB, and using the scale once more to construct LEBF =j 4> + lfI. Lambert then found a point C on the circle as the point of intersection of AD and BF. Repeating this procedure he could determine as many points as he wished. FIGURE

a suitable number of points on a circle. His extremely elegant solution is outlined in figure XII.16. Lambert finished the introductory part of his theory by showing how perspectively horizontal figures in n can be made three-dimensional by adding heights. Since the Renaissance, images of vertical line segments had been constructed directly in the picture plane - although this fact had gone unnoticed. Lambert chose a classical construction (Lambert 1759, §§56-58) similar to one of Piero della Francesca's (figure 11.26). At the very end of his treatment of plane figures Lambert remarked that it was not essential for them to be situated in a horizontal plane or for the picture plane n to be vertical, because he had only made use of the fact that the ground plane was perpendicular to n. Thus, his procedures could be applied in any plane orthogonal to a picture plane of a given direction (ibid., §59).

XII.6

T

Oblique Figures

he material Lambert presented in the first chapter of Freye Perspektive is sufficient to solve ordinary perspective problems and would more than satisfy the needs of most of those performing perspective constructions. However, Lambert also wanted to demonstrate what his further investigations had led to, and he was particularly keen to show how an arbitrary

656

XIL Lambert

polyhedron can be constructed directly in a picture plane fr. He treated this problem in two steps. The first dealt with the problem of constructing the image of a polygon situated in what he called an oblique plane a, meaning that it is neither perpendicular nor parallel to fr - and where fr may have any direction. His second step was to investigate how the image of a polyhedron with one face in a could be built up directly in fr. Taylor had also worked with direct constructions in oblique planes, but he had assumed that the image of one line segment in the figure was known (page 512). Lambert started from scratch, and his idea was to generalize the procedures he had used for planes perpendicular to fr. He first generalized the concepts of a ground line and a horizon, reaching the concepts Taylor had called the intersection and the vanishing line, and which I denote by i a and va (figure XII.17). Lambert does not seem to have cared much about naming these lines. Thus, he gave ia the French name fa ligne d'intersection, but gave it no name in German,16 and he called the vanishing line die Grenzlinie in German and !'horison in French (Lambert 1759, §§165-166). Other authors would reserve the latter term for the vanishing line of a horizontal plane. Lambert also introduced the orthogonal projection, Pa' of 0 upon va and named this fe point de !'amil in French and der Augenpunkt in German (ibid., §162). These were expressions he had used earlier for the principal vanishing

FIGURE XI!.l? Lambert's introduction of perspective concepts belonging to oblique planes. 16Lambert did suggest that one might borrow the term die Knotenlinie (in French fa ligne des noeuds) from astronomy, but he never used it.

6. Oblique Figures

657

point, which he renamed Ie point de ['ami! principal and der Hauptaugenpunkt (ibid.). In working with his new concepts, Lambert stressed that (ibid., §168) PPais perpendicular to the vanishing line va.

(xii.4)

Taylor had also noticed and proved this (result 1, page 506). Lambert's further determination of the point P a involves the angle 8 between the plane a and 1C be (8 being different from O· and 90· and the distance 0 P =d and led to the relations (xii. 5) and

(xii.6) PPa = d cot 8. He used these relations to discuss what must be given in order that a plane a can be uniquely determined in 1C (ibid., §176). Having characterized the position of P a' Lambert looked at the vanishing point of lines in a, such as I (figure XII. 18), that form the angle 4> with the normals to ia . He concluded that the vanishing point VI of I is the point on va determined by PaVI = OPa tan 4> (xii.7) Based on this relation Lambert equipped the vanishing line va with an angle scale similar to the one he had constructed on the horizon. To determine

FIGURE XlI.18.

How to detennine vanishing points on the vanishing line va of the plane a.

658

XII Lambert

lengths on I in 1r, Lambert introduced a measure point, M r He did this by generalizing a method he had used earlier (relation (xii.3», namely that of letting M J be determined by (xii.8) The correctness of this construction follows from an argument similar to the one given in connection with relation (ix.2). Lambert had now supplied the plane a with the means necessary for performing direct constructions of polygons, situated in a, analogous to directly constructing images of horizontal figures in a vertical1r. The next step was to make constructions that involve elements outside a, and to this end Lambert considered the three following problems. Problem 3. Determine in 1r the direction of normals to a given plane a

(Lambert 1759, §188). Problem 4. Given a plane a and a line m in it, determine in 1r the plane 1} that contains m and is perpendicular to a (ibid., §191). Problem 5. Given a plane a and a line I in it, determine in 1r the plane f3 that contains I and forms a given angle ¢> with a (ibid., §195).

When Taylor analysed what he needed for three-dimensional direct constructions he also formulated problem 3 and problem 5 (the non-perspectival versions of which are listed as problems 4' and 7', page 516). He did not treat problem 4, which is a special case of problem 5, but did include a problem concerning the construction of a plane perpendicular to a given line. The solution to problem 3 is fairly obvious, so it is hardly surprising that Lambert's solution is similar to Taylor's (presented page 516). Lambert's solution to problem 4 is outlined in figure XII.l9. Rather than presenting Lambert's solution to problem 5 in detail, I compare it with Taylor's solution, thereby finding an indication that Lambert had not read Taylor when he composed his Freye Perspektive in 1758.

Comparing Some of Taylor's and Lambert's Ideas

T

aylor performed a construction in 1rcorresponding, as we saw on page 518 to the following procedure in the three-dimensional space (figure XII.20). Let 1} be a normal plane to I, m the line of intersection of a and 1}, and n a line in 1} that forms the given angle ¢> with m. Taylor then determined f3 as the plane defined by the lines I and n. Lambert's steps in 1r, on the other hand, correspond to the following operation in space (figure XII.21). First he chose a line m that lies in a and is perpendicular to I, then used the result of problem 4 to determine the plane through· m that is perpendicular to a. This plane matches Taylor's plane 1}, and Lambert also used it to finish the construction. Taylor's and Lambert's constructions do not differ exceedingly, and yet, Taylor's approach, which includes the general problem of finding planes that are perspectively normals to a given line, is considerably more

6. Oblique Figures

659

FIGURE XII.19. Lambert's solution to problem 4. The plane ais given in nby its vanishing line va' its intersection ia, and its direction angle e. The point P is the principal vanishing point, Pa its orthogonal projection upon va' and 0 a the eye point turned into n around the line PP a' In a is given the line m, characterized by its intersection 1m and its vanishing point Vm • It is required to characterize the plane 1] that is perpendicular to a and contains m. To obtain the vanishing line v1) of 1] Lambert used the fact that both Vm and the vanishing point N a of lines perpendicular to a lie on it. He thus constructed the latter point with the help of the solution to problem 3, and then joined Vm and N a . Next he determined the intersection i1) of 1] as the line through 1m parallel to v1)' To construct the angle ljI, which the plane 1] forms with n, he first constructed the centre P 1) of 1] as the orthogonal projection of P upon v1) (cf. (xiiA)), and the point 01) on the perpendicular to PP1) through P determined by P01) =POa' He then obtained ljIas the angle PP1)01)' Finally, Lambert constructed an angle scale on v1) by applying relation (xii.7).

p

FIGURE XII.20. Taylor's solution to problem 5.

660

XII Lambert

a FIGURE XII.21. Diagram to Lambert's solution to problem 5.

refined than Lambert's solution of introducing problem 4. The latter only served as a means to obtain the solution to the more general problem 5. This circumstance tempts me to conclude that Lambert was not familiar with Taylor's work in 1758, because I would take Lambert to be so much the mathematician that he would want to present the most elegant solution he could think of. Turning for a moment to the problem of reversing (section VII.6), I find another argument supporting the view that Lambert wrote Freye Perspektive independently of Taylor - and of 'sGravesande. 17 The two last-mentioned mathematicians had pointed to the fact that their methods of construction implied a reversed image (pages 331 and 511). Lambert's method does not similarly lead to a reversing, and it is therefore very likely that had he read Taylor or 'sGravesande, he would have stressed that his approach had the advantage of avoiding this problem.

The Applicability of the Theory of Oblique Planes

H

aving presented his general solution to problem 5, Lambert dealt with some special cases in which the solution may be found in a simpler way. He also treated the example of constructing a plane f3 parallel to a given plane a (Lambert 1759, §200). Finally he looked at the problem of constructing a plane passing through the eye point, and treated six versions of this problem (ibid., §§204-209). Each of his six solutions is a line whose determination seems to be more of theoretical than practical interest. Lambert, however, claimed that the solution was often useful, specifically for the construction of cylindrical columns (ibid., §202). The contents of Lambert's chapter on oblique figures are mathematically very beautiful, but far too sophisticated for most practitioners. His constructions in this chapter are similar to some of Taylor's constructions presented in section X.8 in that they are mainly mental operations. By this I mean that 17The only mention of 'sGravesande I have noticed in Lambert's works occurs in his historical survey, where he mentions that Christian Wolff praises 'sGravesande's Essai de perspective (Lambert 1774, 28).

7. Shadows

661

they demonstrate it is, in principle, possible to construct certain objects, but that their actual performance is hardly feasible. Thus, if the construction of an arbitrary polyhedron really is to be performed, as described by Lambert, it is quite an operation. First one must construct the image of a polygon situated in a plane a, which has to be provided with an angle scale. Then one must determine another plane f3 perspectively, equip it with an angle scale, go through the process of constructing the image of the face lying in f3 - and subsequently proceed to construct the other faces.

XII.7

Shadows

T

he construction of a perspectival shadow is, as we have seen, a problem of performing a central projection in the picture plane. One might have expected that Lambert would find this type of problem interesting and discuss it in some detail. However, like most of his predecessors, he treated the subject rather briefly, claiming that shadows present no difficulties as long as the position of the light source is known (Lambert 1759, §139, cf. the quotes in note 6). He illustrated his claim by choosing three easy examples in which he constructed the shadow according to the principle illustrated in figures VII.62 and VII.63. Thus, Lambert stayed away from the difficult problem of determining in n the orthogonal projection of a point upon a plane (page 532). In Lambert's first example the source of light is a point (figure XII.22), whereas in his two other examples the light comes from the sun. In the second example (figure XII.23, at left) Lambert let the sun be situated behind

FIGURE XII.22. A shadow cast by a book upon a table in the picture plane, the source of light being in the point L. Lambert explained how he had constructed the shadow point c of the point C: Assuming that L's and Cs orthogonal projections B and A upon the table are known in Jr, he obtained the point c as the point of intersection of LC and BA. Lambert 1759, figure 15.

662

XII Lambert

the picture plane and remarked that this causes the shadow to be cast away from the horizon. In his third example (figure XII.24) he assumed that the sun is in front of the picture plane, pointing out that in this case the vanishing point of the sunrays lies below the horizon and that the shadow is cast towards the horizon. Finally he touched upon the possibility of the sunrays being parallel to the picture plane (Lambert 1759, §149). As can be seen in the right-hand side of figure XII.23, Lambert also took up an example in which an object casts shadows upon two planes. His construction of the shadow the ladder casts upon the wall is correct, but his explanation does not contain all the necessary information (ibid., §l44). What is more, in connection with shadow constructions Lambert even made a mathematical error (figure XII.25), which is a rare phenomenon in his works.

s~ ..........

... '. \,

R

..... \

......... '.

\ ....

'

.

.....~ '

. ............ ' ... ~..

Ii

.M.••.. "'::~::'

....

".

-.,.

FIGURE XII.23. Lambert's second example of constructing shadows in the picture plane n. The source of light is the sun - situated behind n. Lambert let S be the vanishing point of the parallel sunrays and M its orthogonal projection upon the horizon, and assumed they are given. He could then easily construct the shadow cast upon the ground plane by the object to the left because the orthogonal projections in nof points like Band T upon the ground plane are known. He obtained the shadow bA of the edge BA, for instance, by constructing b as the point of intersection of SB and MA. Lambert mentioned that the position of S can either be chosen or constructed in accordance with a real situation. In the latter case he let the position of the sun be given by two angles. The first is the angle between two known vertical planes through the eye point, namely the vertical plane and the one passing through SM. The second angle is the angular height of the sun above the horizon. Making PQ equal to the distance, he constructed the angle PQM equal to the first angle. Having determined R on the horizon by MR = MQ (R lies in the fence), he made the angle MRS equal to the second angle. Lambert 1759, figure 16 with an arrow marking the point R added.

7. Shadows

~

,

-

Q.~/

.

.....

663

.....:u .' ~"1.

./

'-. '. :N ... : ~

FIGURE XII.24. Lambert's example of a shadow cast towards the horizon. N is the vanishing point of the sunrays and M its orthogonal projection upon the horizon. Lambert's construction is similar to the one explained in the previous figure. Lambert 1759, figure 17.

XII.25. Lambert's determination of the perspective shadow cast by the edge tv of the chimney upon the oblique roof (Lambert 1759, §152). I use a to denote the plane of the roof The vanishing point of the sunrays is given as a point S (cf figure XII.50). Since it lies outside the present diagram, only the direction towards it is indicated. The perspectival shadow of the point v is the point of intersection of the line Sv and its orthogonal projection upon a. To obtain the latter line, Lambert first correctly constructed the perspectively orthogonal projection z of the point v upon a. Next he searched for the vanishing point T of the projection of the sunrays upon a. He constructed this point as the orthogonal projection of S upon the vanishing line va of a - which is known (the line rq in figure XII.50). Here he made a mistake, as explained below. Let {3 be the plane determined by a sunray s and its orthogonal projection upon a, and let N a be the vanishing point of lines perpendicular to a. The line SNa is consequently the vanishing line of {3. Since the projection of the sunray s upon a lies in both a and {3, its vanishing point is the point of intersection of va and SNawhich is generally different from the orthogonal projection of S upon va' Lambert completed the construction by determining the point of intersection! of Tz and Sv, which just so happens to coincide with an upper corner of the roof surface. In practice Lambert's erroneous determination of T plays no role, as his chimney shadow looks convincing enough. Redrawn detail from figure 14 in Lambert 1759, reproduced as figure XII. 50. FIGURE

664

XII. Lambert

I mention Lambert's inaccuracy because I want to show that exact constructions of shadows in 1C are not always as easy as Lambert made them sound. Lambert also briefly mentioned the shadow effect of light entering through a window or an open door (figure XII.26). In this connection he dealt with two kinds of shadow: umbra and penumbra (ibid., §154).

XII. 8 Reflections

I

n Freye Perspektive Lambert treated most of the interesting mathematical problems associated with perspective drawing, but ignored the construction of perspectival reflections. Perhaps inspired by Taylor, Lambert took up this topic later and included a section on it in Zusiitze (Lambert 1774, 129-132). To illustrate the degree of perfection to which Lambert honed his perspective geometry, I have included a detailed presentation of his main example of constructing reflections in the picture plane, additionally touching upon a few of his other examples as well. Before beginning, however, let me warn the reader that this section contains many mathematical technicalities. Lambert stated that the entire art of perspectival reflections is based on making the angle of reflection perspectively equal to the angle of incidence. However, his constructions of reflections do not in general involve perspectively equal angles. Instead the constructions are based on a number of observations, which Lambert derived from the rule of the equality of the angles of incidence and reflection, but did not state explicitly. As these observations are crucial to understanding Lambert's procedures I present them together with arguments that I believe to some degree echo Lambert's way of thinking.

....... ......

t ...- . : : · · · · ·

FIGURE XII.26. A situation where light coming through a doorway is stopped by the wall efg. Lambert presented a construction that is the perspective equivalent of constructing the umbra as determined by the lines ae and bf, and the penumbra as determined by the lines afand be. Lambert 1759, figure 18.

8. Reflections

FIGURE

665

XII.27. Lambert's room with two mirrors. Lambert 1774, figure 56.

Lambert's Room

L

ambert's main example concerns the room shown in perspective in figure XII.27, and as a horizontal section in figure XII.28. This room contains two mirrors. The one, let us call it J.l. j , has L, M, and N as its corners and hangs on a wall that is perpendicular to the picture plane n. The other, J.l.2 ' is as wide as the wall EFGK upon which it is placed and makes an angle of 45" with n. Before showing how the mirror images in J.l. j and J.l.2 can be thrown into perspective, Lambert took up the question: Which part of the room can be seen as a mirror image? As far as I am aware, this was a new question in the literature on perspective - and I will return to Lambert's solution later. Lambert's constructions of the mirror images are, as mentioned, based on consequences of the requirement that the angles of reflection and incidence must be perspectively equal. I introduce the results by first looking at the situation in the three-dimensional space (figure XII.29). Let J.l. be a mirror, 0 the eye point, and Ya point that is reflected in J.l.. The light travels from Y to 0

112

9 III

h H

B

FIGURE XII.28. A horizontal section through the eye point in Lambert's room (depicted in figure XlI.27). On this drawing III and 112 represent the two mirrors. Because Lambert's published figure is tiny (6x3 cm) and rather rough, the reconstruction of his room can only be approximative.

666

XII Lambert

° l__~--=~

FIGURE

XII.29. The law of reflection.

via the so-called point of reflection Yr in Il. Its characteristics are that the plane determined by points Y, Yr' and 0 contains the normal n to Il through Y r, and that the two lines YYr and YrO form equal angles with n. To the eye point, the light appears to come from the point I call the mirror point Ym of Y, and which is the point that lies symmetrically to Y with respect to its orthogonal projection Y' upon Il, that is YmY'

= Y'Y.

(xii.9)

When a mirror occurs in a perspective composition, the eye looking in a mirror coincides with the eye point of the composition. The fact that Ym and Yr lie on the same line through 0 implies the following result, which Lambert often used. Observation 2. In 1r the points Y m and Y r coincide.

The point Yr can be characterized as the point of intersection of OYm and the mirror Il, but it can also be defined in another way that involves the mirror point H of the eye point 0 (figure XII.30). The former point is defined by considering the orthogonal projection 0' of 0 upon Il and making H 0' = 0'0 on the normal to Il. A straightforward geometrical consideration leads to: Observation 3. 1',. is the point of intersection of HYand Il.

For the application of this observation in

1r the

following result is useful.

Observation 4. In 1r the points Hand 0' coincide with one another, and with the vanishing point Nil of the normals to Il.

For later use I also note that the following can be concluded from figure XII.30: Observation 5. The points Ywhose mirror images in Il reach the eye point 0 are the points for which HY cuts Il.

8. Reflections

O-I--~~_-=::t-_

FIGURE

667

Ym

XII.30. Another method for determining the point of reflection Yr'

Relation (xii.9) implies that in Jrthe point Ym can be constructed by making YmY' =pY'Y. This was actually the method Taylor had used (page 529). Lambert mentioned it, but he applied it only when the equation, besides being perspectival, is also "geometrical", by which he meant that the line segments have the same length (Lambert 1774, 130). This occurs when Yl" is parallel to Jr, as in the case of the mirror /11 in Lambert's room. He claimed to have constructed the images in this mirror by using the method of equal distances (although his constructions are actually not very precise, as we shall see later). In principle, observations 2, 3, and 4 provide a method for determining Ym in Jr, namely as the point of intersection of HY and /1. As shown earlier, however, it is difficult in practice to determine the point of intersection of a plane and a line in Jr. Hence, Lambert elaborated his technique of determining mirror points by projecting the configuration upon a ground plane r in Jr. The three-dimensional situation is shown in figure XII.31, in which the points Y, Yr, and H are projected upon the points X, X r, and U. Actually, it is not always easy to construct X in Jr either, but in Lambert's example this construction caused no difficulty. For determining H's projection U, Lambert used the fact that U is also the mirror point of the foot F, and besides he involved the point of intersection, S, of OU and /1. Let i be the intersection of the plane containing /1 and the ground plane y; and let T be the point of intersection of i and the vertical line through or. Straightforward geometrical considerations show that S can also be characterized as the midpoint of 07, that is O'S = ST.

When looking at the situation in the picture plane, we get: Observation 6. In Jr the points Sand U coincide.

This implies the following result, which Lambert took for granted:

(xii.10)

668

XIL Lambert

o

FIGURE

XII.31. Relevant lines projected into a ground plane.

Observation 7. In n the point X r is the point of intersection of SX and i.

All these considerations give rise to the following construction of Yr (figure XII.32). In n, let the line of intersection, i, of the plane of the mirror J1 and the ground plane rbe given, and let us assume that the points H, Y, and X are also given. Determine T as the point in which the vertical line through H cuts i, and S as the point that bisects the line segment HT (cf. (xii.lO». Then find X r as the point in which SX meets i (observation 4 and 7), and finally construct Yr = Ym as the point of intersection of HYand the vertical line through X r • For mirror J12 (figure XII.27) Lambert applied the construction just presented, explaining how he had constructed the mirror image v of the point v. He then remarked that the fact that the mirror J12 makes an angle of 45° with n implies that the wall supporting J11' as well as J1] itself, are seen as parallel to n in J12 (Lambert 1774, 131). This statement - which is equivalent to saying that the mirror image in J12 of an orthogonal is a transversal - is easily verified by applying the law of reflection.

H

Yr

y

X

FIGURE

XII.32. Construction of Yr in re.

8. Reflections

669

In his diagram Lambert also included the mirror image in 112 of Ill' although not very distinctly. Even so, he seems to have forgotten the result just mentioned above, for the lower line of his mirror image of III is not horizontal. I am also very puzzled as to what the apparent ovals are supposed to be mirror images of - but perhaps they mirror something hanging on the wall that is not included in the drawing.

Determination of Areas that Can Be Seen Reflected

A

s noted, Lambert began his treatment of reflection by determining the part of the ground plane upon which objects should be placed in order for their reflections to reach the eye point. 18 Before presenting Lambert's solution, I look at the situation in a Euclidean plane (figure XII.33). In a horizontal section through the eye point 0, the area that can be seen reflected in III is determined, according to observation 5, by connecting the mirror point HI of the eye point with the end points g and h of the mirror Ill' and likewise for Ilr In n the construction just described cannot be copied directly, because the entire horizontal section is one single line segment. However, for the mirror 112 Lambert adjusted the construction by projecting the lines upon the ground plane y, applying the fact that in n the mirror point of the foot is the point S (observation 6). Thus (figure XII.27), by connecting S with the points E and K Lambert found the required area in r

HI FIGURE XII.33. Determining the sections of Lambert's room that can be seen reflected in the mirror III and the mirror 1l2 , respectively.

18Lambert must have been aware that the height of a mirror and its vertical position on the wall, and similarly the vertical segment in an object, also playa role (ibid., 129). However, he did not include any specific results relating to this problem.

670

XII Lambert

To determine the corresponding area of the floor for the mirror Pi' Lambert could not apply the same procedure, since in n the mirror point of the eye point with respect to mirror Pi is a point at infinity. Instead he used a result that corresponds to transferring the following observation concerning the three-dimensional situation into n: Observation 8 (figure XII.33). At the level of the eye point, let the mirror Pi have the end points g and h. The line that when reflected in Pi becomes the line Og, is the line gg*, which forms the same angle with the mirror as Og. Since Pi is perpendicular to n, the vanishing point of gg* lies symmetrical to the vanishing point of Og with respect to the principal vanishing point. The same argument applies for the line that is reflected as Oh.

In n the vanishing points of the lines Og and Oh are the points g and h themselves. This made it easy for Lambert to determine the vanishing points of the rays that are reflected as Og and Oh. In his drawing (figure XII.27) 0 is the principal vanishing point, and by making Or = Og and Oq = Oh Lambert obtained these vanishing points as rand q. The latter points are also the vanishing points of the orthogonal projections of the lines gg* and hh* upon yand the ceiling. Hence, by drawing the lines mq, lr, nq, and pr and marking the points a, b, C, and d in which these lines cut the end wall, Lambert determined a prism delimiting the points whose reflections in Pi can reach the eye point. His construction of the four points a, b, C, and d is actually inaccurate; they should all have been placed slightly farther to the right (figure XII.34). It can be noticed (figure XII.27), that in constructing the mirror images of the two portraits hanging on the end wall Lambert nevertheless used the almost-correct position of the lines ac and bd. He thus showed less than half of the left portrait in the mirror and more than half of the right portrait. Before leaving his room, Lambert discussed the following problem (ibid., 131-132): Upon which area in the ground plane must objects be placed in order to be visible from 0 via a double reflection, that is, in order to be

H

FIGURE

precise.

q

r _ -

-=

B

XII.34. The determination of the points a and b in figure XII.27 made more

8. Reflections

671

FIGURE XII.35. Determination of the area from which light rays may reach the eye after a double reflection - first in mirror )1] and then in )12' Lambert's construction in n is based on some results that can be obtained as follows. Let us again consider the horizontal section through the eye point. The rays that are reflected in the mirror )12 and reach the eye are those passing through a point on )12 and through the mirror point H 2 of the eye (observation 5). If rays are to be reflected in )1], they must have a point in common with this mirror. Thus, a ray that is reflected first in )1] and then in )12 will reach the eye if it has a point in common with )1] and its reflection passes through H 2 • This implies that on the horizontal section being considered, the required area can be determined by the following steps. Connect the mirror point H 2 with the end points g and h of the section in )11' then construct the lines that are reflected as gH2 and hH2 , and determine the points e and! in which these lines meet the walls of the room. This will result in the area eghf Lambert's procedure in nconsisted in constructing the perspective projection of this area upon the ground plane, of which the outcome was the area x/my shown in figure XII.27.

reflected first in)11 and then in )12? In figure XII.35 I have explained the principle behind his solution. Lambert did not treat a double reflection in the opposite order - and with good reason, for the solution is an empty space.

Reflection in Curved Surfaces

I

n his Zusiitze Lambert also touched upon objects reflected in water (figure XII.36) and devoted a section to reflection in Gebogene polirte Fliichen (curved polished surfaces). In one example dealing with a marble column, Lambert demonstrated his mastery of solving mathematical problems directly in the picture plane n (figure XII.37). He assumed that a point M is given on the cylindrical column in n and showed how the line SM, having M as its reflection point, should be constructed. As usual, he selected a construction in the three-dimensional space and transferred this to n, and in doing so he made acute observations about how several of the elements coincide in the picture plane (Lambert 1774, 135). There were presumably few, if any, practitioners who would be willing and able to perform Lambert's construction. Nevertheless, his construction does have a practical application, because, as he observed, the colour in M depends on which object in nthe line MS meets first.

672

XII. Lambert

FIGURE XII.36. An example in which Lambert constructed mirror images in water. Lambert 1774, figure 57.

Lambert also took up the inverse problem concerning reflection in the column, meaning the determination of the reflection point M belonging to a given point S. This investigation can hardly have been motivated by practice, but was more likely a result of Lambert's mathematical curiosity. In solving the problem in n he first projected the point S upon the point R in the ground plane, then searched for the reflection point of R. The corresponding problem in the Euclidean plane - known as Alhazen's problem - is the following (figure XII.38): Given the points Rand F (the foot of the eye point) and a circle, determine the point P on the circle in which the ray travelling from R to F via the circle is reflected. There is no simple solution to Alhazen's problem. Lambert referred to a solution that involves a so-called caustic curve. To introduce this curve, I first imagine that for each point X on the circle the line XY which is reflected as XF, has been determined and prolonged into the circle. The caustic, then, is the curve defined by having all the prolonged XY's as tangents. Once the caustic curve is known, the required point P can be determined as the point of intersection of the circle and the tangent drawn from R to the curve. Lambert performed this determination, including the construction of the caustic directly in n, after which he fairly straightforwardly determined the point M shown in figure XII.37. In his drawing Lambert also included a cone; here, as he had for the cylinder, he solved the problem of finding the point that has a given point on the cone as its point of reflection ~ this time refraining from solving the inverse problem. Finally he dealt with a similar problem in connection with a vase that is rotationally symmetric (figure XII.39). The examples presented in this section give the impression that occasionally Lambert amused himself by finding out how specific problems can be

8. Reflections

S r-.... ! '" .

.

I

S ".

. .J

.

'.

..;K

~.._.. L

If.

./'

FIGURE

.'

··C

.•'

'~

..." C

, 4~"~"-~

p

Q.

?

)

673

.....

.

•••.•

XII.37. Reflection in objects made of marble. Lambert 1774, figure 58.

solved in perspective geometry - and in section XII.ll we meet some other examples from Zusiitze that confirm this impression. Apparently when the decision to reprint Freye Perspektive was made, he saw an opportunity to share his observations with others.

FIGURE XII.38. The caustic curve solving Alhazen's problem.

s o

FIGURE XII.39. Lambert's construction of the line MS, which has a given point M on the vase as its mirror point. Lambert 1774, figure 59.

··--· ..·.R ! E

D

674

X/l Lambert

XII.9 Parallel Projections

F

or visual impressions, perspective projection long remained the preferred way to represent three-dimensional objects upon a plane. This technique is not adequate, however, if information about the shape of the original object is required. In such cases it is better to apply a parallel projection, which is a projection determined by a given direction. In this method, a point A is projected upon the point Ai that is the point of intersection of a given picture plane 1r and the line through A parallel to the given direction. If the given direction is perpendicular to 1r, the projection is called an orthogonal projection. A plan and an elevation are two special types of orthogonal projections. Like the history of the art of using plans and elevations, the history of parallel projections is not very well documented. It is unclear where a conscious use of general parallel projections originated and how they were first understood and described. The previous chapters feature illustrations from treatises on perspective whose authors have used parallel projections. In mathematical texts in general, it became increasingly common to draw diagrams based on parallel projections from the end of the sixteenth century onwards. Parallel projections were also often used to illustrate fortifications, a circumstance that seems to have given rise to the term military perspective. The expression cavalier perspective was also used, presumably derived from the fact that an elevated element in a fortification, for instance a watchtower, was occasionally called a cavalier (Steck in Lambert 1943,435). In some traditions the terms military and cavalier perspective have been attached to particular parallel projections, and not always the same ones (ibid., 436). The term axonometry has also been, and still is, used for a parallel projection especially if the drawing includes the projection of a three-dimensional coordinate system. Lambert incorporated a section on parallel projections in Freye Perspektive because he wanted to illustrate how he had derived a technique for treating such projections based on his method of direct constructions in the picture plane. His is an interesting method that deserves to be more widely known, and so I discuss it in some detail - but let me first present Lambert's terminology. He did not use the term parallel projection, but rather orthographic projection,19 and sometimes military and cavalier perspective in the French edition, whereas in the German edition he had no shorter expression than "perspective projection having a point at infinity as an eye point" (Lambert 1759, §251). Lambert was not the first to notice that a parallel projection can be considered to be a perspective projection with an eye point infinitely far away. Many others presumably had this thought, which, as noted, was stated explicitly by Aguilon one and a half centuries earlier (Aguilon 1613, 503). However, Lambert seems to have been the first to draw conclusions from this comparison. 19Vitruvius, as noted in chapter II, used the word orthographia for an elevation (Vitruviuss Arch, book I, chapter 2, §2).

9. Parallel Projections

675

Lambert's description of how to construct images of parallel projections is fairly easy to follow, but an understanding of the connection to perspective requires more arguments than he presented. As done earlier, here I include some of the missing observations. To distinguish an image produced by a parallel projection from a perspective image, I refer to the former as a cylindrical image - thinking of its perspectival counterpart as a conical image. Generally speaking, Lambert's basic idea was to begin by choosing a fixed perspective projection and then relate a given parallel projection to this. In his constructions of cylindrical images he involved the horizontal angle scale of the perspective projection, and hence, to avoid constructing more than one angle scale, it was handy to work with only one perspective projection. He coordinated a given parallel projection to the perspective projection in such a way that the two projections have the image of one point, A o' and the images of all lines passing through this point in common. He then used results concerning the perspective projection to perform a direct construction of the cylindrical images of horizontal polygons. He characterized this procedure as a changeover from the situation where a finite figure is projected from a point at infinity to a situation where an infinitely small figure placed in A o is projected from a point at a finite distance from the picture plane - the infinitely small figure having retained all the relations applying to the elements of the finite figure (Lambert 1759, §253). As we shall see, although Lambert did not use infinitesimal figures in his construction, they played a role in his argument for the correctness of the construction. More specifically Lambert assumed (figure XII.40) that a parallel projection, a horizontal reference plane Yo and a vertical picture plane 1C are given. Any original point A o' situated in rbut not in 1C, and its cylindrical image A determine the direction of the parallel projection - namely that of the line segment AoA. H 1t

P

o

r

FIGURE

XIIAO. Coordinating a parallel projection and a perspective projection.

676

XII Lambert

Lambert let a direction be given by two angles, the first being the angle l/f, which a line m with the given direction forms with its orthogonal projection m' upon Yo and the second being the angie if>, which m' makes with the normals to the ground line GR. When the chosen perspective projection has the eye point 0, determined by the principal vanishing point P and the distance d, and a parallel projection was given by the angles l/f and if>, Lambert wanted to construct the point A in 1r, which is the cylindrical projection of the eye point 0 upon 1r. The point A is interesting because it is a common image for the two projections, the original point A o being the point of intersection of OA and the ground plane Yo To characterize the point A in 1r Lambert considered the point V on the horizon given by LPOV = if>, and he noticed that A is the point situated on the vertical through V below V, so that L VOA = l/f. In other words, he determined A in 1r by the relations

V is the point on the horizon for which PV =d tan if>,

(xii. 11)

and

A is the po int vertically below V for which VA

= co~ if> tan lfI.

(xii. 12)

In his further considerations Lambert applied the fact that lines through A o are mapped upon the same lines by the two projections. That this is true for a line I that cuts 1r in II can be demonstrated as follows. The cylindrical image and the perspectival image of I are determined by the two points II and A, II being its own cylindrical and perspective image and A being a common image of A o' For a line k, which is parallel to 1r and passes through A o' the result is implied by the fact that both images are a line in 1r through A parallel to k. Summing up we get the following result, which was essential for Lambert's constructions: Observation 9. Let the point P be given in the picture plane 1r, let 0 be the point on the perpendicular to 1rthrough P given by OP = d, let A be given by (xii. 11) and (xiL12), and finally let A o be the point of intersection of the line OA and the ground plane Yo The perspective projection with the eye point 0 as well as the parallel projection determined by the angles

Note that although the cylindrical and the perspectival images of a line through A o coincide, the points on the line - apart from A o - are mapped into different points by the two projections. Lambert equipped 1r with the angie scale belonging to his chosen perspective projection. He then constructed the cylindrical images of horizontal lines through A o directly in 1r, and next turned to direct constructions of lines through a given point E in 1r that are the cylindrical images of horizontal lines with a given direction (figure XII.4l). To cut off line segments, Lambert involved his idea of considering infinitesimal figures. Since the cylindrical images of parallel and equal line

9. Parallel Projections FIGURE XII.41. Lambert's solution to the problem of determining the line that passes through E and is the cylindrical image of a horizontal line with a given direction (Lambert 1759, §265). Letting VI be the vanishing point of lines with the given direction, Lambert first constructed the cylindrical image of the line through A with the given direction - which is A ~ - and then found the requested line as the line through E that is parallel to AV/, using the fact that a parallel projection preserves parallelism.

H

677

v

p

A

segments are also parallel and equal, it is sufficient to be able to construct a line segment that has an end point in A. How Lambert did this is explained in the caption of figure XII.42. The procedure described thus far enabled Lambert to construct cylindrical images of horizontal figures directly in Jr. Since the cylindrical image of a

z

a

B

h

FIGURE XII.42. Lambert constructing the cylindrical image of a line segment of length a, with an end point in A, and situated on the line m. First he determined the measure point M m belonging to m and imagined an infinitesimal figure placed at A, in which the line segment AC has been cut off so that it is the perspective image of the length a. His standard way of doing this was to cut off the line segment AD' equal to a on the horizontal line h through A and to construct C as the point of intersection of m and D'Mm • (As Lambert dealt with infinitesimal figures, the unit used for this construction should be an infinitesimal; for obvious reasons it is finite, but small in my figure). Translating this to the finite case, Lambert proceeded as follows. He made AB equal to a, and noticed that passing through M m in the perspective case means that D'Mm is perspectively parallel to AMm • He then reasoned that in the cylindrical case the requested line segment BC should be constructed parallel to AMm - which according to observation 9 is a cylindrical as well as a perspective image. Thus, he determined the point C as the point of intersection of m and the line through B parallel to AMm' Let me add that readers who may feel uncomfortable with Lambert's arguments involving infinitesimals can convince themselves of the correctness of Lambert's construction by proving that the angles ACB and ABC are cylindrical images of equal angles. From this follows that AC and AB are cylindrical images of equal line segments.

678

XII Lambert

vertical line segment has the same length as the original, constructions of heights were completely unproblematic.

A Precursor of Pohlke's Theorem

A

t the end of his chapter on parallel projection Lambert discussed an example of drawing - in some scale - the cylindrical image of a rectangular box (figure XII.43) with two faces parallel to the picture plane 1r (Lambert 1759, §272). Lambert first drew the image face Imts proportional to the corresponding face in the original box, say lomonorosiovozo - thereby choosing the scale. Next he drew the side Ir, claiming that the length of Ir and the angle mlr can be chosen freely. He added, though, that the best impression of a box is obtained when the angle rly is between 400 and 500. Having decided how to represent side Ir and the angle mlr, Lambert finished the box in a straightforward manner. Lambert's procedure is similar to the one usually employed for drawing the parallel projection of a box, which presumably in most cases is done without giving any thought to why it works. However, from Lambert we can learn that the method is correct, because he proved it. In paraphrasing Lambert's proof I use the expression - a bit unconventionally - that two figures are identical when they are congruent and similarly orientated. Lambert's claim can then be expressed as follows.

Result 1. For a given box having a vertical face identical to Imts and parallel to the picture plane, it is always possible to find a parallel projection and a position for the box so that the box is depicted in the drawn box Imnrstvz (Lambert 1759, §274). The correspondence Lambert had established between parallel and perspective projections was absolutely essential to his proof, as we shall see. Since a plane figure parallel to the picture plane is mapped into an identical figure by a parallel projection, Lambert only needed to prove the existence of a parallel projection and a position of the box that result in Vo being mapped upon Ir. His strategy was to let the point I assume the role previously held by the point A, or in other words to let I be a common image of a perspective projection and a parallel projection. He began by defining the perspective projection. As its horizon he chose a line CN parallel to 1m, letting the point C in which the horizon meets the line Ir be the principal vanishing point (because Vo is an orthogonal). To determine the distance Lambert applied the result that the distance points are measure points for orthogonals. To find the measure point G of Ir, he inverted his method for determining the cylindrical lengths of line segments with an end point in I (as described in the caption of figure XII.42). Hence, his steps were to construct lyon the line 1m so it had the same length as Vo' and then to determine G as the point of intersection of the horizon and the line through I parallel to yr. By this procedure Lambert had introduced the perspective projection that has C as its principal vanishing point and CG as its distance.

10. Inverse Problems of Perspective

c

679

ON \

P~'"

~... '

.', .....

.

,,'"

:

0•

"

I:

'1~'" FIGURE

XII.43. A parallel projection of a box. Left part of figure 26 in Lambert 1759.

Lambert obtained the parallel projection defined by the perspective projection and the point I by applying the formulae (xii. 11) and (xii.12). If the original box is moved so that IJ has the direction determining the parallel projection, then it follows from the way G was constructed that the drawn box, lmnrstvz, is indeed the cylindrical image of the box lomonorosJovozo' Lambert's result 1 is a special case of a more general theorem, later called Pohlke's theorem, and rightly so because it was formulated and proved in 1853 by Karl Pohlke, a German painter and teacher of perspective and descriptive geometry (Schwarz s 1864). Pohlke published his result and its proof in 1860 (reissued in Pohlke s 1876, vol. 1, §147) and in 1864 a more elegant proof was provided by the mathematician Hermann Amandus Schwarz, who attended Pohlke's lectures at the Gewerbeinstitut (industrial school) in Berlin (Schwarzs 1864). Pohlke's theorem is equivalent to saying that for any choice of the directions of the three mains axes and the units on the axes in a cylindrical drawing, a corresponding parallel projection exists. Lambert's chapter on parallel projections drew a good bit of attention around the tum of the nineteenth century (Steck in Lambert 1943,434-435, note 277). Mathematicians and historians of mathematics found, in Lambert's considerations, the seeds of ideas that were later used in descriptive and projective geometry, but they did not emphasize his special approach. It is, of course, intellectually and historically significant to know that some of Lambert's ideas resurfaced, but it becomes much more interesting when the context of his achievements is also taken into account. In particular, I find that his imaginative treatment of a parallel projection as a special kind of a perspective projection shows that the history of perspective per se has some aspects that are fascinating even when seen independently of the later development of geometry.

XII.I0

I

Inverse Problems of Perspective

n dealing with inverse problems of perspective, Lambert did not let himself get as carried away by the mathematical possibilities of the theme as Stevin and various other had done - and as Lambert himself did with other aspects

680

J(IL Lalnben

of perspective theory. To some extent he chose his inverse problems so they would have some relevance for an analysis of a perspective composition, and he limited his examples to cases involving a vertical picture plane. Lambert began with seventeen observations that are useful for solving inverse problems of perspective and then treated six specific problems. In some of these he found the eye point by determining the principal vanishing point and the distance. In other cases, the data were not sufficient to determine the eye point uniquely, and he simply found out as much as possible about its position. Generally his solution strategy was to reverse the steps of his procedures for perspective construction. I will not go into detail, but just indicate the nature of Lambert's problems and outline some of his solutions. In his first problem Lambert treated an often-occurring situation, namely one that supposes a given quadrangle in the picture plane tr to be the image of horizontal square, the task being to determine the eye point. Lambert generalized this case, assuming that a given quadrangle in tris the image of a horizontal rectangle whose sides are in a given ratio (Lambert 1759, §296). The caption of figure XII.44 explains how he found the principal vanishing point and the distance. Lambert's sixth and last problem deals with the following question (figure XII.45). Given in tr are the collinear line segments AB and Be having the point _ -__ 0"

B

FIGURE XII.44. One of Lambert's inverse problems of perspective. He assumed that ABCD is the image of a horizontal rectangle whose sides have a given ratio, and wanted to determine the eye point O. Lambert observed that the points of intersection VI and V2 of the opposite sides of the perspective rectangle are the vanishing

points of the two sets of parallel lines in the rectangle, and he concluded that the horizon is the line VI V2 • To characterize 0 he introduced the turned-in eye point 0". Since the sides of the quadrangle ABCD are perspectively perpendicular, Lambert concluded that 0" lies on a circle with the diameter Jj V2 - this circle being the locus of the points from which VI and V2 are seen within the angle 90·. To find out where on the circle 0 "has to be placed, he introduced the vanishing point E of the diagonal BD. From the definition of a vanishing point it follows that LJiO~ is equal to the angle between the originals of BC and BD. This angle, say t/J, is known because the ratio of the sides of the rectangle is given. Knowing the angle t/J, Lambert constructed 0" fairly straightforwardly. Adaptation of figure 28 in Lambert 1759.

10. Inverse Problems of Perspective

681

M

FIGURE XII.45. Lambert's sixth problem of inverse perspective. Lambert 1759, figure 32.

B in common. It is also given that they are images of horizontal line segments whose ratio is known, and finally it is given that AD is vertical. Lambert sought to determine the horizon (Lambert 1759, §304). In three other problems he likewise involved ratios between original line segments, the difference being that he supposed that the horizon was known. In one of these other problems (figure XII.46), he assumed that AlBl and A 2B2 are images of horizontal non-collinear line segments, that AlCl and A2 C2 are vertical line segments, that the original of AlCl has a given ratio to the original of AlB]' and that the same applies to the line segments A 2 C2 and A~2 (Lambert 1759, §299). From this information Lambert determined the principal vanishing point and the distance. In commenting upon another problem in the same category, he admitted that the problem did not occur often in practice, but wrote he had nevertheless included it to provide an interesting use of his theory (ibid., §302). Thus, although by and large he kept away from very theoretical problems when treating inverse problems of perspective, Lambert could not completely resist the temptation of showing a clever application. In connection with some of his examples Lambert touched upon the question of how much can be concluded if the assumptions concerned images of oblique rather than horizontal elements. He did not pursue this matter very far, however. The reason could be that he wanted to finish his book, the chapter on inverse problems of perspective being the last. He may also have lacked any deep interest in inverse problems - a possibility supported by the fact that he did not add anything new on the subject in Zusiitze.

z

H

A1 FIGURE XII.46.

Lambert's fourth problem of inverse perspective. Adaptation of a section of figure 30 in Lambert 1759.

682

XIl Lambert

XII. II

Lambert's Practice of Perspective

A

lthough Lambert chose a style of presentation different from the one usually adopted by mathematicians, he presented many more mathematical observations than are to be found in any other pre-l 800 book on perspective. This resulted in an advanced practice of perspective, which I illustrate in this section - though without explaining all the details. First, however, I want to stress that Lambert's interest in the practice of perspective was genuine. Presumably it arose out of his own drawing experience combined with his strong and comprehensive attraction to applied mathematics. The latter he showed in a voluminous publication on the use of mathematics. In this work he characterized mathematics as "a science of laziness" (LambertS 1765, 3), which reflects, I believe, a strong conviction that mathematics when cleverly applied can ease many tasks - including perspective constructions. Lambert's concern for the practice of perspective manifested itself in, among other things, a lengthy discourse on how to choose the parameters of a painting - a theme most authors avoided or only touched upon briefly. Lambert actually devoted, as noted, an entire chapter to the topic in Freye Perspektive and supplemented his considerations with a dozen pages in his Zusiitze (Lambert 1759, §§60-94; Lambert 1774, 80-92). His discussions include a case in which it is impossible to combine the parameters and a given motif in a way that would enable him to obtain everything he wished (Lambert 1774,90-91).

An Elliptical Scale

A

nother sign of Lambert's interest in the practice of perspective lies in his many remarks about the use of a sector. Freye Perspektive includes several hints on how the constructions he presented can be carried out by means of a sector, and in Zusiitze he added several more observations. As we have seen, Lambert also designed special scales for a sector that was put into production in 1768. The scales on one side of the sector were the five reciprocal ones presented at the end of section XII.3. The other side also had five pairs of scales (figure XII.47): an arithmetical scale, a tangent scale, a secant scale, a sinus scale, and a scale Lambert himself referred to as elliptical. The three trigonometric scales were to be used for various tasks, such as constructing vanishing points and measure points in accordance with the constructions described in sections XII.5 and XII.6. Lambert explained how the elliptical scale was to be used, but not how he had tabulated it. An analysis of its function shows he had designed it so that the distance fin) from the centre of the sector to the point marked n - in a suitable unit - is given by the relation f(n)

= sin (arccos !OrO; n).

(xii. B)

This is confirmed by a remark about the scale in a letter Lambert wrote to Brander in May 1767 (LambertS Briefe, vol. 3, 24). Lambert's idea was to use

11. Lambert's Practice of Perspective

683

.....

L .. ~ ...

FIGURE XII.47. Scales on the sector Lambert designed specifically for use in connection with perspective. Lambert 1768, half of figure 1- the other half is reproduced in figure XII. 10.

the scale for determining points on an ellipse in the Euclidean plane (figure XII.48) and points on perspective circles. He gave one example of throwing a circular arch perpendicular to the picture plane into perspective, and another of drawing the image of a horizontal circle (figure XII.49).

b

x

a

n

FIGURE XII.48. Lambert's procedure for applying his elliptical scale to determine points on the ellipse (x, y) = (a cos v, b sin v) having its centre at (0,0) and the half axes a and b. He imagined that a is divided into 100 equal line segments and marked so that 100 is placed at the centre. This means that when the point (x, 0) is marked by n then x 100 - n cos v = 7i = (1)

-roo

implying that

-roo .

· ( arccos 100-n) y = b sm

The relation (xii.l3) then shows that y

= bf(n), or, since f(100) = I, that

y: b = f(n): f(lOO).

(2)

Lambert constructed the coordinate y determined by this proportion by means of his sector as follows. He opened the sector so that the line segment b lies between the two points marked 100 on the elliptical scales and found y as the line segment between the two points marked n.

684

JfIl Lalnben

It":---t-Hi+Hffl------::::::::~~----------

FIGURE XII.49. In this illustration, Lambert showed how a circular arch perpendicular to the picture plane 1& and a horizontal circle are thrown into perspective. The principle in the two constructions is the same, so I concentrate on the arch, whose diameter AB is given. Lambert first halved the diameter AB perspectively, thereby obtaining the centre C of the arch. By making CD =pCB he determined the top of the arch. His construction of further points on the perspective arch is based on the observation that if we set a =b =r in (1) and (2) in the caption of figure XII.48, we can conceive of a point on a circle with radius r as a point whose abscissa is marked n, and whose ordinate Y is determined by

y:r =f(n):f(IOO).

(1)

In transferring this description to 1& Lambert constructed the image, say n, of the point marked n by making Cn: CB perspectively equal to (100 - n):lOO. His next step was to construct the image Yi of the length y by using the fact that the images of two line segments on the same vertical have the same ratio as the originals. Since CD is the image of r, he deduced from relation (1) that Yi : CD = f(n) : f(100).

By means of the sector and its elliptical scales, Lambert found the length Y i corresponding to n and used this to determine the point In on the arch vertically above n as follows. He placed the length Yi vertically so that it had an end point in C, drew the line connecting the other end point and the principal vanishing point 0, and found In as the point of intersection of this line and the vertical through n (at any place the vertical distance between the former line and AB is perspectively equal to yJ In principle I have only explained how Lambert obtained the half arch above CB, but the procedure for the other part is the same. Lambert 1768, section of figure 4 with the letter n added.

11. Lambert's Practice of Perspective

685

Trompe L'(Bits

L

ambert's concern for practice did not imply that he felt it was his job to teach practitioners the details of how to combine the various geometrical constructions he had described in Freye Perspektive. He limited the number of well-illustrated and carefully explained examples to two - one of which is shown in figure XII. 50. Most of his other illustrations are tiny and concern rather particular cases. One such example deals with creating a trompe l'a:il for the following situation (Lambert 1759, §247). An oblique ceiling (figure XII.5l) above a staircase is to be decorated in such a way that a person standing on the top stair gets the impression of looking into a room through an open door (figure XII.52). Another of Lambert's examples concern the construction of the perspective image of a garden that is partly horizontal and partly oblique. While explaining how the image should be constructed, he once more demonstrated his complete mastery of perspective geometry (figure XII.53). He also mentioned the possibility of laying out a garden on an oblique plot of ground in accordance

rl

FIGURE XII.50. One of Lambert's few illustrative diagrams. In his text he carefully explained how he had used the angle scale on the horizon VW to construct lines with given directions and to determine measure points. He also described how he had constructed the images of line segments of given lengths by means of the measure points. Lambert 1759, figure 14.

686

XII Lambert p 11:

~o

l

I

\I

p

FIGURE XII.5!. A vertical section, perpendicular to the edges of the stairs, of a staircase and an oblique ceiling. Lambert's picture plane was the ceiling, and as his eye point 0 he chose the eye position of an average person standing on the top step. To make this diagram correspond to the following figure drawn by Lambert, I have kept his letters. He chose 1C as the principal vanishing point, and P and p for the vertical and horizontal projections of the eye point 0 upon the picture plane. The point P is the vanishing point of vertical lines, and p that of the horizontal lines parallel to the vertical section.

with the rules of perspective so that it appears to be horizontal (Lambert 1759, §§249-250). This was apparently an idea that appealed to him, for he returned to it in his Zusiitze and explained the factors that should be taken into consideration if the garden was to include pavilions or walls (Lambert 1774, 157).

Rainbows, Fountains, a Starry Sky, and Perspective Pictures n Lambert offered a number of new examples, most of which, Iin mathematically speaking, are considerably more advanced than those found In section XII.8 we already met a couple of examples Zusiitze

Freye Perspektive. from Zusiitze. They concerned perspectival reflection, which Lambert wished to see treated scientifically. He had similar ideas about rainbows, water jets in fountains, the stars, and perspective images of perspectives images. Lambert remarked that throwing a rainbow correctly into perspective requires more optics and physics than practitioners have normally learned, the result in most cases being a rainbow that looks like a "hieroglyph" (Lambert 1774, 108). To remedy this situation, Lambert devoted some nine pages to teaching the basic knowledge needed to draw a rainbow correctly in a perspective composition. Turning to fountains, Lambert made his readers aware of the parabolic paths of the water jets and taught them how to find their images - which most often are hyperbolas (ibid., 117-120). Perspective compositions featuring stars really seem to have upset Lambert, who could see that in most cases no central projection had been

JJ. Lambert's Practice of Perspective

687

FIGURE XII. 52. Lambert's construction of a trompe !'adl on an oblique ceiling (cf. the preceding figure). In his drawing the eye point is turned into the picture plane around the line Pn and marked O. To draw the image hlmn of a door, Lambert used the fact that P is the vanishing point of vertical lines (see the caption of figure XII.51) and that their measure point M is determined by MP = OP (cf.(xii.8» and lies on the horizontalline through P - the vanishing line of the vertical planes parallel to the horizon. Similarly, to draw the room behind the door, he applied his knowledge that the horizontal line through p is the vanishing line of horizontal planes, that the vanishing point of lines perpendicular to the door is p, and that their measure point V is determined by Vp = Op. Lambert 1759, figure 25.

applied to the real constellation. Here, too, he offered a remedy by giving his readers a brief course in astronomy and instruction on how to project the coordinates of the celestial globe upon the picture plane (ibid., 120-123). His illustration is reproduced in figure XII. 54. Interiors drawn in perspective sometimes contain perspective paintings, each of which generally has an eye point different from that of the main composition. If such a painting is to be reproduced according to the rules of perspective, a perspective projection of another perspective projection must be made. Lambert sketched a technique for this (ibid., 124-129). As a real mathematician he saw the possibility of generalizing this procedure and spoke of threefold, fourfold or n-fold perspective (ibid., 128). In his Zusiitze, Lambert was challenged by a problem that may have been discussed already in antiquity, and which was certainly taken up during the Renaissance and later: the creation of stage decorations. He claimed that it has caused considerable headache, since there is no perspectively correct solution that works for all the seats in the theatre. In a typical way, he gave a number of suggestions for obtaining the optimal solution (ibid., 142-152).

688

XII. Lambert

FIGURE XII.53. A perspectival garden in which the part below the line Bb is situated in a horizontal plane and the part above in an oblique plane, say a. Lambert assumed that the two parts of the garden wall bending at the point B lie in the same vertical plane, and that the same applies for the wall bending at b. He let it be given that VW is the horizon, n the principal vanishing point, Aa the ground line, the segment n V is equal to the distance, and that the angle between the horizontal plane and a is equal to L. VPn. He then explained how the different elements in the picture were constructed, among them the vanishing line ML of a, and the orthogonal projection P of the eye point upon ML. He further let F be the point of intersection of the line PB and the normal to the ground line through A; c, D, and E the points in which the line Pn meets the lines Bb, Aa, and the horizontal line through F, subsequently claiming that the line of intersection of the picture plane and a is the line FE. In proving this he applied the observation that a line segment that perspectively and geometrically has the same length - meaning that it has not been foreshortened - must be its own image. Thus, in looking at the line segment FE, which lies in the picture plane and is the image of a line segment in a, he proved (see below) that FE has the same length perspectively and geometrically. Hence FE lies in a as well, implying that this plane cuts the picture plane at FE. Lambert's proof that FE has a common perspective and geometrical length runs as follows. The lines BF and cE in a have P as a common vanishing point and are therefore perspectively parallel. The pair Be and FE are also perspectively parallel, hence FE =pBc. Similarly it can be seen that AD =p Be and FE =pAD. However, FE is also geometrically equal to AD which, as a line segment on the ground line, is represented by its geometrical length in the picture plane. Lambert 1759, figure 23.

J2. Ruler Geometry

FIGURE XII. 54.

689

A sky with constellations drawn in perspective. Lambert 1774, figure 54.

These examples are sufficient, I trust, to give the reader an impression of the special Lambertian approach, which was presumably the result of Lambert combining an interest in practice with his immense and wide-ranging scientific curiosity, as well as an unusually well-developed talent for observing phenomena in nature.

XII.12

Ruler Geometry

The Prehistory n a letter to Karsten dated 6 November 1773, Lambert mentioned that in the I end of he had Zusiitze

690

XIL Lambert

approached geometry again in order to see - under the guidance of the laws of perspective - how far one can go in elementary geometry if one is only allowed to use a ruler. 20

The investigation Lambert referred to is an extremely interesting application of his theory of perspective to traditional geometry, in particular to Euclidean constructions (whereby I understand the constructions that can be performed by the means Euclid allowed in his Elements: a pair of compasses and a ruler). Lambert wanted to find out, as he wrote to Karsten, what would happen if the compasses were left out - an idea he seems to have been toying with for some time, since he mentioned it as early as December 1766 in his Monatsbuch (LambertS 1916). Earlier mathematicians had also tried out the idea of restricting the means of construction. Probably inspired by a practice among artisans, a number of Italian Renaissance mathematicians - including Niccolo Tartaglia, Ludovico Ferrari, and Giovanni Battista Benedetti - found out that all the Euclidean constructions can be performed using only a ruler and compasses with a fixed opening (RoeroS 1997, 54-55).21 This conclusion was confirmed by the Danish mathematician Georg Mohr in 1673. The previous year Mohr had, in fact, published an even more revolutionary observation, namely that the Euclidean constructions - apart from the drawing of a line segment - can be performed with compasses alone. In connection with this latter observation he took up various perspective problems, as we saw in section VII.5. Mohr, who lived in the Northern Netherlands in the early 1670s, may have found the inspiration to look at constructions performed using restricted means in one of Frans van Schooten's works (SchootenS 1656 and SchootenS 1660). Remarking that to avoid the use of compasses, surveyors applied various other instruments, van Schooten suggested what he called a real "geometrical" approach to the topic of constructing without compasses. He replaced the Euclidean postulate concerning the construction of circles with one that permits line segments to be transported from one line to another. Van Schooten then presented ten problems that can by solved using a ruler and his means of moving line segments. The next natural step is to wonder what would happen if one were allowed to use only the ruler. To what extent early modern mathematicians worked on this problem has, as far as I know, not yet been thoroughly investigated, but I am aware that 'sGravesande and Taylor both stressed that they had performed particular constructions using nothing but a ruler. We shall soon meet two examples of this, and more are to be found in 'sGravesande Essai de perspective ('sGravesande 1711, for instance §§31, 35, and 59). 20••• der Geometrie wieder geniihert, um nach Anleitung der perspectivischen Regeln zu sehen, wie weit man reicht, wenn man sich in der Elementargeometrie schlechthin nur den Gebrauch des Lineals erlaubt. [LambertS Brie/e, vol. 4, 325] 21In working with restricted means of construction, the early modern mathematicians showed that the means allowed were sufficient for performing the basic constructions in Euclid's Elements - rather than attempting to prove that all possible Euclidean constructions can be performed.

12. Ruler Geometry

691

It is unclear whether Lambert was acquainted with any of the ideas mentioned here - though we can be fairly sure he did not know about Mohr's books, which were published in Danish and Dutch and soon forgotten. Lambert himself stated in his Zusiitze that perspective had inspired him to look at constructions that can be performed with a ruler alone, because in perspective constructions a ruler is applied much more often than compasses (Lambert 1774, 161). The construction of a line that passes through a given point and is perspectively parallel to a given line, for instance, requires only a ruler. Playing with the words Linearperspektive and Linealperspektive Lambert called his new discipline Linealgeometrie (ibid., 162) - which I translate as ruler geometry.

The Steiner Circle ambert was well aware that not all Euclidean constructions could be performed using nothing but a ruler. In fact, he presented a number of examL ples that called for more than a ruler. He then assumed that a geometrical

configuration was given. In two of his examples he required one circle and its centre to be given, whereas in other cases he required less, apparently applying the principle of assuming that as little as possible was given. If Lambert had instead looked for uniformity in his assumptions, he would undoubtedly have realized that in all his examples a circle and its centre would suffice - and he might then have got the idea that all Euclidean constructions - apart from constructions of circular arcs - can be performed when a circle and its centre are gIven. However, as matters stand, Lambert only anticipated this result. It was first expressly formulated by Poncelet and published in 1822, together with an untraditional proof (PonceletS 1822, §§353-355). His proof was not generally accepted, as exemplified by the German mathematician Jakob Steiner, who in 1833 referred to an earlier conjecture rather than to the theorem about making all Euclidean constructions with ruler alone when a circle is given (Steiners 1833, §l). As it happens, Steiner himself provided a proof that is normally considered to be the one that first established the theorem, and which resulted in the expression the Steiner circle.

Perspective Freedom erspective was not only a source of inspiration for Lambert's investigation of ruler constructions; he also applied the discipline in an important P way in his solutions. He thus interpreted a construction problem as a problem concerning the perspective image of a figure, and he benefited from the following fact. When dealing with a figure in the picture plane 1rfor which no eye point is given, one can make a number of assumptions concerning the original figure - each of which puts some condition upon the position of the eye point. I call this possibility the perspective freedom.

692

XII Lambert

In general, Lambert assumed that 1r is vertical and that a given figure in it is the image of a horizontal figure. He then used the remaining part of the perspective freedom as follows. Among the horizontal figures that can be mapped into the given figure in 1r - depending on the position of the eye point - Lambert chose the one with the nicest properties, which would typically mean the figure had some parallel lines. This technique is similar to one used by later mathematicians who projected a point in a figure into the point at infinity. In most cases Lambert's choice led to the determination of an eye point for the given figure, and of the shape of the horizontal figure mapped upon it. Lambert had previously used his procedure in a mathematical textbook, where he treated a geometrical example in which the information given was inadequate to solve the problem uniquely (LambertS 1765, §158). Lambert's strategy in ruler geometry resembles the one he used to solve inverse problems of perspective, but there is an extremely significant difference between the aims of these two undertakings. In treating inverse problems, Lambert's concern was to reconstruct as much as possible of the original situation uniquely, whereas in ruler geometry his idea was to manoeuvre with a multiple choice. Actually, his way of solving problems in ruler geometry has made mathematicians and historians of mathematics to count him among the protagonists in the history of projective geometry - a theme I will soon come back to.

Lambert's Examples

F

igure XII.55 lists the examples Lambert treated in his section on ruler geometry. In several cases he also looked at variants of a problem, but in the list I have only included his first version. Given specially

Given in the example

Construct

3

a parallelogram

four points in one plane a vanishing line v and a point Von it a point P and a line I

4

a circle and its centre

more points on an ellipse through the four points points on v 'lying 30° away' from V the line through P parallel to I a line perpendicular to I

Example

I 2

5

6

two equal collinear line segments, AB=BC

a line I an inaccessible point of intersection of two lines and a point A a number n

the line through A and the inaccessible point the points that divide A C into n equal line segments

12. Ruler Geometry two sets of collinear equal line segments, A]B] B]C] andA 2B2 = B2 C2 , situated on two nonparallel lines

7

a line I

a line parallel to I

an angle

the Pythagorean theorem in the perspective plane the line halving the angle

693

=

8 a circle and its centre AE= ECon one line, BE = Y1.DE on another line intersecting the first

9

10

a parallelogram

three lines I, m, and n intersecting at the point B

the line through B parallel to the lines from which I, m, and n cut off equal line se.lnllents

B,

a point P

the line through P parallel to I

B,

a point P

the line through P parallel to I

11

12

B,

A1

C

A,

A] A 2 : A]C= B ]B2 : B]Con the line

13

A,

A,

C

B,

A] A 2 : A]C= B JB2 : B]Con the line I

14

B~:

the normal to BC through B

~J

LABC=LCBD

15 A B C

D~~E

the normal to AC through B

A, B, and C collinear and LABD=LEBC FIGURE

XII.55. Table showing the problems Lambert took up in his ruler geometry.

694

XII Lambert

Examples 2 and 8 not only use the laws of perspective, but also concern problems about perspective constructions. Example 8 deals with a perspectival version of the Pythagorean theorem - it being required to construct a right-angled triangle and the squares on its sides directly in n. Examples 1 and 5, to which I return, are rather special. The remaining problems can be characterized as being similar to constructions treated in Euclid's Elements. They concern constructions of parallels, normals, angle bisec;tors, and points that divide a given line segment into a number of equal line segments. Another classic problem - the one of dividing a given line segment into a given ratio occurs implicitly among Lambert's examples, because it can be derived, as he himself pointed out, from his solution to example 3 (Lambert 1774, 171). The rest of this section is devoted to a more thorough presentation of some of Lambert's examples, the aim being to demonstrate that Lambert's technique led to fascinating results. By and large I concentrate on the examples he gave first, since an understanding of his later examples requires familiarity with his solutions to earlier problems.

Points on a Conic

L

ambert's first example was inspired by the following general problem (ibid., 165). Given five points in a plane, use a ruler alone to construct

more points on the conic section passing through the five points (one or two straight lines and a circle being included among the conic sections). However, he started with a different version: Example 1. Given are four points with the property that any three of them form a triangle outside which the fourth point lies. Using a ruler alone, construct more points on one of the ellipses that pass through the four points (ibid., 163).

XII.56. Let the points A o ' Bo' Co' and Do on the circle be the originals of four given points in 1! and let arc AoBo arc BoCo' Lambert introduced a new point Eo on the circle as the point of intersection of the line through Bo parallel to AoDo and the line through Co parallel to BoDo An easy geometrical consideration shows that Eo does indeed lie on the circle. By considering the chords AoEo and BoEo instead of AoDo and BoDo he similarly constructed a sixth point on the circle. This process can go on indefinitely unless the arc AoBo is a rational part of the entire circumference. FIGURE

=

12. Ruler Geometry

695

Lambert's solution strategy was, as explained earlier, to assume that the four points are points in the picture plane 1r and to use the perspective freedom. It is not always straightforward to decide which choices this freedom allows, and Lambert himself did not argue to support his assumptions. However, by reversing the order of his steps it is possible to check that the choices he made are permitted - I shall be more specific on this point when I present some of his examples. In example 1, the data gave Lambert another type of freedom, namely the freedom to choose between the ellipses passing through the four given points. He used this choice, together with the perspective freedom, to suppose that a horizon in 1r is given, and that the four given points are images of four points situated on a circle (figure XII.56) such that three of them determine two equal arcs. These assumptions imply that a fifth point Eo on the circle can be obtained by a construction that, when transferred into 1r, only involves a ruler and hence produces a point E on the ellipse, constructed with a ruler alone. He proceeded by constructing a sixth point by means of the fifth, and so forth. Finally, he discussed the changes to be made if five points on an ellipse had been given instead (ibid., 165).

A Specific Application of the Perspective Freedom n example 2, Lambert dealt with the problem of constructing vanishing Ivanishing points with a ruler alone. In one version of the problem he assumed that a point is given on the horizon and asked for the vanishing points

V of horizontal lines that form an angle of 30· with the lines whose vanishing point is V. To solve this problem he made use of the fact that the perspective freedom implies the following result (ibid., 166).

Result 2. Given are an arbitrary triangle ABC and a point D inside it. In 1r the triangle can be considered to be the image of an equilateral triangle, and D to be the image of the point of intersection of its medians.

Lambert did not indicate how he had come to this conclusion; my understanding - as explained in the captions of figures XII.57 and XII.58 - is that he deduced it by involving a result equivalent to Desargues's theorem.

Constructing a Parallel

W

hile Lambert's first two problems really can be solved with nothing but a ruler, he needed something more when he took up the classical problem of constructing a parallel to a given line through a given point. He formulated this problem three times, in examples 3, 12, and 13 - of which example 3 even covers several versions. Lambert began with the following version (figure XII. 59).

696

XII. Lambert

w

u

A FIGURE XII.5? Given are a triangle ABC and a point D inside it. To prove result 2, I consider the plane defmed by ABC as a vertical perspective plane 1!, and let A', B', and C' be the points in which the lines through the vertices and D meet the opposite sides. The perspective freedom gives us the possibility of assuming that AB and A'B' are perspectively parallel, which means that their point of intersection U is a vanishing point. Similarly we can assume that AC and A'C' are perspectively parallel and their point of intersection V is a vanishing point. Because the triangle A'B' C' is a projection of ABC (with D as the projection centre), it follows from Desargues's theorem that the point of intersection, W, of BC and B' C' lies on the line UV, which means that these lines are also perspectively parallel. (I suggest, in the caption of figure XII.58, a way of concluding this result without involving Desargues's theorem directly.) The assumptions made thus far have fixed the horizon. However, there is still so much freedom left that we can assume that AA' is perspectively perpendicular to BC, and BB' perspectively perpendicular to AG. When X is the vanishing point of AA', the first assumption implies that the eye point 0 lies on the horizontal circle from which XW is seen within a right angle - that is, the horizontal circle with diameter XW. The second assumption similarly fixes a horizontal circle containing 0, and hence 0 is determined. The assumptions concerning orthogonality imply that D is perspectively the point of intersection of the normals of the triangle. This, combined with the fact that the sides of triangle A'B' C' are perspectively parallel to the sides of triangle ABC, lead after some geometrical considerations - to the conclusion that ABC is perspectively an equilateral triangle and D its midpoint.

Example 3. Given are a parallelogram ABeD, a line I, and a point P. Using a ruler alone, construct the line m that passes through P and is parallel to I.

Lambert solved this problem very elegantly (figure XII.59) - indeed so elegantly that in his work on projective geometry Poncelet presented Lambert's solution (PonceletS 1822, §199). As Poncelet also pointed out (ibid.), this problem had been solved earlier by 'sGravesande, but in a manner different from Lambert's. As for

c

A

FIGURE XII.58. An alternative proof. Lambert presumably did not know Desargues's theorem in the form mentioned in the caption of figure XII.57, and I therefore point to another result Lambert could have used.

Result 3. Let ABC be a triangle and D a point inside it, and let A', B', and C be the points in which the lines through the vertices and D meet the opposite sides. Further, let it be given that AB is parallel to A'B' and AC parallel to A'C'. Then the line BC is also parallel to B'C'. This theorem can be proved by classical means, for instance by applying the so-called Menelaus's theorem.

FIGURE XII.59. Lambert's example 3. The line I, the point P, and the parallelogram ABCD are given, and Lambert wanted to construct the line m through P parallel to I. He began with a presentation of his solution and then explained why it is correct. I reverse the order and first paraphrase - with more details than he himself gave - his ideas leading to the solution. He introduced a vertical perspective plane and conceived of I as the ground line, the required line m as the horizon, and the given point P as the vanishing point of the lines AD and Be. A perspective projection is then fixed, in as far as it is known that its eye point 0 must lie on the horizontal line through P parallel to the line obtained by turning AD around I to a horizontal position. There is still freedom to place o on the above-mentioned line. Lambert used this freedom to decide on the direction of the image Ga of the line defined by the diagonal CA. He then effortlessly constructed the image abed of the given parallelogram and in particular, found the vanishing point Q of the lines AB and CD. The point Q, together with P, determines the required line. According to this characterization of the line m, Lambert presented the following construction. Determine the points E, F, G, H, and I in which the lines BC, AD, AC, AB, and CD meet the line I. Connect P with E and F, chose the point a on PF, let e be the point of intersection of PE and Ga, and fmally let Q be the point in which the lines Ha and Ie meet. Draw the line PQ. Lambert 1774, figure 65, with the letters I and madded.

698

XII. Lambert

'sGravesande, he took up the problem in a mathematical textbook from 1727, where its occurrence is a bit surprising because it is presented in a section otherwise dealing with how to solve problems analytically CsGravesandes 1727, §312). The reason 'sGravesande decided to include a construction using a ruler could be, as suggested earlier, that some mathematicians of the early eighteenth century became interested in constructing with a ruler alone. The solution 'sGravesande reached was a traditional geometrical one - without any perspective interpretation. Poncelet's own interest in the problem seems to stem from 1808, when it was formulated by the French mathematician Charles Julien Brianchon (BelhosteS 1998, 23-24). In fact, Poncelet responded to Brianchon by offering a solution that did not involve perspective either (PonceletS 1810,273).22 Lambert elaborated on his example 3 by changing the assumption about the given elements in various ways. First he looked at the situation in which I is parallel to two sides of the parallelogram. He found that in this case, one does not need the entire parallelogram, and he actually solved the problem of constructing, with a ruler alone, the line through P parallel to I when the only given besides I is a line parallel to it (Lambert 1774, 170). Reconsidering his solution to the first version of the problem, Lambert came to the conclusion that it could also have been solved if two equal line segments on I had been given instead of the parallelogram (ibid., 171). Next he assumed that five points E, F, G, H, and I fulfilling EF : FG =HI: GH are given on the line I. This problem is close to the question Lambert took up in example 13 after treating a similar problem in example 12 - the order of the points being different. His solutions to the two last mentioned problems are very simple. He did not prove the correctness of his constructions, but an easy and elegant proof can be obtained by using the same idea as he himself applied in the proof of the last-mentioned version of example 3 (ibid., 170). The construction belonging to example 12 and a proof of it are shown in figures XII.60 and XII.61.

A Line Through an Inaccessible Point final Lambertian example to be treated here is one that only requires (figure XII.62): T aheruler Example 5. Given are a point A, and two lines I and m that meet at an inaccessible point. Using a ruler alone, construct the line that passes through A, and whose extension passes through the inaccessible point (ibid., 172-173).

22For more on the history of this problem and ruler geometry, see Archibalds 1949, 98, note 277.

12. Ruler Geometry

699

H

A1 XII.60. Lambert's example 12. Given are the point P, the line I, and on it the points A l' A}, BI' B}, and C fulfilling

FIGURE

AI A2 : AIC = B I B 2 : BIC

(I)

It is required to construct the line through P parallel to I. Lambert's construction was as follows (Lambert 1774, 178). Connect Aland A} with P, then chose a point H on A}P extended, and draw the lines HB} and He. Let 1 be the point of intersection of the latter and A /p, and let Q be the point of intersection of B/ and B}H, which means PQ is the required line. The proof that this solution is correct is given in the caption of the next figure.

Q

A1

c

FIGURE XII.61. Lambert's example 12, continued. To prove the correctness of Lambert's construction, presented in the caption of the previous figure, I use an idea similar to one Lambert himself presented for his solution to example 3. Keeping the previous figure in mind, let us conceive of I as a ground line and H as the principal vanishing point. This makes the configuration in figure XII. 60 the image of the present diagram where B}Q, Cl, and PAl are parallel. In triangle A/PA} the transversallC is parallel to PAl' and in triangle B/QB} it is parallel to B}Q. Hence, the two pairs of triangles A/C and A/PA}, and B/C and B /QB} are similar, implying that ~A2:AIC=

lC:PA2,

B IB 2 : BIC = IC: QB2,

and hence, based on relation (I) in the caption of the previous figure, lC: PA2= IC: QB2.

This last relation shows that PAl is equal to QB}, and hence that PQ is parallel to A/A} - that is, parallel to I.

700

XIl Lambert F

FIGURE

XII.62. Example 5 for cases in which A lies between I and m.

Lambert solved this problem as follows. First he drew two arbitrary lines through the point A intersecting the lines I and m in B, C, D, and E in such a way that BE and CD intersect each other at a point F. Next he drew a line through F meeting I and m in G and H, respectively. Finally, he determined the point of intersection I of CG and DH, asserting that AI is the required line. In his proof Lambert used the perspective freedom to assume that I and m are perspective images of parallel lines, and that BE, DC, and GH are images of three lines that are perpendicular to I and m. Perspectively, then, the quadrangles BDCE and DCHG are rectangles whose midpoints are the two points A and I. This implies that AI is perspectively parallel to I and m and hence passes through their inaccessible point of intersection (the vanishing point of the two lines in the perspective interpretation). Lambert actually did not need to assume that BE, DC, and GH are perspectively perpendicular to I and m; the proof also works if they are simply considered to be perspectively parallel. In his drawing Lambert only considered the case in which the point A lies between the lines I and m, but his solution also applies if this is not the case (figure XII.63). One comes across Lambert's example 5 in various contexts throughout the history of mathematics. Thus, Taylor included it in his Linear Perspective because he wanted to provide his readers with a technique they could apply when a relevant point, for instance a vanishing point, is situated outside the material upon which a perspective construction is performed. Taylor offered two solutions to the problem and stressed that one of them does not require a compass (Taylor 1715" 30). His solution - which covers both of the two positions of the given point in relation to the given lines - is very close to Lambert's. Contrary to Lambert, however, Taylor did not prove the correctness of his construction. In his New Principles from 1719, which is presumably the only one of Taylor's works on perspective Lambert ever saw, Taylor did not treat Lambert's example 5. It therefore

12. Ruler Geometry

701

A

FIGURE

XII.63. Example 5 for cases in which A does not lie between I and m.

seems unlikely that Taylor inspired Lambert to look at the problem of constructing a line to an inaccessible point of intersection with a ruler alone. At any rate, Lambert added a new angle to the problem by interpreting it as a perspective problem. The problem of constructing a line through an inaccessible point was taken up again, as Roger Laurent has also pointed out, by French mathematicians in the beginning of the nineteenth century (LaurentS 1987, 183-184). In the January 1807 issue of Correspondance sur !'ecole imperiale polytechnique, Louis Poinsot invited the readers to solve the problem with a ruler alone (PoinsotS 1807). Although the editor, Jean Nicolas Pierre Hachette, received several solutions, he published his own anSwer in the following issue of the journal, claiming that his solution was close to Poinsot's - and the most elegant. It is striking how close Hachette's approach is to Lambert's. Like Lambert, Hachette based his arguments on the theory of perspective. When the given point A lies between the given lines, Hachette's solution is, in fact, identical to Lambert's, whereas he applied a slightly different procedure when A does not lie between the two lines (HachetteS 1807, 306). He referred readers unfamiliar with his way of arguing to his article On perspective, which was published in the same issue of Correspondance. In 1908, Gino Loria commented upon Lambert's solution in the following way. The construction recommended by Lambert is still in use. It is based on the harmonic properties of a complete quadrilateral,23

23 Die von Lambert empfohlene Konstruktion ist noch heute in Gebrauch; sie stiitzt sich auf die harmonischen Eigenschaften des vollstiindigen Vierecks. [LoriaS 1908, 613]

702

XII. Lambert

Such translations of certain Lambertian results into projective geometry which Loria was not the only one to do - seem to have induced some mathematicians and historians of mathematics to think that there is also a developmentallink between Lambert's work and projective geometry.

Perspective, Ruler Geometry, and Projective Geometry

B

efore rounding off the presentation of Lambert's work on perspective, I find it relevant to make a few remarks on the relation between the history of perspective, ruler geometry, and projective geometry, focussing upon Lambert's contributions. My first point is that ruler geometry seems to have been cultivated independently of Lambert by some French mathematicians in the beginning of the nineteenth century. In fact, synthetic geometry was revived by a Parisian school of geometers, founded by Monge and joined by a considerable number of French mathematicians. Several of them, more specifically Fran~ois Joseph Servois and some students of the influential mathematician Lazare Camot, became interested in constructing with a ruler alone around 1803 (SommerS 1914, 790; Zachariass 1913, 1090), so when Poinsot took up this theme in 1807, it was not new. It is presumably also through the mentioned Parisian geometers, with whose work he was familiar, that Poncelet, the creator of projective geometry, became aware of ruler geometry. Indirectly, he himself supported this view, for after presenting Lambert's work on ruler geometry in his Traite, Poncelet added that this kind of synthetic geometrical problems had been completely forgotten in France until they were taken up again by Monge and his pupils (PonceletS 1822, xliij-xliv). One may then wonder how and when Poncelet became acquainted with Lambert's work. I tend to believe this happened between 1814 and 1822, that is, after Poncelet returned from Russia - where he developed projective geometry and before he published his Traite. During this period Poncelet made himself acquainted with publications related to his field, seeking, among other things, problems that could be solved more elegantly with his new geometry than by traditional means. It is actually amazing how very thoroughly he studied the mathematical literature and how many references he included in his Traite. My second point is that perspective had no influence on the development that ultimately led to the creation of projective geometry. We saw that Hachette applied perspective to solve the problem of constructing a line through an inaccessible point, but that was a special case. In their work on ruler geometry the French mathematicians generally did not involve perspective, but took an approach that is more likely than perspective to have inspired Poncelet in his work leading to projective geometry. Even though central projections surely played an important role in Poncelet's work and he clearly saw their connection to perspective, and also applied the term perspective broadly, this does not imply that the existing theory of perspective is what inspired him.

13. Lambert's Impact

703

In chapter IX, I argued that in the seventeenth century there was very little interplay between perspective and Desargues's projective ideas because the type of problems taken up in the two disciplines and the methods employed were so different. The same conclusion holds true for the nineteenth century, since only very few of Poncelet's results and construction problems relate to problems in the theory and practice of perspective. It would have been a grand finale to the story on the development of the mathematical theory of perspective to say that it helped give rise to a discipline that achieved great success in the nineteenth and twentieth centuries, but I am afraid the conclusion is that neither Lambert nor perspective contributed in any essential way to the birth of projective geometry. It was only after creating this subject that Poncelet realized that a few of the problems he took up had also been treated by Lambert - and by Desargues. Although Lambert was not influential in the history of projective geometry, he - like Desargues - could have been an isolated figure in the development. However, I do not think this is how he should be thought of. The reason is that he himself saw his use of projections as part of perspective, and did not attempt to isolate his considerations concerning projection, nor did he introduce new concepts or search for invariance properties. A mere three years after his Zusiitze had been published, death stopped Lambert from pursuing his ideas. It is difficult to say whether he had any intention of continuing his work on ruler geometry, and if so, whether he would have recognized the strong potential of his projective techniques.

XII. 13

L

Lambert's Impact

ambert's contribution to the mathematical theory of perspective is indeed impressive. By creating perspective geometry he concluded the process of corning to understand the geometry behind perspective. He himself remarked that he had found general principles by searching for the theory forming the foundation of the rules of perspective (Lambert 1759, preface or Vorrede, 2f-V). Actually, he did this so well that in the theory of perspective per se, little work was left to do after Lambert. In addition to completing the theory of perspective, Lambert also began to apply it in classical geometry by investigating ruler geometry. In Lambert's day, his remarkable contributions went largely unnoticed. In section XI. 6, I have described how a few authors from German-speaking Europe admired his ideas and applied a few of them, but without developing the Lambertian approach to perspective further. And elsewhere, I have found no other signs of his influence. To some extent Lambert himself was responsible for the obscurity that befell his work on perspective. He did not present his insights in a scientific paper written for those who would have been able to understand and appreciate him

704

XII Lambert

- his fellow mathematicians. Instead he chose a style suitable, in principle for people applying perspective. One of the results was that Karsten, who was a real admirer of Lambert, characterized Freye Perspektive as das schone Handbuch (the lovely handbook), stating that it contained all that was needed in connection with perspective and painting (Karsten 1775, 190). Such remarks give no impression that the book does, in fact, contain a wealth of fascinating theoretical material as well. By not adjusting his style to mathematicians, Lambert behaved just like 'sGravesande and Taylor. As mentioned earlier, the fact that these three authors chose this path is a sign that perspective was not considered a subject that was mathematically interesting for its own sake. Moreover, none of them seems to have made any attempt to remedy this situation, although several of the problems they treated were on a par with questions taken up in mathematical journals - one example being the recently presented problem raised by Poinsot. Undoubtedly the three mathematicians also wished that their results reached the practitioners, but success in this endeavour would have required them to be less abstract. As for the form applied by the mathematicians, Monnich wrote that profound scholars like Lambert seldom took sufficient trouble to reach the laymen in their field, but lost themselves in the higher spheres so familiar to themselves (Monnich 1794/1801, fol. A3 V). Lambert's work on perspective is in principle applicable, but only for those with a good mathematical background - or those willing to copy procedures without understanding them. Simple copying has often taken place in the history of perspective, and it generally went well, because it concerned rather simple procedures for which the risk of making mistakes was not that great. In the case of Lambert's examples, however, it would require quite a bit of luck to get correct results without understanding the underlying rules. Taylor's ideas on perspective experienced a renaissance about four decades after he presented them, as we have already seen. On a smaller scale, Lambert's work on perspective apparently had a similar revival: Sabine Siebel has documented that around 1800, a group connected to the academies of art in Berlin and Dresden advocated basing the teaching of perspective on Lambert's ideas (SiebelS 1999, 23, 91, 96, 169-171). She has also pointed to some early nineteenth-century German authors who, to some degree, were inspired by Lambert. 24 My survey of the literature on perspective only includes works composed before 1800, so I have not analysed the Lambert-inspired literature from the nineteenth century in detail. However, my impression is that the titles mentioned consist of works that introduce methods of perspective construction without any attempt to make the reader understand the underlying geometry - contrary to some of the Taylor-inspired literature. It was thus the practice rather than the theory of Lambert's free perspective or perspective 24Among them Weinbrenner 1817 and Hummel 1824-25. I can add that also Eyte1wein 1810, which is a handbook, seems to be inspired by Lambert, or by other authors who took up Lambert's approach.

13. Lambert's Impact

705

geometry that caught people's attention. Still, it is interesting that Lambert had a group of admirers - including the painter Johann Erdmann Hummel, who even used the expression "our Lambert": We have to thank the acumen of our Lambert for the foundation ... of the theory that includes all that which can be called perspective drawing; any attempt to think out something more perfect would be futile. 25

It is noteworthy that, just as Taylor coined the phrase linear perspective, Lambert's use of the expression free perspective seems to have influenced later writers in German. Actually, I have met the expression in a title where it clearly meant linear perspective rather than what Lambert had in mind (Peschka & Koutny 1868). The examples of Lambert's free perspective being used in introductory courses on perspective do not change my impression that his ideas did not have much of a theoretical impact. I therefore repeat what I wrote in the introduction to this chapter: that Lambert's work on perspective stands isolated in the history of the mathematical theory of the discipline. While it is remarkable that none of the eighteenth-century mathematicians treating perspective continued Lambert's work on perspective geometry, it is more understandable that this did not happen in the nineteenth century, as the mathematicians' attention was diverted elsewhere, namely first to descriptive geometry, and later to projective geometry. Perspective was treated within both subjects, but not in ways resembling the Lambertian approach. In the next chapter I illustrate this point for descriptive geometry presenting a construction that is very different from Lambert's. As to projective geometry, it basically incorporated ideas similar to Lambert's, and his theory could be deduced from this discipline. However, the kinds of problems dealt with in the theory of perspective were not the core problems dealt with in projective geometry, which probably explains why the theoretical link to Lambert was not emphasized.

Begriindung ... alles was perspektivische Zeichnung genannt werden kann, umfassenden Lehre, verdanken wir dem Scharfsinn unseres Lamberts; und vergeblich wiirde man versuchen etwas Vollkommeneres auszudenken. [Hummel 1824, Vj

25 Die

Chapter XIII Monge Closing a Circle

A

bout three hundred years after Piero della Francesca presented his perspective construction based on the plan and elevation technique, this perspective method enjoyed a veritable renaissance as part of descriptive geometry. This brief chapter deals with the emergence of descriptive geometry, how the discipline shares one of its basic idea with Lambert's perspective geometry, and how perspective is treated in descriptive geometry.

XIII.!

Monge and Descriptive Geometry

Creation of Descriptive Geometry

T

he name of the French mathematician Gaspard Monge (1746-1818) has cropped up earlier in connection with a description of the further development of applying plans and elevations to make graphical representations. In the late 1760s, while teaching at the Ecole royale du genie (The royal school of engineering) at Mezieres, Monge began to combine the old plan and elevation technique with methods that would later become part of differential geometry. Gradually he developed a new discipline that he called descriptive geometry.l During the French Revolution and the Empire, Monge was given important responsibilities, political as well as administrative. He was, for instance, very influential in renovating the Parisian system of higher education and in deciding the curricula at the Ecole normale and the Ecole polytechnique. At both these schools he made descriptive geometry an essential part of the syllabus. At the Ecole normale he held thirteen lectures on descriptive geometry during 1795, and had his lecture notes duplicated. Under the editorship of Monge's assistant, Jean Nicolas Pierre Hachette, the first nine lectures appeared in 1799 as Geometrie descriptive - the last four lectures were not included, as Monge thought they needed to be refined (TatonS 1951 3,41-42). lPor a brief introduction to the history of descriptive geometry, see Andersen & Grattan-Guinnesss 1994. Por thorough analyses of Monge's work, see TatonS 1951 3 and SakarovitchS 1998. 707

708

XIII Monge Closing a Circle

Hachette's edition was reissued several times, and from 1820 it was enlarged with three additional lectures dealing with shadows, aerial perspective,Z and linear perspective in an edition supervised by Monge's student Barnabe Brisson.

Monge's Descriptive Geometry n innovating the plan and elevation technique, Monge rotated the halfItheplane of the elevation, LMPQ (figure XIII.l), down into the half-plane of plan, LMNO, around their line of intersection, LM, and looked at the geometry in the plane consisting of the plan and the turned-down elevation. I call this plane the descriptive plane - Monge seemingly had no special term for it. In the descriptive plane a point in space, say A, is represented by the two points a and a" lying on a line perpendicular to the line LM. A line segment, say AB, is consequently represented by the pair of line segments ab and a" b" connecting the representations of the two end points (if these points have either a common plan or a common elevation, one of the line segments degenerates into a point). Monge described a number of fundamental connections between elements in three-dimensional space and their representations in the descriptive plane, such as the relationship between the length of a line segment and the lengths of the two line segments representing it. More important for the present story, however, is the fact that Monge also showed how a number of standard Euclidean constructions can be performed directly in the descriptive plane. His very first example concerns the problem of how to construct the line through a given point that is parallel to a given line - where the latter does not contain the given point (Monge 1820, §14). It is interesting to compare Monge's programme with Lambert's. Both applied a plane to represent the three-dimensional space - Lambert working with the perspective plane and Monge with the descriptive plane. Both mathematicians also became engaged in solving the problem of how to perform Euclidean constructions directly in their planes of representation. As Lambert's main work on perspective was published in 1759 - and actually in French - the possibility exists that Monge was inspired by Lambert's idea of working directly with the geometry in a representation. As noted in chapter XII, however, I have found no indications that Monge was familiar with Lambert's ideas or modelled his work on them.

zPerspective aerienne, by which Monge meant the science of light and shadow (SakarovitchS 1998,211).

2. Monge and Linear Perspective , ..,., ~

_~ .

709

p

,.'

/'

..

!

i

:

!

L FIGURE

o

XIII. I. A line segment projected upon the descriptive plane. Monge 1820,

figure 2.

XIII.2

Monge and Linear Perspective

Monge's Presentation of Perspective onge's interest in linear perspective dates back to at least 1768, when he M wrote a small treatise on the subject based on a plan and elevation construction (Sakarovitch 1998,231). Twenty-seven years later, while lecturing S

at the Ecole normale, he took up perspective as a way to apply his descriptive geometry. His approach to perspective seems to be influenced more by the early writers than by authors closer to his own time. Monge thus introduced the idea of the picture plane as a pane of transparent glass, and he described a perspective image as a section in a visual pyramid (Monge 1820, §136). Not only did Monge use a pyramid when introducing perspective, he also dealt with it in an example of throwing an object into perspective (figure XIII.2) - which is, in fact, his only example. Like many of his predecessors, he applied a plan and an elevation of the pyramid and the picture plane to determine what corresponds to the coordinates of the image points of the pyramid. Compared to Piero della Francesca's presentation of a plan and elevation method for constructing perspective images, Monge's example offers

710

XIII Monge Closing a Circle T"

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XIII.2. Monge's example of throwing a pyramid into perspective. Monge 1820, figure 53.

FIGURE

absolutely nothing new. I therefore find it appropriate to say that Monge closed a circle when he incorporated perspective into his descriptive geometry. In fairness, it should be noted that Monge did not stop his treatment of perspective after presenting his pyramid example. He touched upon the theory of perspective, for instance stating the result that the image of a line is generally a line, and presenting the vanishing point theorem. He did not, however, use his descriptive geometry to deduce these results, but obtained

2. Monge and Linear Perspective

711

them by looking at the three-dimensional situation, just as his predecessors had done (ibid., §§136-137). Besides his few theorems, Monge included a number of observations, asserting, for instance, that it was a good idea to determine the visible part of an object before throwing it into perspective. For objects bounded by planes, he found this could be done easily, whereas objects having curved surfaces could present complicated problems. For solving the latter type of problem, he referred readers to his treatment of shadows (ibid., §136). He also held the opinion that students could benefit from their knowledge of descriptive geometry when the picture plane is a curved surface, or when they had to construct cylindrical mirror anamorphoses (ibid., §138). Finally, Monge dealt with the problem of transforming a perspective composition to another picture plane than the one for which it was originally made, while keeping the same eye point.

Monge's Influence on Teaching Perspective

W

e know that, largely thanks to Monge's efforts, descriptive geometry became a topic that was taught at many institutions for more than a hundred years. However, no investigation has been made into how common it was to include perspective in courses on descriptive geometry - and it is outside the scope of this book to undertake such a study. I am therefore unable to say what the return to a plan and elevation approach to perspective meant to the life of perspective geometry, which was partly introduced by Taylor and fully developed by Lambert. It seems fairly safe, however, to conjecture that descriptive geometry did not in any way contribute to the mathematical understanding of perspective.

Chapter XIV Summing Up

XIV 1 Opening Remarks

A

fter all these pages on the history of perspective, the time has come to ask: What have we learned? In giving partial answers I return to the key issues and questions presented in the introduction. For structural reasons I have changed the order and also reformulated some of the questions. Thus, in the next pages I sum up my findings on the following points: • • • • • • •

geographical differences innovations in the mathematical theory of perspective the interplay between perspective and other geometrical disciplines the status of the theory of perspective practitioners' appreciation of the theory of perspective communication between theorists and practitioners the usefulness of the theory of perspective

Finally, I ponder the driving forces behind the theory of perspective. Before addressing the points mentioned above, let me make a few general comments upon my source material. We have seen a few instances of books simply being copied) - and I do not guarantee that there are not other examples of this. We have also met highly original publications that inspired other authors as concerns their content, but not their style. The case of Guidobaldo del Monte and his influence on the works of Simon Stevin (1605), Fran~ois Aguilon (1613), and Samuel Marolois (1614) is the most spectacular example, but the British literature inspired by the work of Brook Taylor (1719) is also noteworthy. A large part of the literature lies between these two extremes, by which I mean that it is rather derivative without containing passages copied verbatim. I assume that an investigation of textbooks on other subjects would lead to a very similar result.

IThus, in 1505 Gregor Reisch copied parts of Viator's De artificialiperspectiva, and in 1509 Jorg Glockendon copied the entire book (page 166). Similarly, in 1790 Robert Bradberry published a book in which most of the content was taken from Benjamin Martin s.a. and the rest from William Hooper 1774 (page 591). 713

714

XIV. Summing Up

My last opening remark concerns women; I have not been able to find a single female author. My impression is that, particularly in the eighteenth century, many women took lessons in painting, and may well have been taught some rudimentary perspective techniques. Apparently, however, none of these women became so keen as to study perspective in such depth that they themselves wished to publish on the subject - although some women did write on science topics during this period. 2 The only woman mentioned in this book in connection with perspective is Mary Somerville - and she falls outside the period considered. She was taught perspective, and in this way she came to learn about Euclid's Elements (page 598), but rather than becoming interested in the mathematical aspect of perspective she turned her interest to other mathematical sciences.

XIV:2 Local Approaches to Perspective nowledge about perspective did cross boarders, but I nevertheless find that by treating contemporary authors from the same area together I have K been able to draw a picture of substantial regional diversities. Within each country there were different approaches to perspective, but we can still distinguish some general trends concerning the interest in the subject and the seriousness with which perspective constructions were treated in various periods. Occasionally repeating some of my earlier conclusions, I will, in the following paragraphs, survey the main lines in the geographically diverse developments.

The Italian Development

E

merging among a circle of Florentine artists, perspective became a muchdesired form of representation in painting during the fifteenth and sixteenth centuries. In the quattrocento two important texts on perspective appeared, written, respectively, by Leon Battista Alberti and Piero della Francesca. Both of these authors were clearly inspired by the painters and architects with whom they were in close contact - actually belonging themselves to these groups. In the cinquecento, a considerable number of publications on perspective appeared in Italy, and in several of these the authors showed a keen interest in coming to understand the geometry behind perspective, as Piero had done earlier. This development led to Guidobaldo's revolutionary approach to perspective, published in 1600. Seen from a mathematical point of view, the Italian literature was dominating until that date. In the seventeenth century, the interest of Italian patrons, painters, and mathematicians in perspective waned considerably, and at the same time the quality of the publications on the subject generally dropped. It is particularly 2To my knowledge, the first textbook on perspective written by a woman is Le Breton 1828.

2. Local Approaches to Perspective

715

remarkable that no Italian scholar followed up on Guidobaldo's ideas. Torricelli would have been more than capable of doing this, but he chose a practical approach, being of the opinion that the theory of perspective was too fatiguing for practitioners, and apparently taking no personal academic interest in the subject. As a result, the Italian tradition of writing on perspective was mainly kept alive by stage designers and painters who specialized in illusionistic paintings.

The French and Belgian Development

J

ust as perspective drawing and painting in Northern Europe was inspired by Italians but found its own expression, so did the literature on perspective. Viator's publications from the early sixteenth century were distinctly different from the Italian texts in style. As the century wore on, some new knowledge emerged in France, demonstrated in Jean Cousin's work from 1560, but probably from an earlier unknown source. Cousin also had a way of expressing himself that was not modelled upon Italian publications. In the seventeenth century, perspective caught the attention of many French and a few Belgian scientists and mathematical practitioners. The first group presented the subject in general textbooks on mathematics and other sciences, while the second group produced separate volumes on perspective in which they took a new approach, introducing perspective scales and performing constructions directly in the picture plane. Later in the seventeenth century, the intense activity in the field of perspective declined. Some textbooks still appeared during the period up to 1800, but they did not show any innovation, nor did they take up the advanced ideas presented by Brook Taylor and Johann Heinrich Lambert.

The German Development

C

learly inspired by Italian sources, Albrecht Durer published the first description in German of a perspective construction, or rather of two constructions. Durer awakened an enthusiasm for perspective in a Nurembergian group of artisans, several of whom published on the topic. It is characteristic for the early German works on perspective that they are geometrically imprecise, for instance by not including descriptions of all the steps needed to construct the perspective image of a figure. These books also show a profound fascination with perspective instruments. Just as the German authors were late in giving distinct presentations of perspective constructions, the mathematical theory of perspective also came late into the German literature. Eventually, at the end of the seventeenth century, it became common to include a section on perspective in German textbooks on mathematics. However, not until the l750s did these presentations reach the level of exactness that had been common in general books on mathematics in France and the Netherlands for almost one and a half centuries.

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XIV Summing Up

The first comprehensive mathematical work on perspective written by a German was published in 1775 - and its author was Wenceslaus Johann Gustav Karsten.

The Dutch Development the seventeenth-century Northern Netherlands, the involvement in prosome geometrical background to perspective ran parallel with interIestsnviding in creating and acquiring paintings with striking perspective compositions, as had been the case during the Renaissance in Italy. In fact, Simon Stevin, Samuel Marolois, and Frans van Schooten produced extremely good texts on the theory and practice of perspective during the golden era of Dutch painting. Both the academic and the practical interest in perspective stayed alive until the beginning of the eighteenth century, culminating with Willem 'sGravesande's work from 1711. Then came a long period without any Dutch publications on perspective. From 1765 the Dutch scene was dominated by the engraver Caspar Jacobszoon Philips, who was an extremely dedicated teacher of the practice of perspective.

The British Development

G

enerally on the Continent, with the notable exception of Germany, the interest in perspective dwindled during the last two thirds of the eighteenth century. Not so in Britain, however, where the writing on perspective actually flourished in this period. As was the case in Italy and the Northern Netherlands, the development in Britain may be taken as an example of the interest in perspective painting and writing on perspective constructions running parallel - though in the opposite sense, so to speak: The British only became interested in perspective drawings very late, and the same holds true for British writing on perspective. On the other hand, the literature then developed quickly within a fairly short span of years, during which perspective became well understood and well presented in Britain. Contrary to what was happening on the Continent, one person dominated the literature in Britain, namely Brook Taylor. Actually, a considerable portion of the numerous British presentations of perspective appearing in the second half of the eighteenth century were directly related to Taylor's work.

XIV: 3 Perspective and Pure Mathematics Innovations in the Mathematical Theory of Perspective

A

lthough perspectivists were mainly inspired by authors from their own geographical circles, some learned from other sources. In the development of the geometrical understanding of perspective at the highest level there was

3. Perspective and Pure Mathematics

717

clearly a continuous development in the sense that when an author took an important new theoretical step, he was, in almost every instance, building upon the most advanced theory of his day. For the main development, I have documented the following trajectory of inspiration: An interest among Italian mathematicians in the latter half of the sixteenth century animated Guidobaldo to take up perspective. Through a gradual process, he came to realize that not only orthogonals and diagonals have vanishing points, and saw the importance of the main theorem of perspective. Guidobaldo's insights travelled from Italy to the Northern Netherlands and were quickly adopted by Stevin. Directly influenced by Stevin, and possibly also by van Schooten, 'sGravesande picked up the theory of perspective. He did not add any decisive new results to the theory, but he appreciated and exploited its potential more strongly than his predecessors, for instance by focussing upon a very elegant and simple visual ray construction. Then, once more, the development crossed a border or, to be more precise, a sea. Taylor was inspired by 'sGravesande's mathematical understanding and added quite a bit of his own thinking. In fact, Taylor provided perspective with a new mathematical life, among other things by introducing and applying the general concept of a vanishing line. I have not been able to trace the sources for Lambert's mathematical knowledge of perspective, but we can be sure he did not reinvent the theory. His work was part of the continuous development of the theory of perspective, and he brought this theory as far as it could presumably be taken as an independent discipline. By this I mean that as for the specific question of how to project three-dimensional figures upon a plane surface, no important question seemed to have been left unanswered. Nor did there seem to be any way of carrying out perspective constructions more elegantly than by the methods advocated by Taylor and Lambert. As a consequence of this success, the theory of perspective became less attractive for mathematicians, to whom a field with no loose ends holds no appeal.

Interplay Between Perspective and Other Geometrical Disciplines

F

rom Commandino onwards, several mathematicians remarked that a perspective projection is just one of several examples of a central projection, and not rarely perspective was presented alongside other projections. However, no general theory of central projection was developed that could be applied to obtain results concerning perspective constructions. In fact, almost all the fundamental results in the theory of perspective were obtained on the basis of knowledge that have been around for almost two thousand years, primarily the theory contained in Euclid's Elements. In connection with investigations of the perspective images of a circle, Apollonius's theory of conics was also applied. In other words, the contents of the perspectivists' mathematical toolbox consisted, first and foremost, of classical geometry.

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XIV. Summing Up

Conversely, the theory of perspective did not offer much inspiration to other fields. Girard Desargues may have been influenced by his theoretical consideration on perspective when he introduced points at infinity - but other mathematicians introduced this concept as well, and did so independently of perspective. Similarly, cross ratios are to be found implicitly in Taylor's work, but it was not through the theory of perspective that mathematicians later realized that a cross ratio is an important concept in projective geometry, and that cross ratios are invariant under a central projection. Lambert came farthest in relating perspective to other disciplines. The elegance of his application of the theory of perspective to obtain a result that was a forerunner of Pohlke's theorem for parallel projections is striking. Even more impressive is his involvement of the theory of perspective in attacking the question of when a traditional construction problem can be solved using a ruler alone. If Lambert had lived longer and had continued his investigation of ruler geometry, he might have anticipated more of Poncelet's work, but as matters stand, the intensification of the work on central projection and the creation of projective geometry during the nineteenth century happened independently of perspective geometry.

The Status of the Theory of Perspective n early modern times a distinction was made between pure and mixed mathematics, and perspective was grouped in the latter category (for a disIcussion of perspective as a mixed science, see Peiffers 2002). This is understandable, as the theory of perspective was created for the explicit purpose of providing a foundation for perspective constructions. In nature, however, theoretical problems on perspective are somewhat similar to classical pure geometrical problems, and could therefore potentially be seen as an interesting part of theoretical mathematics, but that did not happen. Christiaan Huygens found geometrical perspective too easy to be of real interest, and 'sGravesande claimed that the theory was "barren of great discoveries" (pages 319 and 360). In 1758 Jean Etienne Montucla expressed a similar view, characterizing perspective as "a subject in which the greatest difficulties do not go beyond the reach of a mediocre geometer". 3 Comparing problems that were solved in perspective with those solved in classical geometry, I do not find that the perspective problems were significantly easier than the geometrical ones. Hence, the only way I can understand the attitude to perspective expressed by some seventeenth and eighteenthcentury mathematicians is that to be really interesting in those days, a topic had to relate to the two new important mathematical disciplines: analytical geometry and calculus.

3 Un sujet dont les plus grandes difficultes ne sont pas au-dela de la portee d'un Geometre mediocre. [MontuclaS 1758,636]

4. The Theory and Practice of Perspective

719

Regardless of how the mathematicians' sentiments on the theory of perspective should be explained, they had the ironic effect that the more advanced aspect of the theory - the part concerning vanishing lines and direct constructions - only became known to a select few. The reason for this was, as stressed in other chapters, that literature on perspective was in general dedicated to practitioners, who lacked sufficient mathematical education to understand what the mathematicians had written for them. If the theory of perspective had been held in higher regard within mathematics, more publications on perspective would presumably have been addressed to the mathematicians.

XIVA The Theory and Practice of Perspective The Practitioners' Appreciation of the Theory of Perspective

A

s noted, there were three major steps in the development of the theory of perspective: Guidobaldo's introduction of the general notion of a vanishing point and the main theorem of perspective, Taylor's development of the generalized concept of a vanishing line, and Lambert's creation of an abstract perspective geometry. The two last-mentioned innovations did not have any real impact on how perspective was presented in the practitioners' textbooks. The concept of vanishing lines was introduced in most of the Taylor-inspired English literature from the second half of the eighteenth century, but it was hardly ever applied, and outside the British isles it is all but impossible, to find a vanishing line mentioned in a textbook written by a practitioner. In the period examined, Lambert's perspective geometry was almost totally ignored by practitioners and certainly had no influence on the textbooks they composed. As to vanishing points, it is rare to see practitioners writing about other vanishing points than the principal vanishing point and the distance points. Still, it is my impression that Guidobaldo's work was instrumental in making many practitioners aware of the concept of a vanishing point as a convergence point for the images of parallel lines, and a few may even have appreciated that a proof of this property existed. In short, among practitioners of perspective there was no great interest in or appreciation of the theory developed after Guidobaldo, although they did, directly or indirectly, apply the theory of vanishing points.

Communication Between Theorists and Practitioners

T

hroughout this book we have met numerous examples of authors who claimed that the state of communication between the theorists and practitioners of perspective was deplorable - and many more examples can be

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XIV. Summing Up

found. The practitioners tended to blame the theorists for the lack of collaboration, and vice versa. This communication gap is not particular to perspective, but also occurs in several other disciplines where mathematics is used. The problem seems to be that people applying a mathematical discipline are asking to have the contents of mathematics presented without too many mathematical terms, and that the mathematicians take it for granted that the basic elements of mathematics can be quickly understood by readers with no mathematical background. In the case of perspective, a number of attempts were made to bridge the gap, most of which proved ineffective, whereas some of the practitioners who transmitted Taylor's theory in the second part of the eighteenth century to a general British audience were rather more successful.

The Usefulness of the Theory of Perspective

H

aving once understood the contents of Lambert's work on perspective, there is almost no end to what one can draw correctly in perspective: all polyhedra, several solids with curved surfaces, reflections, water jets, rainbows, a starry sky, and perspective pictures in a composition that has itself been thrown into perspective. Lambert also offered the solution for equipping an oblique ceiling with a trompe tadl, or for making an oblique garden appear horizontal. The big question is, however, whether anyone felt the need to put any of this marvellous theory into practice. My feeling is that no one did. Most drawers planning to make a perspectival composition would presumably be well equipped with a three-dimensional coordinate system in the picture plane, already provided by Vredeman de Vries (figure Y.74), supported by a basic understanding of the concept of a vanishing point - including the knowledge that not all vanishing points are situated on the horizon. Actually for his teaching of perspective three hundred years later Joseph Mallord William Turner used a diagram inspired by Vredeman de Vries (figure XIv. I). The elaborate use of the theory of perspective was reserved for those who took pleasure in getting the perspectival details correct, and they were, in all probability, very few. For the sake of completeness, let me add that in the nineteenth century when photometric surveying was introduced, the theory of perspective, more precisely the part concerning inverse problems of perspective, became highly useful - at which point my story had already ended.

XlV. 5 The Driving Forces Behind the Theory of Perspective n this chapter I have stated that in the period I have studied, the theory of perspective did not contribute to the development of other branches of Imathematics, that the discipline in general was not held in high regard by

5. The Driving Forces Behind the Theory of Perspective

1

721

-1-----

s ,( XIY.1. Turner, Lecture Diagram 31: Perspective Method for a Cube by Vredeman de Vries, c. 1810-1827. Tate Britain.

FIGURE

mathematicians, and that only a few of its results were applied. When considering the collective implications of these views, one wonders what the point was of developing the theory of perspective any further than to the concept of a vanishing point. This was not a question the main protagonists asked. The forces that drove them were their interest in and enthusiasm for the theoretical aspects of perspective, and their inclination to give in to the seduction of mathematics. In their publications the theorists claimed that practitioners would benefit from their work, and some of them may have been so naive that they really believed this. However, they may also secretly have hoped that other mathematicians might share their feeling for the beauty of their most elegant results. To what extent this actually happened, we will never know, but we can be sure such a meeting of the minds took place at least once, when Johann Bernoulli expressed his admiration for 'sGravesande's mathematical abilities after reading his Essai de perspective (page 359). A number of mathematical practitioners, especially among the Britons who propagated Taylor's perspective theory, also fell prey to the seductive attraction of the geometrical theory of perspective. Thus, summing up in the words of Thomas Malton: "Of all the Mathematical Sciences, the study of Perspective is perhaps the most entertaining" (T. Malton 1775, ii).

Appendix One On Ancient Roots of Perspective

I

n this appendix, I discuss how some of the ancient Greek sciences may have influenced the convention of drawing in antiquity as well as the development that led to the emergence of linear perspective in the Renaissance. The sciences considered are the geometrical theory of optics, methods of making maps of the earth and sky, and scenography. The protagonists are, in chronological order, Plato, Euclid, Vitruvius, Ptolemy, and Proclus.

Optics The Visual Pyramid and the Angle Axiom

A

s mentioned in the introduction, the discipline now known as perspective got its name from optics (page xx). The two subjects are, however, intertwined by more than mere etymology, most notably by the theory of vision. A basic concept within the latter theory is that of a visual cone whose apex is in the eye point of an observer and contains all the light rays connecting an object and the eye point – the rays being considered as straight lines. When introducing his model for a perspective representation, Alberti, as we have seen, took over this concept, calling it a visual pyramid (page 19). A central part of the theory of vision deals with appearances. For this theory Euclid introduced a fundamental axiom called the angle axiom. Although it has been discussed earlier, I will nevertheless repeat it here:

Magnitudes seen within a larger angle appear larger, whereas those seen within a smaller angle appear smaller, and those seen within equal angles appear to be equal. [Translated from EuclidS Optics/1959, 1]

The angle axiom implies that the size of the image of a line segment can be determined by measuring the length, seen from the eye point and placed at the location of the picture plane, that covers the line segment. Present-day draughtsmen sometimes apply this result by measuring the length of an image with a pencil. It is quite possible that something similar was done in antiquity, and that the procedure was seen as a consequence of the angle axiom. Perspective was introduced by Alberti as a section in a visual pyramid – corresponding to a central projection upon a plane – and the core problem in 723

724

Appendix One

perspective was how to construct such a section. The idea of a section does not occur in Euclid’s optical theory, his aim being to account for visual appearances and not for projections. Irrespective of this, his theory may well have been used for pictorial representations, for instance, as just suggested, to determine the length of the image of a line segment.

The Remoteness Theorem

D

espite the absence of projections in Euclid’s work, the most widely used version of his Optics contains a consideration that, to some degree, resembles a projection. This consideration is part of a proof for the following result, which I call the remoteness theorem:1 The more remote parts in planes situated below the eye appear higher. [Translated from ibid., 8]

Although Euclid formulated this theorem generally about objects situated in planes below the eye point O (figure 1), he restricted his example to deal with three points, A, B, and C, lying on a horizontal line below O, proving that BC appears to be higher up than AB. The currently known proof introduces a line perpendicular to AC intersecting the visual rays OA, OB, and OC in D, E, and F, respectively. It is then argued that because F is situated higher than E, and E higher than D, and because BC is seen between the lines OC and OB, and AB is seen between the lines OB and OA, the line segment BC appears to be situated higher than AB. The line DF can be interpreted as representing a plane of projection, but in the proof its function was to get the rays OA, OB, and OC orientated with respect to above and below. The important thing in the proof was that AB is seen within the angle DOE, not that it is projected upon DE. O

F E D

A

B

C

FIGURE 1. Euclid’s remoteness theorem. 1

In current editions of Euclid’s Optics, this is theorem 10, but according to David C. Lindberg it was number 11 in the edition most likely used in the early fifteenth century (LindbergS 1976, 264, note 23).

Ancient Roots of Perspective

725

Wilbur Knorr has convincingly argued that the intersecting line is a later addition, and that Euclid’s original proof was different (KnorrS 1991, 195–197). Knorr’s findings are, of course, an important support for the idea that Euclid did not think of projections in his Optics. Nevertheless, it is still possible that practitioners and theorists in antiquity thought that the practice of drawing more remote things higher up in a picture was an application of the remoteness theorem. The edition of Euclid’s Optics known in the Renaissance contains the proof of the remoteness theorem that I have just paraphrased. It may be, as suggested by Samuel Edgerton, that the proof, and in particular its accompanying diagram, influenced Alberti in creating his construction of the image of a square (EdgertonS 1966, 373), but he could as well have been inspired by something else.

The Convergence Theorem

T

he optical theory considered thus far is in agreement with perspective in the sense that the optical results also apply for a perspective projection. Throughout history, however, scholars and painters have also used arguments from the theory of appearances that are not valid in the theory of perspective. There seem to be two reasons for this application, the one being that in some cases a discrepancy between the two theories was not observed. The other reason was that, as noted elsewhere, some perspectivists considered perspective to be a science used for reproducing a visual impression (pages 111, 559, 619). The most famous example of an optical result that is not compatible with the theory of perspective is the following result from Euclid’s Optics, which I call the convergence theorem.

Parallel lengths, seen from a distance, appear not to be equally distant from each other. [Translated from EuclidS Optics/1959, 4]

Euclid considered two parallel lines and looked at the cases in which the eye point is either in the plane of the parallel lines or above them. In the first case he implicitly assumed that the eye point lies between the parallel lines, and in the second that its projection upon the plane of the parallel lines falls between them (figure 2). To prove his claim, Euclid introduced line segments lying on normals to the two parallel lines and having their end points on the two lines, and then showed that the visual angles determined by the eye point O and these line segments decrease as the distance between the eye and the normals increases. The angle AOB is thus larger than the angle COD, which is in turn larger than the angle EOF, and so forth. Because the visual angles become smaller and smaller, the two parallel lines seem to converge. The convergence theorem may, at first, seem to be in accordance with the vanishing point theorem. A closer examination shows, however, that this accordance is not complete because Euclid’s result is only valid for situations in which the eye point, or its projection upon the plane of the parallel lines, lies between the two parallel lines. Thus, if we apply Euclid’s method of argumentation to an eye point in another position, we end up with different results for how the

726

Appendix One A

C

E

G

B

D

F

H

O

G O

E H

C A

F D B

FIGURE 2. Euclid’s convergence theorem. In the upper diagram the eye point and the parallel lines are situated in the same plane, and in the lower the eye point lies above the plane of the parallel lines.

parallel lines appear and how they should be drawn in perspective. In the early eighteenth century, Humphry Ditton phrased the visual result as follows: If the Eye be seated anywhere without the Parallels, they will seem to go further from each other (or their Intervals to widen) to a certain Term of Distance; and after that continually to approach each other. [Ditton 1712, 17]

In other words, some sections of the two parallel lines appear to diverge, while other parts appear to converge2 – implying that the lines appear as curves. This conclusion is obviously different from the result obtained by the rules of perspective, according to which the images of the two parallel lines (when neither of them passes through the eye point) are either two converging straight lines or two parallel lines. The convergence theorem is not the only case of non-corresponding results in the theories of vision and perspective, but this one example suffices to show that the different aims of two disciplines may also lead to different results.3 Although it is conceptually important to be aware of the fact that some optical results cannot be applied in perspective, it is historically less relevant because this insight may not have existed in earlier times. It is therefore quite possible that the convergence theorem inspired some drawers in antiquity and other drawers during the Renaissance to depict orthogonals as converging lines. 2

For an argument supporting this conclusion, see AndersenS 19872, note 4. Another often-discussed example is proposition 8 in Euclid’s Optics and its seeming conflict with the law of inverse proportionality, which was discussed in chapter III (page 95).

3

Ancient Roots of Perspective

727

Optics and Perspective in Harmony

S

ome historians have seen optics and linear perspective as two conflicting disciplines. Erwin Panofsky’s influential paper Die Perspektive als symbolische Form (Perspective as symbolic form) has undoubtedly contributed significantly to this opinion. In discussing classical optics, Panofsky concluded that it was antiplanperspektivisch (“antithetical to linear perspective”, PanofskyS 1927/1991, 264/35). Panofsky’s point of view has been questioned by some scholars, among them C.D. Brownson, who, in arguing in favour of the compatibility of the two disciplines, focussed much upon their similarities (BrownsonS 1981). For the present purpose, the arguments on the relationship between optics and linear perspective need not to be discussed further. It is sufficient to be aware that some parts of the theory of appearances were fruitful for perspective by providing, as pointed out by David C. Lindberg, a mathematical skeleton for perspective (LindbergS 1976, 154) – while other parts were of no use for perspective, as I just noted. The circumstance that some optical results are not applicable to perspectival representations does not mean that the two disciplines are in disharmony. As Brownson, too, pointed out, a conflict only arises if one requires perspective to reproduce what the eye sees (BrownsonS 1981, 189). As stressed earlier, this requirement does, indeed conflict with what most perspectivists considered to be the aim of their discipline, namely to produce an image of a tableau that has the same effect on the eye as the tableau itself.

Cartography

T

o construct, on a flat surface, a map of a section of a spherical surface means establishing a correspondence, called a projection or a mapping, from the surface of the sphere to a plane. The ideal mapping would be one that preserves angles and lengths, but such a mapping does not exist. Down through the ages cartographers have searched for mappings with properties near to the ideal. These do not include a central projection, because it distorts either angles, or lengths, or both. Nevertheless, there are two ancient examples of applying a central projection to create maps.

Ptolemy’s Geography

I

n his Geography Ptolemy described three methods of constructing maps. The last of these is technically a very complicated procedure, which, as one of its several steps, involves a central projection (for technical details, see NeugebauerS 1959, AndersenS 19873, BerggrenS 1991, and PtolemyS Geo, 38–40). Edgerton has suggested that the arrival in Florence of the manuscript of Ptolemy’s Geography, and the eagerness with which the quattrocento scholars studied it, may have influenced Brunelleschi and Alberti in their perspective investigations (EdgertonS 1975, 93–120). Considering how difficult Ptolemy’s

728

Appendix One

text is, I do not find this very likely (an opinion also expressed in KempS 1978 and VeltmanS 1980).

Ptolemy’s Planisphaerium

P

tolemy’s oeuvres offer a much more straightforward central projection than the one from his Geography, namely a stereographic projection presented in a work that was translated into Latin from an Arabic manuscript in the midtwelfth century and became known as the Planisphaerium. As described in chapter IV, in this work Ptolemy obtained a representation of the celestial sphere on the plane of the equator by means of a central projection from the south pole (page 139). This could have been a source of inspiration for perspectival representation, but there is no indication that this was the case. As far as I am aware, the first one to comment upon the connection between Ptolemy’s stereographic projection and perspective was Federico Commandino in 1558 (page 140) – long after the emergence of geometrical perspective.

Scenography

I

t has often been discussed whether drawing in ancient Greece and Rome was based on rules, and if so whether these rules were related to any theory. Unfortunately there is very little material that can help us to find an answer. One sign that some ancient theorists reflected on how to draw is the existence of a discipline called scenography. In his De architectura from the first century B.C.E., Vitruvius – having presented ichnography and orthography – wrote:

Item scaenographia est frontis et laterum abscedentium adumbratio ad circinique centrum omnium linearum responsus. [VitruviusS Arch, book I, chapter 2, §2]

Later in the work he hinted that scenography is related to the theatrical stage design and reported that Democritus and Anaxagoras showed how ad aciem oculorum radiorumque extentionem certo loco centro constituto, ad lineas ratione naturali respondere, uti de incerta re incertae4 imagines aedificiorum in scaenarum picturis redderent speciem et, quae in directis planisque frontibus sint figurata, alia abscedentia, alia prominentia esse videantur. [VitruviusS Arch, book VII, preface, §11]

These two descriptions have been interpreted and translated in a variety of different ways.5 Some have taken the centre mentioned in the first quotation 4

Some copies have incertae and others certae. Inspired by Christian Wiener’s and John White’s translations (WienerS 1884, 8; WhiteS 1987, 251), I suggest the following interpretations: Likewise, scenography is the drawing of the outline of the front and of the receding sides and the representation of all the lines to the centre of a circle [literally: a pair of compasses]. ... when the centre is fixed at a certain place, the lines must correspond naturally to the sight of the eyes and the extension of the rays, so that images of an object render the appearance of buildings in the stage paintings, and so that which is shaped upon vertical and frontal planes is seen as partly receding, and partly projecting. For alternative suggestions, see PanofskyS 1927/1991, 38, and notes 18 and 19.

5

Ancient Roots of Perspective

729

to mean a vanishing point, while the certo loco centro introduced in the second quotation has been understood as referring either to an eye point or a vanishing point. Panofsky has criticized his predecessors’ way of reading the text and instead suggested an interpretation that is in accordance with the angle axiom (PanofskyS 1927/1991, 266/38). Despite the various interpretations, it seems safe to assume that Vitruvius described a procedure for making stage sets that looks “natural”. However, as long as we do not have more source material it is fairly hopeless to reconstruct the technique he had in mind. In the fifth century Proclus commented upon the first book of Euclid’s Elements. In a prologue, Proclus described the virtues of mathematics and touched in this connection upon scenography. His description has also been interpreted in rather different ways.6 He presented scenography as a part of optics, and it seems he related the topic to a concern that an image should not seem disproportionate. We cannot be sure that Vitruvius and Proclus meant exactly the same thing by scenography. Proclus apparently did not describe a central projection, but a representation that somehow took the angle axiom into account. This fits in with his great affinity for Plato, and with the fact that in his Sophist, Plato described how, in some large sculptures and paintings, objects are not reproduced in their true proportions, because this would make the upper parts seem too small (PlatoS Soph, 236A). Most likely, Plato referred to the procedure we met in Dürer’s illustration Dürer (figure III.16, page 99), that is, representing highly elevated objects in a larger scale than lower objects of the same height so that the objects will be seen within equal angles. As I have already stressed, it is more difficult to know which technique Vitruvius was describing. It may have involved points of convergence for orthogonals lying in the same plane, or even a common convergence point for all orthogonals. As John White documented, a few examples from antiquity document the practice of using points of convergence for orthogonals in the same plane (WhiteS 1987, 258–267). In general, different points of convergence were used for different planes, but there is one case in which it strongly looks as if almost all orthogonals are converging towards one and the same point (figure 3).

6

The translations include the following: ... die zeigt, wie es zu erreichen ist, dass in den Bildern dargestellten Dinge nicht als verworrene und verzerrte Gebilde erscheinen, sondern der Entfernung und Grösse der dargestellte Objekte entsprechen. [P. Leander Schönberger and Max Steck in ProclusS Comm/1945, 191] ... qui enseigne à faire voir dans des images des aspects non déformés ni disproportionnés par les distances et par la hauteur des dessins. [Paul ver Eecke in ProclusS Comm/1948, 34–35] ... showing how objects can be represented by images that will not seem disproportionate or shapeless when seen at a distance or an elevation. [Glenn R. Morrow in ProclusS Comm/1970, 33]

730

Appendix One

FIGURE 3. Wall painting in the “Room of the Masks”, right-hand wall in the House of Augustus, Palatine Hill, Rome. First century B.C.E.

Conclusion

A

s there is such a paucity of factual knowledge about procedures of drawing in antiquity, it becomes a matter of personal discretion how much theory one ascribes to the ancient art of drawing. In my view, it is more than likely that some results from optics were applied in the practice of drawing. I do not think, however, that practitioners and theoreticians pooled their requirements and insights to create a complete theory of drawing. This task was left to Renaissance artists and scholars. As it turned out, all the concepts they needed, and actually also all the geometry required for developing a mathematical theory of perspective, were available as an inheritance from ancient Greece. The question of what inspired the introduction of a perspectival projection in the Renaissance is as difficult to address as the matters dealt with in this appendix – but the problem is discussed in chapter I.

Appendix Two The Appearance of a Rectangle à la Leonardo da Vinci

In this appendix, I investigate how well Leonardo da Vinci’s answer to the question of “whether a vertical rectangle appears as rectilinear or curvilinear” (page 796) agrees with the theory he applied. In deciding on how a rectangle appears, Leonardo seems to have conceived of it as composed of an infinity of vertical line segments. To handle this idea mathematically, I introduce a movement and imagine that the rectangle ABCD (figure 1) has been produced by a moving line segment that starts in the position AB and ends as DC. I would like to apply Leonardo’s own mathematical tool – that is, the law of inverse proportionality – and to do so I begin with the situation in which the orthogonal projection of the eye point upon the rectangle is the point A. Setting OA = a, AD = x, and OD = y, I deduce from Pythagoras’s theorem that y = a2 + x 2 , and from relation (iii.1) that the apparent ratio between the two sides AB and CD is v (y, a) = v ( a2+ x2 , a) = a : a2+ x2 . Taking the apparent size of AB as a unit, I define the apparent size of DC as a (x) = s (y,a), A x a

D y

O

FIGURE 1. The apparent size of DC as a function of DC’s distance, x, to AB.

B C

731

732

Appendix Two B'

C'

A

D

B

C

FIGURE 2. Introduction of the lines BC and B′C′.

implying that a (x) = a : a2+ x2 .

(1)

This relation shows that when DC’s distance x to AB increases, the apparent size of DC decreases, as claimed by Leonardo, but it also shows that, contrary to what he believed, the decrease is not linear. In the following I elaborate on what relation (1) actually shows. Let us assume (figure 2) that an eye is looking at the lines BC and B′C′ parallel to and at equal distance from the line AD, and that the orthogonal projection of the eye point upon the plane of the figuration falls in A. Following Leonardo, we want to find the apparent shapes of the lines BC and B′C′. When it is assumed that the line AD is seen as a straight line, a(x) in relation (1) provides for each x the apparent distance from AD of a point on BC, or, in other words, the collection of points on BC will be seen as a curve whose equation is given by (1).1 The same argument shows that the apparent shape of B′C′ is a curve symmetrical to the first with respect to the line AD. Equation (1) also shows that the graphs of these curves are dependent on the viewing distance a.

The Curves for Three Different Distances

T

o see how well Leonardo’s solution approximates the mathematical solution, I look at the curves defined by relation (1) for three different values of a. Returning to the situation outlined in figure 1, I have illustrated in figure 3 a horizontal section through the point A containing three eye points O1, O2, and O3 corresponding to the three values of a. Taking, as before, the line segment AB in figure 1 as the unit, I have depicted the situation in which the 1

Provided it is accepted as a premise that the law of inverse proportionality is valid and can be applied as I do.

Leonardo da Vinci’s Rectangle A R

S

733

T

O1 O2

O3

FIGURE 3. Different viewing distances from the line AD, depicted in figure 1.

distances a between A and the three eye points O1, O2, and O3 are 1, 3, and 10, and the maximum viewing angle is 90˚. Thus the points R, S, and T are the points of intersection between the transversal through A and the horizontal diagonals through O1, O2, and O3. In figures 4–6, for points in the interval [−a,a] and for a equal to 1, 3 and 10, I have drawn the curves defined by equation (1) – that is, the supposed apparent graphs of the lines BC and B′C′ both to the right and left of B and B′. Leonardo himself drew these apparent lines as two straight lines sloping towards the line AR at both ends (figure III.12). My figures show, among other things, that: The smaller a is, the more the curve deviates from two straight lines. Each of the considered sections of any of the curves does, however, lie near two straight and symmetrical lines – except in an interval around B'

A

R

B

FIGURE 4. The curvilinear appearance for a = 1.

734

Appendix Two B'

A

R

S

B

FIGURE 5. The curvilinear appearance for a = 3. B' A R B

S

T

FIGURE 6. The curvilinear appearance for a = 10.

the midpoint. Thus, Leonardo’s intuitive solution is, to some extent, in accordance with the mathematical solution.*

The Angle Between the Line Segments

I

n a different manuscript from the one in which he discussed whether a rectangle would appear rectilinear or curvilinear, Leonardo stated that the angle between his two line segments in his figure (reproduced as figure III.12) increases with an increasing viewing distance (Manuscript G, fol. 32r; VeltmanS 19861, 158–159). He did not give any argument for this result. Perhaps he thought that the longer the distance to a rectangle, the less it would appear distorted. At any rate, his conclusion is in accordance with what my curves demonstrate. Thus, when a is equal to 1, the angle between the almost linear parts of the curve is smaller than the corresponding angle for a equal to 3, and this is in turn smaller than the angle for a equal to 10.

*I am thankful to Hans Anton Salomonsen for having drawn the curves in this and the two previous figures.

Appendix Three ’sGravesande Taking Recourse to the Infinitesimal Calculus to Draw a Column Base in Perspective

T

his appendix is devoted to describing how ’sGravesande applied his knowledge of the calculus to solve an intricate problem of determining the visible part of the column base shown in figure VII.61 (page 356, ’sGravesande 1711, §§65–68). Initially (figure 1), ’sGravesande considered the quarter ABCDE of a vertical section of the column base, in which the curve BC is a quarter of a circle whose centre is D. The upper half of the column base is then obtained by rotating ABCDE around the line AE, implying that the horizontal sections in O

B

A

H H'

J C

G

R

G' D

T S

I E

F

U

FIGURE 1. A section in a quarter of the column base. 735

736

Appendix Three

the base are circles with centres on the line AE and radii varying from AB to EC. ’sGravesande placed the eye point O in his chosen vertical plane and set his mind on determining the part of the base that is visible from O. He conceived of the base as consisting of an infinite number of infinitely close horizontal discs or circles and wanted to find out how much of each horizontal circle can be seen from O – the view to each disc being obstructed by the disc lying above it. To be able to solve this problem, ’sGravesande looked at an arbitrary section, for instance the one with GH1 as its radius, and constructed on its circumference two points, to be defined later, which I call the boundary points of visibility – although ’sGravesande did not give them a name.

The First Step

B

efore introducing the boundary points of the circle GH, ’sGravesande looked at how the upper of two discs, having a finite distance, blocks the view to the lower disc. For this purpose, together with the circle GH he considered the circle IJ, and, from O projected the lower circle upon the plane of the upper circle. Since the calculations become a bit more straightforward by projecting, instead, the upper circle on the plane of the lower, and since this procedure does not change the final result,2 I choose the latter. Therefore, let the projection of the upper circle GH on the plane of the lower circle IJ be the circle G′H′ (figure 2), and let the points of intersection of the two latter K

H'

J

G'

I

M

N

P

L

FIGURE 2. The upper circle projected upon the plane of the lower circle.

1 2

In the following I call a circle with the radius XY ‘the circle XY’. Because at the end ’sGravesande took a step corresponding to taking a limit.

’sGravesande’s Column Base

737

circles be K and L. These two points determine the ‘crescent moon’ KPLN of the circle IJ that can be seen from O. Let M be the point of intersection of KL and IJ. ’sGravesande set IJ = a, G′H′ = b, and the distance between the two centres I and G′ as IG′ = c. By a simple calculation he then found that 2 2 IM = b - a - c . 2c 2

(1)

Based on this equation it is easy to determine the points K and L on the circle IJ.

The Infinitesimal and Limit Situation

L

et us now, with ’sGravesande, turn to the situation in which the distance between circles GH and IJ is infinitesimal. Taking a step that corresponds to taking the limit, ’sGravesande obtained a situation (figure 1) in which the three points G, G′, and I coincide, as do the two circles GH and IJ, and in which the points K and L in figure 2 determine the arc of circle GH that can be seen from the eye point. In this limit situation I refer to the points K and L as the boundary points of visibility for the circle GH. To determine the position of the boundary points with the aid of the point M, ’sGravesande first calculated the a, b, and c corresponding to the infinitesimal situation. To present his calculations (figure 1), I let F be the point of intersection of the vertical line through O and the line CE, and S and T the points of intersection of OF and the lines IJ and GH. Finally, I let R be the point of intersection of the vertical through the point H and the line IJ. ’sGravesande set FE = GT = e, OT = x, TS = GI = HR = dx, GH = y, and RJ = dy. Since IR = GH, the last assumption implies that IJ = y + dy.

By looking at similar triangles, ’sGravesande found results corresponding to y+e IG l= e dx x and RH l= x dx , and hence y+e y G lH l= IR - G lI + RH l= y - e dx x + x dx = x dx Inserting these three expressions instead of a, c, and b in (1) results in IM =

(y + (y/x) dx)2- (y + dy)2 edx . 2x 2edx/x

Taking a limit step, ’sGravesande obtained the following result: y2 xy dy GM = IM = e - e . dx

738

Appendix Three

Constructing the right-hand side geometrically, ’sGravesande first constructed the length dy y (2) dx as the subnormal – without using the word – to the section of the base in the point H with respect to the axis AE. When U is the point of intersection of the normal HD to the circle and the line AE, the above-mentioned subnormal is the projection GU of the normal in H upon the line AE. With the length (2) at his disposal, ’sGravesande straightforwardly constructed the length GM, thereby obtaining the boundary points K and L.

The Perspective Image of the Visible Part of the Column Base

T

o perform his construction of the perspective image of the column base, ’sGravesande suggested to determine the boundary points for a few more horizontal sections, then throwing the visible arcs between the boundary points into perspective and making smooth curves between these arcs. His result is shown in figure VII.61.

Appendix Four The Perspective Sources Listed Countrywise in Chronological Order

Introduction

F

ollowing is a list of the literature containing presentations of perspective constructions up to 1800, arranged chronologically and countrywise in the following order: Italy; France and the Southern Netherlands; Germany, Austria and Switzerland; the Northern Netherlands; and Britain. To make it easier to get an overview, abbreviated titles have been used, and in general only the first edition of a publication is included.1 Consequently translations have also been left out.

Italy 1435 Leon Battista Alberti, De pictura; first printing 1540. ~1460s Filarete, byname of Antonio Averlino, Trattato di architettura; first printing 1890. ~1470s Piero della Francesca, De prospectiva pingendi; first printing 1899. ~1480s Francesco di Giorgio Martini, Trattato di architettura civile e militare; first printing 1841. 1504 Pomponio Gaurico, De sculptura. 1530s Jacopo Barozzi da Vignola, Perspective manuscript; printed as Le due regole della prospettiva pratica con i commentarii del R. P. M. Egnatio Danti, 1583. 1545 Sebastiano Serlio, D’Architettura/L’Architecture. Il secondo libro di perspettiva/Le second livre de perspective. 1558 Federico Commandino, In planisphaerium Ptolemaei commentarius. 1567 Pietro Cataneo, L’architettura. 1568 Daniele Barbaro, La pratica della perspettiva. 1583 Jacopo Barozzi da Vignola, Le due regole della prospettiva pratica con i commentarii del Egnatio Danti. 1584 Giovanni Paolo Lomazzo, Trattato dell’arte della pittura, scoltura, et architettura. 1

Also in cases where I have not seen the first edition, I have listed this here and then referred to the edition included in the first bibliography. Books which have no year of publication I have placed where I considered it likely they appeared. 739

740

Appendix Four

1585

Giovanni Battista Benedetti, Diversarum speculationum mathematicarum et physicarum liber. Lorenzo Sirigatti, La pratica di prospettiva. Guidobaldo del Monte, Perspectivae libri sex. Scipione Chiaramonti, Delle scene, e teatri; published posthumously in 1675. Lodovico Cardi (called il Cigoli), Trattato della prospettiva pratica; first printing 1992. Matteo Zaccolini, Prospettiva lineale; manuscript. Pietro Antonio Barca, Avvertimenti e regole circa l’architettura ... prospettiva. Pietro Accolti, Lo inganno degl’occhi, prospettiva pratica. Ferdinando di Diano, L’occhio errante dalla ragione emendate, prospettiva. Giuseppe Viola-Zanini, Della architettura ... con ... regole prospettiva. Niccolò Sabbatini, Pratica di fabricar scene e machine ne’ teatri. Mario Bettini, Apiaria universae philosophiae mathematicae. Bernardino Contino, La prospettiva pratica; cf. Contino 1645. Evangelista Torricelli, Prospettiva pratica; unfinished, undated manuscript. Leonardo da Vinci, Trattato della pittura. A collection of texts gathered from various Leonardo manuscripts from the end of fifteenth and the beginning of sixteenth century. Christoph Scheiner, Prattica del parallelogrammo da disegnare del P. Christoforo Scheiner, ed. Giulio Troili. Guarino Guarini, Euclides adauctus et methodicus mathematicaque universalis. Giulio Troili, Paradossi per pratticare la prospettiva senza saperla; cf. Troili 1683. Andrea Pozzo, Perspectiva pictorum et architectorum/prospettiva di pittori e architetti, vol. 1. Andrea Pozzo, Perspectiva pictorum et architectorum/prospettiva di pittori e architetti, vol. 2. Ferdinando Galli-Bibiena, L’architettura civile preparata sul la geometria e ridotta alla prospettiva. Paolo Amato, La nuova pratica di prospettiva. Ferdinando Galli-Bibiena, Direzioni al giovani studenti nel disegno dell’architettura civile. Giuseppe Galli-Bibiena, Architettura & prospettiva. Giovanni Lodovico Quadri, La prospettiva pratica delineata in tavola a norma della secondo regola di Giacomo Barozzi da Vignola. Giovanni Francesco Costa, Elementi di prospettiva per uso degli architetti e pittori. Giovanni Biago Amico, L’architetto pratico. Giovanni Battista Piranesi, Opere varie de architettura ... prospettiva. Ferdinando Galli-Bibiena, Direzioni della prospettiva teorica. Eustachio Zanotti, “De perspectiva in theoremam unum redacta”. Bernardo Antonio Vittone, Istruzioni elementari per indirizzo de’giovani allo studio dell’Architettura civile. Eustachio Zanotti, Trattato teorico-pratico di prospettiva. Giovanni Battista Spampani & Carlo Antonini, Il Vignola illustrato proposto Giambattista Spampani e Carlo Antonini studenti d’ architettura. Baldassare Orsini, Della geometria e prospettiva pratica. Giuseppe Torelli, Elementorum prospectivae libri duo.

1596 1600 ~1600 ~1610 ~1620 1620 1625 1628 1629 1638 1642 1643 ~1645 1651

1653 1671 1672 1693 1700 1711 1714 1731 1740 1744 1747 1750 1750 1753 1755 1760 1766 1770 1773 1788

The Perspective Sources Countrywise

741

France and the Southern Netherlands 1505 1531 1560 1576 1612 1613 1628 1630 1631 1636

1637 1638 1638 1640 1642 1642 1642

1643

1643 1644 1644 1644 1646 1647 1648 1648 1648

1649 1653 1660 1661 1661

Viator (Jean Pélerin), De artificiali perspectiva. Joachim Fortius Ringelberg, Opera. Jean Cousin, Livre de perspective. Jaques Androuet du Cerceau, Lecons de perspective positive. Salomon de Caus, La perspective avec la raison des ombres et miroirs. François Aguilon, Opticorum libri sex. Jacques Aleaume, Introduction à la perspective, printed posthumously, but not published and now lost. Jean Louis Vaulezard, Perspective cilindrique et conique ou traicté des apparences veuës par le moyen des miroirs. Jean Louis Vaulezard, Abrégé ou racourcy de la perspective. Girard Desargues, Exemple de l’une des manieres universelles de S.G.D.L. touchant la pratique de la perspective sans emploier aucun tiers point, de distance ny d’autre nature qui soit hors du champ de l’ouvrage. Pierre Hérigone, Cursus mathematicus/Cours mathématique, vol. 5. Claude Mydorge, Examen du livre des recreations mathématiques. Jean François Niceron, La perspective curieuse, ou magie artificielle des effets merveilleux. Henry Guenon, Pratique ... de la perspective sur les seules parties egales du compas de proportion sans y adiouster aucune ligne d’optique. Pierre Hérigone, Supplementum cursus mathematici. Jean Dubreuil, La perspective pratique.... Jean Dubreuil, Diverses methodes universelles et nouvelles ... Tirées pour la pluspart du contenu du livre de la Perspective pratique. Ce qui servira de plus de response aux deux affiches du Sieur Desargues contre ladite Perspective pratique. Jacques Aleaume, La perspective speculative et pratique ... Ensemble la maniere universelle de la pratiquer non seulement sans plan géométral & sans tiers poinct, dedans ni dehors de la champ du tableau. Mais encore par le moyen de la ligne communément appelée horisontale. Girard Desargues, Livret de perspective adressé aux théoriciens; lost. Jacques Curabelle, Examen des oeuvres du Sieur Desargues. Nicolas Baytaz, Abbreviations des plus difficiles operations de perspective pratique ... principalement aux vrais peintres. Marin Mersenne, Universae geometriae mixtae. Jean François Niceron, Thaumaturgus opticus seu admiranda optices. Jean Dubreuil, Second partie de la perspective pratique. Emmanuel Maignan, Perspectiva horaria. Abraham Bosse, Maniere universelle de Mr Desargues pour pratiquer la perspective. René Gaultier de Maignannes, Invention nouvelle et brieve pour reduire en perspective par le moïen du quarré ... sans se servir d’autres points soit tiers, ou accidentaux, que de ceux qui peuvent tomber dans le tableau.. Jean Dubreuil, Troisiesme et derniere partie de la perspective pratique. Abraham Bosse, Moyen universel de pratiquer la perspective sur les tableaux, ou surfaces irrégulières. Jacques Le Bicheur, Traité de perspective. Charles Bourgoing, La perspective affranchie. Pierre Bourdin, Le cours de mathematique.

742

Appendix Four

1665

Abraham Bosse, Traité des pratiques géométrales et perspectives enseignées dans l’Académie royale de la peinture et sculpture. Andreas Tacquet, Opera mathematica. Grégoire Huret, Optique de portraiture et peinture. Claude François Milliet Dechales, Cursus seu mundus mathematicus. Jacques Rohault, Oeuvres posthumes. Sébastien Leclerc, Discours touchant le point de veue. Claude François Milliet Dechales, Cursus seu mundus mathematicus; revised edition of Dechales 1674. Jacques Ozanam, Cours de mathématique. Bernard Lamy, Traité de perspective. Louis Bretez, La perspective practique de l’architecture. Jean Courtonne, Traité de la perspective pratique. Abbé Deidier, Traité de perspective théorique et pratique. Abbé Deidier, Elemens generaux des principales parties des mathematiques neccesaires à l’artellerie et au génie. Edme Sébastien Jeaurat, Traité de perspective à l’usage des artistes. Nicolas Louis Lacaille, Leçons élémentaires d’optiques; second edition. Claude Roy, Essai sur la perspective pratique par le moyen du calcul. Jacques Silvabelle, “Méthode generale, pour trouver la perspective d’un objet donné”. Ennemond Alexandre Petitot, Raisonnement sur la perspective pour en faciliter l’usage aux artistes/Ragionamento sopra la prospettiva. Johann Heinrich Lambert, La perspective affranchie de l’embaras du plan géometral. Nicolas François de Curel, Essai sur la perspective linéaire. Guillaume Germain Guyot, Nouvelles récréations physiques et mathématique. S. N. Michel, Traité de perspective linéaire. Charles Dupuis, Cours de géométrie pratique, ..., de perspective. Aléxandre Sobro, Traité de perspective à l’usage des artistes. Pierre Henri Valenciennes, Elémens de perspective pratique à l’usage des artistes.

1669 1670 1674 1682 1689 1690 1693 1701 1706 1725 1744 1745 1750 1756 1756 1757 1758 1759 1766 1769 1771 1773 s.a. 1800

Germany, Austria, and Switzerland 1508 1509 1525 1531 1538 1543 1547 1564 1567 1567 1568

Gregor Reisch, Margarita Philosophica Nova. Jörg Glockendon, Von der Kunnst Perspectiva. Albrecht Dürer, Underweysung der Messung mit dem Zirckel und Richtsheyt. Johann II von Simmern, Eyn schön nützlich Büchlin und Underweisung der Kunst des Messens mit dem Zirckel, Richtscheidt oder Lineal. Erhard Schön, Underweissung der Proportzion unnd Stellung der Possen. Augustin Hirschvogel, Ein aigentliche und grundtliche Anweysung, in die Geometria. Walther Hermann Ryff, Der furnembsten, notwendigsten, der ganzen Architectur angehörigen mathematischen und mechanischen Künst. Heinrich Lautensack, Des Circkels unnd Richtscheyts, auch der Perspectiva. Lorenz Stör, Geometria et Perspectiva. Hans Lencker, Perspectiva literaria. Wenzel Jamnitzer, Perspectiva corporum regularium. Das ist, ein fleyssige Fürweysung.

The Perspective Sources Countrywise 1571 1583 1599 1610 1615 1616 1623 1625 1626 1630 1631 1633 1646 1652 1657 1675 1677 1680 1683 1699 1713 1715 1717 1717 1718 1719 1724 1725 1727 1733 1737

2

743

Hans Lencker, Perspectiva. Hierinnen auffs kürtzte beschrieben ... wie allerley Ding ... in die Perspectyf gebracht werden mag. Georg Hass, Künstlicher und zierlicher ... perspectifischer Stück. Paul Pfinzing, Ein schöner kurtzer Extract der Geometriae unnd Perspectivae. Johann Faulhaber, Newe geometrische und perspectivische Inventiones. Lucas Brunn, Praxis perspectivae, das ist von Verzeichnungen ein aussführlicher Bericht. Paul Pfinzing, Optica, das ist gründtliche doch kurtze Anzeigung wie notwendig die löbliche Kunst Geometriae seye inn der Perspectiv. Andreas Albrecht, Zwey Bücher. Das erste von der ohne und durch die Arithmetica gefundenen Perspectiva. Das andere von dem darzu gehörigen Schatten. Peter Halt, Perspektivische ... Reisskunst. Peter Halt, Drey wichtige newe Kunststück in underschidlichen perspectivischen Instrumenten inventiert und erfunden. Benjamin Bramer, Beschreibung eines sehr leichten Perspectiv und Grundreissenden Instruments. Christoph Scheiner, Pantographice. Johann Faulhaber, Ingenieurs Schul. Athanasius Kircher, Ars magna lucis et umbrae. Theodosius Haesell, Geistliche perspectiva. Gaspar Schott, Magia universalis naturae et artis, Pars I Optica. Joachim Sandrart, L’Academia todesca. Caspar Schott, Cursus mathematicus. Jacob Johann Füllisch, Compendium artis ... Das ist kurzer leichter ... Unterricht von der geometrisch-ignographischen Zeugnungs und Baukunst. Daniel Hartnack, Perspectiva mechanica und eigentliche Beschreibung derer vornehmsten Instrumenten. Johann Christoph Sturm, Mathesis juvenilis. Jakob Leupold, Anamorphosis mechanica nova, oder Beshreibung dreier neuen Maschinen. Christian Wolff, Elementa matheseos universae. Johann Wenceslaus Kaschube, Cursus mathematicus, oder deutlicher Begriff der mathematischen Wissenschaften. Anton Bernhard Lauterbach, Clavis perspectivae verticalis geometrica. Johann Friedrich Weidler, Institutiones matheseos; cf. Weidler 1736. Johann Jacob Schübler, Perspectiva Pes Pictura. Das ist kurze und leichte Verfassung der practicabelsten Regul zur perspectivischen ZeichnungsKunst. Johann Jacob Schübler & Johann Leonhard Rost, Mathematischer Lust und Nutzgarten ... sammt einer Anleitung zur Perspectiv. Johann Bernhard Wiedeburg, Einleitung zu denen mathematischen Wissenschaften. Paul Heinecke, Lucidum prospectivae speculum. Das ist ein heller Spiegel der Perspective. Friedrich WilhelmWeidemann, Kurtze Einleitung zu der optischen Perspectiv; cf. Weidemann 1746. Christian Wolff, Anfangsgründe aller mathematischen Wissenschaften; fifth edition.2

Perspective is presumably also treated in some of the earlier editions.

744

Appendix Four

1741 1747 1747 1752

Johann Christoph Bischoff, Kurtzgefasste Einleitung zur Perspectiv. Georg Erhard Hamberger, Dissertatio sistens leges perspectivae. Joachim Georg Darjes, Erste Gründe der gesammten Mathematik. Abraham Gotthelf Kästner, Perspectivae et projectionum theoria generalis analytica. Johann Heinrich Lambert, Anlage zur Perspektive; manuscript. Albrecht Ludwig Friederich Meister, Instrumentum scenographicum. Abraham Gotthelf Kästner, Anfangsgründe der Arithmetik, Geometrie ... und Perspectiv; cf. Kästner 1774. Johann Heinrich Lambert, La perspective affranchie ... and Die freye Perspektive. Georg Heinrich Werner, Die Erlernung der Zeichenkunst durch Geometrie und Perspectiv. Johann Heinrich Lambert, Kurzgefasste Regeln zu perspectivischen Zeichnungen vermittelst eines ... Proportional-Zirkels. Johann Friedrich Hennert, Elementa optices, perspectivae, catoptrices et phaometria. Johann Heinrich Lambert, Die freye Perspective ... mit Anmerkungen und Zusätzen vermehrt. Wenceslaus Johann Gustav Karsten, Lehrbegrif der gesamten Mathematik. Der siebende Theil. Die Optik und Perspectiv. Johann Andreas Segner, Gründe der Perspectiv. Johann Heinrich Lambert, Grundsätze der Perspectiv, aus Betrachtung einer perspectivisch gezeichneten Landschaft abgeleitet. Johann Leonhard Hoffmann, Anweisung zur Vertigung und Gebrauch des allgemeinen Zeichnen-Instruments ohne Gläser. Lukas Voch, Abhandlung von der Perspektivkunst. Karl Scherffer, Beyträge zur Mathematik. Johann Michael Rödel, Abhandlungen von den zufälligen Punkten in der Perspectivkunst für Werkmeister; cf. Rödel 1796. Johann Georg Sulzer, Allgemeine Theorie der schönen Künste. Johann Friedrich Lorenz, Grundriss der reinen und angewandten Mathematik; cf. Lorenz 1799. Bernhard Friedrich Mönnich, Versuch die mathematischen Regeln der Perspektive für den Künstler ohne Theorie anwendbar zu machen; cf. Mönnich 1801. Abel Bürja, Der mathematische Maler oder gründliche Anweisung zur Perspektive. Georg Heinrich Werner, Gründliche Anweisung zur Zeichenkunst. Karl Gottlieb Horstig, Briefe über die mahlerische Perspektive. Johann Heinrich Lambert, “Grundsätze der Perspectiv”.

1752 1753 1758 1759 1763 1768 1770 1774 1775 1779 1779 1780 1780 1781 1784 1787 1791 1794

1795 1796 1797 1799

The Northern Netherlands 1560 1560 1604 1605 1614

Johan Vredeman de Vries, Artis perspectivae. Johan Vredeman de Vries, Scenographiae sive perspectivae. JohanVredeman de Vries, Perspective. Simon Stevin, Derde Stuck der Wisconstighe ghedachtnissen. Van de Deursichtighe. Samuel Marolois, La perspective contenant la theorie et la pratique.

The Perspective Sources Countrywise 1623 1660 1676 1678 1699 1703 1705 1707 1711 1765 1769 1773 1775 1786 1786 1788

745

Hendrik Hondius, Onderwijssinge in de perspective conste. Frans van Schooten, Mathematische Oeffeningen. Abraham de Graaf, De geheele mathesis of wiskonst. Samuel van Hoogstraten, Inleydung tot de hooge schoole der schilderkonst. Nicolaas Hartsoeker, Proeve der deursicht-kunde. Dirk Bosboom, Perspectiva of doorzicht-kunde. Hendrik van Houten, Verhandelinge van de grontregelen der doorzigtkunde of tekenkonst (perspectief). Gerard de Lairesse, Het groot schilderboek. Willem Jacob ’sGravesande, Essai de perspective. Caspar Jacobszoon Philips, Uitvoerig onderwys in de perspectiva of doorzichtkunde. Rienk Jelgerhuis, Nauwkeurige aanmerkingen op een vornaam gedeelte van de perspectiva of doorzigtkunde van Casper Philips Jacobsz. Jacob de Vlaming, Kort zaamenstel der perspectief. Caspar Jacobszoon Philips, Handleiding in de spiegelperspectief. Caspar Jacobszoon Philips, Wis-meet-en doorzichtkundige handleiding. Caspar Jacobszoon Philips, Zeemans onderwijser in de tekenkunst ... doorzichkundige en perspectivische regelen. Caspar Jacobszoon Philips, Handleiding om ... als ook de der perspectivische regelen.

Britain 1669

1670 1672 1712 1715 1719 1731 1738 1743 1746 1751 1754 1754 1755 1756

Henry Oldenburg (presumably), The Description of an Instrument, Invented Divers Years ago by Dr. Christopher Wren, for Drawing the Out-Lines of any Object in Perspective. Joseph Moxon, Practical Perspective, or, Perspective Made Easie. William Salmon, Polygraphice or the Art of Drawing. Humphry Ditton, A Treatise of Perspective, Demonstrative and Practical. Brook Taylor, Linear Perspective or, a New Method of Representing justly all Manner of Objects. Brook Taylor, New Principles of Linear Perspective. William Halfpenny, Perspective Made Easy or a New Method for Practical Perspective. John Hamilton, Stereography, or a Compleat Body of Perspective. James Hodgson, “The Theory of Perspective”; printed as introduction to Dubreuil 1743. Patrick Murdoch, Neutoni genesis curvarum per umbras seu perspectivae universalis elementa. Matthew Darly, A New Book of ... Chairs with the Manner of Putting them in Perspective According to Brook Taylor. Joseph Highmore, A Critical Examination of those two Paintings ... in which Architecture is introduced so far as relates to the Perspective. John Joshua Kirby, Dr. Brook Taylor’s Method of Perspective Made Easy, Both in Theory and Practice. Godfrey Smith, The Laboratory; or School of Arts ... and Easy Introduction to the Art of Drawing in Perspective. Thomas Bardwell, The Practice of Painting and Perspective Made Easy.

746

Appendix Four

1757?

John Joshua Kirby, Dr. Brook Taylor’s Method of Perspective, compared with Examples lately publish’d on this Subject as Sirigatti’s; published s.a. John Joshua Kirby, The Perspective of Architecture ... deduced from the Principles of Dr. Brook Taylor. Daniel Fournier, A Treatise of the Theory and Practice of Perspective. Wherein the Principles ... by Dr. B. Taylor are fully and clearly explained. Joseph Highmore, The Practice of Perspective. On the Principles of Dr. Brook Taylor, written many years since, but now first published. Benjamin Martin, A New and Comprehensive System of Mathematical Institutions. John Muller, Elements of Mathematics; third edition. John Lodge Cowley, The Theory of Perspective, Demonstrated in a Method Entirely New. William Emerson, Cyclomathesis Or an Easy Introduction to the several Branches of the Mathematics. William Emerson, Perspective Or the Art of Drawing the Representations of all Objects upon a Plane. Joseph Priestley, A Familiar Introduction to the Theory and Practice of Perspective. Benjamin Martin, The Principles of Perspective. Benjamin Martin, The Description and Use of a Graphical Perspective. Edward Noble, The Elements of Linear Perspective. Joseph Priestly, History and Present State of Discoveries Relating to Vision, Light and Colours. John Wright, Elements of Trigonometry. William Hooper, Rational Recreations. James Ferguson, The Art of Drawing in Perspective Made Easy. Thomas Malton, A Compleat Treatise on Perspective in Theory and Practice on the True Principles of Dr. Brook Taylor, Made Clear. Henry Clarke, Practical Perspective. Thomas Malton, An Appendix or Second Part to the Compleat Treatise on Perspective. Robert Bradberry, The Principles of Perspective. Explained in a Genuine Theory and Applied in an Extensive Practice. George Adams, Geometrical and Graphical Essays. Thomas Sheraton, The Cabinet-maker and Upholsterer’s Drawing Book. A. Cobin, Short and Plain Principles of Linear Perspetive Adapted to Naval Architecture. John Wood, Elements of Perspective, Containing the Nature of Light and Colours and the Theory and Practice of Perspective. James Malton, The Young Painter’s Maulstick being a Practical Treatise on Perspective founded on the process of Vignola and Sirigatti, ... united with the theoretic principles of ... B. Taylor.

1761 1761 1763 1764 1765 1765 1765 1768 1770 s.a. 1771 1771 1772 1772 1774 1775 1775 1776 1783 1790 1791 1793 1794 1799 1800

First Bibliography Pre-Nineteenth Century1 Publications on Perspective

Accolti, Pietro 1625 Lo inganno degl’occhi, prospettiva pratica, Firenze. Facsimile reprint by William Clowes, s.l. 1972. Adams, George 1791 “An Essay on Perspective, and a Description of some Instruments for Facilitating the Practice of that Useful Art”, in Geometrical and Graphical Essays, London, 461–485. Second edition London 1797. Aguilon, François 1613 “De scenographice”, part of “Liber VI. De proiectionibus” in Opticorum libri sex, Antwerpen, 637–681. Alberti, Leon Battista 1435 De pictura, manuscript containing some sections on perspective, first published Basel 1540. In 1436 Alberti wrote an Italian version, Della pittura. There are numerous editions and translations of the two manuscripts (for a survey, see Alberti 1992, 253–255), of which three recent ones are listed as the following items. 1972 On Painting and On Sculpture. The Latin Texts of De Pictura and De Statua, ed. and tr. Cecil Grayson, London. The English translation of De pictura is reedited in Alberti 1991. 1991 On painting, ed. and tr. Cecil Grayson and Martin Kemp, London. 1992 De la peinture. De Pictura (1435), tr. Jean Louis Schefer, Paris. Albrecht, Andreas 1623 Zwey Bücher. Das erste von der ohne und durch die Arithmetica gefundenen Perspectiva. Das andere von dem darzu gehörigen Schatten, Nürnberg. Later posthumous editions, some in Latin, among them the next item. 1671 Duo libri. Prior de perspectiva ... Posterior de umbra, Nürnberg. Aleaume, Jacques 1628 Introduction à la perspective, ensemble l’usage du compas optique et perspective, Paris.* The book seems to have existed in a print which was never published, at present no copies of it are known. 1

The bibliography contains a few nineteenth century titles related to discussions about the further development in particular fields. A star * at a title indicates that I have not seen the publication, and a (*) that the illustrations were missing in the copy I studied. 747

748 1643

First Bibliography La perspective speculative et pratique ... Ensemble la maniere universelle de la pratiquer non seulement sans plan géométral & sans tiers poinct, dedans ni dehors de la champ du tableau. Mais encore par le moyen de la ligne communément appelée horizontale, ed. Étienne Migon, Paris. The same book with a new title page Paris 1663.

Amato, Paolo 1736 La nuova pratica di prospettiva, Palermo.(*) Amico, Giovanni Biago 1750 “Compendio della prospettiva pratica” in L’architetto pratico, Palermo, vol. 2, 122–150. Androuet du Cerceau, Jacques, see Cerceau Antonini, Carlo, see Spampani & Antoni Averlino, Antonio, see Filarete Barbaro, Daniele 1569 La pratica della perspettiva, Venezia. First edition 1568. Facsimile of the 1569 edition, Sala Bolognese 1980. Barca, Pietro Antonio 1620 “Prospettiva” in Avvertimenti e regole circa l’architettura ... prospettiva ..., Milano, 25–27. Bardwell, Thomas 1756 “The Principles of Perspective” in The Practice of Painting and Perspective Made Easy, London, 42–64. Barozzi da Vignola, Giacomo, see Vignola Baytaz, Nicolas 1644 Abbreviations des plus difficiles operations de perspective pratique ... principalement aux vrais peintres ..., Annecy. Benedetti, Giovanni Battista 1585 “De rationibus operationum perspectivae”, in Diversarum speculationum mathematicarum et physicarum liber, Torino, 119–140. “De rationibus ...” was also printed in two later editions of Benedetti’s work, Speculationum mathematicarum et physicarum ... tractatus, Venezia 1586 and Speculationum liber, Venezia 1599. Bettini, Mario 1642 “Apiarium V” in Apiaria universae philosophiae mathematicae, Bologna. Later editions. The section has the heading optics and scenography, but it mainly treats anamorphoses. Bicheur, Jacques Le 1660 Traité de perspective faict par un peintre de l’Académie Royale, dédié à Monsiuer Le Brun ..., Paris.* Bischoff, Johann Christoph 1741 Kurtzgefasste Einleitung zur Perspectiv, darinnen nebst dem wahren Fundamente derselben ... dem noch beygefüget eine neue Erfindung eines Instruments, Halle.

Publications on Perspective

749

Blacker, George O. 1885 John Heywood’s Second Grade Perspective ... adapted from Dr. Brook Taylor, Manchester 1885–1888. Bosboom, Dirk 1703 Perspectiva of doorzicht-kunde, Amsterdam. Later edition Amsterdam 1729. Bosse, Abraham 16482 Maniere universelle de Mr Desargues pour pratiquer la perspective par petit-pied, comme le geometral ..., Paris. Facsimile reprint Alburgh 1987. Dutch translation Bosse 16641. Also a Japanese edition (TatonS 1951, 57). 1653 Moyen universel de pratiquer la perspective sur les tableaux, ou surfaces irregulieres, Paris. Dutch translation Bosse 16642. 16641 Algemeene manier van de Hr Desargues, tot de practyk der perspectiven, gelyk tot die der meet-kunde met de kleine voet-maat ..., tr. J. Bara, Amsterdam. Second edition Amsterdam 1686. 16642 Algemeen middel tot de practijk der doorsicht-kunde op tafereelen of regel-lose buyten gedaanten, Amsterdam. Second edition Amsterdam 1686. 1665 Traité des pratiques géométrales et perspectives enseignées dans l’Académie Royale de la Peinture et Sculpture ..., Paris. Bourdin, Pierre 1661 Le cours de mathematique, third edition, Paris. In the text to plate 172 Bourdin touched upon perspective. Bourgoing, Charles 1661 La perspective affranchie, contenant la vraye et naturele pratique jusques icy inconnue ..., par laquelle l’on peut representer toutes sortes de figures ... sans tracer ny supposer le plan geometral ordinaire, Paris. Bradberry, Robert 1790 The Principles of Perspective. Explained in a Genuine Theory and Applied in an Extensive Practice, Edinburgh. By and large a copy of Martin s.a. Bramer, Benjamin 1630 Beschreibung eines sehr leichten Perspectiv und Grundreissenden Instruments auff einem Stande: Auff Joh. Faulhabers, Ingenieurs zu Ulm, weitere Continuation seines mathematischen Kunstspiegels geordnet, Kassel. Bretez, Louis 1706 La perspective practique de l’architecture ..., Paris. Later edition Paris 1751. Brunn, Lucas 1615 Praxis perspectivae, das ist von Verzeichnungen ein aussführlicher Bericht, Leipzig. Bürja, Abel 1795 Der mathematische Maler oder gründliche Anweisung zur Perspektive, Berlin. Cardi, Lodivico, see Cigoli

2

According to Jean Pierre Le Goff the first copies appeared in 1647 (Le GoffS 1994).

750

First Bibliography

Cataneo, Pietro 1567 “Libro ottavo dove si mostra a operare praticamente nelle cose di prospettiva ...” in L’architettura, Venezia. An earlier edition of L’architettura (Venezia 1554) does not contain book eight. Cerceau, Jaques Androuet du 1576 Lecons de perspective positive, Paris. Later editions, facsimile Paris 1978. Chiaramonti, Scipione 1675 Delle scene, e teatri opera postuma, Cesena.* 3 Cigoli, il byname of Ludovico Cardi Pros Prospettiva pratica, manuscript published as Trattato pratico di prospettiva di Ludovico Cardi detto il Cigoli, ed. Rudolfo Profumo, Roma 1992. Clarke, Henry 1776 Practical Perspective being a Course of Lessons Exhibiting Easy and Concise Rules for Drawing Justly all Sorts of Objects, London. Cobin, A. 1794 Short and Plain Principles of Linear Perspetive Adapted to Naval Architecture, London.* Fourth edition. Commandino, Federico 1558 In planisphaerium Ptolemaei commentarius; in quo universa scenographices ratio quam brevissime traditur, ac demonstrationibus confirmatur, Venezia. Fol. 2r–19r deal with perspective. Contino, Bernardino 1645 La prospettiva pratica, Venezia. First issue 1643, later edition Venezia 1684. Costa, Giovanni Francesco 1747 Elementi di prospettiva per uso degli architetti e pittori, Venezia. Courtonne, Jean 1725 Traité de la perspective pratique avec des remarques sur l’architecture ..., Paris. Cousin, Jean 1560 Livre de perspective, Paris. Facsimile reprint Unterschneidheim 1974. Cowley, John Lodge 1765 The Theory of Perspective, Demonstrated in a Method Entirely New ... Invented, and now Published for the Use of the Royal Academy at Woolwich, London. Curabelle, Jacques 1644 “L’examen de l’une des manieres universelles de Sieur Desargues touchant la pratique de la perspective” in Examen des œuvres du Sieur Desargues, Paris, 66–81. Curel, Nicolas François de 1766 Essai sur la perspective linéaire et sur les ombres, Strasbourg. Danti, Egnazio, see Vignola

3

According to G. Benzoni the book was composed before 1610 (BenzoniS 1980, 542).

Publications on Perspective

751

Darjes, Joachim Georg 1747 “Elementa perspectivae” in Erste Gründe der gesamten Mathematik, Jena, 587–600. Later editions. Darly, Matthew 1751 A New Book of Chinese, Gothic, and Modern Chairs with the manner of putting them in Perspective according to Brook Taylor, London* (RococoS 1984, 167–168). De Caus, Salomon 1612 La perspective avec la raison des ombres et miroirs, London. Dechales, Claude François Milliet 1674 “Perspectiva” in Cursus seu mundus mathematicus, Lyon, vol. 2, 465–532, reprinted in revised version in Dechales 1690, vol. 3, 491–566. 1690 Cursus seu mundus mathematicus, 4 vols., ed. Amati Varcin, Lyon. Deidier, (Abbé) 1744 Traité de perspective théorique et pratique, Paris. Second edition Paris 1770. Despite the different titles the contents of this book and the next item are very close. 1745 “Pratique de la perspective” in Elemens generaux des principales parties des mathematiques neccesaires à l’artellerie et au génie, Paris, vol. 2., 279–379. Second edition Paris 1773. Desargues, Girard 1636 Exemple de l’une des manieres universelles de S.G.D.L. touchant la pratique de la perspective sans emploier aucun tiers point, de distance ny d’autre nature qui soit hors du champ de l’ouvrage, Paris. Reprinted in Bosse 1648, 321–334, and in Desargues Œuvres, vol. 1, 55–84. Facsimile of the first printing together with an English translation in Field and GrayS 1987, 190–201 and 144–160. 1643 Livret de perspective adressé aux théoriciens.* This booklet is mentioned by Curabelle (1644, 70), but it is lost and its exact title is unknown. Its content was presumably presented in the section Aux théoriciens in Bosse 1648. This section is not so easy to find because its text claims to be comments to plates 112–119 which Bosse later renumbered so they became plates 141–148. Moreover the page numbers are missing, but according to the order of the book they should be 313–320. Bosse’s presentation is reprinted in Œuvres, vol. 1, 439–462. Œuvres Œuvres de Desargues réunies et analysées ..., 2 vols., ed. N. Poudra, Paris 1864. Diano, Ferdinando di 1628 L’occhio errante dalla ragione emendate, prospettiva, Venezia. The section covering the pages 153–182 deals with perspective. Ditton, Humphry 1712 A Treatise of Perspective, Demonstrative and Practical, London. Dubreuil, Jean 16421 La perspective pratique ... par un religieux de la compagnie de Jesus, Paris 1642. Published anonymously. For a second and third part, see Dubreuil 1647 and 1649. Particularly the first part of La perspective pratique, known as the “Jesuit’s perspective”, became very popular, it was reissued Paris 1651, and thereafter often. Two English and one German translations are listed separately.

752

First Bibliography

16422 Diverses methodes universelles et nouvelles, en tout ou en partie pour faire des perspectives ... Tirées pour la pluspart du contenu du livre de la Perspective pratique. Ce qui servira de plus de response aux deux affiches du Sieur Desargues contre ladite Perspective pratique, Paris. 1647 Second partie de la perspective pratique ..., Paris. Second edition Paris 1657. 1649 Troisiesme et derniere partie de la perspective pratique ..., Paris. Second edition Paris 1659. 1672 Perspective practical, tr. Robert Pricke, London. Second edition London 1698. 1710 Perspectiva practica, oder Vollständige Anleitung zu der Perspectiv-Reia-Kunst ..., tr. Johann Christoph Rembold, Augsburg. Facsimile reprint Hannover 1977. 1743 The Practice of Perspective. Or an Easy Method of Representing Natural Objects According to the Rules of Art ..., tr. E. Chambers. Dürer, Albrecht 1525 Underweysung der Messung mit dem Zirckel und Richtsheyt in Linien, Ebnen, unnd gantzen Corporen, Nürnberg. Enlarged, posthumous edition Nürnberg 1538. Latin editions with the title Institutiones geometricae, Paris 1532 and later. Abbreviated edition in modernised language in Albrecht Dürer’s Unterweisung der Messung, ed. Alfred Peltzer, München 1908, facsimile Vaduz 1970. Facsimile reprint of the 1525 text, eds. Alvin Jaeggli & Christine Papesch, Zürich 1966, further facsimile reprints Portland, Oregon 1972, Unterschneidlin 1972, in Dürer 1977, and Nördlingen 1983. English and French translations in the next items. 1977 The Painter’s Manual. A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler, ed. and tr. Walter L. Strauss, New York. Besides an English translation,the book contains a facsimile of the 1525 edition and of the changes and additions in the 1538 edition of Underweysung der Messung. 1995 “Instructions pour la mesure à la regle et au compas des lignes, plans et corps solides” in Géométrie, ed. and tr. Jeanne Peiffer, Paris, 132–353. Dupuis, Charles 1773 Cours de géométrie pratique, ..., de perspective. Paris.* Edwards, Edward 1803 A Practical Treatise of Perspective, on the Principles of Dr. B. Taylor, London. Emerson William 1765 “Perspective” in Cyclomathesis: Or an Easy Introduction to the several Branches of the Mathematics, London, vol. 6* (JonesS 1947, 221). 1768 Perspective: Or the Art of Drawing the Representations of all Objects upon a Plane, London. Appeared as a separate part of Emerson’s The Elements of Optics, London. Eytelwein, Johann Albert 1810 Handbuch der Perspektive, Berlin. Faulhaber, Johann 1610 Newe geometrische und perspectivische Inventiones, Frankfurt am Main. 1633 Ingenieurs Schul. Dritter Theil ... die irregular Figuren zu fortificern, Ulm. Ferguson, James 1775 The Art of Drawing in Perspective Made Easy ..., London.

Publications on Perspective

753

Filarete, byname of Antonio Averlino Arch Book 23 of Trattato di architettura, manuscript from around 1460. The tract was first published together with a German translation in Antonio Averlino Filarete’s Tractat über die Baukunst, ed. and tr. Wolfgang von Oettingen, Wien 1890. The following titles are later editions. References are to Filarete 1972. 1965 Filarete’s Treatise on Architecture, 2 vols., ed. and tr. John R. Spencer, New Haven. 1972 Trattato di architettura, 2 vols., ed. Anna Maria Finoli & Liliana Grassi, Milano. Fortius, see Ringelberg Fournier, Daniel 1761 A Treatise of the Theory and Practice of Perspective. Wherein the Principles ... by Dr. B. Taylor are fully and clearly explained by the means of moveable Schemes, London. Later editions London 1762, 1763, 1764. Francesco di Giorgio Martini Arch “Geometria e modi di misurare distanze altezze e profundità” in Trattati di architettura ingegneria e arte militare, this section contains a page on perspective. Manuscript from around 1480, first published in Torino 1841. References are to Francesco 1967. 1967 Trattati di architettura ingegneria e arte militare, 2 vols., ed. Corrado Maltese & Livia Maltese Degrassi, Milano. Füllisch, Johann Jacob 1680 “Von Perspectiven” part 6 of Compendium artis ... Das ist kurzer leichter ... Unterricht von der geometrisch-ignographischen Zeugnungs und Baukunst, Nerolingen. Galli-Bibiena, Ferdinando 1711 “Della prospettiva in generale” part 3 of L’architettura civile preparata sul la geometria e ridotta alla prospettiva, Parma, 77–114. A lavish edition, the text reappeared slightly revised in a more economical edition being the next item. 1731 Direzioni al giovani studenti nel disegno dell’architettura civile ..., 2 vols., Bologna 1731 and 1732. The second volume of this work was edited posthumously with yet another title, namely the next item. 1753 Direzioni della prospettiva teorica ..., Bologna. Galli-Bibiena, Giuseppe 1740 Architettura & prospettiva, Augusta. Gaultier de Maignannes, René 1648 Invention nouvelle et brieve pour reduire en perspective, par le moïen du quarré, toutes sortes de plans ... sans se servir d’autres points de tiers, ou accidentaux, que de ceux qui peuvent tomber dans le tableau, La Fleche. Gaurico, Pomponio 1504 “De perspectiva” in De sculptura, Firenze. Numerous later editions. In modern times edited by Heinrich Brockhaus with a German translation, Leipzig 1886 and in the following title. 1969 “De perspectiva” in De sculptura, Latin text and French translation, ed. and tr. André Chastel & Robert Klein, Genève, 182–201.

754

First Bibliography

Glockendon, Jörg 1509 Von der Kunnst Perspectiva, Nürnberg. A pirated German version of Viator 1505. Graaf, Abraham de 1676 “Perspectief of tekenkonst” the tenth book in De geheele mathesis of wiskonst, herstelt in zijn natuurlijke gedaante, Amsterdam, 213–226. A revised version in 1694, where perspective occurs on the pages 213–222. Several later editions. ’sGravesande, Willem Jacob 1711 Essai de perspective, Den Haag. Later edition Rotterdam 1717. Reprint ’sGravesande 1774. An English translation in the next item and a free Dutch translation in ’sGravesande 1837. 1724 An Essay on Perspective ..., tr. E. Stone, London. 1774 “Essai de perspective” in Œuvres philosophiques et mathématiques de Mr G.J. ’sGravesande, ed. Nicolas Sébastien Allamand, Amsterdam, vol. 1, 1–88. 1837 Beginselen der doorzigtkunde door G.J. ’sGravesande. Vrij vertaald uit het fransch en met bijvoegselen vermeerderd, ed. and tr. H. van Blanken, Zwolle. Guarini, Guarino 1671 “Tractatus XXVI de stereographia” in Euclides adauctus et methodicus mathematicaque universalis, Torino, 452–462. Guenon, Henri 1640 Pratique nouvelle et universelle de la perspective sur les seules parties egales du compas de proportion, sans y adiouster aucune ligne d’optique, Paris. This is a booklet of 11 pages. Guidobaldo del Monte 1600 Perspectivae libri sex, Pesaro. Reproduction of the Latin text and an Italian translation in the following title. 1984 I sei libri della prospettiva di Guidobaldo dei marchesi Del Monte, tr. Rocco Sinisgalli, Roma. Guyot, Guillaume Germain 1769 Nouvelles récréations physiques et mathématiques ..., Paris,* numerous other editions. According to Hooper, Guyot treated anamorphoses, and it seems most likely that he did so in this work (Hooper 1774, 172). Haesell, Theodosius 1652 Geistliche perspectiva, Dresden. Halfpenny, William 1731 Perspective Made Easy: Or a New Method for Practical Perspective, London. Halt, Peter 1625 Perspektivische allen Bauleuten dienende Reisskunst ..., Augsburg. 1626 Drey wichtige newe Kunststück in underschidlichen perspectivischen Instrumenten inventiert und erfunden, Augsburg.* Hamberger, Georg Erhard 1747 Dissertatio mathematica sistens leges perspectivae ad situm plani transparentis mutatum adplicatas, Jena; mathematical dissertation defended 1719.

Publications on Perspective

755

Hamilton, John 1738 Stereography, or a Compleat Body of Perspective ..., London. Later editions London 1740, 1748, 1749. Hartnack, Daniel 1683 Perspectiva mechanica und eigentliche Beschreibung derer vornehmsten Instrumenten ... zum perspectivischen Reissen bissher erfunden worden, Lüneburg. Hartsoeker, Nicolaas 1699 Proeve der deursicht-kunde in het frans beschreeven ... en vertaald door A. Block, Amsterdam.* Hass, Georg 1583 Künstlicher und zierlicher newer vor nie gesehener funffzig perspectifischer Stück ..., Wien. Heinecke, Paul 1727 Lucidum prospectivae speculum. Das ist ein heller Spiegel der Perspective, Augsburg. Hennert, Johann Friedrich 1770 “Perspectivae” in Elementa optices, perspectivae, catoptrices, dioptrices et phaometriae, Utrecht, 37–51. Hérigone, Pierre 1637 “Perspective” in Cursus mathematicus/Cours mathématique, Paris, vol. 5, 190–21. Later edition Paris 1654. 1642 “De la perspective” in Supplementum cursus mathematici, 99–116, Paris. Highmore, Joseph 1754 A Critical Examination of those two Paintings on the Ceiling of the BanquetingHouse at Whitehall, in which Architecture is introduced, so far as relates to the Perspective, London. 1763 The Practice of Perspective on the Principles of Dr. Brook Taylor, ... Written many years since, but now first published, London. Hirschvogel, Augustin 1543 “Anfang des Perspectiva” in Ein aigentliche und grundtliche Anweysung, in die Geometria, sonderlich aber, wie alle regulierte, und unregulierte Corpora in den Grund gelegt, und in das Perspecktiff gebracht ... sollen werden, fol. fiiv– hiiir, Nürnberg. Hodgson, James 1743 “The Theory of Perspective” printed as introduction to Dubreuil 1743, 1–16. Hoffmann, Johann Leonhard 1780 Anweisung zur Vertigung und Gebrauch des algemeinen Zeichnen-Instruments ohne Gläser, Anspach. Hondius, Hendrik 1623 Onderwijssinge in de perspective conste, Den Haag. Without many changes the book was reedited as Grondige onderrichtinge in de optica of te perspective konste, one edition is from Amsterdam without a year of printing, another appeared in Amsterdam 1647 and a third in den Haag 1647 (where the title is written

756

1625

First Bibliography Grondighe onderrichtinghe ...). All the editions – including the French mentioned below – have the same frontispiece decorated with the title Institutio artis perspectivae, hence references to such a work by Hondius occur frequently. Instruction en la science de perspective, den Haag. Facsimile reprint Alburgh 1987.

Hoogstraten, Samuel van 1678 Inleyding tot de hooge schoole der schilderkonst, Rotterdam. Facsimile reprint Davaco Publishers, s.l. 1969. Later editions. Hooper, William 1774 “Perspective Recreations” in Rational Recreations, London, vol. 2, 168–188. This section contains an instruction in making anamorphoses. Horstig, Karl Gottlieb 1797 Briefe über die mahlerische Perspektive, Leipzig. Houten, Hendrik van 1705 Verhandelinge van de grontregelen der doorzigtkunde of tekenkonst (perspectief), Amsterdam. Hummel, Johann Erdmann 1824 Die freie Perspektive, erlaütert durch praktische Aufgaben und Beispiele, hauptsächlich für Maler und Architekten, 2 vols., Berlin 1824 and 1825. Huret, Grégoire 1670 Optique de portraiture et peinture, Paris. Later edition Paris 1672. Jamnitzer, Wenzel 1568 Perspectiva corporum regularium. Das ist, ein fleyssige Fürweysung, wie die fünff regulirten Cörper ... inn die Perspectiva gebracht ... werden mügen, Nürnberg. Reprint Graz 1973. Facsimile reprint together with a French translation in FloconS 1964. The work seems to be copied in Sintagma, in quo variae eximiaque corporum diagrammata ex praescriptio opticae exibentur ..., Amsterdam 1608 and 1618 (JonesS 1947, 155 and VagnettiS 1979, 335). Jeaurat, Edme Sébastien 1750 Traité de perspective à l’usage des artistes, Paris. Jelgerhuis, Rienk 1769 Nauwkeurige aanmerkingen op een vornaam gedeelte van de perspectiva of doorzigtkunde van Casper Philips Jacobsz, Leeuwarden. Johann II, Simmern von 1531 Eyn schön nützlich Büchlin und Underweisung der Kunst des Messens mit dem Zirckel, Richtscheidt oder Lineal. Zu Nutz allen ... denen so sich der Kunst ... Perspectiva ... zugebrauchen Lust haben, Simmern. Facsimile reprint Graz 1970. Later edition Frankfurt 1546. Kästner, Abraham Gotthelf 1752 Perspectivae et projectionum theoria generalis analytica, Leipzig. 1774 “Die Perspective” in Anfangsgründe der Arithmetik, Geometrie ... und Perspectiv, third revised edition Göttingen, 460–469. First edition appeared in Göttingen 1752 and the fifth enlarged in Göttingen 1792.

Publications on Perspective

757

Karsten, Wenceslaus Johann Gustav 1775 “Die Perspectiv” in Lehrbegrif der gesamten Mathematik, Greifswald, vol. 7, 110–928. Kaschube, Johann Wenceslaus 1717 “Die Perspectiv-Kunst” in Cursus mathematicus, oder deutlicher Begriff der mathematischen Wissenschaften ..., Jena, 353–361. Kirby, John Joshua 1754 Dr. Brook Taylor’s Method of Perspective Made Easy, Both in Theory and Practice, Ipswich. Second edition Ipswich 1755, this contains an appendix which was issued separately in 1754. For later editions, see Kirby 1765. s.a. Dr. Brook Taylor’s Method of Perspective, compared with Examples lately publish’d on this Subject as Sirigatti’s by Isaac Ware, London. According to De MorganS 1861 this book was published in 1757. 1761 The Perspective of Architecture ... Deduced from the Principles of Dr. Brook Taylor, Part First Contains the Description and Use of a new Instrument Called the Architectonic Sector, Part the Second, a New Method of Drawing ... in Perspective, London. The first part also appeared as a separate book London 1761. 1765 Dr. Brook Taylor’s Method of Perspective Made Easy, Both in Theory and Practice, third revised and enlarged editon of Kirby 1754, in folio as well as in quarto both of which were reissued London 1768. Kircher, Athanasius 1646 “De arte scenographica” in Ars magna lucis et umbrae, Roma, 161–196. Second edition Amsterdam 1671, 124–143. Koutny, E., see Peschka & Koutny Lacaille, Nicolas Louis 1756 “Traité de perspective” in Leçons élémentaires d’optiques, second edition, Paris, 128–198. In the first edition of his book on optics, Lacaille did not treat perspective. Later edition Paris 1808. Lairesse, Gerard de 1707 Het groot schilderboek, 2 vols., Amsterdam. Several later editions, among them Amsterdam 1712, Haarlem 1740. English translation next title. 1738 The Art of Painting in all its Branches, tr. John Frederick Fritsch, London. Lambert, Johann Heinrich 1752 Manuscript entitled Anlage zur Perspektive edited in Lambert 1943, 161–186, and translated into French in Lambert 1981. 1759 Lambert 1759 refers to both the two following items. 17591 La perspective affranchie de l’embaras du plan géométral, Zürich. Facsimile reprints Paris 1977 and Alburgh 1987. German edition below. 17592 Die freye Perspektive, oder Anweisung jeden perspektivischen Aufriss von freyen Stücken und ohne Grundriss zu verfertigen, Zürich. Second edition as part of Lambert 1774. Reprinted in Lambert 1943, 192–301. 1768 Kurzgefasste Regeln zu perspectivischen Zeichnungen vermittelst eines ... Proportional-Zirkels, Augsburg. 1774 Die freye Perspective ... mit Anmerkungen und Zusätzen vermehrt, Zürich. Reprinted in Lambert 1943, 309–380. French translation of the Anmerkungen

758

First Bibliography

in LaurentS 1987, 193–284. Page references to Lambert 1774 are to the part containing the Zusätze. 1799 “J.H. Lamberts Grundsätze der Perspectiv, aus Betrachtung einer perspectivisch gezeichneten Landschaft abgeleitet”, ed. Johann Bernoulli, Archiv der reinen und angewandten Mathematik, vol. 9, 1–21. 1943 Johann Heinrich Lambert, Schriften zur Perspektive, ed. Max Steck, Berlin. 1981 Essai sur la perspective, ed. Roger Laurent and tr. Jeanne Peiffer, Coubron. Translation of Anlage zur Perspektive (Lambert 1752). Lamy, Bernard 1701 Traité de perspective, Paris. Later edition Amsterdam 1734. English translations in the next items. 1702 A Treatise of Perspective, tr. A. Forbes, London* (SothebyS 2002, 238). 1710 Perspective made Easie, tr. A. Forbes, London. Facsimile reprint Alburgh 1987. Lautensack, Heinrich 1564 Des Circkels unnd Richtscheyts, auch der Perspectiva, und Proportion der Menschen und Rosse, kurtze doch gründtliche Underweisung, des rechten Gebrauchs, Frankfurt. Lauterbach, Anton Bernhard 1717 Clavis perspectivae verticalis geometrica, Jena.* Le Breton, Adèle 1828 Traité de perspective simplifiée (linéaire) ..., Paris.* Second edition Paris 1832. Leclerc, Sébastien 1679 Discours touchant le point de veue, dans lequel il est prouvé que les choses qu’on voit distinctement, ne sont veuës que d’un oeil, Paris. Lencker, Hans 1567 Perspectiva literaria. Das ist ein clerliche Fürreyssung, wie man alle Buchstaben ... in die Perspectif ... bringen mag, Nürnberg. Several later editions. Reprint Frankfurt 1972. 1571 Perspectiva. Hierinnen auffs kürtzte beschrieben ... wird, ein newer ... Weg, wie allerley Ding ... in Grund zulegen ist, ... ferner in die Perspectyf gebracht werden mag ..., Nürnberg. Later edition Nürnberg 1595. Leonardo da Vinci Codex Atlanticus

Biblioteca Ambrosiana, Milano, published in Leonardo 1894. Codex Urbinas 1270 compiled by Francesco Melzi around 1530, Biblioteca Vaticana, Roma, published in Leonardo 1956 and in Leonardo 1995. Manuscript A 1492, Institut de France, Paris, published in Leonardo 1881, vol. 1. Manuscript Arundel 1480–1518, British Library, London, published in Leonardo 1923. Manuscript Ash I 1487–1490, Institut de France, Paris, published in Leonardo 1881, vol. 6. Manuscript C 1490–1491, Institut de France, Paris, published in Leonardo 1881, vol. 3. Manuscript E 1513–1514, Institut de France, Paris, published in Leonardo 1881, vol. 3.

Publications on Perspective

759

Manuscript Forster

around 1505, Victoria and Albert Museum, London, published in Leonardo 1923. Manuscript G 1510–1515, Institut de France, Paris, published in Leonardo 1881, vol. 5. Manuscript W 1478–1518, Royal Library, Windsor, published in Leonardo 1957, Leonardo 1968, and Leonardo, 1978. 16511 Trattato della pittura di Lionardo da Vinci, ed. Raphael du Fresne, Paris. Numerous later editions in several languages, the first translations being the next items. For a list of editions from the period 1651–1956, see SteinitzS 1958. 16512 Traité de la peinture, tr. Roland Fréart de Chambray, Paris. 1721 A Treatise of Painting, tr. John Senex, London. 1724 Tractat von der Mahlerey, tr. Johann Georg Böhm, Nürnberg. Second edition Nürnberg 1747. 1802 A Treatise on Painting, tr. John Francis Rigaud, London. Reprinted New York 2002. 1881 Les manuscripts de Léonard de Vinci ... de la bibliothèque de l’Institut, 6 vols., ed. and tr. Charles Ravaisson-Mollien, Paris 1881–1891. 1894 Il codice Atlantico di Leonardo da Vinci nella Biblioteca Ambrosiana di Milano, 35 vols., ed. G. Piumati, Firenze, 1894–1904. 1923 I manoscritti e i disegni di Leonardo da Vinci, Roma 1923–1941. 1956 Treatise on Painting, ed. and tr. A. Philip McMahon, Princeton New Jersey. 1957 Leonardo da Vinci: Fragments at Windsor Castle from Codex Atlanticus, ed. Carlo Pedretti, London. 1964 Leonardo da Vinci on Painting. A Lost Book (Libro A) Reassembled from the Codex Vaticanus Urbinas 1270 and from the Codex Leiceister, ed. and tr. Carlo Pedretti, Berkeley. 1968 A Catalogue of Drawings by Leonardo da Vinci in the Collection of Her Majesty the Queen at Windsor Castle, 3 vols., ed. Kenneth Clark and Carlo Pedretti, London 1968–1969. 1970 The Notebooks of Leonardo da Vinci, 2 vols., ed. and tr. Jean Paul Richter, New York. Reprint of The Literary Works of Leonardo da Vinci, London 1883. 1977 Carlo Pedretti, The Literary Works of Leonardo da Vinci. A Commentary to Jean Paul Richter’s Edition, 2 vols., London. 1978 Leonardo da Vinci. Corpus of the Anatomical Studies in the Collection of Her Majesty the Queen at Windsor Castle, 3 vols., ed. Kenneth D. Keele and Carlo Pedretti, London 1978–1980. 1989 Leonardo on Painting. An Anthology of Writings by Leonardo da Vinci, ed. and tr. Martin Kemp and Magaret Walker, New Haven. 1995 Libro di pittura: Codice urbinate lat. 1270 nella Biblioteca apostolica Vaticana / Leonardo da Vinci, ed. Carlo Pedretti, tr. Carlo Vecce, Firenze. Leupold, Jakob 1713 Anamorphosis mechanica nova, oder Beschreibung dreyer neuen Maschinen mit welchen ... mancherley Bilder und Figuren können gezeichnet werden dass sie ganz ungestalt und unkäntlich fallen ..., Leipzig. First published in Acta eruditorum 1712. Lomazzo, Giovanni Paolo 1584 “Libro quinto della prospettiva” in Trattato dell’arte della pittura, scoltura, et architettura. Milano. Several later editions. Fascimile of the 1584 edition Hildesheim 1968. English translation next title.

760 1598

First Bibliography A Tracte Containing Artes of Curious Paintinge, Carvinge & Buildinge, ed. Richard Haydock, Oxford.

Lorenz, Johann Friedrich 1799 “Von den ersten Gründen der Perspectiv” in Grundriss der reinen und angewandten Mathematik. Zweyter Theil. Die angewandte Mathematik, second edition Helmstedt, 132–138. First edition Helmstedt 1791. Maignan, Emmanuel 1648 Perspectiva horaria ..., Roma. Proposition 77 in the third book, 438–449, deals with anamorphoses. Malton, James 1800 The Young Painter’s Maulstick: being a Practical Treatise on Perspective ... Founded on the Process ... of Vignola and Sirigatti, ... United with the Theoretic Principles of ... Dr. Brook Taylor ... Addressed to Students in Drawing, London. Malton, Thomas 1775 A Compleat Treatise on Perspective in Theory and Practice on the True Principles of Dr. Brook Taylor, Made Clear ..., London. Also London 1776, 1778 and 1779. 1783 An Appendix or Second Part to the Compleat Treatise on Perspective, London. Second edition London 1800. Marolois, Samuel 1614 La perspective contenant la theorie et la pratique, Den Haag. This work was reprinted often, among the years of publication are 1628, 1638, 1647 and 1662; it appeared in Dutch, German, French, and Latin separately and as part of Marolois’s Opera mathematical Œuvres mathematiques which appeared from 1614 onwards. There is some language confusion in several of the editions their title page being in one language and the main text in another. Martin, Benjamin 1764 “Universal Perspective” in A New and Comprehensive System of Mathematical Institutions, Agreeable to the Present State of Newtonian Mathesis, vol. 2, London, 148–228. s.a. The Principles of Perspective explained in a Genuine Theory and Applied in an Extensive Practice. With the Construction and Uses of all such Instruments as are Subservient to the Purpose of this Science, London. 1771 The Description and Use of a Graphical Perspective and Microscope for Drawing all Kinds of Objects in True Perspective ..., London. A booklet of a dozen pages of which nine deals with a perspective instrument. Mayer, Johann Tobias 1786 “Perspectivische Zeichnungen” & “Katoptriche Zeichnungen” in Unterricht zur praktischen Rechenkunst, zu geometrischen, perspectivischen und optischen Zeicnungen ... ehemals durch Schübler und Rost verfasset, nun aber umgeändert und mit Zusätzen vermehrt, Nürnberg, 211–262 and 265–272. Meister, Albrecht Ludwig Friedrich 1753 Instrumentum scenographicum, Göttingen.

Publications on Perspective

761

Mersenne, Marin 1644 “De arte perspectivae” in Universae geometriae mixtae ..., Paris, 541–548. Michel, S.N. 1771 Traité de perspective linéaire, Paris. Mönnich, Bernhard Friedrich 1794 Versuch die mathematischen Regeln der Perspektive für den Künstler ohne Theorie anwendbar zu machen, Berlin. Second edition Berlin 1801. Mohr, Georg 1672 Euclides Danicus, Amsterdam. One edition in Danish and another in Dutch. The problems 19–22 in part two deal with perspective constructions. Monge, Gaspard 1820 “Théorie de la perspective” in Géométrie descriptive,4e édition augmentée d’une théorie des ombres et de la perspective, extraite des papiers de l’auteur, par M. Brisson, Paris, §§ 136–139, later edition Paris 1827. Reprinted in L’École normale de l’an III. Leçons de mathématiques, ed. Jean Dhombres et. al, Paris 1992. Moxon, Joseph 1670 Practical Perspective, or, Perspective Made Easie, London. Muller, John 1765 “Elements of Perspective” in Elements of Mathematics. Third edition which is “improved with an addition of a new treatise on perspective”, London, 304–312. Murdoch, Patrick 1746 “Perspectivae linearis principia”, Section I in Neutoni genesis curvarum per umbras seu perspectivae universalis elementa; exemplis coni sectionum et linearum tertii ordinis illustrata, London, 1–18. French translation in Taylor 1757. Mydorge, Claude 1638 Examen du livre des recreations mathématiques et des problemes en géometrie, mechanique, optique & catoptrique, Paris. Revised second edition with the title Recreations mathematiques ..., Paris 1661. Problem 2 touches upon perspective. Niceron, Jean François 1638 La perspective curieuse, ou magie artificiele des effets merveilleux ..., Paris. Later enlarged editions Paris 1652, Paris 1663, and Paris 1679. Revised Latin edition below. 1646 Thaumaturgus opticus ..., Paris. Noble, Edward 1771 The Elements of Linear Perspective, London. Oldenburg, Henry 1669 “The Description of an Instrument, Invented Divers Years ago by Dr. Christopher Wren, for Drawing the Out-Lines of any Object in Perspective”, Philosophical Transactions of the Royal Society, vol. 4, 898–899.4 4

The text is not signed, but probably written by the secretary Henry Oldenburg (BennettS 1982, 75).

762

First Bibliography

Orsini, Baldassare 1771 Della geometria e prospettiva pratica, 2 vols., Roma 1771* and 1773* (SothebyS 2002, 293). Ozanam, Jacques 1693 La perspective théorique et pratique ..., vol. 4 of Ozanam’s Cours de mathématique, 5 vols., Paris. The Cours was reissued Paris 1697 and Amsterdam 1699 and appeared in English as Cursus mathematicus: Or a Compleat Course of the Mathematicks, 5 vols., London 1712. Moreover La perspective was edited separately Paris, 1711, 1720, and 1769. Pélerin, Jean, see Viator Peschka, Gustav A. & Koutny, E. 1868 Freie Perspektive in ihrer Begründung und Anwendung, Hannover. Petitot, Ennemond Alexandre 1758 Raisonnement sur la perspective pour en faciliter l’usage aux artistes/ Ragionamento sopra la prospettiva ..., Parma; parallel French and Italian text. Pfinzing, Paul 1599 Ein schöner kurtzer Extract der Geometriae unnd Perspectivae, Nürnberg. Appeared in a second revised edition under the following title. 1616 Optica, das ist gründtliche doch kurtze Anzeigung wie notwendig die löbliche Kunst Geometriae seye inn der Perspectiv. Augsburg. Philips, Caspar Jacobszoon 1765 Uitvoerig onderwys in de perspectiva of doorzichtkunde, Amsterdam. Second printing, Amsterdam 1781. Edited in German as Ausführlicher Unterricht in der Perspective, Degen 1803. 1775 Handleiding in de spiegelperspectief, om door de regulen der doorzichtkunde alle voorwerpen in vlakke spiegels over te brengen, Amsterdam. Later editions Amsterdam 1780 and 1803. 17861 Wis-meet-en doorzichtkundige handleiding volgende welke men ten allen tyden en plaatse der stand der zonne en maane ..., Amsterdam. 17862 Zeemans onderwijser in de tekenkunst of handleiding om door geometrische, doorzichkundige en perspectivische regelen ..., Amsterdam. 1788 Handleiding om in de kunst-tafereelen den afstand het oog des zienders tot de zelve ... als ook de der perspectivische regelen ..., Amsterdam. Piero della Francesca Pros De prospectiva pingendi, manuscript from before 1492. Exists in an Italian and a Latin version – both with the Latin title. The earliest Italian version, a Parma codex, is partially an autograph. This was first printed together with a German translation in Piero 1899, and published again in 1942 as the first edition of Piero 1974. References are to Piero 1974. 1899 Petrus Pictor Burgensis: De prospectiva pingendi, tr. Constantin Winterberg, Strasbourg. 1974 De prospectiva pingendi, 2 vols., ed. Giusta Nicco Fasola, Firenze. First edition Firenze 1942 and third edition Firenze 1984. 1998 De la perspective en peinture, ed. Jean Pierre Le Goff, Paris.

Publications on Perspective

763

Piranesi, Giovanni Battista 1750 Opere varie de architettura, prospettiva ..., ed. Giovanni Bouchard, Roma. Pozzo, Andrea 1693 Perspectiva pictorum et architectorum/Prospettiva de’ pittori e architetti, 2 vols., Roma 1693 and 1700. As the title indicates the work has parallel Latin and Italian text. Dozens of later editions of the first part in several languages and various combinations of languages (KerberS 1971, 267–270). In fact both volumes were available in French, German, and English rather early in the 18th century (ibid.). 1707 Rules and Examples of Perspective, in Latin and English, ed. John James with new engravings by John Sturt, London. This edition exists in two printings, one by B. Motte and one by J. Senex and J. Osborn which is undated. Facsimile reprint of the Motte edition New York 1991 and facsimile of the Senex and Osborn edition, in reduced size, New York 1989. Priestley, Joseph 1770 A Familiar Introduction to the Theory and Practice of Perspective, London. Second edition 1780. Reprinted New York 1970. 1772 “A Short History of Perspective”, in The History and Present State of Discoveries Relating to Vision, Light and Colours, 2 vols., London, 91–96. Quadri, Giovanni Lodovico 1744 La prospettiva pratica delineata in tavola a norma della secondo regola di Giacomo Barozzi da Vignola, Bologna. Reisch, Gregor 1508 “Introductio architecturae et perspectivae” in Margarita philosophica nova, Strassbourg. Copy without reference of parts of Viator 1505. Many later editions. Ringelberg, Joachim Fortius 1531 “Optice” in Opera, Lyon, 459–480. Facsimile reprint Niewkoop 1967. The section on optics is completely devoted to perspective. Rödel, Johann Michael 1796 Abhandlungen von den zufälligen Punkten in der Perspektivkunst für Werkmeister, Leipzig. The book has a preface by Abraham Gotthelf Kästner. First edition Coburg 1784. Rohault, Jacques 1682 “La perspective” in Œuvres posthumes, ed. Claude de Clerselier, Paris. In the second edition Den Haag 1690, “La perspective” appears in vol. 2, 259–284. Rost, Johann Leonhard (see also Schübler & Rost) 1745 Mathematischer Lust und Nutzgarten ... darinnen das Nothwendigste von der Arithmetica vulgari ... Sammt einer Einleitung zur Perspectiv, wie sie in des Herrn Desargues Anfangsgründe ..., Nürnberg.* 5

5

Most likely, this is a second edition of Schübler & Rost 1724 as the two books have the same title, but I have not had the occassion to examine Rost 1745.

764

First Bibliography

Roy, Claude 1756 Essai sur la perspective pratique par le moyen du calcul, Paris. Ryff, Walther Hermann (also known as Gualtherus Rivius) 1547 Der furnembsten, notwendigsten, der ganzen Architectur angehörigen mathematischen und mechanischen Künst ..., Nürnberg. Later editions Nürnberg 1558 and Basel 1582. Sabbatini, Niccolò 1638 Pratica di fabricar scene e machine ne’ teatri, Ravenna, second enlarged edition. First edition 1637. Facsimile of the 1638 edition together with a German translation by Willi Flemming, Weimar 1926. Part of it translated into English in HewittS 1958. Salmon, William 1672 “The General Practice of Perspective” in Polygraphice or the Art of Drawing ..., London, 73–79. The book reappeared frequently in the following decades. Sandrart, Joachim 1675 L’Academia todesca ... oder teutsche Academie der edlen Bau-Bild-und Mahlerey-Künste ... Darinn enthalten ein gründlicher Unterricht ... von der Perspectiv, Nürnberg. Edited by A.R. Peltzer, München 1925. Scheiner, Christoph 1631 Pantographice ... prior epipedographicen, sive planorum, posterior stereographicen, seu solidorum ..., Roma. Later editions in Italian among them the next item. 1653 Prattica del parallelogrammo da disegnare del P. Christoforo Scheiner, ed. Giulio Troili, Bologna. Scherffer, Karl 1781 “Perspektivische Aufgaben durch welche der Gebrauch des Tangentenmaassstabes ... erleichtert wird” in Beyträge zur Mathematik, Wien, 191–225. Schön, Erhard 1538 Underweissung der Proportzion unnd Stellung der Possen ... wie man das vor Augen sicht ..., Nürnberg. Enlarged editions Nürnberg 1540–1543 (SchülingS 1973, 40–42). Facsimile reprint ed. Leo Baer, Frankfurt a. M. 1920. Schooten, Frans van 1660 “Tractaet der perspective, ofte schynbaere teycken-konst. Waer in de fondamenten derselbe konst op het kortste verhandelt en betoont worden”, published in Mathematische Oeffeningen, Amsterdam, 501–544. Schott, Gaspar 1657 “De magia anamorphosi optica”, Liber III of Magia universalis naturae et artis, Pars I: Optica, Würzburg, 99–169. Later edition Bamberg 1677. 1677 “Opticae practicae sive perspectivae” in Cursus mathematicus, Bamberg, 470. Posthumous edition. Schübler, Johann Jacob (see also Schübler & Rost) 1719 Perspectiva pes picturae. Das ist kurze und leichte Verfassung der practicabelsten Regul zur perspectivischen Zeichnungs-Kunst, 2 vols., Nürnberg 1719 and 1720. Several later editions with the title Perspectiva geometrico-practica.

Publications on Perspective

765

Schübler, Johann Jacob & Rost, Johann Leonhard 1724 “Von der Perspectiv” and “Von der Anamorphosi Catoptrica” in Mathematischer Lust und Nutzgarten ... darinnen das nothwendigste von der Arithmetica vulgari ... sammt einer Anleitung zur Perspectiv wie sie in des Herrn Desargues Anfangsgründe ..., Nürnberg, 204–249 and 249–262. The title page only reveals that the book is written by S.R. On page 369, I have explained why I consider S. R. to be an abbreviation of Schübler and Rost. Segner, Johann Andreas 1779 Gründe der Perspectiv, Berlin. Serlio, Sebastiano 1545 D’Architettura/L’Architecture. Il secondo libro di perspettiva/Le second livre de perspective, Paris. This second book in Serlio’s work on architecture was, as the title indicates, printed with parallel Italian and French text. It became very popular, went through many editions and was translated into Dutch, English, German, Latin and Spanish. The first Dutch and English translations are listed separately. The Italian text appeared again in Serlio 1584. 1553 Den tweeden boeck van architecturen Sebastiani Serlii tracterende van perspectyven, tr. Pieter Coecke van Aelst, Antwerpen. 1584 “Il secondo libro di prospettiva”, in Tutte l’opere d’architettura, Venezia. Facsimile reprint Sala Bolognese 1987. Reprints 1600 and 1619, fascimile edition of the latter Ridgewood, New Jersey, 1966. 1611 The Second Booke of Architecture ... Entreating of Perspective which is Inspection or Looking into by Shortening of the Sight, tr. Robert Peake, London. This book was a translation of the Dutch edition. Sheraton, Thomas 1793 “Of Perspectives” in The Cabinet-maker and Upholsterer’s Drawing Book, London, 177–350. Silvabelle, Jacques 1757 “Méthode generale, pour trouver la perspective d’un objet donné, par le moyen des lignes perpendiculaires & paralleles” in Taylor 1757/1759, xxix–xxxvi. Sirigatti, Lorenzo 1596 La pratica di prospettiva, Venezia. Later edition Venezia 1625. English translation mentioned below. 1756 Practice of Perspective with Figures Engraved by Isaac Ware, London. Smith, Godfrey 1755 The Laboratory; or School of Arts ... and Easy Introduction to the Art of Drawing in Perspective ... 2 vols., London*. Sobro, Aléxandre s.a. Traité de perspective à l’usage des artistes ..., Paris.* Presumably published in the 1790s. Spampani, Giovanni Battista & Antonini, Carlo 1770 “Prospettiva pratica di M. Giacomo Barozzi da Vignola” in Il Vignola illustrato proposto Giambattista Spampani e Carlo Antonini studenti d’ architettura ... , Roma.

766

First Bibliography

Stevin, Simon 16051 “Van de verschaeuwing. Eerste bouck der deursichtighe” in Derde Stuck der Wisconstighe Ghedachtnissen, Leiden. Most of it is reprinted in Stevin 1978, on the even pages from 796 to 964. Translations listed below. 16052 “De optica” in Hypomnemata mathematica a Simone Stevino. Tomus tertius, ed. and tr. Willebrord Snellius, Leiden. Reprinted in SinisgalliS 1978. 16053 “Des perspectives” in Mémoires mathématiques par Simon Stevin. Livre trois, ed. and tr. Jean Tuning, Leiden. Slightly revised version of this translation in Stevin 1634. 1634 Les œuvres mathématiques de Simon Stevin ..., ed. and tr. Albert Girard, Leiden. 1958 The Principal Works of Simon Stevin, ed. and tr. Dirk J. Struik, Amsterdam, vol. II.B. In this volume are included a facsimile of most of Stevin 16051 and an English translation of it. 1978 SinisgalliS 1978 contains a reprint of Stevin 16052 together with an Italian translation. Stör, Lorenz 1567 Geometria et Perspectiva. Hierinn etliche zerbrochne Gebew den Schreinern in eingelegter Arbait dienstlich ... Augsburg. Reprint Frankfurt 1972. Sturm, Johann Christoph 1701 “Optica” in Mathesis juvenilis, Nürnberg, vol. 2., second edition, perspective and anamorphoses are dealt with on pages 154–164. First edition Nürnberg 1699. Many later editions also in German. Sulzer, Johann Georg 1787 “Perspectiv” in Allgemeine Theorie der schönen Künste, Leipzig, 552–569. Tacquet, Andreas 1669 “De perspectiva sive projectionis scenographicae theoria et praxi” in the posthumously published Opera mathematica, Antwerpen, 158–177. Later edition Antwerpen 1707. Taylor, Brook 17151 Linear Perspective: or, a New Method of Representing justly all Manner of Objects, London. Facsimile reprint in AndersenS 19921. 17152 “Accounts of Books: Linear Perspective ... by Brook Taylor ... London 1715”, Philosophical Transactions, vol. 29, 300–304. Published anonymously but its contents clearly show that Taylor is the author. 1719 New Principles of Linear Perspective: or the Art of Designing on a Plane the Representations of all sorts of Objects, in a more General and Simple Method than has been done before, London. Facsimile reprint in AndersenS 19921. Reissued as the third edition (because Taylor 1715 was counted as the first edition) by J. Colson 1749. Edited as Dr. Brook Taylor’s Method of Perspective by I. Ware, London 1767. A revised editon London 1811. Edited as Brook Taylor’s Principles of Linear Perspective by J. Jopling, London 1835. Translations listed below. 1755 Elementi di perspettiva secondo li principii di Brook Taylor, con varii aggiunti, ed. and tr. François Jacquier, Roma. 1757 Nouveaux principes de la perspective linéaire, traduction de deux ouvrages, l’un anglois de Docteur Brook Taylor, l’autre latin de M. Patrice Murdoch, Amsterdam.

Publications on Perspective

1782

767

Second edition 1759. The translator’s name is not mentioned, but there is a general agreement that the translation was made by Antoine Rivoire. The book also contain a section written by Jacques Silvabelle (Silvabelle 1757). Nuovi principii della prospettiva lineare, ed. and tr. Giacopo Stellini, published in StelliniS 1782.

Torelli, Giuseppe 1788 Elementorum prospectivae libri duo, ed. J.B. Bertolini, Verona. Torricelli, Evangelista Pros “Prospettiva pratica”, unfinished, undated manuscript (Gal. 134 T.XXIV della Div. IV, Mss. Galileiani, Biblioteca Nazionale di Firenze), presumably from the 1640s; part of it is printed in Torricelli Opere, vol. 2, 313–320. Opere Opere di Evangelista Torricelli, 4 vols., ed. Gino Loria & Giuseppe Vassura, Faenza, 1919–1944. Troili, Giulio 1683 Paradossi per pratticare la prospettiva senza saperla, Bologna. First edition Bologna 1672. Valenciennes, Pierre Henri 1800 Elémens de perspective pratique à l’usage des artistes, Paris. An VIII. Vaulezard, Jean Louis 1630 Perspective cilindrique et conique ou traicté des apparences veuës par le moyen des miroirs ..., Paris. 1631 Abrégé ou racourcy de la perspective ..., Paris. Later editions Paris 1633, and Paris 1643. Viator byname of Jean Pélerin 1505 De artificiali perspectiva, Toul with Latin and French text. Facsimile in IvinsS 1975. Part of De artificiali perspectiva was pirated by Gregor Reisch under the name Introductio architecturae et perspectivae (Reisch 1508). Another pirated edition by the printer Jörg Glockendon appeared as Von der Kunnst Perspectiva (Glockendon 1509). Later editions, see below. 1509 De artificiali perspectiva, Toul; revised version of the 1505 edition with parallel Latin and Italian text. Facsimile reprint Paris 1860 and in IvinsS 1975. Republished in a slightly revised version Toul 1521. Later edition: La perspective positive de Viator, ed. Mathurin Jousse,6 La Fleche 1635. Critical edition of the text in Brion-GuerryS 1962. For a comparison of the four editions, see Brion-GuerryS 1962, Table de Concordance. Vignola, Giacomo Barozzi da 1583 Le due regole della prospettiva pratica con i commentarii del R. P. M. Egnatio Danti, Roma. Facsimile reprints Vignola 1974 and Alburgh 1987. During the seventeenth and eighteenth centuries a dozen editions of this book were issued. Facsimile of the Venezia 1743 edition Sala Bolognese 1985.

6

Liliane Brion-Guerry has pointed out that Estienne Marlange helped with the preparation of this edition, but his name disappeared for unknown reasons from the title page (Brion GuerryS 1962, 160–162).

768

First Bibliography

Viola-Zanini, Giuseppe 1629 Della architettura ... con ... regole prospettiva, Padova. Later edition 1677. Vittone, Bernardo Antonio 1760 “Della prospettiva ...” in Istruzioni elementari per indirizzo de’giovani allo studio dell’Architettura civile, Lugano, 527–545. Vlaming, Jacob de 1773 Kort zaamenstel der perspectief op eene geheele nieuwe wyze afgeleid uit de gronden der driehoeksmeetinge, Amsterdam. Voch, Lukas 1780 Abhandlung von der Perspektivkunst. Worinnen nicht allein die algemeine, als auch die Sirigatische ... und ihre Gründen gelehret wird ... Zum Nützen derer Baumeister, Ingenieurs. Augsburg.* Vredeman de Vries, Johan 1560 Artis perspectivae ..., Antwerpen. Several later enlarged editions, some with the text in Dutch and at least one with the text in German. 1560 Scenographiae sive perspectivae, Antwerpen. Later enlarged editions. 1604 Perspective, 2 vols., Den Haag 1604 and 1605. For some later editions, see Vredeman de Vries 1615. Facsimile of the plates New York 1968. Facsimile of the entire work with comments in Dutch and English, a transcription of de Vries’s text into modern Dutch and a translation of it into English, ed. and tr. Peter Karstkarel, Mijdrecht 1979. 1615 Perspective ... augmentée et corrigée ... par Samuel Marolois, Den Haag. Several later editions in Marolois’s Opera mathematica (Marolois 1614) Weidemann, Friedrich Wilhelm 1746 Kurtze Einleitung zu der optischen Perspectiv nebst deren ersten Grund und Lehrsätzen, Berlin. First edition Berlin 1733. Weidler, Johann Friedrich 1736 “Ars perspectiva” in Institutiones matheseos, Wittenberg, 254–263. First edition Wittenberg 1718, several later editions. Weinbrenner, Friedrich 1817 Perspectivische Zeichnungslehre, Architektonisches Lehrbuch Teil 2, Tübingen.* Werner, Georg Heinrich 1763 Die Erlernung der Zeichenkunst durch Geometrie und Perspectiv, Erfurt. The second edition appeared under the title shown in the next item. 1796 “Von der Perspectiv” in Gründliche Anweisung zur Zeichenkunst ..., Erfurt, 61–122. Wiedeburg, Johann Bernhard 1735 “Von der Perspectiv und mancherley Verstellungen der Figuren” in Einleitung zu denen mathematischen Wissenschaften, Jena. First edition Jena 1725. Wolff, Christian 1715 “Elementa perspectivae” in Elementa matheseos universae, Halle, vol. 2, 89–115. Second edition in Elementa matheseos universae, Halle 1735, vol. 3, 103–134, facsimile of the latter Hildesheim 1968.

Publications on Perspective 1737

769

“Perspectiv” in Anfangsgründe aller mathematischen Wissenschaften, fifth edition, Halle, vol. 3, 1065–1078.

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