Families of Curves and the Origins of Partial Differentiation

FAMILIES OF CURVES AND THE ORIGINS OF PARTIAL DIFFERENTIATION

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICS STUDIES

Families of Curves and the Origins of Partial Differentiation STEVEN B. ENGELSMAN Museum Boerhaave Leiden

1984

NORTH-HOLLAND-AMSTERDAM

NEW YORK OXFORD

93

0 Elsevier

Science Publishers B.V.. 1984

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, o r transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, o r otherwisc, without the prior permission of the copyright owner.

ISBN: 0 444 86897 6

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V P.O. BOX 1991 1OOOBZ AMSTERDAM T H E NETHERLANDS Sole distributorsfor the U.S . A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A.

PRINTED IN THE NETHERLANDS

ACKNOWLEDGEMENTS

Most of t h e r e s e a r c h r e p o r t e d on the f o l l o w i n g pages h a s been c a r r i e d o u t a t t h e Mathematical I n s t i t u t e of t h e U n i v e r s i t y of Utrecht d u r i n g t h e p e r i o d

1975-1979.

I am v e r y g r a t e f u l t o t h a t I n s t i t u t i o n and t o my s u p e r v i s o r

A.F. Monna f o r t h e i r generous a s s i s t a n c e and encouragement. c e a s i n g r e a d i n e s s of H.J.M.

The n e v e r

Bos t o s h a r e knowledge and e x p e r t i s e h a s been of

paramount importance f o r my f i n d i n g a way through t o an u n d e r s t a n d i n g o f t h e mathematics of t h e p a s t . However, t h i s book would n e v e r have been w r i t t e n without b o t h t h e material and t h e moral s u p p o r t of t h e d i r e c t o r of and my c o l l e a g u e s a t t h e Museum Boerhaave i n Leiden. They provided t h e r i g h t atmosphere and t h e n e c e s s a r y s p a r e t i m e t o s e t down i n t o c o h e r e n t form what had o n l y been f a s c i n a t i n g thoughts and p i l e s of n o t e s b e f o r e . My g r a t i t u d e a l s o e x t e n d s t o numerous o t h e r s , who have

-

i n ways too d i v e r s e t o l i s t h e r e

c o n t r i b u t e d t o t h e completion of t h i s e n t e r p r i s e : C h r i s t o p h e r Burch, F l o r i s Cohen, Marc E d e l s t e i n , Aemilius Fellmann, S h e i l a McNab, Lenore de Leeuw, J e s p e r Lctzen, J e r r y Ravetz, Jade Seow and Margot S i t e u r . Permission t o p u b l i s h h i t h e r t o unpublished t e x t s by Nicolaus I B e r n o u l l i and by Leonhard E u l e r h a s been g r a n t e d by t h e u f f e n t l i c h e B i b l i o t h e k d e r U n i v e r s i t s t Base1 and by t h e Euler-Kommission

d e r Schweizer Naturforschenden G e s e l l s c h a f t .

S.B.

Engelsman,

December 1983.

-

vi

TABLE OF CONTENTS

V

ACKNOWLEDGEMENTS

vi

TABLE O F CONTENTS CHAPTER 1: INTRODUCTION

52.1 52.2

"Monde inconnu" Criteria for partial differentiation

1 1

2

5 1.2.2

Forma Z coincidence

2

52.2.2

Elements of Leibnizian calculus

4

5I. 3

'i%o dimensionaz problem s i t u a t i o n s

7

51.4

D i f f e r e n t i a l s versus d e r i v a t i v e s and v a z i d i t y o f theorems

51.5

PoZicy of t r a n s c r i p t i o n and i n t e r p r e t a t i o n

9 13

51.6

TranscendentaZ curves and trmscendentaz expressions

18

52.7

Conventions

20

CHAPTER 2 : F A M I L I E S O F CURVES I N THE 1690s

22

5 2.1

Enve Lopes

22

52.2.1

L e i b n i z ' s "new application of the caZcuZus"

22

52.1.2

The variable parameter

23

52.1.3

The enveZope aZgorithm

25

52.1.4

The s a f e t y parabola

27

52.1.5

Conclusion

29

52.2

The brachystochrone and i t s aftermath The problem

30

52.2.2

The brachystochrone and t h e syrrchrone

31

52.2.3

Genealogy of problems derived from the brachystochrone

35

52.2.4

Solutions for s i m i l a r c u m e s

37

52.2.1

30

52.2.5

The tangent problems for dissimiZar curves

41

52.2.6

L e i b n i z ' s construction

43

52.2.7

Interchangeability of d i f f e r e n t i a t i o n and integration

44

52.2.8

Leibniz ' s reaction

46

§2.2.9

Johann BernoulZi's reaction

48

52.2.10

Jakob B e r n o u l l i ' s s o l u t i o n s

51

52.3

ConcZusions

57

Table of Contents

vii

59

CHAPTER 3: ORTHOGONAL T R A J E C T O R I E S 1694- 1720

53.1

Introduction

59

53. 2

The problem posed

60

53.3

Orthogonal t r a j e c t o r i e s of the brachystochrones

62

53.4

The l i m i t s of L e i b n i z ' s method

63

53.5

Logarithmic curves

65

53.6

The break-through t o transcendental Curves

67

53. 7

Jakob Bernoulli 's reaction

69

53.8

Renascence of the problem

71

53.9

F i r s t reactions t o t h e challenge

73

53.10

The f i n a l test-case

75

53.11

Johann B e r n o u l l i ' s a l t e r n a t i v e s

79

83.11.1

The s i m i l a r i t y method

80

53.11.2

The generalised synchrone method Johann Bernoulli 's comparison of methods

86

53.12

CHAPTER 4 : NICOLAUS I BERNOULLI AND ORTHOGONAL T R A J E C T O R I E S

87 92

54.1

Biogruphy and bibliography

92

54.1.1

Biographical sketch

92

54.1.2

95

54.2

Sources Nico1au.s I B e r n o u ~ ~ i p' sa r t i a l d i f f e r e n t i a l c a h d u s

54.2. 1

Principles of reconstruction

97

54.2.2

Analytic and geometric data of f a m i l i e s of curves

54.2.3

The completion problem

100

97 97

54.2.4

Partia 1 and t o t a l d i f f e r e n t i a l s

100

54.2.5

Equality o f mixed second order d i f f e r e n t i a l s

105

54.2.6

?'he interchangeability theorem for d i f f e r e n t i a t i o n and i n t e g r a t i o n

106

family o f curves

107

54.2. 7

I n t e g r a t i o n along t r a j e c t o r i e s i n

54.2.8

General s o l u t i o n of t h e completion problem

110

54.2.9

Concluding remarks

111

§4.3

Nicolaus I B e r n o u l l i ' s r e s o l u t i o n of t h e variable parameter equation

112

§ 4 *3.1

Introduction

112

54.3.2

Rationale of Nieolaus I Bernoulli 's t r a j e c t o r y construction The test-case: t r a j e c t o r i e s of generalised cycloids

1 I3

54.3.3 54.3.4 54.3.5

G

I I5

Analysis of t h e variable parameter equation i n the Demonstratio

1 I6

Synthesis of r e s u l t s i n t h e Tentamen

120

Table of Contents

viii

54.3.6

Concluding remarks

CHAPTER 5: EULER'S THEORY OF MODULAR EQUATIONS I N THE 1730s

i22 124

Introduction Euler's expos6 o f p a r t i a l d i f f e r e n t i a l calculus i n De d i f f e r e n t i a t i o n e

124

55.2.1

Problem and method

126

55.2.2

The equality o f mixed second order d i f f e r e n t i a l s

128

55.2.3

Interchangeability of d i f f e r e n t i a t i o n and i n t e g r a t i o n

130

55. 2 . 4

Homogeneous f u n c t i o n s

131

55.2.5

Solutions t o the completion problem

132

15.1 55.2

126

55.3

Early applications of p a r t i a l d i f f e r e n t i a t i o n

133

55.3.1

Orthogonal t r a j e c t o r i e s

133

55.3.2

Equal area t r a j e c t o r i e s

138

55.4

Euler 's theory of modular equations

140

55.4.1

The s h i f t from t r a j e c t o r i e s t o d i f f e r e n t i a l equations

140

15.4.2

The concept o f a modular equation

142

55.4.3

The method of i n t e g r a l reduction

144

55.4.4

Homogeneous and generalised homogeneous functions

145

55.4.5

The c o e f f i c i e n t lemma f o r t o t a l d i f f e r e n t i a l s

148

55.4.6

Modular equations and p a r t i a 2 d i f f e r e n t i a 2 equations

149

55.5

Modular equations and ordinary d i f f e r e n t i a 2 equations

150

55.5.1

Equal arcs t r a j e c t o r i e s i n a f a m i l y o f ellipses

150

55.5.2

Ordinary d i f f e r e n t i a l equations and t h e method o f modular equations

154

55.6

E u l e r ' s view of t h e i n f i n i t e s i m a l calculus around 1740

156

EPILOGUE

161

FOOTNOTES CHAPTER 1

163

FOOTNOTES CHAPTER 2

166

FOOTNOTES CHAPTER 3

176

FOOTNOTES CHAPTER 4

185

FOOTNOTES CHAPTER 5

190

APPENDIX 1: NICOLAUS I BERNOULLI'S "DEMONSTRATIO ANALYTICA CONSTRUCTIONIS CLJRVARUM, QUAE A L I A S P O S I T I O N E DATAS AD ANGULOS RECTOS SECANT, TRADITAE I N A C T I S LIPS. 1719 PAG 295 ET SEQQ."

199

Introduction

199

Text Trans l a t i o n

200 202

Table of Contents APPENDIX 2 : LEONHARD EULER'S "DE DIFFERENTIATIONE FUNCTIONUM DUAS PLURESVE VARIABILES QUANTITATES INVOLVENTIUM"

ix 204

Introduction

204

Text Marginalia

205

Trans Zation

214

APPENDIX 3: NEWTON'S RULE FOR THE RADIUS OF CURVATURE OF MAY 2 1 S T , 1665 BIBLIOGRAPHY

213 223

227

Prearnb l e

227

L i s t of Zetters

228

Books, a r t i c l e s , manuscripts

230

This Page Intentionally Left Blank

1 CHAPTER 1

INTRODUCTION

5 I . 1 "Monde inconnu l r "Here i s an e n t i r e l y new way of c a l c u l a t i o n ; t h e r u l e s of t h e d i f f e r e n t i a l c a l c u l u s known up t i l l now a r e u s e l e s s , and i t was n e c e s s a r y t o i n v e n t f o r i t a new type of d i f f e r e n t i a l

and a l s o t o c o n s t r u c t new r u l e s . M r . L e i b n i z

and I have a l r e a d y p e n e t r a t e d q u i t e f a r i n t o t h i s unknown world [ c e monde inconnu]; M r . L e i b n i z has found t h e e n t r a n c e w h i l e I provided him w i t h t h e o p p o r t u n i t y and i n d i c a t e d t h e f i r s t tracks'" This i s how Johann B e r n o u l l i on December 24th, 1697, informed h i s c o n t i n u a l correspondent and former p r o t e c t o r Guillaume F r a n v o i s Marquis de l ' H 6 p i t a l e t c . about t h e r e c e n t d i s c o v e r y of p a r t i a l d i f f e r e n t i a l c a l c u l u s . However, 1'HEpit a l was n o t made an i n i t i a t e of t h i s new way of c a l c u l a t i o n , c o n s i s t i n g s o l e l y of t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n ; L e i b n i z and Johann B e r n o u l l i had decided t o keep t h e i r d i s c o v e r y s e c r e t f o r a w h i l e , s o t h a t they themselves could e x p l o i t t h e promising r i c h mines of t h i s new world. And indeed they s u c c e s s f u l l y avoided a g o l d r u s h . I t was n o t unt i l h a l f a c e n t u r y l a t e r t h a t Jean l e Rond d'Alembert, a countryman of t h e Mar-

q u i s , t u r n e d the e x p l o i t a t i o n of t h i s new world i n t o a money-making e n t e r p r i s e . I n 1746 t h e newly i n s t a l l e d P r u s s i a n Academy of S c i e n c e s awarded d'Alembert w i t h a medal worth 50 Ducats f o r h i s e s s a y concerning t h e cause of winds'.

Together

w i t h a n a r t i c l e concerning t h e i n f i n i t e l y s m a l l v i b r a t i o n s of a s t r i n g , p u b l i s h e d s l i g h t l y l a t e r , t h i s e s s a y c o n t a i n e d t h e f i r s t a p p l i c a t i o n t o p h y s i c a l phenomena of t h e r e s u l t s i n p a r t i a l d i f f e r e n t i a l c a l c u l u s t h a t had emerged s i n c e t h e end of t h e 17th c e n t u r y . d'Alembert's p r i z e winning work on t h e cause of t h e winds was t h e beginning of an i n c r e a s i n g l y r a p i d development of t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s . E u l e r q u i c k l y caught up w i t h d'Alembert, and he developed i n t o a most p r o l i f i c w r i t e r i n t h i s new a r e a ; E u l e r ' s f i r s t textbook p r e s e n t a t i o n of t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s appeared w i t h i n twenty-five y e a r s . The developments r e s u l t i n g from d ' A l e m b e r t ' s work of 1746 have been s t u d i e d q u i t e thoroughly by h i s t o r i a n s of mathematics3. However, t h e p e r i o d from t h e

2

Introduction

1690s through 1746 has r e c e i v e d b u t s c a n t a t t e n t i o n . How, i n p a r t i c u l a r , d i d

t h e f i r s t c o l o n i s a t i o n of Johann B e r n o u l l i ' s and L e i b n i z ' s unknown world t a k e p l a c e ? What e x a c t l y were t h e s o u r c e s from which d'Alembert drew h i s knowl e d g e , and what d i d t h i s knowledge c o n s i s t o f ? What was t h e n a t u r e of t h e problems t h a t mathematicians had f a c e d , and t h a t had f o r c e d them t o develop p a r t i a l differential calculus? I t i s t o t h e s e q u e s t i o n s t h a t t h i s book i s devoted.

51.2 Criteria for pur-tial differentiation

51.2.1 Formal ooincidenee Obviously, t h e h i s t o r y of p a r t i a l d i f f e r e n t i a l c a l c u l u s depends on what parts

of mathematics one chooses t o c h a r a c t e r i s e a s " p a r t i a l " . Although t h e

s u b j e c t has r e c e i v e d b u t c a s u a l i n t e r e s t , o p i n i o n s about t h e most adequate d e f i n i t i o n of t h i s concept d i f f e r c o n s i d e r a b l y , and h i s t o r i o g r a p h i c a l o p i n i o n s a r e f r e q u e n t l y a t v a r i a n c e w i t h each o t h e r . Hence we must f i r s t s e t t l e t h i s m a t t e r of d e f i n i t i o n b e f o r e we can proceed t o develop our own v e r s i o n of t h e h i s t o r y of p a r t i a l d i f f e r e n t i a l c a l c u l u s . I s h a l l r e f r a i n from g i v i n g a comp r e h e n s i v e account of a l l p o i n t s of view t h a t have been a d o p t e d 4 . Recent claims a l l e g e t h a t t h e o r i g i n s of p a r t i a l d i f f e r e n t i a t i o n must be sought i n Newton's r e s e a r c h concerning normals, c u r v a t u r e , and t h e r e s o l u t i o n of t h e g e n e r a l problem of t a n g e n t s i n t h e y e a r 16655. These c l a i m s a l l d e r i v e from D.T.

W h i t e s i d e ' s t r u l y i m p r e s s i v e e d i t i o n of and comments upon Newton's

mathematical p a p e r s ; t h e i r p u r p o r t may be s u m a r i s e d a s f o l l o w s : I n 1665 Newton h i t upon a g e n e r a l r u l e f o r t h e r a d i u s of c u r v a t u r e f o r any a l g e b r a i c c u r v e ; i n f o r m u l a t i n g t h i s r u l e , h e i n t r o d u c e d a s e r i e s of e x p r e s s i o n s X , X , 3 2 , X

32 which have t o be e v a l u a t e d from t h e

polynomial

,

e q u a t i o n X = O of t h e given

curve. I n t h e t r a n s c r i p t i o n of t h e s e s i d e - d o t t e d X ' s , f i r s t and second o r d e r p a r t i a l d e r i v a t i v e s of t h e e x p r e s s i o n X = f f x , y )

occur. Hence Newton a l r e a d y i n

1665 "had a concept, i n c l u d i n g a n o t a t i o n , c o r r e s p o n d i n g t o p a r t i a l d e r i v a t i v e s " , a s some have i t 6 . Indeed, Newton's s i d e - d o t t e d e x p r e s s i o n s a r e t o some e x t e n t r e l a t e d t o p a r t i a l d e r i v a t i v e s , a s i s shown by t h e i r t r a n s c r i p t i o n ( c f . Appendix 3 , cont a i n i n g t h e s e t r a n s c r i p t i o n s and a d i s c u s s i o n o f t h e meaning of t h e s i d e - d o t t e d Z's);

but t o what e x t e n t ? What e x a c t l y d i d Newton seek t o c o d i f y by h i s s i d e -

d o t t e d e x p r e s s i o n s ? They r a t h e r s e r v e d

-

I b e l i e v e - t o make p o s s i b l e t h e e l e g a n t

3

Criteria f o r partial differentiation

f o r m u l a t i o n of a r u l e of c a l c u l a t i o n , than t o denote Newtonian concepts t h a t a r e e q u i v a l e n t t o p a r t i a l d e r i v a t i v e s . They c o d i f y c e r t a i n a l g o r i t h m i c a l subr o u t i n e s , which o c c u r i n t h e c a l c u l a t i o n of t h e r a d i u s of c u r v a t u r e of an a l g e b r a i c curve, given i t s polynomial e q u a t i o n . I n t h i s way, Newton's s i d e - d o t t e d e x p r e s s i o n s a r e no more c l o s e t o p a r t i a l d e r i v a t i v e s t h a n t h e e x p r e s s i o n s which Johannes Hudde d e f i n e d i n 1659, and which o c c u r i n h i s g e n e r a l r u l e of t a n g e n t s . Hudde's r u l e i s o f t e n t r a n s c r i b e d i n t h e f o l l o w i n g way: I f f(x,yJ=O

denotes the

polynomial e q u a t i o n o f a c u r v e , then t h e s u b t a n g e n t of t h i s curve i s provided by t h e e x p r e s s i o n yf (~,yY)/f~(x,y)~.

Y

The r e l a t i o n between p a r t i a l d e r i v a t i v e s and Hudde's o r Newton's express i o n s o n l y i s a formal one. I n t h e modern t r a n s c r i p t i o n of t h e s e e x p r e s s i o n s we use p a r t i a l d e r i v a t i v e s of f ( x , y l ,

thus i n t r o d u c i n g a l l t h o s e m u l t i d i m e n s i o n a l

c o n n o t a t i o n s which p a r t i a l d e r i v a t i v e s i n e v i t a b l y c a r r y a l o n g : t h e f u n c t i o n

z=f(x,y)

and i t s d e r i v a t i v e s

a az ax and ay as e n t i t i e s w i t h a w e l l d e f i n e d meaning

i n themselves; t h e i d e a of a curve f(x,y)=O

as embedded i n a s u r f a c e z=f(x,y).

All t h e s e c o n n o t a t i o n s f a i l t o o b t a i n i n t h e o r i g i n a l Newtonian o r Huddenian p i e c e s of mathematics. I n f a c t , only a formal coincidence e x i s t s between t h e polynomial e x p r e s s i o n s which Newton and Hudde d e f i n e , and t h e e x p r e s s i o n s which we nowadays

d e r i v e by a p p l y i n g p a r t i a l d i f f e r e n t i a t i o n . T h i s p o i n t may be

i l l u m i n a t e d somewhat f u r t h e r i n t h e f o l l o w i n g way: I n 17th and e a r l y 18th c e n t u r y mathematics t h e r e was no concept of f u n c t i o n . The u s u a l way t o r e p r e s e n t a curve, t h e r e f o r e , w a s n o t t o w r i t e y=f(xl, b u t r a t h e r t o g i v e the two v a r i a b l e s x and y: V(x,y)=O.

Such an e q u a t i o n d i d n o t r i n g a m u l t i -

dimensional b e l l ; t h e e x p r e s s i o n V(x,yl e q u a t i o n , t o g e t h e r w i t h "="

an e q u a t i o n i n

w a s o n l y one of t h e c o n s t i t u e n t s of t h e

and "0"; i t was n o t t h e r e p r e s e n t a t i v e of a f u n c t i o n

of two independent v a r i a b l e s , h a v i n g a meaning a l s o i n i t s e l f . I n t h i s way, d i f f e r e n t i a l o r f l u x i o n a l c a l c u l u s was n o t concerned w i t h f u n c t i o n s and t h e i r d e r i v a t i v e s , b u t r a t h e r w i t h v a r i a b l e s , r e l a t e d by e q u a t i o n s , and d i f f e r e n t i a l s . The q u e s t i o n t h e n emerges whether t h e r e were o t h e r , c o n c e p t u a l r a t h e r than formal, occurrences of p a r t i a l d i f f e r e n t i a t i o n i n t h e 17th and e a r l y 1 8 t h cent u r y . Which type

of problem

could have produced such o c c u r r e n c e s ? I n o r d e r

t o answer t h i s q u e s t i o n , I s h a l l f i r s t give a b r i e f s k e t c h of t h e elements of Leibnizian calculus.

Introduction

4

91.2.2

Elements of L eibniz ian calculus

L e i b n i z ' s f i r s t p u b l i c p r e s e n t a t i o n of h i s d i f f e r e n t i a l c a l c u l u s i n 1684 was s e v e r e l y determined by h i s a t t e m p t t o avoid t h e l o g i c a l d i f f i c u l t i e s conn e c t e d w i t h t h e i n f i n i t e l y small. In h i s a r t i c l e 1684

L e i b n i z took t h e d i f -

f e r e n t i a l to be a f i n i t e l i n e segment r a t h e r t h a n t h e i n f i n i t e l y s m a l l q u a n t i t y t h a t w a s used i n p r a c t i c e . L e i b n i z ' s own p r e s e n t a t i o n of t h e d i f f e r e n t i a l c a l c u l u s , t h e r e f o r e , does n o t g i v e much i n s i g h t i n t o t h e i d e a s and concepts underlying t h i s calculus: A s u i t a b l e p r e s e n t a t i o n of the Leibnizian d i f f e r e n t i a l c a l c u l u s a s i t was employed i n p r a c t i c e , and i n which i n f i n i t e l y s m a l l q u a n t i t i e s a r e accepted a s genuine mathematical o b j e c t s ,

i s provided by t h e a r t i c l e

1974a of Bos. For t h e f o l l o w i n g account I s h a l l draw h e a v i l y from t h e e x p o s i t i o n

of t h e L e i b n i z i a n c a l c u l u s a s given i n t h i s a r t i c l e . L e i b n i z i a n c a l c u l u s was an a n a l y t i c a l machinery f o r t h e s t u d y of c u r v e s , a s becomes c l e a r a l r e a d y from t h e t i t l e of L e i b n i z ' s a r t i c l e of 1684: "A new method f o r maxima and minima, as w e l l a s t a n g e n t s , which i s n e i t h e r impeded by f r a c t i o n a l nor i r r a t i o n a l q u a n t i t i e s , and a remarkable type of c a l c u l u s f o r them"; t h e i n t i m a t e l i n k w i t h c u r v e s becomes even c l e a r e r i n t h e t i t l e of t h e f i r s t textbook on t h e d i f f e r e n t i a l c a l c u l u s , v i z . l ' H 8 p i t a l ' s "Analysis of t h e i n f i n i t e l y s m a l l , f o r t h e understanding of curved l i n e s " , published i n 1696. The v e r y o b j e c t s of t h i s c a l c u l u s were t h e c u r v e , and t h e v a r i a b l e s d e f i n e d on a curve. Such v a r i a b l e s a r e f o r i n s t a n c e : t h e a b s c i s s a x, t h e o r d i n a t e y, t h e subtangent t, t h e t a n g e n t (see f i g .

T,

t h e a r c l e n g t h s , t h e a r e a &, t h e normal n e t c .

1).

None of t h e s e v a r i a b l e s m a i n t a i n s a p r e f e r e n t i a l p o s i t i o n as an independent v a r i a b l e ; i n p r i n c i p l e , a l l v a r i a b l e s have e q u a l r i g h t s , and t h e i r r e l a t i o n s a r e embodied i n t h e curve. D i f f e r e n t i a l s of t h e s e v a r i a b l e s e n t e r through t h e conception of a " p r o g r e s s i o n of t h e v a r i a b l e " ,

t h a t i s , an ordered sequence of

5

Criteria f o r partial differentiation

v a l u e s , i n f i n i t e l y c l o s e t o each o t h e r , over which t h e v a r i a b l e ranges; f o r i n s t a n c e , i f t h e p r o g r e s s i o n of t h e v a r i a b l e y i s denoted by y , y ' , y", y ' " ,

.. . ,

( s e e f i g . 2 ) t h e f i r s t o r d e r d i f f e r e n t i a l s of y a r e d e f i n e d a s t h e

s u c c e s s i v e d i f f e r e n c e s i n t h i s p r o g r e s s i o n : dy=y '-y,

...,

idyl '=y"-y

',

(dy) "-7' "-y",

and, l i k e w i s e , second o r d e r d i f f e r e n t i a l s of y a r e d e f i n e d a s s u c c e s s i v e

d i f f e r e n c e s i n the p r o g r e s s i o n of f i r s t o r d e r d i f f e r e n t i a l s : ddy=fdyl '-dy,

(ddyl '=(dy)"-fdyl

. ..

I ,

Now a given p r o g r e s s i o n of t h e v a r i a b l e y induces an

ordered sequence of p o i n t s P, P', P", P"', P r r r, r

.. .

on t h e c u r v e , and t h e c u r v e ,

a c c o r d i n g l y , i s regarded a s i d e n t i c a l with the i n f i n i t a n g u l a r polygon

PP'PrrPrffP"''

. . . . By

means of t h i s sequence of p o i n t s on t h e curve, t h e pro-

g r e s s i o n of t h e v a r i a b l e y induces p r o g r e s s i o n s f o r a l l o t h e r v a r i a b l e s (see fig. 2).

fig. 2

YY Y" Y"

Y" Y' Y

I n t h i s way, t h e p r o g r e s s i o n of t h e v a r i a b l e y does n o t o n l y f i x t h e d i f f e r e n t i a l s of y , but a l s o , by t r a n s m i s s i o n through t h e c u r v e , t h e d i f f e r e n t i a l s of a l l o t h e r v a r i a b l e s . Mutatis mutandis one can s t a r t w i t h any o t h e r v a r i a b l e f o r determining the i n f i n i t a n g u l a r

polygon

PPrPr'P' frP'rfr

.. .

and t h e d i f -

f e r e n t i a l s of t h e remaining v a r i a b l e s . I t w i l l be c l e a r from t h i s account t h a t t h e d e f i n i t i o n of t h e d i f f e r e n t i a l s

i n v o l v e s an a r b i t r a r i n e s s , i n t h a t t h e curve can be approximated by d i f f e r e n t s o r t s of i n f i n i t a n g u l a r of

polygons.

For i n s t a n c e , a t t h e o u t s e t a p r o g r e s s i o n

x can be taken such t h a t &=eonstant o r , e q u i v a l e n t l y , ddx=O; b u t , e q u a l l y

w e l l , one might s t a r t with a p r o g r e s s i o n of s such t h a t d s = e o n s t a n t o r dds=O. Hence, t h e r e remains a degree of freedom i n t h a t an e x t r a c o n d i t i o n may be i m posed i n o r d e r t o e l i m i n a t e the indeterminacy of t h e p r o g r e s s i o n s of v a r i a b l e s ; such c o n d i t i o n s need n o t n e c e s s a r i l y be of t h e form t h a t one of t h e f i r s t o r d e r d i f f e r e n t i a l s i s c o n s t a n t . This a r b i t r a r i n e s s i n t h e choice of t h e i n f i n i t e angular polygon i s r e f l e c t e d i n t h e d i f f e r e n t i a l e q u a t i o n s o r d i f f e r e n t i a l exp r e s s i o n s t h a t can be d e r i v e d . Apart from f i r s t o r d e r d i f f e r e n t i a l e x p r e s s i o n s

6

Introduction

o r e q u a t i o n s , which a r e unique and v a l i d i r r e s p e c t i v e of t h e s p e c i a l c h o i c e of p r o g r e s s i o n s of v a r i a b l e s , i n p r i n c i p l e a l l h i g h e r o r d e r e q u a t i o n s o r express i o n s a r e v a l i d only f o r one s p e c i a l p r o g r e s s i o n of t h e v a r i a b l e s . For i n s t a n c e , t h e r a d i u s of c u r v a t u r e R i s given by: ds f o r ddx = 0, by (a) R = -

dxddy

(b)

R =-

ds 3

(c)

Y dxds R =ddg

(d)

R = &ddy-dyd&

f o r ddy = 0, by f o r d d s = 0, and by

-*ds

f o r a l l p r o g r e s s i o n s of v a r i a b l e s .

An e x p r e s s i o n of type ( d ) , v a l i d f o r a l l p r o g r e s s i o n s was c a l l e d a "complete" d i f f e r e n t i a l e x p r e s s i o n . A s Bos h a s shown, t h i s indeterminacy of e x p r e s s i o n s i n v o l v i n g h i g h e r o r d e r d i f f e r e n t i a l s h a s e v e n t u a l l y been one of t h e major reasons f o r h i g h e r o r d e r d i f f e r e n t i a l s t o be banned from t h e c a l c u l u s and f o r t h e d e r i v a t i v e t o emerge as a b a s i c concept of t h e c a l c u l u s ' . From t h i s s k e t c h i t w i l l b e c l e a r t h a t t h e one d i m e n s i o n a l i t y of t h e curve

was of eminent importance f o r t h e concept of d i f f e r e n t i a l s . E s s e n t i a l l y , i t i s t h i s one d i m e n s i o n a l i t y t h a t g u a r a n t e e s a unique t r a n s m i s s i o n of t h e given prog r e s s i o n of one v a r i a b l e o n t o t h e o t h e r v a r i a b l e s . The p r o g r e s s i o n s of t h e v a r i a b l e s , l i n k e d through t h e curve, a l s o provided t h e means t o d e r i v e t h e r u l e s of c a l c u l a t i o n f o r t h e d i f f e r e n t i a l o p e r a t o r . This p o i n t can be i l l u s t r a t e d by t h e d e r i v a t i o n of t h e product r u l e ' : Consider a curve A P P ' , t h e p o i n t s P and P ' r e p r e s e n t i n g two s u c c e s s i v e p o i n t s polygon. Now the v a r i a b l e q r e p r e s e n t s t h e a r e a of t h e

of t h e i n f i n i t a n g u l a r

quadrangle ABPC (see f i g . 3)

Y

fig. 3

C'

C

A The quadrangle AB'P'C'

X

t h a t corresponds w i t h t h e p o i n t P ' h a s a r e a

i ~ ~ c y l ' = f c c + d x ~ f y + dhence y); t h e d i f f e r e n t i a l of q can b e found i n t h e f o l l o w i n g way : (1. I )

d f q ) = ( x + d x lfy+dy)-q=ydx+xdy+dxdy.

Two dimensional problem situations Since

I

dxdy i s i n f i n i t e l y s m a l l w i t h r e s p e c t t o xdy and ydx, t h i s summand may

be n e g l e c t e d t o y i e l d :

d(xy)-ydz+xdy.

(1.2)

C a j o r i , i n h i s 2928, c i t e s t h e product r u l e a s an example of t h e omnipresence of p a r t i a l d i f f e r e n t i a t i o n techniques i n L e i b n i z i a n c a l c u l u s , s i n c e "one v a r i a b l e i s f o r t h e moment assumed t o be c o n s t a n t , then t h e o t h e r " . However, t h e d e r i v a t i o n of the product r u l e given above c l e a r l y shows t h a t v a r i a b l e s a r e only considered t o vary s i m u l t a n e o u s l y , s i n c e only t h o s e combinations of v a r i a b l e s a r e c o n s i d e r e d t h a t correspond t o p o i n t s on t h e given curve. The same remark a p p l i e s t o t h e L e i b n i z i a n p r a c t i c e of d i f f e r e n t i a t i n g e q u a t i o n s , t o which I now t u r n . Let a curve be given by t h e e q u a t i o n

and l e t i t s t a n g e n t be r e q u i r e d , then the L e i b n i z i a n procedure was t o d i f f e r e n t i a t e t h e e q u a t i o n (1.3) a c c o r d i n g t o t h e r u l e s of d i f f e r e n t i a t i o n . Obviously, t h e f i n a l r e s u l t of t h i s procedure can be t r a n s c r i b e d as:

where f

X

(x,c,yiand f fx,yi r e p r e s e n t t h e e x p r e s s i o n s emerging from p a r t i a l d i f Y f(x,y) w i t h r e s p e c t t o x o r y. The index no-

f e r e n t i a t i o n of t h e e x p r e s s i o n

t a t i o n i s very convenient, i n t h a t i t c l e a r l y shows where t h e e x p r e s s i o n s o r i g i n a t e from; t h e r e f o r e , I s h a l l o f t e n t r a n s c r i b e d i f f e r e n t i a l e q u a t i o n s i n t h i s way. However, they a g a i n c o i n c i d e w i t h p a r t i a l d e r i v a t i v e s only i n a formal way, s i n c e t h e s e c o e f f i c i e n t s of

dx

and

dy i n t h e d i f f e r e n t i a l e q u a t i o n

were n o t d e f i n e d a s p a r t i a l d e r i v a t i v e s ; they r a t h e r emerge from d i f f e r e n t i a t i n g an e q u a t i o n term by t e r m , where t h e v a r i a b l e s a r e c o n s i d e r e d t o vary s i m u l t a neously, and from subsequent c o l l e c t i n g of a l l terms t h a t i n v o l v e e i t h e r dx o r

dY.

51.3 !l'wo dimensional problem situations We can now r e t u r n t o t h e q u e s t i o n posed a t t h e end of 11.2.1: Where does one have t o look f o r occurrences of p a r t i a l d i f f e r e n t i a t i o n t h a t are c o n c e p t u a l r a t h e r t h a n formal? I t w i l l have become clear that t h e curve - t h e model of a one dimensional problem s i t u a t i o n

-

a c t s a s t h e L e i b n i z i a n e q u i v a l e n t of our

f u n c t i o n of a s i n g l e independent v a r i a b l e . Now genuine p a r t i a l d i f f e r e n t i a t i o n

8

Introduction

would a t l e a s t r e q u i r e some s o r t of a two dimensional problem s i t u a t i o n , a L e i b n i z i a n e q u i v a l e n t t o o u r concept of a f u n c t i o n of two independent v a r i a b l e s . Already i n t h e e a r l y 1690s such an e q u i v a l e n t was s u p p l i e d by f m i l i e s o f curves.

Families of curves were t o become the paradigm problem s i t u a t i o n f o r p a r t i a l d i f f e r e n t i a l calculus, and problems about t r a j e c t o r i e s i n families of curves were t o become the paradigm problems which required the development of p a r t i a l d i f f e r e n t i a l techniques. P a r t i a l d i f f e r e n t i a l c a l c u l u s d i d n o t h i s t o r i c a l l y develop i n what we may r e g a r d as t h e more n a t u r a l type o f problem, v i z . d i f f e r e n t i a l g e o m e t r i c a l problems concerning s u r f a c e s . This had a d e c i s i v e inf l u e n c e on t h e s t y l e of e a r l y p a r t i a l d i f f e r e n t i a l c a l c u l u s : i t was c a l l e d " d i f f e r e n t i a t i o n from curve t o curve"",

and i t d i d n o t d e a l w i t h t h r e e space

v a r i a b l e s of t h e same c h a r a c t e r , but w i t h two space v a r i a b l e s J: and y and w i t h t h e parameter a , o r "modulus" of a f a m i l y of c u r v e s . Extension of t h e d i f f e r e n t i a l c a l c u l u s t o f a m i l i e s of curves made i t n e c e s s a r y t h a t t h e modulus could a l s o be viewed as a d i f f e r e n t i a b l e q u a n t i t y , a s a q u a n t i t y r a n g i n g over an o r d e r e d s e t of v a l u e s . Such a p r o g r e s s i o n of t h e modulus a induces an o r d e r e d sequence of curves: k a , k,,,

k a r , e t c . , and t h i s s i t u a t i o n was thoroughly ana-

l y s e d by Leibniz i n h i s a r t i c l e s 2692 and 2694. I n t h e s e a r t i c l e s , L e i b n i z provided t h e t e c h n i c a l term f o r such a family of c u r v e s , t h e term t h a t remained i n use through t h e h a l f of t h e 18th c e n t u r y : " i n f i n i t a e curvae o r d i n a t i m p o s i t i o n e datae" ( " i n f i n i t e l y many c u r v e s , given by p o s i t i o n i n o r d e r e d sequence"). Of course t h e r e a r e some e x c e p t i o n s t o t h e supremacy of f a m i l i e s of c u r v e s . The problem t o f i n d geodesics on c e r t a i n s u r f a c e s was posed as e a r l y a s 1697", and i n some of the s o l u t i o n s t o t h i s problem p a r t i a l d i f f e r e n t i a l techniques occur".

However, t h e s e a r e r a t h e r t r i v i a l a p p l i c a t i o n s of techniques a l r e a d y

a v a i l a b l e by then, and they d i d n o t e x e r t any i n f l u e n c e upon t h e development o f p a r t i a l d i f f e r e n t i a l c a l c u l u s . Furthermore, i n t h e l a t e 1730s a p a r a l l e l

development of p a r t i a l d i f f e r e n t i a l c a l c u l u s emerged e s p e c i a l l y among French mathematicians i n connection w i t h o r d i n a r y and t o t a l d i f f e r e n t i a l e q u a t i o n s and i n t e g r a t i n g f a c t o r s 1 3 . To a v e r y l a r g e e x t e n t t h i s development was independent of t h e t r a d i t i o n of d i f f e r e n t i a t i o n from curve t o curve. Also i n Newtonian f l u x i o n a l c a l c u l u s some occurrences of p a r t i a l d i f f e r e n t i a t i o n have been i d e n t i f i e d I 4 . Rather than p r o v i d i n g a l a r g e l y unconnected enumeration of a l l o c c u r r e n c e s of p a r t i a l d i f f e r e n t i a l c a l c u l u s , however

unimportant or t r i v i a l , I s h a l l

r e s t r i c t my i n v e s t i g a t i o n and my n a r r a t i v e t o t h e t r a d i t i o n of f a m i l i e s of curves i n L e i b n i z i a n d i f f e r e n t i a l c a l c u l u s . I n o t h e r words, t h i s book p r o v i d e s

Differentials versus derivatives

9

t h e biography of t h e concept o f t h e v a r i a b l e modulus. Here a r e t h e main e v e n t s i n the c a r e e r of t h e modulus: (a) 1692: Leibniz e l e v a t e s t h e modulus t o t h e rank of d i f f e r e n t i a b l e quantities. (b)

1697: Leibniz d i s c o v e r s t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n

and i n t e g r a t i o n : d a s s : . ( x , a ) ~ = ~ x d a p ( x J a ) I . I n 1698 t h i s p r o p e r t y i s a l s o XO

used by Jakob B e r n o u l l i .

( c ) +1718: Nicolaus I B e r n o u l l i d i s c o v e r s t h e e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s : dxday=dadg, which he u s e s f o r f i n d i n g i n t e g r a t i n g f a c t o r s of d i f f e r en t i a 1 e q u a t i o n s

.

(d) f1730: E u l e r d i s c o v e r s t h e theorem on homogeneous f u n c t i o n s : y - y ( x , a ) homogeneous of degree n, then ~ ~ y = z+uyu. g X

( e ) +1734: E u l e r uses t h e c o e f f i c i e n t lemma f o r t o t a l d i f f e r e n t i a l s : i f dy=sdt

i s a t o t a l d i f f e r e n t i a l , then y = $ ( t j and s=$'(t), where @ i s an a r b i t r a r y f u n c t i o n (t i s some given f u n c t i o n i n x and a ) . (f)

1747: d'Alembert i n t e g r a t e s t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n of t h e

v i b r a t i n g s t r i n g , i n which t h e modulus occurs i n d i s g u i s e a s a time v a r i a b l e . The e v e n t s ( a ) and (b) a r e d i s c u s s e d i n c h a p t e r 2 ; ( c ) i s d i s c u s s e d i n c h a p t e r 4, and f i n a l l y , (d) and ( e ) a r e d i s c u s s e d i n c h a p t e r 5. Chapter 3 does n o t c o n t a i n any of t h e h i g h l i g h t s from t h i s l i s t , b u t r a t h e r p r o v i d e s t h e c h r o n i c l e of some h i g h l y s e t e x p e c t a t i o n s about d i f f e r e n t i a t i o n from curve t o curve which f a i l e d t o m a t e r i a l i s e ; o r , e l a b o r a t i n g t h e metaphor:

i t i s de-

voted t o t h e w h i m s i c a l i t y of t h e v a r i a b l e modulus's puberty and e a r l y adol e s c e n c e , i n which h i g h e x p e c t a t i o n s and grave d e c e p t i o n s go hand i n hand.

5 1.4 D i f f e r e n t i a l s v e r s u s d e r i v a t i v e s and v a l i d i t y o f theorems Today, t h e concept of t h e t o t a l d i f f e r e n t i a l

(1.5)

dflx,y)=fx(z,yYI+f

Y

(x,y)dy

of a f u n c t i o n z = f ( x , y ) , t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and integration

and t h e e q u a l i t y of mixed p a r t i a l d e r i v a t i v e s (1.7)

f

XY

f x J y ) = f (x,y) YX

Introduction

10

c o n s t i t u t e t h e elementary c o n c e p t , and t h e elementary theorems of p a r t i a l d i f f e r e n t i a l c a l c u l u s . However, t h e i r v a l i d i t y no l o n g e r e x t e n d s t o a l l f u n c t i o n s i n v o l v i n g two v a r i a b l e s , and they a r e formulated a s l o c a l theorems r a t h e r than g l o b a l theorems ( i . e .

f o r t h e neighbourhoodof a c e r t a i n p o i n t

r a t h e r than f o r a l l v a l u e s of t h e arguments). The c o n d i t i o n s a f u n c t i o n

z=f(x,y) must s a t i s f y i n o r d e r f o r t h e s e theorems t o h o l d o r t h e concept of t o t a l d i f f e r e n t i a l t o e x i s t a r e d i f f e r e n t from c a s e t o c a s e . Normally, such c o n d i t i o n s - as given i n a c a l c u l u s course o r i n a textbook

-

are sufficient

c o n d i t i o n s only. Rarely a r e they a l s o n e c e s s a r y . A common s e t of c o n d i t i o n s f o r is:

example

- f o r t h e t o t a l d i f f e r e n t i a l t o e x i s t a t a p o i n t (xo,yo): b o t h f and f X

continuous i n aneighbourhood of fxo,y )

-

0

f o r t h e i n t e r c h a n g e a b i l i t y theorem t o h o l d f o r and y ranging between c and d : f and f

[a,b 1

X

Y

Ic,dl

are

x ranging between a and b ,

are continuous i n t h e b l o c k

- f o r t h e e q u a l i t y theorem t o hold i n a p o i n t (x ,y ) : b o t h f continuous i n a neighbourhood of

Y

(xo,yo).

0

0

XY

and f

YX

are

I n t h e p e r i o d we c o n s i d e r i n t h i s book, and f o r an even l o n g e r s p a n of time up t o t h e middle of t h e 19th c e n t u r y , t h e s e theorems were supposed t o be v a l i d u n i v e r s a l l y , and t h e concept of t o t a l d i f f e r e n t i a l t o e x i s t f o r a l l functions

z=f(x,y). This s i t u a t i o n begs t h e q u e s t i o n s i n which r e s p e c t t h e

p r o o f s t h a t were given were regarded s a t i s f a c t o r y , and whether any paradoxes occurred due t o the f a c t t h a t t h e s e theorems s u f f e r from e x c e p t i o n s . What s o r t of assumptions were made i n o r d e r t o prove t h e s e theorems a s u n i v e r s a l l y v a l i d ones? I s h a l l d i s c u s s t h e s e q u e s t i o n s h e r e , b u t l i m i t myself t o t h e e q u a l i t y theorem f o r mixed

derivative^!^

T h i s d i s c u s s i o n may s e r v e t o a p p r e c i a t e t h e 18th

c e n t u r y p o s i t i o n towards p r o o f s by comparison w i t h 19th c e n t u r y r i g o r o u s proofana 1ys i s . L e t me s t a r t w i t h E u l e r ' s p r o o f , d a t i n g from around 1730 ( t o be d i s c u s s e d

Z=P(x,y) ( i . e . an e x p r e s s i o n i n t h e v a r i a b l e s x and y ) . Now dxPfx,y)=P(x+dx, y)-P(x, y) -where d d e n o t e s t h e X

a g a i n i n ch. 5 ) . E u l e r c o n s i d e r e d a f u n c t i o n

d i f f e r e n t i a l when y i s taken c o n s t a n t ( c f . 9 1 . 5 ) . Hence

d d P (2,y i = (P fx+&, y+dy )-P {x,y+dy ) I - (P fx+&, y i -P (x,y i I. Analogous 1y ,

Y X XY

d d P (x,y ) = (P(x+dx,y +dy) -P (x+dx,y i I - (P(x,y +dy I -P (2,y ) )

. R e shuf f 1i n g

t h e r ight

d P(x,y) and d d P(x,yl X Y Y X can be r e a d i l y p e r c e i v e d . This i s a l l . E u l e r ' s proof i s i n f a c t n o t h i n g e l s e

hand s i d e of t h i s l a s t d i f f e r e n t i a l , t h e e q u a l i t y of d

than t h e e q u a l i t y f o r p a r t i a l d i f f e r e n c e s , w r i t t e n w i t h d i f f e r e n t i a l n o t a t i o n . I t i s a mere e x t r a p o l a t i o n from f i n i t e t o i n f i n i t e l y s m a l l d i f f e r e n c e s . Now

11

Differentials versus derivatives

t h i s type of e x t r a p o l a t i o n was a v e r y common procedure i n d i f f e r e n t i a l c a l c u l u s ; hence E u l e r ' s e q u a l i t y theorem f o r mixed p a r t i a l d i f f e r e n t i a l s was i n no way l e s s v a l i d than t h e concept of d i f f e r e n t i a l i t s e l f : i t w a s r e g a r d e d a s u n i v e r s a l l y v a l i d . However, i m p l i c i t assumptions emerge when t h e e q u a l i t y of mixed d i f f e r e n t i a l s i s t r a n s l a t e d i n t o t h e e q u a l i t y of d i f f e r e n t i a l c o e f f i c i e n t s E u l e r ' s e q u i v a l e n t t o p a r t i a l d e r i v a t i v e s . I t was taken f o r g r a n t e d

t h a t any

d i f f e r e n t i a l dP(x, y ) could always be expressed i n t h e form Q ( x , y ) d x + R ( x , y l d y ; Q and R h e r e denote t h e e x p r e s s i o n s t h a t a r e d e r i v e d from P ( x , y ) by d i f f e r e n t i a -

t i o n w i t h r e s p e c t t o z o r y . No l i m i t s were involved i n the d e f i n i t i o n of t h e s e d i f f e r e n t i a l c o e f f i c i e n t s ; t h e i r e x i s t e n c e was regarded an undoubtable t r u t h . On t h e b a s i s of such an assumption, t h e e q u a l i t y of mixed d i f f e r e n t i a l s immediately i m p l i e s e q u a l i t y of mixed d i f f e r e n t i a l c o e f f i c i e n t s o r mixed p a r t i a l d e r i v a t i v e s : d d P(x,y)=P X Y

Y"

(x,yidxdy and dydzP(x,yY)=Pxy(x,yyldxdy.

Hence d i v i s i o n by dxdy immediately y i e l d s :

or

Here a g a i n , t h e e q u a l i t y theorem a p p e a r s a s a u n i v e r s a l l y v a l i d one. Not u n t i l 1867 was t h e t r u t h of t h i s n a i v e form of t h e e q u a l i t y theorem doubted e f f e c t i v e l y . I n t h a t y e a r , t h e F i n n i s h mathematician L i n d e l s f p u b l i s h e d a review (2867) of some of t h e p r o o f s then c u r r e n t

-

n o t a b l y a proof by

Schlsmilch of 1862,16 and a proof by B e r t r a n d of 1864.

Lindel6f pinned down

l o g i c a l flaws i n both of them. B e r t r a n d , i n h i s Traite' d e Calcul Efferentie'Z.. of 1864, had p r e s e n t e d t h e e q u a l i t y theorem s t i l l i n i t s n a i v e form, v a l i d f o r a l l f u n c t i o n s of two v a r i a b l e s ; h i s "proof" was based on t h e f o l l o w i n g "lemma": I f a f u n c t i o n F(x,al i s i n f i n i t e l y

s m a l l f o r i n f i n i t e l y s m a l l v a l u e s of a ,

then [ f o r t h e s e i n f i n i t e l y small v a l u e s of a ] F f x , a ) i s e q u a l t o Fx(2JaY)'7. L i n d e l s f r e f u t e d t h i s lemma by means of t h e f o l l o w i n g counterexample:

( I . 10)

F(x,aa)=a.sin(x/a).

Since Fx(x,aa)=cos(x/a) sweeps up and down between +I and -2 f o r v a l u e s of a tending t o zero, F

X

cannot be e q u a l t o

F i t s e l f , s i n c e F indeed t e n d s t o zero.

L i n d e l s f p r e s e n t e d t h i s counterexample as a l o c a l one ( i . e .

a counterexample

r e f u t i n g t h e proof and n o t the theorem), and he himself s e t o u t t o g i v e a n o t h e r p r o o f , which, however, was a l s o i n v a l i d . He d i d n o t n o t i c e t h a t example (1.10) also refutes the

n a i v e v e r s i o n of t h e e q u a l i t y theorem g l o b a l l y , f o r t h e

simple r e a s o n t h a t b o t h F

xu

and Fax do n o t e x i s t f o r a=O. Hence (1.10)

is a l s o

Introduction

12

a g l o b a l counterexample f o r the n a i v e v e r s i o n of t h e e q u a l i t y theorem. Lindel s f ' s example provokes t h e q u e s t i o n whether perhaps t h e e q u a l i t y theorem h o l d s a s soon a s both mixed p a r t i a l d e r i v a t i v e s F

xu

answered by H.A.

and F

ax

e x i s t . This q u e s t i o n was

Schwarz i n h i s 1 8 7 3 , p r o v i d i n g a g l o b a l counterexample of t h e

r e f i n e d e q u a l i t y theorem ( i . e . t h e theorem t h a t t h e mixed d e r i v a t i v e s a r e e q u a l when they e x i s t ) . Schwarz's example was: (I. I I )

$ix,yI=x2arctan(y/x)-y2arctun(x/y)

and h e r e f

QJ

(O,O)=-l

and f

e q u a l i t y theorem under t h e derivatives f

QJ

and f

YX

YX

-

(O,O)=+l.

Schwarz then provided a proof of t h e

proofgenerated

-

conditions, t h a t both p a r t i a l

a r e continuous.

A f t e r Schwarz's work was p u b l i s h e d , an impressive amount of e f f o r t was s p e n t on proving t h e e q u a l i t y theorem under less r e s t r i c t i v e c o n d i t i o n s ' 8 . These a c t i v i t i e s concerning r i g o r o u s p r o o f s f o r t h e e q u a l i t y theorem and a n a l y s i s of t h e p r e c i s e e x t e n t of the theorem can c o n v e n i e n t l y be d i s c u s s e d i n terms of t h e concepts p u t f o r t h by Lakatos i n h i s b r i l l i a n t Proofs and

Refutations, The Logic of Mathematical Discovery. They a r e v e r y h e l p f u l t o d i s c e r n t h e d i f f e r e n t a t t i t u d e s towards t h e p r o o f s of a theorem, and t h e y s e r v e t o a n a l y s e t h e p r e c i s e s t a t u s of counterexamples, a u x i l i a r y lemmas e t c . However, when one t r i e s t o apply t h e L a k a t o s i a n terminology t o 18th c e n t u r y a c t i v i t i e s a s w e l l , i t does n o t y i e l d very much. A t b e s t , they a r e t o be r a t e d "naive",

and t h e p r o o f s proposed then a r e a t b e s t "explanations".

This i s

q u i t e a s t r a n g e s i t u a t i o n f o r The philosophy of mathematical discovery. The reason i s t h a t t h e mathematicians Lakatos p r e s e n t s c a r e d f o r r i g o u r of p r o o f s , and cared f o r p r e c i s e d e f i n i t i o n s of concepts. They a r e c h a r a c t e r i s e d i n terms of t h e i r a t t i t u d e towards t h e l o g i c a l components of mathematics: p r o o f s and d e f i n i t i o n s . Consequently, a uniform and n a i v e s o r t of mathematician emerges when one t r i e s t o c h a r a c t e r i s e t h o s e 18th c e n t u r y mathematicians t h a t d i d n o t b o t h e r about such baroque s o p h i s t i c a t i o n .

18th Century mathematicians - w i t h

a few e x c e p t i o n s l i k e Nicolaus I B e r n o u l l i

-

doxes":

were happy t o l i v e w i t h para-

they were i n t e r e s t e d i n f i n d i n g m a n i f e s t r e s u l t s i n t h e enormous

t e r r i t o r y they were e x p l o i t i n g , and they had n e i t h e r l e i s u r e n o r good reason t o p o l i s h a l l r e s u l t s i n t o l o g i c a l l y impeccable and r i g o r o u s form. O f c o u r s e , a l l t h i s i s s l i g h t l y e x a g g e r a t e d , but a s a g e n e r a l r u l e one may c l a i m t h a t 18th century mathematicians were i n t e r e s t e d i n expanding t h e i r knowledge r a t h e r than c o n s o l i d a t i n g t h e i r r e s u l t s . Hence, t h i s c e n t u r y of g r e a t d i s c o v e r i e s i n mathematics r e q u i r e s a n o t h e r type of L o g i c o f m a t h e m a t i c a l d i s c o v e r y than t h e one t h a t o n l y c h a r a c t e r i s e s t h e c e n t u r y a s " n a i v e " 2 o .

Policy of transcription and interpretation

13

Another f e a t u r e of 18th c e n t u r y mathematics may i l l u m i n a t e t h e p o i n t somewhat more. 18th Century mathematicians were d e a l i n g w i t h e x p r e s s i o n s and f u n c t i o n s almost always i n a g l o b a l way. That i s , they were n o t s o much concerned w i t h t h e behaviour o f t h e s e e x p r e s s i o n s f o r c e r t a i n v a l u e s of t h e arguments, b u t r a t h e r they were concerned w i t h the e x p r e s s i o n a s a whole. Hence l o c a l d i f f i c u l t i e s were no r e a s o n s f o r concern, u n l e s s they a f f e c t t h e r e s u l t a s a whole. Schwarz’s counterexample ( l . I l ) ,

fq =

x2$$= fyx 2

y i e l d i n g mixed p a r t i a l d e r i v a t i v e s

f o r x#O y#O would n o t have been regarded a counterexample,

b u t a c o r r o b o r a t i o n of t h e e q u a l i t y theorem, s i n c e g l o b a l l y t h e two mixed

d e r i v a t i v e s a r e obviously e q u a l , W e have t o a p p r e c i a t e such g l o b a l a t t i t u d e s t o mathematics when w e t r y t o

understand ?he l o g i c of 18th c e n t u r y mathematical d i s c o v e r y .

51.5 Policy of transcription and interpretation The h i s t o r i a n of mathematics, d i s c u s s i n g a given p i e c e of e a r l y L e i b n i z i a n c a l c u l u s , can f a i r l y w e l l use t h e o r i g i n a l d i f f e r e n t i a l symbolism i n o r d e r t o r e n d e r t h e argument he i s concerned w i t h . This i s because t h e r a t h e r uniform system of n o t a t i o n i n t h i s c a l c u l u s i s n o t d i f f i c u l t t o u n d e r s t a n d f o r anyone w i l l i n g t o g e t acquainted w i t h t h e concept of d i f f e r e n t i a l . The o r i g i n a l n o t a t i o n , t h e n , has one g r e a t advantage o v e r a f u l l y modernized t r a n s c r i p t i o n , i n t h a t i t i s n o t burdened by t h o s e c o n n o t a t i o n s which a modernized t r a n s c r i p t i o n i n e v i t a b l y c a r r i e s along, and which f r e q u e n t l y f a i l t o o b t a i n i n t h e o r i g i n a l p i e c e of work. Such c o n n o t a t i o n s a r e many; f o r i n s t a n c e , t r a n s c r i p t i o n of a d i f f e r e n t i a l e q u a t i o n M(x,y)dx+N(x,y)dy=O a s Mfx,y)+N(x,y)y’=O immediately i n t r o d u c e s t h e i d e a of x a s t h e independent and y a s t h e dependent v a r i a b l e , and i t s u g g e s t s t h a t t h i s d i f f e r e n t i a l e q u a t i o n should be s o l v e d by a f u n c t i o n

y=f(x). These i d e a s however a r e q u i t e a s e v e r e d i s t o r t i o n of t h e o r i g i n a l meaning, s i n c e t h e absence of independent v a r i a b l e s and

t h e absence of

f u n c t i o n s w a s p r e c i s e l y one of t h e c h a r a c t e r i s t i c s of e a r l y d i f f e r e n t i a l c a l culus. I n g e n e r a l , t h e meaning of a d i f f e r e n t i a l e q u a t i o n o r e x p r e s s i o n as w r i t t e n i n t h e o r i g i n a l L e i b n i z i a n n o t a t i o n i s determined u n i q u e l y , s a v e f o r t h e p r o g r e s s i o n of t h e v a r i a b l e s t h a t must be s p e c i f i e d s e p a r a t e l y

-

e.g.

by

p u t t i n g ddx=O or ddy=O. Once t h e p r o g r e s s i o n i s known, no f u r t h e r a m b i g u i t i e s with r e s p e c t t o t h e meaning of such a d i f f e r e n t i a l e q u a t i o n o r e x p r e s s i o n e x i s t . However, a s soon a s t h e d i f f e r e n t i a l c a l c u l u s i s a p p l i e d t o more-dimensional problems many more a m b i g u i t i e s emerge, which make i t d i f f i c u l t t o m a i n t a i n t h e

14

Introduction

P u r i t a n way of r e n d e r i n g an argument i n i t s o r i g i n a l d i f f e r e n t i a l n o t a t i o n . Let me make t h i s p o i n t c l e a r e r . The c h a r a c t e r i s t i c s i t u a t i o n o c c u r r i n g i n a two dimensional

problem i s

t h a t of two i n f i n i t e l y c l o s e c u r v e s from a given f a m i l y , t h a t a r e i n t e r s e c t e d i n some way by a t h i r d curve, a curve d e f i n e d g e o m e t r i c a l l y i n terms of t h e given f ami l y

.

fig. 4

a+da

Y + dY

-a Y

A

x

x+dx

Figure 4 r e p r e s e n t s such a s i t u a t i o n , where APP' and AQQ' a r e taken t o b e two i n f i n i t e l y c l o s e c u r v e s of t h e given f a m i l y , and where PQ' i s a t r a j e c t o r y

-

d e f i n e d f o r i n s t a n c e a s t h e curve i n t e r s e c t i n g a l l curves of t h e given f a m i l y a t a given a n g l e . Denote t h e a b s c i s s a AM by

2,

t h e o r d i n a t e MP

by y and the parameters of APP' and A&&' by a and a+&

r e s p e c t i v e l y . Then any

one of t h e s e t h r e e v a r i a b l e s ~ , y , a can i n p r i n c i p l e be regarded a s dependent on t h e two remaining o t h e r s ; phrased o t h e r w i s e , a c e r t a i n r e l a t i o n V(x,y,a)=O h o l d s , d e f i n i n g t h e r e l a t i o n between t h e v a r i a b l e s z,a,y. p e r t a i n i n g t o t h e curve PQ'

-

the d i f f e r e n t i a l equation f o r

e.g.

Now i n any argument

on c o n s t r u c t i n g t h e t a n g e n t , o r on f i n d i n g

P&' - t h e f o l l o w i n g t h r e e d i f f e r e n t i a l s of t h e

v a r i a b l e y w i l l occur: (1.12)

y(P'ky(P1,

t h e d i f f e r e n c e between t h e v a l u e s of y i n two s u c c e s s i v e p o i n t s on a s i n g l e curve. This i s t h e d i f f e r e n t i a l of y f o r c o n s t a n t a and v a r i a b l e ( 1.13)

z,

y (Ql-y (P),

t h e d i f f e r e n c e between t h e v a l u e s of y i n two corresponding p o i n t s on two s u c c e s s i v e curves. This i s t h e d i f f e r e n t i a l of y f o r c o n s t a n t z and v a r i a b l e a, and d i f f e r e n t i a t i o n under t h e s e c o n d i t i o n s w a s c a l l e d " d i f f e r e n t i a t i o n from curve t o curve". F i n a l l y a t h i r d d i f f e r e n t i a l of y w i l l emerge, v i z . ( 1 . 14)

y(Q')-yiPI,

15

Policy of transcription and interpretation the d i f f e r e n c e between t h e v a l u e s of y i n two s u c c e s s i v e p o i n t s a l o n g t h e t r a -

j ectory

.

Obviously, t h i s s i t u a t i o n i n which a t l e a s t t h r e e d i f f e r e n t forms of d i f f e r e n t i a l s of y occur can no l o n g e r be met by a n o t a t i o n merely employing "d" f o r d i f f e r e n t i a l s . Indeed, from t h e 17th c e n t u r y on a l a r g e number of n o t a t i o n a l d e v i c e s have been adopted i n o r d e r t o d i s t i n g u i s h between t h e s e d i f f e r e n t c a s e s of t h e d i f f e r e n t i a l of y . These n o t a t i o n s d i f f e r from a u t h o r t o a u t h o r , and f r e q u e n t l y even work. C a j o r i f o r i n s t a n c e

-

w i t h i n t h e oeuvre of one a u t h o r

fromwork t o

i n h i s t r e a t i s e on n o t a t i o n s used i n mathematics-

a l r e a d y l i s t s about f i f t e e n d i f f e r e n t types of n o t a t i o n f o r p a r t i a l and t o t a l d i f f e r e n t i a l s used i n t h e 18th c e n t u r y . Here i s a l i s t of t h o s e n o t a t i o n s which appear i n t h e s o u r c e s t o be d i s c u s s e d i n t h i s book. (a)

L e i b n i z , i n h i s l e t t e r t o Johann B e r n o u l l i d a t e d 3/8/1697

containing the

i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n , r e f r a i n e d from u s i n g t h e symbol

"d" a l t o g e t h e r except f o r t h e v a r i a b l e s a and x and t h e

d i f f e r e n t i a l of y a l o n g t h e curves of t h e given family. The o t h e r d i f f e r e n t i a l s , i n c l u d i n g mixed second o r d e r d i f f e r e n t i a l s , he wrote i n t h e g e o m e t r i c a l f a s h i o n a s linesegments: f o r i n s t a n c e QP f o r y ( Q ) - y ( P I , and &'P'-QP f o r t h e mixed second o r d e r d i f f e r e n t i a l (y(Q')-y(P')-ly(Q)-y(P)). (b)

I n h i s B e i l a g e o f 1697, Leibniz w r o t e : d y f o r t h e d i f f e r e n t i a l a l o n g a

given curve, and d ( s e c u n d . a l y f o r t h e d i f f e r e n t i a l y ( & j - y ( P ) . (c)

Johann B e r n o u l l i , i n h i s answer t o L e i b n i z ' s l e t t e r c i t e d i n ( a ) imme-

d i a t e l y e x p r e s s e d t h e p a r t i a l d i f f e r e n t i a l s of y i n terms of t h e d i f f e r e n t i a l s

da and dx i n t h e f o l l o w i n g way: y ( Q ) - y f P ) = & x d a , and y f P ' I - y f P ) = ~ x & 2 1 , where hX and X r e p r e s e n t e x p r e s s i o n s i n t h e v a r i a b l e s a and x . (d)

Johann and/or Nicolaus I1 B e r n o u l l i around 1720 used t h e symbol dy f o r

a l l t h r e e v a r i a b l e s , w h i l e s p e c i f y i n g c o n d i t i o n s about t h e v a r i a b i l i t y of y i n t h e c o n t e x t . I n t h e c a s e of y(Q)-y(P) they would f o r i n s t a n c e s t a t e "x manente,

a f l u e n t e " ("for c o n s t a n t x, f o r v a r i a b l e a " ) . P a r t i a l d i f f e r e n t i a l s f u r t h e r more were expressed i n terms of t h e d i f f e r e n t i a l s way: y ( Q I - y ( P ) = q d a , y(P')-y(P)-dx, (e)

cb and da i n t h e f o l l o w i n g

and y ( & ' ) - ~ ( P J = p & + q d a ~ ~ .

Nicolaus I B e r n o u l l i from about 1719 onwards used t h e f o l l o w i n g very ex-

p l i c i t and c l e a r n o t a t i o n i n h i s p r i v a t e n o t e s and correspondence:

y ( Q ) - y (P)=6y, y ( P ' I - y (P)=dij, y ( Q ' l - y (P)=dy. Furthermore h e g e n e r a l l y s t a t e d : &=&y+by,

Gy=qda and &---p&.

The use of d i f f e r e n t i a l c o e f f i c i e n t s p and q had

become f a i r l y g e n e r a l l y a c c e p t e d around 1 7 2 O Z 3 . (f)

E u l e r i n t h e 1730s i n d i c a t e d a l l d i f f e r e n t i a l s by t h e s t r a i g h t d , and a l -

ways s p e c i f i e d constancy o r v a r i a b i l i t y i n t h e c o n t e x t . Like Johann and Nico-

Introduction

16

l a u s I I B e r n o u l l i , he u s u a l l y r e f e r r e d t o t h e p a r t i a l d i f f e r e n t i a l s a s Q&

and

Rda, having i n t r o d u c e d t h e t o t a l d i f f e r e n t i a l dy=Qdz+Rdu a t t h e o u t s e t of h i s d i s c o u r s e . Not u n t i l t h e 1750s would E u l e r i n t r o d u c e a d i f f e r e n t i a l n o t a t i o n for the differential coefficients:

(2)

d i f f e r e n t i a l of y , and analogously

($?

(9) (2) >T

....

f o r t h e c o e f f i c i e n t of & i n t h e t o t a l f o r t h e c o e f f i c i e n t of

da.

Euler's

c o i n c i d e with our 3 an: &. aa az From t h i s enumeration i t w i l l b e c l e a r t h a t a d i s t i n c t i o n - more o r l e s s and

i m p l i c i t - between independent and dependent v a r i a b l e s i s p r e s e n t i n a l l ins t a n c e s . The independent v a r i a b l e s a r e those t h a t can do w i t h t h e simple s t r a i g h t d only, t h e dependent v a r i a b l e s a r e t h o s e t h a t r e q u i r e s p e c i a l a t t e n t i o n , and a s e r i e s of d i f f e r e n t symbols o r e x p l a n a t i o n s . By t h i s d i s t i n c t i o n , t h e v a r i a b l e s z and a o b t a i n a pre-eminent p o s i t i o n among t h e v a r i a b l e s o c c u r r i n g i n a given p r o b l e m I n b e i n g v a r i a b l e , c o n s t a n t o r somehow dependent

on each o t h e r , t h e s e v a r i a b l e s

z and

u determine t h e v a r i a t i o n s of a l l o t h e r

v a r i a b l e s . Thus f o r c o n s t a n t a t h e v a r i a b l e z determines t h e s u c c e s s i o n of p o i n t s on t h e curves of t h e given family and i n consequence t h e s u c c e s s i v e v a l u e s of y along such a curve. And t h e v a r i a b l e a , f o r c o n s t a n t

X,

determines

t h e corresponding p o i n t s on t h e s u c c e s s i v e c u r v e s of t h e family. I n t h i s way, a l r e a d y i n t h e 1690s, t h e i d e a of independent v a r i a b l e s emerged, b e i n g n e c e s s i t a t e d by t h e need t o i d e n t i f y d i f f e r e n t s e r i e s of s u c c e s s i v e p o i n t s i n two dimensional problem s i t u a t i o n s . However, i t i s n o t u n t i l t h e 1730s t h a t t h e corresponding concept of a f u n c t i o n of two v a r i a b l e s e m e r g e s i n t h e work o f Leonhard E u l e r . E u l e r , from t h e v e r y beginning of h i s r e s e a r c h , c o n s i s t e n t l y used a f u n c t i o n y=Pfz,a) t o f i x t h e r e l a t i o n s between t h e v a r i a b l e s i n a two dimensional problem. P r i o r t o E u l e r , the pre-eminent r o l e of the independent v a r i a b l e s w a s r e s t r i c t e d t o t h e d e f i n i t i o n of t h e p a r t i a l d i f f e r e n t i a l s , and e q u a t i o n s l i k e Vfz,y,u)=O were used r a t h e r than f u n c t i o n s of two independent v a r i a b l e s t o denote t h e r e l a t i o n between t h e v a r i a b l e s . Hence, p r i o r t o E u l e r t h e c o e f f i c i e n t s p and q i n t h e t o t a l d i f f e r e n t i a l dy=p&+qda i n a l l three variables

2,

were i n g e n e r a l assumed t o be e x p r e s s i o n s

y and a.

Returning now t o t h e problem of t r a n s c r i p t i o n , a P u r i t a n t r a n s c r i p t i o n j u s t copying the o r i g i n a l s i s obviously o u t of t h e q u e s t i o n . There i s no s e n s i b l e p o i n t i n f o r c i n g t h e r e a d e r t o work h i s way through a l l s o r t s of ad hoc n o t a t i o n s . Moreover, a c l e a r t r a n s c r i p t i o n of a given argument a l s o r e q u i r e s t h a t t h e i m p l i c i t assumptions about independent and dependent v a r i a b l e s a r e made e x p l i c i t ; i t i s e s s e n t i a l f o r a thorough u n d e r s t a n d i n g of an argument t o be c l e a r l y informed about such assumptions. I s h a l l t h e r e f o r e t r a n s c r i b e p a r t i a l

Policy of transcription and interpretation

17

and t o t a l d i f f e r e n t i a l s i n t h e f o l l o w i n g way: r e p r e s e n t s t h e d i f f e r e n t i a l of y f o r v a r i a b l e x and c o n s t a n t a ,

- d$ -

dg

-

dy r e p r e s e n t s t h e t o t a l d i f f e r e n t i a l of y , and hence dy=d$+d

r e p r e s e n t s t h e d i f f e r e n t i a l of y f o r v a r i a b l e a and c o n s t a n t x,

y=p&+qda.

This symbol dy w i l l a l s o be used f o r t h e d i f f e r e n t i a l of y when y i s c o n s i d e r e d t o vary a l o n g a given t r a j e c t o r y curve, such a s t h e curve PQ’ i n f i g u r e 4 . Before concluding t h i s s e c t i o n , some more m a t t e r s of t r a n s c r i p t i o n must be s e t t l e d . F i r s t of a l l , e x p r e s s i o n s i n v o l v i n g a s e r i e s of v a r i a b l e s were u s u a l l y denoted by c a p i t a l l e t t e r s , and t h e v a r i a b l e s o c c u r r i n g i n t h e e x p r e s s i o n s were enumerated i n t h e ambient t e x t . For c l a r i t y , I s h a l l denote t h e s e v a r i a b l e s between b r a c k e t s behind t h e l e t t e r t h a t r e p r e s e n t s t h e e x p r e s s i o n : e . g .

V(x,y,al

i n s t e a d of V , o r p ( x l i n s t e a d of p . Furthermore, t h e t r a n s c r i p t i o n of

i n t e g r a l s r e q u i r e s some a t t e n t i o n . I n v a r i a b l y , i n t e g r a l s were w r i t t e n w i t h o u t bounds b e i n g i n d i c a t e d i n t h e n o t a t i o n : e . g . I p d ~ . I f a t a l l , t h e bounds, a g a i n , were d e f i n e d i n t h e c o n t e x t ; f r e q u e n t l y

however

i t was l e f t t o the reader

t o decide about t h e n e c e s s a r y bounds of i n t e g r a t i o n , o r e q u i v a l e n t l y , about t h e c o n s t a n t t h a t had t o be added t o t h e i n t e g r a t i o n . I n g e n e r a l i t h a s been my p o l i c y t o i n d i c a t e t h e bounds of i n t e g r a t i o n e x p l i c i t l y i n t h e n o t a t i o n : e.g.

3:

Ip(d& x0

or

X

J p(x,aa)dx. Modem n o t a t i o n would s t i l l p r e f e r t h a t t h e

x0

n o t a t i o n a l d i f f e r e n c e b e made between t h e dummy v a r i a b l e of i n t e g r a t i o n , s a y

t, and t h e v a r i a b l e bound of t h e i n t e g r a l , say

I Xp ( t , a l d t .

5:

e.g.

X

J p f t ) d t or 20

However, save f o r a few e x c e p t i o n s , I have c o n s c i o u s l y r e f r a i n e d

b

from u s i n g such dummy v a r i a b l e s , f o r t h e f o l l o w i n g reason: I f one t r a n s c r i b e s an i n t e g r a l such a s / p & , p i n v o l v i n g x and a , and bounds of i n t e g r a t i o n b e i n g X x0 and x, i n t h e form I p ( t , a ) d t , then t h e r e appears t o be no d i f f i c u l t y i n XO

s u b s t i t u t i n g a=A(x) i n t o t h e i n t e g r a n d :

X

I p ( t , A l x ) ) d t . However, such a subX

s t i t u t i o n was regarded p o s s i b l e o n l y when t h e i n t e g r a l c o u l d a c t u a l l y be c a l c u l a t e d , and i t was always d e s c r i b e d a s “ s u b s t i t u t i o n a f t e r t h e i n t e g r a t i o n has been performed”. This p r o v i s o was n e c e s s a r y p r e c i s e l y because t h e r e w a s no concept of a dummy v a r i a b l e of i n t e g r a t i o n ; hence, s u b s t i t u t i n g ~ = A ( x ) i n t h e i n t e g r a n d p ( x , a ) i t s e l f would produce a r e s u l t completely d i f f e r e n t from sub3:

s t i t u t i o n a f t e r t h e i n t e g r a t i o n h a s been performed, namely

I pfx,A(x)l&, XO

and

i n t h i s c a s e t h e A(xl would a l s o p a r t i c i p a t e i n t h e i n t e g r a t i o n . I n t h i s way, t r a n s c e n d e n t a l e x p r e s s i o n s d e f i n e d by an i n t e g r a l l i k e y=Jp(x,a)& garded inadequate f o r such subs t i t ~ t i o n s ~ ~ .

were r e -

Introduction

18

5 I. 6 T'ranscendental Curves and transcendentaz expressions

Most of t h e problems which we s h a l l d i s c u s s i n t h i s book concern t r a n s c e n d e n t a l curves or f a m i l i e s of t r a n s c e n d e n t a l curves. To q u i t e an e x t e n t t h e dev elopment of p a r t i a l d i f f e r e n t i a l c a l c u l u s was motivated p r e c i s e l y by t h e d i f f i c u l t i e s which 17th and 18th c e n t u r y mathematicians encountered i n d e a l i n g with t r a n s c e n d e n t a l c u r v e s o r t r a n s c e n d e n t a l e x p r e s s i o n s . T h e i r use of t h e term " t r a n s c e n d e n t a l " , however, i s o f t e n confusing and sometimes q u i t e p r o b l e m a t i c a l . Hence i t w i l l be convenient t o c o l l e c t t h e b a s i c f a c t s h e r e and c l a r i f y t h e meaning of the term " t r a n s c e n d e n t a l " a s i t was used t h e n , and a s i t w i l l be used h e r e . A l l non-algebraic

c u r v e s , by d e f i n i t i o n , a r e c a l l e d t r a n s c e n d e n t a l ; a l -

g e b r a i c c u r v e s , i n t u r n , a r e t h o s e curves t h a t can be r e p r e s e n t e d by an e q u a t i o n i n terms of t h e r e c t i l i n e a r c o o r d i n a t e s z and y , which o n l y i n v o l v e s t h e f i v e o p e r a t i o n s i,-,z,:,d.

These d e f i n i t i o n s were a l s o agreed upon i n t h e 17th

century. However, t h e r e i s one g r e a t d i f f e r e n c e between o u r modern use of t h e concept of t r a n s c e n d e n t a l c u r v e s and t h e way i t was used i n t h e 17th c e n t u r y . Nowadays one f i r s t p r o v i d e s a w e l l d e f i n e d g e n e r a l concept of c u r v e , b a s e d on t h e g e n e r a l concept of f u n c t i o n , which i n a c e r t a i n s e n s e p r o v i d e s an a - p r i o r i demarcation of t h e u n i v e r s e of d i s c o u r s e : one knows

what a curve i s , and hence

t h e e x t e n t of t h e realm of t r a n s c e n d e n t a l c u r v e s i s c l e a r l y

marked.

17th cen-

t u r y mathematicians on t h e c o n t r a r y d i d n o t have such a g e n e r a l concept of curve. By consequence i t was n o t a t a l l c l e a r how many n o n - a l g e b r a i c c u r v e s d i d i n f a c t e x i s t . However, i n the p e r i o d 1690-1740 w i t h which we a r e d e a l i n g h e r e t h i s absence of a g e n e r a l concept of curve d i d n o t w i t h h o l d mathematicians from f r e e l y u s i n g t h e term " t r a n s c e n d e n t a l " o r from p o s i n g problems f o r " a l l " t r a n s c e n d e n t a l curves. I n p r a c t i c e , t r a n s c e n d e n t a l curves were taken t o b e t h o s e curves t h a t could b e r e p r e s e n t e d by i n t e g r a l s t h a t could n o t be c a l c u l a t e d exp l i c i t l y , such a s t h e brachystochrone c y c l o i d s

Hence, the s t a n d a r d way of r e p r e s e n t i n g a t r a n s c e n d e n t a l curve a n a l y t i c a l l y was t o w r i t e X

(1.16)

y = l p(x)d;c

.

XO

Now t h e f o l l o w i n g i s s u e s a r i s e : F i r s t o f a l l , whether o r n o t a curve was t o be c a l l e d t r a n s c e n d e n t a l depended on t h e a n a l y t i c s k i l l of t h e mathematician i n

Transcendental curves and expressions

19

q u e s t i o n : a l g e b r a i c i t y o r transcendency of a curve was determined by h i s a b i l i t y o r i n a b i l i t y t o c a l c u l a t e an i n t e g r a l . So i n p r i n c i p l e t h e s i t u a t i o n c o u l d occur where a curve which i n f a c t was a l g e b r a i c was c a l l e d t r a n s c e n d e n t a l because i t was known only by an e q u a t i o n i n v o l v i n g t h e t r a n s c e n d e n t a l e x p r e s s i o n Jp(x)&, which t h e mathematician i n q u e s t i o n was unable t o c a l c u l a t e e x p l i c i t l y . I n f a c t , I have n e v e r come a c r o s s such a s i t u a t i o n . Furthermore, i t m u s t have been q u i t e c l e a r a t t h a t time t h a t t h e p r a c t i c a l o r o p e r a t i o n a l concept o f a t r a n s c e n d e n t a l curve, v i z . a curve d e f i n e d by an e q u a t i o n such a s (1.16) d i d n o t c o i n c i d e w i t h the t h e o r e t i c a l concept, v i z . t h a t of a n o n - a l g e b r a i c a l curve. L e i b n i z , f o r i n s t a n c e , when d i s c u s s i n g t h e curve t h a t c u t s o f f e q u a l a r c s from a f a m i l y of l o g a r i t h m i c curves provided by t h e e q u a t i o n

came up w i t h t h e c o n d i t i o n

d e f i n i n g t h i s e q u a l a r c s t r a j e c t o r y . Since no a l g e b r a i c e q u a t i o n between

y could be found from t h e s e two c o n d i t i o n s ( I . 17) and ( I . 18),

x and

t h e curve had t o

be c a l l e d t r a n s c e n d e n t a l . But s t i l l i t was a l s o impossible t o p r o v i d e an equat i o n of t h e form ( 1 . 1 6 ) f o r t h i s e q u a l a r c s t r a j e c t o r y . Hence t h e r e a p p e a r s t o e x i s t a t e n s i o n between t h e p r a c t i c a l and t h e t h e o r e t i c a l d e f i n i t i o n of a t r a n s c e n d e n t a l curve i n 17th and 18th c e n t u r y mathematics. T h i s p o i n t may be i l l u m i n a t e d somewhat more by having a c l o s e r look a t t h e way i n which t r a n s c e n d e n t a l curves emerged i n 17th c e n t u r y mathematics. I n h i s Ge'ometrie of 1637 D e s c a r t e s had i n t r o d u c e d t h e c l a s s of a l g e b r a i c c u r v e s , which h e c a l l e d "geometric curves". These curves D e s c a r t e s a c c e p t e d a s genuine g e o m e t r i c a l obj e c t s , and h e could d e a l w i t h them by a l g e b r a i c methods. O u t s i d e t h e realm of geometric curves t h e r e were o t h e r c u r v e s , c a l l e d "mechanical curves", which d i d n o t occur i n c l a s s e s , b u t a s i n d i v i d u a l s ; t h e y were n o t d e f i n e d i n a uniform way, b u t each one i n i t s own s p e c i f i c way, f o r i n s t a n c e by way of g e n e r a t i o n ( c y c l o i d : t h e p a t h of a p o i n t on t h e circumference

of a c i r c l e

r o l l i n g along an a x i s ) , o r by s p e c i f i c d e f i n i n g p r o p e r t i e s ( l o g a r i t h m i c a : a r i t h m e t i c a l sequence maps onto a geometric sequence). These curves d i d n o t f i t i n t o D e s c a r t e s ' s scheme, and, t h e r e f o r e , they were banned from geometry. The f i r s t e x t e n s i o n of t h e realm of manageable curves by a n e n t i r e class of new curves took p l a c e i n t h e e a r l y 1690s, when L e i b n i z and Johann B e r n o u l l i i n t r o d u c e d t h e e x p o n e n t i a l s . Exponential curves a r e those t h a t can be

20

Introduction

expressed b y an e q u a t i o n t h a t i n v o l v e s n o t o n l y t h e o p e r a t i o n s +,-,x,:,J, b u t a l s o v a r i a b l e exponents such a s a

X

, z?

e t c . Both L e i b n i z and Johann B e r n o u l l i

showed how t h e d i f f e r e n t i a l c a l c u l u s can be a p p l i e d t o t h o s e e x p o n e n t i a l c u r v e s z 5 . Hence t h e s e c u r v e s could a l s o be d e a l t w i t h , n o t by a l g e b r a i c means b u t by means of t h e d i f f e r e n t i a l c a l c u l u s . The c a l c u l u s i t s e l f a l s o l e d t o a new c l a s s of curves; d i f f e r e n t i a t i o n of a l g e b r a i c o r e x p o n e n t i a l e x p r e s s i o n s d i d n o t l e a d t o anything new, b u t i n t e g r a t i o n f r e q u e n t l y l e d

t o c a s e s where

t h e i n t e g r a l could n o t a c t u a l l y be found. The c u r v e s d e f i n e d by such i n t e g r a l s y=Jp ix)dx, however, were f a i r l y a c c e p t a b l e . They r e p r e s e n t q u a d r a t u r e s of a l g e b r a i c curves. Hence one could d e f i n e a new c l a s s of c u r v e s i n terms of t h e o p e r a t i o n of i n t e g r a t i o n : those c u r v e s t h a t can b e r e p r e s e n t e d by an e q u a t i o n of t h e form X

(1.16)

y = s p(xldx. 50

Thus t h e u n i v e r s e of d i s c o u r s e a t t h e end of t h e 17th c e n t u r y c o n s i s t e d of t h r e e c l a s s e s of c u r v e s , t h e a l g e b r a i c s , t h e e x p o n e n t i a l s , and t h o s e t r a n s c e n d e n t a l s t h a t could be e x p r e s s e d by means of i n t e g r a l s . Other t r a n s c e n d e n t a l curves were known, o f course, b u t t h e y remained o u t s i d e t h e f i e l d of manageable c u r v e s ; they might emerge a s s o l u t i o n s t o s p e c i f i c problems, b u t they were l e f t o u t when problems were posed f o r curves o r f a m i l i e s of c u r v e s i n g e n e r a l . Here i s t h e g r e a t d i f f e r e n c e between t h e approach common nowadays and t h e approach

of 17th and 18th c e n t u r y mathematicians. I n s t e a d of g i v i n g an a - p r i o r i demarcation of t h e s e t of " a l l curves" they g r a d u a l l y extended t h e realm of manageable c u r v e s , a l o n g w i t h t h e e x t e n s i o n of t h e methods f o r i n v e s t i g a t i n g those curves. I n t h i s way 17th and 18th c e n t u r y mathematicians e f f e c t i v e l y avoided t h e f r u s t r a t i n g s i t u a t i o n of having t o b o t h e r about a l l those c u r v e s t h a t e x i s t by f o r c e of a g e n e r a l d e f i n i t i o n b u t which can not b e d e a l t w i t h by means o f t h e a l g e b r a i c o r a n a l y t i c means a v a i l a b l e .

5 I. 7 Conventions References t o primary o r secondary s o u r c e s a r e made e i t h e r by i t a l i c i s e d d a t e o r by i t a l i c i s e d s h o r t t i t l e ; f o r example: L e i b n i z 1692, o r Nicolaus I B e r n o u l l i Tentamen. F u l l b i b l i o g r a p h i c i n f o r m a t i o n i s t o be found i n t h e b i b l i o g r a p h y under t h e r e s p e c t i v e a u t h o r s ' names. A s a r u l e , t h e d a t e s r e f l e c t t h e y e a r of p u b l i c a t i o n . References t o l e t t e r s a r e provided by an i t a l i c i s e d

Conventions s t r i n g of t h e f o l l o w i n g form: (sender): ( r e c i p i e n t ) (date o f Z e t t e r ) , f o r example GWL:JohB 3 / 8 / 1 6 9 7 .

A l i s t of a b b r e v i a t i o n s i s given i n t h e preamble

t o t h e b i b l i o g r a p h y . P l a c e s of p u b l i c a t i o n of l e t t e r s a r e g i v e n i n t h e l i s t of l e t t e r s i n t h e b i b l i o g r a p h y . A l l q u o t e s from f o r e i g n

languages have been t r a n s l a t e d . Mathematical

symbols and conventions i n quotes have f r e q u e n t l y been adapted t o my own system. Remarks between square

b r a c k e t s a r e always my own, remarks between

curved b r a c k e t s a r e o r i g i n a l . The u n a l t e r e d o r i g i n a l t e x t of a quote can a l ways be found i n a f o o t n o t e . Footnotes have been numbered p e r c h a p t e r , and they a r e p r i n t e d as one l o t behind t h e Epilogue.

21

22

CHAPTER 2

FAMILIES OF CURVES I N THE 1690s

5 2 . 1 EnveZopes

L e i b n i z ' s "new appZication of the c a l c u l u s "

52.1.1

I n 1692, e i g h t y e a r s a f t e r h i s f i r s t p u b l i c a t i o n about t h e d i f f e r e n t i a l c a l c u l u s , L e i b n i z published a s h o r t a r t i c l e i n t h e Acta Erudftorwn t h e t i t l e of which promised "a new u s e of t h e a n a l y s i s of t h e i n f i n i t e " . Two y e a r s l a t e r he followed up t h i s a r t i c l e w i t h y e t a n o t h e r "new a p p l i c a t i o n and u s e of t h e d i f f e r e n t i a l calculus"'.

Both a r t i c l e s were devoted t o one and t h e same problem,

namely t o f i n d an a l g o r i t h m f o r t h e curve which i n each of i t s p o i n t s touches a curve from a given family of c u r v e s : t h e s o - c a l l e d envelope. The a l g o r i t h m t h a t L e i b n i z produced i s e q u i v a l e n t t o t h e e l i m i n a t i o n scheme:

V (2, y, al=0

a -V

aa

(2,y ,

a ) =O

d

I

e l i m i n a t i o n of a

r e l a t i o n between z and

y of t h e envelope

where V(x,y,al=O i s t h e e q u a t i o n of t h e given f a m i l y of c u r v e s . Now envelopes were n o t new. The concept had appeared a s e a r l y a s 1644 i n

T o r r i c e l l i ' s s t u d i e s on e x t e r n a l b a l l i s t i c s . T o r r i c e l l i had demonstrated t h a t a l l b a l l i s t i c p a r a b o l a s which a r e t h e t r a j e c t o r i e s of cannon-balls s h o t a t d i f f e r e n t a n g l e s of e l e v a t i o n ( i n i t i a l v e l o c i t y and v e r t i c a l plane remaining) touch one f i x e d p a r a b o l a . This p a r a b o l a i s c a l l e d t h e " s a f e t y parabola" s i n c e i t d e f i n e s t h e range of t h e cannon. I t i s t h e envelope of t h e b a l l i s t i c para-

bolas'.

Huygens i n h i s t h e o r y of l i g h t 3 had a l s o made e x t e n s i v e use of en-

v e l o p e s . Thus t h e concept was not new. What was r e a l l y n o v e l , however, was t h e a p p l i c a t i o n of t h e d i f f e r e n t i a l c a l c u l u s a t t h i s problem about f a m i l i e s of curves. L e i b n i z ' s programme of e x t e n d i n g t h e d i f f e r e n t i a l c a l c u l u s t o f a m i l i e s of curves was f a r from simple. A s d i s c u s s e d i n c h a p t e r 1, t h e d i f f e r e n t i a l calcul u s was i n i t s conceptual f o u n d a t i o n s a s w e l l a s i n i t s main a p p l i c a t i o n s ( t a n g e n t s ) i n t i m a t e l y l i n k e d with t h e s i n g l e curve. T h e c u r v e - c o n s i d e r e d an i n f i n i t e a n g u l a r polygon

-

t o be

provided t h e p r o g r e s s i o n s of t h e v a r i a b l e s and

Envelopes

23

thus made these variables susceptible of differentiation. Thus it was not at all clear conceptually how a quantity such as the parameter of a family of curves

-

o f a nature so different from the variables such as abscissa, ordinate,

arclength pertaining to a single curve

-

could also be susceptible of different-

iation. It is precisely this problem which Leibniz studied and which makes his article of interest for u s . Here was a conceptual barrier which the calculus had to overcome if partial differentiation were to become possible at all. Leibniz's article 1692 is for the most part devoted to the conceptual preparation of the new field of families of curves for treatment by means of the calculus, and it contains Leibniz's first answers to the problems: how can the parameter of a family of curves be conceived as a differentiable quantity, and what is the common rationale ofthe well-known application of the differential calculus to tangent problems and its new application in envelope problems?

52.1.2 The variable parameter

Leibniz's article commences with a careful introduction of the concept of a family of curves, termed "infinitae curvae ordinatim positione datae" (infinitely many curves given by position in ordered sequence)4. This concept is presented as a generalisation of the idea of the ordered set of ordinates as it occurs in ordinary differential calculus. fig. 1

The order in the family of curves is based on a correspondence between the curves and the points on a line of reference called the "ordinatrix". Consider e.g. (this is Leibniz's main example) a mirror and incoming light rays (see fig. 2). In each point of the mirror one can by means of the law of reflection construct the reflected light ray. Now the family of these reflected light rays is an ordered family of straight lines, and the order in this family is derived from the order of the points on the mirror.

Families of Curves in the 1690s

24

fig. 2

Thus the m i r r o r a c t s a s the o r d i n a t r i x of t h e f a m i l y of r e f l e c t e d l i g h t r a y s . Let me quote L e i b n i z : "For example, i f some m i r r o r through the a x i s

-

-

o r r a t h e r i t s i n t e r s e c t i o n with a plane

of any shape whatsoever given by p o s i t i o n r e f l e c t s sun

r a y s which a r r i v e e i t h e r d i r e c t l y o r a f t e r some o t h e r r e f l e c t i o n o r ref r a c t i o n , then the r e f l e c t e d r a y s w i l l be i n f i n i t e l y many s t r a i g h t l i n e s drawn i n ordered sequence, and a t any p o i n t of t h e m i r r o r ( t h e r e s t remaining) t h e corresponding r e f l e c t e d r a y w i l l be given."' G e n e r a l i s i n g from s t r a i g h t l i n e s t o c u r v e s , L e i b n i z a r r i v e d a t h i s concept of " i n f i n i t e l y many curves g i v e n by p o s i t i o n i n o r d e r e d sequence": "But I accept a s "drawn i n o r d e r e d sequence" n o t only s t r a i g h t l i n e s b u t a l s o any c u r v e s , provided t h e law i s known a c c o r d i n g t o which i n a g i v e n point

of some given curve ( t h e o r d i n a t r i x ) t h e [curved] l i n e corresponding

t o t h a t p o i n t can be drawn, which [ l i n e ] w i l l be one of t h o s e t h a t a r e t o be drawn i n ordered sequence, o r t h a t are given by p o s i t i o n . By p a s s i n g through t h e p o i n t s of t h e o r d i n a t r i x ( e . g . t h e curve by t h e r o t a t i o n of which t h e m i r r o r d i s c u s s e d above i s g e n e r a t e d , o r i t s i n t e r s e c t i o n w i t h t h e a x i s ) those l i n e s w i l l emerge which a r e given i n o r d e r e d sequence."6

I n t h i s new a p p l i c a t i o n of t h e c a l c u l u s t h e o r d i n a t r i x t a k e s on t h e r o l e of t h e s i n g l e curve i n t h e o r d i n a r y d i f f e r e n t i a l c a l c u l u s . J u s t a s t h e p o i n t s of t h e curve determine t h e p r o g r e s s i o n s of t h e v a r i a b l e s i n t h e o r d i n a r y d i f f e r e n t i a l c a l c u l u s , s o t h e p o i n t s of t h e o r d i n a t r i x determine t h e p r o g r e s s i o n of t h e c u r v e s i n t h e family. I n i n t r o d u c i n g t h e term "ordinatim" ( i n o r d e r e d sequence) Leibniz made it very c l e a r t h a t i t was t h i s very analogy which he wanted t o formulate. However, t h e m i s s i n g concept h e r e i s t h e parameter of the family of c u r v e s ; t h e family of curves being p a r a m e t r i s e d by t h e p o i n t s o f t h e o r d i n a t r i x one would expect a second s t e p , namely t h e i n t r o d u c t i o n of t h e parameter as a v a r i a b l e d e f i n e d on t h e p o i n t s o f t h e o r d i n a t r i x . But t h e r e i s n o t h i n g of t h a t k i n d . I n f a c t , t h e r e i s q u i t e a paradox between L e i b n i z ' s c a r e -

25

Envelopes

f u l geometric i n t r o d u c t i o n of f a m i l i e s of curves by means of t h e o r d i n a t r i x and t h e almost complete absence of t h e o r d i n a t r i x a s soon a s t h e a p p l i c a t i o n of t h e d i f f e r e n t i a l c a l c u l u s t o e q u a t i o n s of f a m i l i e s of c u r v e s i s e x p l a i n e d . A seemingly u n r e l a t e d d i s c u s s i o n then follows about e q u a t i o n s which r e p r e s e n t f a m i l i e s of c u r v e s . The occurrence of a v a r i a b l e parameter a i n an e q u a t i o n

V(x,y,a)=O of a f a m i l y of curves is p r e s e n t e d a s an e m p i r i c a l f a c t r a t h e r than a consequence of t h e o r d i n a t r i x concept. This becomes v e r y c l e a r i n L e i b n i z ' s second a r t i c l e about envelopes ( 2 6 9 4 ) , which i s e s s e n t i a l l y i d e n t i c a l t o t h e f i r s t one, b u t formulated much more e x p l i c i t l y : "The c o e f f i c i e n t s a,b,c which a r e employed i n t h e e q u a t i o n t o g e t h e r w i t h z and y denote q u a n t i t i e s which a r e c o n s t a n t i n one and t h e same c u r v e ;

[...I

By comparing t h e c u r v e s of t h e s e r i e s , o r by c o n s i d e r i n g a t r a n s i t i o n from one curve t o a n o t h e r one, some of t h e c o e f f i c i e n t s a r e v e r y c o n s t a n t o r permanent ( t h o s e t h a t remain f i x e d n o t only i n one curve b u t i n a l l c u r v e s of t h e s e r i e s ) and o t h e r s a r e v a r i a b l e . Namely when t h e law o f t h e s e r i e s i s given, n e c e s s a r i l y o n l y a s i n g l e v a r i a b i l i t y remains i n t h e Thus t h e parameter i s i n t r o d u c e d i n a very m a t t e r - o f - f a c t

coefficient^."^

way by t h e s t a t e m e n t

t h a t a s i n g l e v a r i a b i l i t y must remain i n t h e c o e f f i c i e n t s of a n e q u a t i o n p e r t a i n i n g t o a family o f c u r v e s . Leibniz d i d n o t even imply t h a t one of the coe f f i c i e n t s i n such an e q u a t i o n must be i d e n t i f i e d a s

the parameter;

i n s t e a d he

allowed f o r a s e t of c o e f f i c i e n t s t o remain v a r i a b l e , provided s u f f i c i e n t a d d i t i o n a l r e l a t i o n s between t h e s e c o e f f i c i e n t s were given t o reduce t h e degree of freedom t o one. Hence, a family of curves can be r e p r e s e n t e d by a s e t of equations l i k e

Vix,y,a,b)=O

(2.1)

and A(a,b)=O

o r by a s i n g l e e q u a t i o n of t h e form

Vix,y,al=O.

(2.2)

I n my d i s c u s s i o n of L e i b n i z ' s argument, I s h a l l presume t h a t t h e e q u a t i o n of the f a m i l y of curves has form ( 2 . 2 )

82.1.3

The envelope a l g o r i t h m

Like a l l h i s contemporaries L e i b n i z regarded t h e envelope of a f a m i l y of curves a s being formed by i n t e r s e c t i o n s of s u c c e s s i v e c u r v e s . Likewise, h e regarded t h e t a n g e n t t o a s i n g l e c u r v e as a l i n e i n t e r s e c t i n g t h e given c u r v e

Families of Curves in the 1690s

26

in two successive points. Leibniz derived the envelope algorithm by singling out this idea of successive intersection as the common rationale in both types of problems, and by identifying as "differentiable" those variables that reflect the successive intersections. In discussing the envelope, Leibniz argued as follows: "It is clear that the intersecting [curves], which are tangent to the line formed by their intersection [the envelope] are twofold, and the point of intersection is unique, as is the corresponding ordinate. On the other hand, in the usual investigation of lines - either straight or curved

[...I

-

which touch a given curve and which are to be sought from the ordinates of the given curve, these ordinates are conceived of as twofold and the tangents are considered to be unique. Therefore, in the present calculation where these ordinates are investigated from the tangent curves or straight lines given by position (contrary to what is common), the coordinates x and y remain invariable in this transition from a curve to the next, and thus

they are indifferentiable. And the coefficients (which in the ordinary calculation are judged to be indifferentiable because they are constant) are differentiated as far as they are variable."' Leibniz's argument as presented here is hardly convincing. In a few lines he introduces the four new concepts "unique'', "twofold", "differentiable" and "indifferentiable", he describes the well known algorithm for the determination of a tangent to a given curve in terms of these new concepts, he describes the new algorithm for calculating the envelope of a family of curves and finally presumes that he has made everything plausible. In fact, there is not much of a proof here, and Leibniz's argument is a highly heuristic one, begging for a sympathetic understanding. I shall try to elaborate Leibniz's ideas here: fig. 3

fig. 4

Envelopes

27

In both the ordinary tangent problem and the envelope problem one can observe a dichotomy of the quantities involved: they happen to be either unique or twofold. I n the ordinary tangent problem (see figure 3 ) , a curve k is given by an equation V(x,y,c)=O,

where z and y represent the coordinates of the points

of the curve and c represents the coefficients occurring in the equation. A s the tangenl: intersects the curve in both P and the infinitely close point P' the point of intersection is twofold, or consists of two points, coinciding

except for an infinitesimal difference. Thus the abscissa x and the ordinate

y pertaining to this twofold point of intersection are twofold. The coefficients in the equation, like c , being invariable along the curve, are considered to be unique. On the other hand, in the envelope problem (see figure 4 ) the point of intersection P of two infinitely close curves k and k ' is unique, and thus the coordinates x and y are unique. But the curves k and k ' are obviously twofold, and this is reflected in the fact that the parameter a in the equation

V(z,y,a)=O

is considered t o be twofold, or to consist of two values a and a'

coinciding except for an infinitesimally small difference. Now Leibniz takes

it for granted that twofold is the criterion for differentiable and unique is the criterion for indifferentiable. Differentiation of the given equation with respect to those quantities that are differentiable yields a differential equation by means of which the tangent problem under consideration can b e solved: In the ordinary tangent problem x and y are twofold, and thus differentiable. Therefore the resulting differential equation in that case is: (2.3)

Vxfx,y,eldx+V (x,y,c)dy=O.

Y

In the envelope problem a is considered twofold, thus a is differentiable and differentiation of equation (2.2) yields the condition (2.4)

V ix,y,alda=O.

a

Elimination of a from (2.2) and ( 2 . 4 ) then yields the equation of the envelope.

§2.1.4 The s a f e t y paraboZa

The most interesting example of the application of Leibniz's method was provided by Johann Bernoulli, who - at de l'H8pital's request

-

dealt with

Torricelli's safety parabola. At the end of the year 1692, Johann Bernoulli showed l'H8pital how Leibniz's method could be applied to find the safety parabola, and l'H8pital shortly afterwards communicated this solution as his own

Families o f Curves in the 1690s

28

t o L e i b n i z . The example a l s o found i t s way i n t o l ' H 8 p i t a l ' s textbook AnaZyse

des infiniment p e t i t s . .

, of

1 6 9 6 ' . This example i s e s p e c i a l l y i n t e r e s t i n g s i n c e

n o t long a f t e r w a r d s Jakob B e r n o u l l i showed how i t could be handled more e a s i l y by c l a s s i c a l methods. I s h a l l b r i e f l y i n d i c a t e both Johann and Jakob B e r n o u l l i ' s

s o l u t i o n t o the problem. fig. 5

Consider t h e f a m i l y of p a r a b o l a s ( s e e f i g u r e 5 ) , a l l of which p a s s through t h e o r i g i n C, have v e r t i c e s on t h e semi e l l i p s e ANMC, and have v e r t i c a l a x e s . Let CD=s r e p r e s e n t t h e a b s c i s s a of a p o i n t i t s o r d i n a t e . The e l l i p s e SM=a

-

(2.5)

-

D on t h e e l l i p s e , and DN=t r e p r e s e n t

having semi minor a x i s SA=ka

and semi major a x i s

i s then r e p r e s e n t e d by t h e e q u a t i o n

S(s,t)=s2+4t2-4at=0.

Let CF=x be t h e a b s c i s s a of t h e p o i n t

P on one of t h e p a r a b o l a s and l e t FP-7

be t h e corresponding o r d i n a t e . Then t h e p a r a b o l a w i t h v e r t e x N i s d e f i n e d by the e q u a t i o n

(2.6)

V ( X , ~ S, , t)=~js~-2tx~s+t~~=O.

Here t h e e l l i p s e ANMC a c t s a s o r d i n a t r i x of t h e family of p a r a b o l a s , s i n c e i n each of i t s p o i n t s N t h e p a r a b o l a p a s s i n g through N i s w e l l d e f i n e d . Johann B e r n o u l l i ' s d e t e r m i n a t i o n of t h e s a f e t y p a r a b o l a family of parabolas j u s t d e f i n e d

-

-

b e i n g t h e envelope o f t h e

proceeded a l o n g t h e f o l l o w i n g l i n e s :

D i f f e r e n t i a t e e q u a t i o n ( 2 . 6 ) w i t h r e s p e c t t o s and t, t h e parameters o f t h e f a m i l y of p a r a b o l a s . This y i e l d s : (2.7)

.

Vs (x,Y,s, t)ds+Vt(x,y,s, t ) d t = 2 ~ y d ~ - 2 t ~ d ~ - 2 ~ ~ d t + ~ ~ d t = O

E l i m i n a t i o n of y from e q u a t i o n ( 2 . 7 ) by means of (2.6) y i e l d s t h e i n t e r m e d i a t e result

29

Envelopes

~=2sisdt-tdsi/isdt-Ztd~i.

(2.8)

By combining the e q u a t i o n s ( 2 . 5 ) ,

( 2 . 6 ) and ( 2 . 8 )

some s t r a i g h t f o r w a r d

c a l c u l a t i o n leads t o t h e e q u a t i o n :

4ay-4a2+x2=0

(2.9)

f o r che envelope. Hence t h e s a f e t y

p a r a b o l a h a s v e r t e x i n A and focus i n C'O.

This c a l c u l a t i o n does indeed f o l l o w t h e l i n e s s e t o u t i n L e i b n i z ' s a r t i c l e s of IG92 and 1694, and i t c l e a r l y shows t h a t no need was f e l t t o r e duce t h e e q u a t i o n f o r t h e family o f c u r v e s t o a n e q u a t i o n i n v o l v i n g o n l y one v a r i a b l e parameter. A few y e a r s l a t e r Jakob B e r n o u l l i showed t h a t no highbrow c a l c u l u s w a s

needed t o c a l c u l a t e t h e envelope o f t h e b a l l i s t i c parabolas".

He proved

e x a c t l y t h e same by means of a simple double r o o t argument and w i t h a c a l c u l a t i o n much s h o r t e r than t h e one Johann B e r n o u l l i and l ' H 8 p i t a l had needed. By e l i m i n a t i n g t from ( 2 . 6 ) by means of (2.5) h e found t h e f o l l o w i n g one para-

meter e q u a t i o n f o r t h e b a l l i s t i c p a r a b o l a s : (2. I 0 )

s '-5 i 2 a q + x ) / i x 2 + y ) + i a y x 2 + ~ xI/ 4 (x2+y I =o.

Now c o n s i d e r i n g t h a t t h e envelope i s c h a r a c t e r i s e d by t h e f a c t t h a t i t s e p a r a t e s t h e p o i n t s through which two p a r a b o l a s pass and t h e p o i n t s through which no p a r a b o l a p a s s e s a t a l l , i t i s c l e a r t h a t e q u a t i o n (2.10) must have a double r o o t i n s on t h e envelope. S t r a i g h t f o r w a r d c a l c u l a t i o n of t h i s double r o o t then y i e l d s e q u a t i o n ( 2 . 9 ) .

92.3.5 Conclusion

L e i b n i z ' s new a p p l i c a t i o n of t h e d i f f e r e n t i a l c a l c u l u s was a remarkable achievement, i n t h a t i t demonstrated t h a t t h e c a l c u l u s was a p p l i c a b l e n o t o n l y t o a s i n g l e c u r v e , b u t a l s o t o f a m i l i e s of c u r v e s . It showed t h a t t h e o p e r a t i o n of d i f f e r e n t i a t i o n c o u l d be e m p l o y e d n o t o n l y t o t h e c l a s s i c a l v a r i a b l e s such a s a b s c i s s a and o r d i n a t e p e r t a i n i n g t o a s i n g l e c u r v e , b u t a l s o t o t h e parameter of a family of c u r v e s , o r as L e i b n i z put i t : "It f o l l o w s t h a t one and t h e same e q u a t i o n c a n have d i f f e r e n t d i f f e r e n t i a l

e q u a t i o n s , o r is d i f f e r e n t i a b l e i n a v a r i e t y o f w a y s , j u s t a s t h e scope of the i n v e s t i g a t i o n requires."" However, t h e envelope a r t i c l e s 1692 and 1694 o n l y c o n s t i t u t e an i s o l a t e d e p i s o d e i n t h e development of p a r t i a l d i f f e r e n t i a t i o n . They f a i l e d t o have any

30

1:amiIies of' Curves in the 1690s

s i g n i f i c a n t e f f e c t on t h e c h o i c e of problems, o r more g e n e r a l l y , o n t h e work of t h e l i t e r a t i of t h e c a l c u l u s . Even Leibniz himself h a r d l y r e f e r r e d t o them again. T hus t h e r e w a s no s i g n i f i c a n t f o l l o w up a t a l l . The o n l y l a s t i n g e l e ments i n t h e s e a r t i c l e s were t h e envelope a l g o r i t h m i t s e l f and t h e concept of " i n f i n i t e l y many c u r v e s given by p o s i t i o n i n o r d e r e d sequence". S t r i p p e d of i t s p u r e l y geometrical c h a r a c t e r , t h i s concept became t h e t e c h n i c a l term t o denote a family of c u r v e s r e p r e s e n t e d by t h e same e q u a t i o n . I t was

used a s

such up t i l l t h e t i m e of E u l e r . The reasons f o r t h i s l a c k of i n f l u e n c e of L e i b n i z ' s envelope a r t i c l e s a r e manifold. F i r s t of a l l , t h e g e n e r a l i s a t i o n o f t h e d i f f e r e n t i a l c a l c u l u s t o " d i f f e r e n t i a b l e " q u a n t i t i e s w a s r a t h e r ad hoc, and t h e c r i t e r i o n f o r d i f f e r e n t i a b i l i t y r a t h e r obscure and l i m i t e d ; e . g .

it

i s n o t p o s s i b l e t o employ t h i s c r i t e r i o n t o argue t h a t a l o n g a t r a j e c t o r y i n a f a m i l y of curves t h e parameter a a l s o c o n s t i t u t e s a d i f f e r e n t i a b l e q u a n t i t y . I n f a c t , t h e d i s t i n c t i o n between dependent and independent v a r i a b l e s i s s t i l l e n t i r e l y missing f r o m t h e s e envelope a r t i c l e s , and t h e e x t e n s i o n of t h e d i f f e r e n t i a l o p e r a t o r t o " d i f f e r e n t i a b l e " q u a n t i t i e s i s s t i l l a f a r c r y from t h e simultaneous u5e of d i f f e r e n t i a t i o n w i t h r e s p e c t t o two independent v a r i a b l e s . Furthermore, the envelope problem d i d n o t c o n s t i t u t e a r e a l c h a l l e n g e f o r L e i b n i z ' s contemporaries. A s Jakob B e r n o u l l i showed, a l l int e r e s t i n g examples of envelopes could a l s o be t r e a t e d w i t h c l a s s i c a l double r o o t arguments. Thus t h e a p p l i c a t i o n s were n o t v e r y promising and d i d n o t encourage

further study.

52.2 The b r a c h y s t o c h r o n e and i t s aftermath 5 2 . 2 . 1 The probZem F a m i l i e s of curves became t h e focus of a t t e n t i o n a few y e a r s l a t e r when Johann B e r n o u l l i i n 1696 posed t h e famous brachystochrone

problem t o h i s

f e 1low ma t h e m t i c i a n s : "Let two p o i n t s fl and B be given i n a v e r t i c a l plane. Determine t h e p a t h AMB of a moving body M, along which, descending by i t s own g r a v i t y , and s t a r t i n g t o move a t p o i n t A , i t a r r i v e s a t t h e o t h e r p o i n t B i n t h e s h o r t e s t t i m e " 1 3 (cf. figure 6). Posed as a c h a l l e n g e problem t h e brachystochrone problem a t t r a c t e d f u l l a t t e n t i o n of a l l l e a d i n g mathematicians of t h e 1690s: L e i b n i z , Newton, l ' t l ^ o p i t a l , Tschirnhaus, Jakob B e r n o u l l i . All of them, e x c e p t 1'H^opital, s o l v e d

The bruchystochrone und its aftermath

31

t h e problem s u c c e s s f u l l y and showed t h a t t h e b r a c h y s t o c h r o n e

c o n n e c t i n g A and

B i s a c y c l o i d having i t s o r i g i n i n A and p a s s i n g through B14. Both Johann and

Jakob B e r n o u l l i extended t h e i r i n v e s t i g a t i o n s t o t h e f a m i l y of a l l c y c l o i d s w i t h o r i g i n i n A , and they both concluded t h e i r a r t i c l e s by posing new q u e s t i o n s about t h i s family of c y c l o i d s . These q u e s t i o n s t r i g g e r e d o f f an exp l o s i o n of i n v e s t i g a t i o n s about f a m i l i e s of c u r v e s . Becoming involved i n an i n c r e a s i n g l y h a r s h c o n t r o v e r s y , Johann and Jakob B e r n o u l l i were c o n s t a n t l y i n s e a r c h of d i f f i c u l t problems with which they c o u l d c h a l l e n g e each o t h e r p u b l i c l y . F a m i l i e s of curves provided such problems, and t h u s became f i r m l y e s t a b l i s h e d a s a demanding and d i f f i c u l t f i e l d of mathematical a c t i v i t y . Only a few months a f t e r t h e p u b l i c a t i o n of t h e i r s o l u t i o n s t o t h e brachystochrone problem i n 1697,

b o t h Johann B e r n o u l l i and Jakob B e r n o u l l i became aware of

the need t o develop new techniques i n o r d e r t o d e a l w i t h c e r t a i n f a m i l i e s of t r a n s c e n d e n t a l c u r v e s . The e s s e n t i a l new element i n t r o d u c e d t o overcome t h e s e d i f f i c u l t i e s was t h e theorem of i n t e r c h a n g e a b i l i t y of d i f f e r e n t i a t i o n and int e g r a t i o n . This theorem was found and communicated t o Johann B e r n o u l l i by L e i b n i z i n August 1697 and d i s c o v e r e d o r a t l e a s t used i m p l i c i t l y by Jakob Bern o u l l i b e f o r e the end of 1698. S e c t i o n 9 2 . 2 i s devoted t o t h e s e problems which stemmed from t h e brachystochrone problem and which l e d t o t h e r e c o g n i t i o n of t h e d e f e c t s i n t h e d i f f e r e n t i a l c a l c u l u s and t o t h e break-through

provided by

t h e i n t e r c h a n g e a b i l i t y theorem of d i f f e r e n t i a t i o n and i n t e g r a t i o n .

92.2.2 The brachystochrone and t h e synchrone

The c h a i n o f problems t h a t l e d t o t h e d i s c o v e r y of t h e i n t e r c h a n g e a b i l i t y theorem was t o a l a r g e e x t e n t

determined by Johann B e r n o u l l i ' s

construction

of t h e brachystochrones and h i s subsequent d i s c o v e r y of t h e o r t h o g o n a l t r a j e c t o r i e s of t h e brachystochrone c y c l o i d s . T h e r e f o r e I s h a l l f i r s t p a r a p h r a s e Johann B e r n o u l l i ' s i d e a s h e r e , s t a r t i n g w i t h h i s brachystochrone c o n s t r u c t i o n (see f i g u r e 6 ) . Consider t h e v e r t i c a l p l a n e

a s a medium b u i l t up of h o r i z o n t a l

l a y e r s of e q u a l d e n s i t y , b u t w i t h v a r y i n g d e n s i t y a l o n g t h e v e r t i c a l a x i s . Then a l i g h t r a y , e m i t t e d from A and a r r i v i n g i n B w i l l be propagated a l o n g a curved p a t h , which

-

according t o Fermat's p r i n c i p l e

-

i s t h e quickest path

f o r t h e given d e n s i t y d i s t r i b u t i o n . According t o t h e law of r e f r a c t i o n , t h e v e l o c i t y z1 of t h e l i g h t r a y w i l l v a r y a c c o r d i n g t o t h e r e l a t i o n : (2.11)

s i n ct = constant V

Families of Curves in the 1690s

32

fig. 6

where a i s t h e a n g l e between t h e l i g h t r a y and t h e v e r t i c a l a t p o i n t C ( s e e f i g . 6 ) . S i n c e , according t o Huygens's t h e o r y of l i g h t , t h e v e l o c i t y U i s inv e r s e l y p r o p o r t i o n a l t o t h e d e n s i t y of t h e medium t r a v e r s e d , one can imagine a d e n s i t y d i s t r i b u t i o n such t h a t t h e v e l o c i t y U of a l i g h t ray i n every p o i n t

C

i s p r o p o r t i o n a l t o t h e v e l o c i t y t h a t a heavy body a c q u i r e s i n f a l l i n g under

t h e i n f l u e n c e of i t s own g r a v i t y through t h e corresponding a l t i t u d e AD. I n t h i s c a s e , t h e p a t h of t h e l i g h t ' r a y - b e i n g t h e q u i c k e s t p a t h a c c o r d i n g t o Fermat's p r i n c i p l e - i s c o i n c i d e n t w i t h t h e brachystochrone.

Since, according

t o Galilee's law, t h e v e l o c i t y of a heavy p a r t i c l e f a l l i n g through t h e a l t i t u d e J:

is proportional t o

s t i t u t i o n of s i n a =

&,

%

i n s e r t i o n of t h i s v a l u e f o r

2,

i n ( 2 . 1 1 ) and sub-

y i e l d s t h e following d i f f e r e n t i a l equation f o r the

l i g h t r a y s and b r a c h y s t o c h r o n e s : (2.12)

dy/ &ds = constant.

P u t t i n g t h e c o n s t a n t e q u a l t o l / & one a r r i v e s a t : ( 2 . 13)

d

y

=

~

&

.

Johann B e r n o u l l i immediately r e c o g n i s e d t h i s d i f f e r e n t i a l e q u a t i o n a s b e i n g t h e one t h a t p e r t a i n s t o c y c l o i d s w i t h a g e n e r a t i n g c i r c l e having d i a m e t e r a and r o l l i n g along t h e h o r i z o n t a l y-axis.

By way of c o r o l l a r y t h i s brand of o p t i c a l

and mechanical models f o r t h e brachys tochrone produced a n o t h e r important r e s u l t . I n t h e family of a l l brachystochrone c y c l o i d s , o r i g i n a t i n g i n A , Johann B e r n o u l l i c o n s i d e r e d t h e curves formed by simultaneous p o s i t i o n s of heavy p a r t i c l e s , which a r e r e l e a s e d a t A a t t h e same i n s t a n t . These c u r v e s h e termed t h e "synchrones" ( s e e f i g u r e 7 ) . Turning back t o t h e o p t i c a l model i t i s c l e a r t h a t t h e synchrones c o i n c i d e with t h e curves formed by simultaneous p o s i t i o n s of t h e l i g h t p u l s e s , e m i t t e d from A a t t h e same i n s t a n t . Hence, t h e synchrones c o i n c i d e w i t h t h e wave f r o n t s . Since, a c c o r d i n g t o Huygens's t h e o r y of l i g h t , wave f r o n t s and l i g h t

33

The brachystochrone and its aftermath

fig. 7

rays are everywhere perpendicular the synchrones are the orthogonal trajectories of the brachystochrone cycloids. Without any further proof Johann Bernoulli

produced the following construction for the synchrones (at the end of his 169 7a):

fig. 8

E a

B

Families of Curves in the 1690s

34

Consider t h e synchrone DCT which meets t h e v e r t i c a l a x i s a t t h e p o i n t D w i t h a b s c i s s a xo. B e r n o u l l i showed how t o c o n s t r u c t t h e p o i n t o f i n t e r s e c t i o n C of t h i s synchrone DCT w i t h t h e c y c l o i d ACB, g e n e r a t e d by t h e c i r c l e ESB which has diameter a. On t h i s c i r c l e t a k e S such t h a t t h e c i r c u l a r a r c mean p r o p o r t i o n a l between a and

x

Z=c(i s

the

: hence a:a=cI:x , Then t h e h o r i z o n t a l l i n e

through S meets t h e c y c l o i d ACB i n i t s p o i n t of i n t e r s e c t i o n C w i t h t h e synchrone. I s h a l l suggest

h e r e how Johann B e r n o u l l i probably found t h i s c o n s t r u c t i o n .

Let 2' be the time a heavy p a r t i c l e t a k e s t o f a l l through t h e a l t i t u d e

A t o D. Taking t h e v e l o c i t y - l a w

x0 from

where c i s a c o n s t a n t , T can be e x p r e s s e d

7-)=G,

as follows:

T=j" $ =joO %=2 2

(2.14)

0

Along a c y c l o i d w i t h g e n e r a t i n g c i r c l e of diameter a, a p a r t i c l e i n t h i s amount of time f a l l s over a v e r t i c a l d i s t a n c e x', which s a t i s f i e s :

JxirL

( h e r e ds i s taken from t h e eq. ( 2 . 13)). Thus x' i s determined by 2 J ( x

/c$=L JF

0

o r , e q u i v a l e n t l y , by

v

ax-22

d;C

Now Johann B e r n o u l l i must have r e c o g n i s e d t h e i n t e g r a l i n t h e right-hand s i d e

of (2.16) as t h e a r c over which t h e g e n e r a t i n g c i r c l e w i t h diameter a h a s moved t o produce a p o i n t on t h e c y c l o i d w i t h a b s c i s s a 2'. Thus e q . (2.16) i m p l i e s t h a t t h i s a r c i s t h e mean p r o p o r t i o n a l between t h e diameter a of t h e g e n e r a t i n g c i r c l e and t h e c o n s t a n t xo which p a r a m e t r i s e s

t h e synchrone. This i s e x a c t l y

what Johann B e r n o u l l i ' s c o n s t r u c t i o n of t h e synchrone i s up t o 1 6 . This c o n s t r u c t i o n i n f a c t i s h i g h l y complicated. I n o r d e r t o f i n d one p o i n t of t h e synchrone i t i s n e c e s s a r y t o suppose t h a t t h e r e c t i f i c a t i o n of a c i r c l e w i t h diameter a i s given, and t h a t t h e c y c l o i d g e n e r a t e d by t h i s c i r c l e has been drawn a l r e a d y . Johann B e r n o u l l i ' s c o n s t r u c t i o n of t h e synchrone i s an example of a c o n s t r u c t i o n "by r e c t i f i c a t i o n of curves",

t h a t i s , f o r each p o i n t

on t h e synchrone t h e r e c t i f i c a t i o n of a n o t h e r c i r c l e m u s t b e g i v e n . Hence such a c o n s t r u c t i o n presupposes a n i n f i n i t y of r e c t i f i c a t i o n s . This c o n s t r u c t i o n was n o t c o n s i d e r e d t o be very e l e g a n t , and it was r a t e d worse than a c o n s t r u c t i o n of a curve which presupposed only one r e c t i f i c a t i o n , a s o - c a l l e d

35

The brachystochrone and its aftermath c o n s t r u c t i o n "by r e c t i f i c a t i o n of a curve". A s we s h a l l s e e i n 1 2 . 2 . 5 ,

a

c o n s t r u c t i o n of a curve i n v o l v i n g i n f i n i t e l y many r e c t i f i c a t i o n s was regarded as i n s u f f i c i e n t f o r c o n s t r u c t i n g t h e t a n g e n t s t o such a c u r v e .

5 2.2.3 Genealogy o f problems derived from t h e brachys tochrone

Three d i f f e r e n t types o f problems about f a m i l i e s of c u r v e s can be ident i f i e d i n t h e c a t ' s c r a d l e of problems occasioned by t h e brachystochrone problem: 1 ) A t t h e end of h i s brachystochrone a r t i c l e 7697a Johann B e r n o u l l i had posed

the o r t h o g o n a l t r a j e c t o r y problem f o r f a m i l i e s of t r a n s c e n d e n t a l c u r v e s . This problem was d e f i n i t e l y i n s p i r e d by h i s f i n d i n g t h e synchrone, which i s t h e o r t h o g o n a l t r a j e c t o r y of t h e c y c l o i d s which s o l v e t h e brachystochrone problem. I s h a l l d i s c u s s t h e o r t h o g o n a l t r a j e c t o r y problem i n d e t a i l i n c h a p t e r 3 .

2 ) Jakob B e r n o u l l i ' s s o l u t i o n of t h e brachystochrone problem"

contained the

r o o t s of t h e c a l c u l u s of v a r i a t i o n s : i t w a s founded on t h e g e n e r a l p r i n c i p l e t h a t a curve which minimizes a c e r t a i n q u a n t i t y a s a whole a l s o minimizes t h a t q u a n t i t y i n i t s i n f i n i t e l y small p a r t s . Jakob B e r n o u l l i w a s c e r t a i n l y aware of t h e s u p e r i o r i t y of h i s method o v e r h i s b r o t h e r ' s ad hoc method d e r i v e d from Huygens's theory of l i g h t and Fermat's p r i n c i p l e . Thus Jakob B e r n o u l l i s e t some i s o p e r i m e t r i c a l problems where h e knew h i s b r o t h e r t o be v u l n e r a b l e , and where he d i d n o t r e f r a i n from e x p l o i t i n g h i s own s u p e r i o r i t y t o t h e g r e a t e s t p o s s i b l e e x t e n t . Johann and Jakob B e r n o u l l i ' s d i s c u s s i o n about t h e i s o p e r i m e t r i c a l problems q u i c k l y developed i n t o a v e r y h a r s h p u b l i c c o n t r o v e r s y . This c o n t r o v e r s y , and t h e o r i g i n s of v a r i a t i o n a l c a l c u l u s , have a t t r a c t e d a c o n s i d e r a b l e amount of i n t e r e s t from h i s t o r i a n s " .

I s h a l l n o t dwell on t h i s t o p i c h e r e , f o r i t had

no b e a r i n g on t h e development of p a r t i a l d i f f e r e n t i a t i o n . 3 ) I t i s a t h i r d group of problems, a l s o o r i g i n a t i n g from t h e b r a c h y s t o c h r o n e ,

t h a t gave r i s e t o t h e development of p a r t i a l d i f f e r e n t i a t i o n . These problems were motivated by Jakob B e r n o u l l i ' s "problem o f q u i c k e s t approach" ("problema de c e l e r r i m o appulsu") : "On which of t h e i n f i n i t e l y many c y c l o i d s ( o r c i r c l e s , p a r a b o l a s o r o t h e r c u r v e s ) p a s s i n g through A with t h e same b a s e

AH can a heavy body f a l l from

A t o a given v e r t i c a l l i n e ZK i n t h e s h o r t e s t t i m e ? " ' q (See f i g u r e 9 ) . By means of h i s synchrone Johann B e r n o u l l i could easily

s o l v e t h e problem of q u i c k e s t approach f o r t h e c y c l o i d s . The p o i n t of

q u i c k e s t c o n t a c t on ZK i s o b v i o u s l y t h e p o i n t where a synchrone touches ZK,

Families of Curves in the 1690s

36

A fig. 9

and t h u s because of t h e o r t h o g o n a l i t y of c y c l o i d s and synchrones i t i s t h e p o i n t where t h e l i n e ZK i s i n t e r s e c t e d o r t h o g o n a l l y by a c y c l o i d . Thus t h e req u i r e d c y c l o i d h a s a g e n e r a t i n g c i r c l e with a p e r i m e t e r which i s double t h e

d i s t a n c e AZ."

B u t t h i s of c o u r s e w a s n o t t h e f u l l s o l u t i o n t o t h e problem of

q u i c k e s t approach: t h e problem had a l s o been s e t f o r c i r c l e s , p a r a b o l a s and o t h e r f a m i l i e s of c u r v e s . For o t h e r f a m i l i e s of c u r v e s t h e r e s o l u t i o n of t h e problem i s much h a r d e r . The p o i n t of q u i c k e s t c o n t a c t i s s t i l l t h e p o i n t where a synchrone touches t h e l i n e ZK, b u t i t i s no l o n g e r t h e p o i n t where ZK i s i n t e r s e c t e d o r t h o g o n a l l y by one of t h e curves of t h e f a m i l y . Thus i n a l l o t h e r f a m i l i e s of curves t h e synchrone BB'B" i s only known through i t s d e f i n i t i o n t h a t t h e f a l l i n g times a l o n g each of t h e c u r v e s A B , A B ' ,

AB" are equal:

The method t h a t Johann B e r n o u l l i adopted f o r s o l v i n g t h e problem of q u i c k e s t approach i n t h e s e c a s e s was t o f o r m u l a t e t h e problem a s a t a n g e n t problem f o r t h e synchrones: determine on each of t h e synchrones t h e p o i n t C where t h e t a n g e n t i s p a r a l l e l t o t h e given l i n e ZK. Then t h e p o i n t of i n t e r s e c t i o n o f ZK with t h e curve connecting a l l t h e s e p o i n t s C i s t h e r e q u i r e d p o i n t of q u i c k e s t c o n t a c t . This t a n g e n t problem f o r t h e synchrones, however, i s only a s p e c i f i c i n s t a n c e of a more g e n e r a l t a n g e n t problem, namely: given a curve B B ' B " which i n t e r s e c t s a l l c u r v e s of a given family AB,AB',AB"

e t c . a c c o r d i n g t o a given

law; f i n d i t s t a n g e n t i n any p o i n t B . Johann B e r n o u l l i must have r e c o g n i s e d a s much, s i n c e i n the Journai! des Savans of August 1697 he s e t t h i s t a n g e n t problem f o r (a) equaZ area t r a j e c t o r i e s ( i . e . c u r v e s BB'B" d e f i n e d by areaABD=

=areaAB'D'; s e e f i g u r e 10) i n a f a m i l y of e l l i p s e s over t h e same axis, and f o r (b) equaZ ares trajectom'es

( i . e . c u r v e s BB'B" d e f i n e d by arc AE=arc A % ' )

31

The bractiystochrone and its aftermath

i n any f a m i l y of (what he c a l l e d ) "curves of t h e same s o r t " .

f i g . 10

Johann B e r n o u l l i d i d n o t c l a r i f y t h e meaning of t h e term "curves of t h e same s o r t " when he p u b l i s h e d t h e s e problems; i n f a c t

-

by mentioning a f a m i l y of

s i m i l a r p a r a b o l a s a s s p e c i f i c example - he s u g g e s t e d t h a t

h e had f a m i l i e s of

s i m i l a r curves i n mind. Half a y e a r l a t e r , i n h i s correspondence w i t h l'H8pit a l , Johann B e r n o u l l i was f o r c e d t o r e v e a l t h a t t h i s was n o t t h e c a s e and t h a t t h e term had a c t u a l l y been used t o denote any f a m i l y of c u r v e s one could imagine. I s h a l l d i s c u s s t h i s p o i n t and t h e d e l i b e r a t e vagueness of Johann B e r n o u l l i i n 52.2.9.

52.2.4

S o l u t i o n s f o r similar c o v e s

I n two l e t t e r s , b o t h d a t e d 15/10/1697, Johann B e r n o u l l i communicated t o l ' H 8 p i t a l h i s s o l u t i o n of t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s i n any f a m i l y of s i m i l a r c u r v e s , and s u p p l i e d Varignon w i t h t h e s o l u t i o n of t h e problem of q u i c k e s t approach i n a family of s i m i l a r c u r v e s z 1 . I n b o t h l e t t e r s , t h e l a t t e r of which was intended f o r p u b l i c a t i o n , Johann B e r n o u l l i d i d n o t rev e a l t h e a n a l y s i s which had l e d him t o h i s s o l u t i o n . However, h e had a l r e a d y communicated t o L e i b n i z i n August 1697 t h e a n a l y s i s p e r t a i n i n g t o t h e problem of q u i c k e s t approach".

The argument g i v e n t h e r e can e a s i l y be adapted t o t h e

equal a r c s t r a j e c t o r i e s and i s used h e r e t o r e c o n s t r u c t t h e s o l u t i o n of t h i s prob lem23 . I s h a l l f i r s t d i s c u s s Johan B e r n o u l l i ' s s o l u t i o n of t h e problem of q u i c k e s t

approach f o r s i m i l a r c u r v e s , a s p r e s e n t e d i n h i s l e t t e r t o L e i b n i z z 2

(see

f i g u r e 11). Two curves ABB' and ACC'

a r e called "similar with respect t o the

p o l e A" when f o r any two t r i p l e s of c o l l i n e a r p o i n t s A,B,C

and A , B ' , C ' ,

with

38

Families of Curves in the 1690s

A

fig. 1 1

B,B'

on the f i r s t c u r v e and C,C' on t h e o t h e r curve, t h e f o l l o w i n g p r o p o r t i o n -

a l i t y h o l d s : AB:AC=AB':AC'.

Johann B e r n o u l l i and h i s contemporaries c o n s i d e r e d

a family of s i m i l a r curves

o be b u i l t up from one f i x e d c u r v e , f o r example

A B B ' , c a l l e d t h e " p r i n c i p a l s " , m u l t i p l i c a t i o n of which w i t h r e s p e c t t o t h e p o l e A by d i f f e r e n t f a c t o r s y i e l d s t h e o t h e r c u r v e s i n t h e family. Due t o t h e f a c t t h a t t h e f a l l i n g times

tA2,tAzof

heavy p a r t i c l e s , r e l e a s e d a t A w i t h

zero i n i t i a l v e l o c i t y and f a l l i n g a l o n g similar a r c s A% and A?, (2.18)

a r e r e l a t e d by

-=m:m=m:a,

tAg.tAC

t h e synchrones of a family of s i m i l a r curves a g a i n form a f a m i l y of s i m i l a r curves themselvesz4.

f i g . 12

Now l e t t h e curve ABN ( s e e f i g u r e 12) be t h e p r i n c i p a l i s of t h e g i v e n family of similar c u r v e s . Johann B e r n o u l l i then s e t o u t t o f i n d a method f o r c o n s t r u c t i n g t h e t a n g e n t s t o t h e synchrones i n t h e i r p o i n t s of i n t e r s e c t i o n w i t h t h i s p r i n c i p a l i s ABN. L e t AB'P be a n o t h e r curve from t h e given f a m i l y , in-

f i n i t e l y c l o s e t o t h e p r i n c i p a l i s ABN, and t a k e t h e p o i n t

P t o be c o l l i n e a r

w i t h A and B. Considering t h e synchrones BB' and PN through B and P

39

The brachystochrone and its aftermath

r e s p e c t i v e l y , t h e t a n g e n t t o t h e synchrone i n B i s o b v i o u s l y p a r a l l e l t o t h e 1

arcsegment PN. The t r i a n g l e ABNP a c t s a s t h e c h a r a c t e r i s t i c t r i a n g l e , b e i n g s i m i l a r t o the f i n i t e t r i a n g l e ABRA, where R i s on t h e t a n g e n t t o t h e p r i n c i p a l i s i n B such t h a t (2.19)

BR:BA=B%:BP.

Johann B e r n o u l l i s e t o u t t o f i n d an e x p r e s s i o n f o r t h e l i n e segment BR: S i n c e

-

tAF:tAz=m: JAB and

tB3=tAT-tA2 and (2.20)

tA-p=tA3 one

because

(tAT-tA%) :t A % = ( f i - f i )

:&+(AP-ABI

AP-AB=BP

tB-,=(BP:2AB).tA-j

has :

-=m: t -*t ~ AB ~ . m; thus

. Because

:2AB (with % / +m=2m)

one a r r i v e s a t t B j : t A z = B P : 2 A B ,

or

.

tG c a n be c a l c u l a t e d from: t - = % / J & and tA%can be c a l c u l a t e d BN ds from: At=,-J - I n s e r t i o n of t h e s e v a l u e s i n (2.20) y i e l d s L%/&=(BP/ZAB)?-

Now

A

or

&

.

ds A&?

Combination of (2.21) and (2.19) f i n a l l y y i e l d s :

Thus by means of e q u a t i o n (2.22) BR i s e x p r e s s e d i n terms of q u a n t i t i e s which can be c a l c u l a t e d f o r each p o i n t B of t h e p r i n c i p a l i s . Equation (2.22) makes p o s s i b l e t h e f o l l o w i n g c o n s t r u c t i o n of t h e t a n g e n t t o t h e synchrone i n a p o i n t

B of t h e p r i n c i p a l i s : Given a p o i n t B on t h e p r i n c i p a l i s ; c o n s t r u c t t h e p o i n t R on the t a n g e n t t o t h e p r i n c i p a l i s i n B such t h a t BR=-&? J 2 A q u i r e d t a n g e n t i n B i s p a r a l l e l t o AR.

fig.

13

ds Then t h e r e x'

Families of Curves in the 1690s

40

Carrying o u t t h i s procedure f o r a l l p o i n t s B , a new curve AOR emerges, which can be used t o s o l v e t h e problem of q u i c k e s t approach ( s e e f i g u r e 13): Draw a l i n e through A p a r a l l e l t o t h e given l i n e ZK, which i n t e r s e c t s AOR i n R . C o n s t r u c t t h e t a n g e n t t o t h e p r i n c i p a l i s through R , which touches t h e p r i n c i p a l i s i n B . Then t h e t a n g e n t t o t h e synchrone i n B i s p a r a l l e l t o A R and thus p a r a l l e l t o ZK. D r a w t h e s t r a i g h t l i n e through A and B which i n t e r s e c t s t h e given l i n e ZK i n a p o i n t K. This p o i n t K i s then t h e r e q u i r e d p o i n t of q u i c k e s t approach. Johann B e r n o u l l i only communicated t h i s f i n a l c o n s t r u c t i o n t o Varignon, who published i t i n t h e Journal des SavansZ1. A s mentioned above, Johann Bern o u l l i on t h e same day communicated a c o n s t r u c t i o n f o r t h e t a n g e n t s t o e q u a l a r c s t r a j e c t o r i e s t o l ' H ^ o p i t s l , a g a i n n o t p r o v i d i n g t h e a n a l y s i s t h a t had produced t h i s c o n s t r u c t i o n . However, t h e argument paraphrased h e r e can e a s i l y be adapted t o y i e l d t h e c o n s t r u c t i o n of t a n g e n t s t o equal a r c s t r a j e c t o r i e s i n a family of s i m i l a r curves a s w e l l : Consider t h e c o n f i g u r a t i o n of curves drawn i n f i g u r e 1 2 , where BB' and NP a r e now taken t o r e p r e s e n t e q u a l a r c s t r a j e c t o r i e s ; hence by d e f i n i t i o n one h a s : a r c A B = a r c A B ' t o BB'

and u r e A N = u r c A P .

The t a n g e n t

i n B i s r e q u i r e d . ABNP i s a g a i n taken as t h e c h a r a c t e r i s t i c t r i a n g l e ,

b e i n g s i m i l a r t o t h e f i n i t e t r i a n g l e ABRA, where R i s on t h e t a n g e n t t o t h e curve AB i n B such t h a t (2.2 3)

BR: BA=BN: BP

--

B R can then be c a l c u l a t e d a s f o l l o w s : Since AP:AB=AP:AB d e f i n i t i o n AP=AN one has

-AN:AB=AP:AB;

of BN=AN-AB

one a r r i v e s a t

-

A

- - - and BP=AP-AB

( 2.2 4

and because by

thus: (A?-G)/A%=(AP-AB)/AB

and because

BT:A%=B P :AB

Comparison of (2.23) and (2.24) y i e l d s : (2.25)

i3R=A%

Thus by means of (2.25) t h e t a n g e n t t o t h e e q u a l a r c s t r a j e c t o r y i n B c a n be c o n s t r u c t e d a s f o l l o w s : on t h e t a n g e n t t o t h e c u r v e A B i n B c o n s t r u c t t h e p o i n t R such t h a t BR=A%.

Then t h e r e q u i r e d t a n g e n t t o t h e equal a r c s t r a j e c t o r y

i n B i s p a r a l l e l t o AR. This c o n s t r u c t i o n c o i n c i d e s e x a c t l y w i t h t h e one communicated t o l ' H 8 p i t a l by Johann B e r n o u l l i i n t h e above mentioned l e t t e r of 15/10/1697, and, t h e r e f o r e , we may assume t h a t he found i t i n t h e way s k e t c h e d above.

41

The brachystochrone and its aftermath

52.2.5 The t a n g e n t probZems for d i s s i m i Z a r curves The problem of q u i c k e s t approach was a l s o d i s c u s s e d i n t h e correspondence between Johann B e r n o u l l i and L e i b n i z , and both men c l e a r l y understood t h e r e d u c t i o n of t h i s problem t o t h e t a n g e n t problem f o r t h e synchrones. Touching upon t h e problem s u p e r f i c i a l l y , L e i b n i z had claimed t h a t by t h i s r e d u c t i o n t h e problem of q u i c k e s t approach could be regarded a s e s s e n t i a l l y s o l v e d , s i n c e a s he s a i d

-

-

t h e synchrones were always known by q u a d r a t u r e s and t h e t a n g e n t s

t o t h e synchrones could t h e r e f o r e always be c o n s t r u c t e d , " a t l e a s t t r a n s c e n d e n t a l l y " ( i . e . by assuming t h a t q u a d r a t u r e s o r r e c t i f i c a t i o n s a r e g i v e n ) . Johann B e r n o u l l i took s t r o n g e x c e p t i o n t o L e i b n i z ' s judgement, and e x p r e s s e d h i s doubts a s t o whether t h e c o n s t r u c t i o n of t h e synchrones by means of int e g r a l s a l o n g d i f f e r e n t curves would indeed a l s o provide a t a n g e n t c o n s t r u c t i o n f o r t h e s e synchrones: "I can e a s i l y b e l i e v e t h a t a r a s h c o n s i d e r a t i o n d u r i n g w r i t i n g s u g g e s t s t o

you t h a t t h e synchrones can always be c o n s t r u c t e d by means of q u a d r a t u r e s : f o r , t h i s i s t h e f i r s t i d e a which p r e s e n t s i t s e l f i n t h e c o n s i d e r a t i o n of t h e s e c u r v e s , namely t h a t f o r a given i n t e r v a l of t i m e one can d e t e r m i n e t h e p o i n t i n a given curve a t which a moving p a r t i c l e a r r i v e s , and t h a t t h i s can be done f o r t h e same i n t e r v a l of time i n any of t h e g i v e n c u r v e s , and t h a t t h u s t h e e n t i r e synchrone can be c o n s t r u c t e d . However, i n i t s e l f a c o n s t r u c t i o n of t h i s kind i s n o t t o be valued much, because i t i s n o t executed by a continuous q u a d r a t u r e of one and t h e same undetermined [curved] l i n e segment, and because, by consequence, from h e r e no method r e s u l t s t o draw t a n g e n t s t o t h e synchrone, a s i s a b s o l u t e l y n e c e s s a r y . So I ask you t o i n v e s t i g a t e t h i s m a t t e r a l i t t l e more thoroughly; perhaps you

w i l l withdraw your words: " t a k e one of t h e synchrones, and draw a t a n g e n t t o i t p a r a l l e l t o t h e given s t r a i g h t l i n e , a s can always be done, a t l e a s t transcendentally". I do n o t y e t s e e how, e i t h e r t r a n s c e n d e n t a l l y o r a l g e b r a i c a l l y , t h e t a n g e n t can be drawn by means of t h i s c o n s t r u c t i o n through q u a d r a t u r e s of d i f f e r e n t [ c u r v e d ] l i n e segments. I r e a l l y t h i n k t h a t t h e main a r t i f i c e i s t o reduce t h e s e q u a d r a t u r e s t o one undetermined q u a d r a t u r e of one continuous [ c u r v e d ] l i n e segment, a s I have been s o f o r t u n a t e t o achieve"'

'.

Johann B e r n o u l l i had indeed a p o i n t when h e r a i s e d t h e m a t t e r of whether a c o n s t r u c t i o n of a curve by means of q u a d r a t u r e s of curves would a l s o be s u f f i c i e n t t o a l l o w d e t e r m i n a t i o n of t h e t a n g e n t t o t h i s c u r v e . H i w own s o l u t i o n of t h e t a n g e n t problem f o r t h e synchrones ( a s w e l l a s f o r t h e e q u a l

Families of Curves in the 1690s

42

a r c s t r a j e c t o r i e s ) i n a family of s i m i l a r c u r v e s drew h e a v i l y upon t h e f a c t t h a t i n t h i s c a s e t h e c o n s t r u c t i o n by means of q u a d r a t u r e s of curves c o u l d be reduced t o a c o n s t r u c t i o n r e q u i r i n g t h e q u a d r a t u r e of one s i n g l e curve o n l y . The time i n t e g r a l a l o n g any arc AB’ could always be reduced t o t h e t i m e i n t e g r a l a l o n g an a r c AB, s i m i l a r t o A B ‘ and taken on t h e p r i n c i p a l i s :

Once t h e i n t e g r a l JB @ was assumed t o be known i n a l l p o i n t s B of one s i n g l e A ” c u r v e , t h e p r i n c i p a l i s , a l l synchrones could be c o n s t r u c t e d . Furthermore t h e r e l a t i o n ( 2 . 2 6 ) made i t f a i r l y e a s y t o c o n s t r u c t t h e t a n g e n t s t o t h e synchrone. The same s i t u a t i o n a r o s e i n t h e c a s e of equal a r c s t r a j e c t o r i e s , where t h e relation

allowed f o r t h e r e d u c t i o n of t h e c o n s t r u c t i o n by r e c t i f i c a t i o n s of c u r v e s t o a c o n s t r u c t i o n r e q u i r i n g only t h e r e c t i f i c a t i o n of t h e p r i n c i p a l i s . Formulated a n a l y t i c a l l y , c o n s t r u c t i o n s by means of q u a d r a t u r e s of c u r v e s involve i n t e g r a l s of t h e type J

X

p ( x , a ) d x , whereas c o n s t r u c t i o n s by means of a X

ZO

q u a d r a t u r e of a s i n g l e curve only i n v o l v e a n i n t e g r a l of t h e type J D i f f e r e n t i a t i o n of t h e s e i n t e g r a l s d e t e r m i n a t i o n of t a n g e n t s

-

d i f f i c u l t i e s i n the c a s e of

-

p(z)dcc.

XO

as i s obviously necessary f o r t h e

i s simple i n t h e l a t t e r c a s e , b u t i n v o l v e s J

X

p(x,a)dcc, s i n c e t h e i n t e g r a l occurs i n a

20

s i t u a t i o n where b o t h i t s upperbound x and t h e parameter a i n t h e i n t e g r a n d a r e v a r i a b l e . Hence t h e d i f f i c u l t i e s which Johann B e r n o u l l i met i n c o n s t r u c t i o n s by “ q u a d r a t u r e s of c u r v e s ” r e l a t e t o t h e a n a l y t i c a l problem o f d i f f e r e n t i a t i n g an i n t e g r a l with r e s p e c t t o a parameter o c c u r r i n g under t h e i n t e g r a l s i g n . Johann B e r n o u l l i i l l u s t r a t e d h i s c r i t i c i s m of L e i b n i z ’ s remarks by s e t t i n g t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s i n a f a m i l y of e l l i p s e s having, a common h o r i z o n t a l a x i s and v a r i a b l e v e r t i c a l a x e s . Such e l l i p s e s a r e no longer s i m i l a r , and no such r e d u c t i o n a s (2.27) of t h e i n t e g r a l s a l o n g d i f f e r e n t e l l i p s e s t o an i n t e g r a l a l o n g one f i x e d e l l i p s e i s p o s s i b l e h e r e . “So f a r I have been unable t o f i n d any way t o a r r i v e a t t h e t a n g e n t s “ wrote

Johan B e r n o u l l i , “ I f you could show m e one g r a t e f u l n e s s t o you w i l l be f a r from

-

however t r a n s c e n d e n t a l

-

my

43

The brachystochrone and its aftermath 52.2.6 L e i b n i z ' s construction

Only a few days l a t e r L e i b n i z provided t h e c o n s t r u c t i o n Johann B e r n o u l l i had r e q u i r e d ; a t t h e beginning of h i s l e t t e r he gave some n i c e i n f o r m a t i o n on t h e whereabouts of h i s d i s c o v e r y : "You w i l l have r e c e i v e d my most r e c e n t l e t t e r . I n t h e meanwhile I have v i s i t e d t h e Monarch o f the Russians and h i s Delegation i n t h e v i c i n i t y . [ .

..I

On my way back, m e d i t a t i n g d u r i n g t h e t r i p a s i s my h a b i t , I have found t h e g e n e r a l method you asked f o r . " 2 7 What t h e n d i d Leibniz i n v e n t d u r i n g h i s c o a c h - t r i p back t o Hannover? I n s t e a d of t h e e l l i p s e s proposed by Johann B e r n o u l l i , L e i b n i z c o n s i d e r e d a n o t h e r f a m i l y of d i s s i m i l a r c u r v e s , namely l o g a r i t h m i c c u r v e s ABF, A B ' F ' ,

a l l p a s s i n g through

one p o i n t A and a l l h a v i n g t h e same asymptote; t h e s e l o g a r i t h m s are d e s c r i b e d by t h e e q u a t i o n

S,

3:

(2.28)

yix,~l=a

$

- = a.logx

.

D I

Y

For t h e s e l o g a r i t h m i c c u r v e s , t h e a r c l e n g t h d i f f e r e n t i a l i s d s-

x

t h e equal a r c s t r a j e c t o r y BB'B"

(2.29)

Jx$

s(x,a)=

dx x

/-2-2 3:

+a , and

i s t h e r e f o r e determined by t h e c o n d i t i o n

@G2= constant.

Considering two i n f i n i t e l y c l o s e c u r v e s ABI: and AB'F'

L e i b n i z took t h e in-

f i n i t e l y small t r i a n g l e ABQB' a s t h e c h a r a c t e r i s t i c t r i a n g l e . This c h a r a c t e r i s t i c t r i a n g l e i s s i m i l a r t o t h e f i n i t e t r i a n g l e ABDE, where E i s t h e p o i n t of i n t e r s e c t i o n of t h e r e q u i r e d t a n g e n t through B and t h e l i n e DE, p a r a l l e l t o t h e t a n g e n t t o t h e given curve ABC i n B; Thus: ABQB'

%

ABDE, and consequently:

Families of Curves in the 1690s

44

Once t h e r a t i o B&:&B'

i s known, t h e p o i n t E can be c o n s t r u c t e d and thus pro-

v i d e s t h e r e q u i r e d t a n g e n t BE. Both t h e c a l c u l a t i o n of BQ and

6'i n v o l v e

d i f f e r e n t i a t i o n with r e s p e c t

t o the parameter a ; BQ can be expressed a n a l y t i c a l l y t h u s : ( 2 . 3 1)

and

BQ=d$ (zoJa) ,

Q2',being

--

e q u a l t o AB-A& because A%'=k%,

can be expressed t h u s :

Now t h e c a l c u l a t i o n of BQ is f a i r l y easy i n t h e c a s e of t h e s e l o g a r i t h m i c c u r v e s and can be c a r r i e d o u t by s t r a i g h t f o r w a r d d i f f e r e n t i a t i o n of y(zo,u)=a.%ogx (2.33)

w i t h r e s p e c t t o a, t o y i e l d

BQ=da,Zogzo.

Q?' i n v o l v e s more d i f f i c u l t i e s , s i n c e s ( z ,a) does n o t have a form t h a t a l l o w s s t r a i g h t f o r w a r d d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e parameter a . It was a t t h i s p o i n t t h a t Leibniz faced t h e need t o c l a r i f y t h e meaning of d

U

/p(z,a)&,

and t h a t he d i s c o v e r e d the i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and integration :

Applying t h i s theorem i n t h e c a s e of ( 2 . 3 2 ) , L e i b n i z found:

By means of ( 2 . 3 3 ) and ( 2 . 3 5 ) t h e r a t i o BQ:@'

could be c o n s t r u c t e d , t h u s pro-

v i d i n g t h e c o n s t r u c t i o n of t h e t a n g e n t t o t h e e q u a l a r c s t r a j e c t o r y BB'B"

in

t h e p o i n t B.

82.2.7 Interchangeability of differentiation and integration X

q - 2 - 2 x +a L e i b n i z solved t h e problem of i n t e r p r e t i n g t h e e x p r e s s i o n d J 0 a 1 2 by going r i g h t back t o t h e v e r y b a s i c i d e a s of t h e c a l c u l u s , a c c o r d i n g t o which

t h e d i f f e r e n t i a l i s t h e e x t r a p o l a t i o n of " d i f f e r e n c e " and t h e i n t e g r a l i s t h e e x t r a p o l a t i o n of "sum" from t h e realm of f i n i t e q u a n t i t i e s t o t h e realm o f t h e i n f i n i t e l y s m a l l . I n t h e 1670s i t w a s t h i s k i n d of e x t r a p o l a t i o n from t h e

45

The brachystochrone and its aftermath

theory of sequences, sum- and d i f f e r e n c e - s e q u e n c e s which had provided L e i b n i z w i t h the b a s i c i d e a s and r u l e s f o r t h e calculus''. 3:

expression d

f a 1

d"a22 Leibniz

For t h e e v a l u a t i o n of t h e

came up w i t h a v e r y g e n e r a l i n t e r -

c h a n g e a b i l i t y p r i n c i p l e , which i s obviously t r u e f o r f i n i t e sums and d i f f e r e n c e s : the sum uf the d i f f e r e n c e s of the p a r t s i s equa2 t o the d i f f e r e n c e I

of the swns of t h e p a r t s . Considering t h e a r c s AB and two wholes, b u i l t up as sums of t h e " p a r t s "

6'and

A3

C%'

( s e e f i g u r e 14) as t h e

respectively, t h i s

i n t e r c h a n g e a b i l i t y p r i n c i p l e immediately y i e l d s : -

(2.36)

c

AB-AQ=z(G-cG~J.

E x p r e s s i n g sums and d i f f e r e n c e s a s i n t e g r a l s and d i f i e r e n t i a l s , t h i s e q u a l i t y

(2.36) t r a n s l a t e s i n t o

o r , by e x p r e s s i n g s ( x a ) as t h e sum of t h e p a r t s : 0 '

Although L e i b n i z developed t h i s theorem f o r t h e s p e c i f i c i n s t a n c e of a r c l e n g t h i n t e g r a l s , h e was v e r y w e l l aware of i t s g e n e r a l i t y . T h i s becomes c l e a r f o r i n s t a n c e when he e x p l a i n e d t o Johann B e r n o u l l i how t o c a l c u l a t e d y ( x o , a ) i f a y ( x , a ) i s a l s o given as an i n t e g r a l which i n v o l v e s t h e parameter a under t h e integral sign: " I f DB o r DQ [ - y ( x O J a ) ;- y ( x ,a+da)] had a l s o been g i v e n by some q u a d r a t u r e , 5

where a i s involved under t h e i n t e g r a l - s i g n , one would have had t o proceed i n t h e same manner t o f i n d [ . the integral-sign

.. I

B&[=d$(xo,a) 1 [ .

. . 1;

t h e q u a n t i t y under

should have been d i f f e r e n t i a t e d w i t h r e s p e c t t o a , and

should t h e n have been i n t e g r a t e d a g a i n , b u t w i t h r e s p e c t t o

I do n o t s e e

2.

what could e v e r h i n d e r t h i s p r ~ c e d u r e . " ~ ~ I n a s h o r t memoir (which we w i l l r e f e r t o a s t h e BeiZage3') which L e i b n i z composed f o r p r i v a t e u s e a f t e r having mailed h i s l e t t e r t o Johann B e r n o u l l i , he s t a t e d t h e g e n e r a l i t y of t h e i n t e r c h a n g e a b i l i t y theorem even more c l e a r l y by f o r m u l a t i n g i t i n a n ad hoc n o t a t i o n a s f o l l o w s : (2.39)

d(secund.a)

1 s

d x . x ~ a =d x d ( s e c u n d . a ) x ~ a .

Here L e i b n i z ' s d(secund.a) c o i n c i d e s w i t h my d

a' and

3cTa

i s a symbol r e -

p r e s e n t i n g an a r b i t r a r y ( a l g e b r a i c a l ) e x p r e s s i o n i n t h e l e t t e r s

2

and a .

Some remarks a r e c a l l e d f o r h e r e : According t o L e i b n i z ' s d e r i v a t i o n , t h e

Families of Curves in the 1690s

46

i n t e r c h a n g e a b i l i t y theorem appears t o be v a l i d i r r e s p e c t i v e of t h e type of i n t e g r a n d s . Not u n t i l a c e n t u r y and a h a l f l a t e r , and n o t u n t i l t h e c a l c u l u s had undergone a complete f a c e - l i f t i n t h e 19th c e n t u r y was i t r e c o g n i s e d t h a t t h e v a l i d i t y of t h e i n t e r c h a n g e a b i l i t y theorem i s l i m i t e d t o c e r t a i n w e l l behaved i n t e g r a n d s ( c f . c h a p t e r I ) .

This s i t u a t i o n i n f a c t i s very common i n

17th and 18th c e n t u r y a n a l y s i s , where almost a l l theorems w i l l f a i l t o h o l d i n c e r t a i n s p e c i a l s i t u a t i o n s . P r o o f s were never r i g o r o u s i n modern terms, b u t r a t h e r they were arguments t o render a r e s u l t g e n e r a l l y a c c e p t a b l e . One can now a l s o understand why Johann B e r n o u l l i d i d n o t f i n d t h e i n t e r c h a n g e a b i l i t y theorem h i m s e l f . From t h e very beginning Johann B e r n o u l l i had adopted t h e d e f i n i t i o n of i n t e g r a t i o n a s t h e formal i n v e r s e of d i f f e r e n t i a t i o n . To him an i n t e g r a l J p d z by d e f i n i t i o n was a q u a n t i t y I.’ such t h a t dV=pdx. A l -

though t h i s was a x i o m a t i c a l l y perhaps a more e l e g a n t i n t r o d u c t i o n of t h e conc e p t of ” i n t e g r a l ” , t h i s formal view blocked t h e way back t o i n t e g r a l s as sums, and t h u s blocked t h e a c c e s s t o t h o s e h e u r i s t i c r u l e s and g e n e r a l p r i n c i p l e s stemming from t h e t h e o r y of sumnation of sequences,

5 2 . 2 . 8 Leibniz’s reaction

I n h i s BeiZage L e i b n i z summarised t h e e v e n t s t h a t had l e d t o h i s d i s covery a s follows: “ I n t h e month of J u l y 1697 M r . Johann B e r n o u l l i asked me about a s o l u t i o n t o t h i s problem and s i m i l a r ones [ t h e t a n g e n t problems f o r c u r v e s d e f i n e d p o i n t w i s e by q u a d r a t u r e s ] , which a r e very d i f f i c u l t , and which have up t o now n o t been amenable t o o u r methods. Having c o n s i d e r e d t h e m a t t e r f o r a w h i l e I b e l i e v e t o have found what w e looked f o r . This c e r t a i n l y i s of g r e a t im-

p o r t a n c e , and e l i m i n a t e s a g r e a t d e f e c t i n o u r d i f f e r e n t i a l c a l c u l u s . We have come t o c o n s i d e r such q u e s t i o n s a t t h e o c c a s i o n of t h o s e which M r . Jacob B e r n o u l l i , p r o f e s s o r i n B a s e l , posed t o h i s b r o t h e r Mr. Johann Bern o u l l i , p r o f e s s o r i n Groningen, and which t h e l a t t e r could o n l y s o l v e because they involved curves of t h e same s p e c i e s , t h a t i s t o say s i m i l a r and s i m i l a r l y posed o n e s . ” 3 ’ Leibniz h e r e a s s i g n e d t h e i n t e r c h a n g e a b i l i t y theorem i t s p l a c e a s a n e c e s s a r y supplement t o t h e e x i s t i n g r u l e s of t h e d i f f e r e n t i a l c a l c u l u s ; by means of i t t h e t a n g e n t problem could now a l s o be d e a l t w i t h f o r t h o s e c u r v e s , which a r e d e f i n e d p o i n t w i s e by q u a d r a t u r e s o f given c u r v e s . I n t h i s way, t h e i n t e r c h a n g e a b i l i t y theorem d i d indeed e l i m i n a t e a grave - and unexpected

-

defect

41

The brachystochrone and its aftermath

of t h e d i f f e r e n t i a l c a l c u l u s , s i n c e t h e t a n g e n t problem always had been i t s paradigm problem.Apart from being a supplement, however i m p o r t a n t , t o t h e " d i r e c t method of t a n g e n t s " t h e i n t e r c h a n g e a b i l i t y theorem had o t h e r i m p o r t a n t a s p e c t s i n t h a t i t a l s o provided L e i b n i z with a v i s t a over t h e h i t h e r t o unknown world of multidimensional c a l c u l u s . This becomes c l e a r from a s h o r t remark which L e i b n i z

made about i n t r o d u c i n g m u l t i p l e i n t e g r a t i o n ; t h e s e remarks

can be summarised and e x p l a i n e d as f o l l o w s : The i n t e r c h a n g e a b i l i t y theorem shows t h a t a n i n t e g r a l l i k e of d i f f e r e n t i a l s gral: d

JxO

a.1

X

2

involves a product J d d s= J ~da-~which 1 a x 1 X v 5 +a can be i n t e r p r e t e d as t h e d i f f e r e n t i a l of an i n t e -

da and dx J X &? G2Z2. Hence: d

d s=d

a I

Integration of both

w i t h lower bound a

adadx JX &? &*G2=X O m-.

a 1 s i d e s of t h i s e q u a l i t y w i t h r e s p e c t t o t h e parameter a ,

and upper bound a y i e l d s : 1

Since t h e i r t e g r a l J a l

a

and t h e d i y f e r e n t i a l d

a

c a n c e l each o t h e r , t h e l e f t -

&? & 'Gi - 1". I " l which r e p r e s e n t s t h e a r c l e n g t h d i f f e r e n c e slxo,al)-s(x , ao). 0 hand s i d e of ( 2 . 4 0 ) gan i n f a c t b e viewed a s

fig.

15

Jx0

X

&*+a20'

1

Y

XO

By consequence, t h e double i n t e g r a l

alal

adadx

--in

X-.

t h e right-hand s i d e of

(2.40) c a n b e i n t e r p r e t e d geometricalPy i n terms of a r c l e n g t h d i f f e r e n c e s i n

t h e f a m i l y of l o g a r i t h m i c c u r v e s ( s e e f i g u r e 15): (2.41)

J a l J ; Xo xadadx 22-2

= s l x ,a I-slxo,ao). 0

1

aO

L e i b n i z concluded t h i s e x c u r s i o n w i t h t h e f o l l o w i n g r a t i o n a l e : "thus one a r r i v e s a t double i n t e g r a t i o n s , which were h i t h e r t o unknown;

[...I

t r u l y , s o f a r we were only a b l e t o i n t e g r a t e and d i f f e r e n t i a t e w i t h r e s p e c t t o t h e v a r i a t i o n of a s i n g l e l e t t e r , o r w i t h r e s p e c t t o d i f f e r e n t l e t t e r s v a r y i n g s i m u l t a n e o u s l y everywhere; b u t n o t [were we a b l e t o d i f f e r e n t i a t e

48

Families of Curves in the 1690s

o r i n t e g r a t e ] when d i f f e r e n t [ l e t t e r s ] o c c u r r e d , which a r e p a r t l y v a r i a b l e , partly invariable

." *

Having r e f l e c t e d upon t h e s e consequences of t h e i n t e r c h a n g e a b i l i t y theorem, L e i b n i z wrote a n e x t l e t t e r t o Johann B e r n o u l l i : "You w i l l have r e c e i v e d my l a s t two l e t t e r s

[...I,

t h e l a s t of which c o n t a i n e d

t h e new method of d i f f e r e n t i a t i o n you d e s i r e d . I a m w r i t i n g t h i s one, i n o r d e r t o add something I f o r g o t t o mention when w r i t i n g t h e l a s t l e t t e r . F o r i t i s my o p i n i o n t h a t , f o r t h e time b e i n g , we would do w e l l by keeping t h i s new method somewhat s e c r e t , u n t i l

we o u r s e l v e s have made s u f f i c i e n t u s e of i t ,

f o r i t h i d e s q u i t e a few m a t t e r s of g r e a t e r importance t h a n one would susp e c t a t f i r s t s i g h t . T h e r e f o r e , I b e l i e v e i t t o be b e s t t h a t we n e i t h e r propose t o o t h e r s t o f i n d t h i s new method of d i f f e r e n t i a t i o n , o r of drawing t a n g e n t s , nor t a l k about our i n v e n t i o n , l e t a l o n e expose what t h e a r t i f i c e c o n s i s t s o f , u n t i l we o u r s e l v e s a r e i n a p o s i t i o n t o d e a l w i t h i t i n accordance w i t h i t s v a l u e . " 3 3 The e x t e n s i o n of t h e d i f f e r e n t i a l c a l c u l u s t o more dimensional problem sit u a t i o n s and t h e e x p l o i t a t i o n of t h e s e important consequences of t h e i n t e r c h a n g e a b i l i t y theorem a t which L e i b n i z h i n t e d i n h i s l e t t e r t o Johann Bernoull remained a v i s t a . For example, t h e l o g i c a l c o u n t e r p a r t

t o t h e interchange-

a b i l i t y theorem, namely t h e theorem of mixed d i f f e r e n t i a t i o n d d s=d d s was a x xu I n s t e a d t h i s theorem was n o t found then, although i t was w i t h i n easy only d i s c o v e r e d about twenty y e a r s l a t e r by Nicolaus I B e r n o u l l i . N e i t h e r i n L e i b n i z ' s correspondence nor i n h i s published papers i s t h e r e any e v i d e n c e t h a t t h e s e l i n e s of thought were developed any f u r t h e r . T h i s i s n o t s o s u r p r i s i n g i f one r e c a l l s t h a t L e i b n i z ' s busy l i f e a s a d i g n i t a r y a t t h e Hanoverian Court l e f t him l i t t l e l e i s u r e t o pursue h i s mathematical i d e a s . One would have expected Johann B e r n o u l l i , t h e f u l l - t i m e mathematician, t o t a k e

up t h e s e i d e a s

and e x p l o i t t h e new world of multidimensional c a l c u l u s , b u t a s we s h a l l s e e he a l s o had o t h e r matters t o a t t e n d t o .

52.2.9 Johann Bernoulli's reaction Johann B e r n o u l l i ' s r e a c t i o n t o L e i b n i z ' s d i s c o v e r y of t h e interchangea b i l i t y theorem was abundantly e n t h u s i a s t i c : "I f r a n k l y c o n f e s s now t h a t I had n o t thought about t h i s type of d i f f e r e n t i a t i o n [ w i t h r e s p e c t t o t h e parameter] i n c o n n e c t i o n w i t h t h e t r a n s i t i o n of v a r i a b l e s from curve t o curve. I cannot s t o p t o marvel how t r u l y i n g e n i o u s l y ,

The brachystochrone and its uftermuth

49

how a c u t e l y you have adapted i t t o t h i s m a t t e r ; c e r t a i n l y n o t h i n g more e l e gant e x i s t s o r can b e invented than t h i s method of yours of d i f f e r e n t i a t i n g a curve by means of a sum of a n i n f i n i t e number of d i f f e r e n t i a l l e t s [ i . e . 2nd o r d e r d i f f e r e n t i a l s d,d,cs]."35 But then Johann B e r n o u l l i c h a n n e l l e d h i s enthusiasm t o t h e p r a c t i c a l and s t r a t e g i c a l p o s s i b i l i t i e s opened up by t h e new way of d i f f e r e n t i a t i o n , r a t h e r than going on t o r e f l e c t and e l a b o r a t e upon t h e fundamental i s s u e , v i z . m u l t i dimensional a p p l i c a t i o n s and e x t e n s i o n s of t h e c a l c u l u s . To him, t h e main achievement of t h e i n t e r c h a n g e a b i l i t y theorem l a y i n t h e f a c t t h a t i t made p o s s i b l e d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e parameter a f o r a l l t r a n s c e n d e n t a l e x p r e s s i o n s of t h e form ipI;t.,a)&.

T h i s type of d i f f e r e n t i a t i o n could now be

a p p l i e d f r e e l y and u n i v e r s a l l y t o both a l g e b r a i c and t r a n s c e n d e n t a l e x p r e s s i o n s . I t provided a powerful t o o l f o r a l l s o r t s of problems i n v o l v i n g f a m i l i e s of

c u r v e s , as Johann B e r n o u l l i demonstrated by d e r i v i n g a l o n g s o u g h t - a f t e r g e n e r a l d i f f e r e n t i a l e q u a t i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s of any a l g e b r a i c o r t r a n s c e n d e n t a l f a m i l y of c u r v e s

( c f . c h . 3 ) . Thus t h e d i s c o v e r y of t h e

i n t e r c h a n g e a b l i t y theorem s t i m u l a t e d B e r n o u l l i ' s i n t e r e s t i n problems a b o u t f a m i l i e s of c u r v e s , and he wrote: "Could n o t problems be e l i c i t e d , such a s I have a l r e a d y g i v e n f o r t h e E l l i p s e s , by means of which we can torment t h e Geometers, however w e l l - v e r s e d they may b e i n t h e very c o r e of Geometry? They would c e r t a i n l y see t h a t a l l t h e i r a t t e m p t s a r e f r u s t r a t e d a s long a s t h e y f a i l t o push through t o o u r a r t i f i c e , and they would marvel about t h e i r own i n f i r m i t y t h e more, s i n c e t h i s type of problem

seems s o e a s y and seems t o be taken o n l y from t h e

d i r e c t method of t a n g e n t s . " 3 6 The k i n d of a c t i o n proposed h e r e by Johann B e r n o u l l i runs c o u n t e r completely t o L e i b n i z ' s r e q u e s t t h a t t h e new method be k e p t s e c r e t , and t h a t t h e y would r e f r a i n from s e t t i n g problems t h a t might p u t o t h e r s on t h e t r a c k towards t h e i n t e r c h a n g e a b i l i t y theorem. However, when t h i s r e q u e s t reached Johann B e r n o u l l i ,

i t d i d n o t r e a l l y make him change h i s mind; h e w a s b a d l y i n want of a d i f f i c u l t by means of which he c o u l d c h a l l e n g e h i s b r o t h e r and p u t h i s back up,

problem

and h e r e was a f i e l d t h a t could provide him w i t h such problems. Already i n t h e same month (August 1697) h e proposed t h e t a n g e n t problem f o r equal a r c s t r a j e c t o r i e s i n f a m i l i e s of "curves o f t h e same s o r t " ( c f . § 2 . 2 . 3 ) 3 7 . He d i d n o t c l a r i f y t h e meaning of t h e t e r m "curves of t h e same s o r t " a t t h a t t i m e , b u t instead

-

by mentioning a f a m i l y o f similar p a r a b o l a s by way of example

-

he

s u g g e s t e d t h a t t h e term was used t o denote s i m i l a r c u r v e s . However, i n t h i s c a s e t h e t a n g e n t problem w a s f a i r l y e a s y , a s w e have seen i n 52.2.4. A s w i l l

Families of Curves in the 1690s

50

become c l e a r s h o r t l y , t h i s was n o t t h e c a s e , and Johann B e r n o u l l i i n f a c t used t h e term t o denote any family of c u r v e s whatsoever. U n f o r t u n a t e l y , I know of no documentation i n which B e r n o u l l i makes c l e a r h i s motives f o r t h i s a p p a r e n t amb i g u i t y . Yet i t may p l a u s i b l y be i n f e r r e d from t h e e x i s t i n g m a t e r i a l t h a t t h i s ambiguity was d e l i b e r a t e l y i n t r o d u c e d i n o r d e r t o keep two o p t i o n s open: i n c a s e Jakob B e r n o u l l i f a i l e d t o s o l v e t h e t a n g e n t problem f o r s i m i l a r c u r v e s , t h e r e was no need t o v i o l a t e L e i b n i z ' s r e q u e s t of keeping t h e new method s e c r e t ; i n t h a t c a s e "curves of t h e same s o r t " could be taken t o denote s i m i l a r curves. On t h e o t h e r hand, should Jakob B e r n o u l l i s o l v e the problem f o r s i m i l a r c u r v e s , Johann B e r n o u l l i could s t i l l s t a n d up and j e e r i n g l y exclaim t h a t of course "curves of t h e same s o r t " was a much b r o a d e r term t h a n " s i m i l a r curves". Thus t h e ambiguity i n t r o d u c e d i n t h e f o r m u l a t i o n of t h e problem made i t p o s s i b l e t o take advantage of t h e new method of d i f f e r e n t i a t i o n immediately w i t h o u t a t t h e same time f r u s t r a t i n g L e i b n i z ' s r e q u e s t . However, b e f o r e t h e end of t h e y e a r Johann B e r n o u l l i had been asked by 1 ' H b p i t a l t o e x p l a i n t h e p r e c i s e meaning of t h e term "curves of t h e same s o r t " . 1 ' H b p i t a l had a l r e a d y solved t h e t a n g e n t problem f o r t h e p a r a b o l a s i n Septemb e r 1697, and i n r e t u r n Johann B e r n o u l l i had conveyed h i s own c o n s t r u c t i o n f o r s i m i l a r curves t o 1 ' H b p i t a l . I n November l ' H 8 p i t a l r e t u r n e d t o t h e m a t t e r and wrote: "I m u s t confess t o you t h a t i f t h e curves a r e d i s s i m i l a r something a d d i t i o n a l

i s r e q u i r e d , b u t I have understood t h a t by "curves of t h e same s o r t " you mean

s i m i l a r c u r v e s , a s moreover t h e example of t h e p a r a b o l a s which you gave seems t o confirm t h i s o p i n i o n . Anyway, I do n o t p r e t e n d t o have s o l v e d your l a s t problem i n any o t h e r b u t t h i s s e n s e , and I e x p e c t t o h e a r from you i n c a s e t h e curves a r e di ssi mi l ar . ''3 8 Thus t h e r e was no way o u t f o r Johann B e r n o u l l i , who i n a r a t h e r a g g r e s s i v e tone wrote back a t t h e end of December 1697: "By curves of t h e same s o r t I have understood a l l c u r v e s having t h e same name, o r which a r e given i n a n o r d e r e d way

[..,I,

as a r e f o r example a l l e l l i p s e s

over t h e same a x i s . Had I meant t o denote o n l y s i m i l a r and s i m i l a r l y posed c u r v e s , I would have s a i d s o [ . . . ] . Y o u have r e a s o n t o s a y t h a t f o r d i s s i m i l a r c u r v e s something a d d i t i o n a l i s r e q u i r e d , b u t t h i s something additionaZ which you might t h i n k of a s being of minor importance i s i n f a c t s o c o n s i d e r a b l e , t h a t I r e g a r d t h e d i f f i c u l t y one encounters f o r s i m i l a r c u r v e s a s n o t h i n g compared t o t h e d i f f i c u l t y which one encounters when t h e y a r e d i s s i m i l a r . " 3 9 Johann B e r n o u l l i d i d n o t g i v e t h e show away, and d i d n o t inform 1 ' H b p i t a l about t h e p r e c i s e form of t h e "something a d d i t i o n a l " . However, h e could n o t r e f r a i n

The brachystochrone and its aftermath

51

from v o i c i n g h i s enthusiasm about t h e new way of d i f f e r e n t i a t i o n : "Here i s an e n t i r e l y new way of c a l c u l a t i o n ; t h e r u l e s of t h e d i f f e r e n t i a l c a l c u l u s known up t i l l now a r e u s e l e s s , and i t was n e c e s s a r y t o i n v e n t f o r i t a new type of d i f f e r e n t i a l

and a l s o t o c o n s t r u c t new r u l e s . M r . L e i b n i z

and I have a l r e a d y p e n e t r a t e d q u i t e f a r i n t o t h i s unknown world, M r . L e i b n i z has found t h e e n t r a n c e while I provided him w i t h t h e o p p o r t u n i t y and i n d i c a t e d t h e f i r s t t r a c k s . I have a l s o drawn a n easy and g e n e r a l method from i t t o f i n d t h e curve t h a t c u t s a l l curves g i v e n i n a n o r d e r e d way, be t h e y geom e t r i c o r mechanical, s i m i l a r o r d i s s i m i l a r according t o a p r e s c r i b e d cond i t i o n , e.g. a t a r i g h t angle."40 Here a g a i n t h e a p p l i c a t i o n of t h e new type o f d i f f e r e n t i a t i o n was d e s c r i b e d a s a powerful t o o l f o r t h e s t u d y of f a m i l i e s of c u r v e s ; t h e r e i s no h i n t a t a l l of more g e n e r a l multi-dimensional e x t e n s i o n s of t h e d i f f e r e n t i a l c a l c u l u s .

52.2.20

Jakob B e r n o u l l i ' s s o l u t i o n s

Jakob B e r n o u l l i d i d indeed s o l v e t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s i n a f a m i l y of s i m i l a r c u r v e s ( i n h i s 1 6 9 8 ~ ) i n a way t h a t c l o s e l y resembled t h e c o n s t r u c t i o n which Johann B e r n o u l l i had communicated t o l'H8pit a l . N o t u n t i l 1744, when h i s @era were p u b l i s h e d by Cramer, d i d i t become c l e a r t h a t Jakob B e r n o u l l i had h i t upon t h e s o l u t i o n t o t h i s t a n g e n t problem f o r d i s s i m i l a r curves a s w e l l , and t h a t he had a l s o found a s o l u t i o n f o r t h e r e l a t e d problem of q u i c k e s t approach i n f a m i l i e s of d i s s i m i l a r c u r v e s . The

@era included an e x t e n s i v e s e c t i o n of p r e v i o u s l y unpublished m a t e r i a l , ent i t l e d "Varia Posthuma", which had been e x t r a c t e d from h i s mathematical d i a r y , the'bfeditationes" and e d i t e d by Jakob Hermann and Nicolaus I B e r n o u l l i 4 ' . A r t i c l e s I V and V of t h e s e Varia Posthuma ( J a k . B e r n o u l l i 1744a, 2744b) cont a i n e d t h e s o l u t i o n s of two anagrauuns,which Jakob B e r n o u l l i had p u b l i s h e d a t t h e end of h i s paper 1698d on t h e i s o p e r i m e t r i c a l problems. These anagramms happened t o h i d e t h e r e s o l u t i o n of b o t h t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s and t h e problem of q u i c k e s t approach i n c e r t a i n f a m i l i e s of d i s s i m i l a r c u r v e s . These anagramms r e v e a l e d t h a t Jakob B e r n o u l l i had a l s o h i t upon t h e i d e a of comparing corresponding a r c l e n g t h d i f f e r e n t i a l s a l o n g i n f i n i t e l y c l o s e curves.. I s h a l l r e s t r i c t my d i s c u s s i o n t o Jakob B e r n o u l l i ' s r e s o l u t i o n of t h e

t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s ( c o n t a i n e d i n h i s 1 7 4 4 b ) , w h i l e a d a p t i n g t h e n o t a t i o n used i n 52.2.6.

Jakob B e r n o u l l i commenced h i s argument by

Families of Curves in the 1690s

52

considering a family of affine curves, that is, a family of curves that arises from multiplication of one given curve with respect to the horizontal axis (see figure 1 6 ) . Consider two infinitely close curves ACHB and A C ' H ' B '

and the

equal arcs trajectory B B ' . Like Leibniz, Jakob Bernoulli set out to calculate the infinitely small line segment BQ=d zj(z ,a) and the infinitely small arc a 5 segment QB'=das(x5,a). Due to the affinity o f the curves ACHB and A C ' H ' B ' one I

may assume that the parameter a satisfies the relation y ( x , u ) : y ( x , a ' ) = u : a ' all abscissas

for

by consequence, BQ can be expressed in the following way:

2;

BQ=day(xo, a1-y (x5, alda/a.

(2.42)

The evaluation of

@'

is again based on the consideration that QB' is built up

-

-

as the sum of differences of corresponding arclength differentials CH and C'H' respectively; thus

- - 'I.

&?'=I (CH-C'H

(2.43)

-

R

Since Q B ' d J

aA

d,s

and C%C%'=dad,s,

relation ( 2 . 4 3 ) is in fact equivalent

to the interchangeability theorem:

but Jakob Bernoulli did not show any special interest for this point. Instead

- -

he went on to express the difference CH-C'H'

more explicitly in terms of da

and the differential operator d,.

fig. 16a

Consider the quadrangle C'CHH' (see figure 16a),and draw the line C ' I parallel to CH, take R on C'I such that C'R=CTH' in length, and take L on HI such that

C'L is parallel to the *axis.

Let K denote the common point of intersection

with the paxis of the tangents to both curves ACE and AC'H'

in C and C'

respectively; that these tangents KC and KC' do indeed meet each other on the x-axis is clear from the affinity of the curves ACH and A C ' H ' . By definition

53

The brachystochrone and its aftermath

-

R one h a s : G-C'H'=RI i n l e n g t h , o r RI=dadxs . S i n c e t h e t r i a n g l e s ALC'H' and ARH'I a r e s i m i l a r , RI can be e x p r e s s e d a s of t h e p o i n t

( 2 .44)

RI=H 'I. (LH' :H';C

'1.

Considering t h e p a i r of s i m i a r t r i a n g l e s

ACKC' and AIC'H', H'I can be ex-

pressed as: (2.45)

H'I=C7H'. (KC':C'C)

I n s e r t i n g t h e v a l u e s C'C=d$, C'H'=d s, KC'=t=(ydxs:dgl i n ( 2 . 4 5 ) y i e l d s : X H'I=daydg:y. S u b s t i t u t i o n of t h i s v a l u e f o r H'I i n ( 2 . 4 4 ) , and t a k i n g LH '=dg, H'C'=dXs y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r RI: RI=Idgj2day/ydxs, and r e c o l l e c t i n g t h a t day=yda/a by f o r c e of ( 2 . 4 2 ) f i n a l l y l e a d s t o : da i d 2 1 (2.46) R I az dxs = dadxs and t h e r e f o r e :

(2.47)

6 '=das(xoJaj=ida/alJ

B

( d g )'/dxs.

A Jakob B e r n o u l l i t r a n s l a t e d t h e s e r e s u l t s ( 2 . 4 2 ) and ( 2 . 4 7 ) i n t o t h e f o l l o w i n g construction f o r the tangent t o t h e equal a r c s t r a j e c t o r y B B ' : "The t a n g e n t t o t h e c u r v e , which c u t s o f f e q u a l a r c s from i n f i n i t e l y many curves of t h e same genus i s found i n t h e f o l l o w i n g way: Draw one of t h e inf i n i t e l y many c u r v e s through a p o i n t given on t h e c u t t i n g curve [ t h e t r a j e c t o r y ] , t o g e t h e r w i t h i t s t a n g e n t [ t ] and o r d i n a t e [ y ] . S e t up t h e prop o r t i o n a l i t y : a s t h e e x c e s s of t h i s t a n g e n t o v e r t h e sum o f t h e t h i r d prop o r t i o n a l s of t h e elements of t h e curve of t h e o r d i n a t e

[dXs ] which i s c u t and t h e elements

[ d g ] , [ a s t h i s e x c e s s ] i s t o t h e t a n g e n t , s o t h e sub-

t a n g e n t [ a ] t o t h e f o u r t h . T h i s [ f o u r t h ] w i l l denote t h e segment [ a + u ] of t h e a x i s i n t e r c e p t e d by t h e t a n g e n t s of b o t h c u r v e s , t h e c u t t i n g curve and t h e curve c u t

'I4'

(see figure 1 7 ) .

b

e

-

/"-

Families of Curves in the 1690s

54

T r a n s l a t i n g t h i s r e c i p e i n t o an e q u a t i o n , w e f i n d : f t - C t h i r d p r o p . l : t =

=u:(u+u).

The

:(third prop.

third prop.

) , which y i e l d s :

Z(third prop.)= J proportion:

i s d e f i n e d by t h e p r o p o r t i o n a l i t y d x s : d g = d 2 :

A

B

Idg)'/dxs.

( t h i r d prop. I=ld2)':dZs

and hence:

Thus Jakob B e r n o u l l i ' s r e c i p e l e a d s t o t h e

u

Since t ( - y d X s / d g ) , o ( = y d x / d g l and

f d g ) ' / d X s can a l l be c a l c u l a t e d f o r A any p o i n t B on any of t h e given c u r v e s , t h i s p r o p o r t i o n a l i t y f o r any such J

p o i n t B d e f i n e s t h e v a l u e of (a+u), which r e p r e s e n t s t h e l i n e segment i n t e r cepted b e t w e e n t h e p o i n t s of i n t e r s e c t i o n of t h e t a n g e n t t o t h e given curve through B and the r e q u i r e d t a n g e n t through B w i t h t h e h o r i z o n t a l a x i s . Thus by means of t h i s v a l u e of ( a + p ) t h e r e q u i r e d t a n g e n t can indeed be c o n s t r u c t e d . 4 3 The most remarkable f e a t u r e of Jakob B e r n o u l l i ' s r e c i p e as p r e s e n t e d above i s t h a t t h e c o n s t r u c t i o n i s s o normal; i t i s a c o n s t r u c t i o n t h a t c l o s e l y resembles o t h e r c o n s t r u c t i o n s o c c u r r i n g a t t h e t i m e ; formulated i n terms of a p r o p o r t i o n a l i t y , i n v o l v i n g only such q u a n t i t i e s a s can be determined by means of o r d i n a r y (one-dimensional) d i f f e r e n t i a l - and i n t e g r a l - c a l c u l u s , no glimpse i s given a t a l l i n t o t h e e n t i r e l y novel a n a l y s i s t h a t has produced t h i s c o n s t r u c t i o n . None of t h e terms i n t h e p r o p o r t i o n a l i t y shows t h a t h e r e simultaneous d i f f e r e n t i a t i o n w i t h r e s p e c t t o two independent v a r i a b l e s x and

a had been n e c e s s a r y . The r e a s o n , of c o u r s e , i s t h a t Jakob B e r n o u l l i presupposed a family of affiine curves. This s p e c i a l p r o p e r t y of a f f i n i t y made i t p o s s i b l e t o go a l l t h e way through and e x p r e s s d i f f e r e n t i a t i o n s w i t h r e s p e c t t o a i n terms of t h e u s u a l d i f f e r e n t i a l o p e r a t o r d

X

and t h e d i f f e r e n t i a l

da,

which v a n i s h e s i n t h e f i n a l p r o p o r t i o n s . From Jakob B e r n o u l l i ' s a n a l y s i s pert a i n i n g t o t h i s c o n s t r u c t i o n one g e t s t h e impression t h a t he himself d i d n o t r e c o g n i s e t h e n o v e l t y of t h e procedures e i t h e r , and t h a t he w a s j u s t c a r r y i n g o u t what t h e c a l c u l a t i o n s d i c t a t e d t h a t h e should do. There i s no s i g n of any

--

awareness t h a t t h e i d e a of c o n s i d e r i n g t h e d i f f e r e n c e i n a r c l e n g t h AB-A& a s t h e accumulation of t h e d i f f e r e n c e s of corresponding a r c l e n g t h d i f f e r e n t i a l s

C'H'

and

i s a p o i n t of fundamental importance, and t h a t t h i s i d e a might s e r v e a

broader purpose than merely b e i n g t h e v e h i c l e l e a d i n g t o t h e r e q u i r e d t a n g e n t c o n s t r u c t i o n . I t seems as i f Jakob B e r n o u l l i was o n l y i n t e r e s t e d i n t h e f i n a l r e s u l t , and t h a t he was n e i t h e r w i l l i n g nor i n t e r e s t e d i n r e f l e c t i n g upon h i s r e s u l t s a t a conceptual r a t h e r than a computational l e v e l . We s h a l l have a n o t h e r o c c a s i o n t o p o i n t towards such a c h a r a c t e r i s t i c of Jakob B e r n o u l l i ' s

55

The brachystochrone and its aftermath mathematics i n c h a p t e r 3 when we d i s c u s s h i s i d e a s about t h e g e n e r a l problem

of o r t h o g o n a l t r a j e c t o r i e s . Hofmann h a s a l s o drawn a t t e n t i o n t o t h i s c h a r a c t e r i s t i c i n t e r e s t i n s p e c i f i c problems and d i s l i k e of g e n e r a l programmes: "Jakob B e r n o u l l i ' s d e c i s i v e s c i e n t i f i c achievement l a y n o t i n t h e f o r m u l a t i o n of e x t e n s i v e theor e s , b u t i n t h e c l e v e r and pre-eminently a n a l y t i c a l

treat-

ment of i n d i v i d u a l problems. '''' Jakob B e r n o u l l i s s k i l l s a t t h i s a n a l y t i c a l , computational l e v e l become even more a p p a r e n t i n h i s subsequent t r e a t m e n t of t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s of what a r e c a l l e d i n t h e Upera "any c u r v e s whatsoever, which are expressed by a common e q u a t i o n i n v o l v i n g a v a r i a b l e parameter".

I n t h i s case

t h e f a m i l y of c u r v e s i s s t r i p p e d of a l l s p e c i f i c g e o m e t r i c a l p r o p e r t i e s and i s merely d e f i n e d by an a l g e b r a i c e q u a t i o n

V(x,y,a)=O,

i n v o l v i n g t h e parameter a .

Jakob B e r n o u l l i modelled h i s d i s c u s s i o n of t h i s more g e n e r a l c a s e a l o n g t h e l i n e s s e t o u t i n the c a s e of a f f i n e c u r v e s ; a g a i n t h e i n f i n i t e l y small l i n e segment BQ(=d y ( z o , a ) ) and t h e i n f i n i t e s i m a l a r c segment Q%'(=d s ( x , a ) ) have a o t o be c a l c u l a t e d ( c f . f i g u r e 16), b u t now t h e r e was no way t o go through and e x p r e s s e v e r y t h i n g i n terms of da and t h e d i f f e r e n t i a l o p e r a t o r d,.

Hence

Jakob B e r n o u l l i immediately proceeded from t h e e q u a t i o n

to the

V(x, y,a)=O

t o t a l d i f f e r e n t i a l e q u a t i o n of t h e f a m i l y of curves: " D i f f e r e n t i a t e t h e e q u a t i o n of t h e s e c u r v e s , b o t h w i t h r e s p e c t t o t h e coord i n a t e s x and y and w i t h r e s p e c t t o t h e parameter a , and t h e r e f o l l o w s :

(2.49)

fdx+gdy+hda=O"

45

.

Taking x c o n s t a n t i n ( 2 . 4 9 ) l e a d s t o gd y+hda=O and immediately y i e l d s : a

.

BQ=d y=-h/g.da a

(2.50)

Taking a c o n s t a n t i n t h e t o t a l d i f f e r e n t i a l e q u a t i o n i n t h e same way y i e l d s t h e d i f f e r e n t i a l e q u a t i o n fdx+gdg=O, and hence dg=-fdx/g. 3-

3-

d s = ( d g 2 + d x 2 )'=(f2+g2) 'dx/g X

Therefore:

--

( r e p r e s e n t e d by CH i n f i g . 1 6 ) . Now @'=AB-A&

i s a g a i n considered t o be t h e sum of t h e d i f f e r e n c e s of t h e corresponding a r c length d i f f e r e n t i a l s

and C';fi',

and hence Jakob B e r n o u l l i c a l c u l a t e d

&?-C';ir'=d d s i n t h i s case a s f o l l o w s : a x "Taking x and

dx c o n s t a n t , ($2+921' i s d i f f e r e n t i a t e d , t o y i e l d g

+ nda=(because dy i s C'C)=-hmda/g+nda; thus dds o r G-C%'= = (-mhdadx+gndadx)/g, and hence I [ i-hm&dx+gndadx) /g) =AB-AQ=AR-AB ' '' d W ) a' = . n d y Y

This i n t e g r a l /ii-hmdadx+gndadxl/g/

r e p r e s e n t s Idadxs

i n t e r m s of t h e

d i f f e r e n t i a l c o e f f i c i e n t s emerging when t h e d i f f e r e n t i a l p l i c i t l y determined i n terms of da.

(f2

21%

d -2%-i s a

g

ex-

Families of Curves in the 1690s

56

As always, the implicit assumption here is that all curves pass through one and the same point A on the horizontal axis. This assumption is necessary for the rule @r=Z(C??-CTHr)

to hold. Here Jakob Bernoulli clearly applied the

differential operators d and d simultaneously to the same variable s . a X Notationally, however, he makes no difference between these differential operators, and this leads to the rather uncommon second order differential "dds": which coincides with my d d s. But the different meanings of d are accurately a x accounted for in the context, as they are accompanied by statements like:

"a constant", " x constant" or "z and dx constant". Certainly this last statement was at variance with the usual rules of the differential calculus, according to which "z constant" immediately implies &=O;

in the context of

a two-dimensional problem situation it merely implies that the variable

(m2+g2T/gis )dx considered

only with respect to the variation of a and the

variation of y induced by the independent variable a . Again there is no sign that Jakob Bernoulli has grasped the importance of the interchangeability principle d J=Jd at a broader level than at which it actually occurs, viz. a a the calculation of d s ( x o , a ) . Apparently he did not see what bearing it had on a the differentiation of transcendental expressions of the type Jp(x,a)& with respect to a in general. This is clear from the fact that the "general family of curves" considered by Jakob Bernoulli is still only an algebraic family, for which the total differential equation ( 2 . 4 9 ) f&+gdy+hda=O

can be found by

straightforward differentiation (of the equation) with respect to the three variables x , y and a . Jakob Bernoulli had all the means at his disposal here to extend his argument a l s o to those cases where the family of curves is given by a transcendental expression y= Jp(x,a)&.

In that case, the total

differential equation ( 2 . 4 9 ) would have taken the form (2.5 I )

dy-p&

( Jpa&) da=O

dg,d,y,

dxs, d d s could have been expressed easily a x in terms of & and da; the resulting construction would not have been more

and all differentials

difficult than the one already given. It is quite certain that Jakob Bernoulli has not considered this case, since he makes no mention of the interchangeability principle in connection with the calculation of the coefficient of in the total differential equation ( 2 . 4 9 ) .

da

57

Conclusions 92.3 ConcZusions

In this chapter we have scrutinised two instances of differentiation with respect to the parameter of a family of curves in the 1690s: Leibniz's derivation of the envelope algorithm in 1692 and 1694, and the discovery and use of the interchangeability theorem for differentiation and integration in 1697 by Leibniz and Johann Bernoulli and in 1698 by Jakob Bernoulli. These two instances are quite unconnected. The work that Leibniz did in the early 1690s remained isolated and was not very influential. His extension of the applicability of the differential calculus was guided by an ad hoc reciprocity between ordinary tangent problems and the envelope problem and led to the distinction between "differentiable" and "indifferentiable" quantities occurring in an equation. This was not yet genuine partial differentiation, since Leibniz considered a special type of one dimensional problem-situation rather than a two dimensional one. Leibniz's main achievement in the early 1690s was to show how the differential calculus could be employed with respect

to other variables than the classical geometric ones, defined in points o f a single curve; of course, the envelope algorithm itself was a fine result as well. The discovery of the interchangeability theorem in 1697 came as a surprise to both Leibniz and Johann Bernoulli; unfortunately Jakob Bernoulli's reaction is not documented, but he seems not to have grasped the full conceptual implication of his discovery. While studying a complicated version of the standard problem of the differential calculus, viz. the determination of the tangent to a single curve, Leibniz and Johann Bernoulli suddenly found themselves confronted with a defect of the calculus. This was the more surprising since such defects were known to exist in inverse tangent problems, i.e. in the integration of differential equations, but they were certainly not expected in direct tangent problems, which seemed

to require straightforward

differentiation only. This surprise was voiced for example by Johann Bernoulli, who expected fellow-mathematicians to be greatly astonished at their inability to solve a problem as easy as a direct tangent problem. This expectation certainly reveals Johann Bernoulli's own feelings in this situation. The elimination of this defect of the calculus led to the consideration of a hitherto unknown kind of second order differential, viz. dadxs, emerging from the comparison of corresponding arclength differentials along two infinitely close curves. Here is a truly

two dimensional application of the differential

calculus, since the variable s is considered to depend on two independent variables x and a, and i s differentiated simultaneously with respect to both

Families of Curves in the 1690s

58

of them. Both Leibniz and Johann Bernoulli were aware of the conceptual novelty of such an application of the calculus, which they termed "differentiation from curve to curve" in contrast to the ordinary curve.

differentiation along a single

Despite Leibniz's prospective views into the unknown world of multi-

dimensional calculus and his ambitious plans to cultivate this world, the main effect of the interchangeability theorem was to provide a technical means for using differentiation with respect to the parameter for transcendental expressions as well, thus bringing transcendental expressions and curves more firmly under control of the calculus. No more conclusive results were reached until around 1719 when Nicolaus I Bernoulli hit upon the logical counterpart of the interchangeability theorem d Jp(x,aa)dx = J d p ( x , a ) d x , namely the equality theorem dad$

d ' a y. x a

a

a

59

CHAPTER 3

ORTHOGONAL TRAJECTORIES 1694-1720

53.1 I n t roduct i on

The problem how t o c o n s t r u c t t h e o r t h o g o n a l t r a j e c t o r i e s of a f a m i l y of curves was one of t h e b i g p o i n t s a t i s s u e which mathematicians faced a t t h e end of t h e 17th and in t h e beginning of t h e 18th c e n t u r y . An e x c e p t i o n a l amount of time and energy was s p e n t on t r y i n g t o s o l v e t h i s o r t h o g o n a l t r a j e c t o r y problem. Johann B e r n o u l l i was t h e f i r s t t o r a i s e t h e problem p u b l i c l y i n h i s a r t i c l e of 1697 about t h e brachystochrone. The impetus t o s t u d y t h e problem and t o pose i t a s a c h a l l e n g e f o r o t h e r mathematicians came mainly from L e i b n i z ' s d i s c o v e r y , i n t h e same y e a r , of t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n . A f t e r a 15-year p e r i o d of s i l e n c e t h e problem was r a i s e d a g a i n i n 1716 by L e i b n i z who put i t t o t h e E n g l i s h . L e i b n i z ' s motive f o r s e l e c t i n g t h i s problem f o r c h a l l e n g e was h i s c o n v i c t i o n t h a t i t could o n l y be d e a l t w i t h p r o p e r l y by t h o s e mathematicians who were a c q u a i n t e d with t h e method of d i f f e r e n t i a t i o n from curve t o curve, i . e . p a r t i a l d i f f e r e n t i a t i o n . But L e i b n i z had been too o p t i m i s t i c , and t h e t e s t f a i l e d t o have the e f f e c t expected. The method of d i f f e r e n t i a t i o n from curve t o curve, which had bestowed L e i b n i z and Johann B e r n o u l l i w i t h a f e e l i n g of s a f e s u p e r i o r i t y , f a i l e d t o l i v e up t o t h e e x p e c t a t i o n s voiced s o e n t h u s i a s t i c a l l y by b o t h of them i n 1697. Although Johann B e r n o u l l i d i d have some s u c c e s s i n i t i a l l y ( h i s o n l y c o n t r i b u t i o n to p a r t i a l d i f f e r e n t i a l c a l c u l u s ) , t h e follow-up w a s blocked by some s e v e r e b a r r i e r s . Such was Johann B e r n o u l l i ' s d i s i l l u s i o n i n t h i s r e s p e c t t h a t i n 1720, when f i n a l l y r e v e a l i n g what methods d i f f e r e n t i a t i o n from curve t o curve had produced f o r c o n s t r u c t i n g o r t h o g o n a l t r a j e c t o r i e s , he b l u n t l y r e f e r r e d t o them a s g e n e r a l a b s t r a c t nonsense. However, one y e a r e a r l i e r Nicolaus I B e r n o u l l i had made a n o t h e r important break-through

in partial

d i f f e r e n t i a l c a l c u l u s , a g a i n i n connection w i t h t h e o r t h o g o n a l t r a j e c t o r y problem. But t h e d i s c o v e r y remained unrecognised, i n o b s c u r i t y , and i t d i d n o t e x e r t any i n f l u e n c e on t h e course of t h e d i s c u s s i o n between C o n t i n e n t a l and B r i t i s h mathematicians.

Orthogonal Trajectories 1694-1 720

60

I n t h i s c h a p t e r I s h a l l look a t what r e s u l t s p a r t i a l d i f f e r e n t i a t i o n methods were expected t o produce and comment on t h e o t h e r methods which were developed when t h e e x p e c t a t i o n s f a i l e d t o m a t e r i a l i s e . T h i s c h a p t e r w i l l t h e r e f o r e t r e a t f a m i l i e s of curves i n a f a i r l y wide c o n t e x t and r e v e a l o t h e r mathem a t i c a l techniques with which p a r t i a l d i f f e r e n t i a t i o n had t o complete'.

I shall

c e n t e r my d i s c u s s i o n around t h e work of Johann B e r n o u l l i . Readers who a r e int e r e s t e d p r i m a r i l y i n t h e emergence of p a r t i a l d i f f e r e n t i a l c a l c u l u s w i l l prof i t mainly from t h e s e c t i o n s 53.6, 53.8, and 53.12.

53.2 The probZem posed

The orthogonal t r a j e c t o r y problem was f i r s t posed by Johann B e r n o u l l i i n t h e y e a r 1694, when asking L e i b n i z t o c o n s i d e r t h e q u e s t i o n : "Given i n f i n i t e l y many curves by p o s i t i o n ; f i n d t h e curve t h a t i n t e r s e c t s them a l l a t r i g h t angles."' B e r n o u l l i claimed t o have been a c q u a i n t e d t o the problem f o r a long time, and commented t h a t he had begun t o t h i n k of i t a g a i n when he came a c r o s s L e i b n i z ' s a r t i c l e 1694 about envelopes of f a m i l i e s of c u r v e s . He m o t i v a t e d t h e t r a j e c t o r y problem w i t h a r e f e r e n c e t o Huygens's wave t h e o r y of l i g h t a s developed i n t h e

T r a i t 6 de Zu Zumie're ( 1 6 9 0 ) ; h e r e l i g h t r a y s a r e viewed a s t h e o r t h o g o n a l t r a j e c t o r i e s of the wave f r o n t s , and t h u s , Johann B e r n o u l l i s u g g e s t e d , methods t o f i n d orthogonal t r a j e c t o r i e s t o f a m i l i e s of c u r v e s would be of importance f o r f i n d i n g l i g h t r a y s . He e x p l i c i t l y r e f e r r e d t o a very n i c e p i c t u r e i n t h e T r a i t 6 which shows t h i s r e l a t i o n between l i g h t r a y s and wave f r o n t s 3 :

fig. 1

61

The problem posed

Although a l a r g e number of methods f o r t h e c o n s t r u c t i o n of o r t h o g o n a l t r a j e c t o r i e s had been developed between 1694 and 1720, none of them seems e v e r t o have been employed f o r a c t u a l l y f i n d i n g l i g h t r a y s from given wave f r o n t s . On t h e c o n t r a r y , i n 1720 one of Johann B e r n o u l l i ' s b e s t methods f o r c o n s t r u c t i n g t r a j e c t o r i e s was s t i l l based on i n t e r p r e t i n g t h e given c u r v e s a s l i g h t r a y s and t h e t r a j e c t o r i e s as wave f r o n t s . Thus t h e f i e l d of a p p l i c a t i o n envisaged i n 1694 i n f a c t t u r n e d o u t t o remain t h e f i e l d of i n s p i r a t i o n . Johann B e r n o u l l i could s o l v e t h e o r t h o g o n a l t r a j e c t o r y problem f o r some p a r t i c u l a r c a s e s o n l y , such a s f o r i n s t a n c e f o r some f a m i l i e s of p a r a b o l a s . Thus, more p r e c i s e l y , he asked L e i b n i z t o produce a g e n e r a l a n a l y t i c r u l e f o r f i n d i n g t r a j e c t o r i e s , a r u l e analogous t o t h e a l g o r i t h m L e i b n i z had j u s t developed f o r e n v e l o p e s 4 . Apparently Johann B e r n o u l l i had n o t y e t found such a r u l e h i m s e l f ; h i s own r e s u l t s were based upon t h e g e o m e t r i c a l c o n s i d e r a t i o n t h a t t h e subtangent u of a curve i s equal t o t h e s u b n o r m a l o f t h e t r a j e c t o r y i n a p o i n t of i n t e r s e c t i o n P.

L e i b n i z provided t h e method Johann B e r n o u l l i had asked f o r b e f o r e t h e end of t h e y e a r 5 . I t was an a l g o r i t h m indeed, which was based on t h e p r i n c i p l e t h a t i n a p o i n t of i n t e r s e c t i o n of a given curve and t h e t r a j e c t o r y one h a s :

(h:dY) curve=(-dY:

(3.1)

did trajectory'

L e i b n i z ' s method can be sunnnarised a s f o l l o w s : L e t t h e f a m i l y of c u r v e s be given by t h e e q u a t i o n

Vlx,y,a)=O,

(3.2)

where a i s t h e parameter of t h e family. I n t h e p o i n t of i n t e r s e c t i o n P o f t h e curve (having parameter a ) and t h e o r t h o g o n a l t r a j e c t o r y t h e r e l a t i o n between the d i f f e r e n t i a l s

(3.1) (3.3)

-

dcc

and dy along the t r a j e c t o r y i s

given by t h e e q u a t i o n

Vx (x,y, a I dy-V (x,y, a 1&=O. Y

-

by f o r c e of c o n d i t i o n

Orthogonal Trajectories 1694-1720

62

E x p l i c i t s o l u t i o n of ( 3 . 2 ) f o r a and i n s e r t i o n of t h e v a l u e found i n ( 3 . 3 ) then y i e l d s a f i r s t order d i f f e r e n t i a l equation i n

z and

y f o r the orthogonal t r a -

j e c t o r i e s . With h i n d s i g h t i t w i l l be c l e a r t h a t L e i b n i z ' s method cannot be a p p l i e d u n i v e r s a l l y ; a t l e a s t i t i s n e c e s s a r y t h a t t h e e q u a t i o n V(z,y,a)=O allows e x p l i c i t s o l u t i o n f o r t h e parameter a .

53.3 Orthogonal t r a j e c t o r i e s o f t h e brachystochrones

L e i b n i z ' s method seemed t o have s a t i s f i e d Johann B e r n o u l l i ' s demands q u i t e w e l l , s i n c e the i s s u e was n o t taken up f o r t h e n e x t two y e a r s . Only i n 1696 d i d Johann B e r n o u l l i come back t o t h e m a t t e r , i n connection w i t h h i s lucky f i n d of t h e orthogonal t r a j e c t o r i e s of t h e family of brachystochrone c y c l o i d s ( c f . 52.2.2).

A s we i n d i c a t e d i n t h e preceding c h a p t e r , B e r n o u l l i had, by way of a

c o r o l l a r y t o h i s analogy between b r a c h y s t o c h r o n e s and l i g h t r a y s , been a b l e t o f i n d a c o n s t r u c t i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s of t h e brachystochrone c y c l o i d s . These c y c l o i d s a r e given by the e q u a t i o n

Now i t was immediately c l e a r t o B e r n o u l l i t h a t L e i b n i z ' s method could never produce the orthogonal t r a j e c t o r i e s of curves t h a t a r e given by a t r a n s c e n d e n t a l equation l i k e ( 3 . 4 )

-

f o r the simple reason t h a t t h i s e q u a t i o n cannot be s o l v e d

e x p l i c i t l y f o r t h e parameter a. Thus Johann B e r n o u l l i had good reason t o w r i t e to Leibniz: "I r e c a l l t h a t I once proposed t o you i n g e n e r a l t o f i n d t h e curve t h a t

i n t e r s e c t s o t h e r s , given by p o s i t i o n , p e r p e n d i c u l a r l y , which [problem] I had solved f o r q u i t e a few c a s e s myself. You indeed e n t r u s t e d me w i t h your gen e r a l method. But i f you would c a r e t o r e c o n s i d e r i t a g a i n , y o u ' l l s e e t h a t

i t f r e q u e n t l y does n o t work,[namely] when t h e c u r v e s g i v e n by p o s i t i o n are t r a n s c e n d e n t a l , a s w i l l become c l e a r i n t h i s very example [ t h e c y c l o i d s ] . For t h e o t h e r example which I propose i n my t r e a t i s e , and which concerns t h e p e r p e n d i c u l a r i n t e r s e c t i n g of l o g a r i t h m i c [ c u r v e s ] by a c u r v e , a d m i t t e d l y , I have n o t been a b l e t o f i n d e i t h e r a c o n s t r u c t i o n , o r a f i r s t o r d e r d i f -

f e r e n t i a l e q u a t i o n ; n e v e r t h e l e s s I can produce an extremely simple s e r i e s for it.8s6 Thus i n t h e c a s e of f a m i l i e s of curves r e p r e s e n t e d by an e q u a t i o n of t h e 3:

type y=

1 pl'r,a)& t h e o r t h o g o n a l t r a j e c t o r y problem appeared t o be u n s o l v a b l e xL7

The limits of Leibniz's method

63

by known methods. A t t h e end of h i s brachystochrone a r t i c l e (2697a) Johann B e r n o u l l i t h e r e f o r e i n v i t e d h i s f e l l o w mathematicians t o employ t h e i r s k i l l s i n t a c k l i n g t h e f o l l o w i n g problem: " I f someone would l i k e t o t r y o u t h i s method i n o t h e r c a s e s [ o t h e r than t h e c y c l o i d s ] , l e t him seek t h e curve which i n t e r s e c t s a t r i g h t a n g l e s o t h e r (not a l g e b r a i c , t h a t i s n o t d i f f i c u l t , b u t ) t r a n s c e n d e n t a l curves given by p o s i t i o n i n ordered sequence such a s f o r i n s t a n c e l o g a r i t h m i c c u r v e s above a common a x i s and drawn through a f i x e d p o i n t . " 7

53.4 The l i m i t s o f L e i b n i z ' s method

By s e t t i n g t h e t r a j e c t o r y problem f o r t r a n s c e n d e n t a l c u r v e s s p e c i f i c a l l y , Johann B e r n o u l l i made i t c l e a r t h a t i n h i s o p i n i o n t h e main and e s s e n t i a l d i f f i c u l t y of the problem was t o come t o terms w i t h t r a n s c e n d e n t a l c u r v e s . Bef o r e I proceed t o show how B e r n o u l l i soon a f t e r w a r d s indeed came t o g r i p s w i t h t h i s problem, l e t me f i r s t a n a l y s e t h e i m p l i c a t i o n s of Johann B e r n o u l l i ' s c h o i c e . T h i s d i s c u s s i o n w i l l e n a b l e me t o i d e n t i f y t h e s t a n d p o i n t s which were occupied by t h e d i f f e r e n t p a r t i c i p a n t s i n t h e d e b a t e s over t h e t r a j e c t o r y problem. By i t s very n a t u r e , t h e o r t h o g o n a l t r a j e c t o r y problem i s a geometric

problem. I t i s concerned w i t h a given f a m i l y of c u r v e s , and i t r e q u i r e s t h a t a new s e t of curves be found, which a l l c u t t h e given c u r v e s a t r i g h t a n g l e s . T h e r e f o r e , i n good 17th c e n t u r y f a s h i o n , t h i s problem must b e s o l v e d by f i n d i n g a geometric c o n s t r u c t i o n f o r t h e r e q u i r e d t r a j e c t o r i e s . Hence t h e f i n a l 17th c e n t u r y answer t o t h e q u e s t i o n whether o r n o t a c e r t a i n s o l u t i o n i s s a t i s f a c t o r y o r a c c e p t a b l e depends on geometric c r i t e r i a . Now L e i b n i z ' s method l e a s t globally

-

-

at

provided t h e f o l l o w i n g approach t o t h e t r a j e c t o r y problem:

Produce a n e q u a t i o n V(x,y,a)=O

f o r t h e given f a m i l y of c u r v e s ,

s o l v e t h i s e q u a t i o n e x p l i c i t l y f o r t h e parameter a , t h a t i s , f i n d a n a l g e b r a i c e x p r e s s i o n A(x,c,)

such t h a t a=A(x,y).

By means of t h i s e x p r e s s i o n A(x,y) e l i m i n a t e t h e parameter a from

Vx(x,y,a)dy-V

Y

(x,y,a)dx=O

t o a r r i v e a t a d i f f e r e n t i a l e q u a t i o n i n x and

y , p e r t a i n i n g t o t h e orthogonal t r a j e c t o r i e s . Solve t h e r e s u l t i n g d i f f e r e n t i a l e q u a t i o n e i t h e r a l g e b r a i c a l l y o r by quad r a t u r e s , and f i n a l l y t r a n s l a t e t h e s o l u t i o n found i n (d) i n t o a geometric c o n s t r u c t i o n f o r the t r a j e c t o r i e s .

Orthogonal Trajectories 1694-1 720

64

I n f a c t , each of t h e s e s t e p s ( a ) , ( b ) , ( c ) , ( d ) , and ( e ) h a s i t s own d i f f i c u l t i e s and could i n p r i n c i p l e o b s t r u c t t h e a p p l i c a t i o n of L e i b n i z ' s method. A s we s h a l l s e e i n the f o l l o w i n g s e c t i o n , Johann B e r n o u l l i was unable t o produce a s u i t a b l e e q u a t i o n f o r t h e family of l o g a r i t h m i c curves which h e proposed a s a s p e c i f i c example of t h i s t r a j e c t o r y p r o b l e m H e r e , t h e r e f o r e , t h e d i f f i c u l t y was i n s t e p ( a ) . For o t h e r t r a n s c e n d e n t a l c u r v e s given by an e q u a t i o n l i k e

y= J

X

p ( x , a l d z t h e d i f f i c u l t y i s i n s t e p ( b ) , s i n c e such an e q u a t i o n cannot be

"0

solved e x p l i c i t l y f o r t h e parameter a , a s i s r e q u i r e d i n s t e p (b)'.

Hence, w i t h

t r a n s c e n d e n t a l c u r v e s i t seemed even impossible t o make a s t a r t w i t h L e i b n i z ' s method. However, b a r r i e r s do n o t

o n l y a r i s e f o r t r a n s c e n d e n t a l c u r v e s ; alge-

b r a i c curves may produce d i f f i c u l t i e s as w e l l , even i n s t e p ( b ) , s i n c e now know

-

n o t a l l a l g e b r a i c e q u a t i o n s V'ix,y,u)=O

-

a s we

can b e s o l v e d e x p l i c i t l y f o r

a l l v a r i a b l e s t h a t occur i n them. I f , f o r i n s t a n c e , V(z,y,a)=O would be a q u i n t i c e q u a t i o n i n t e r m s of t h e parameter a , then no e q u a t i o n a=A(z,y) e x i s t s which would s a t i s f y t h e requirements s e t i n ( b ) . Should one indeed be a b l e t o p u t through t o a d i f f e r e n t i a l e q u a t i o n M(x,y)&+N(x,y)dy=O

f o r the orthogonal

t r a j e c t o r i e s , then t h e r e might a w a i t another ambush; i n t e g r a t i o n of d i f f e r e n t i a l e q u a t i o n s was o f t e n impossible, and hence one might s t i l l g e t s t u c k a t s t e p ( d ) . F i n a l l y , t r a n s l a t i o n of an a n a l y t i c s o l u t i o n of a d i f f e r e n t i a l e q u a t i o n i n t o a geometric c o n s t r u c t i o n ( s t e p ( e ) ) was i n i t s e l f q u i t e a problem. This i n t e g r a l p a r t of any

s o l u t i o n of a geometric problem c o u l d , however, draw

from a v a s t number o f h i g h l y developed methods, c a l l e d " c o n s t r u c t i o n s o f d i f f e r e n t i a l e q u a t i o n s " ; a s f a r as I know, t h e s e methods never f a i l e d f o r t r a j e c t o r y c o n s t r u c t i o n s . I s h a l l r e f r a i n from going i n t o d e t a i l s about such c o n s t r u c t i o n s here'. Summarising, t h e r e were a number of o b s t a c l e s t h a t could i n p r i n c i p l e o b s t r u c t t h e s u c c e s s f u l a p p l i c a t i o n of L e i b n i z ' s method: t h e problem of prov i d i n g an adequate e q u a t i o n f o r f a m i l i e s of t r a n s c e n d e n t a l

c u r v e s , such a s

l o g a r i t h m s , t h e i m p o s s i b i l i t y of e l i m i n a t i n g parameters from t r a n s c e n d e n t a l X

e x p r e s s i o n s y=

pfx,a)dx, t h e e l i m i n a t i o n o f parameters from a l g e b r a i c ex-

X

p r e s s i o n s , and t g e s o l u t i o n of d i f f e r e n t i a l e q u a t i o n s . Hence, when Johann Bern o u l l i s i n g l e d o u t t h e t r a j e c t o r y problem f o r t r a n s c e n d e n t a l curves as t h e main problem, while pushing a s i d e t h e e l i m i n a t i o n problem f o r a l g e b r a i c exp r e s s i o n s and t h e i n t e g r a t i o n problem f o r d i f f e r e n t i a l e q u a t i o n s , h e made a choice which r e f l e c t e d h i s p e r s o n a l a p p r e c i a t i o n s and p r e f e r e n c e s , Johann Bern o u l l i ' s o p i n i o n d i d break w i t h t h e c u r r e n t o p i n i o n t h a t a geometric problem had t o be solved by a geometric c o n s t r u c t i o n , and, t h e r e f o r e , h i s f o r m u l a t i o n of the t r a j e c t o r y problem w a s open t o c r i t i c i s m . Indeed, i t d i d n o t t a k e long

Logarithmic curves

65

b e f o r e such c r i t i c i s m was voiced, l o u d l y and a c c u r a t e l y , by Jakob B e r n o u l l i . However, Johann B e r n o u l l i ' s o p i n i o n was a l l b u t u n r e a s o n a b l e : a t t h a t time

i t seemed a s though t h e t r a j e c t o r y problem f o r t r a n s c e n d e n t a l curves d i d n o t even allow a n a l y t i c f o r m u l a t i o n ; i t could n o t even be reduced t o an e l i m i n a t i o n problem o r a d i f f e r e n t i a l e q u a t i o n . For a l g e b r a i c curves one could a t l e a s t come t h a t f a r . Furthermore, e l i m i n a t i o n from a l g e b r a i c e q u a t i o n s o r s o l u t i o n of d i f f e r e n t i a l e q u a t i o n s a r e not a c t i v i t i e s which a r e s p e c i f i c f o r s o l v i n g t r a j e c t o r y problems. They occur everywhere i n a n a l y s i s . Hence, f i n d i n g methods t o c a r r y o u t e l i m i n a t i o n s o r t o s o l v e d i f f e r e n t i a l e q u a t i o n s i s one of t h e g e n e r a l problems i n mathematics, and any s p e c i f i c problem might r e a s o n a b l y b e regarded solved as soon a s i t can be reduced t o one of t h e s e g e n e r a l problems. But then geometric c r i t e r i a can no longer b e a p p l i e d t o d e c i d e about t h e a c c e p t a b i l i t y of s o l u t i o n s ; they must be r e p l a c e d by a n a l y t i c c r i t e r i a . I t i s t h i s i n t e r p l a y

between a n a l y t i c and geometric a s p e c t s which can be i d e n t i f i e d i n the 1 7 t h and 18th c e n t u r y d i s c u s s i o n s about o r t h o g o n a l t r a j e c t o r i e s .

53.5 Logarithmic curves Johann B e r n o u l l i had made an a s t o n i s h i n g choice when h e proposed a family of

logarithmic curves i n order t o i l l u s t r a t e h i s point t h a t the t r a j e c t o r y

problem w a s d i f f i c u l t f o r t r a n s c e n d e n t a l c u r v e s and simple f o r a l g e b r a i c curves. Since t h e s e l o g a r i t h m i c curves a r e d e s c r i b e d by t h e e q u a t i o n

(3.5)

x = a.logiyl,

i t i s e a s y t o s o l v e t h i s e q u a t i o n f o r a e x p l i c i t l y : a = x/logfy). Furthermore, L e i b n i z ' s method immediately produces t h e n i c e l y s e p a r a t e d d i f f e r e n t i a l equation (3.6)

X&

= -yZogfy)dg

f o r t h e orthogonal t r a j e c t o r i e s . What then,one a s k s , w e r e Johann B e r n o u l l i ' s d i f f i c u l t i e s w i t h t h e s e l o g a r i t h m i c c u r v e s ? The answer i s s i m p l e , b u t unexpected: i n e a r l y 1696, Johann B e r n o u l l i d i d n o t know t h e s e l o g a r i t h m s w e l l enough t o have a n o t a t i o n by means of which he could e x p r e s s t h e given f a m i l y by an e q u a t i o n such a s ( 3 . 5 ) . L e i b n i z ' s method, t h e r e f o r e , a l r e a d y f a i l e d a t the f i r s t s t e p i n t h i s case. L e i b n i z r e p l i e d t o Johann B e r n o u l l i ' s remarks about t h e t r a j e c t o r y problem f o r t r a n s c e n d e n t a l curves i n August 1696:

66

Orthogonal Trajectories 1694-1720

"I no longer remember what I wrote t o you about a Method of mine f o r f i n d i n g t h e p e r p e n d i c u l a r t o curves given by p o s i t i o n i n o r d e r e d sequence, which method you deny o b t a i n i n g f o r t r a n s c e n d e n t a l s ; I ask you t o i n d i c a t e t o me what i t c o n s i s t s of"'', and t h e n , obviously w i t h o u t knowing p r e c i s e l y what h i s method was, L e i b n i z immediately continued: " I t can a t l e a s t be accomodated t o t r a n s c e n d e n t a l c u r v e s given by an ex-

p o n e n t i a l l y p e r c u r r e n t e q u a t i o n ; r e a l l y , I have always r a t e d t h e s e exp r e s s i o n s f o r [ t r a n s c e n d e n t a l c u r v e s ] t o be t h e most p e r f e c t . P e r c u r r e n t e x p r e s s i o n s I c o n s i d e r t o be a gender, and e x p o n e n t i a l s , indeed, a r e t h e i r most p e r f e c t s p e c i e s " . l o " P e r c u r r e n t e x p r e s s i o n s " was the t e c h n i c a l term which L e i b n i z employed a t t h a t time t o denote t r a n s c e n d e n t a l e x p r e s s i o n s , and e x p o n e n t i a l l y p e r c u r r e n t exp r e s s i o n s t o him were those e x p r e s s i o n s which o n l y involved e x p o n e n t i a l s , such as f o r instance a

x , xx , y x and s o f o r t h " .

Such e x p r e s s i o n s he regarded t o be

so c l o s e l y r e l a t e d t o a l g e b r a i c e x p r e s s i o n s t h a t h e t r u s t e d any a l g o r i t h m f o r a l g e b r a i c e x p r e s s i o n s a l s o t o h o l d f o r them. Motivated by L e i b n i z ' s remarks, Johann B e r n o u l l i soon succeeded i n f i n d i n g a s o l u t i o n i n terms of e x p o n e n t i a l e x p r e s s i o n s f o r t h e t r a j e c t o r i e s i n h i s family of l o g a r i t h m i c curves'

2:

Consider a l l such c u r v e s w i t h common asymptote AB and common p o i n t C; one curve of r e f e r e n c e

CE i s then chosen, such t h a t i t s s u b t a n g e n t

segment AC. Hence, t h i s curve i s t h e one w i t h a 45' X

a i s equal t o t h e

t a n g e n t i n C and t h u s p l a y s

t h e r o l e of y=e ; o n l y f o r t h i s curve of r e f e r e n c e d i d Johann B e r n o u l l i employ a s p e c i a l n o t a t i o n e q u i v a l e n t t o x=Zogfyl. fig. 3

67

The break-through to transcendental curves

The o t h e r curves l i k e CD and Cd a r e r e l a t e d t o t h e curve of r e f e r e n c e by means of t h e g e o m e t r i c a l p r o p e r t y t h a t t h e a b s c i s s a e a t corresponding o r d i n a t e s y i n t h e curve CD and CE r e s p e c t i v e l y a r e i n t h e same p r o p o r t i o n a s t h e s u b t a n g e n t s of t h e s e c u r v e s 1 3 ; t h u s , d e n o t i n g t h e s u b t a n g e n t of CD by y d x / d y , t h i s y i e l d s :

(ydx:dyI:a=x:Zogly).

(3.7)

Now, s i n c e according t o L e i b n i z ' s method (&:dyi

curve

=f-dy:dx)

trajectory

( c f . (3. 1)) , t h e t r a j e c t o r i e s a r e d e s c r i b e d by t h e d i f f e r e n t i a l e q u a t i o n

-y Zog ( y )dy=oxdx.

(3.8)

Johann B e r n o u l l i could s o l v e t h i s d i f f e r e n t i a l e q u a t i o n b y m e a n s of the exponential expression b

2 2y 2y2-by2 , where b i s taken such t h a t Zogb=o.

I t i s i n t e r e s t i n g t h a t Johann B e r n o u l l i s t i l l d i d n o t u s e t h e l o g - n o t a t i o n

h e r e f o r a l l l o g a r i t h m i c c u r v e s ; i t o n l y a p p l i e d t o t h e c u r v e of r e f e r e n c e . Only i n 1 7 1 7 d i d Jakob Hermann d e a l w i t h t h e s e l o g a r i t h m i c c u r v e s by s t r a i g h t forward a p p l i c a t i o n of L e i b n i z ' s method t o t h e e q u a t i o n (3.5)

".

However, f o r t h o s e simple t r a n s c e n d e n t a l c u r v e s , which could be e x p r e s s e d by means of l o g a r i t h m s and e x p o n e n t i a l s , Johann B e r n o u l l i had now come t o g r i p s w i t h the t r a j e c t o r y problem. The b i g problem t h a t remained was how t o develop an approach t o t r a n s c e n d e n t a l s i n g e n e r a l , v i z . those t r a n s c e n d e n t a l s given only by an e q u a t i o n l i k e y= rone c y c l o i d s .

53.6 The break-through

X

J p ( x , a ) & such a s f o r i n s t a n c e t h e brachystoch20

t o transcendental, c u r v e s

Johann B e r n o u l l i ' s g r e a t break-through f o r t r a n s c e n d e n t a l c u r v e s came i n August 1697, and was an immediate consequence of L e i b n i z ' s d i s c o v e r y e a r l i e r t h a t month of t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e gration

15

. When

he r e c e i v e d L e i b n i z ' s l e t t e r c o n t a i n i n g t h i s theorem, Ber-

n o u l l i a t once r e c o g n i s e d t h a t i t opened t h e way t o d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e parameter f o r any t y p e of e x p r e s s i o n . There had been no d i f f i c u l t y i n interpreting V (x,y,al

a

as f a r as a l g e b r a i c e x p r e s s i o n s V(z,y,a) a f x p(x,aId;c had been

were concerned, and now t h e problem of i n t e r p r e t i n g -

aa

solved a s w e l l .

x0

B e r n o u l l i f i r s t adopted an ad hoc n o t a t i o n f o r t h e p a r t i a l d i f f e r e n t i a l s of any v a r i a b l e y ( e q u i v a l e n t t o o u r day and d 2 ) l 6 . He t h e n approached t h e t r a j e c t o r y problem w i t h t h i s extended i d e a of d i f f e r e n t i a t i o n .

Orthogonal Trajectories 1694-1 720

68

fig. 4

l

a

A

X

Y

I

Consider a family of curves r e p r e s e n t e d i n f i g . 4 by AF, AE, AC and one of i t s orthogonal t r a j e c t o r i e s FEC. L e t AH=x and HB-y.

Along t h e t r a j e c t o r y FEC t h e

parameter a of t h e f a m i l y of i n t e r s e c t e d c u r v e s can be c o n s i d e r e d a s depending on the a b s c i s s a cc. L e t the curve A G , drawn above t h e h o r i z o n t a l a x i s AH, r e p r e s e n t how t h i s parameter v a r i e s w i t h cc a l o n g FEC, thus G H = a f s ) . Johann Bern o u l l i then d e r i v e d a d i f f e r e n t i a l e q u a t i o n f o r t h i s r e l a t i o n a ( s ) d e f i n e d by the curve A G ; such a d i f f e r e n t i a l e q u a t i o n can indeed be regarded a s a d i f f e r e n t i a l e q u a t i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s , s i n c e f o r example FEC can be c o n s t r u c t e d as soon a s t h i s curve AG r e p r e s e n t i n g a=a(x) i s known. I n f i g . 4 take DC=&,

a ( E ) < a (B)=a (C)I , BD=d$=pdx,

BE=-day=-qda

(minus s i g n because y(E)>y(B) and

t h u s DE=BE-BD-qda-pdx.

Since ED. DE=DC2 (because

ABEC i s r e c t a n g u l a r ) , one has pdx(-qda-pdx/=dx2, and hence - p q d a - p 2 d x = d x

or

(1+p2 ) dx+pqda=o.

(3.9)

I s h a l l c a l l t h i s d i f f e r e n t i a l e q u a t i o n (3.9)

t h e v a r i a b l e parameter e q u a t i o n ;

i t d e f i n e s t h e r e l a t i o n between t h e v a r i a b l e parameter a and t h e a b s c i s s a

3:

along t h e orthogonal t r a j e c t o r i e s of the f a m i l y of c u r v e s A F , AE, A C . The v a r i a b l e parameter e q u a t i o n ( 3 . 9 ) provided an a l t e r n a t i v e approach t o t h e orthogonal t r a j e c t o r y problem such a s Johann B e r n o u l l i had been l o o k i n g f o r . I t had one g r e a t advantage over L e i b n i z ' s method i n t h a t i t s a p p l i c a b i l i t y was

n o t r e s t r i c t e d t o a l g e b r a i c c u r v e s a l o n e . The q u a n t i t i e s p and q o c c u r r i n g i n t h e v a r i a b l e parameter e q u a t i o n can n o t o n l y b e found f o r a l g e b r a i c c u r v e s , b u t also

-

due t o t h e i n t e r c h a n g e a b i l i t y theorem - f o r t r a n s c e n d e n t a l c u r v e s given

by an e q u a t i o n of t h e form

69

Jakob Bernoulli's reaction

For such f a m i l i e s of t r a n s c e n d e n t a l c u r v e s , t h e v a r i a b l e parameter e q u a t i o n t a k e s on t h e form:

Hence, f a m i l i e s of t r a n s c e n d e n t a l curves d i d no l o n g e r w i t h s t a n d a n a l y t i c a l f o r m u l a t i o n of t h e t r a j e c t o r y problem: The g e n e r a l problem of c o n s t r u c t i n g o r t h o g o n a l t r a j e c t o r i e s had now been reduced t o t h e a n a l y t i c a l problem of int e g r a t i n g d i f f e r e n t i a l e q u a t i o n s . However, t h i s was only a f i r s t s t e p towards X

complete mastery o f t h e problem; u n l e s s t h e i n t e g r a l 1 p algebraically i n

"0

5

dx

a

could be e x p r e s s e d

and a , t h e r e were no methods a v a i l a b l e t o a c t u a l l y s o l v e

d i f f e r e n t i a l e q u a t i o n s l i k e (3.11)

which i n v o l v e t r a n s c e n d e n t a l c o e f f i c i e n t s .

53.7 Jakob Bernoulli's reaction

Jakob B e r n o u l l i r e a c t e d t o h i s b r o t h e r ' s o r t h o g o n a l t r a j e c t o r y problem i n

1698, and he immediately took e x c e p t i o n t o Johann's view t h a t t h e e s s e n t i a l d i f f i c u l t y of the t r a j e c t o r y problem l a y i n t h e t r e a t m e n t of t r a n s c e n d e n t a l curves : "The q u e s t i o n h e r e i s about such c u r v e s t h a t i n t e r s e c t a l l given c u r v e s a t r i g h t a n g l e s . T h i s problem depends on t h e i n v e r s e method of t a n g e n t s [ i . e .

on s o l v i n g d i f f e r e n t i a l e q u a t i o n s ] , and, t h e r e f o r e , i t does n o t permit a g e n e r a l s o l u t i o n ; f o r d i f f e r e n t p o s i t i o n s of t h e given c u r v e s i t i s of a wonderful d i v e r s i t y . N e i t h e r t h e degree nor t h e s p e c i e s of t h e c u r v e s i s a c r i t e r i o n f o r t h e s i m p l i c i t y o r d i f f i c u l t y of t h e problem, s i n c e sometimes t h e m a t t e r succeeds o n l y w i t h d i f f i c u l t y f o r a l g e b r a i c c u r v e s b u t , on t h e contrary, with ease f o r transcendental Jakob B e r n o u l l i c o r r o b o r a t e d h i s p o i n t i n two ways: f i r s t he provided cons t r u c t i o n s f o r t h e o r t h o g o n a l t r a j e c t o r i e s of f i v e d i f f e r e n t f a m i l i e s of c u r v e s , t h r e e of them f a m i l i e s of p a r a b o l a s , and two of them f a m i l i e s of l o g a r i t h m i c curves, i n c l u d i n g t h e example of l o g a r i t h m i c c u r v e s which Johann B e r n o u l l i had proposed. Although Jakob B e r n o u l l i d i d n o t r e v e a l h i s methods, i t seems obv i o u s t h a t h i s a n a l y s i s had run a l o n g t h e l i n e s of L e i b n i z ' s method. Secondly, Jakob B e r n o u l l i proposed a t r a j e c t o r y problem f o r an a l g e b r a i c f a m i l y of curves i n r e t u r n : "The s o l u t i o n of a l l t h e s e c a s e s was e a s y indeed; however, o t h e r p o s i t i o n s of t h e c u r v e s can b e given which r e n d e r t h e problem much more d i f f i c u l t , and

Orthogonal Trajectories 1694-1 720

I0

which l e a d t o unexplored c a s e s of t h e i n v e r s e method of t a n g e n t s [ a l r e a d y ] f o r a simple p a r a b o l a , f o r example when t h e curve i s sought t h a t i n t e r s e c t s

perpendicuzar Zy aZZ parabolas above the same a x k , having l a t e r a r e c t a which are equal t o the r e s p e c t i v e distances of t h e v e r t e x t o a given p o i n t . The family of p a r a b o l a s mentioned by Jakob B e r n o u l l i i s n o t determined u n i q u e l y , s i n c e both the p a r a b o l a s given by

y2=a(x-a!

(3.12)

and t h e p a r a b o l a s given by

y'=a(z+al

(3. 13)

s a t i s f y Jakob's d e s c r i p t i o n . I n e i t h e r of t h e s e c a s e s L e i b n i z ' s method l e a d s t o a d i f f e r e n t i a l e q u a t i o n f o r t h e orthogonal t r a j e c t o r i e s , v i z .

ydy2+2x&dy+4ydrC2=O,

(3. 14) or

ydy2+2~&dy-4y&'=0

(3.15)

r e s p e c t i v e l y . Apparently, n e i t h e r of t h e s e d i f f e r e n t i a l e q u a t i o n s could be solved a t t h a t t i m e " .

Johann B e r n o u l l i ' s s o l u t i o n , a geometric one a s r e -

q u i r e d by 17th c e n t u r y s t a n d a r d s , appeared i n 1698, and was n e i t h e r based on L e i b n i z ' s method n o r on the v a r i a b l e parameter e q u a t i o n ; i n s t e a d Johann Bern o u l l i had t o develop another method f o r s o l v i n g t h i s problem, a method based on the s i m i l a r i t y o f t h e parabolas".

Jakob B e r n o u l l i ' s r e a c t i o n was as simple

a s powerful; h i s view of t h e e s s e n t i a l d i f f i c u l t y of t h e t r a j e c t o r y problem was completely a t v a r i a n c e with h i s b r o t h e r s , who had d e c l a r e d t h e t r a j e c t o r y problem t o be e a s y f o r a l g e b r a i c curves. The c l u e t o t h e d i f f e r e n c e i n o p i n i o n about t h e e s s e n t i a l d i f f i c u l t y of t h e orthogonal t r a j e c t o r y problem w i l l be c l e a r from our d i s c u s s i o n of t h e l i m i t s t o L e i b n i z ' s method i n J3.4. Jakob B e r n o u l l i adopted t h e r i g i d , though c u r r e n t and w e l l e s t a b l i s h e d p o s i t i o n t h a t a geometric problem i s s o l v e d o n l y by a geometric c o n s t r u c t i o n ; he o b v i o u s l y d i d n o t a c c e p t t h e r e d u c t i o n of t h e t r a j e c t o r y problem t o d i f f e r e n t i a l e q u a t i o n s a s a s o l u t i o n . Given such a p o i n t of view, t h e r e i s indeed no reason t o d e c l a r e t h e t r a j e c t o r y problem f o r t r a n s c e n d e n t a l s t o be more d i f f i c u l t than f o r a l g e b r a i c curves. Furthermore, Jakob B e r n o u l l i ' s p o s i t i o n was only c o n s o l i d a t e d by h i s l i m i t a t i o n of t h e concept of t r a n s c e n d e n t a l c u r v e s t o simple t r a n s c e n d e n t a l s l i k e l o g a r i t h m s ; he seemed unwilling

-

for

t a c t i c a l reasons?

-

t o extend h i s realm of i n t e r e s t i n g t r a n s -

c e n d e n t a l curves t o t h e c l a s s of curves r e p r e s e n t e d by an e q u a t i o n of t h e type

Renascence of the problem

71

X

(3.10):

I p(X,a)dx.

y=

"CO

Johann

Thus f o r Jakob B e r n o u l l i t h e game w a s o v e r - and won?

B e r n o u l l i t r i e d once more t o make h i s p o s i t i o n c l e a r :

"What I have adduced may s u f f i c e t o show t h a t a method f o r f i n d i n g the equat i o n [of the o r t h o g o n a l t r a j e c t o r i e s ] - p e r t a i n i n g n o t o n l y t o j u s t one o r a n o t h e r a l g e b r a i c c a s e , b u t t o a l l - has a l r e a d y been f a m i l i a r t o us [ i . e . t o h i m s e l f , and t o L e i b n i z ] f o r long. B u t s i n c e i t does n o t succeed i n t r a n s c e n d e n t a l s , u n l e s s they

a r e s i m i l a r such as t h e c y c l o i d s [ t h i s i s b l u f f ;

B e r n o u l l i could o n l y d e a l w i t h them by means of h i s optico-mechanical a r guments], o r u n l e s s they can be reduced t o one c o n s t a n t curve such a s t h e Logarithms [of 93.51

, [...I

a n o t h e r method had t o be i n v e n t e d t h a t i s g e n e r a l

and can be a p p l i e d t o whatever [ f a m i l y of c u r v e s ] given by p o s i t i o n i n ordered sequence, b o t h t r a n s c e n d e n t a l and a l g e b r a i c ones [ . . . I . I have f i n a l l y digged up, [ .

for b e t t e r perfection

-

.. 1 and n o t

Such a method

even t h e l e a s t b i t can be added t o i t

a s Leibniz can w i t n e s s - i f o n l y , because i t always

l e a d s t o an e q u a t i o n [ t h a t i s : t h e v a r i a b l e parameter e q u a t i o n ] ; i f i n t h i s e q u a t i o n the v a r i a b l e s happen t o be i n s e p a r a b l e t h i s w i l l n o t make t h e method l e s s p e r f e c t , s i n c e i t i s n o t p a r t of t h i s , b u t of a n o t h e r method t o s e p a r a t e t h e v a r i a b l e s . T h e r e f o r e , I ask my b r o t h e r t o apply h i s s k i l l s i n a m a t t e r of such importance a l s o . " 2 1 Jakob B e r n o u l l i never responded t o t h i s challenge",

and t h e problem of ortho-

gonal t r a j e c t o r i e s w a s r e l e g a t e d t o o b l i v i o n .

53.8 Renascence of t h e problem When t h e p r i o r i t y d i s p u t e o v e r t h e d i s c o v e r y of t h e c a l c u l u s was r e a c h i n g i t s summit L e i b n i z a t t h e end of 1714 informed Johann B e r n o u l l i how he intended to r e t a l i a t e against the English

mathematicians:

"I s h a l l a l s o do my b e s t t o produce some [problems] i n which I know Newton w i l l falter."23 For t h e n e x t y e a r L e i b n i z and B e r n o u l l i mathematical

-

-

h i s c o n t i n u a l c o n f i d a n t i n matters

d i s c u s s e d t h e choice of a d i f f i c u l t problem t h a t would s u i t t h e i r

purpose. Their d i s c u s s i o n i s v e r y i n t e r e s t i n g , s i n c e i t p r o v i d e s us w i t h a good p i c t u r e of what b o t h men considered t o be t h e most d i f f i c u l t p a r t s o f t h e i r d i f f e r e n t i a l c a l c u l u s ; we s e e on which i s s u e s they judged t h e d i f f e r e n t i a l c a l c u l u s could b e s t contend f o r t h e mastery w i t h i t s f l u x i o n a l c o u n t e r p a r t . B e r n o u l l i r e p l i e d t o L e i b n i z ' s remark a l r e a d y i n h i s n e x t l e t t e r :

12

Orthogonal Trajectories 1694-1 720

“No doubt t h e r e a r e a l o t of such [problems] which we have once d i s c u s s e d

and which a r e by no means easy t o t r e a t w i t h t h e common method of d i f f e r e n t i a l s : of t h i s type a r e t h o s e [problems] about t h e t r a n s i t i o n from curve t o c u r v e , which w e have d e a l t w i t h by h a r n e s s i n g a c e r t a i n s i n g u l a r t y p e of differentiation.

‘I2

Most of t h e problems which Johann B e r n o u l l i went on t o mention a r e c l o s e l y rel a t e d t o the ones w e know from t h e 1690s, such a s , f o r example, t h e problem of q u i c k e s t approach i n a family of a f f i n e e l l i p s e s . These a r e p r e c i s e l y t h e problems which had n e c e s s i t a t e d t h e e x t e n s i o n o f t h e d i f f e r e n t i a l c a l c u l u s t o v a r i a b l e parameters and had l e d t o t h e d i s c o v e r y of t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n . I t was r e a d i l y decided t h a t t h e c h a l l e n g e problem should indeed involve f a m i l i e s of curves and t h e procedure of d i f f e r e n t i a t i o n from curve t o curve; t h i s choice was n o t an unreasonable one, s i n c e b o t h men s t i l l had a v i v i d r e c o l l e c t i o n of the o b s t a c l e s they had encountered i n 1697 b e f o r e they had made t h e break-through

t o t h i s type of

d i f f e r e n t i a t i o n . The English mathematicians would almost c e r t a i n l y e n c o u n t e r such d i f f i c u l t i e s should they come t o t r y t h e i r s k i l l s on problems concerning t r a j e c t o r i e s i n f a m i l i e s of curves. L e i b n i z f i n a l l y chose t h e o r t h o g o n a l t r a j e c t o r y problem. I n November 1715 Johann B e r n o u l l i had mailed a copy of h i s 1697-derivation of t h e v a r i a b l e parameter e q u a t i o n t o HannoverZ5, and L e i b n i z immediately a f t e r w a r d s confirmed t h a t he had put t h i s problem t o t h e E n g l i s h mathematicians. However, h i s c h o i c e was an unlucky one, and L e i b n i z had n o t informed himself p a r t i c u l a r l y w e l l about t h e p r o g r e s s made i n t h i s problem. I n 1697 t h e i n t e r c h a n g e a b i l i t y theorem had indeed provided t h e g r e a t break-through

t o t h e v a r i a b l e parameter

equation (3. 11)

L:

il+p2(x,al)dx+(p(x,al

.

p,(x>a)dzlda=O

f o r t h e t r a j e c t o r i e s i n f a m i l i e s of t r a n s c e n d e n t a l curves. But n e i t h e r Johann B e r n o u l l i nor L e i b n i z himself had achieved any f u r t h e r r e s u l t s i n t h i s f i e l d . Thus by 1715 t h e approach t o o r t h o g o n a l t r a j e c t o r i e s a l o n g t h e l i n e s of v a r i a b l e parameters was by no means so f u l l y developed and understood t h a t i t could be used w i t h confidence and g u a r a n t e e s u c c e s s a g a i n s t t h e E n g l i s h : t h e n e c e s s a r y complement t o t h e v a r i a b l e parameter e q u a t i o n w a s s t i l l b a d l y missing. There were s t i l l no methods f o r i n t e g r a t i n g such d i f f e r e n t i a l e q u a t i o n s w i t h t r a n s c e n d e n t a l c o e f f i c i e n t s . Thus no geometric c o n s t r u c t i o n s f o r o r t h o g o n a l t r a j e c t o r i e s could y e t be expected from t h e v a r i a b l e parameter e q u a t i o n .

13

First reactions to the chullenge 53.9 First reactions t o the chaZZenge

L e i b n i z had t r a n s m i t t e d h i s c h a l l e n g e problem t o t h e E n g l i s h i n December

1715 through h i s i n t e r m e d i a r y t h e Abb6 Conti i n P a r i s and h e had phrased i t i n t h e f o l l o w i n g form: "Find t h e l i n e BCD which i n t e r s e c t s a t r i g h t a n g l e s a l l c u r v e s given i n o r d e r e d p o s i t i o n and of the same k i n d , f o r example a l l hyperbolas w i t h t h e same v e r t e x and t h e same c e n t r e A B , AC, AD e t c . , and t h i s i n a g e n e r a l way.1126 Here a g a i n he had made a m i s t a k e ; although L e i b n i z had e x p l i c i t l y asked f o r a g e n e r a l approach t o t h e problem,the s p e c i f i c example of t h e hyperbolas was f a r too simple t o i l l u s t r a t e t h e degree of g e n e r a l i t y which L e i b n i z had i n mind. This example could be solved e a s i l y by b o t h E n g l i s h mathematicians and t h o s e c o n t i n e n t a l mathematicians who d i d n o t have t h e s l i g h t e s t i d e a about d i f f e r e n t i a t i o n from c u r v e t o curve. I t w a s Johann B e r n o u l l i who r e v e a l e d t h i s t a c t i c a l mistake t o L e i b n i z : "I admit t h a t t h i s problem i n i t s f u l l g e n e r a l i t y w i l l n o t e a s i l y b e s o l v e d

by t h e s e A n a l y s t s u n l e s s t h e y f i n d o u t about our way of d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e parameter o r o t h e r l i n e s t h a t can b e taken i n s t e a d , o r

about

t r a n s i t i o n by d i f f e r e n t i a t i o n from curve t o curve. But I s h o u l d l i k e t o c a l l your a t t e n t i o n t o t h e f a c t t h a t n o t a l l p a r t i c u l a r c a s e s of t h i s [problem] a r e e q u a l l y d i f f i c u l t : f o r t h e r e a r e those c a s e s which do n o t r e q u i r e t h i s pec u l i a r method of d i f f e r e n t i a t i o n , such as t h o s e c u r v e s , where t h e v a r i a b l e parameters need n o t be c o n s i d e r e d f o r t h e d e t e r m i n a t i o n of t h e t a n g e n t s . The example which you proposed about t h e h y p e r b o l a s i s of such a t y p e , s i n c e they, l i k e a l l c o n i c s w i t h common c e n t r e and v e r t e x , have t h e same subtangent f o r common a b s c i s s a s whatever t h e parameter i s : f o r a l l h o l d s OF:OA=OA:OE [see f i g u r e 5 I

fig. 5

0

14

Orthogonal Trajectories 1694-1 120

T h e r e f o r e , we must f e a r t h a t t h e English A n a l y s t s w i l l s o l v e t h i s c a s e w i t h common methods; and then, when they n o t i c e t h a t they have succeeded s o e a s i l y they w i l l emerge even more puffed-up w i t h p r i d e and even more confirmed i n t h e i r b e l i e f t h a t they are B e r n o u l l i enclosed w i t h h i s l e t t e r a s h o r t t r e a t i s e w r i t t e n by h i s son Nicolaus

11, i n which t h e hyperbola problem was s o l v e d , thus demonstrating t h e p o i n t he had made about t h e p a r t i c u l a r s i m p l i c i t y of t h e hyperbola problem. This t r e a t i s e appeared i n t h e Acta of May 1716 and was t h e f i r s t of a long s e r i e s of a r t i c l e s t h a t responded t o L e i b n i z ' s challenge". I s h a l l render Nicolaus I1 B e r n o u l l i ' s s o l u t i o n h e r e , which i l l u m i n a t e s Johann's remarks v e r y w e l l . For a l l hyperbolas c o n t a i n e d i n t h e e q u a t i o n

x2/a2-y2/b2=l,where a i s f i x e d and b i s t h e p a r a m e t e r , t h e subtangent a t abs c i s s a z i s given by o(x)=fx2-a2)/z;hence, t h e o r t h o g o n a l t r a j e c t o r i e s w i l l a l l have the same subnormal v ( r ) = o ( z )

a t a given a b s c i s s a 2 ; thus t h e i r d i f -

f e r e n t i a l e q u a t i o n can immediately be w r i t t e n down: -ydy/&=(x2-a2)/x,

and

s i n c e t h e v a r i a b l e s a r e n e a t l y s e p a r a t e d h e r e , t h e e q u a t i o n can be i n t e g r a t e d s t r a i g h taway

.

The c h a l l e n g e was of course p r i m a r i l y i s s u e d a t Newton, b u t he showed an expressed d i s l i k e f o r t h e t r a j e c t o r y problem and, i n f a c t , d e c l a r e d i t t o b e almost u s e l e s s . Indeed, t h e r e l a t i o n of t h e t r a j e c t o r y problem t o t h e Hugenian wave t h e o r y of l i g h t d i d n o t s e r v e t o m o t i v a t e t h e a u t h o r of t h e competing p a r t i c l e theory of l i g h t . Newton's s o l u t i o n appeared a s an anonymous s h o r t n o t e z 9 e a r l y 1716 i n the PhiZosophicaZ Transactions. He p o i n t e d o u t t h a t t h e problem had a l r e a d y been d i s c u s s e d i n t h e 1690s, p a r t i c u l a r l y by Johann Bern o u l l i i n h i s 2698, and took t h e p o s i t i o n t h a t agreement had a l r e a d y t h e n been reached t h a t t h e e s s e n t i a l p a r t o f t h e problem c o n s i s t e d of f i n d i n g t h e r e l e v a n t d i f f e r e n t i a l e q u a t i o n . H e then proceeded t o g i v e

an e n t i r e l y v e r b a l

d e s c r i p t i o n of how t o f i n d such d i f f e r e n t i a l e q u a t i o n s , which was a programme r a t h e r than a method. Other E n g l i s h mathematicians, f o r example John K e i l l succeeded i n s o l v i n g the t r a j e c t o r y problem f o r t h e hyperbolas q u i t e e a s i l y 3 ' , t h e r e b y e f f e c t i v e l y demonstrating the t r u t h of the remarks t h a t Johann B e r n o u l l i had made t o Leibniz. Newton's programme was h e a v i l y c r i t i c i s e d by Jakob Hermann

-

one of t h e

s t u d e n t s of t h e B e r n o u l l i ' s and by then p r o f e s s o r of mathematics i n F r a n k f u r t an d e r Oder

-

i n h i s f i r s t a r t i c l e about t h e t r a j e c t o r y problem of 2727. Her-

mann's main o b j e c t i o n was t h a t Newton's method would have r e c o u r s e t o second o r d e r d i f f e r e n t i a l e q u a t i o n s , whereas t h e problem c o u l d , and t h e r e f o r e s h o u l d

I5

The final test-case

be s o l v e d by means of f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s . Hermann demonstrated t h a t f i r s t o r d e r e q u a t i o n s d i d i n f a c t s u f f i c e by g i v i n g a c l e a r r u l e t h a t was a p p l i c a b l e t o a l g e b r a i c and simple t r a n s c e n d e n t a l c u r v e s . This r u l e became known a s t h e Canon Hermanni; i t was, i n f a c t , n o t h i n g e l s e b u t a v e r y c l e a r d e s c r i p t i o n o f t h e a l g o r i t h m discovered by L e i b n i z i n 1694: E l i m i n a t i o n of t h e parameter a from t h e e q u a t i o n V(x,y,a)=O and the d i f f e r e n t i a l e q u a t i o n

of t h e given f a m i l y of c u r v e s

VY (x, y , a ) d . ~ VX (x,y,a)dy=O y i e l d s a f i r s t o r d e r

d i f f e r e n t i a l e q u a t i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s . Hermann

demonstrated

the a p p l i c a t i o n of h i s Canon with f o u r examples, i n c l u d i n g t h e h y p e r b o l a s proposed by Leibniz and t h e l o g a r i t h m i c c u r v e s proposed by Johann B e r n o u l l i i n 1697 ( c f . 53.3). Such was t h e f a t e of L e i b n i z ' s c h a l l e n g e : Newton d i d n o t b o t h e r about i t , minor E n g l i s h mathematicians such a s K e i l l could s o l v e i t e a s i l y , and Jakob Hermann provided a f a i r l y g e n e r a l approach which w a s i n f a c t Leibn i z ' s own i n t e l l e c t u a l p r o p e r t y , b u t i n a new g u i s e . The t e s t had indeed f a i l e d t o have t h e e f f e c t expected.

53.10 The f i n a l test-ease

L e i b n i z answered Johann B e r n o u l l i ' s c r i t i c i s m about t h e choice of t h e hyperbolas f o r a s p e c i f i c example of the t r a j e c t o r y problem e a r l y i n 1716 a s follows: "I e x p l i c i t l y added t h a t a g e n e r a l method i s r e q u i r e d . But i f you would c a r e

t o provide me w i t h one example t h a t you t h i n k cannot be t r e a t e d by any part i c u l a r t r i c k b u t which r e q u i r e s a g e n e r a l t r e a t m e n t , I s h a l l be g r a t e f u l t o you.''3' Johann B e r n o u l l i r e p l i e d t o L e i b n i z ' s q u e s t i o n i n March 1716 by proposing a problem t h a t became t h e f i n a l t e s t - c a s e f o r a l l methods of c o n s t r u c t i o n f o r o r t h o g o n a l t r a j e c t o r i e s . The problem, however, was d i f f e r e n t from e a r l i e r t r a j e c t o r y problems i n t h a t now t h e c u r v e s t o be i n t e r s e c t e d were n o t given b u t had f i r s t t o be found from a g e o m e t r i c a l d e f i n i n g p r o p e r t y . The problem w a s formulated thus ( s e e f i g u r e 6 ) : (a) "Above t h e s t r a i g h t l i n e A G , which i s taken a s t h e a x i s , an i n f i n i t e number of curves l i k e ABD have t o be c o n s t r u c t e d through t h e p o i n t A which a r e of such a n a t u r e t h a t t h e r a d i u s of c u r v a t u r e a t each s i n g l e p o i n t B of each s i n g l e curve i s i n t e r s e c t e d by t h e a x i s AG i n C i n a given r a t i o , t o w i t

BO: BC=l :n"

Orthogonal Trujectories 1694-1 720

76

(b) The t r a j e c t o r i e s , such a s ENF, have t o be c o n s t r u c t e d which i n t e r s e c t t h e p r e v i o u s curves ABD a t r i g h t a n g l e s . " 3 2 fig.

I n h i s l e t t e r , Johann B e r n o u l l i provided c o n s t r u c t i o n s f o r both p a r t s o f the problem; b u t he added n e i t h e r a proof nor an a n a l y s i s f o r them. He c h a r a c t e r i s e d t h e f a m i l y of curves t o b.e i n t e r s e c t e d , and r e q u i r e d i n p a r t (a) of t h e problem by (3.16)

X

dx

Since e q u a t i o n (3.16) y i e l d s t h e well-known brachystochrone c y c l o i d s i n t h e case n=$, I s h a l l c a l l the curves d e f i n e d by (3.16) t h e " g e n e r a l i s e d c y c l o i d s " . Johann B e r n o u l l i c o n s t r u c t e d t h e orthogonal t r a j e c t o r i e s by means of a n a u x i l i a r y curve A M 3 3 , t h e o r d i n a t e s PM=z of which a r e given by z= Choose a c o n s t a n t c, which c o n s t a n t r e p r e s e n t s t h e parameter of

ay1

xn ~ p _ z 2 n - '

the t r a j e c t o r y E " F ; hence, a n o t h e r choice of c y i e l d s a n o t h e r t r a j e c t o r y . Now determine P (with o r d i n a t e x ) on t h e v e r t i c a l a x i s such t h a t t h e a r e a under t h e curve AM between the p o i z t s A and P is e q u a l t o c ( t h u s

JXo

Then xo w i l l be t h e o r d i n a t e of t h e p o i n t of i n t e r s e c t i o n N

0

&=c)

an

XnJP_521(

of t h e t r a j e c t o r y EflF and t h e given curve AN having parameter a . Put i n another way, the e q u a t i o n

an

JXO

dx=O

i m p l i c i t l y d e f i n e s t h e r e l a t i o n xo=x f a )

x n " J 7 along t h e t r a j e c t o r y ENF w i t h parameter c . 0

0

By way of i n t r o d u c t i o n , Johann B e r n o u l l i c h a r a c t e r i s e d t h e d i f f i c u l t i e s o c c u r r i n g i n t h i s twin problem as f o l l o w s : "I s h a l l c o m u n i c a t e

-

though w i t h o u t a n a l y s i s

-

the s o l u t i o n of a c a s e

The final test-case

I7

which has a l l the r e q u i s i t e s you d e s i r e ; i t cannot indeed be solved by any particular t r i c k , but rather requires a certain p a r t i c u l a r straightforwardn e s s , which w i l l n o t be obvious t o everybody. Furthermore i t i s a l s o n e c e s s a r y t o have r e c o u r s e t o second o r d e r d i f f e r e n t i a l s , which, by o u r method, can be reduced t o f i r s t o r d e r . F i n a l l y i t i s of such a n a t u r e t h a t , having c a r r i e d o u t t h e s e s t e p s , i t can be reduced t o q u a d r a t u r e s by means of a s i n g u l a r a r t i f i c e , whereas o t h e r w i s e , i f n o t t r e a t e d p r o p e r l y ,

an e q u a t i o n

emerges i n which t h e v a r i a b l e s and t h e i r d i f f e r e n t i a l s a r e found t o be s o e n t a n g l e d and mixed up t h a t t h e y seem t o be i n ~ e p a r a b l e . " ~ ~ This c h a r a c t e r i s a t i o n i n f a c t a p p l i e s only t o p a r t ( a ) of t h e problem, as may

become c l e a r from t h e f o l l o w i n g - r e c o n s t r u c t e d

-

argument: The p r o p o r t i o n

BO:BC=I:n l e a d s t o t h e second o r d e r d i f f e r e n t i a l e q u a t i o n (3.17)

-n(dx2+dy2)=xd&,

o r i n modern n o t a t i o n : -n11+(x'12)=zx". Now t h i s second o r d e r d i f f e r e n t i a l e q u a t i o n - b e i n g what i s c a l l e d "autonomous" because t h e independent v a r i a b l e

y does n o t appear - can be reduced t o a f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n f o r x and z by means of t h e s u b s t i t u t i o n z=x'. This s u b s t i t u t i o n zdy=dx Johann B e r n o u l l i presumably a l l u d e d t o i n the phrase "which can by o u r method b e r e duced t o f i r s t o r d e r " ; i t y i e l d s t h e n i c e l y s e p a r a b l e d i f f e r e n t i a l e q u a t i o n (3.18)

n ( z ' + I ) d x = -zzdz.

I n t e g r a t i o n of t h i s d i f f e r e n t i a l e q u a t i o n y i e Ids Zogx-2n+const=Zogf 2+z2 ),

and

by s u i t a b l y choosing t h e c o n s t a n t ("the s i n g u l a r a r t i f i c e " ? ) e q u a l t o 2 0 g a * ~ t h i s r e s u l t can be r e w r i t t e n as z=(aZn-x2")/xn. R e c o l l e c t i n g t h a t z=c?x/dy one arrives a t M

This f i n a l r e s u l t (3.19) c o i n c i d e s e x a c t l y w i t h t h e e q u a t i o n (3.16) g i v e n by Johann B e r n o u l l i i n h i s l e t t e r t o L e i b n i z . For t h e s u c c e s s f u l r e s o l u t i o n of p a r t ( a ) of t h e problem c o n s i d e r a b l e e x p e r i e n c e i n t r e a t i n g d i f f e r e n t i a l e q u a t i o n s i s r e q u i r e d , and, s o f a r , Johann B e r n o u l l i ' s c h a r a c t e r i s a t i o n of t h e problem seems t o be j u s t i f i e d . The argument t h a t had produced Johann B e r n o u l l i ' s r e s o l u t i o n of t h e second p a r t of t h e problem was r e v e a l e d o n l y i n 1720, when Nicolaus I1 B e r n o u l l i , Johan B e r n o u l l i ' s e l d e s t son, published a survey of methods f o r c o n s t r u c t i n g o r t h o g o n a l t r a j e c t o r i e s . The i d e a behind t h i s c o n s t r u c t i o n was f r i g h t f u l l y d u s t y and c l a s s i c a l ; i n f a c t , i t was n o t h i n g e l s e b u t a d i r e c t consequence o f

I8

Orthogonul Trajectories 1694-1 720

B e r n o u l l i ' s c o n s t r u c t i o n of t h e synchrone t o t h e b r a c h y s t o c h r o n e s , a c o n s t r u c t i o n p r e s e n t e d a l r e a d y i n 1697 ( c f . 1 2 . 2 . 2 ) .

Johann B e r n o u l l i e v e n t u a l l y t u r n e d t h i s

i d e a i n t o a method, which I s h a l l d i s c u s s i n t h e n e x t s e c t i o n . The t r a j e c t o r y problem f o r t h e g e n e r a l i s e d c y c l o i d s proposed by Johann B e r n o u l l i f o r the f i n a l t e s t was i n f a c t a w h i t e l i e , which d i d n o t r e a l l y f u l f i l t h e requirements Leibniz had s e t f o r a s u i t a b l e c h a l l e n g e . I t was anything b u t a c l e a r - c u t , d i f f i c u l t t r a j e c t o r y problem n e c e s s i t a t i n g a g e n e r a l approach; Johann B e r n o u l l i ' s own s o l u t i o n was a mere brand of s p e c i a l t r i c k s i n t h e f i e l d of d i f f e r e n t i a l e q u a t i o n s mixed w i t h a s i n g u l a r i d e a a l ready twenty y e a r s o l d . Thus i t i s n o t s u r p r i s i n g t h a t Johann B e r n o u l l i d i d n o t c o n f i d e h i s a n a l y s i s of t h e problem t o L e i b n i z , and o n l y provided him w i t h t h e f i n a l c o n s t r u c t i o n s . He had been caught i n a dilemma: t h e v a r i a b l e parameter e q u a t i o n had given b o t h Leibniz and himself a f e e l i n g of s u p e r i o r i t y which now f a i l e d t o m a t e r i a l i s e i n c o n c r e t e r e s u l t s , and he obviously d i d n o t d a r e conf e s s t o Leibniz t h a t h e had f a i l e d t o f i n d i n t e g r a t i o n methods f o r t h e v a r i a b l e parameter e q u a t i o n . Only once, s h o r t l y b e f o r e L e i b n i z ' s d e a t h , d i d Johann Bern o u l l i express h i s d i f f i c u l t i e s on t h i s p o i n t :

"I indeed e x p e r i e n c e something i n t h e c a s e of t r a n s c e n d e n t a l s t h a t t h w a r t s my d e s i g n s , and t h a t causes t h a t I cannot y e t b o a s t of having found a s o l u t i o n a s g e n e r a l a s I would d e s i r e ; t h i s , however, i s o n l y t h e c a s e f o r c e r t a i n types of t r a n s c e n d e n t a l s , s i n c e f o r v e r y many o t h e r s the m a t t e r succeeds q u i t e b e a u t i f u l l y

.

"

I n t h e following s e c t i o n I s h a l l d i s c u s s t h e a l t e r n a t i v e methods which Johann B e r n o u l l i had developed f o r t r e a t i n g t h e t r a j e c t o r y problem, and t o which he a l l u d e s i n t h i s quote. L e i b n i z t r a n s m i t t e d the t r a j e c t o r y problem f o r t h e g e n e r a l i s e d c y c l o i d s t o t h e E n g l i s h , immediately a f t e r he had r e c e i v e d i t from B e r n o u l l i ; a g a i n he

was s o l v e d successused h i s P a r i s connection, t h e Abb6 C ~ n t i The ~ ~ problem . f u l l y by Brook Taylor i n 1727. Newton himself d i d n o t a t t e m p t a s o l u t i o n , a l though he knew of t h e problem and c l e a r l y n o t i c e d t h a t t h e t r a j e c t o r y problem had now been supplemented by a problem concerning d i f f e r e n t i a l e q u a t i o n s ; t h i s becomes c l e a r from a remark i n h i s p r i v a t e p a p e r s : "And by a l l t h i s [ r e f e r r i n g t o t h e t r a j e c t o r y problem i n i t s f i r s t f o r m u l a t i o n ] t h e s e r i e s of curves t o be c u t i s given and n o t h i n g more i s t o be found then [ s i c ] t h e o t h e r s e r i e s which i s t o c u t i t a t r i g h t a n g l e s . But M r . L e i b n i t z [ s i c ] b e i n g t o l d t h a t h i s Probleme was s o l v e d , he changed i t i n t o a new one of f i n d i n g b o t h t h e s e r i e s t o be c u t and t h e o t h e r s e r i e s w [ h i ] c h i s t o c u t

i t . And the p a r t i c u l a r Probleme proposed i n t h i s L e t t e r i s a s p e c i a l c a s e ,

Johann BernoulliS alternatives

79

n o t of t h e g e n e r a l Probleme f i r s t proposed, b u t of t h i s new double g e n e r a l Problem. And t h e f i r s t p a r t of t h i s double Probleme ( v i z t [ v i d e l i c e t ] by any given p r o p e r t y of a s e r i e s of Curves t o f i n d t h e Curves) i s a g e n e r a l Problem h a r d e r then t h e former & of w [ h i ] c h M r L e i b n i z had a g e n e r a l s o l u t i o n . " 3 7 However, t h e f i n a l t e s t problem was n o t u n s o l v a b l e f o r Johann B e r n o u l l i ' s and Newton's contemporaries. I n 171 7 Jakob Hermann3' and Brook Taylor3') published c o n s t r u c t i o n s of t h e o r t h o g o n a l t r a j e c t o r i e s of t h e g e n e r a l i s e d c y c l o i d s ; i n 1718, Nicolaus I1 B e r n o u l l i i s s u e d two d i f f e r e n t c o n s t r u c t i o n s which were i n f a c t h i s f a t h e r ' s i n t e l l e c t u a l p r o p e r t y , and i n 1719 Nicolaus I B e r n o u l l i , a nephew of Johann B e r n o u l l i , c l o s e d t h e ranks w i t h y e t a n o t h e r s o l u t i o n . N e i t h e r Jakob Hermann n o r Nicolaus I B e r n o u l l i d i s c u s s e d how they had found t h e e q u a t i o n ( 3 . 1 6 ) f o r t h e g e n e r a l i s e d c y c l o i d s ; they c o n c e n t r a t e d s o l e l y on t h e problem of c o n s t r u c t i n g i t s f a m i l y of o r t h o g o n a l t r a j e c t o r i e s Jakob Hermann's and Brook T a y l o r ' s s o l u t i o n s were both based on t h e s i m i l a r i t y of t h e f a m i l y of g e n e r a l i s e d synchrones; t h i s s i m i l a r i t y based approach t o t h e t r a j e c t o r y problem was a l s o developed by Johann B e r n o u l l i - t o a v e r y h i g h degree of p e r f e c t i o n - and i t had provided one of t h e two c o n s t r u c t i o n s which Nicolaus

I1 B e r n o u l l i published i n 1718. I s h a l l d i s c u s s t h i s method t o g e t h e r w i t h t h e one t h a t had provided Johann B e r n o u l l i w i t h t h e c o n s t r u c t i o n d i s c u s s e d above i n t h e f o l l o w i n g s e c t i o n , and t h i s d i s c u s s i o n may a l s o s e r v e t o convey t h e f l a v o u r of Hermann's and T a y l o r ' s s o l u t i o n . N i c o l a u s I B e r n o u l l i ' s s o l u t i o n , published i n 1719, was t h e only one t h a t made u s e of t h e v a r i a b l e parameter e q u a t i o n ; i t d e f i n i t e l y marks a l a n d s l i d e i n t h e development of p a r t i a l d i f f e r e n t i a t i o n ; c h a p t e r 4 i s devoted e n t i r e l y t o Nicolaus I B e r n o u l l i ' s work.

§3.11 Johann B e r n o u l l i ' s a l t e r n a t i v e s The two c o n s t r u c t i o n s f o r t h e o r t h o g o n a l t r a j e c t o r i e s o f t h e f a m i l y of g e n e r a l i s e d c y c l o i d s which Nicolaus I1 B e r n o u l l i i s s u e d i n 1718 were i n t e n d e d i n the f i r s t p l a c e a s a show; nobody should t h i n k t h a t he could b e a t t h e B e r noulli's,

and t h u s n o t only one b u t two c o n s t r u c t i o n s were r e l e a s e d . However,

n e i t h e r of t h e s e c o n s t r u c t i o n s was accompanied by an e x p l a n a t i o n of how i t was found. The methods t h a t had produced t h e s e c o n s t r u c t i o n s were n o t r e l e a s e d u n t i l 1720, when Nicolaus I1 B e r n o u l l i p u b l i s h e d a long review about a l l t h e d i f f e r e n t methods f o r c o n s t r u c t i n g orthogonal t r a j e c t o r i e s . T h i s a r t i c l e had been w r i t t e n i n c l o s e c o l l a b o r a t i o n w i t h Johann B e r n o u l l i , whose pen one can r e c o g n i s e i n s u b s t a n t i a l s e c t i o n s of t h e a r t i c l e ; i t a p p e a r s t h a t t h e des-

Orthogonal Trajectories 1694-1 720

80

c r i p t i o n s of t h e d i f f e r e n t methods - t h e p u r e l y mathematical p a r t s

-

were w r i t t e n

by Johann B e r n o u l l i h i m s e l f , w h i l e the more polemical s e c t i o n s had been composed by Nicolaus I14'. The reason why Johann B e r n o u l l i decided t o appear i n d i s g u i s e can f a i r l y w e l l be t r a c e d i n volume 6 of Newton's Correspondence: he was a l l too eager n o t t o d e s t r o y h i s r e l a t i o n s w i t h t h e B r i t i s h which a t l e a s t a t a superf i c i a l l e v e l s t i l l looked good4'. So h e l e t h i s son engage i n the polemics. Given such a d i v i s i o n of l a b o u r between f a t h e r and son B e r n o u l l i , I s h a l l n o t b o t h e r about N i c o l a u s ' s formal a u t h o r s h i p of t h e survey and I s h a l l r e f e r t o the methods r e v e a r e d t h e r e a s Johann B e r n o u l l i ' s methods. These methods a r e : the s i m i l a r i t y method ( c f . 53.11.1)

( c f . 53.11.2),

,

t h e g e n e r a l i s e d synchrone method

and t h e v a r i a b l e parameter method ( c f . § 3 . 1 2 ) .

83.11.1 The s i m i l a r i t y method

The pre-eminent way of l o o k i n g a t f a m i l i e s of s i m i l a r curves adopted by Johann B e r n o u l l i was t o c o n s i d e r such f a m i l i e s t o be g e n e r a t e d by m u l t i p l i c a t i o n of a f i x e d given curve, say ABD, w i t h r e s p e c t t o t h e s i m i l a r i t y p o l e A.

This

curve ABD was termed t h e " p r i n c i p a l i s " .

fig. 7

I f one r e g a r d s the curve ABD i n f i g . 7 a s t h e p r i n c i p a l i s , o t h e r c u r v e s l i k e

AB'D' i n t h e f a m i l y a r e then d e f i n e d by t h e p r o p e r t y t h a t t h e r a t i o A B ' : A B i s c o n s t a n t f o r any p o i n t B on t h e p r i n c i p a l i s and i t s corresponding p o i n t B' on A B ' D ' , t h i s r a t i o AB':AB r e p r e s e n t i n g the m u l t i p l i c a t i o n f a c t o r . Obviously d i f f e r e n t curves of t h e family emerge f o r d i f f e r e n t v a l u e s of t h e m u l t i plication factor. Now t h e c l u e t o t h e s i m i l a r i t y approach t o o r t h o g o n a l t r a j e c t o r i e s i s t h e f o l l o w i n g c o n s i d e r a t i o n (see f i g u r e 8) :

Johann Bernoulli’s alternatives

81

fig. 8

An o r t h o g o n a l t r a j e c t o r y l i k e B”C‘E can a l s o be regarded a s b e i n g g e n e r a t e d by m u l t i p l i c a t i o n of t h e p r i n c i p a l i s ABD under t h e q u a l i f i c a t i o n t h a t t h e m u l t i p l i c a t i o n f a c t o r a i s no longer c o n s t a n t b u t d i f f e r s from p o i n t t o p o i n t on the p r i n c i p a l i s ; thus a(B)=AB”:AB, a(C)=AC‘:AC, a(E)=AE:AE=l. I n t r o d u c i n g t h e convention AB=r and AB‘=R,

the m u l t i p l i c a t i o n f a c t o r i s a=R:r, and a l o n g t h e

t r a j e c t o r y B”C’E we have R = R ( r l . Once one h a s d e r i v e d a method f o r e l i c i t i n g the v a l u e o f R ( r ) f o r each r one can indeed c o n s t r u c t t h e o r t h o g o n a l t r a j e c t o r y

B”C’E. Johann B e r n o u l l i ’ s method of f i n d i n g the r e l a t i o n R = R ( r ) , which had been employed i n t h e t r a j e c t o r y problem f o r t h e g e n e r a l i s e d c y c l o i d s i s d e s c r i b e d i n 534 o f t h e review (Nic.11 B e r n o u l l i 1 7 2 0 ) ; t h i s i s c l e a r from an e x p l i c i t r e f e r e n c e i n the t e x t . The argument proceeded as f o l l o w s ( t h e n o t a t i o n s b e i n g adapted t o mine):

fig. 9

A

82

Orthogonal Trajectories 1694-1 720

Consider diagram 9 , being an e l a b o r a t e d d e t a i l of f i g u r e 8, where B and C a r e taken t o be two i n f i n i t e l y c l o s e p o i n t s on t h e p r i n c i p a l i s , and B ' and C' consequently a r e i n f i n i t e l y c l o s e p o i n t s on t h e curve AB'D'

similar to the

p r i n c i p a l i s . Let B'IVE be an o r t h o g o n a l t r a j e c t o r y . I n t r o d u c e t h e p o i n t s F and G on t h e r a d i u s AC' such t h a t t h e segments AF and AG a r e e q u a l i n l e n g t h t o the segments AB and A B ' r e s p e c t i v e l y . Now FC=&

and GN=-&

(minus s i g n because

GN i s p o s i t i v e whereas & i s n e g a t i v e ) ; furthermore Johann B e r n o u l l i w i t h o u t f u r t h e r j u s t i f i c a t i o n put BF=X(r)dr; by X he understood "a q u a n t i t y somehow composed by r and c o n s t a n t s , t h a t i s , some f u n c t i o n of P

~ I ~s h a l~l d i~s c u s.s

the i m p l i c a t i o n s and j u s t i f i c a t i o n of t h i s s u p p o s i t i o n l a t e r . Two p a i r s of s i m i l a r t r i a n g l e s can be d i s c e r n e d i n t h e c o n f i g u r a t i o n of f i g u r e 9 : A(BFC)%A(B'GC') and A(B'GC)$AfNGB') and t h e r e f o r e a l s o : A(RFC)%A(NGB'). Hence the f o l l o w i n g p r o p o r t i o n a l i t y h o l d s : BE':E'C=NG:GB

'.

I n s e r t i n g FC=&,

NG=

-dR and

BF=XIri& and c o n s i d e r i n g t h a t GB'=(R:P)BF due t o s i m i l a r i t y , one a r r i v e s a t the f i n a l d i f f e r e n t i a l equation (3.20)

dR -- x 2 ( r ) dr - R

P

which d e s c r i b e s t h e r e l a t i o n between t h e r a d i u s r of a p o i n t on t h e p r i n c i p a l i s and t h e r a d i u s R of t h e corresponding p o i n t on t h e o r t h o g o n a l t r a j e c t o r y . Johann B e r n o u l l i d i d n o t go on t o show how (3.20) l e a d s t o the c o n s t r u c t i o n f o r t h e t r a j e c t o r i e s of t h e f a m i l y of g e n e r a l i s e d c y c l o i d s which N i c o l a u s I1 B e r n o u l l i had p u b l i s h e d i n h i s 1718. But t h e argument can b e r e c o n s t r u c t e d q u i t e e a s i l y . The main p o i n t i s t o f i n d a handy c h a r a c t e r i s a t i o n of t h e quant i t y X(P), which o c c u r s i n e q u a t i o n (3.20). I n t h e d e r i v a t i o n of (3.20) X(r) i s i n t r o d u c e d by way of t h e d e f i n i t i o n BF=X(rldr; i n t h e c o n s t r u c t i o n p u b l i s h e d

i n 1718, BF i s i d e n t i f i e d a s BF=tan($)dr, where $ i s t h e a n g l e between t h e t a n g e n t t o t h e p r i n c i p a l i s i n B and t h e r a d i u s AB ( s e e f i g u r e 10).

Johann Bernoulli’s alternatives Since t h e t r i a n g l e A f B F C ) i s i n f i n i t e l y s m a l l ,

+

83

can a l s o b e taken t o r e -

p r e s e n t t h e angle < ( B C F I , and thus i t i s c l e a r t h a t tan+ can indeed be taken f o r X ( r ) . By t h i s means, e q u a t i o n (3.20) t u r n s i n t o (3.21)

-

dR -- tan2+ dr R r

However, s o l u t i o n of t h i s e q u a t i o n seems t o be i m p o s s i b l e a n a l y t i c a l l y , as long a s tani$l cannot be e x p r e s s e d e x p l i c i t l y i n terms of r; i t w i l l o f t e n be d i f f i c u l t t o f i n d such an e x p l i c i t e x p r e s s i o n , and hence one might wonder i n what way (3.21) could l e a d t o a s o l u t i o n of t h e t r a j e c t o r y problem. Neverthel e s s , Johann B e r n o u l l i could s o l v e the e q u a t i o n (3.21) w i t h o u t having any exp l i c i t e x p r e s s i o n f o r tan+ o r X f r ) a v a i l a b l e . H e s o l v e d (3.21) by a g e o m e t r i c c o n s t r u c t i o n , t h a t i s t o say, he provided a geometric c o n s t r u c t i o n f o r t h e r e l a t i o n between R and

P

d e f i n e d by e q u a t i o n ( 3 . 2 1 ) , o r , e q u i v a l e n t l y , by i t s

i n t e g r a l equation

fig. 1 1

Consider t h e c o n f i g u r a t i o n drawn i n f i g u r e 1 1 , t h e f i r s t quadrant of which depicts

t h e given f a m i l y of g e n e r a l i s e d c y c l o i d s w i t h p r i n c i p a l i s ABD and

o r t h o g o n a l t r a j e c t o r y B’E. I n t h e second quadrant an a u x i l i a r y curve AVP w i t h a b s c i s s a e AM i s c o n s t r u c t e d as f o l l o w s : t a k e AM=r ( t h u s M i s t h e p o i n t of i n t e r s e c t i o n of a c i r c l e w i t h c e n t r e A and r a d i u s AB w i t h t h e v e r t i c a l a x i s ) ; take

K on t h e h o r i z o n t a l a x i s such t h a t t h e a n g l e
between t h e t a n g e n t t o the p r i n c i p a l i s i n B and t h e r a d i u s A B ( t h u s , AK:AM i n f a c t r e p r e s e n t s tan(+)); t a k e a p o i n t

I on t h e a x i s AM a t w i l l ( t h u s

84

Orthogonal Trajectories 1694-1 720

Al~=constant=c),and furthermore t a k e T on t h e h o r i z o n t a l a x i s a g a i n such t h a t

MK; t a k e L on t h e v e r t i c a l a x i s such

t h e segment IT i s p a r a l l e l t o t h e segment

t h a t t h e segment TL i s p e r p e n d i c u l a r t o MT, and f i n a l l y erect t h e o r d i n a t e (now h o r i z o n t a l s i n c e t h e a b s c i s s a e AM a r e v e r t i c a l ) PM of the a u x i l i a r y curve AVP a t p o i n t M such t h a t PM i s t h e same i n l e n g t h a s A L . The o r d i n a t e t h u s c o n s t r u c t e d r e p r e s e n t s t h e i n t e g r a n d of t h e right-hand s i d e i n t e g r a l i n e q u a t i o n ( 3 . 2 2 ) up t o the c o n s t a n t e2, a s can be seen by v e r i f y i n g t h e c o n s t r u c t i o n of PM(=AL):

AT i s t h e p e r p e n d i c u l a r i n t h e r e c t a n g u l a r t r i a n g l e A(MTL), one h a s

Since

AT2=AL. AM, o r AL=AT2:AM; (3.23)

hence, because AL=PM:

PM = AT2:AM

Due t o t h e s i m i l a r i t y of t h e t r i a n g l e s A(ATI) and A(AKM) one h a s AT:AI=AK:AM, and thusATcan be e x p r e s s e d as AT=AI.AK:AM.

Due t o t h e d e f i n i t i o n of

K one h a s :

AK:AM=tan(@), and c o n s i d e r i n g t h a t A I = e , one can e x p r e s s AT as AT=e.tan($); f i n a l l y i n s e r t i n g t h i s v a l u e i n ( 3 . 2 3 ) w h i l e r e c a l l i n g t h a t AM?

one a r r i v e s

at: (3.24)

PM = c2.tan2i@)/r

The i n t e g r a n d l/R of t h e l e f t - h a n d s i d e i n t e g r a l i n ( 3 . 2 2 )

- up t o t h e same c o n s t a n t c2

-

is r e p r e s e n t e d

by an a u x i l i a r y curve WS i n t h e f o u r t h q u a d r a n t :

c o n s t r u c t t h e hyperbola WS such t h a t i t s o r d i n a t e CW a t a b s c i s s a AC(=R) i s equal t o c2/R, o r i n o t h e r words, such t h a t AC.CW=c2 f o r a l l p o i n t s C. Take t h e p o i n t H a s f i x e d , AH r e p r e s e n t i n g R

0

(R0 then a c t s as t h e parameter f o r

t h e t r a j e c t o r y B 'E). Given t h e s e a u x i l i a r y curves A L P and WS, t h e r e l a t i o n R=R(r) can b e c o n s t r u c t e d a s f o l l o w s : For given r

(=AM and thus depends on t h e p o i n t B on

t h e p r i n c i p a l i s ) determine t h e p o i n t C such t h a t t h e a r e a under t h e h y p e r b o l a R e2& i s e q u a l t o t h e a r e a under t h e segment segment between W and S (=I -) of t h e a u x i l i a r y curve AW

Ro

(=

o

c2tan2'drl. Then t h e p o i n t r

C determines the

r a d i u s R(=AC) which determines t h e p o i n t B ' on t h e o r t h o g o n a l t r a j e c t o r y B ' E corresponding t o t h e p o i n t B on t h e p r i n c i p a l i s . By choosing o t h e r v a l u e s of

r , o r o t h e r p o i n t s B on t h e p r i n c i p a l i s , o t h e r p o i n t s of t h e t r a j e c t o r y can be found. Johann B e r n o u l l i ' s c o n s t r u c t i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s of t h e g e n e r a l i s e d c y c l o i d s i s a p i e c e of browbeating, i n t i m i d a t i n g f i r e w o r k . However, i n which s e n s e i s t h i s c o n s t r u c t i o n indeed a s o l u t i o n of t h e t r a j e c t o r y problem? I t i s c e r t a i n l y n o t a g e o m e t r i c a l c o n s t r u c t i o n t h a t can be c a r r i e d o u t de f a c t o , l e t alone t h a t i t could b e performed by r u l e r and compass. Too many d a t a a r e taken f o r g r a n t e d , f o r i n s t a n c e t h e a u x i l i a r y curve AW and i t s q u a d r a t u r e ;

Johann Bernoulli's alternatives

85

t h e c o n s t r u c t i o n of t h e p o i n t s P of t h i s curve AVP i t s e l f d i d a l r e a d y r e q u i r e t h a t the angle between t h e r a d i u s and t h e t a n g e n t could be determined i n a l l p o i n t s of t h e p r i n c i p a l i s . I n f a c t , how "given" must t h e p r i n c i p a l i s b e f o r t h i s angle t o be d e t e r m i n a b l e ? More such p o i n t s may b e r a i s e d , a l l l e a d i n g up t o the g e n e r a l q u e s t i o n : why should one r e g a r d t h e f i n a l c o n s t r u c t i o n f o r t h e o r thogonal t r a j e c t o r i e s a s being more simple o r more a c c e p t a b l e than t h e o r i g i n a l problem i t s e l f . Why n o t j u s t s o l v e the problem by c o n s t r u c t i n g curves t h a t i n t e r s e c t t h e given curves p e r p e n d i c u l a r l y , o r r a t h e r , by supposing t h a t t h e t r a j e c t o r i e s a r e "given"? Well, t h e r e must of c o u r s e remain some work t o b e done. When i s a problem s o l v e d , what methods a r e a c c e p t a b l e f o r s o l v i n g problems, and what methods a r e n o t ? Apparently t h e r e were r a t i o n a l e s behind t h e r e d u c t i o n of geometric problems t o geometric c o n s t r u c t i o n s , and t h e r e were taxonomies of methods of s o l u t i o n , r a n g i n g from c o n s t r u c t i o n s i n v o l v i n g r u l e r and compasses o n l y t o c o n s t r u c t i o n s presupposing many t r a n s c e n d e n t a l d a t a . L e t me quote Johann B e r n o u l l i , o r perhaps Nicolaus I1 B e r n o u l l i , on t h e i s s u e

of

a c c e p t a b i l i t y of t h e c o n s t r u c t i o n d i s c u s s e d above : "Now I should l i k e t h e b e n e v o l e n t r e a d e r t o compare t h i s most u n i v e r s a l s o l u t i o n f o r s i m i l a r [ c u r v e s ] - b e i n g reduced t o t h e most extreme grade of p e r f e c t i o n , namely q u a d r a t u r e s , and being e q u a l l y s i m p l e , obvious and e a s y w i t h t h e s o l u t i o n s of t h e p a r t i c u l a r example [ t h e g e n e r a l i s e d c y c l o i d s ] given by t h e gentlemen Hermann and Taylor, and then [I s h o u l d l i k e him] t o judge f o r himself whether t h o s e a r e n o t i n t r i c a t e , long and d i f f i c u l t ( a t l e a s t i n respect t o ours).433 Leaving a s i d e t h e r e f e r e n c e t o Hermann and T a y l o r , i t i s c l e a r from t h i s remark t h a t Johann B e r n o u l l i argued t h e e l e g a n c e and p e r f e c t i o n of h i s s o l u t i o n by way of t h e simple d i f f e r e n t i a l e q u a t i o n ( 3 . 2 0 ) ; t h i s e q u a t i o n , indeed, h a s n i c e l y s e p a r a t e d v a r i a b l e s and thus can b e c o n s t r u c t e d by means of q u a d r a t u r e s . The c o n s t r u c t i o n i t s e l f i s a l l b u t simple. Here i s Johann B e r n o u l l i ' s t r i c k ; a l l d i f f i c u l t i e s involved i n f i n d i n g t h e e x p r e s s i o n X l r ) a r e s h i f t e d over t o t h e f i n a l c o n s t r u c t i o n , and thus t h e e l e g a n c e and p e r f e c t i o n of Johann Bern o u l l i ' s e q u a t i o n (3.20) a r e bought a t t h e expense of t h e f i n a l c o n s t r u c t i o n . This c o n s t r u c t i o n , u n e l e g a n t a s i t may b e , g u a r a n t e e s t h a t t h e geometric problem has indeed been s o l v e d . I n t h i s way, t h e p e r f e c t i o n and e l e g a n c e of a s o l u t i o n a r e judged by t h e d i f f e r e n t i a l e q u a t i o n and hence by a n a l y t i c c r i t e r i a , whereas t h e a c c e p t a b i l i t y of t h e s o l u t i o n i s judged by t h e c o n s t r u c t i o n , a geometric c r i t e r i o n .

Johann B e r n o u l l i e l a b o r a t e d t h e approach a l o n g t h e l i n e s of s i m i l a r i t y a s f a r a s he could; he s e t o u t t o g e n e r a l i s e t h e concept of s i m i l a r i t y t o what he

86

Orthogonal Trajectories 1694-1 720

c a l l e d " e x p o n e n t i a l s i m i l a r i t y " and " f u n c t i o n a l s i m i l a r i t y " 4 4 , w h i l e p r o v i d i n g d i f f e r e n t i a l e q u a t i o n s f o r t h e r e l a t i o n between t h e m u l t i p l i c a t i o n f a c t o r and the c o o r d i n a t e s of t h e p o i n t s on t h e p r i n c i p a l i s . No f u r t h e r methods of c o n s t r u c t i o n f o r t h e s e d i f f e r e n t i a l e q u a t i o n s a r e given, o n l y t h e i r e l e g a n c e

is stressed.

5 3 . 1 1 . 2 The generazised synchrone method A t the very end of t h e s u r v e y a r t i c l e of 1720

t o t h e s i m i l a r i t y method and i t s g e n e r a l i s a t i o n s

-

-

a f t e r 30 s e c t i o n s devoted

Johann B e r n o u l l i f i n a l l y

r e v e a l e d t h e method by which he had f i r s t s o l v e d t h e o r t h o g o n a l t r a j e c t o r y problem f o r the f a m i l y of g e n e r a l i s e d c y c l o i d s . As we have s e e n i n 52.2.2, Johann B e r n o u l l i had solved h i s brachystochrone problem i n 1696 by c o n s i d e r i n g the brachystochrones a s l i g h t r a y s i n a medium w i t h v a r i a b l e d e n s i t y , g i v i n g r i s e t o the v e l o c i t y law v=&;

t h e concept of wave f r o n t had immediately

v o l u n t e e r e d t h e concept of t h e synchrone and i t s p r o p e r t y of o r t h o g o n a l i t y w i t h r e s p e c t t o t h e brachystochrones.

Now t h i s argument could b e g e n e r a l i s e d

by c o n s i d e r i n g o t h e r v e l o c i t y laws (see f i g u r e 1 2 ) .

Take v = v ( x ) ; t h e n t h e brachystochrones or l i g h t r a y s

(cf. 52.2.2)

s a t i s f y t h e c o n d i t i o n sina/vIx)=constant=c. S u b s t i t u t i n g dy/ds f o r s i n a y i e l d s dy=cvds, and i n s e r t i n g ds=(clx2+dy2)s l e a d s t o t h e d i f f e r e n t i a l e q u a t i o n (3.25)

dy=

v(doh

i i / e 2 - u 2( 2 )

f o r t h e family of brachystochrones s u b j e c t t o t h i s v e l o c i t y l a w v - v f x ) . Again t h e synchrones can b e c o n s t r u c t e d p o i n t w i s e by choosing i n each of t h e ds i s e q u a l t o 1 -

brachystochrones t h e p o i n t B such t h a t t h e time i n t e g r a l T=

A ' a p r e v i o u s l y given c o n s t a n t T o ; i n o t h e r words, t h e r e l a t i o n between t h e

Johunn Bernoulli's compurison of methods

87

a b s c i s s a x and t h e parameter c of t h e brachystochrones a l o n g t h e synchrone i s d e f i n e d i m p l i c i t l y by t h e e q u a t i o n

Now one can f r e e l y choose e x p r e s s i o n s X(x) f o r v and e x p r e s s i o n s A ( a ) f o r

c. For example, t a k i n g

V=Z

n

-n l e a d s t o t h e e q u a t i o n

and c=a

f o r t h e b r a c h y s t o c h r o n e s , and y i e l d s t h e e q u a t i o n

n (3.27)

&=T

f o r t h e synchrones. Here, (3.16) r e p r e s e n t s t h e g e n e r a l i s e d c y c l o i d s , and (3.27) t h e i r o r t h o g o n a l t r a j e c t o r i e s . This i s p r e c i s e l y t h e r e s u l t which Johann B e r n o u l l i had communicated t o L e i b n i z e a r l y 1716 ( c f . 53.10).

An even more g e n e r a l c l a s s of curves can be found by p u t t i n g v=I:&) and c = l : m ) . Then t h e family of curves i s d e s c r i b e d by t h e e q u a t i o n

and t h e orthogonal t r a j e c t o r i e s a r e d e f i n e d by t h e e q u a t i o n (3.29) I n a f i n a l remark Johann B e r n o u l l i o r Nicolaus I1 B e r n o u l l i mentioned t h a t t h i s l a s t r e s u l t had a l s o been found by t h e i r nephew Nicolaus I Bern o u l l i i n h i s t r a j e c t o r y a r t i c l e o f 1719, "be i t t h a t h e found t h i s i n a d i r e c t way, and no doubt from some s o u r c e i n d i c a t e d above.

'I4'

This blunderbuss indeed h i t i t s t a r g e t , b u t i t a l s o r e v e a l e d t h a t n e i t h e r Johann n o r Nicolaus I1 B e r n o u l l i knew what t h e y were t a l k i n g about.

§ 3 . 1 2 Johann Bernoulli's comparison of methods A s a preamble t o h i s survey of methods f o r t h e s o l u t i o n of t h e o r t h o g o n a l

t r a j e c t o r y problem, Johann B e r n o u l l i c l a r i f i e d what h e meant by "solving", and he provided a taxonomy of t h e p o s s i b l e ways of s o l v i n g a problem:

88

Orthogonal Trajectories 1694-1 720

"lo.

A problem i s solved i n some way by a r r i v i n g a t l e a s t a t a d i f f e r e n t i a l

e q u a t i o n which c o n t a i n s more than two v a r i a b l e s and t h e i r d i f f e r e n t i a l s . This method of s o l u t i o n i s s a i d t o b e of t h e f i r s t o r lowest d e g r e e . 2'.

It i s solved a l i t t l e more p e r f e c t l y , i f i t can be reduced t o a d i f -

f e r e n t i o - d i f f e r e n t i a l equation [ i . e . a higher order d i f f e r e n t i a l e q u a t i o n ] , i n which only two v a r i a b l e s and t h e i r d i f f e r e n t i a l s o c c u r ; t h i s i s t h e second degree of p e r f e c t i o n .

3'.

I t i s solved p e r f e c t l y by f i n d i n g a d i f f e r e n t i a l e q u a t i o n which does n o t

c o n t a i n h i g h e r o r d e r d i f f e r e n t i a l s , and which c o n s i s t s of o n l y two v a r i a b l e s . Such a s o l u t i o n o b t a i n s t h e t h i r d degree of p e r f e c t i o n .

4'.

I t i s solved most p e r f e c t l y , i f , moreover, i t can be reduced t o an

e q u a t i o n c o n s i s t i n g of f i n i t e terms by i n t e g r a t i o n , o r , a t l e a s t , t o quad r a t u r e s by s e p a r a t i o n of v a r i a b l e s , s o t h a t i t can b e c o n s t r u c t e d ; t h i s i s t h e f o u r t h and h i g h e s t degree of p e r f e ~ t i o n . " ~ ~ Here p e r f e c t i o n o f methods i s judged e x p l i c i t l y by way of t h e d i f f e r e n t i a l e q u a t i o n s which they involve. Obviously i t was o n l y f o r d i f f e r e n t i a l e q u a t i o n s with s e p a r a t e d v a r i a b l e s (case 4')

t h a t Johann B e r n o u l l i expected t h a t a geo-

m e t r i c c o n s t r u c t i o n f o r the t r a j e c t o r i e s could be found. According t o t h i s taxonomy, t h e s i m i l a r i t y method, l e a d i n g up t o d i f f e r e n t i a l e q u a t i o n (3.20) i s c l e a r l y t h e winner; t h e g e n e r a l i s e d synchrone method, t o which t h e taxonomy

-

s t r i c t l y speaking

-

does n o t a p p l y f o r t h e absence of a d i f f e r e n t i a l equa-

t i o n , a l s o l e a d s t o a c o n s t r u c t i o n of t h e t r a j e c t o r i e s and t h e r e f o r e might reasonably a l s o be c a l l e d a s o l u t i o n of t h e t r a j e c t o r y problem i n t h e most p e r f e c t sense. The e x t r a c t i o n of r o o t s from a l g e b r a i c e q u a t i o n s taken f o r g r a n t e d , Leibn i z ' s method f o r f a m i l i e s of a l g e b r a i c curves i s d e c l a r e d t o b e a s o l u t i o n of t h e t r a j e c t o r y problem i n a t l e a s t t h e t h i r d degree of p e r f e c t i o n . Indeed

-

a s d i s c u s s e d i n 53.4

-

on t h i s assumption i t i s always p o s s i b l e t o a r r i v e

a t a d i f f e r e n t i a l e q u a t i o n i n terms of x and y f o r t h e o r t h o g o n a l t r a j e c t o r i e s ; d i f f i c u l t i e s a r i s e only f o r f a m i l i e s of t r a n s c e n d e n t a l c u r v e s , given by an e q u a t i o n of t h e form y=

X

1 pfx,a)&,

0

duces the c o n d i t i o n -&=pfx,a)dy,

i n which c a s e L e i b n i z ' s method o n l y pror e p r e s e n t i n g a s o l u t i o n of t h e f i r s t o r

lowest degree of p e r f e c t i o n according t o Johann B e r n o u l l i ' s taxonomy. Here then, f i n a l l y , i s t h e occasion t o p r e s e n t t h e v a r i a b l e parameter e q u a t i o n (3.9)

( 1+ p

I dx+pqda=O

and demonstrate i t s s u p e r i o r i t y over L e i b n i z ' s method i n c a s e of a f a m i l y of t r a n s c e n d e n t a l c u r v e s :

Johann Bernoulli's comparison of methods

89

"This e q u a t i o n i s of no use a s long a s t h a t what h a s t o be s u b s t i t u t e d f o r q i s unknown, s i n c e i t does n o t even s o l v e the problem i n t h e lowest degree of p e r f e c t i o n [...

1.

But i f by some t r i c k q can b e determined

-

namely i n t h e

t r a n s c e n d e n t a l s about which we speak h e r e - t h e e q u a t i o n r e s u l t i n g from subs t i t u t i o n of t h i s v a l u e i n - d a = ( I + p 2 ) & / p q more e x c e l l e n t than t h e o t h e r &=-pdy

s o l v e s t h e problem i n a degree f a r

[ i . e . L e i b n i z ' s method], because i t

then c o n t a i n s two v a r i a b l e s o n l y , supposed p i s given i n a and

2.

y would a l s o occur i n p , then t h e y would walk s i d e by s i d e [ i . e .

For, i f the variable

parameter e q u a t i o n and L e i b n i z ' s method]. So it w i l l b e worthwhile t o show i n which way i n t r a n s c e n d e n t a l s t h e v a l u e of t h i s q i t s e l f can be f ~ u n d . ' ' ~ ' The t r i c k t o f i n d q i n t h e c a s e of t r a n s c e n d e n t a l curves i s provided by t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n . I t was o n l y on t h i s o c c a s i o n t h a t Johann B e r n o u l l i made t h i s theorem known p u b l i c l y . I s h a l l b r i e f l y d e a l h e r e w i t h Johann B e r n o u l l i ' s p r e s e n t a t i o n of t h e i n t e r c h a n g e a b i l i t y theorem and i t s a p p l i c a t i o n f o r e s t a b l i s h i n g t h e v a r i a b l e parameter e q u a t i o n . Johann B e r n o u l l i ' s argument proceeded i n t h e same g e o m e t r i c f a s h i o n a s L e i b n i z ' s argument over 20 y e a r s e a r l i e r . The g u i d i n g p r i n c i p l e , a g a i n , i s the s t a t e m e n t t h a t t h e sum of t h e d i f f e r e n c e s of t h e p a r t s is e q u a l t o t h e d i f f e r e n c e of the sums of the p a r t s .

A Consider two c u r v e s ( s e e f i g u r e 1 3 ) , b o t h p a s s i n g through t h e p o i n t A o n t h e a x i s , which p o i n t i s taken t o be t h e o r i g i n . It was B e r n o u l l i ' s purpose t o d e r i v e an a n a l y t i c e x p r e s s i o n f o r t h e l i n e segment Bb, r e p r e s e n t i n g

day=qda.

Now t h i s l i n e segment Bb can be regarded a s t h e sum of s u c c e s s i v e d i f f e r e n c e s of a series of p a r a l l e l (3.30)

l i n e segments, s i t u a t e d between A and t h e segment Bb:

Bb=(Bb-EeJ+(Ee-Hh)+(Hh-Gg)+.

.. .

These d i f f e r e n c e s , l i k e fBb-Eel, can b e r e w r i t t e n i n t h e f o l l o w i n g way:

(3.31)

Bb-Ee=Bb-mn=Bmbn.

Orthogonal Trajectvries 1694-1 720

90

This is a c r u c i a l s t e p ! Expressing Bb-Ee and Bm-bn i n terms of d i f f e r e n t i a l s , e q u a t i o n ( 3 . 3 1 ) can be i d e n t i f i e d as:

(3.32)

dxday=dadg.

Hence, ( 3 . 3 1 ) i s i n f a c t e q u i v a l e n t t o t h e e q u a l i t y theorem f o r mixed p a r t i a l d i f f e r e n t i a l s . However, t h e r e i s no s i g n a t a l l t h a t Johann B e r n o u l l i r e c o g n i s e d the importance of t h i s s t e p , l e t alone t h a t h e was aware of t h e f a c t t h a t he had h i t upon a theorem, o r a t l e a s t a p r i n c i p l e , a s important as t h e i n t e r theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n i t s e l f . To Johann

changeability Bernoulli

( 3 . 3 1 ) w a s j u s t one of the s t e p s l e a d i n g up t o t h e proof of t h i s

i n t e r c h a n g e a b i l i t y theorem. Considering t h e s i t u a t i o n i n which t h e c u r v e s a r e t r a n s c e n d e n t a l , and given by an e q u a t i o n of t h e form

(3.33)

Y=JxPix,n!aj: 0

Johann B e r n o u l l i p u t the t o t a l d i f f e r e n t i a l of p ( x , a ) t o be

(3.34)

dpix,a)=R(x,alda+Sir,aldx,

and he argued t h a t Em-bn can be e x p r e s s e d a n a l y t i c a l l y by:

(3.35)

Bm-bn=da ( p d x l =Rdadx.

Hence by combining ( 3 . 3 0 ) and ( 3 . 3 1 ) , h e found: X

(3.36)

Eb=da/

R(x,aldx, 0

a r e s u l t i d e n t i c a l w i t h t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n . This v a l u e of Bb l e a d s t o t h e f o l l o w i n g form of t h e v a r i a b l e parameter e q u a t i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s of t h e c u r v e s d e f i n e d by equation (3.33) : X

(3.37)

R f x , a )dxl

-da=(l i p (2, a) ) d x / ( p (x,al/ 0

Johann B e r n o u l l i a l s o c o n s i d e r e d t h e c a s e

-

a l l u d e d t o i n t h e quote given

above - t h a t t h e f a m i l y of c u r v e s i s d e f i n e d by an e q u a t i o n of t h e form

(3.38)

g=JBplx,y,a)dx. A

The i n t e r p r e t a t i o n of such an e q u a t i o n - g o c c u r r i n g b o t h on t h e l e f t - h a n d s i d e

and under

the i n t e g r a l s i g n

-

i s q u i t e a problem i n i t s e l f ; w e s h a l l e x t e n s i v e -

l y d e a l w i t h t h i s problem i n § 4 . 2 . 7 . Along an argument which c l o s e l y resembles t h e one provided by Nicolaus I B e r n o u l l i ( c f . § 4 . 2 . 8 ) , Johann B e r n o u l l i

Johann Bernoulli's comparison of methods

91

arrived a t

(3.39)

B k d a {Re-'

Tdx&) /e-' Tdx

where t h e e x p r e s s i o n s R and T a r e taken from t h e t o t a l d i f f e r e n t i a l

(3.40)

dp ( 2 ,y , a ) =R ( 2 , y , a ) da+T ( x ,y , a)dy+S (x,y , a)dx.

I n s e r t i n g the v a l u e f o r 9 which r e s u l t s from ( 3 . 3 9 ) i n t o t h e v a r i a b l e parameter equation y i e l d s the following d i f f e r e n t i a l equation f o r the orthogonal t r a j e c t o r i e s of the c u r v e s d e f i n e d by t h e e q u a t i o n

( 3 . 4 1)

-da= ( 2 +p2 ) cJTdxdx/(pJRc-'

Thdx)

And i n t h i s argument, a g a i n , t h e e q u a l i t y of mixed p a r t i a l d i f f e r e n t i a l s was used; however, a s i n the preceding argument, i t remained e n t i r e l y i m p l i c i t . The v a r i a b l e parameter e q u a t i o n i n i t s form ( 3 . 4 1 ) i s a r e s u l t f a r from b e a u t i f u l ; t h e occurrence o f t r a n s c e n d e n t a l c o e f f i c i e n t s and t h e f a c t t h a t t h r e e v a r i a b l e s z,y,a

a r e involved must have d e s t r o y e d any hopes f o r Johann

B e r n o u l l i t h a t i t could e v e n t u a l l y l e a d t o a c o n s t r u c t i o n f o r the o r t h o g o n a l t r a j e c t o r i e s of t h e curves given by t h e e q u a t i o n ( 3 . 3 8 ) . Hence t h i s s o l u t i o n of t h e t r a j e c t o r y problem w a s r a t e d as t h e lowest grade of p e r f e c t i o n ,

mentioned o n l y f o r t h e sake of completeness, and much l e s s handy than t h e two champion methods, v i z . t h e s i m i l a r i t y method and t h e g e n e r a l i s e d synchrone method: "But, having given t h i s most u n i v e r s a l formula f o r t h e t r a j e c t o r i e s , which i n i t s e l f , I admit, e x c e l l s i n c u r i o s i t y r a t h e r than u t i l i t y when a p p l i e d t o examples, I s h a l l f i n a l l y proceed t o v a r i o u s o t h e r methods; they do n o t s o l v e t h e problem i n t h e most u n i v e r s a l way, b u t , n e v e r t h e l e s s , they a r e a l l g e n e r a l i n t h e i r own way."48 I n t h i s way, t h e accomplishments of 1697 concerning v a r i a b l e parameters and d i f f e r e n t i a t i o n from curve t o curve were e v e n t u a l l y judged t o be a form of " g e n e r a l a b s t r a c t nonsense".

92 CHAPTER 4 NICOLAUS I BERNOULLI AND ORTHOGONAL TRAJECTORIES

§ 4 . 1 Biography and bibliography

54.1.1 Biographical s k e t c h In this chapter I shall discuss the contributions to partial differential calculus and the orthogonal trajectory problem made by Nicolaus I Bernoulli, the son of Johann and Jakob Bernoulli's brother Nicolaus, the painter. Together with Johann Bernoulli's sons Nicolaus I1 (1695-1759), 1782) and Johann I1 (1710-1790) Nicolaus I (1687-1759)

Daniel (1700-

belongs to that second

generation of Bernoulli mathematicians who had to find a mode of living under the depressing example and auspices of their father or uncle Johann Bernoulli'. Only Daniel Bernoulli eventually succeeded in establishing a firm and independent position for himself in the history of mathematics. Nicolaus I, highly respected in his time, remained on the misty fringe of history, where he is frequently mistaken for or mixed up with his cousin Nicolaus 11. Hence it will be useful to clarify Nicolaus's biography to some extent2. Born in Basel on October the 21st 1687, Nicolaus I received his mathematical training from his uncle Jakob, with whom he obtained a masters degree in 1704 by defending a thesis on infinite series. Four years after Jakob Bernoulli

died, Nicolaus I was granted a doctorate of jurisprudence in 1709 with a dissertation concerning the application of probability theory in matters of law. This thesis was explicitly presented as an elaboration of the results of Jakob Bernoulli in probability theory, which, by then, however had not yet seen the light.

A

few years later, in 1713 Nicolaus I would see his uncles

great book De arte conjectandi through the press, thus supplementing the basis of his own work. Probability theory remained one of Nicolaus's favourite subjects. Pierre RLmond de Montmort (1678-1719) for instance, when reporting to Johann Bernoulli about Nicolaus's visit to him in 1713 wrote: "Your Mr. nephew is awfully clever and very indefatigable. I can't work for two hours in succession, he works six without being weary. He proposed to me several quite jolly problems, from which I got off with sufficient honours

93

Biography and bibliography

and luck. This will serve to enrich my b00k.l'~ The problems Montmort and Nicolaus I Bernoulli discussed at that time were concerned with games of chance and the theory of probability. A lively correspondence on this matter had already developed between them from 1 7 1 0 onward, in which Montmort certainly was the receiving part. Nicolaus's letters to Montmort were eventually added as an appendix to the second edition of Montmort's Essay d ' a n a l y s e sur l e s j e u x de hazard of 1713. This appendix also contains Nicolaus's Petersburg paradox. In his comments on this correspondence J. Henny recently characterised Nicolaus's most striking talent as his "drive towards generalisation and the tough power of demonstration, oriented towards c tarity"

.'

In 1716 Nicolaus I was called to the chair of mathematics of the university of Padua, the chair formerly held by Jakob Hermann and, much earlier, by Galileo. He returned to Basel in 1719; the professorship of logic was first bestowed upon him in 1722, followed by the professorship of law in 1731. He died in Basel on November 29th, 1 7 5 9 . Nicolaus's fringe position in history is to a large extent the result of his own choice. Instead of publishing his results in journals and allowing world wide exposure, Nicolaus elevated the writing of letters into his sole channel of communication, thus selecting his own specialist audience. A huge stock of letters (approximately 5 6 0 ) and manuscripts, some of

them ready for

the printer, are kept at the university library in Basel. The Nicolaus Bernoulli papers form a most valuable and essential part of the Bernoulliana collection preserved there, neatly organised and selected by Nicolaus himself before he transferred his Nachlass to that library. Among these literary remains a considerable number of Jakob Bernoulli's papers were also found; they had been put under Nicolaus's custody by Jakob's family so as to secure them from intellectual robbery by Johann Bernoulli, and these papers were used by Nicolaus I for his edition of Jakob's Varia Posthwna in the Opera of 1 7 4 4 . Furthermore the heritage contained a long sought copy of Johann Bernoulli's lectures to 1'HZpital on differential calculus, brought to light in 1922 by Schafheitlin. It is generally assumed that Nicolaus had made this copy when visiting his uncle Johann in Groningen in 1705. In sifting his papers presumably Jakob's papers as well

-

-

and

Nicolaus made sure that only those papers

he considered relevant to mathematics and the sciences remained; personal papers and business papers were left out and presumably were destroyed. Thus in a manner rather pedantic Nicolaus himself determined what posteriority was allowed to know about his person. No comprehensive edition of the Nicolaus

94

Bernoulli and Orthogonal Trajectories

Bernoulli papers and correspondence has been published or prepared as yet. However, from what has transpired of Nicolaus's correspondences and from those parts which have been published in the mean time, the contemporary high esteem o f his mathematical ability seems only justified. His favourite mathematical

activity was to evaluate and criticise the foundations and logical organisation of a theory rather than to exploit

the obvious and to apply known techniques.

A l l published fragments of Nicolaus's correspondence reveal a shrewd and acute

mind, criticising the logical flaws and insufficiencies in arguments of others. For example, both in his correspondence with Leibniz and in his correspondence with Euler he took exception to their indiscriminate u s e of series; he urged that the convergence of series be tested and he pointed out that the derivation of a series for a given expression in

itself is no guarantee for the equality

of the given expression and the sum of the series5. Another example of such unusually acute - at least in the 18th century

-

criticism occurs in the correspondence with Euler around 1 7 4 3 in connection a m an with the criterion -=for a differential rnt;kc+ndy t o be total6. Bernoulliz6 ay ax agreed with Euler on the necessity of this condition, but vigorously raised the issue that this did not imply the converse to be true as well. Such matters as the convergence of a series o r the sufficiency of a necessary condition

were usually passed over in silence in the 18th century; only in 19th century rigorous analysis did these questions attract attention and were they regarded as essential steps in an argument which required a proof. Historiography of analysis usually credits Nicolaus I Bernoulli with the discovery of the equality of mixed second order partial derivatives

-

and

rightly s o , as we shall see in 5 4 . 2 , This theorem again was not sent to the press by Nicolaus himself; he had communicated it in a letter t o Johann Bernoulli, and an excerpt of this letter was eventually published in Nicolaus I1 Bernoulli's 1720 survey article about the trajectory problem. Why did Nicolaus I Bernoulli remain in the backyard of mathematical activity? Why did he watch this roaring development from such a peripheral position? And why did he not actively participate in this enterprise by publishing his results and material in such a way that others could read, understand and appreciate his findings? All around him his fellow mathematicians were taking their share in the glory of discovery, and he alone refused his share. These questions must remain largely unanswered. Perhaps full publication of Nicolaus's literary remains will yield some answers, while other answers were contained in those private parts of his papers which Nicolaus himself carefully sifted out and destroyed. At the moment these questions form what one might call "the Nicolaus

Biography and bibliography

95

I Bernoulli enigma".

These questions are not new. Montmort for instance raised them in 1718 in one of his letters to Nicolaus I Bernoulli. He was answered: "You ask me why I do not appeal to arms with those who have solved the

famous problem of Mr. Leibniz [i.e. the orthogonal trajectory problem], and why I refuse the glory that could fall to my share; I respond to you: because I am not ambitious, and because I have judged it to be sufficient that

I have assured you in my great letter of 3 1 / 3 / 1 7 1 6 that I have solved it."' Modesty and lack of ambition, they may provide an answer to the Nicolaus I enigma. But questions remain: Why did Nicolaus not reveal his great finding in the orthogonal trajectory problem to any of his correspondents? Why did he select precisely this matter for normal publication in the Aeta Eruditorum, and why did he then choose a form of presentation that effectively concealed the methods that had produced the break-through? From Nicolaus's great letter of 31/3/1716 to Montmort it becomes clear that he was actively interested in the orthogonal trajectory problem from the very beginning. The letter contained Nicolaus's rediscovery of Leibniz's method and its application to the examples treated by Jakob Bernoulli in 1698. In 1718 Nicolaus communicated his solution of the trajectory problem for generalised cycloids to Montmort; this solution was based on the similarity of the generalised cycloids, the clue that had provided Johann Bernoulli with one of his solutions of this problem as well. Nicolaus I explicitly added there:

"I urge you, Sir, not to communicate my solution to anybody without the consent of my uncle, whom I shall give notice of what I have just written to you."* Why?

5 4 . 1 . 2 Sources

Nicolaus I Bernoulli published only one article in connection with the orthogonal trajectory problem: his "Tentamen solutionis generalis problematis de construenda curva, quae alias ordinatim positione datas ad angulos rectos secat" ("Attempt at a general solution of the problem of constructing the curve that intersects at right angles other curves, given in ordered sequence by position"), printed in Aeta Eruditorwn of June 1719. I shall refer to this article as the Tentamen. It contained only a set of rules for constructing

96

Bernoulli and Orthogonal Trajectories

orthogonal trajectories, which were clarified by means of a series of examples; a proof or an analysis pertaining to these rules was lacking. However, this defect is made good by a three page manuscript, entitled: "Demonstratio analytica constructionis curvarum, quae alias positione datas ad angulos rectos secant, traditae in Actis Lips. 1719 pag 2 9 5 et seqq." ("Analytical demonstration of the construction of curves which intersect others given by position at right angles, communicated in the Acta of Leipzig 1719 page 295 ff."). The Demonstratio as I shall term this manuscript for the sake of brevity is kept in the university library Base1 in volume LIa48, p p . 24-25. It is immediately preceded by the manuscript of the Tenlamen (pp. 2 0 - 2 3 ) . A copy of the llemonstratio was mailed to Leonhard Euler by Nicolaus I Bernoulli in his letter of 6 / 4 / 1 7 4 3 ; it was introduced as follows: "I shall communicate to you here the analytical demonstration of the construction that I have once developed and never made public property."' Eventually this letter to Euler, and hence the Demonstratio was published by F u s s in his 1843b. This Demonstratio forms the main source for my reconstruction

of Nicolaus Bernoulli's partial differential calculus; for easier reference I have attached an edition of this manuscript a s Appendix I. This text differs from the one published by Fuss only in that F u s s used different symbols to denote the partial differential operators d

Y

and d,.

The third source for my arguments in this chapter is a fragment of a letter from Nicolaus I Bernoulli to Johann Bernoulli, dated 1/1/1718. The letter itself seems not to have survived, but an "Excerptum ex epistola ad patruum d. l Jan. 1718" ("Excerpt from a letter to uncle dated 1st of January 1718") exists in the Basel Bernoulliana collection (number LIaZI,f.23-14). Among other reflections, this excerpt contains Nicolaus's solution of the completion problem (cf. 9 4 . 2 . 3 ) which has been quoted by Nicolaus I1 Bernoulli in his survey article 2720 about the orthogonal trajectory problem. This quote has been termed the "Modus inveniendi aequationem differentialem completam ex data aequatione differentiali incompleta" ("Method to find the complete differential equation from a given incomplete differential equation") by Poggendorff; I shall stick t o thistitle,and refer to it as Modus Inueniendi for short. Bibliographically, the Modus Inveniendi is a monster! l o In 5 4 . 2 I shall provide a reconstruction of Nicolaus 1 Bernoulli's partial differential calculus based on all three sources: the T e n t m e n , the

Demonstratio and the Modus Inveniendi; f 4 . 3 will be devoted to Nicolaus's solution of the variable parameter equation; it is based on the Tentamen and the Demonstratio only.

91

Bernoulli's partial differential calculus §4.2 Nicolaus I Bernoulli's partial differential calculus

54.2.1 Principles of reconstruction In this section I shall give a coherent account of the concepts, methods and ideas that can be identified in Nicolaus I Bernoulli's use of partial differentiation. No such account by the hand of Nicolaus himself exists. This section, therefore, is a reconstruction of the logical organisation of Nicolaus's partial differential calculus, based on close examination of the few sources available (Demonstratio, Tentamen and Modus Inveniendi) and on interpretation of the explicit statements and implicit assumptions which they contain. The reconstruction complies with the policy set forth in 9 1 . 5 , that is, it is as close as possible to Nicolaus's original texts but deviates from his notations in that additional information given in the context or assumed implicitly is made explicit: e.g. a quantity involving the letters z, y , a which Bernoulli simply denoted as R shall be transcribed as R ( z , y , a ) .

One more

general warning note should be sounded: in order to comply with Nicolaus

1's

conventions, in 9 5 4 . 2 and 4 . 3 x always denotes the vertical ordinate of a point on a curve, whereas y denotes its abscissa.

94.2.2 Analytic and geometric data of families of curves Nicolaus I Bernoulli's partial differential calculus was firmly rooted in the tradition of differentiation from curve to curve. The main motive for his developing such a calculus came from the problem to construct the orthogonal trajectories to a family of curves. Moreover, the specific way in which the partial differentials were defined, or in which the equality of mixed second order differentials was proved, for short, the conceptual structure of Bernoulli's partial differential calculus, was severely determined by its being developed for families of curves. Before I proceed to describe this conceptual structure I have to discuss in some more detail how Nicolaus I Bernoulli considered families of curves to be given. Geometrically, families of curves always occur as "curves given by position". That is, the single curves are all determined uniquely, for example by means of a method of construction that guarantees that each of the curves ka may actually be regarded as drawn on a sheet of paper. In particular this implies that the position of the curves with respect to the axes is known, and

Bernoulli und Orthogonal Trajectories

98

t h a t f o r i n s t a n c e t h e o r d i n a t e s of t h e p o i n t s of i n t e r s e c t i o n of t h e d i f f e r e n t curves w i t h the axes can be expressed i n terms o f t h e parameter of t h e c u r v e s . This becomes c l e a r , f o r example, a t one i n s t a n c e i n t h e Demonstratio, where B e r n o u l l i considered a c o n f i g u r a t i o n a s drawn i n f i g u r e 1 , and where he s t a t e d :

“Since t h e curves which a r e t o be i n t e r s e c t e d a r e given by p o s i t i o n , t h e l i n e segment AD i s given i n terms of a and c o n s t a n t s , w h i c h , i f w e put i t t o be equal t o E , y i e l d s [ .

.. ]

q=&’/da.””

Hence, t h e f a c t t h a t t h e curves a r e given by p o s i t i o n i s t a k e n t o imply t h a t the l i n e segments DA (=x ) can be expressed by means of a known E ( a ) i n t h e f o l l o w i n g way:

zo=E(a).

( 4 . 1)

A n a l y t i c a l l y , B e r n o u l l i considered two s t a n d a r d forms of d i f f e r e n t i a l e q u a t i o n s f o r a f a m i l y of c u r v e s , d e f i n e d t h u s : ” I c a l l such a d i f f e r e n t i a l e q u a t i o n of t h e curves which a r e t o be i n t e r -

s e c t e d complete which e x p r e s s e s t h e r e l a t i o n which t h e d i f f e r e n t i a l s have t o each o t h e r , n o t o n l y those of t h e c o o r d i n a t e s of t h e s e c u r v e s , b u t a l s o [ t h e d i f f e r e n t i a l ] of t h e v a r i a b l e parameter, o r l i n e segment c a l l e d modulus by t h e c e l e b r a t e d Hermann. I c a l l t h a t d i f f e r e n t i a l e q u a t i o n incompZete, which e x p r e s s e s only t h e r e l a t i o n e x i s t i n g between t h e d i f f e r e n t i a l s of t h e coord i n a t e s of one of t h e curves t h a t a r e t o be i n t e r s e c t e d , the parameter o r modulus remaining c o n s t a n t . The complete d i f f e r e n t i a l e q u a t i o n of t h e curves t h a t a r e t o be i n t e r s e c t e d I g e n e r a l l y denote i n t h i s way dx--pdy+qda, t h e incomplete [ d i f f e r e n t i a l e q u a t i o n ] i n t h i s way: &--pdy,

and

i n which equa-

z and y denote t h e c o o r d i n a t e s of t h e c u r v e s t o be i n t e r a t h e parameter, and p and q q u a n t i t i e s somehow given i n x J y J a and

tions the l e t t e r s sected,

c o n s t a n t s . ‘” Thus B e r n o u l l i assumed t h a t a f a m i l y of curves can always be r e p r e s e n t e d a n a l y t i c a l l y by a d i f f e r e n t i a l e q u a t i o n which i s e i t h e r a compZete one, of t h e form

99

Bernoulli’s partial differential calculus

dx-pdyiqda, where p=P(x, y,a) and q=Q(x,y,a),

(4.2)

o r by an incomplete one, of t h e form

dyx=pdy, where again p=Pix, y,al

(4.3)

.

However, none of t h e s e two d i f f e r e n t i a l e q u a t i o n s determines t h e family of c u r v e s uniquely. I n both c a s e s a d d i t i o n a l i n f o r m a t i o n is n e c e s s a r y t o f i x t h e i r e x a c t p o s i t i o n . The p r e c i s e form of such n e c e s s a r y a d d i t i o n a l i n f o r m a t i o n can be found thus: Without f u r t h e r r e s t r i c t i o n t h e complete d i f f e r e n t i a l e q u a t i o n ( 4 . 2 ) can i n p r i n c i p l e be s o l v e d i n t h e form

x=X(y, a,c),

(4.4)

where c i s an a r b i t r a r y c o n s t a n t of i n t e g r a t i o n . Hence, t h e complete d i f f e r e n t i a l e q u a t i o n determines t h e f a m i l y of c u r v e s up t o a c o n s t a n t . I n o r d e r t o determine t h i s c o n s t a n t i t i s s u f f i c i e n t i f one p o i n t on one of t h e given curves i s known e x p l i c i t l y . Given t h e f a c t t h a t t h e p o i n t (y,,x

,

curve ka

) i s on t h e

then t h e c o n s t a n t c can be found from t h e c o n d i t i o n xo=X(yO,aO,c).

0

The incomplete d i f f e r e n t i a l e q u a t i o n ( 4 . 3 ) can be viewed a s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n which i n v o l v e s a parameter a . I n t e g r a t i o n of such a f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n l e a d s t o a g e n e r a l s o l u t i o n of t h e form

(4.5)

x=X(y, a,c ( a ) ) ,

i n v o l v i n g an a r b i t r a r y f u n c t i o n c i a )

of t h e parameter a. Hence, t h e incomplete

d i f f e r e n t i a l e q u a t i o n determines t h e f a m i l y of c u r v e s up t o an a r b i t r a r y f u n c t i o n c(a). This a r b i t r a r y f u n c t i o n can be f i x e d , however, by assuming t h a t i n each of t h e curves k

a

a point D f y

0

,x0 ) i s known, o r , i n p a r t i c u l a r , by

assuming t h a t t h e p o i n t of i n t e r s e c t i o n of each of t h e c u r v e s k

a

with t h e ver-

t i c a l a x i s i s known. A s we have s e e n , t h e mere f a c t t h a t t h e c u r v e s were cons i d e r e d t o be given by p o s i t i o n was regarded s u f f i c i e n t g u a r a n t e e t h a t an exp r e s s i o n E ( a ) could b e found, e x p r e s s i n g t h e o r d i n a t e z

0

s e c t i o n of t h e curve k

of t h e p o i n t of i n t e r -

w i t h t h e v e r t i c a l a x i s . Hence, t h e a r b i t r a r y f u n c t i o n

a c i a ) can always be found by means of t h e c o n d i t i o n E(a)=Xlo,a,cla)l. I n c a s e a l l c u r v e s would p a s s through t h e same p o i n t on t h e v e r t i c a l a x i s a s l i g h t

a d a p t i o n of t h i s argument, v i z . a s h i f t of t h e a x i s , a l s o s e t t l e s t h e m a t t e r . To conclude t h i s s e c t i o n , t h e s e a r e t h e forms of t h e a n a l y t i c d a t a which completely determine a f a m i l y of c u r v e s :

-

a complete d i f f e r e n t i a l e q u a t i o n drc=pdy+qda and one given p o i n t i n one given curve, o r

Bernoulli and Orthogonal Trajectories

100

-

an incomplete d i f f e r e n t i a l e q u a t i o n d x=qda and one given p o i n t i n each of

Y

t h e curves.

54.2.3

The completion problem

One of t h e c e n t r a l problems i n Nicolaus I B e r n o u l l i ' s p a r t i a l d i f f e r e n t i a l c a l c u l u s was i n t i m a t e l y connected w i t h t h e d i f f e r e n t forms of t h e a n a l y t i c d a t a f o r a f a m i l y of curves. I t i s what I have c a l l e d t h e "completion problem": Given an incomplete d i f f e r e n t i a l e q u a t i o n d x=pdy of a f a m i l y of c u r v e s , f i n d i t s complete d i f f e r e n t i a l e q u a t i o n

Y &=pdy+qda.

Nicolaus B e r n o u l l i gave h i s

g e n e r a l s o l u t i o n of t h i s completion problem i n h i s Modus Inveniendi; t h i s r a t h e r foggy s o l u t i o n of t h e completion problem has up t o now been t h e o n l y source used f o r a s s e s s i n g N i c o l a u s ' s c o n t r i b u t i o n s t o p a r t i a l d i f f e r e n t i a l c a l c u l u s and f o r c r e d i t i n g him w i t h t h e d i s c o v e r y of t h e e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s " .

However, a s becomes c l e a r from t h e Demonstratio,

the s o l u t i o n of t h e completion problem i n t h e Modus Innveniendi i s t o be judged an a p p l i c a t i o n of a c l e a r and w e l l d e f i n e d p a r t i a l d i f f e r e n t i a l c a l c u l u s r a t h e r than t h e only appearance of such a c a l c u l u s . T h e r e f o r e , I s h a l l f i r s t d i s c u s s t h e elements of N i c o l a u s ' s p a r t i a l d i f f e r e n t i a l c a l c u l u s and p r e s e n t h i s s o l u t i o n of t h e completion problem as an a p p l i c a t i o n of t h i s c a l c u l u s i n

54.2.

a.

54.2.4

Partial and t o t a l d i f f e r e n t i a l s

Nicolaus I B e r n o u l l i c l e a r l y recognised t h a t two d i f f e r e n t t y p e s o f d i f f e r e n t i a l s have t o be i d e n t i f i e d i n a two dimensional problem s i t u a t i o n , and t h a t t h e s e two t y p e s of d i f f e r e n t i a l s correspond t o t h e choice of two i n dependent v a r i a b l e s . He d e f i n e d : "[.

..] l e t

Ct be

the symbol of t h e d i f f e r e n t i a l s when a i s taken c o n s t a n t , and

6 be t h e symbol of t h e d i f f e r e n t i a l s when y i s taken ~ o n s t a n t . " ' ~ N i c o l a u s ' s d i f f e r e n t i a l o p e r a t o r s & a n d 6 correspond e x a c t l y w i t h o u r d and d,,

Y

and hence y and a a r e chosen a s independent v a r i a b l e s . B e r n o u l l i con-

s i d e r e d these two p a r t i a l d i f f e r e n t i a l o p e r a t o r s t o a c t upon t h e dependent v a r i a b l e s e x c l u s i v e l y : thus f o r example he wrote d x--pdy and d x=qda, b u t he

Y

never used them i n any of t h e f o l l o w i n g combinations: d y(=dy),

d y(=O),

Y

a

d af=01,

Y

d a f = d a l . The d i f f e r e n t i a l s of t h e independent v a r i a b l e s a and y were

101

Bernoulli's partial differential calculus always denoted by means of t h e s t r a i g h t d : da and d y . Besides t h e p a r t i a l d i f f e r e n t i a l s d z and d

a

Y

3:

of a v a r i a b l e z i t s t o t a l d i f f e r e n t i a l

dx a l s o o c c u r s

i n N i c o l a u s ' s work. The main occurrence i s of c o u r s e i n t h e complete d i f f e r e n t i a l e q u a t i o n d;c=pdy+qdu of a f a m i l y of curves. T h i s e q u a t i o n was d e f i n e d i n t h e f o l l o w i n g way: " [ i t ] e x p r e s s e s t h e r e l a t i o n which t h e d i f f e r e n t i a l s have t o each o t h e r , n o t only t h o s e of t h e c o o r d i n a t e s of the c u r v e s , b u t a l s o [ t h e d i f f e r e n t i a l ] of t h e v a r i a b l e parameter [.

.

J''l

dx o b v i o u s l y c o n s i s t s of t h e sum of t h e two p a r t i a l

The t o t a l d i f f e r e n t i a l

d i f f e r e n t i a l s d z and d z. Such a r e l a t i o n between t o t a l and p a r t i a l d i f -

a

Y

f e r e n t i a l s occurs e x p l i c i t l y i n t h e D e m o n s t r a t i o , where Nicolaus w i t h o u t f u r t h e r comment wrote dn=d n+d n f o r a v a r i a b l e n o c c u r r i n g i n

Y

a

problem s i t u a t i o n under c o n s i d e r a t i o n . Arguments

t h e two dimensional

f o r t h e s e d e f i n i t i o n s of

t o t a l and p a r t i a l d i f f e r e n t i a l s o r a m o t i v a t i o n f o r t h e r e l a t i o n between p a r t i a l and t o t a l d i f f e r e n t i a l s a r e l a c k i n g i n t h e Demonstratio. I s h a l l supply two d i f f e r e n t i n t e r p r e t a t i o n s of N i c o l a u s ' s d e f i n i t i o n s h e r e , one of them based on t h e formal r u l e s of d i f f e r e n t i a t i o n , t h e o t h e r based on t h e more g e o m e t r i c a l concept of p r o g r e s s i o n s of v a r i a b l e s . Which of t h e s e i n t e r p r e t a t i o n s i s t h e most a c c u r a t e o r t h e c l o s e s t t o Nicolaus B e r n o u l l i ' s i d e a s i s d i f f i c u l t t o a s s e s s w i t h o u t f u r t h e r s o u r c e s . However, i t might w e l l be t h a t h e had pursued both l i n e s of thought. S t a r t i n g p o i n t f o r t h e formal i n t e r p r e t a t i o n i s N i c o l a u s ' s d e f i n i t i o n of t h e complete d i f f e r e n t i a l e q u a t i o n f o r a f a m i l y of curves. A f a m i l y of c u r v e s c o n s i d e r e d given by a n e q u a t i o n T(x,y,a)=O, d i f f e r e n t i a l s of a l l t h r e e v a r i a b l e s z,y,a

then t h e r e l a t i o n between t h e can of course be found by a p p l y i n g

the d i f f e r e n t i a l operator d t o t h i s equation. This y i e l d s

Tz(x,y,a)dx+T

(4.6)

Y

(x,y,a)dy+Ta(x, y,a)da=U

by s t r a i g h t f o r w a r d a p p l i c a t i o n of t h e r u l e s of d i f f e r e n t i a t i o n . This d i f f e r e n t i a l e q u a t i o n ( 4 . 6 ) can t h e n be solved e x p l i c i t l y f o r one of t h e d i f f e r e n t i a l s , f o r example

dx and y i e l d s :

(4.2) where p=-T

dx=pdy+qda

(x,y,a)/Tz(x,y,a) and q=-Ta(x,y,a)/Tx(x, y , a ) . For example, f o r t h e f a m i l y of hyperbolas pay=U t h i s procedure immediately l e a d s to:dx-ady-yda=U Y

o r dx=ady+yda. I n t h i s way, the complete d i f f e r e n t i a l e q u a t i o n immediately r e s u l t s from d i f f e r e n t i a t i n g t h e e q u a t i o n of t h e f a m i l y of c u r v e s w i t h r e s p e c t t o a l l v a r i a b l e s and subsequent s o l v i n g of t h e d i f f e r e n t i a l e q u a t i o n f o r t h e

102

Berrtoulli and Orthogonal Trajectories

d i f f e r e n t i a l &. Now t h e e q u a t i o n of t h e f a m i l y of c u r v e s may a l s o be d i f f e r e n t i a t e d under t h e c o n d i t i o n a=constant, t o y i e l d dz--pdy, o r under t h e c o n d i t i o n y=eonstunt, t o y i e l d dx=qda. Obviously, t h e d i f f e r e n t i a l s of z a r i s i n g from such d i f f e r e n t i a t i o n under s p e c i a l c o n d i t i o n s do n o t d e s c r i b e t h e f u l l v a r i a t i o n of

z;hence i t i s only n a t u r a l t o d e n o t e them by means of a s p e c i a l

n o t a t i o n which t a k e s t h e s e c o n d i t i o n s imposed on t h e v a r i a b i l i t y of 2 i n t o account: a s we know, Nicolaus B e r n o u l l i wrote &=pdy and Sz=qda. Comparison of t h e s e two d i f f e r e n t i a l s w i t h t h e complete d i f f e r e n t i a l e q u a t i o n ( 4 . 2 ) immed i a t e l y y i e l d s the e q u a l i t y of t h e t o t a l d i f f e r e n t i a l dx and t h e sum of p a r t i a l d i f f e r e n t i a l s d z+d z. Thus i n t h i s i n t e r p r e t a t i o n t h e choice of y and a a s the

Y

a

p a i r of independent v a r i a b l e s i s t h e mere r e s u l t of t h e f a c t t h a t d z h a s been e l i c i t e d a s t h a t d i f f e r e n t i a l f o r which t h e d i f f e r e n t i a l e q u a t i o n ( 4 . 6 ) had t o be solved e x p l i c i t l y . Once such a choice i s made i t i s o b v i o u s l y e f f i c i e n t t o s t i c k to i t , s i n c e i t a l s o a f f e c t s t h e d e f i n i t i o n of t h e p a r t i a l d i f f e r e n t i a l o p e r a t o r s d and 6 . The geometrical i n t e r p r e t a t i o n of N i c o l a u s ’ s d e f i n i t i o n s t o which I now t u r n a s s i g n s a r o l e somewhat more e s s e n t i a l t o t h e independent v a r i a b l e s y and

a. A s d e s c r i b e d i n c h a p t e r 1 , i n o r d i n a r y one dimensional L e i b n i z i a n c a l c u l u s t h e r e always was a freedom of choice f o r t h e p r o g r e s s i o n s of v a r i a b l e s i n a one dimensional problem s i t u a t i o n . This indeterminacy was s u c h , t h a t t h e prog r e s s i o n f o r one of t h e v a r i a b l e s

... - could be

-

say y , and l e t the p r o g r e s s i o n be y,y’,y”

chosen a t w i l l , f o r i n s t a n c e such t h a t dy=eonst <=> ddyZ0, o r

ydy=const <==> ddy=dy2/y. Given such a p r o g r e s s i o n y,y ’,y”.

.. of

the variable

y t h e d i f f e r e n t i a l dy of y was d e f i n e d a s t h e d i f f e r e n c e of two s u c c e s s i v e values occurring i n the progression, f o r instance dy-y’-y. a p r o g r e s s i o n of

y induced t h e

Furthermore, such

p r o g r e s s i o n s of a l l o t h e r v a r i a b l e s o c c u r r i n g

i n t h e problerr. s i t u a t i o n ; t h e p r o g r e s s i o n of t h e a r c l e n g t h v a r i a b l e s f o r i n s t a n c e i s determined by t h e v a l u e s of s corresponding t o t h e v a l u e s y,y’,y” of t h e v a r i a b l e y . By analogy, i n a two dimensional problem s i t u a t i o n t h e p r o g r e s s i o n s of two of t h e v a r i a b l e s may be p r e s c r i b e d : Nicolaus B e r n o u l l i o b v i o u s l y took t h e s e v a r i a b l e s t o b e y and a. Thus t h e d i f f e r e n t i a l s of t h e s e two v a r i a b l e s again a r e defined a s t h e i n f i n i t e s i m a l d i f f e r e n c e s of s u c c e s s i v e v a l u e s of y and a i n t h e i r r e s p e c t i v e p r o g r e s s i o n s of v a r i a b l e s : dy-y’-y

and du=a’-a;

it

i s only n a t u r a l then t o denote t h e s e d i f f e r e n t i a l s w i t h t h e same symbol a s used i n one dimensional c a l c u l u s , namely t h e s t r a i g h t

d. The e s s e n t i a l d i f f e r e n c e

between one and two dimensional problem s i t u a t i o n s o n l y becomes c l e a r when one c o n s i d e r s t h e o t h e r v a r i a b l e s , f o r i n s t a n c e z ( s e e f i g u r e 2 ) .

103

Bernoulli's partial differential calculus

fig. 2

Now both the progression of y and the progression of a are projected upon the variable

2.

If a is considered to be constant, then the variable x will take

values in a progression determined by the progression of y exclusively:

.

x ( y , a ) , x i y ' , a l , x(y",a), . . Analogously, when y is considered to be constant, x will take values in a progression determined by the progression of a exclusively: x ( y , a ) , x ( y , a '),

x(y,a"),

...

Thus dx is no longer determined uniquely, contrary to da and d y ,

and a notation is called for to distinguish between the differentials of z arising from the variations of y and a respectively. This notation is Nicolaus I Bernoulli's &and 6. In practice, the precise farm of the progressions of the variables is o n l y relevant in case higher order differentials of y and a are considered; equations and expressions involving only first order differentials are not affected by the choice of the progression of variables. Since in Nicolaus's partial differential calculus no second or higher order differentials of the independent variables y and a occur the entire formalism of this calculus is valid for all progressions of these variables. In particular, the mixed second order differentials d d x and d d x only involve first order differentials of Y a aY y and a , and, therefore, they are not affected by the indeterminacy in the progressions of y and a. These remarks provide a clue for the interpretation of Nicolaus Bernoulli's complete differential equation of a family of curves, or of the total differential of the dependent variable x . Consider a family of curves k a , k,,,

karr,

... with an arbitrary, but

fixed progression of the parameter a

(see

figure 3), and imagine an arbitrary, but fixed curve intersecting these curves (thus not necessarily an orthogonal trajectory): Now the points of intera section P, P', Prr, . of this trajectory with the curves k a , k,,, karr, . .

k

..

determine the progression y , y ' , y " , gression x , x ' , x " ,

... of

... of

.

the abscissa y, and the pro-

the ordinate x . The differential dx=x'-x=x(P')-x(P)

Bernoulli and Orthogonal Trajectories

104

fig. 3

fig. 4

can be calculated as follows (see figure 4 ) :

dx=x(P I-x (P)=(x(P )-x (Q ' /+ (x (Q ) -x (PII=d (x+d xi +dax=d x+dax+d d x. Since Y a Y Y O d d x is infinitely small with respect to both d x and dux, this term might Ya Y be neglected, and hence one arrives at: dx=d x+dax

(6.7)

Y

Now since the trajectory PPrPr' was chosen arbitrarily and since the equation ( 4 . 7 ) only involves first order differentials of a and y, this equation is

valid

for all progressions of y and a , and all progressions x(y,a), z(y',a'i,

x(y",a"),

.. .

of x. Hence equation ( 4 . 7 ) describes the general relation be-

tween the differentials of the variables x, y and a , and therefore & is a genuine total differential of x. Nicolaus's choice of the terms "complete" and "incomplete" for the two types of differential equations of families of curves can also be explained now. In ordinary one dimensional Leibnizian calculus, a differential equation or a differential expression was called complete, if it was valid for all progressions of the variables involved; e.g. the right hand side in the c(&c2+d

formula R =- .

')%

for the radius of curvature of

a

curve is a complete

differential expression, whereas the right hand side in the formula R =

, valid only under the assumption

ddx=O, was called an in-

105

Bernoulli's partial differential calculus complete differential expression' 6 . By analogy, the differential equation

&=d x+d x or dx=pdy+qda, valid for all progressions of y andais a "complete" Y Q differential equation, and the differential equation a' ~ = p a ' y , valid only under

Y

the restriction a=constant, is an "incomplete" differential equation.

§ 4 . 2 . 5 Equality o f mixed second order d i f f e r e n t i a l s

The prominent preliminary result in the Demonstratio is Nicolaus's discovery of the equality of mixed second order differentials. It is presented as the first step towards the solution of the completion problem, that is, the problem of finding qi=d q'dai for families of transcendental curves, being introduced by the remark:

a

"As for transcendental curves which have to be cut the quantity

q is not

given, one must attempt its elimination by means of the following consideration.

((

'

fig. 5

Nicolaus's consideration pertained to diagram 5, "in which the value of the little line segment IF is found in two different ways: evidently, IF=HE+d HE=dax+d d x, but it also is Y Y a IF=da CF=daBEid a d y BE=dax+dadyx; subtracting dUx in both cases yields: d d x=d d x , t t l B Y a aY He then immediately went on to prove the interchangeability theorem for differentiation and integration as a consequence of this equality of mixed second order differentials. We shall discuss this proof in the next section, and first assess the role Nicolaus Bernoulli attributed to this equality of mixed differentials. As becomes clear further on in the Demonstratio, Nicolaus distinctly recognised this equality to be universally valid for all dependent variables. When considering a variable S depending on y and a he, for instance, wrote:

106

Bernoulli and Orthogonal Trajectories

"Let i n g e n e r a l dS=zdy+uda; t h e n a s a b o v e d d x=d d x

d d S=daduS. ~a

y a

'

one h a s h e r e

a Y

Over twenty y e a r s l a t e r , i n h i s correspondence w i t h Leonhard E u l e r , Bern o u l l i was t o e x p r e s s himself on t h e l o g i c a l s t a t u s of t h e e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s . E u l e r had informed him about a q u a r r e l i n t h e P a r i s Academy between Bouguer and F o n t a i n e , both of whom claimed t h e d i s c o v e r y of t h i s e q u a l i t y f o r themselves, and Nicolaus r e p l i e d : "I myself, f o r s u r e , do n o t c l a i m t h e g l o r y of any i n v e n t i o n i n t h i s m a t t e r ,

s i n c e I have n o t proposed t h i s p r o p e r t y of d i f f e r e n t i a l gave r i s e t o the c o n t r o v e r s y

-

formulas

-

which

by way of a theorem; r a t h e r have I supposed

i t t o be an axiom, which I thought t o b e obvious t o anybody from t h e mere n o t i o n of d i f f e r e n t i a l s , even w i t h o u t i n s p e c t i o n of a diagram."2o Thus a t l e a s t i n r e t r o s p e c t , Nicolaus B e r n o u l l i d i d n o t r e g a r d t h e argument c i t e d above a s a proof of a theorem, b u t r a t h e r as an i l l u s t r a t i o n of an axiom. However, i n terms of d i f f e r e n t i a t i o n from curve t o curve i t was t h e b e s t s o r t of proof one could f i n d ; i n no way can t h i s argument be judged i n f e r i o r than f o r i n s t a n c e L e i b n i z ' s proof of t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n i n 1697.

54.2.6

The i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n m d i n t e g r a t i o n

A s mentioned, t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and int e g r a t i o n i s p r e s e n t e d i n t h e Demonstratio a s an immediate consequence of t h e e q u a l i t y of mixed d i f f e r e n t i a l s ; d d x=d d x y i e l d s : d q d a z d g d y , and t h u s : (4.8)

dr, a dy q = - da d y e

Y a

aY

Y

B e r n o u l l i then s t a t e d : " t h e i n t e g r a l of which can be found [ i . e . of ( 4 . 8 ) ] , a t l e a s t i n t e r m s

of

q u a d r a t u r e s , i f x does n o t e n t e r t h e q u a n t i t y dz. I t i s n e c e s s a r y t o add a q u a n t i t y composed of a and o t h e r c o n s t a n t s i n t h e i n t e g r a t i o n such t h a t i n

GD=daAD. S i n c e t h e c u r v e s which are t o be i n t e r s e c t e d a r e given by p o s i t i o n t h e linesegment AD i s given i n terms t h e c a s e y=U HE o r qda changes i n t o

of a and c o n s t a n t s , which, i f we put i t e q u a l t o E, y i e l d s f o r t h e c a s e y=U:

q=dE/da.

I'

*'

N i c o l a u s ' s argument p e r t a i n s t o f i g u r e 6 and h i s r e s u l t may be w r i t t e n as: f o r p=Pfy,al.

107

Bernoulli’s partial differential calculus fig. 6

The i n t e r c h a n g e a b i l i t y theorem a s p r e s e n t e d h e r e i s a remarkable e x t e n s i o n of L e i b n i z ’ s theorem. I t i s no l o n g e r l i m i t e d only t o those c u r v e s which a l l p a s s through t h e o r i g i n ; by c a r e f u l a n a l y s i s Nicolaus B e r n o u l l i a l s o took t h e bounds of i n t e g r a t i o n i n t o c o n s i d e r a t i o n , and e x p l i c i t l y denoted t h e n e c e s s a r y c o n s t a n t of i n t e g r a t i o n i n terms of t h e o r d i n a t e s E l a ) of t h e p o i n t s of i n t e r s e c t i o n of the curves k

a

with t h e v e r t i c a l a x i s . Thus h i s r e s u l t may be summarised a s

follows: f o r f a m i l i e s of curves given by t h e e q u a t i o n

( 4 . 10)

x=L’Ply, a , dy+E ( a ) ,

the p a r t i a l d e r i v a t i v e q=d x/da i s given by e q u a t i o n ( 4 . 9 ) . Equation ( 4 . 9 ) p r o v i d e s t h e s o l u t i o n t o t h e completion problem f o r f a m i l i e s of c u r v e s given by an incomplete d i f f e r e n t i a l e q u a t i o n d x=pdy, where p does

Y

involve t h e two independent v a r i a b l e s y and a o n l y . But b e f o r e we can proceed t o N i c o l a u s ’ s g e n e r a l s o l u t i o n of t h e completion problem, t h a t

is,

his

s o l u t i o n f o r t h e case p=P(z,y,a), we must s t i l l d i s c u s s h i s s i n g u l a r use of l i n e i n t e g r a l s i n two dimensional problem s i t u a t i o n s .

54.2.7 I n t e g r a t i o n along t r a j e c t o r i e s i n a fam;Ly of curves Both i n t h e Modus Inveniendi and i n t h e T e n t m e n p e c u l i a r i n t e g r a l s o c c u r , which a r e n o t d e f i n e d o r c l a r i f i e d i n any one of our s o u r c e s ; o b v i o u s l y t h e y were judged t o be s e l f - e v i d e n t and d i d n o t r e q u i r e any s p e c i a l e x p l a n a t i o n . Such an i n t e g r a l f o r i n s t a n c e i s :

o c c u r r i n g i n t h e Tentamen, example V I ; i n t h e c o n t e x t i t i s s t a t e d t h a t t h i s i n t e g r a l should be e v a l u a t e d under t h e c o n d i t i o n

108

Bernoulli and Orthogonal Trajectories

and t h e f i n a l r e s u l t given t h e r e i s :

(4.13)

I ( y , a)=log(y-a)-logfy-I)

I s h a l l a t t e m p t a d e f i n i t i o n of such i n t e g r a l s h e r e , which covers Nicolaus B e r n o u l l i ' s use of them. T h i s d e f i n i t i o n i s a n e x t e n s i o n of Johann B e r n o u l l i ' s concept of i n t e g r a t i o n a s t h e formal i n v e r s e of d i f f e r e n t i a t i o n . Consider a f a m i l y of curves k

a

and one of i t s o r t h o g o n a l t r a j e c t o r i e s t

(see f i g u r e 7 ) .

fig. 7

X

Y Along t h e t r a j e c t o r y , t h e d i f f e r e n t i a l s of x and y a r e r e l a t e d t o t h e d i f f e r e n t i a l da by means of t h e v a r i a b l e parameter e q u a t i o n s

(4. 4 )

dy = S d a

and

( 4 . 5)

dx = +da.

Now t h e i n t e g r a l

( 4 . 16)

lV(x,y,a)da

which has t o be e v a l u a t e d along t h e t r a j e c t o r y t can b e d e f i n e d a s a q u a n t i t y

I ( x , y , a ) such, t h a t ( 4 . 17)

dI(x,y,a)=Vfx,y,al&

where &(x,y,al

(4.1 8)

h a s t o be e v a l u a t e d i n t h e f o l l o w i n g way:

& (x,y, a ) =Ix( 2 ,y , a)dx+I (x,y, a ) dy+Ia (x,y , a)da= ( I x( x , y,

Y

+IY ( x , y , a ) s + r a ( x , y , a ) / h

Thus I ( x , y , a )

.

a k -42 1+P

+

i s t h e i n t e g r a l e x p r e s s i o n r e q u i r e d when t h e r i g h t hand s i d e of

109

Bernoulli's partial differential calculus

( 4 . 1 8 ) c o i n c i d e s w i t h V(cc,y,a)da.

In t h i s definition i t is i n f a c t irrelevant t h a t the integration is carried o u t a l o n g an o r t h o g o n a l t r a j e c t o r y . I n s t e a d any o t h e r curve could a l s o b e taken, provided r e l a t i o n s such a s ( 4 . 1 4 ) and (4.15) a r e a v a i l a b l e , which desc r i b e t h e r e l a t i o n between t h e d i f f e r e n t i a l s d y o r k r e s p e c t i v e l y and

da

along t h i s curve. The main p o i n t i n t h i s d e f i n i t i o n i s t o demonstrate t h a t t h e i n t e g r a l s o c c u r r i n g i n N i c o l a u s ' s c a l c u l a t i o n s do indeed make s e n s e . However, i t does n o t provide any i n s i g h t i n t o how such i n t e g r a l e x p r e s s i o n s I ( x , y , u )

can

a c t u a l l y be found. The r a t i o n a l e b e h i n d the method of f i n d i n g t h e i n t e g r a l exp r e s s i o n s may be summarised t h u s : along t h e t r a j e c t o r y t ( o r any o t h e r curve r e l a t i n g t o s u i t a b l y chosen c o n d i t i o n s ( 4 . 1 4 ) and ( 4 . 1 5 ) ) a d i f f e r e n t i a l excan t a k e on a l l s o r t s of d i f f e r e n t forms, s i n c e da can

pression V(x,y,u)du

dx and d y . Moreover, t h e r e may be an o v e r a l l

a l s o be expressed i n terms of r e l a t i o n T(x,y,a)=U

d e f i n i n g how t h e v a r i a b l e s z , y ,

and a are connected i n t h e

family of curves under c o n s i d e r a t i o n . For c a r r y i n g o u t t h e i n t e g r a t i o n

/ V ( z , y , a ) d a a l o n g t h e t r a j e c t o r y t, a d i f f e r e n t i a l e x p r e s s i o n e q u i v a l e n t t o V ( x , y , a ) d a must then be chosen o u t of a l l e q u i v a l e n t forms such t h a t immediate i n t e g r a t i o n i s allowed. Let me i l l u s t r a t e t h i s procedure by e l a b o r a t i n g t h e example ( 4 . 1 1 ) : The i n t e g r a n d i n ( 4 . I I ) , which we now denote by V(y,a)da, can be r e w r i t t e n a s follows : (4.19)

--da

du

V(Y=a)da=(y-a, ( ] + ( y - a ) 2 )

*

Furthermore, t h e c o n d i t i o n e q u a t i o n ( 4 . 1 2 ) s o l v e d f o r du t a k e s on t h e form: (4.20)

du

=

(1-a) (I+(y-a)2)

(2-y)

I n s e r t i o n of t h i s v a l u e of

da

dY

.

i n t h e f i r s t summand of t h e r i g h t hand s i d e of

(4.19) y i e l d s : (4.21)

Since (4.22)

V(y,aidu =

(I-a)

( y - a ) (1-yl

1-a I ( y - a ) (I-y) y-a

I

_.--

y-1

dy

-

1 da . Y-a

t h e f o l l o w i n g e q u a t i o n f o r V ( y , a ) d a emerges:

V ( y , a ) d a = _dy-da-.& y-a y-1

This l a s t e q u a t i o n ( 4 . 2 2 ) i s of such a form t h a t i t can r e a d i l y be i n t e g r a t e d t o produce t h e e x p r e s s i o n I(y,a)=Zog(y-a)-Zogfy-l) a l r e a d y given i n ( 4 . 1 3 ) .

110

Bernoulli and Orthogonal Trajectories

54.2.8

General solution

01t h e completion

probZem

W e can now proceed t o d i s c u s s and i n t e r p r e t t h e g e n e r a l s o l u t i o n of t h e

completion problem p u b l i s h e d by Nicolaus I B e r n o u l l i i n h i s Modus Inveniendi. I

L e t an incomplete d i f f e r e n t i a l e q u a t i o n of a f a m i l y of curves be g i v e n :

(4.3)

d x=pdy, where p=P(x,y,a).

Y

The complete d i f f e r e n t i a l e q u a t i o n

(4.2)

dx=pdy+q&

i s r e q u i r e d . This t a s k t h e r e f o r e c o n s i s t s of f i n d i n g an a c c e p t a b l e e x p r e s s i o n f o r q , given p=P(x,y,aa). Nicolaus B e r n o u l l i s o l v e d t h e problem i n t h e f o l l o w i n g way: Put

(4.23)

dP(x, y, al=T (x,y, a)&+S (2,y, a)dy+R (x,y, a)da,

hence T=P S=P R=P s’ y’ a’ Making y c o n s t a n t and dy=O i n e q u a t i o n ( 4 . 2 3 ) y i e l d s

(4.24)

dap=T Is,y, aldax+R(x, y, alda,

and i n s e r t i o n of

(4.25)

dax=qdu i n t h i s e q u a t i o n l e a d s t o

dap=(T(x,y,a)q+R(x,y,all&.

Now i n g e n e r a l ( i . e . order d i f f e r e n t i a l s

(4.26)

w i t h o u t r e s t r i c t i o n s on dy), t h e e q u a l i t y of mixed second

d d s=d d Y e

aY

LC

yields:

d qda=dapdy,

Y

and hence by means of t h i s r e l a t i o n ( 4 . 2 6 ) t h e e q u a t i o n ( 4 . 2 5 ) can b e w r i t t e n as

(4.27)

d qda=(T(x,y,alq+R(x,y,a))da$y.

Y

E l i m i n a t i o n of du from ( 4 . 2 7 ) produces

(4.28)

d q=(T(x,y, alq+R (2,y, a l I dy, Y

which e q u a t i o n d e f i n e s d q i n terms of dy, and hence can b e regarded a s a. f i r s t

Y

o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r q i n terms of y , c o n t a i n i n g t h e v a r i a b l e a merely a s a parameter. Nicolaus B e r n o u l l i then s e t o u t t o i n t e g r a t e t h i s e q u a t i o n under t h e c o n d i t i o n

that a i s constant.

Define t h e v a r i a b l e z by means of t h e e q u a t i o n d z (4.29 T (x,y, a ) dy= - 2which e q u a t i o n can b e i n t e g r a t e d i n t h e way d e s c r i b e d i n 5 4 . 2 . 7

under t h e

Bernoulli's partial differential calculus

111

conditions

&=Pix, y , a ) d y

(4.30)

and

du=O

(4.31)

and y i e l d s :

-Zogz=JTfX, y , a ) d y

(4.32)

o r with c such t h a t Zogfc) = -1

(hence c = l / e ) :

z=c J T { x , y , a ) d y

(4.33)

Here t h e d e f i n i t i o n of i n t e g r a t i o n a l o n g a curve i s adapted i n such a way t h a t i t applies to integrals

-

i n v o l v i n g a l l t h r e e v a r i a b l e s x J y and a

c u r v e s of t h e given family. I n s e r t i o n of

z a s defined i n

(4.29)

-

along the

i n the equation

(4.28) yields

d q=-qd z / z + R ( x , y , a ) d y . Y Y

(4.34)

Rewriting t h i s e q u a t i o n f i n a l l y y i e l d s :

d (qzl=zR(x,y,a)dy

(4.35)

Y

This e q u a t i o n can be solved by i n t e g r a t i n g both s i d e s , a g a i n under t h e cond i t i o n s s t a t e d i n ( 4 . 3 0 ) and ( 4 . 3 1 ) ; t h e f o l l o w i n g e x p r e s s i o n f o r q then r e sults:

hi.,

q=c-JT (2,y J a )dy

(4.36)

y , a ) cJ T f x , y , a ) d y dY

This i s Nicolaus B e r n o u l l i ' s s o l u t i o n of the g e n e r a l completion problem; i t coincides

e n t i r e l y w i t h t h e s o l u t i o n given by Johann B e r n o u l l i i n t h e survey

a r t i c l e of 1 7 2 0 , d i s c u s s e d i n 5 3 . 1 2 .

54.2.9

ConcZuding remarks

Nicolaus B e r n o u l l i p u b l i s h e d none of t h e t e c h n i q u e s and r e s u l t s d i s c u s s e d above, a p a r t from h i s s o l u t i o n of t h e completion problem i n t h e Modus I n v e n i e n d i . However, t h i s s o l u t i o n d i d n o t r e v e a l much of t h e f u l l y developed t h e o r y behind i t ; i t w a s merely t h e t i p of the i c e b e r g . The e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s upon which t h e s o l u t i o n of t h e completion problem was e s s e n t i a l l y based was s t a t e d a s an axiom, and t h e two d i f f e r e n t d i f f e r e n t i a l o p e r a t o r s were n o t d i s t i n g u i s h e d n o t a t i o n a l l y . This e q u a l i t y t h e r e f o r e appeared as: d d x d d x .

112

Bernoulli and Orthogonal Trajectories

Thus Nicolaus B e r n o u l l i ' s work d i d n o t have any i n f l u e n c e upon t h e development of p a r t i a l d i f f e r e n t i a t i o n i n mathematics a s a whole. No one p r o f i t e d from i t , no one drew i n s p i r a t i o n from i t , no one continued a l o n g t h e l i n e s s e t o u t by Nicolaus I B e r n o u l l i . Why

B e r n o u l l i a c t e d a s he d i d remains p a r t of

the Nicolaus I enigma. Nonetheless, I t h i n k N i c o l a u s B e r n o u l l i ' s work i s h i g h l y r e l e v a n t f o r our purpose, t h e h i s t o r i o g r a p h y o f p a r t i a l d i f f e r e n t i a l c a l c u l u s . I t c l e a r l y shows t h a t a f u l l fledged p a r t i a l d i f f e r e n t i a l c a l c u l u s was p o s s i b l e

w i t h i n t h e L e i b n i z i a n paradigm of d i f f e r e n t i a l c a l c u l u s and curves. It shows t h a t f a m i l i e s o f . c u r v e s provided a f i e l d t h a t was both c h a l l e n g i n g and wide enough f o r p a r t i a l d i f f e r e n t i a t i o n t o be developed up t o a l e v e l of a s m a l l b u t c o h e r e n t and l o g i c a l l y o r g a n i s e d s e t of concepts and t e c h n i q u e s . And f i n a l l y i t shows t h a t t h i s type of p a r t i a l d i f f e r e n t i a l c a l c u l u s i s d i f f e r e n t from i t s l a t e r forms. Here we have p a r t i a l d i f f e r e n t i a t i o n i n t h e appearance of d i f f e r e n t i a t i o n from curve t o curve. The d i f f e r e n t i a l o p e r a t o r s a c t on v a r i a b l e s which a r e r e l a t e d i n a family of c u r v e s , and two of t h e s e v a r i a b l e s a r e given a p o s i t i o n of p r e f e r e n c e . However, t h i s choice of independent v a r i a b l e s only concerns t h e d e f i n i t i o n of t h e d i f f e r e n t i a l o p e r a t o r s ; i t does n o t concern t h e way i n which f a m i l i e s of c u r v e s a r e given a n a l y t i c a l l y . Rather than f u n c t i o n s of two v a r i a b l e s , e x p r e s s i o n s and e q u a t i o n s i n t h r e e v a r i a b l e s a r e employed i n d e s c r i b i n g a f a m i l y of c u r v e s a n a l y t i c a l l y . A s w e have s e e n t h e s e e x p r e s s i o n s and e q u a t i o n s allow a c e r t a i n a r b i t r a r i n e s s of form, which was e x p l o i t e d f o r i n s t a n c e i n m a r s h a l l i n g d i f f e r e n t i a l e x p r e s s i o n s i n t o a form readily integrable.

J 4 . 3 Nicolaus I B e r n o u l l i ' s r e s o l u t i o n of the variable parameter equation 14.3. I Inttroduction

I s h a l l s p l i t my d i s c u s s i o n of Nicolaus I B e r n o u l l i ' s method of c o n s t r u c t -

i o n f o r t h e o r t h o g o n a l t r a j e c t o r i e s of a f a m i l y of c u r v e s i n t o two p a r t s . I n s e c t i o n 4 . 3 . 2 I s h a l l p r e s e n t a s l i g h t l y modernised and s i m p l i f i e d account of t h e r a t i o n a l e behind B e r n o u l l i ' s method; s e c t i o n 4 . 3 . 3 then demonstrates i t s a p p l i c a b i l i t y i n t h e t e s t c a s e of t r a j e c t o r y methods, t h e f a m i l y of g e n e r a l i s e d c y c l o i d s . S e c t i o n s 4 . 3 . 4 and 4 . 3 . 5 t h e r e a f t e r provide a d e t a i l e d d i s c u s s i o n of b o t h N i c o l a u s ' s a n a l y s i s of t h e v a r i a b l e parameter e q u a t i o n i n t h e Demon-

s t r a t i o and t h e s y n t h e s i s of t h e r e s u l t s achieved t h e r e i n t h e s e t of r u l e s put f o r t h i n t h e Tentamen. I n t h i s way, the f i r s t s e c t i o n s s e r v e t o g i v e an

113

Bernoulli's resolution

o v e r a l l view of t h e main t e c h n i q u e s , i d e a s and t h e e x t e n t of N i c o l a u s ' s method, whereas t h e l a t t e r s e c t i o n s s e r v e as a j u s t i f i c a t i o n of t h e account i n 5 § 4 . 3 . 2 and 4 . 3 . 3 and a t t h e same time i l l u m i n a t e t h e o r g a n i s a t i o n of Nicolaus I Bern o u l l i ' s arguments.

1 4 . 3 . 2 Rationale of NicoZaus I B e r n o u l l i ' s t r a j e c t o r y construction Consider a given f a m i l y of c u r v e s , which a r e denoted by ka (a b e i n g t h e p a r a m e t e r ) , having a complete d i f f e r e n t i a l e q u a t i o n

dx=pdy+qda

(4.2)

where p=P(y,a)and q=&(y,a) a r e given e x p r e s s i o n s i n y and a ( t h i s r e s t r i c t i o n on p and 4 i s mine, and s e r v e s h e r e t o s i m p l i f y the argument). Furthermore,

l e t t h e o r d i n a t e s z of t h e p o i n t s of i n t e r s e c t i o n of t h e c u r v e s k vertical

a x i s be given in terms of the parameter a by way of

(4.1)

z =Efa)

with the

fig. 8 X

t x,:E(a)

1 A s we have seen i n 5 3 . 6 , (4.37)

Y t h e v a r i a b l e parameter e q u a t i o n

*y+qda=o

P

determines t h e r e l a t i o n between t h e parameter a and t h e a b s c i s s a y of a p o i n t of i n t e r s e c t i o n H of t h e o r t h o g o n a l t r a j e c t o r y

HF w i t h curve k

(see f i g u r e 8 ) .

The g e n e r a l s o l u t i o n of t h e v a r i a b l e parameter e q u a t i o n i s of t h e form

(4.38)

Vfy,a)=constant=e,

where t h e c o n s t a n t of i n t e g r a t i o n c a c t s a s t h e parameter of t h e f a m i l y of o r t h o g o n a l t r a j e c t o r i e s (which we s h a l l denote by t ) . Given such a s o l u t i o n ( 4 . 3 8 ) , t h e o r t h o g o n a l t r a j e c t o r y tc can be c o n s t r u c t e d p o i n t w i s e by i n t e r -

114

Bernoulli and Orthogonal Trajectories

s e c t i n g t h e curve k

a

w i t h t h e v e r t i c a l l i n e a t a b s c i s s a y , while a and y

s a t i s f y equation ( 4 . 3 8 ) . of i n t e g r a t i o n of t h e v a r i a b l e parameter e q u a t i o n

N i c o l a u s ' s method (4.37)

i s based on a s i n g u l a r l y profound and i n t r i c a t e d e t e r m i n a t i o n ( t o be

discussed i n 54.3.4)

of an i n t e g r a t i n g f a c t o r f o r t h i s e q u a t i o n , which h e took

t o be of t h e form l / n . This c a l c u l a t i o n involved n e a r l y a l l of t h e techniques of p a r t i a l d i f f e r e n t i a t i o n d i s c u s s e d i n 1 4 . 2 . Given such an i n t e g r a t i n g f a c t o r , t h e v a r i a b l e parameter e q u a t i o n t u r n s i n t o t h e t o t a l d i f f e r e n t i a l e q u a t i o n

22 Pn

(4.39)

dy + z d a = O

i n which the c o e f f i c i e n t of dy corresponds w i t h t h e p a r t i a l d e r i v a t i v e V ( y , a )

Y

and t h e c o e f f i c i e n t of da w i t h t h e p a r t i a l d e r i v a t i v e V (y,a) of an e x p r e s s i o n

a

V ( y , a ) . Thus t h e l e f t hand s i d e of ( 4 . 3 9 ) then r e p r e s e n t s t h e t o t a l d i f f e r e n t i a l of t h i s e x p r e s s i o n V ( y , a ) and hence ( 4 . 3 9 ) i s s o l v e d i n a g e n e r a l way by

.

V (y ,a I =c

Nicolaus i n t e g r a t e d t h e e q u a t i o n ( 4 . 3 9 ) i n a way c l o s e l y r e l a t e d t o t h e f o l l o w i n g form of s o l u t i o n : (4.40)

sy*y+ 0 pn

la(:)

ly,oda=constart=c

aO

The f i r s t q u a d r a t u r e i n t h i s e q u a t i o n r e p r e s e n t s t h e a r e a

-

say S(y,a)

-

under a 'curve w i t h a b s c i s s a y and o r d i n a t e z , d e f i n e d by (4.41)

z

=

z+p2

pn The i n t e g r a n d ($ly=o o c c u r r i n g i n t h e second i n t e g r a l allows t h e f o l l o w i n g geo-

metric interpretation:

dax

s i n c e q = da hence qlyI0 = ~

daxo and da

thus because x0= E ( a ) , q

a

P u t t i n g n ( o , a ) = rn(a), t h e second i n t e g r a l t u r n s i n t o

&(a) lyre = 7 .

and m(a)

w i l l be

0

denoted by A ( a ) . Thus ( 4 . 4 0 ) can now be w r i t t e n i n t h e form:

(4.42)

S(y,a)=e-A(a)

which e q u a t i o n immediately l e a d s t o t h e f o l l o w i n g c o n s t r u c t i o n f o r t h e p o i n t of i n t e r s e c t i o n H of t h e t r a j e c t o r y t and t h e curve k

a' I n t h e f o u r t h

quadrant of f i g u r e 9 LMN r e p r e s e n t s t h e curve w i t h a b s c i s s a e y and o r d i n a t e s

z ; t h e a r e a ABML then r e p r e s e n t s S ( y , a ) . Now determine y such, t h a t S(y,a)=e-A(al; t h i s y then i s t h e a b s c i s s a of t h e p o i n t of i n t e r s e c t i o n H .

115

Bernoulli's resolution

fig. 9

54.3.3

The test-case: t r a j e c t o r i e s o f generaZised cycZoids

The method of s o l u t i o n d i s c u s s e d i n t h e p r e v i o u s s e c t i o n seems n o t t o involve t h e c o e f f i c i e n t q o c c u r r i n g i n t h e complete d i f f e r e n t i a l e q u a t i o n ( 4 . 2 ) of the f a m i l y of c u r v e s k a . Only i n t h e d e f i n i t i o n of A ( a ) does q appear f o r m a l l y , b u t i n t h e a c t u a l c a l c u l a t i o n of A ( a ) , q)y=o i s r e p l a c e d by & l a ) / & , and thus A l a ) can be found w i t h o u t a c t u a l l y knowing

the expression Q(y,a)

which we have taken t o d e f i n e q . However, i n g e n e r a l t h e d e t e r m i n a t i o n of t h e i n t e g r a t i n g f a c t o r l / n can o n l y be c a r r i e d o u t when &(y,a) is known e x p l i c i t l y ; s t i l l , t h e r e i s one s p e c i a l case i n which N i c o l a u s ' s method of f i n d i n g t h e

i n t e g r a t i n g f a c t o r l / n does n o t r e q u i r e t h e knowledge of Q(y,a); t h i s i s t h e c a s e when t h e q u a n t i t y p=P(y,a) i s o f t h e form:

(4.43)

.

P ( y , a )=l:JY (y I B ( ~ ~ 1 - 1

Under t h i s c o n d i t i o n on t h e form of p , t h e i n t e g r a t i n g f a c t o r t a k e s on t h e form l / n = l / m , and consequently f o r any f a m i l y of curves h a v i n g an e q u a t i o n of t h e type

t h e o r t h o g o n a l t r a j e c t o r i e s a r e d e f i n e d by t h e f o l l o w i n g e q u a t i o n :

This method, e x p l i c i t l y p r e s e n t e d by Nicolaus I B e r n o u l l i a s a c o r o l l a r y of h i s g e n e r a l method i n t h e Tentamen, l e a d s t o e x a c t l y t h e same r e s u l t a s d i d

116

Bernoulli and Orthogonal Trajectories

Johann B e r n o u l l i ' s g e n e r a l i s e d synchrone method (cf 1 3 . 1 1 . 2 ) ;

i t i s t h e method

a t which Johann B e r n o u l l i had h i n t e d when he s t a t e d : "be i t t h a t he found t h i s i n a d i r e c t way, and no doubt from a s o u r c e i n d i c a t e d somewhere above. ' I 2 ' I n a sense Johann B e r n o u l l i was c o r r e c t s i n c e he had s w i f t l y mentioned t h e

v a r i a b l e parameter e q u a t i o n "above", b u t Nicolaus I B e r n o u l l i ' s way of d e a l i n g with t h i s e q u a t i o n was e n t i r e l y a t v a r i a n c e w i t h t h e approach suggested by Johann B e r n o u l l i ; moreover, t h e r e i s convincing evidence t h a t Johann B e r n o u l l i d i d n o t even know t h a t Nicolaus I B e r n o u l l i ' s r e s u l t s were d e r i v e d from t h e v a r i a b l e parameter e q u a t i o n , which Johann B e r n o u l l i had e v e n t u a l l y c h a r a c t e r i s e d a s g e n e r a l a b s t r a c t nonsense. Given Nicolaus I B e r n o u l l i ' s c o r o l l a r y method, s o l u t i o n of t h e t r a j e c t o r y problem f o r the g e n e r a l i s e d c y c l o i d s i s a mere m a t t e r of a d a p t i n g Y ( y ) and

B(a) t o t h e s p e c i a l form of t h e e q u a t i o n

Taking Y ( y l = ~ - ' ~and B(a)=a2n, and c o n s i d e r i n g t h a t i n t h e c a s e of t h e g e n e r a l i s € c y c l o i d s E(a)=O, t h e f o l l o w i n g f a m i l i a r c o n d i t i o n f o r t h e o r t h o g o n a l tra( c f . 5 5 3 . 1 1 . 2 and 3.10) :

j e c t o r i e s emerges

-2n n a

dy =const o r

54.3.4 Analysis of the variable parameter equation i n the Demonstratio

B e r n o u l l i ' s a n a l y s i s of t h e v a r i a b l e parameter e q u a t i o n i s l a r g e l y determined by t h e f i n a l c o n s t r u c t i o n he had i n mind: For given a , t h e corresponding v a l u e of y i s t o b e c o n s t r u c t e d by c u t t i n g o f f an a r e a C-Afa) from t h e curves LMN ( c f . f i g . 9 ) ; thus the form of t h e s o l u t i o n of t h e v a r i a b l e parameter e q u a t i o n

(4.37)

*'dy+qda=O P

i s presupposed t o be

This i d e a , j u s t a s t h e i d e a of t h e i n t e g r a t i n g f a c t o r l/n, i s simply p r e s e n t e d ;

117

Bernoulli’s resolution

i t was given no f u r t h e r e x p l a n a t i o n . The Demonstratio only p e r t a i n s t o

f i n d i n g s u i t a b l e e x p r e s s i o n s f o r t h e i n t e g r a t i n g f a c t o r l / n and t h e q u a n t i t y

A=A (aI

.

Applying t h e e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s t o t h e complete d i f f e r e n t i a l equation

&=pdy+qdu

(4.2)

( i n general)

of t h e f a m i l y of c u r v e s y i e l d s t h e g e n e r a l r e l a t i o n (where g e n e r a l s t a n d s f o r : no c o n d i t i o n s imposed on t h e d i f f e r e n t i a l s ) : d pdy=d qdu, which can be w r i t t e n

a

Y

as : (in general). The v a r i a b l e parameter e q u a t i o n ( 4 . 3 7 ) y i e l d s :

=*

(4.51)

(along t r a j e c t o r y ) .

Bv combination of ( 4 . 5 0 ) and ( 4 . 5 1 ) one f i n d s :

dyq -PduP q =I+p’

(4.52)

(along t r a j e c t o r y )

v a l i d o n l y f o r those v a l u e s of t h e v a r i a b l e s p and q p e r t a i n i n g t o p o i n t s on

Y

t h e t r a j e c t o r y . The a r e a S= J zdy, w i t h z d e f i n e d by (4.53)

z

0

= I+p2 Pn

( i n general)

has a t o t a l d i f f e r e n t i a l of t h e form:

dS=zdy+uda.

(4.54)

( i n general).

E q u a l i t y of t h e mixed second o r d e r d i f f e r e n t i a l s then g i v e s r i s e t o :

duzdy=d u d u , which c o n d i t i o n can be w r i t t e n as:

Y

(4.55)

d u $=&

(in general).

D i f f e r e n t i a t i n g e q u a t i o n ( 4 . 4 9 ) , w h i l e i n s e r t i n g t h e v a l u e of dS from ( 4 . 5 4 1 , y i e l d s : zdy+uda=-bda, (4.56)

which e q u a t i o n can be w r i t t e n a s :

&zU+b da -2

(along t r a j e c t o r y ) .

Here bda i s t h e d i f f e r e n t i a l of t h e q u a n t i t y A ( u ) , thus by d e f i n i t i o n : (4.57)

dA(a)=bdu

( i n general).

Equating ( 4 . 5 6 ) and ( 4 . 5 1 ) y i e l d s (4.58)

z=

lu+bl (1+p21 P4

(along t r a j e c t o r y )

and i n s e r t i n g t h e v a l u e f o r z a s d e f i n e d i n ( 4 . 5 3 ) i n t o ( 4 . 5 8 ) p r o v i d e s t h e relation :

Bernoulli and Orthogonal Trajectories

118

(4.59)

u+b = 4

(along t r a j e c t o r y ) .

Nicolaus used e q u a t i o n ( 4 . 5 9 ) f o r determining t h e q u a n t i t y A i a ) i n t h e f o l l o w i n g way: s i n c e y=O i m p l i e s u=O (because u=d S/du and S

a

(4.60)

b=iq/nlyrO

=O),

one f i n d s :

.

Since q y=o =dEial/du ( c f . 5 4 . 2 . 6 , b can a l s o be e x p r e s s e d a s : (4.61)

y=o

eq. ( 4 . 9 ) ) and p u t t i n g nyZO=rn by d e f i n i t i o n

.

b=dEial/mdu

Equation ( 4 . 6 I ) g i v e s r i s e t o (4.62)

.

%$A

Thus one of t h e two q u a n t i t i e s n e c e s s a r y f o r N i c o l a u s ' s t r a j e c t o r y c o n s t r u c t i o n has been found now. The r e s t of t h e argument p e r t a i n s t o f i n d i n g an e x p r e s s i o n f o r t h e i n t e g r a t i n g f a c t o r I / n . E l i m i n a t i o n of (u+b) from ( 4 . 5 6 ) by means of (4.59) yields: (4.63)

g=z

(along t r a j e c t o r y ) .

By combination of ( 4 . 6 3 ) and ( 4 . 5 4 ) one f i n d s : (4.64)

duz -3 2 4 Y

--

(along t r a j e c t o r y ) .

Now d u can be found by s t r a i g h t f o r w a r d d i f f e r e n t i a t i o n of ( 4 . 5 9 )

u

that

,

recollecting

;d b=O, and i s found t o s a t i s f y Y nd q-qd n

(4.65)

d u =

(4.66)

L=x+y

y Y n2 I n s e r t i o n of t h i s v a l u e f o r d u i n ( 4 . 6 4 ) y i e l d s :

d z

-dq q

2

(along t r a j e c t o r y ) .

d n

(along t r a j e c t o r y ) .

n

Nicolaus then c a l c u l a t e d d z by simply d i f f e r e n t i a t i n g t h e e q u a t i o n ( 4 . 5 3 ) , U

d e f i n i n g z w i t h r e s p e c t t o a; thus t h e f o l l o w i n g r e l a t i o n f o r d-z emerges: U

( i n general). Equating t h e s e two e q u a t i o n s f o r d_z ( i . e .

( 4 . 6 6 ) and ( 4 . 6 7 ) ) Y i e l d s :

(along t r a j e c t o r y )

.

Now s i n c e d n+d n=dn t h i s e q u a t i o n can b e w r i t t e n a s : Y

(4.69)

O

$-=duZoq+

J 2 2

(along t r a j e c t o r y ) .

This i s Nicolaus B e r n o u l l i ' s d e f i n i n g e q u a t i o n f o r t h e i n t e g r a t i n g f a c t o r l / n . Here obviously dn d e n o t e s t h e d i f f e r e n t i a l of n along t h e t r a j e c t o r i e s , where-

119

Bernoulli's resolution

as the d i f f e r e n t i a l operator d JI+2

i n t h e r i g h t hand s i d e s t a n d s f o r d i f f e r e n t i a t i o n

a

of the e x p r e s s i o n l o g 2 under t h e c o n d i t i o n y=eonstant. T h i s i s q u i t e s t r a n g e

P

a s i t u a t i o n : i n t h e l e f t hand s i d e t h e v a r i a b l e s a and y a r e bound t o t h e t r a j e c t o r y and i n t h e r i g h t hand s i d e they seem t o be f r e e ; however, t h e paradox i s r e s o l v e d when one r e c o l l e c t s t h a t t h e d i f f e r e n t i a l o p e r a t o r d

a

occurring

i n the r i g h t hand s i d e has a l r e a d y been i n t r o d u c e d i n e q u a t i o n s ( 4 . 5 0 ) and ( 4 . 6 7 ) , which a r e v a l i d i n g e n e r a l . The

Demonstratio ends w i t h c o n d i t i o n ( 4 . 6 9 ) f o r t h e i n t e g r a t i n g f a c t o r ;

no i n d i c a t i o n i s g i v e n about how t o f i n d a f i n i t e e x p r e s s i o n f o r t h i s f a c t o r . A s y n t h e s i s o f t h e r e s u l t s f r o m t h e Demonstratio i s provided o n l y i n t h e Tentamen.

Before we proceed t o d i s c u s s t h e s e t of r u l e s given t h e r e , l e t us review t h e d e r i v a t i o n of c o n d i t i o n ( 4 . 6 9 ) f o r t h e i n t e g r a t i n g f a c t o r ; t h i s d e r i v a t i o n i s r a t h e r unsurveyable, meandering through many d i f f e r e n t e q u a t i o n s and c o n d i t i o n s a s i t does. Some l i g h t might be shed upon the r a t i o n a l e of B e r n o u l l i ' s d e r i v a t i o n by comparing i t with t h e approach t o i n t e g r a t i n g f a c t o r s which i s s t a n d a r d now. Consider a d i f f e r e n t i a l e q u a t i o n (4.70)

M(y,a)dy+N(y,alda=O,

and l e t l / n w i t h n =n(y,a) be an i n t e g r a t i n g f a c t o r of ( 4 . 7 0 ) . Then n(y,a) must s a t i s f y the condition:

which l e a d s t o a f i r s t o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n (4.72)

e-&+nl'N

-M )=O aY Y a f o r n. S i n c e i t i s s u f f i c i e n t t o know o n l y one p a r t i c u l a r s o l u t i o n of t h i s p a r t i a l d i f f e r e n t i a l e q u a t i o n , t h e r e i s no need f o r an e l a b o r a t e t h e o r y t o s o l v e ( 4 . 7 2 ) . S p e c i a l t r i c k s and ad hoc methods a r e u s u a l l y a p p l i e d t o e l i c i t one such s o l u t i o n of ( 4 . 7 2 ) . Nicolaus B e r n o u l l i on t h e c o n t r a r y would reduce (4.72)

t o an o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r t h e i n t e g r a t i n g f a c t o r a l o n g

s o l u t i o n s of t h e given d i f f e r e n t i a l e q u a t i o n ( 4 . 7 0 ) . I f one e l i m i n a t e s t h e e x p r e s s i o n N from ( 4 . 7 2 ) by means of ( 4 . 7 0 ) , t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 4 . 7 2 ) t a k e s on t h e form:

M(E+$

$)+nliv -M )=O (along s o l u t i o n s of ( 4 . 7 0 ) ) . Y a This e q u a t i o n o n l y h o l d s f o r y and a along s o l u t i o n s o f ( 4 . 7 0 ) , and t h e r e f o r e (4.73)

t h e combination

(*+an a)i n f a c t r e p r e s e n t s aa ay da

( 4 . 7 0 ) . Hence: (4.74)

M n-& - a

-N

M

Y.

&/da a l o n g t h e s o l u t i o n s of

120

Bernoulli and Orthogonal Trajectories

I f one e q u a t e s

M and N t o t h e c o e f f i c i e n t s

z+p2 and P

q

r e s p e c t i v e l y of t h e

v a r i a b l e parameter e q u a t i o n , c o n d i t i o n ( 4 . 7 4 ) indeed t r a n s f o r m s i n t o Nicolaus B e r n o u l l i ' s c o n d i t i o n ( 4 . 6 9 ) . For a modern mathematician e q u a t i o n ( 4 . 7 4 ) i s a p p a l l i n g r a t h e r than a p p e a l i n g . Why should i n t e g r a t i o n of e q u a t i o n ( 4 . 7 4 ) , i n v o l v i n g t h e v a r i a b l e y a s a f u n c t i o n of t h e v a r i a b l e a , i m p l i c i t l y d e f i n e d by t h e o r i g i n a l d i f f e r e n t i a l e q u a t i o n ( 4 . 7 0 ) , be s i m p l e r than i n t e g r a t i o n of ( 4 . 7 0 ) i t s e l f ? Nicolaus B e r n o u l l i ' s answer t o t h i s q u e s t i o n would have been t h a t by means of ( 4 . 7 0 ) t h e r i g h t hand s i d e of ( 4 . 7 4 ) could perhaps be m a r s h a l l e d i n t o t h e form of a t o t a l d i f f e r e n t i a l and thus be i n t e g r a t e d . A s we have s e e n i n

54.2.7,

h i s concept of and t e c h n i q u e s f o r i n t e g r a t i o n were v e r s a t i l e enough f o r

a p p l y i n g them t o e q u a t i o n s

l i k e ( 4 . 7 4 ) w i t h a r e a s o n a b l e chance of s u c c e s s .

Synthesis of r e s u l t s in the Tentamen

54.3.5

The s e t of r u l e s provided i n t h e Tentamen a p p l i e s t o f a m i l i e s of c u r v e s , which a r e given by p o s i t i o n , and which a r e known a n a l y t i c a l l y e i t h e r by t h e i r complete d i f f e r e n t i a l e q u a t i o n

dx =pdpqda

(4.2)

o r by t h e i r incomplete d i f f e r e n t i a l e q u a t i o n

d x =pdy. Y

(4.3)

The phrase "given by p o s i t i o n " g u a r a n t e e s t h a t s u f f i c i e n t g e o m e t r i c a l i n formation i s a v a i l a b l e t o f i x t h e curves e n t i r e l y ; t h i s i m p l i e s e s p e c i a l l y t h a t t h e p o i n t s of i n t e r s e c t i o n of t h e curves w i t h t h e v e r t i c a l x-axis a r e known ( c f . J 4 . 2 . 2 ) .

I s h a l l d i s c u s s t h e s e t of r u l e s s t e p by s t e p .

The d e t e r m i n a t i o n of t h e i n t e g r a t i n g f a c t o r l / n i s p r e s e n t e d a s f o l l o w s : "The l o g a r i t h m of t h e q u a n t i t y J l l + p p ) : p b e d i f f e r e n t i a t e d , t a k i n g y c o n s t a n t and t h e parameter a v a r i a b l e , and s u b s t i t u t i n g f o r

dx

i t s v a l u e qdz; t h i s

d i f f e r e n t i a l having been found, i t be i n t e g r a t e d a g a i n , w h i l e y i s taken t o be v a r i a b l e a s w e l l and where, i f n e c e s s a r y , t h e v a l u e s of t h e e q u a t i o n s

&+ppdx=qda

and dy+ppdy=-pqda a r e b e i n g s u b s t i t u t e d ; t h e i n t e g r a l ( i f i t can

be found) i s taken a s a l o g a r i t h m t h e argument of which be c a l l e d n.''23 Thus n i s t o be found by i n t e g r a t i n g d a Z o q E along t h e t r a j e c t o r i e s , i n t h e way d e s c r i b e d i n 9 4 . 2 . 7 ,

(4.75)

P

and by p u t t i n g

n = e r d, 2og ( J l + p l / p I

121

BernoulliS resolution Next A(a) i s determined:

" t h e q u a n t i t y q be d i v i d e d by t h i s q u a n t i t y n ; having s u b s t i t u t e d f o r x i t s v a l u e expressed i n y , a and c o n s t a n t s , a l l terms from t h e q u o t i e n t [ q / n ] i n which y occurs be dropped, and t h e r e s i d u e m u l t i p l i e d by da s h a l l be int e g r a t e d and c a l l e d

A."

This r u l e f o r f i n d i n g A(a) i s a t v a r i a n c e w i t h t h e r e s u l t i n t h e Demonstratio. According t o e q u a t i o n ( 4 . 6 0 ) , A i s t o be found by i n t e g r a t i n g t h e e x p r e s s i o n

(q/n)y=o, considered a s a f u n c t i o n of a, o r by i n t e g r a t i n g dE(al/mda as shown i n ( 4 . 6 1 ) . However, h e r e i n t h e Tentamen Nicolaus took A t o be d e f i n e d by (4.76)

A=jz)'da k

where ( q / n )

r e p r e s e n t s those summands i n ( q / n ) t h a t do n o t c o n t a i n y , pro-

vided x has been e l i m i n a t e d a l r e a d y from ( q / n ) . This s h i f t of d e f i n i t i o n i s miraculous. Why d i d h e r e p l a c e t h e e l e g a n t and u n c o n t r o v e r s i a l i n t e g r a l J-

dE f a ) rn

by t h e i n t e g r a l ( 4 . 7 6 ) , thus l i m i t i n g t h e a p p l i c a b i l i t y of h i s r u l e t o t h o s e c a s e s of f a m i l i e s of curves f o r which an e x p l i c i t e x p r e s s i o n z=X(y,a) c a n be provided. Moreover, why d i d he make a mistake h e r e ? For simple polynomial forms of ( q / n ) t h e two d e f i n i t i o n s fq/n)" and (q/n)y=o do indeed c o i n c i d e ; b u t i n o t h e r c a s e s they e a s i l y l e a d t o c o n t r a d i c t i o n s z 4 . Was t h i s a mere s l i p of t h e pen o r an o v e r s i g h t ? O r a mephistophelian t r i c k t o mislead t h o s e r e a d e r s who t r i e d t o f i n d t h e i r way through t o t h e r a t i o n a l e behind t h e s e t of r u l e s ?

A(a) and n having been determined, t h e f i n a l c o n s t r u c t i o n i s p h r a s e d as follows : "Having done s o , t h e a r e a C-A s h a l l be c u t o f f i n t h e curve whose a b s c i s s a e a r e y and whose o r d i n a t e s , which I w i l l c a l l z, a r e (l+pp)/pn, where I unders t a n d by C an a r b i t r a r y q u a n t i t y which i s c o n s t a n t f o r a l l p o i n t s of t h e same t r a j e c t o r y ; t h e a b s c i s s a y of t h i s a r e a w i l l g i v e t h e a b s c i s s a of t h e req u i r e d t r a j e c t o r y i n i t s p o i n t of i n t e r s e c t i o n w i t h a c u r v e t h a t i s t o be intersected

.

"

This c o n s t r u c t i o n c o i n c i d e s p r e c i s e l y with t h e one e n v i s a g e d i n t h e Dernonstra-

t i o ( c f . e q u a t i o n ( 4 . 4 9 ) ) . However, i t i s f o r m u l a t e d i n a r a t h e r vague way, i n t h a t i t i s n o t a t a l l c l e a r where t h e bounds f o r t h e a r e a under t h e curve should be. Again t h i s i s a s t o n i s h i n g , s i n c e t h i s p o i n t had been d e a l t w i t h i n a v e r y p r e c i s e way i n t h e Demonstratio. Nicolaus I continued h i s set of r u l e s w i t h a d e s c r i p t i o n of t h e c o r o l l a r y method which we have met i n 9 4 . 3 . 3 . "The o t h e r r u l e , which i s a c o r o l l a r y of the f i r s t one, o n l y h o l d s f o r t h o s e t r a n s c e n d e n t a l curves t o be i n t e r s e c t e d f o r which t h e incomplete d i f f e r e n t i a l

122

Bernoulli and Orthogonal Trajectories

e q u a t i o n dx--pdy i s arranged i n such a way, t h a t t h e q u a n t i t y m

:

p

c o n s i s t s of two f a c t o r s , which I s h a l l c a l l B and Y, where t h e f i r s t i s given i n a and c o n s t a n t s , and t h e o t h e r i n terms of y and c o n s t a n t s . " Under t h e c o n d i t i o n s t a t e d h e r e , t h e i n t e g r a t i n g f a c t o r can indeed b e found immediately and i s l/B(a). I n terms of p t h e c o n d i t i o n t a k e s on t h e form:

1

p=rB-

. Thus

t h i s i s e x a c t l y t h e case of f a m i l i e s of t r a n s c e n d e n t a l c u r v e s

which Johann B e r n o u l l i could d e a l w i t h by means of h i s g e n e r a l i s e d synchrone method ( c f . 13.1 I . 2 ) .

Nicolaus c o n t i n u e d :

"In t h e s e c a s e s t h e curve h a s t o be c o n s t r u c t e d i n which t h e o r d i n a t e s z a t a b s c i s s a e y a r e Il+ppl/pB; t h e r e s t has t o b e c a r r i e d o u t a s b e f o r e , w i t h t h i s q u a l i f i c a t i o n t h a t i n s t e a d of t h e f r a c t i o n qda/n t h a t s e r v e d f o r f i n d i n g the q u a n t i t y A , h e r e t h e f r a c t i o n

&/B must b e a p p l i e d ; E denotes t h e

v a r i a b l e d i s t a n c e between t h e p o i n t of i n t e r s e c t i o n of t h e curves which a r e t o be i n t e r s e c t e d with t h e a x i s and an a r b i t r a r i l y given p o i n t , from which t h e o r i g i n of t h e a b s c i s s a e of t h e r e q u i r e d t r a j e c t o r y i s taken. T h i s d i s t a n c e

E i s given by a and c o n s t a n t s : t h e r e f o r e h e r e t h e a r e a t o b e c u t o f f [from t h e curve w i t h o r d i n a t e s z ] w i l l always b e C - l f d E / B ) ;

t h i s a r e a w i l l be c o n s t a n t

i f t h e curves t o be i n t e r s e c t e d a l l p a s s through t h e same p o i n t of t h e a x i s . " Herewith t h e a l g o r i t h m s provided i n t h e Tenturnen end, and what follows a r e examples i n o r d e r t o c l a r i f y how t h e s e a l g o r i t h m s could b e a p p l i e d . The f i r s t of t h e s e examples w a s concerned with t h e f a m i l y of g e n e r a l i s e d c y c l o i d s , t h e u l t i m a t e t e s t c a s e f o r t r a j e c t o r y methods.

54.3.6

ConcZuding remarks

Nicolaus I B e r n o u l l i ' s s o l u t i o n of t h e v a r i a b l e parameter e q u a t i o n w a s a s p l e n d i d p i e c e of a n a l y t i c f i r e w o r k . It involved new and f e r t i l e i d e a s througho u t , and i t c e r t a i n l y c o n s t i t u t e d a major break-through i n t h e t h e o r y o f d i f f e r e n t i a l e q u a t i o n s . One can o n l y guess what t h e e f f e c t of a more c l e a r and complete account of N i c o l a u s ' s i d e a s would have been. C e r t a i n l y t h e i n t r o d u c t i o n of the i n t e g r a t i n g f a c t o r a s t h e second important method f o r s o l v i n g d i f f e r e n t i a l e q u a t i o n s (next t o s e p a r a t i o n of v a r i a b l e s ) would have been advanced by some 20 y e a r s . However, Nicolaus I B e r n o u l l i chose t o h i d e h i s i d e a s , and i t was l e f t t o C l a i r a u t and F o n t a i n e t o develop t h e concept o f t h e i n t e g r a t i n g f a c t o r and t h e techniques f o r f i n d i n g such f a c t o r s a g a i n a t t h e end of t h e 1730s. But i n 1743, when he had a l r e a d y picked up t h e i d e a s of F o n t a i n e and C l a i r a u t , E u l e r could s t i l l l e a r n from Nicolaus B e r n o u l l i ' s i n t r i c a t e

123

Bernoulli's resolution a n a l y s i s of t h e v a r i a b l e parameter e q u a t i o n from 1719. E u l e r r e p l i e d t o Nicolaus Bernoulli's

l e t t e r c o n t a i n i n g t h e Demonstratio w i t h t h e f o l l o w i n g remarks:

"I acknowledge my g r e a t d e b t t o you, most d i s t i n g u i s h e d S i r , f o r t h e

demonstration of your extremely e l e g a n t c o n s t r u c t i o n of o r t h o g o n a l t r a j e c t o r i e s , which you p u b l i s h e d i n t h e Acta of L e i p z i g of 1719. Already long ago I tormented myself w i t h d i g g i n g up a demonstration f o r i t ; however I have n o t e l i c i t e d a n o t h e r c a s e b u t t h e one i n which --P-of t h e parameter

a a l o n e . Although I was l o o k i n g

is a function

Jl+pp for a quantity

n , divided

b y which the e q u a t i o n O = d y ( l + P p ) + qda i s r e n d e r e d i n t e g r a b l e , y e t i t s t i l l

P

d i d n o t occur t o me t h a t i n t h e s e a r c h of t h i s n t h e proposed d i f f e r e n t i a l e q u a t i o n i t s e l f can be c a l l e d t o h e l p . Thus, by means of your method q u i t e a few d i f f e r e n t i a l e q u a t i o n s can be i n t e g r a t e d e ~ p e d i t i o u s l y . " ~ ~

124

CHAPTER 5 EULER‘S THEORY OF MO DULAR EQUATIONS IN THE 1730s

5 5 . 1 Introduuctiori

In 1 7 2 7 , the 20 year old Leonhard Euler travelled to St. Petersburg, where he joined Nicolaus I1 and Daniel Bernoulli, and Jakob Hermann, who had been appointed to the newly founded Petersburg Academy of Sciences a few years earlier. Euler had received a thorough mathematical training from Johann Bernoulli, through the ordinary curriculum at Base1 University, through regular Saturday-afternoon privatissima, and through intensive private study and reading. Euler was well informed about the state of the art in mathematics, and he had already published a short treatise concerning the problem

of re-

ciprocal trajectories. This problem had been raised by Johann and Nicolaus I1 Bernoulli at the end of their 1720 survey of methods for solving the orthogonal trajectory problem. We may well assume that Euler was familiar with all those methods for constructing orthogonal trajectories which Johann Bernoulli had presented in that article; and it only seems to be very likely that Euler knew of the technique of differentiation from curve to curve and the interchangeability theorem for differentiation and integration. However, he definitely knew nothing of the extensive partial differential calculus that Nicolaus I Bernoulli had developed around 1 7 1 9 . This becomes clear from the correspondence between Euler and Nicolaus I in the year 1743. Euler remained in Petersburg till 1741, when he changed over to the Prussian Academy of Sciences in Berlin. This first Petersburg period

-

Euler

was to move back to Russia in 1766 - was one of the most fruitful in Euler’s career, though not the most productive if measured in output of printed pages. In those years prior to 1 7 4 0 , Euler formed most of his ideas and programmes, which were to be elaborated and polished in the later periods of his life’. His plans for the great textbooks, his ideas in number theory, many ideas on integrating differential equations et

. date

from these years.

Specifically, there are abundant historiographical references to Euler’s partial differential calculus of the

7 3 0 s . Two articles i n particular, en-

titled: “On infinitely many curves of the same genus, or a method of finding

Introduction equations for infinitely many curves

of

125

the same genus", together with an

"Addition to the dissertation about infinitely many curves of the same genus"

(De i n f i n i t i s eurvis and Additwnentwn) are cited over and over again in connection with partial differential calculus: they contain Euler's proof of the equalitytheorem for mixed partial derivatives, his theorem on homogeneous functions, partial differential equations, and methods to solve partial differential equations. However, it is a singular feature of Euler's work of the 1730s that he did not integrate partial differential equations, although his

methods were fully capable of doing so. Instead, Euler sought to solve other problems by means of these equations and the concomitant integration techniques. This specific context in which partial differential equations emerge in Euler's articles D e i n f i n i t i s c u r v i s and Additamentm has caused much confusion in historiography. Cousin, for instance, stated that Euler in 1 7 3 4 was in full possession of the theory of partial differential equations, but that he had obviously forgotten about the issue until d'Alembert took it up again around 1750'.

This statement makes the enigma connected with Euler's partial dif-

ferential calculus of the 1 7 3 0 s very clear. If Euler had indeed arrived at the theory of partial differential equations already in 1734, how could he possibly have forgotten about such an important matter? It is impossible to believe that Euler, with his fantastic memory and his enormous ability to build general theories upon a few inspiring examples would have forgotten anything of such importance. It was this field which he took up with such vigour in the 1750s and which he had ordered, and presented in a most influential textbook within twenty years. Others have been more scrupulous in their characterisation of Euler's dealing with partial differential equations. Demidov, for instance, in his 1 9 7 5 , provided a thorough analysis of the partial differential equations occurring in the Additamentwn and in De i n f i n i t i s euruis, and he concluded that, with the present perspective, we can identify partial differential equations and their integration in Euler's work; but, he continued, Euler's problem was a different one which is difficult to understand nowadays. In this chapter I shall attempt a broader approach to Euler's partial differential calculus of the 1730s, being convinced that a more profound understanding of Euler's achievements can only be reached through an appreciation of his own motives and his own programmes. My analysis will be based on the following sources: a hitherto unpublished manuscript, De d i f f e r e n t i a t i o n e (published in Appendix 2) dating from about 1730, the articles De i n f i n i t i s

curvis and the Additamentwn, which were read to the Petersburg Academy in 1 7 3 4 but were published only in 1740, and two more articles, De construetione and

Euler's Theory of Modular Equations in the 1730s

126

and S o l u t i o problematurn, composed in 1736 and 1735 respectively, in which Euler provided applications of the theory which he had developed in De infinitis

curvis and the Additamentwn. The following picture arises from this analysis: in the 1 7 3 0 s Euler was engaged in a programme to find what he called "modular equations", and to apply these modular equations to trajectory problems and to the integration problem for ordinary differential equations. These modular equations were in fact an elaboration of Nicolaus I Bernoulli's concept of complete differential equations f o r families of curves; all the applications of and contributions to partial differential calculus which Euler developed in the 1 7 3 0 s were closely related to families of curves. Hence, as far as motives and types of problems are concerned, Euler's research in this decade is fully within that tradition of partial differential calculus and families of curves, which we have been reviewing in the previous chapters of this book. This picture, then, also explains why Euler's partial differential calcul u s in the 1730s has been so enigmatic. Since the theory of modular equations

did not survive after the 1750s

and had receded far into the background even

in the 1 7 4 0 s , there is no direct descendant of this theory in later and modern mathematics. Modular equations simply died away. Hence, it is impossible to understand the rationale of Euler's research prior to 1 7 4 0 by merely tracing back ideas from modern mathematics. Such a procedure can be applied only to those historical concepts that still survive today in some recognisable way. The "modular equation" is not such a concept.

55.2 EuZer Is 55.2.1

expos6 of partial differential calculus in De differentiatione

Problem and method

The manuscript De differentiatione consists of two distinct parts; the first half of the manuscript contains exactly what the title promises: the elementary calculus of partial differentiation, consisting of the equality theorem, the interchangeability theorem and Euler's theorem on homogeneous functions. The second half of the manuscript is devoted to problems concerning families of curves, notably the orthogonal trajectory problem and the problem of equal arc or equal area trajectories. These problems are treated explicitly as examples of the application of the partial differential calculus as developed in the first half of the manuscript. This clear disentanglement of the differential calculus and its applications is the fascinating aspect of

Euler's expost! in De differentiatione

127

De differentiatione. Here i s t h e g r e a t d i f f e r e n c e between E u l e r ' s work and t h a t of h i s p r e c u r s o r s

-

n o t a b l y Nicolaus I B e r n o u l l i - i n t h e y e a r s b e f o r e

1720. I t i s the d i f f e r e n c e between a c o n c e p t u a l l y independent c a l c u l u s based on t h e key concept of f u n c t i o n s o r e x p r e s s i o n s i n v o l v i n g s e v e r a l v a r i a b l e s and a c a l c u l u s which i s c o n c e p t u a l l y t i e d up w i t h t h e concept o f a f a m i l y of curves. This i s a n o t h e r example of t h e well-known p r o c e s s of t h e d e g e o m e t r i s a t i o n of t h e c a l c u l u s , t a k i n g p l a c e i n t h e f i r s t h a l f o f t h e 18th c e n t u r y . However, t h i s does n o t i m p l y t h a t f a m i l i e s of c u r v e s no l o n g e r p l a y an i m p o r t a n t r o l e i n E u l e r ' s p a r t i a l d i f f e r e n t i a l c a l c u l u s . On t h e c o n t r a r y : they s t i l l p r o v i d e t h e main m o t i v a t i o n f o r E u l e r t o develop such a c a l c u l u s , and they s t i l l p r o v i d e t h e i n t e r e s t i n g problems. Only t h e i r r o l e a s t h e conceptual b a s i s of t h e c a l culus has now given way t o t h e more formal concepts of f u n c t i o n s and exp r e s s i o n s of s e v e r a l v a r i a b l e s . The important r o l e f a m i l i e s of c u r v e s s t i l l play i n E u l e r ' s p a r t i a l d i f f e r e n t i a l c a l c u l u s becomes c l e a r from t h e v e r y beginning of De differentiatione. E u l e r i n t r o d u c e d t h e g o a l s f o r h i s t r e a t i s e i n the f o l l o w i n g way: "For t h e s o l u t i o n of problems i n v o l v i n g i n f i n i t e l y many c u r v e s of t h e same s o r t t h e d i f f e r e n t i a l must be d e f i n e d of a q u a n t i t y composed of two v a r i a b l e s , when t h e d i f f e r e n t i a l of t h i s q u a n t i t y i s known f o r t h e c a s e t h a t one of t h e s e v a r i a b l e s i s taken t o be c o n s t a n t . Various problems of t h i s type conc e r n i n g orthogonal t r a j e c t o r i e s o r i n t e r s e c t i n g a s e r i e s of c u r v e s a c c o r d i n g t o a given law have been proposed and solved. I f P i s a f u n c t i o n composed of

a and

CC,

and Ydcc i s i t s d i f f e r e n t i a l i n c a s e a i s h e l d c o n s t a n t , then from

t h e s e d a t a the d i f f e r e n t i a l of P i s r e q u i r e d when a i s a l s o c o n s i d e r e d as v a r i a b l e . This d i f f e r e n t i a l w i l l have a form of t h i s type: &&+Rda, has t o be shown i n what way R can be found from

P i s n o t given o t h e r w i s e than by I&&,

and i t

P and &. But i f t h i s q u a n t i t y

s o t h a t b e s i d e s 6? n o t h i n g e l s e i s

known, i t seems t o be d i f f i c u l t indeed t o determine R from Q ~ n l y . ' ' ~ E u l e r h e r e p r e s e n t e d t h e completion problem a g a i n : t h e problem t h a t had occupied L e i b n i z i n 1697 and Johann B e r n o u l l i and Nicolaus I B e r n o u l l i around 1719, t h e problem r e s p o n s i b l e f o r most of t h e developments i n p a r t i a l d i f f e r e n t i a l c a l c u l u s up t i l l then. Without r e f e r e n c e t o t h e s e e a r l i e r

c o n t r i b u t i o n s , some of

which E u l e r had d e f i n i t e l y been informed about by Johann B e r n o u l l i , E u l e r pres e n t e d a c l e a r and e l e g a n t e x p o s i t i o n of t h e s e r e s u l t s i n De differentiatione. To t h i s he added h i s own d i s c o v e r y , v i z . t h e s o l u t i o n of the completion problem

by means of h i s theorem on homogeneous f u n c t i o n s ( c f . 1 5 . 2 . 4 ) .

128

EulerS Theory of Modular Equations in the 1730s

95.2.2

The equality of mixed second order differentials

A s Nicolaus I B e r n o u l l i had done, E u l e r gave t h e prominent p l a c e i n h i s

p a r t i a l d i f f e r e n t i a l c a l c u l u s t o the e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s . He continued i n De differentiatione a s f o l l o w s : "As t o the v a r i a b l e s of which P and Q c o n s i s t , the e n t i r e t a s k can be s e t t l e d b y means of t h e f o l l o w i n g p r i n c i p l e . I f a q u a n t i t y composed of two v a r i a b l e s

i s d i f f e r e n t i a t e d twice, where one of t h e v a r i a b l e s i s t r e a t e d a s a c o n s t a n t i n t h e f i r s t d i f f e r e n t i a t i o n , and t h e o t h e r v a r i a b l e i n t h e second d i f ferentiation

-

a s can be done i n two ways

-

then t h e r e s u l t s i n both c a s e s

w i l l be equal.'"* A d i d a c t i c w r i t e r t o t h e backbone, E u l e r c o r r o b o r a t e d t h i s p r i n c i p l e i n t h r e e

d f f e r e n t ways. He f i r s t demonstrated i t s v a l i d i t y by means of an example,

d d (x3y-y2x2)=3x2dxdy-4yxdxdy=d d (x3y-y2x2)and i n v i t e d h i s r e a d e r s t o c a l XY y x c u l a t e through some more examples:

"He who would be w i l l i n g t o apply t h e s e o p e r a t i o n s i n s e v e r a l examples w i l l n o t o n l y always observe t h i s concurrence, b u t he w i l l a l s o g r a s p t h a t t h e t r u t h of t h i s p r i n c i p l e must n e c e s s a r i l y h o l d i n a l l c a s e s . However, f o r the sake o f those who a r e l e s s a b l e t o g r a s p t h i s n e c e s s i t y , t h e f o l l o w i n g geom e t r i c a l demonstration can be p r ~ v i d e d . " ~ E u l e r ' s f i r s t of t h e two p r o o f s of t h e e q u a l i t y of mixed d i f f e r e n t i a l s was a proof by d i f f e r e n t i a t i o n from curve t o curve. I t i s almost i d e n t i c a l w i t h a p a r t of Johann B e r n o u l l i ' s proof of t h e i n t e r c h a n g e a b i l i t y of d i f f e r e n t i a t i o n and i n t e g r a t i o n , which indeed - as we have d i s c u s s e d i n 53.12

-

implicitly

employed t h e e q u a l i t y of mixed d i f f e r e n t i a l s . Although E u l e r does n o t r e f e r t o Johann B e r n o u l l i ' s argument, i t seems t o b e v e r y l i k e l y t h a t he had been i n s p i r e d by i t .

Proof by differentiation from curve to curve: Consider a curve AMm above an a x i s AP, and a curve AM, i n f i n i t e l y c l o s e t o

AMm a r i s i n g from AMm by i n c r e a s i n g i t s parameter a by t h e increment da (see figure I ) .

AP=x, and l e t PM r e p r e s e n t t h e e x p r e s s i o n y = P ( x , a ) . Now t h e mixed d i f f e r e n t i a l s d d P and d d P can b e given g e o m e t r i c a l i n t e r p r e t a t i o n s i n t h e x a a x

Put

f o l l o w i n g way: (5. I )

dX P=rm=S(x,aldx

and analogously

(5.2)

ns=S(x,a+da)dx;

129

Euler's expos4 in De differentiatione

fig. 1

hence

d d P = ns-rm. a x

(5.3)

Furthermore,

daP = MN = T(x,aa)da,

(5.4)

and analogously

mn = T(x+dx,a)da;

(5.5) hence

d d P = LVZ-MN. xa

(5.6)

Since n s = n r - s r and mr = n r - m n , t h e e q u a l i t y ( 5 . 3 ) t u r n s i n t o

d ad xP = r~ - r m = ( n r - s r ) - (nr-rim) = mn-sr = mn-MN

(5.7)

dx da P;

t h i s proves t h e a s s e r t e d e q u a l i t y of mixed second o r d e r d i f f e r e n t i a l s . The o t h e r proof provided by E u l e r was i n t r o d u c e d w i t h t h e f o l l o w i n g a r gumen t : " B u t s i n c e t h i s demonstration i s drawn from an a l i e n s o u r c e , I s h a l l d e r i v e

an o t h e r demonstration from t h e very n a t u r e of t h i s d i f f e r e n t i a t i o n . ' I 6

Proof from the "nature of d i f f e r e n t i a t i o n " : Consider t h e e x p r e s s i o n P(x,a) and perform t h e f o l l o w i n g s u b s t i t u t i o n s : Put P(x+dx,a)=Q, P(x,a+da)=R and P(x+&,a+da)=S. Then c l e a r l y S i s t h e r e s u l t of t h e s u b s t i t u t i o n of a+& f o r a i n t h e e x p r e s s i o n &, and a l s o t h e r e s u l t of the substitution

x+&

f o r x i n t h e e x p r e s s i o n R . By means of t h e s e e x p r e s s i o n s

t h e d i f f e r e n t i a l dxP can be e x p r e s s e d a s the d i f f e r e n c e Q-P, and dadx,? can b e expressed as

d d P=fS-R)-(&-P)=S-Q-R+P. a x Analogously, t h e d i f f e r e n t i a l d P can be e x p r e s s e d as t h e d i f f e r e n c e 8-P, and a (5.8)

Euler's Theory of Modular Equations in the 1730s

130

d,daP as

dxdaP=(S-Ql-(R-PI=S-&-R+P.

(5.9)

Hence obviously:

dxdaP=dadxP.

(5.10)

Euler would repeat this proof from the nature of differentiation over and over again in the next forty years'. Euler's treatment of the equality of mixed differentials is highly illuminating for his view on partial differentiation. Obviously, families of curves have no intrinsic relation with the basic conceptions of partial differentiation, since the proof of the equality based on differentiation from curve to curve is characterised as "drawn from an alien source". Instead, the nature of differentiation is limited to variables and expressions only, and differentiation is nothing else than taking differences of variables and differences of expressions. Once differentiation is regarded in such a way and taking differences i s granted, Euler's proof of the equality of mixed differentials is indeed valid.

5.2.3

I n t e r c h a n g e a b i Z i t y of d i f f e r e n t i a t i o n and i n t e g r a t i o n

The interchangeability theorem for differentiation and integration is an easy consequence of the equality of mixed partial differentials, and Euler indeed presented the theorem in this way. Taking the differential of P to be (5.1 1 )

dP=&&+Rda

and furthermore putting (5.12)

d&=K&+Lda and

(5.13)

dR=M&+Nda,

the equality of the mixed second order partial differentials of P immediately leads to: (5.14)

d,dxP=Ldadx=M&da=dxdaP;

hence : (5.15)

L=M. X

Now since R=

Mdx, and M=L=d

5 0

Euler finds: a&/A,

Eufer's expos6 in De differentiatione

131

I n f a c t , (5. 16) i s e q u i v a l e n t t o t h e i n t e r c h a n g e a b i l i t y theorem; s i n c e R=X

and P=/ Qdx one immediately f i n d s :

The

dap da

r e s u l t (5.16) provides t h e f i r s t s o l u t i o n t o t h e completion problem

which E u l e r had formulated i n t h e i n t r o d u c t o r y s e c t i o n of De d i f f e r e n t i a t i o n e :

E u l e r phrased t h i s r e s u l t i n the f o l l o w i n g r u l e :

"Q be d i f f e r e n t i a t e d while p u t t i n g

dx/&

J: c o n s t a n t ,

the r e s u l t be multiplied with

and i n t e g r a t e d a g a i n w h i l e a i s c o n s i d e r e d c o n s t a n t . The r e s u l t i n g in-

t e g r a l i s t h e v e r y q u a n t i t y which m u l t i p l i e d by da t o g e t h e r w i t h Qdx forms t h e r e q u i r e d d i f f e r e n t i a l of p."'

J5.2.4

Homogeneous f u n c t i o n s

Up t o t h i s p o i n t , none of E u l e r ' s r e s u l t s w a s new. The e q u a l i t y of mixed p a r t i a l d i f f e r e n t i a l s was known a l r e a d y t o Nicolaus I

B e r n o u l l i , who had a l s o

employed i t t o d e r i v e t h e i n t e r c h a n g e a b i l i t y of d i f f e r e n t i a t i o n and i n t e g r a t i o n , and t h e l a t t e r p r o p e r t y had a l r e a d y been found by L e i b n i z i n 1697. But what followed then was brand new: i t i s E u l e r ' s famous theorem on homogeneous functions : I f P(x,a) i s an homogeneous f u n c t i o n of degree n i n x and a , and dP=Qdx+R&, then nP=Qx+Ra. This theorem p r o v i d e s t h e c l u e t o a l l f u r t h e r r e s u l t s i n p a r t i a l d i f f e r e n t i a l c a l c u l u s a t which E u l e r a r r i v e d i n t h e 1730s. We s h a l l , t h e r e f o r e , pay cons i d e r a b l e a t t e n t i o n t o E u l e r ' s d e a l i n g w i t h homogeneous f u n c t i o n s . E u l e r ' s c r i t e r i o n f o r a f u n c t i o n P=P(x,a) t o be homogeneous of degree

n=O i s t h a t by p u t t i n g a=zx i n P ( x , a ) , P ( x , z x ) t u r n s i n t o a f u n c t i o n of z a l o n e . n Furthermore, P(x,a) i s homogeneous of degree n i f P ( x , a ) / x i s homogeneous of degree 0. These c r i t e r i a immediately y i e l d t h e proof of t h e theorem: For P(x,a) homogeneous of degree 0 , t h e d i f f e r e n t i a l dP=Qdx+Rda t u r n s i n t o dP=Qdx+Rzdx+

+Rxdz by t h e s u b s t i t u t i o n of z z f o r a . Since P ( x , 2 ~ ) i t s e l f i s a f u n c t i o n of z o n l y , t h e c o e f f i c i e n t of dx i n t h e t o t a l d i f f e r e n t i a l of P must be e q u a l t o

132

Euier's Theory of Modulur Equations in the 1730s

0. Thus: Q+Rz=O, or by inserting z=a/x again: Qx+Ru=O. Application of this n for homogeneous P of degree n immediately yields the reresult to P(X,U)/X sult stated in the theorem above: nP=Qx+Ra. The result onhomogeneous functions provided Euler with another method to s o l v e the completion problem, a method applicable to those cases in which the

integrand Q(z,ai is a homogeneous function of degree n-1. As Euler put it, homogeneity of Q(x,a)

of degree

n-1 implies homogeneity of P(x,a)=/Q(x,a)&

of degree n . Thus Euler could then solve the completion problem in the following way: For Q(x,u)

homogeneous of degree n-1, P(x,a)

gree n, and hence nP=Qz+Ra, or R=(nP-QxJ/u.

is homogeneous of de-

Thus the complete differential of

P ( x , a ) in this case is of the form: (5.1 9)

dP=Qdx+ (da/a)(nP-Qx).

However, in general the implication "Q homogeneous of degree n-1, then P homogeneous of degree n" is false; it does not hold as long as P(x:,a) is regarded

to represent the indefinite integral / & ( z , a ) d z . Since Euler never in-

dicated bounds of his integrals this remains a problematical issue. The implication can, nevertheless, be saved by assuming that the integral defining

P ( z , a i has lower bound equal to zero": (5.20)

P(x,a)=

LX

Q(t,a)dt

I shall always make this assumption.

5 5 . 2 . 5 Solutions to the completion problem

To sum up, Euler derived two solutions to the completion problem in De

dilferentiatione. For any expression Pix,&

defined by

OX

(5.21)

Pix,u)=jx Q (2,uldx 0

the complete differential of P ( x , a ) is provided by (5.18)

dP=Q&+da(J5

"2

(daQ/du)dx).

0

This solution I shall term Euler's general solution of the completion problem. The other solution to the completion problem holds only for those expressions P ( x , a ) , which are defined by 5

(5.20)

P(z,u)=L Q(x,a)dx,

Early applications

133

where & ( z , a ) is a homogeneous function of degree n-3. In this case the complete differential of Plx,a) is provided by dP=Qdx+ ( d a / a ) (nP-Qxl.

(5.19)

Both solutions are, of course, correct only in the global 18th century sense discussed in 51.4.

55.3 E a r l y app2ication.s of partial differentiation

"In order that the use of these theorems will be more fully appreciated, we shall apply them for the solution of certain problems. Let this, in the first place, be the problem of orthogonal trajectories, which has been dealt with very often."" This is how Euler introduced the first application of the partial differential calculus which he had developed in the first half of De differentiatione. The second application which he provided was concerned with determining differential equations for equal area trajectories. Hence both types of classical problems that had motivated the first development of partial differential calculus appear again in De differentiatione. Their status, however, has changed essentially. They have now become applications of an independent partial differential calculus. Here is another difference between Euler and his precursors. Nicolaus

I Bernoulli, for instance, had developed his partial differential calculus with the sole aim to tackle the orthogonal trajectory problem and to deal with the variable parameter equation. We shall first discuss Euler's achievements in the orthogonal trajectory problem, and deal with the equal area trajectories in 85.3.2; both applications will yield reasons why Euler may have withheld

De differentiatione from publication.

55.3.1

Orthogonal. trajectories

Let a family of curves be given by the equation (5.22)

y=P(x,al,

and let ( 5.2 3 )

dy=Q (x,a)dx+R(x,a )da

represent the total differential of the variable y defined in (5.22).

By means

Euler’s Theory of Modular Equations in the 1730s

134

of a geometric argument, which i s e q u i v a l e n t t o (dy:&Jcume=(-&:dy) Euler d e r i v e d t h e f o l l o w i n g c o n d i t i o n f o r t h e d i f f e r e n t i a l s

trajectory,

dx anddy t o s a t i s -

f y along t h e orthogonal t r a j e c t o r i e s : (5.24)

&:dy

= -Q(x,u).

The b a s i c i d e a of Eul.er’s approach t o t h e t r a j e c t o r y problem w a s t o combine t h e c o n d i t i o n ( 5 . 2 4 ) w i t h a s u i t a b l y chosen e q u a t i o n o r d i f f z r e - n t i a l e q u a t i o n pertaining A)

t h e given f a m i l y of c u r v e s . Three o p t i o n s were open t o him:

t3

I f P(x,a) i s a l g e b r a i c , then a d i f f e r e n t i a l equation f o r t h e orthogonal

t r a j e c t o r i e s can be found by e l i m i n a t i n g a from t h e two e q u a t i o n s (5.22)

y=P(z,u) and

(5.24)

dx:dy = -Q(x,a).

This i s p r e c i s e l y t h e same procedure which had produced L e i b n i z ’ s method i n 1694 and t h e Canon Hermanni i n 1717.

B)

I f t h e f a m i l y of curves is given by an e q u a t i o n of t h e form

(5.25)

.y=JxQ(x,a)h, 0

where Q(z,a) i s homogeneous of d e g r e e n-1 i n x and a , t h e n t h e o r t h o g o n a l t r a j e c t o r i e s m u s t b e found by combining t h e complete d i f f e r e n t i a l e q u a t i o n equivalent to (5.25): (5.26)

dy=Q(z,aldxi(&/a) (ny-xQ(x,a1 I w i t h

(5.24)

dx:dy = -Q(x,a).

C)

I f t h e family of c u r v e s i s given by an e q u a t i o n of t h e form

then t h e orthogonal t r a j e c t o r i e s must be found by combining t h e complete d i f f e r e n t i a l equation equivalent t o ( 5 . 2 7 ) : (5.28)

Lx

dy=Q(x,a)dx+da

Qa(x,u)dx w i t h

0

(5.24)

&:dy

= -Q(x,u).

Hence, both i n c a s e B) and c a s e C) t h e o r i g i n a l e q u a t i o n of t h e f a m i l y of c u r v e s must be r e p l a c e d by t h e r e l e v a n t complete d i f f e r e n t i a l e q u a t i o n , which Euler had taught t o f i n d i n t h e f i r s t p a r t of De differentiatione. E u l e r d e a l t w i t h c a s e B) i n 516 of De differentiatione. H i s argument may be summarised thus: e l i m i n a t i o n of t h e e x p r e s s i o n &(x,a) from e q u a t i o n ( 5 . 2 6 )

135

Early applications

by means of ( 5 . 2 4 ) and some rearrangement of the resulting equation yields: (5.29)

da/a= ( d x 2-dy2 ) / ( n y d y + x & ) .

This differential equation ( 5 . 2 9 ) still involves a and d a ; both a and da can, however, be eliminated again by means of the condition ( 5 . 2 4 ) ; since, solving ( 5 . 2 4 ) explicitly for a in terms of x,y,&,dy

and differentiating again yields

an expression for da in terms of x,y,dx,dy,d&,ddy taking d&=O

(which can be simplified by

for instance). Hence, this approach leads down to a second order

differential equation for the orthogonal trajectories. Euler's differential equation ( 5 . 2 9 ) is identical with the differential equation which Johann Bernoulli had found for the trajectories of a family of curves which possess exponential similarity (cf. J 3 . 1 2 footnote 4 4 ) . Euler knew about this coincidence, as becomes clear from a marginal note in which he stated: "for functionally similars he found da/a= (dx2+dy')/((Aa/a) ydy+x&l,

da=Ada;

it will be of help to make this comparison.""

Undoubtedly, Euler's reference is to 9549-61 of Nicolaus I1 Bernoulli's 2720; these sections contain the expos5 of Johann Bernoulli's method for finding the orthogonal trajectories of families of exponentially and functionally similar curves. The interesting point in Euler's derivation of ( 5 . 2 9 ) is that he arrived at this differential equation by assuming a special type of equation for the family of curves, viz. homogeneous P(x,a), whereas Johann Bernoulli had found the equation by assuming that the curves satisfy the geometrical requirement of exponential similarity. We shall return to this coincidence shortly. Let us now turn to the third option, case C), not yet discussed. In 915 of De differentiatione Euler had briefly discussed the case in which the coefficient & ( x , a ) of dcc in equation ( 5 . 2 8 ) is eliminated by means of equation (5.24).

Such elimination leads to the equation

X

which may be of use once the integral I Q (x,a)dx can be calculated explicitly 20

a

in terms of x and a. However, the result is rather inconclusive and Euler immediately turned his attention to case B). But in a long marginal calculation (viz. (m2), see Appendix 2 ) Euler attempted another solution based on the equations ( 5 . 2 8 ) and ( 5 . 2 4 ) , which brought him within very close reach of Nicolaus I Bernoulli's corollary method (cf. 5 4 . 3 . 3 ) and Johann Bernoulli's generalised synchrone method (cf. 5 3 . 1 1 . 2 ) . Eliminating d y from ( 5 . 2 8 ) by means of ( 5 . 2 4 ) Euler arrived at the

136

Euler's Theory of Modular Equations in the 1730s

v a r i a b l e parameter e q u a t i o n

which d e s c r i b e s t h e r e l a t i o n between

3:

and a a l o n g t h e o r t h o g o n a l t r a j e c t o r i e s .

H e t h e n t r i e d t o f i g u r e o u t under which c o n d i t i o n s , t o be imposed on Q ( z , a ) ,

t h i s e q u a t i o n (5.31) can be i n t e g r a t e d by means of an i n t e g r a t i n g f a c t o r A(a). Given such an i n t e g r a t i n g f a c t o r t h e s o l u t i o n of (5.31) would t a k e on t h e f o l l o w i n g form: (5.32)

+l

S_wx$j-Adz * 0

= constant.

By comparing t h e t o t a l d i f f e r e n t i a l of t h e i n t e g r a l i n t h e l e f t hand s i d e of (5.32) w i t h t h e l e f t hand s i d e of (5.31) E u l e r found t h e f o l l o w i n g c o n d i t i o n for the integrating factor A ( a ) : (5.33)

daQ QdaQ d A -- - __ A Q Q2+1

I n t e g r a t i n g e q u a t i o n (5.33) w i t h r e s p e c t t o a w h i l e i n t r o d u c i n g Y ( 3 : ) a s t h e c o n s t a n t of i n t e g r a t i o n E u l e r a r r i v e d a t : (5.34)

A(al rQ y(x)

J&2+1

I f handled p r o p e r l y e q u a t i o n (5.34) immediately y i e l d s t h e f o l l o w i n g g e n e r a l form of Q ( x , a ) : (5.35)

&(x,al= J ( Y 2 (xl-A2 f a )

By p u t t i n g Z(3:)=Y2(3:1 (5.36)

Q(x,a)=

and B ( a ) = l / A 2 ( a ) (5.35) transforms i n t o 1

JB f a ) ~ ( x 1 - 2 However, Euler had made a mistake i n h i s c a l c u l a t i o n s , and i n s t e a d of a r r i v i n g a t e q u a t i o n (5.36) he found: (5.37)

Q(x,a)=

A(a)

Y ( x ) (Y2 (x)-A2( a l l

Some f a i n t s c r i b b l i n g s a t t h e end of t h e marginal n o t e (m2) show t h a t E u l e r then t r i e d t o reduce e q u a t i o n (5.37) t o t h e form ( 5 . 3 6 ) , b u t t h a t , of c o u r s e , was impossible. T h i s marginal c a l c u l a t i o n (m2) i s h i g h l y f a s c i n a t i n g and may p r o v i d e t h e c l u e t o understanding E u l e r ' s i n t e n t i o n s w i t h t h e o r t h o g o n a l t r a j e c t o r y problem. E u l e r obviously knew which r e s u l t s he was l o o k i n g f o r ; he w a s e a g e r t o

Early applications

137

f i n d h i s way through t o the r e s o l u t i o n o f t h e t r a j e c t o r y problem f o r c u r v e s given by e q u a t i o n -5

(5.38)

y=j x

0

1

dB(0.1 Z ( X ) - 1

&.

This becomes even more e v i d e n t from a n o t h e r marginal n o t e , v i z . (d), i n which t h e i n t e g r a l of (5.38) appears a g a i n , and where Euler a g a i n w a s u n s u c c e s s f u l . . Why was E u l e r so preoccupied with t h i s e q u a t i o n (5.38)? From r e a d i n g Nicolaus

I1 B e r n o u l l i ' s survey 2720 Euler d e c i d e d l y knew t h a t t h e o r t h o g o n a l t r a j e c t o r i e s of t h e c u r v e s given by e q u a t i o n (5.38) could be c o n s t r u c t e d by means of

a c o n d i t i o n of t h e form (5.32). T h i s had been t h e main r e s u l t of Johann Bern o u l l i ' s g e n e r a l i s e d synchrone method ( c f . J3.11.2).

And perhaps E u l e r had a l -

so come a c r o s s Nicolaus I B e r n o u l l i ' s c o r o l l a r y method ( c f . 14.3.3) a s p r e s e n t e d i n t h e Tentamen, which method a l s o a p p l i e d t o t h e c a s e (5.38). I t seems t h a t E u l e r had i n f a c t s e t o u t i n De d i f f e r e n t i a t i o n e t o d e r i v e a l l those methods f o r c o n s t r u c t i n g o r t h o g o n a l t r a j e c t o r i e s which had emerged i n t h e y e a r s 1716-1720, b u t now i n a novel way, based e n t i r e l y on complete d i f f e r e n t i a l e q u a t i o n s and e l i m i n a t i o n procedures. I f s u c c e s s f u l ,

such a u n i f i e d p r e s e n t a -

t i o n of t h e well-known methods would indeed h a v e made good p u b l i c i t y f o r t h e p a r t i a l d i f f e r e n t i a l c a l c u l u s developed i n Ee d i f f e r e n t i a t i o n e . But t h e a c t u a l r e s u l t s must have been d i s a p p o i n t i n g t o E u l e r . Although he had found t h e d i f f e r e n t i a l e q u a t i o n (5.26),

and i n d i c a t e d i t s c o i n c i d e n c e w i t h Johann B e r n o u l l i ' s

e q u a t i o n f o r f a m i l i e s of e x p o n e n t i a l l y s i m i l a r c u r v e s , t h e f a i l u r e t o p u t through t o t h e m y s t e r i o u s i n t e g r a l (5.38) may w e l l have been one of t h e r e a s o n s why E u l e r decided t o withhold De d i f f e r e n t i a t i o n e from p u b l i c a t i o n . To conclude t h i s d i s c u s s i o n , l e t u s q u i c k l y glance over E u l e r ' s l a t e r a l t e r c a t i o n s w i t h t h e o r t h o g o n a l t r a j e c t o r y problem. They a l l show t h e same preoccupation w i t h t h e f a m i l y of c u r v e s given by e q u a t i o n (5.38).

I n 1743 Nico-

l a u s I B e r n o u l l i r e v e a l e d t o E u l e r h i s a n a l y s i s behind the methods p r e s e n t e d i n t h e Tentamen, among them t h e r e s o l u t i o n of e q u a t i o n (5.31) and t h e s o l u t i o n of t h e t r a j e c t o r y problem f o r t h e c u r v e s given by e q u a t i o n (5.38). E u l e r w a s duly impressed and e n t h u s i a s t i c ( c f . 94.3.6). occur a g a i n i n Euler ' s Adversaria Mathematics t h e y e a r s 1749-1753l

3,

The c u r v e s d e f i n e d by (5.38) ( h i s mathematical d i a r y ) f o r

and h e r e E u l e r c a l c u l a t e d through completely and success-

f u l l y , f o r t h e f i r s t time a s f a r as I know. Then i n 1768, 1771, and 1775, back i n P e t e r s b u r g and d i c t a t i n g h i s mathematical r e c o l l e c t i o n s , E u l e r composed h i s f i r s t a r t i c l e s about t h e t r a j e c t o r y p r ~ b l e m ' ~ And . i n a l l t h r e e of them t h e case (5.38) was d i s c u s s e d and c a l c u l a t e d through e x t e n s i v e l y ; i t was given VIP

Euler's Theory of Modular Equations in the 1730s

138

t r e a t m e n t i n a l l t h r e e a r t i c l e s . I n t h e l a s t one, p u b l i s h e d posthumously ( 1 7 8 7 1 , Euler remarked: " I t i s abundantly c l e a r how profound a t a s k t h e i n v e s t i g a t i o n of t h e e q u a t i o n A was, s i n c e i t was deduced from t h e n a t u r e of f u n c t i o n s o f two

y = J m &

v a r i a b l e s o n l y , which [ n a t u r e ] by then was almost e n t i r e l y unknown. Now t h e f i r s t t o b r i n g t o l i g h t t h i s most e x t r a o r d i n a r y specimen of such type of a n a l y s i s , a l r e a d y 60 y e a r s ago, was the v e r y a c u t e Nicolaus B e r n o u l l i , t h e son of Nicolaus and p r o f e s s o r of law a t t h e u n i v e r s i t y of Basel. T h e r e f o r e , i t is t o him t h a t we owe the enormous a m p l i f i c a t i o n s which have been i n t r o duced and, what i s even more, a c c e p t e d s i n c e i n a n a l y ~ i s . " ' ~

95.3.2 E q u a l area trajectories E s s e n t i a l l y , t h e problem t o d e r i v e a d i f f e r e n t i a l e q u a t i o n f o r an e q u a l a r e a t r a j e c t o r y of a family of c u r v e s c o n s i s t s of t h e f o l l o w i n g e l i m i n a t i o n procedure: L e t t h e family of c u r v e s be r e p r e s e n t e d by e q u a t i o n (5.39)

y=P(x,a)

then e l i m i n a t e a from ( 5 . 3 9 ) and t h e c o n d i t i o n e q u a t i o n (5.40)

L:

P(x,aldx=constant.

S(z,al=

When t h e i n t e g r a l

X

1 P(x,a)&

can be c a l c u l a t e d e x p l i c i t l y , one i s done. I f

20

the c u r v e s , however, do not admit a l g e b r a i c q u a d r a t u r e , o t h e r methods a r e c a l l e d f o r . The c o n d i t i o n e q u a t i o n ( 5 . 4 0 ) i t s e l f i s i n s u f f i c i e n t f o r such e l i mination,

f o r t h e simple reason t h a t i n s e r t i o n of a=A(x) i n t o t h e i n t e g r a n d

was regarded impossible. This p o i n t , which w e have d e a l t w i t h i n §1.5, b a s i c a l l y d e r i v e s from t h e f a c t t h a t t h e 17th and 18th c e n t u r y mathematicians d i d n o t make the c o n c e p t u a l d i s t i n c t i o n between a dummy v a r i a b l e of i n t e g r a t i o n and t h e v a r i a b l e bounds of t h e i n t e g r a l . Hence, a s soon a s t h e q u a n t i t y u i n t h e i n t e g r a n d would be r e p l a c e d by A(xcl, t h i s A f s l would a l s o p a r t i c i p a t e i n the a c t u a l i n t e g r a t i o n ; t h e r e f o r e , a s u b s t i t u t i o n of t h i s type was always d e s c r i b e d c a r e f u l l y as " s u b s t i t u t i o n a f t e r t h e i n t e g r a t i o n h a s been c a r r i e d out". X

I P(x,a)dx can n o t be c a l c u l a t e d e x p l i c i t l y , one can "CO X t u r n t o t h e complete d i f f e r e n t i a l e q u a t i o n e q u i v a l e n t t o Sfx,ua)=I Pfx,a)dx, I n case the i n t e g r a l

which i n i t s g e n e r a l form i s

3 3 0

Early applications

139

P u t t i n g Slz,a)=const. y i e l d s t h e c o n d i t i o n e q u a t i o n

xd P (5.42)

.

-&?XI

O=P(;c,ddx+da< 0

Now a g a i n , i f t h e i n t e g r a l can be c a l c u l a t e d e x p l i c i t l y , one i s done, s i n c e i n t h a t c a s e a and

da

can be e l i m i n a t e d from ( 5 . 4 2 ) by means of t h e e q u a t i o n ( 5 . 3 9 ) ,

and a d i f f e r e n t i a l e q u a t i o n i n terms o f x and

y f o r the equal area t r a j e c t o r y

would emerge. However, when t h e i n t e g r a l cannot be c a l c u l a t e d , one must go on and apply ad hoc t r i c k s which might l e a d t o a c o r n p l e t e d i f f e r e n t i a l e q u a t i o n c o n t a i n i n g a l g e b r a i c c o e f f i c i e n t s only. E u l e r d e s c r i b e d such procedures i n 922 of De d i f f e r e n t i a t i o n e . I n view of such t r o u b l e w i t h t r a n s c e n d e n t a l c o e f f i c i e n t s one r e a d i l y a p p r e c i a t e s those c a s e s where t h e complete d i f f e r e n t i a l e q u a t i o n does n o t c o n t a i n t r a n s c e n d e n t a l c o e f f i c i e n t s , a s i t happens, f o r i n s t a n c e , when t h e a r e a S(z,al

i s e x p r e s s e d by a homogeneous f u n c t i o n of x and a. Then

the complete d i f f e r e n t i a l of S(x,a) can be w r i t t e n down immediately, a s E u l e r had shown i n t h e f i r s t p a r t of De d i f f e r e n t i a t i o n e : I f P(x,a) i s homogeneous of degree n,and i f S(x,a)=

X

J

P(z,a)dx, then

0

(5.43)

dS lx, aj=P(x, aj&+(da/a) i (n+l)Six,aj-xPix,a)) .

E u l e r i n v e s t i g a t e d t h i s s i t u a t i o n i n § 2 3 and t h e f i n a l § 2 4 of De d i f f e r e n t i a -

t i o n e . For such homogeneous P(x,a), t h e e q u a l a r e a t r a j e c t o r i e s can be found by e l i m i n a t i n g a and

da

from

( r e s u l t i n g from t h e s u b s t i t u t i o n S(x,a)=constant i n ( 5 . 4 3 ) ) by means of t h e equation (5.39)

y=Pix,a).

E u l e r c a l c u l a t e d through t h e example (5.45)

y=Piz,al=rZGG2 ,

d e s c r i b i n g a f a m i l y of e l l i p s e s , and he found t h e f o l l o w i n g d i f f e r e n t i a l equat i o n f o r the equal ar ea t r a j e c t o r i e s : (5.46)

4Cydy+4Cx& y2+x2

- -2 C d ~- Zy2xdy-2y3& X

xZiy2

A t t h i s p o i n t , t h e manuscript breaks o f f ; we a r e l e f t w i t h a p u z z l i n g

s i t u a t i o n , which c a l l s f o r some s p e c u l a t i o n . E u l e r ' s t r e a t m e n t of t h e e q u a l a r e a t r a j e c t o r i e s c l e a r l y showed t h e advantage of complete d i f f e r e n t i a l equat i o n s w i t h a l g e b r a i c c o e f f i c i e n t s o v e r t h o s e having t r a n s c e n d e n t a l c o e f f i c i e n t s ; f o r , i n t h o s e c a s e s one can immediately e l i m i n a t e t h e parameter a and i t s

EulerS Theory of Modular Equations in the 1730s

140

differential da. Furthermore, it becomes clear that such a complete differential equation with algebraic coefficients is a very convenient intermediary between the geometrical definition of a trajectory and its differential equation: insertion of S(x,a)=constant

in this equation immediately paves the way to the

differential equation of the trajectory. In De znfinitis curvis Euler would extensively elaborate precisely this point. The geometrical definition of the equal area trajectory is in fact a very elegant definition, which would serve well as the solution of a problem. We have met such constructions in previous chapters, though never as a problem to be solved, but rather as the solution to a differential equation. It seems highly probable that Euler recognised this issue

as

well when he arrived at the differential equation ( 5 . 4 6 ) ; I guess

that at this point the idea struck him that problem and solution should instead be reversed: the geometrical definition of the equal area trajectory should be regarded the solution rather than the problem, and the differential equation that is found should be regarded the problem instead of the solution. Hence, Euler had sufficient reason to reconsider the entire situation, and here may be a final reason why he withheld De differentiatione from publication.

55.4 EuZer's theory of modular equations 55.4.1 The shift from trajectories to differential equations The Euler-Daniel Bernoulli correspondence of the years 1734-1735 clearly corroborates the conjecture concerning Euler's inversion of "problem" and "solution" in his research on trajectories in families of curves. The subject emerged in Euler's letter of February 18th, 1 7 3 4 , in which Euler informed Bernoulli that he had found, in an indirect way, a construction for the differential equationI6: (5.47)

r2+b2 ddu = r ( b 2 - r 2 1 drdu+

&d r 2 .

In this equation, the variable r represents the variable vertical axis in a family of ellipses with horizontal axis b , and the variable u represents the circumference of the ellipse with vertical axis r. This identification of the variables u and r indeed provides a construction for the differential equation ( 5 . 4 7 ) in terms of rectification of ellipses.

In November of the same year, Euler took up the issue again, when he wrote:

Euler's theory of modular equations

141

"I have solved the following problems, about which I should like to hear the opinion of you yourself, of your father [Johann Bernoulli]

, and

of other

mathematicians. The first concerns finding a curve that cuts off equal arcs from infinitely many ellipses, all of which are erected over one [and the same] transverse axis; [the other]

-

in the same way

-

[is about] a curve

that cuts off equal arcs from infinitely many ellipses having one common vertex and a common conjugate axis. The construction of these curves by means of the rectification of ellipses is easy, but I require an equation for these curves, which will be of such a nature, that one cannot find a construction from it without my method; by way of this method I have also constructed the Riccati equation."17 Politely Daniel Bernoulli wrote back: "Your problem about cutting off equal arcs in a series of ellipses is very profound and cannot, I believe, be solved otherwise than indirectly (litt.: a posteriori), by means of your method of series."" However, it was not the method of series",

developed by Euler around 1733

which had yielded these constructions, but a new, and much more powerful method. This is what Euler pointed out to Daniel Bernoulli in his reply of June 2nd, 1735:

"I have not been able to solve the problem concerning cutting off equal arcs in a series of ellipses by means of my a posteriori method of series; therefore, I do not believe that it can be solved in the same way in which I have given a construction of the Riccati equation. However, I have hit upon another direct method of much broader scope, by means of which I cannot only achieve much more than hitherto done in the problem of orthogonal trajectories, but also have I found, without series, the same construction for the Riccati equation which I had invented earlier. By means of this very method

I have arrived at the equation for the curve that cuts off equal arcs from infinitely many ellipses; this is a second order differential equation, and, moreover, of so complicated a nature that I have in no way been able to reduce it to a first order differential equation. I have read the principles of this new method in our conventions [i.e. of the Petersburg Academy] already last year, and I have prepared the solution of the problem of these ellipses, which I am going to read in the near future. However, many more dissertations will be needed in order to pursue this matter appropriately."' Exactly one week later, on the 9th of June 1735 Euler indeed read his S o l u t i o

probZernatwn to the Petersburg Academy. This article dealt with the problem of finding differential equations for equal arcs trajectories in families of

Euler's Theory of Modular Equations in the 1730s

I42

e l l i p s e s by means o f "modular e q u a t i o n s " . It was followed up by De construc-

t i o n e , read on the 7 t h of February 1737, i n which modular e q u a t i o n s were a p p l i e d f o r f i n d i n g s o l u t i o n s of the R i c c a t i e q u a t i o n . The new method was E u l e r ' s theory of modular e q u a t i o n s , which had been submitted t o t h e Academy a l r e a d y i n 1734 i n the a r t i c l e s De i n f i n i t i s curvis (read on t h e 5 t h of May) and t h e

Additamentum ( r e a d on t h e 12th of J u l y , 1734). I s h a l l d i s c u s s E u l e r ' s concept of modular e q u a t i o n s i n t h i s s e c t i o n , and proceed t o i t s a p p l i c a t i o n s i n 55.5.

The concept o f a modular equation

55.4.2

The f u l l t i t l e of E u l e r ' s De i n f i n i t i s curvis i s : "On i n f i n i t e l y many curves of t h e same genus, o r a method of f i n d i n g t h e e q u a t i o n s f o r i n f i n i t e l y many curves of t h e same genus".

I n t h i s a r t i c l e Euler i n t r o d u c e d t h e concept

of a "modular equation" of a f a m i l y of c u r v e s , which he d e f i n e d i n t h e f o l l o wing way: " I n f i n i t e l y many c u r v e s of t h e same genus a r e a l l e x p r e s s e d by one s i n g l e e q u a t i o n , i n which t h e modulus, which I s h a l l always i n d i c a t e by t h e l e t t e r a , i s contained. F o r , i f t o t h i s modulus d i f f e r e n t v a l u e s a r e a s s i g n e d s u c c e s s i v e l y , t h e e q u a t i o n w i l l c o n t i n u a l l y produce o t h e r c u r v e s , which a r e a l l contained i n one e q u a t i o n . Following Hermann",

we s h a l l c a l l t h i s equa-

t i o n which c o n t a i n s t h e modulus, t h e modular e q u a t i o n . I n t h i s [modular equation]

,

t h e r e f o r e , b e s i d e s o t h e r c o n s t a n t s and q u a n t i t i e s having t h e same

v a l u e i n a l l of t h e curves, t h e modulus a and two v a r i a b l e s o c c u r , which pert a i n t o an a r b i t r a r y curve, such a s a b s c i s s a and o r d i n a t e , o r a b s c i s s a and a r c l e n g t h , o r t h e a r e a under t h e curve and t h e a b s c i s s a , j u s t a s i s r e q u i r e d by t h e problem t o be This e x p l a n a t i o n alone does n o t d e f i n e t h e concept; f o r , a modular e q u a t i o n seems t o be n o t h i n g e l s e t h a n an e q u a t i o n i n v o l v i n g t h e parameter o r modulus

a and two o t h e r geometric v a r i a b l e s which c h a r a c t e r i s e t h e s i n g l e c u r v e s . Other requirements f o r modular e q u a t i o n s f o l l o w ; however, t h e y a r e n o t v e r y exp l i c i t and must be d i s t i l l e d from s c a t t e r e d remarks. T h i s i s q u i t e c u r i o u s i n an a r t i c l e of E u l e r ' s ; a s a r u l e , E u l e r always e x p l a i n s h i s problems p r e c i s e l y and w i t h d i d a c t i c s k i l l . The crux of a modular e q u a t i o n i s the requirement t h a t t h e modulus a must be a v a r i a b l e ; t h i s requirement r u l e s o u t t r a n s c e n d e n t a l e q u a t i o n s of t h e form

143

Euler’s theory of modular equations

s i n c e , a s E u l e r s t a t e d , i n t h i s i n t e g r a l “a must be c o n s i d e r e d t o be c o n s t a n t , and t h i s runs counter t o t h e n a t u r e of a modular e q u a t i o n , i n which, namely,

a must be a s v a r i a b l e a s

5

and y r r 2 3 .For e x a c t l y t h e same r e a s o n an incomplete

d i f f e r e n t i a l equation (5.49)

d$=P(x,a)dx

i s i n s u f f i c i e n t f o r a modular e q u a t i o n , because h e r e a g a i n a must be taken c o n s t a n t i n t h e p r o c e s s of d i f f e r e n t i a t i o n . E u l e r i l l u m i n a t e d t h i s p o i n t somzwhat f u r t h e r ; an e q u a t i o n of type ( 5 . 4 8 ) , he w r o t e , “ s u f f i c e s f o r knowing and c o n s t r u c t i n g the

s e p a r a t e l y . Indeed, f o r each a t h e e q u a t i o n ( 5 . 4 8 )

provides a c o n s t r u c t i o n i n terms of q u a d r a t u r e s f o r t h i s curve. However, a s the equation ( 5 . 4 8 )

soon a s one “must a s s i g n c e r t a i n p o i n t s on t h e s e

i s no l o n g e r s u f f i c i e n t . The reason f o r t h i s i n s u f f i c i e n c y , I b e l i e v e , i s a g a i n t h e m a t t e r of s u b s t i t u t i o n i n i n t e g r a n d s . A s we have n o t i c e d i n 8 5 . 3 . 2 ,

sub-

s t i t u t i o n of A ( x ) f o r a i n t h e i n t e g r a n d of ( 5 . 4 8 ) was regarded i m p o s s i b l e a t t h a t t i m e . Hence i t was impossible t o d e r i v e an e q u a t i o n f o r a t r a j e c t o r y curve

-

d e f i n e d , f o r i n s t a n c e , by t h e p r o p e r t y t h a t it i n t e r s e c t s t h e curve w i t h para-

meter a=A(x) a t a p o i n t with a b s c i s s a 5

-

by s t r a i g h t f o r w a r d s u b s t i t u t i o n of

A(x) i n t o e q u a t i o n ( 5 . 4 8 ) . I f a family of curves i s given by a t r a n s c e n d e n t a l e q u a t i o n of type ( 5 . 4 8 1 , one must l o o k f o r an e q u a t i o n e q u i v a l e n t t o ( 5 . 4 8 ) i n which a occurs a s a v a r i a b l e w i t h o u t f u r t h e r r e s t r i c t i o n s . The modular e q u a t i o n e q u i v a l e n t t o ( 5 . 4 8 ) i s of t h e form (5.50)

when

dy=P (z, a)&+Qda

Q can b e determined a l g e b r a i c a l l y i n terms of z,y and a. I f such a 6? cany must be employed, andone must

n o t be found, h i g h e r o r d e r d i f f e r e n t i a l s of

i n v e s t i g a t e whether f o r example ddy can b e e x p r e s s e d i n t h e f o l l o w i n g way: (5.5 1)

ddy=AZgebraic E q r . in (x,y, a,dx,dy,da, ddx,dda)

.

I f such an a l g e b r a i c e x p r e s s i o n cannot be found one must proceed t o h i g h e r o r d e r d i f f e r e n t i a l s of y, o r even conclude t h a t no modular e q u a t i o n e x i s t s i n t h e given c a s e . E u l e r ’ s concept of modular e q u a t i o n i s i n f a c t a r e f i n e d v e r s i o n of t h e concept of a complete d i f f e r e n t i a l e q u a t i o n a s i n t r o d u c e d by Nicolaus I Bern o u l l i and a s used by himself i n De differentiatione; t h e problem t o f i n d mod u l a r e q u a t i o n s i s a r e f i n e d v e r s i o n of t h e completion problem, which h a s a l s o been d e a l t w i t h i n De differentiatione. Hence, modular e q u a t i o n s are complete d i f f e r e n t i a l e q u a t i o n s (though n o t . n e c e s s a r i l y of f i r s t o r d e r ) w i t h a l g e b r a i c

Euler's Theory o f Modular Equations in the 1730s

144

c o e f f i c i e n t s . A l l methods which E u l e r had d e r i v e d i n De d i f f e r e n t i a t i o n e t o s o l v e t h e completion problem occur again i n De i n f i n i t i s curvis, b u t now f o r f i n d i n g modular e q u a t i o n s . Here, f o r t h e f i r s t time, E u l e r p u b l i c l y p r e s e n t e d h i s proof of the e q u a l i t y theorem, h i s subsequent d e r i v a t i o n of t h e i n t e r c h a n g e a b i l i t y theorem f o r d i f f e r e n t i a t i o n and i n t e g r a t i o n , and, f i n a l l y , t h e theorem on homogeneous f u n c t i o n s . The p r e s e n t a t i o n of t h e s e theorems i n D e i n -

f i n i t i s curvis d i f f e r s from t h e one i n De d i f f e r e n t i a t i o n e i n i r r e l e v a n t d e t a i l s o n l y , and I s h a l l , t h e r e f o r e , r e f r a i n from r e p e a t i n g them a g a i n .

5 5 . 4 . 3 The method of i n t e g r a Z reduction

Given a family of curves by t h e e q u a t i o n

(5.48)

y=JxP(z,a)dx, X

t h e g e n e r a l s o l u t i o n of t h e completion problem De d i f f e r e n t i a t i o n e then p r o v i d e s t h e f o l l o w i n g complete d i f f e r e n t i a l of y : 2

(5.52)

dy=P ( x , a I d x + d a s Pa (a, n I dx.

Xo

x

Unless t h e i n t e g r a l

I Pa(x,a)&

can a c t u a l l y b e c a l c u l a t e d , ( 5 . 5 2 ) i s n o t a

20

modular equation. However, i t may y i e l d a modular e q u a t i o n i f t h e i n t e g r a l X

I P (x,a)dx

can be expressed a l g e b r a i c a l l y i n terms of t h e i n t e g r a l

xo a

fXP(x,a)d.z XO

i n t h e f o l l o w i n g way: X

(5.53)

~ X P a ( ~ , a ) d x = A (P(x,a)dx+R(x,a). a)~ X

0

Replacing t h e i n t e g r a l

2 O

J P(x,a)dx

i n ( 5 . 5 3 ) by y and combining ( 5 . 5 3 ) w i t h

?O

( 5 . 5 2 ) y i e l d s t h e followirig modular e q u a t i o n : (5.54)

dy=P(x, a)dx+da ( A (a)y+B(2,a)1 .

The same procedure can b e a p p l i e d f o r f i n d i n g h i g h e r o r d e r modular equat i o n s , a s w i l l be c l e a r from t h e f o l l o w i n g example: i n g e n e r a l , ddy i s of t h e form: t

(5.55)

ddy=P(x, a)ddx+PX(x,a)dx2+2Pa(x,a)dxda+{

+(Lx

Paa(x,a)dx)da2.

0

Pa 3:

0

(2,a)dx)dda+

Euler’s theory of modular equations I f an a l g e b r a i c r e l a t i o n of t h e form

(5.56)

Lx?,, (2,

0

a)&=*

s::

(a)

sx

P(x, a)dx+B(a)

0

Pa

(3,

145

a)dx+C(x,a )

““0

can be found, then (5.55) c a n b e turned i n t o a modular e q u a t i o n by means of t h e following s u b s t i t u t i o n s :

lxPaa(x,a)dx be r e p l a c e d by t h e r i g h t hand s i d e of (5.56)

(a)

s,”.

(b)

a

( z , a ) d x be r e p l a c e d by (dy-?(x,a!dxI/&,

0

which e x p r e s s i o n can be found from e q u a t i o n ( 5 . 5 2 ) . A s we s h a l l s e e i n 55.5.2,

E u l e r ‘ s d e r i v a t i o n of t h e d i f f e r e n t i a l e q u a t i o n

f o r e q u a l a r c s t r a j e c t o r i e s i n a f a m i l y of e l l i p s e s , about which he had informed Daniel B e r n o u l l i i n June 1735, was based on e x a c t l y t h i s method of i n t e g r a l red u c t i o n f o r second o r d e r modular e q u a t i o n s .

55.4.4 Homogeneous and generalised homogeneous functions E u l e r ’ s second method f o r f i n d i n g modular e q u a t i o n s c o i n c i d e s e n t i r e l y

with h i s method of f i n d i n g complete d i f f e r e n t i a l e q u a t i o n s f o r homogeneous X

f u n c t i o n s y= J P(x,a)dx, a s p u t f o r t h i n De differentiatione: I f t h e i n t e g r a n d 0

Pix,a) i n (5.5 7)

Y=/%x,

a,)&

0

i s homogeneous of degree n-1 i n a and

x, then t h e modular e q u a t i o n p e r t a i n i n g

t o (5.57) i s

(5.48)

dy=P(x,a)dx+ida/a) iny-x?cP(x,al).

Hence, f o r homogeneous P ( x , a ) t h e modular e q u a t i o n always i s a f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n . E u l e r was d e l i g h t e d by t h e ease w i t h which modular equat i o n s could be found i n such c a s e s . This s u c c e s s motivated h i s s e a r c h f o r o t h e r r e c o g n i s a b l e c l a s s e s of i n t e g r a n d s P ( z , a ) , which would a l l o w t h e same e a s y way of f i n d i n g modular e q u a t i o n s . I t i s t h i s g o a l which he pursued i n t h e A d d i t a -

mentwn, as E u l e r s t a t e d e x p l i c i t l y a t t h e o u t s e t of t h i s a r t i c l e : ” I n t h e preceding t r e a t i s e I have d i s c o v e r e d t h a t q has an a l g e b r a i c v a l u e whenever p i s such a f u n c t i o n of

x and a, t h a t t h e number of dimensions which

x and a c o n s t i t u t e i s e v e r y w h e r e e q u a l t o -1 [indeed, then q - p x / a ] . [ . . . I

Euler's Theory of Modular Equations in the 1730s

146

f u r t h e r I have a l s o observed t h a t whenever i n p t h e l e t t e r s a and x everywhere c o n s t i t u t e t h e same number of dimensions, q depends on t h e i n t e g r a t i o n of p&

[ i . e . y ; indeed, i n t h a t case q=(ny-px)/u]. Since from t h i s such ex-

c e l l e n t e x p e d i e n t s f o l l o w f o r f i n d i n g modular e q u a t i o n s , i t w i l l be v e r y h e l p f u l t o i n v e s t i g a t e whether perhaps t h e r e are o t h e r such f u n c t i o n s p which enjoy t h e same p r e r o g a t i v e s . T h e r e f o r e I have decided t o i n v e s t i g a t e such [ f u n c t i o n s ] on t h e b a s i s e s t a b l i s h e d b e f o r e , and a s a r e s u l t a method h a s emerged f o r f i n d i n g such

function^."^^

The p r e r o g a t i v e s of a homogeneous i n t e g r a n d P(x,a) i n e q u a t i o n (5.57) c o n s i s t of t h e f a c t t h a t from t h e mere form of P(x,a) one can immediately conclude:

Q(x,ai=(ny-zP(z,a))/a. Hence, w i t h o u t any f u r t h e r c a l c u l a t i o n one can inmed i a t e l y w r i t e down the modular e q u a t i o n . E u l e r ' s key i d e a t o t h e Additamentun was t o f i n d o u t whether t h e r e e x i s t o t h e r r e a d i l y r e c o g n i s a b l e c l a s s e s of integrands P(x,a) which a r e a l s o r e l a t e d t o &(x,a) by means of a r e l a t i o n (5.59)

Flx,y,u, P, Q)=O

Such r e l a t i o n s a s (5.59) are i n f a c t p a r t i a l d i f f e r e n t i a l e q u a t i o n s , and t h i s i s p r e c i s e l y how p a r t i a l d i f f e r e n t i a l e q u a t i o n s e n t e r i n t o E u l e r ' s c a l c u l u s of t h e 1730s: they emerge a s g e n e r a l i s a t i o n s of t h e r e l a t i o n t h a t e x i s t s f o r homogeneous f u n c t i o n s . E u l e r ' s method of s i n g l i n g o u t such c l a s s e s of f u n c t i o n s , a l l u d e d t o i n t h e l a s t sentence of the q u o t e , i s t h e v e r y method t h a t would produce n e a r l y a l l of t h e r e s u l t s i n h i s 1770-textbook ICI 111 on p a r t i a l d i f f e r e n t i a l e q u a t i o n s : a l s o , i t i s t h e v e r y same method t h a t i n 1746 enabled d ' A l e m b e r t t o s o l v e t h e v i b r a t i n g s t r i n g e q u a t i o n . But t h e r e i s one g r e a t d i f f e r e n c e between E u l e r ' s t r e a t m e n t of p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n t h e 1730s and i n 1770, i n t h a t E u l e r

did not solve p a r t i a l d i f f e r e n t i a l e q u a t i o n s

i n t h e 1730s i n any proper s e n s e . The f i r s t o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s of t h e type (5.59) s t u d i e d i n t h e Additarnentwn range from t h e simple, well-known p a r t i a l d i f f e r e n t i a l e q u a t i o n of homogeneous f u n c t i o n s of degree 0 (5.60)

q = -pda

and i t s g e n e r a l i s a t i o n

f o r homogeneous f u n c t i o n s of degree n t o t h e e q u a t i o n (5.62)

4 = R(x,a)p +F(a)y

(or:

3 = R(x,a) $ + F(a)y aa

).

Euler's theory of modular equations

147

A l l of t h e s e e q u a t i o n s a r e g e n e r a l i s a t i o n s of t h e r e l a t i o n s f o r homogeneous f u n c t i o n s , and a l l a r e comprised by t h e f i n a l e q u a t i o n ( 5 . 6 2 ) . A l i s t of t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s s t u d i e d i n the Additamenturn, as w e l l a s a det a i l e d survey of the methods of i n t e g r a t i o n i s provided by Demidov, i n h i s

1975. I s h a l l l i m i t my d i s c u s s i o n h e r e t o t h e most simple c a s e ( 5 . 6 0 ) , and t h e a l l - c o m p r i s i n g case ( 5 . 6 2 ) . E u l e r s t a r t e d h i s i n v e s t i g a t i o n s i n t h e Additamenturn i n known t e r r i t o r y ; he f i r s t demonstrated h i s method by showing t h a t i t would y i e l d t h e w e l l known r e s u l t i n t h e c a s e of homogeneous f u n c t i o n s : E l i m i n a t i o n of q from t h e total differential (5.63)

dy = p d x + q d a

by means of t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n (5.60)

q = -px/a

yields :

dy = p&

- (px/a)da;

some r e w r i t i n g f i n a l l y y i e l d s : (5.64)

dy = pad(x/a).

Now from t h i s f i n a l r e l a t i o n ( 5 . 6 4 ) E u l e r a t once concluded t h a t t h e c o e f f i c i e n t pa must be an a r b i t r a r y f u n c t i o n of t h e argument x / a , hence: (5.65)

p =

l x a$(;).

The hidden lemma employed h e r e may be made e x p l i c i t i n t h e f o l l o w i n g way: I f t h e t o t a l d i f f e r e n t i a l dy can be w r i t t e n i n t h e form dy=tdu, then t must be some f u n c t i o n of u. I s h a l l d e a l with t h i s c o e f f i c i e n t lemma, a s I have termed i t , f o r t o t a l

d i f f e r e n t i a l s , separately in 55.4.5.

Hence, E u l e r ' s f i r s t r e s u l t was i n s p i r i n g

and showed t h a t one might put o n e ' s f a i t h i n t h e new method, s i n c e t h e r e s u l t ( 5 . 6 5 ) o b v i o u s l y c o i n c i d e s with what was known a l r e a d y .

E u l e r ' s a n a l y s i s of t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n ( 5 . 6 2 ) i s t r u l y a m a s t e r p i e c e . E l i m i n a t i o n of q from the t o t a l d i f f e r e n t i a l ( 5 . 6 3 ) by means of (5.62) yields: dy-l&i(R(x,a)p+F(a)ylda,

which d i f f e r e n t i a l e q u a t i o n can be re-

w r i t t e n i n t o t h e form: (5.66)

dy-F(a)ydu = pfdx+R(x,a)da).

E u l e r then assumed t h a t t h e r e e x i s t s an i n t e g r a t i n g f a c t o r S(x,a) such, t h a t

S ( x , a ) (dx+R(x,a)da) i s t h e t o t a l d i f f e r e n t i a l of an e x p r e s s i o n T(x,u); f u r t h e r more, he observed t h a t t h e l e f t hand s i d e of ( 5 . 6 6 ) can b e w r i t t e n i n t h e form:

Euler’s Theory of Modular Equations in the 1730s

148

eJFia’dad(ye-’F(a)da). (5.67)

Hence, e q u a t i o n (5.66) transforms i n t o

d(ye-JF(a)da) =

s (2,a )

e-rFfa)dadT(,Ja).

Again employing t h e c o e f f i c i e n t lemma f o r t o t a l d i f f e r e n t i a l s , E u l e r could conclude t h a t

SlxJai

e-JF(aida must be an a r b i t r a r y f u n c t i o n of T ( x , a ) , and

hence h e a r r i v e d a t t h e f o l l o w i n g g e n e r a l form f o r p: (5.68)

p=S(x,a)$fT(x,aal) e -/F ( a )da

E u l e r ’ s r e s u l t i s remarkable, b u t t h e programme i t s e l f was c l o s e t o comp l e t e f a i l u r e . I t t u r n e d o u t t h a t t h e c l a s s e s of i n t e g r a n d s a t which E u l e r a r r i v e d by t h e s e c a l c u l a t i o n s were by f a r l e s s e l e g a n t and l e s s e a s y t o r e cognise than the homogeneous i n t e g r a n d s . Given an e x p r e s s i o n P(x,a), t h e t e s t f o r homogeneity was n o t d i f f i c u l t , b u t t h e t e s t whether t h i s P(x,a) was cont a i n e d i n t h e g e n e r a l form (5.68) would r e q u i r e a f a i r amount of d i v i n a t i o n . Accordingly, t h e s e g e n e r a l i s e d homogeneous f u n c t i o n methods were never employed by E u l e r t o a c t u a l l y d e r i v e modular e q u a t i o n s . I n t h e few i n s t a n c e s where he r e a l l y employed modular e q u a t i o n s f o r s o l v i n g a s p e c i f i c problem, E u l e r a l ways used t h e method of i n t e g r a l r e d u c t i o n o u t l i n e d i n 5 5 . 4 . 3 . I t a p p e a r s t h a t Euler was c a r r i e d away i n t h e A d d i t m e n t w n by h i s d e l i g h t o v e r homogeneous f u n c t i o n s , and over the new method of f i n d i n g g e n e r a l forms of p a r t i a l d e r i v a t i v e s p from a given p a r t i a l d i f f e r e n t i a l e q u a t i o n . I n t h i s way, the Addita-

mentwn c o n s t i t u t e s a f a i r l y i s o l a t e d e x c u r s i o n i n t o t h e realm of f u n c t i o n s of two independent v a r i a b l e s , which f a i l e d t o have any s p e c i f i c b e a r i n g on t h e o r i g i n a l m o t i v a t i o n a l problem of f i n d i n g modular e q u a t i o n s ,

5 5 . 4 . 5 The c o e f f i c i e n t l e m a f o r total d i f f e r e n t i a l s

The hidden lemma i n E u l e r ’ s argument on g e n e r a l i s e d homogeneous f u n c t i o n s can be formulated i n the f o l l o w i n g way: I f a t o t a l d i f f e r e n t i a l dy=p&+qda can be manipulated i n t o t h e form dy=tdu, where u i s an e x p r e s s i o n i n t h e o r i g i n a l v a r i a b l e s x and a, then t and y are f u n c t i o n s of u : y=$(u) and t=$’(u). The t r u t h of t h i s lemma and t h e v a l i d i t y of t h e c o n c l u s i o n were obvious t o E u l e r as w e l l a s t o h i s contemporaries. E u l e r s p e n t no e f f o r t s a t a l l on exp l a i n i n g why h i s conclusions are j u s t i f i e d ; d’Alembert, i n h i s e s s a y of t h e cause of t h e winds ( 1 7 4 7 ) and h i s a r t i c l e on t h e v i b r a t i n g s t r i n g ( 1 7 4 9 ) merely s t a t e d t h a t t h e c o e f f i c i e n t B , and t h e v a r i a b l e y i n a t o t a l d i f f e r e n t i a l

dy=tdu must be f u n c t i o n s of u , because o t h e r w i s e t h e i n t e g r a t i o n would n o t

Euler's theory of modular equations

149

succeed. The f o l l o w i n g e x p l a n a t i o n of t h e c o e f f i c i e n t lemma might i l l u m i n a t e t h i s c o n c l u s i o n : The t o t a l d i f f e r e n t i a l dy of y r e p r e s e n t s the d i f f e r e n t i a l of y when no c o n d i t i o n s have been s t a t e d on t h e v a r i a b i l i t y of any of t h e v a r i a b l e s t h a t occur i n a given problem. Hence, dy=tdu i m p l i e s t h a t whenever u=eonstant ( t h u s du=O and by consequence dy=O) y must be c o n s t a n t a s w e l l . Thus l o c a l l y a s one would p r e f e r t o say nowadays

-y

-

a t least

can be regarded a f u n c t i o n of u ,

say 4 l u ) . When no f u r t h e r c o n d i t i o n s have t o be met b u t t h e mere c o n d i t i o n

d y z t d u , where t i s undetermined, t h e n t h e f u n c t i o n 4 i s a r b i t r a r y . By consequence, t = $ ' ( u ) . This argument can be made c l e a r e r by c o n t r a s t i n g t h e t o t a l d i f f e r e n t i a l dy with t h e p a r t i a l d i f f e r e n t i a l of y , say d$. v a r i a b l e y , one h a s by d e f i n i t i o n d$--pdx,

Obviously, f o r any

a c o n d i t i o n much

l i k e the condition

dy=tdu. However, d g r e p r e s e n t s t h e s o l e v a r i a t i o n i n y induced by t h e v a r i a b l e

x, and hence d x = O o n l y i m p l i e s t h a t dg=O. Now dg=O does n o t imply t h a t y = c o n s t a n t , f o r t h e simple reason t h a t y may s t i l l v a r y a c c o r d i n g t o t h e v a r i a t i o n of t h e second independent v a r i a b l e a . Hence, i t w i l l be c l e a r t h a t t h e coe f f i c i e n t lemma e s s e n t i a l l y r e q u i r e s t h a t the t o t a l d i f f e r e n t i a l of y i s w r i t t e n i n the form dy=tdu. I n f a c t , t h i s c o e f f i c i e n t lemma i s t h e 18th c e n t u r y c o u n t e r p a r t of t h e theorem on f u n c t i o n a l dependence. This theorem may be formulated t h u s : I f two continuously d i f f e r e n t i a b l e f u n c t i o n s y=y fx,a ) and u=uIs,a ) i d e n t i c a l l y annih i l a t e t h e determinant y u -yaux

xa

i n a r e g i o n G, then a f u n c t i o n F = F ( s , t )

exists,

having continuous p a r t i a l d e r i v a t i v e s of t h e f i r s t o r d e r and n o t b e i n g ident i c a l l y e q u a l t o z e r o i n any r e g i o n , such t h a t F(y(x,al,u(x,a))=O

f o r a l l (x,a)

i n G. The c o n d i t i o n dy=tdu o c c u r r i n g i n t h e c o e f f i c i e n t lemma immediately l e a d s

da=t(ux&+uada), and hence: y :u =y : u a , o r y u -y u =O. Except f o r a x x a x a a x t h e c o n d i t i o n s on d i f f e r e n t i a b i l i t y e t c . t h e v a r i a b l e s y and u, regarded as t o y &+y

x

f u n c t i o n s of x and a , do indeed s a t i s f y t h e r e q u i r e m e n t s of t h i s theorem on f u n c t i o n a l dependence, and t h e f i n a l r e s u l t of t h e c o e f f i c i e n t lemma, v i z . there i s a

0

such t h a t y = $ ( u l , almost c o i n c i d e s w i t h t h e c o n c l u s i o n of t h i s

theorem.

95.4.6

ModuZar e q u a t i o n s and p a r t i a l d i f f e r e n t i a 2 e q u a t i o n s To conclude t h i s s e c t i o n , l e t us review E u l e r ' s achievements i n t h e f i e l d

of p a r t i a l d i f f e r e n t i a l e q u a t i o n s . Can one r e a s o n a b l y i d e n t i f y t h e b e g i n n i n g s of t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n E u l e r ' s s e a r c h f o r g e n e r a l i s e d

Euler's Theory of Modular Equations in the 1730s

150

homogeneous f u n c t i o n s i n t h e Additamentwn? The answer t o such q u e s t i o n s i s l a r g e l y a m a t t e r of p e r s o n a l t a s t e , s i n c e it depends on t h e c r i t e r i a one chooses t o employ. I n my opinion t h e r e a r e a t l e a s t t h r e e independent cond i t i o n s which determine t h e beginning of t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s as a mathematical d i s c i p l i n e i n i t s e l f : a) p a r t i a l d i f f e r e n t i a l equat i o n s must occur e x p l i c i t l y , b ) the i n t e g r a t i o n problem f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s must be s t a t e d e x p l i c i t l y , and c ) methods must be a v a i l a b l e t o s o l v e t h e i n t e g r a t i o n problem, a t l e a s t f o r a c e r t a i n c l a s s of p a r t i a l d i f f e r e n t i a l e q u a t i o n s . E u l e r ' s Additamenturn c l e a r l y meets c o n d i t i o n s a) and c ) ; e q u a t i o n (5.62) i s a genuine p a r t i a l d i f f e r e n t i a l e q u a t i o n , and t h e c o e f f i c i e n t lemma p r o v i d e s a genuine method f o r i n t e g r a t i n g t h i s e q u a t i o n . Applying t h e c o e f f i c i e n t lemma t o (5.67) immediately y i e l d s ye- ' F ( a ) d a = $ ( T ( x , a ) l and hence (5.69)

'

y=e F ( a ) d a $ (T(x,a)).

This i s p r e c i s e l y how Euler t a u g h t t o i n t e g r a t e such p a r t i a l d i f f e r e n t i a l equat i o n s i n h i s IC1- 111 of 1770. However, i n t h e Additamenturn E u l e r d i d n o t draw t h i s conclusion a t a l l , f o r t h e simple reason t h a t he was n o t i n t e r e s t e d i n t h e g e n e r a l s o l u t i o n s of the p a r t i a l d i f f e r e n t i a l e q u a t i o n s h e considered. The i n t e g r a t i o n problem f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s i s completely a b s e n t i n t h e Additurnentun. Although E u l e r had t r a v e l l e d f a r from h i s o r i g i n a l motivat i o n a l problems t h e r e , h i s b a s i c q u e s t i o n s were s t i l l determined by t h e concept of modular e q u a t i o n s : f i n d t h e g e n e r a l form of t h e d e r i v a t i v e p , when a p a r t i a l d i f f e r e n t i a l e q u a t i o n f ( x , y,a,p, ql=O i s given.

55.5 Modular equations and ordinary d i f f e r e n t i a l equations 55.5.1

Equal arcs t r a j e c t o r i e s in a f k l y of e l l i p s e s

I n h i s a r t i c l e S o l u t i o problematiim ( r e a d t o t h e P e t e r s b u r g Academy i n June 1735) Euler r e t u r n e d t o t r a j e c t o r i e s i n f a m i l i e s of e l l i p s e s and those problems with which he had been occupied when he i n t e r r u p t e d t h e manuscript De d i f f e r e n -

t i u t i o n e . He now p r e s e n t e d h i s new method of f i n d i n g d i f f e r e n t i a l e q u a t i o n s f o r such curves a s e q u a l a r c s t r a j e c t o r i e s i n f a m i l i e s of e l l i p s e s , about which method he had corresponded w i t h Daniel B e r n o u l l i i n 1734 and 1735 ( c f . 55.4.1). E u l e r introduced h i s aims as f o l l o w s : "Already i n t h e p r e v i o u s c e n t u r y problems of t h i s type were t r e a t e d by Geom e t e r s , i n which a curved l i n e i s r e q u i r e d t h a t c u t s o f f e q u a l a r c s from

Ordinary differential equations

151

i n f i n i t e l y many c u r v e s given by p o s i t i o n . A t t h a t time, t o o , c e l e b r a t e d Geom e t e r s d i d communicate e l e g a n t s o l u t i o n s f o r t h e c a s e i n which t h e c u r v e s given by p o s i t i o n a r e s i m i l a r , a s f o r i n s t a n c e when from i n f i n i t e l y many c i r c l e s o r p a r a b o l a s e q u a l a r c s have t o be c u t o f f . But nobody, f o r s u r e , h a s p r o g r e s s e d any f u r t h e r , and no one has s o l v e d the problem f o r d i s s i m i l a r c u r v e s , even though a l r e a d y by then t h e problem had been posed f o r i n f i n i t e l y many e l l i p s e s . And even a t p r e s e n t , when I mentioned i n a l e t t e r t o a famous mathematician [Daniel B e r n o u l l i ] t h a t I had found t h e e q u a t i o n of t h e curve t h a t c u t s o f f e q u a l a r c s from i n f i n i t e l y many d i s s i m i l a r e l l i p s e s , he r e p l i e d t o me t h a t t h e s o l u t i o n of t h i s problem w a s beyond h i s power and a t t h e same time he asked me t o communicate my s o l u t i o n f o r t h e b e n e f i t of a n a l y s i s . " 2 6 E u l e r ' s h i s t o r i c a l survey i s s l i g h t l y d i s t o r t e d

-

and adapted t o h i s own i s s u e ;

s i n c e , i n 1697 t h e main problem had been t o c o n s t r u c t t h e t a n g e n t s t o e q u a l a r c s t r a j e c t o r i e s r a t h e r than t o produce d i f f e r e n t i a l e q u a t i o n s f o r them. But E u l e r was c e r t a i n l y r i g h t when he s t a t e d t h a t nobody had a s y e t been a b l e t o f i n d a d i f f e r e n t i a l e q u a t i o n f o r such t r a j e c t o r i e s . E u l e r ' s d e r i v a t i o n was based on t h e concept of a modular e q u a t i o n ; and Euler i n t h i s connection r e f e r r e d t o De i n f i n i t i s curvis e x p l i c i t l y .

fig. 2

Consider t h e f a m i l y o f e l l i p s e s , a l l above t h e same h o r i z o n t a l semi-axis A C ( = e l , and having v a r i a b l e c o n j u g a t e semi-axes CF, CG, CH, which w i l l be denoted by

a ( c f . f i g u r e 2 ) . Furthermore, p u t : AP=t, P M a , arc AM=z, and l e t t h e curve MNO r e p r e s e n t t h e t r a j e c t o r y , which c u t s o f f a r c s of l e n g t h A W f . The e l l i p s e s AMF, ANG, AOH a r e d e s c r i b e d by t h e e q u a t i o n (5.70)

a u = - d 2ct

- t2,

and t h e a r c l e n g t h z a l o n g t h e e l l i p s e s can be e x p r e s s e d a s :

EulerS Theory of Modular Equations in the 1730s

152

S u b s t i t u t i o n of a new v a r i a b l e y , d e f i n e d by u=ay, i n t o (5.71) y i e l d s a more elegant expression f o r the arclength:

' d y t 2 'I-yz' "y2

(5.72)

0

Hence, the e q u a l a r c s t r a j e c t o r y MNO i s d e f i n e d by the e q u a t i o n :

fqYdy$x=

(5.73)

l-y2

0

which d e s c r i b e s t h e r e l a t i o n between y and a a l o n g t h e t r a j e c t o r y . E u l e r then s e t o u t t o f i n d t h e modular e q u a t i o n e q u i v a l e n t t o e q u a t i o n ( 5 . 7 2 ) ; such a mod u l a r e q u a t i o n i s indeed a modular e q u a t i o n f o r t h e given e l l i p s e s , s i n c e E u l e r had s t a t e d e x p l i c i t l y i n De i n f i n i t i s curvis ( c f . 15.4.21,

t h a t any e q u a t i o n

between two v a r i a b l e s t h a t d e s c r i b e t h e s i n g l e c u r v e s and t h e modulus a can be a c a n d i d a t e f o r t h e modular e q u a t i o n . To t h i s end, E u l e r a p p l i e d h i s method of i n t e g r a l r e d u c t i o n ( c f . 55.4.3):

f o r z=Z(y,a)

a s defined i n equation (5.72),

one f i n d s

and

Now t h e f o l l o w i n g a l g e b r a i c r e l a t i o n between 2, Z a ,

and Z

aa e x i s t s :

By means of t h i s r e l a t i o n ( 5 . 7 5 ) , t h e f o l l o w i n g modular e q u a t i o n can be produced:

ccydy'

-

1

da2 ( l - y y l q J ( a 2 ( l - y y ) + c c y y ) - ~

i n which a,y,

d

a2f1-yyl+ccyy I-YY

J

and z a r e v a r i a b l e , and where y i s s t i l l r e l a t e d t o u by means of

daddz-dzdda u=ay. The symbols l i k e d dz. denote ~ the d i f f e r e n t i a l expression , da2 a r i s i n g from d i f f e r e n t i a t i n g d z / d a f o r m a l l y . Now, by p u t t i n g z=f i n t h i s

153

Ordinary differential eyuutions modular equat on ( 5 . 7 6 ) ,

which d e s c r i b e s t h e e q u a l a r c s t r a j e c t o r i e s l i k e MNO:

o r d e r emerges

$-

a h i g h l y complicated d i f f e r e n t i a l e q u a t i o n of second

2adyil-yy) (l-yy)+c'cyy)

&(a2

+

ccydy'

du2 (1-yy) l a 2 Il-yy)+ccyy)

+

d . 2 =O.

Without knowing i t s provenance, i t would be a t a s k of s h e e r i m b e c i l i t y t o t r y t o i n t e g r a t e t h i s d i f f e r e n t i a l e q u a t i o n . However, knowing t h a t i t d e s c r i b e s t h e r e l a t i o n between a and y(=u/al

along e q u a l a r c s t r a j e c t o r i e s i n a f a m i l y of

e l l i p s e s with h o r i z o n t a l semi-axis

c',

i t s geometrical construction i s f a i r l y

easy. E u l e r produced y e t another d i f f e r e n t i a l e q u a t i o n t h a t can be c o n s t r u c t e d by means of t h e s e e l l i p s e s : I f one s e t s y=Z (hence u=a) i n t h e modular equat i o n ( 5 . 7 6 ) the f o l l o w i n g much more a t t r a c t i v e d i f f e r e n t i a l e q u a t i o n emerges: (5.78)

azda2=iu2+c21dadz+aia2-c2)ddz .

This d i f f e r e n t i a l e q u a t i o n , by consequence, d e s c r i b e s t h e r e l a t i o n between t h e q u a r t e r of t h e circumference z of an e l l i p s e w i t h h o r i z o n t a l semi-axis c , and conjugate semi-axis a. This d i f f e r e n t i a l e q u a t i o n i s i d e n t i c a l w i t h e q u a t i o n ( 5 . 4 7 ) which Euler had c o m u n i c a t e d t o Daniel B e r n o u l l i i n h i s l e t t e r of Fe-

bruary 1734 ( c f . 5 5 . 4 . 1 ) . In h i s S o l u t i o problematwn E u l e r had i n t e n d e d more than t h e mere s o l u t i o n of an almost 40 y e a r o l d c h a l l e n g e problem of t h e B e r n o u l l i b r o t h e r s . I n one of h i s i n t r o d u c t o r y s e c t i o n s t o t h i s a r t i c l e , E u l e r had made i t c l e a r t h a t t h e i n t e r e s t i n g p o i n t of such an e n t e r p r i s e a s f i n d i n g t h e d i f f e r e n t i a l e q u a t i o n of an e q u a l a r c s t r a j e c t o r y was i n f a c t t h e i n v e r s i o n of problem and s o l u t i o n afterwards : "Concerning the curve t h a t c u t s o f f e q u a l a r c s from i n f i n i t e l y many e l l i p s e s , i t s c o n s t r u c t i o n i n i t s e l f i s e a s y , and can b e c a r r i e d o u t by means of t h e

r e c t i f i c a t i o n of c u r v e s t h a t can v e r y e a s i l y be d e s c r i b e d . And t h i s c o n s t r u c t i o n , I t h i n k , i s by f a r t o b e p r e f e r r e d over o t h e r c o n s t r u c t i o n s which a r e performed by q u a d r a t u r e s of c u r v e s 2 7 . Thus i t i s n o t t h e c o n s t r u c t i o n of t h i s curve t h a t i s r e q u i r e d , b u t r a t h e r i t s e q u a t i o n , from which i t w i l l become c l e a r which e q u a t i o n s can be c o n s t r u c t e d s o e a s i l y . For t h i s reason a n a l y s i s

w i l l r e c e i v e no l i t t l e enrichment i f those e q u a t i o n s a r e b r o u g h t t o l i g h t which allow a c o n s t r u c t i o n by means of t h e r e c t i f i c a t i o n of e l l i p s e s . " 2 8

Euler's Theory of Modular Equations in the 1730s

154

55.5.2 Ordinary d i f f e r e n t i a l equations and t h e method of modular equations I n h i s SoZutio problematurn Euler had shown how one could determine a c l a s s of d i f f e r e n t i a l e q u a t i o n s t h a t can be c o n s t r u c t e d by means of r e c t i f i c a t i o n s of e l l i p s e s . One and a h a l f y e a r s l a t e r , i n 1737, he f i n a l l y p r e s e n t e d h i s new method f o r s o l v i n g d i f f e r e n t i a l e q u a t i o n s i n i t s f u l l g e n e r a l i t y . I n h i s i n t r o d u c t i o n t o t h e a r t i c l e De construetione ("On t h e c o n s t r u c t i o n of e q u a t i o n s " ) , Euler provided a c l e a r survey of t h e s t a t e of t h e a r t i n t h e t h e o r y of d i f f e r e n t i a l e q u a t i o n s , and h e p o i n t e d o u t what t h e e s s e n t i a l s of h i s new method c o n s i s t e d o f . I s h a l l quote t h e s e i n t r o d u c t o r y s e c t i o n s i n f u l l h e r e : "Whenever t h e r e s o l u t i o n of problems l e a d s t o d i f f e r e n t i a l e q u a t i o n s , one must p r i m a r i l y i n v e s t i g a t e whether t h e s e e q u a t i o n s admit i n t e g r a t i o n . For, a problem must be judged s o l v e d i n the most p e r f e c t way i f i t i s reduced t o t h e c o n s t r u c t i o n of an a l g e b r a i c e q u a t i o n . But i f an e q u a t i o n can i n no way be transformed i n t o an a l g e b r a i c form

-

a s i t happens f r e q u e n t l y

-

then e i t h e r

q u a d r a t u r e s o r r e c t i f i c a t i o n s of curves whose c o n s t r u c t i o n i s known have t o be employed i n t h e r e s o l u t i o n of t h e problem. Now i n o r d e r t o achieve t h i s the e q u a t i o n which c o n t a i n s t h e s o l u t i o n of t h e problem must be of t h e f i r s t o r d e r and, furthermore, must a l l o w s e p a r a t i o n of v a r i a b l e s , a t l e a s t i f one wishes t o apply t h e common and s u f f i c i e n t l y known r u l e s . B u t t h e s e r u l e s s u f f e r from a d e f e c t , because by means of them n e i t h e r d i f f e r e n t i a l equat i o n s of h i g h e r o r d e r nor i n s e p a r a b l e d i f f e r e n t i a l e q u a t i o n s of t h e f i r s t o r d e r can be c o n s t r u c t e d . For t h i s reason t h e c o n s t r u c t i o n of an e q u a t i o n i s sought i n v a i n by t h e s e r u l e s , u n l e s s i t can be reduced t o a d i f f e r e n t i a l e q u a t i o n of t h e f i r s t o r d e r and a s e p a r a t i o n of v a r i a b l e s can be d e t e c t e d a t t h e same time.'Iz9 According t o t h i s d e c l a r a t i o n , t h e prime method f o r s o l v i n g d i f f e r e n t i a l equat i o n s was s t i l l t h e method of s e p a r a t i o n of v a r i a b l e s ; o t h e r methods - t h e l e s s well-known,

uncommon ones

- were

r a t e d a d hoc and, o b v i o u s l y , of even more

r e s t r i c t e d a p p l i c a b i l i t y . I t i s a f t e r such a r a t h e r p e s s i m i s t i c s k e t c h t h a t Euler o u t l i n e d h i s new method: "In the p a s t 1 have given examples of a c e r t a i n p e c u l i a r and much f u r t h e r r e a c h i n g method a l r e a d y a few times. By means of i t I have n o t o n l y cons t r u c t e d s e v e r a l i n s e p a r a b l e d i f f e r e n t i a l e q u a t i o n s [of t h e f i r s t o r d e r ]

,

b u t a l s o d i f f e r e n t i a l e q u a t i o n s of t h e second o r d e r t h a t could n o t be reduced t o f i r s t o r d e r d i f f e r e n t i a l s . A t f i r s t I have u s e d i n f i n i t e s e r i e s i n t o which I transformed t h e given e q u a t i o n s and I reduced t h e i r sums t o q ~ a d r a t u r e s ~ ' .

But t h e n , j u d g i n g

t h i s road n o t s u f f i c i e n t l y genuine, I have looked f o r a

155

Ordinary diffeerentiut equations

d i r e c t method b y means of which I could r e a c h t h e same c o n s t r u c t i o n s . To t h i s t a s k I have n o t n e e d l e s s l y e x e r t e d myself, f o r I have h i t upon a method of f i n d i n g modular e q u a t i o n s , by means of which t h e road i s p r e p a r e d towards c o n s t r u c t i o n s of t h e most d i f f i c u l t e q u a t i o n s . I have a l r e a d y exposed t h i s method a t l e n g t h [ i n De i n f i n i t i s c u r u i s ] , b u t a t t h a t t i m e t h e r e was no o p p o r t u n i t y t o demonstrate i t s t i o n s . Yet i n t h e meantime

enormous use f o r t h e c o n s t r u c t i o n of equa-

I have q u i t e r e c e n t l y given an example of t h o s e

e q u a t i o n s t h a t can be c o n s t r u c t e d by t h e r e c t i f i c a t i o n of the e l l i p s e . I s h a l l now develop some s p e c i a l c a s e s from which t h e c o n s t r u c t i o n s of s e v e r a l e q u a t i o n s f o l l o w , so t h a t t h e use of t h i s new method can be more f u l l y perceived. I s h a l l t a k e the p r i n c i p l e s from t h e d i s s e r t a t i o n about i n f i n i t e l y many c u r v e s of t h e same genus which I have r e a d l a s t y e a r . " 3 1 Having e x p l a i n e d t h e n a t u r e of modular e q u a t i o n s a g a i n , E u l e r summarised t h e r a t i o n a l e behind h i s new method of c o n s t r u c t i o n f o r d i f f e r e n t i a l e q u a t i o n s i n the f o l l o w i n g p h r a s e :

"b ..I

such an e q u a t i o n , which I have c a l l e d modular [ .

t h r e e v a r i a b l e s z,x,

.. ]

w i l l contain the

and a ; b u t i t t u r n s i n t o an e q u a t i o n of two v a r i a b l e s

when e i t h e r z o r x a r e given a v a l u e which i s e i t h e r determined o r which depends on a. Such an e q u a t i o n

-

whatever i t s form o r o r d e r

-

can always be

c o n s t r u c t e d by means of t h e e q u a t i o n z=/Pdx [ e q u i v a l e n t t o t h e modular equat i o n ] . For, i f t h e i n t e g r a t i o n /P&

i s performed f o r any g i v e n a , as can a l -

ways b e done by q u a d r a t u r e , and e i t h e r z o r z i s taken t o be e q u a l t o i t s a s s i g n e d v a l u e , then t h e remaining one of z and x i s determined, and t h e r e f o r e i t s value w i l l become known. Hence, i n t h i s way f o r a given v a l u e o f one of t h e i n d e t e r m i n a t e s t h e v a l u e of t h e o t h e r can always b e found; and t h i s i s p r e c i s e l y what the c o n s t r u c t i o n of any e q u a t i o n c o n s i s t s o f . " 3 2 E u l e r ' s programme may be c l a r i f i e d t h u s : Given a t r a n s c e n d e n t a l e q u a t i o n of t h e form PX

and i t s modular e q u a t i o n , which, f o r i n s t a n c e , h a s t h e form (5.80)

ddz=AZg. Expr. in (x,a, z, dx,ddx,da, dda,dz).

I f one s e t s x = X ( a ) , f o r i n s t a n c e , then t h e modular e q u a t i o n (5.80) transforms into: (5.81)

ddz=AZg. E x p . i n ( X ( a ) , a , R , X ' ialda,X"fa)dda,da, dda,dz),

and t h i s , o b v i o u s l y , i s a 2nd o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n i n z and a .

Euler's Theory of Modulur Equations in the 1730s

156

On t h e o t h e r hand, g r a n t e d t h e c o n s t r u c t i o n o f the curves provided by ( 5 . 7 9 ) , the t r a j e c t o r y emerging from t h e c o n d i t i o n x=X(a) can be c o n s t r u c t e d by means of t h e s e curves (5.79). Hence, t h e c o n d i t i o n x=X(ctl p r o v i d e s a c o n s t r u c t i o n i f

combined with ( 5 . 7 9 ) , and i t p r o v i d e s a d i f f e r e n t i a l e q u a t i o n i f combined with (5.80),

the modular e q u a t i o n e q u i v a l e n t t o (5.79).

E u l e r i l l u s t r a t e d t h i s technique by e l a b o r a t i n g upon i n t e g r a l s of t h e type J e a x X ( x ) d x , and he showed t h a t f o r a s u i t a b l y chosen X ( x ) and s u i t a b l e s u b s t i t u t i o n s t h e c o n s t r u c t i b l e c a s e s of the R i c c a t i e q u a t i o n could a g a i n be d e r i v e d .

55.6 EuZer's view of the infinitesirnu2 eaZeuZus around 1140 The manuscript De d i f f e r e n t i a t i o n e , and t h e a r t i c l e s De i n f i n i t i . s eurvis,

Additamenturn, SoZutio problematwn, and De construetione, a l l d i s c u s s e d i n t h i s c h a p t e r , form a r a t h e r coherent chain of problems and i d e a s . They r e v e a l a v e r y i n t i m a t e l i n k between E u l e r ' s t h e o r y of modular e q u a t i o n s and h i s p a r t i a l d i f f e r e n t i a l c a l c u l u s . I t appears t h a t modular e q u a t i o n s a r e t h e e x c l u s i v e a r e a of a p p l i c a t i o n f o r E u l e r ' s p a r t i a l d i f f e r e n t i a l c a l c u l u s . A s we have s e e n i n

55.4.6,

Euler had i n f a c t come very c l o s e t o t h e theory of p a r t i a l d i f f e r e n t i a l

e q u a t i o n s i n 1734: p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e t h e r e , and s o i s t h e main i n t e g r a t i o n technique, v i z . t h e c o e f f i c i e n t lemma. The o n l y m i s s i n g l i n k i s t h e r i g h t q u e s t i o n . Due t o t h e s p e c i f i c requirements of modular e q u a t i o n s E u l e r was only i n t e r e s t e d i n t h e g e n e r a l form of p = a s a t i s f y i n g a given p a r t i a l

ax

d i f f e r e n t i a l e q u a t i o n flx,a,y,p,q)=U,

and n o t i n t h e g e n e r a l s o l u t i o n y - y ( x , a ) .

Now one f a c e s t h e q u e s t i o n , whether t h e i n t e g r a t i o n problem was a b s e n t because E u l e r had n o t y e t r e c o g n i s e d i t , o r whether i t was a b s e n t o n l y because i t had n o t h i n g t o do w i t h modular e q u a t i o n s . I n o t h e r words: d i d E u l e r perhaps s t u d y p a r t i a l d i f f e r e n t i a l e q u a t i o n s , i n t h e i r own r i g h t and w i t h t h e obvious i n t e g r a t i o n problem, a t some o t h e r time i n t h e 1730s, o u t s i d e the c o n t e x t of modular e q u a t i o n s ? We s h a l l answer t h i s q u e s t i o n t o t h e n e g a t i v e . T h i s answer i s made p o s s i b l e by a r a t h e r d e t a i l e d survey which E u l e r made i n 1740 of t h e e x t e n t of t h e i n f i n i t e s i m a l c a l c u l u s . From t h i s survey i t becomes v e r y c l e a r t h a t E u l e r had n o t y e t recognised the i n t e g r a t i o n problem f o r p a r t i a l d i f f e r e n t i a l equat i o n s i n 1740. This survey i s an index of c h a p t e r s f o r a major t r e a t i s e on t h e i n f i n i t e simal c a l c u l u s , e n t i t 1 e d " I n d e x capitum t r a c t a t u s cujusdam m a j o r i s de a n a l y s i

Euler's view of the infinitesimal calculus

157

infinitorurn" ("Index of c h a p t e r s of a c e r t a i n major t r e a t i s e on t h e i n f i n i t e simal a n a l y s i s . " )

' '. U n f o r t u n a t e l y ,

the beginning of t h i s Index capitwn i s

missing. According t o t h i s p l a n , E u l e r intended t o t r e a t t h e e n t i r e d i f f e r e n t i a l , i n t e g r a l and v a r i a t i o n a l c a l c u l u s , t o g e t h e r w i t h t h e i r g e o m e t r i c a l a p p l i c a t i o n s i n a s e r i e s of s i x books. Each book was t o be d i v i d e d i n t o two s e c t i o n s , the f i r s t of which was t o c o n t a i n a n a l y s i s p r o p e r , and t h e second i t s a p p l i c a t i o n s t o g e o m e t r i c a l m a t t e r s . I s h a l l l i s t t h e t i t l e s of t h e d i f f e r e n t books h e r e and i n d i c a t e t h e i r c o n t e n t . Book 1 was t o c o n t a i n t h e e n t i r e d i f f e r e n t i a l c a l c u l u s . However, most of t h e index of t h i s book i s m i s s i n g , and o n l y t h e t i t l e s of t h e l a s t f o u r c h a p t e r s ( n r s . 8-1 I ) of s e c t i o n two have s u r v i v e d : they mention elementary d i f f e r e n t i a l geometry of s u r f a c e s : t a n g e n t p l a n e s , normals, c u r v a t u r e and g e o d e s i c s . Hence, c o n s i d e r i n g the o r g a n i s a t i o n of each book, t h e f i r s t s e c t i o n of book 1 must have c o n t a i n e d t h e elements of p a r t i a l d i f f e r e n t i a l c a l c u l u s , such a s t h e t o t a l d i f f e r e n t i a l , the e q u a l i t y of mixed p a r t i a l d i f f e r e n t i a l s and t h e r e l a t i o n between t h e p a r t i a l d e r i v a t i v e s of homogeneous f u n c t i o n s . Book 2 i s e n t i t l e d : "About t h e i n t e g r a t i o n of d i f f e r e n t i a l formulas i n v o l v i n g a single variable".

S e c t i o n one was devoted t o i n t e g r a t i n g f u n c t i o n s of a

s i n g l e v a r i a b l e : r a t i o n a l f u n c t i o n s , logarithms, exponentials, trigonometric and c y c l o m e t r i c f u n c t i o n s ; s e c t i o n two l i s t s : r e c t i f i c a t i o n and q u a d r a t u r e of c u r v e s , s u r f a c e s and volumes of r e v o l u t i o n . Book 3 i s e n t i t l e d : "About d i f f e r e n t i a l e q u a t i o n s i n v o l v i n g two v a r i a b l e s " . S e c t i o n one was planned t o c o n t a i n i n t e g r a t i o n techniques f o r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s : i n t e g r a t i n g f a c t o r s , s e p a r a t i o n of v a r i a b l e s , s u b s t i t u t i o n of v a r i a b l e s , i n t e g r a t i o n by s e r i e s , R i c c a t i e q u a t i o n . The g e o m e t r i c a l a p p l i c a t i o n s i n s e c t i o n two concern i n v e r s e t a n g e n t problems, d e t e r m i n a t i o n of t h e e q u a t i o n i n C a r t e s i a n c o o r d i n a t e s when some r e l a t i o n between two v a r i a b l e s along a curve i s g i v e n , and curves on s u r f a c e s , e s p e c i a l l y g e o d e s i c s . Book 4 i s e n t i t l e d : "About d i f f e r e n t i a l e q u a t i o n s i n v o l v i n g t h r e e v a r i a b l e s " . We s h a l l d i s c u s s t h e c o n t e n t s of t h i s book i n d e t a i l s h o r t l y . Book 5 i s e n t i t l e d : "Containing q u i t e some p e c u l i a r methods which a r e accomod a t e d t o t h e s o l u t i o n of a number of important problems".

Not a v e r y i l l u m i n a -

t i n g t i t l e ! With book 4 , t h e obvious sequence one v a r i a b l e , two v a r i a b l e s , t h r e e v a r i a b l e s had been completed, and now E u l e r t u r n s t o t h e r e s t of analy-

sis. S e c t i o n one was i n t e n d e d t o c o n t a i n some more i n f o r m a t i o n about t o t a l (or P f a f f i a n ) d i f f e r e n t i a l e q u a t i o n s which cannot be i n t e g r a t e d by means of i n t e g r a t i n g f a c t o r s ( t h e s e e q u a t i o n s a l s o emerge i n book 4 ) , and methods a r e mentioned t o reduce q u a d r a t u r e s of curves t o r e c t i f i c a t i o n s of a l g e b r a i c c u r v e s .

158

Euler's Theory of Modular Equations in the 1730s

S e c t i o n two of book 5 i s e n t i r e l y devoted t o t h e c a l c u l u s of v a r i a t i o n s . Book 6 i s e n t i t l e d : "About t h e u s e of what preceded i n t h e d o c t r i n e of s e r i e s a s w e l l a s i n t h e d e t e r m i n a t i o n of t h e n a t u r e of curves by means of t h e i r way of g e n e r a t i ~ n " ~S~e .c t i o n one promises summation methods f o r s e r i e s based on i n f i n i t e s i m a l c a l c u l u s , v i z . r e d u c t i o n of sums t o q u a d r a t u r e s , and t h e det e r m i n a t i o n of d i f f e r e n t i a l e q u a t i o n s which can be s o l v e d by means of a given s e r i e s . S e c t i o n two i s devoted t o mechanical c u r v e s , which can be of h e l p i n the i n t e g r a t i o n of d i f f e r e n t i a l e q u a t i o n s , f o r example t h e t r a c t r i x . E u l e r concluded h i s Index capiturn with t h e f o l l o w i n g innuendo: "So t h e s e a r e t h e c h a p t e r s i n which, a s seems t o me, t h e e n t i r e i n f i n i t e s i m a l

a n a l y s i s can be included t o t h a t l i m i t of p e r f e c t i o n t o which i t h a s been brought ahead a t p r e s e n t . I have n o t y e t been a b l e t o f i n d anything t h a t could n o t be i n c o r p o r a t e d i n any of t h e s e c h a p t e r s . T h e r e f o r e , i t w i l l be a work of the g r e a t e s t u t i l i t y , i f someone would e x p l a i n t h i s most r e l e v a n t p i e c e of mathematics i n the o r d e r d e s c r i b e d h e r e . " 3 5 This f i n a l remark makes i t v e r y c l e a r t h a t t h e Index capiturn indeed p r o v i d e s a comprehensive survey of t h e e x t e n t of a n a l y s i s as E u l e r saw i t around 1740. Hence, i f E u l e r had achieved t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n the 1730s, then some v e s t i g e s of p a r t i a l d i f f e r e n t i a l e q u a t i o n s should have found t h e i r way i n t o t h i s Index capitwn. The obvious p l a c e t o look f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n t h i s index i s book 4 , s i n c e i n t h i s book d i f f e r e n t i a l e q u a t i o n s with t h r e e v a r i a b l e s were t o be d i s c u s s e d . However, t h e r e i s no t r a c e of p a r t i a l d i f f e r e n t i a l e q u a t i o n s h e r e . L e t me give t h e e n t i r e l i s t of c h a p t e r s f o r t h i s book: "Section one: On f i n d i n g and s o l v i n g e q u a t i o n s which involve t h r e e v a r i a b l e s . 1. Concerning d i f f e r e n t i a l formulas i n t h r e e v a r i a b l e s . I n which c h a p t e r a

c r i t e r i o n i s sought f o r

d i s t i n g u i s h i n g between determined formulas, o r those

t h a t admit i n t e g r a t i o n , and indetermined ones. 2. Concerning d i f f e r e n t i a l e q u a t i o n s which involve t h r e e v a r i a b l e s . Where i t i s shown again t h a t some of t h e s e e q u a t i o n s a r e determined, b u t t h a t o t h e r s

a r e indetermined, [namely t h o s e ] which do n o t admit i n t e g r a t i o n w i t h o u t an a d d i t i o n a l hypothesis. 3. On modular e q u a t i o n s and how t o f i n d them. I n which c h a p t e r a method i s produced t o f i n d from a [ g i v e n ] e q u a t i o n c o n t a i n i n g two v a r i a b l e s a n o t h e r [ e q u a t i o n ] i n which b e s i d e s t h e s e two v a r i a b l e s a new v a r i a b l e o c c u r s which had been k e p t c o n s t a n t h i t h e r t o . 4. On the use of modular e q u a t i o n s f o r c o n s t r u c t i n g d i f f e r e n t i a l e q u a t i o n s of h i g h e r o r d e r . I n t h i s c h a p t e r c o n s t r u c t i o n s b y m e a n s o f modular e q u a t i o n s

Euler's view of the infinitesimal calculus a r e d i s p l a y e d f o r many d i f f e r e n t i a l e q u a t i o n s of both f i r s t and h i g h e r o r d e r which exceed p r i o r methods. S e c t i o n two: About t r a j e c t o r i e s of a l l s o r t s . 1 . Concerning t r a j e c t o r i e s t h a t c u t o f f e q u a l a r e a s o r a r c s o r o t h e r

f u n c t i o n s 3 6 from i n f i n i t e l y many c u r v e s given by p o s i t i o n . I n which c h a p t e r a method w i l l be r e v e a l e d f o r f i n d i n g modular e q u a t i o n s f o r i n f i n i t e l y many c u r v e s , and f o r s o l v i n g problems of t h i s type w i t h t h e i r h e l p . 2. On t r a j e c t o r y c u r v e s , which i n t e r s e c t i n f i n i t e l y many g i v e n curves a t given a n g l e s . I n which c h a p t e r b e s i d e s t h e e q u a t i o n s f o r t h e s e t r a j e c t o r i e s also t h e i r construction i s investigated.

3 . On t h e s o l u t i o n of s e v e r a l problems of r e c i p r o c a l s . I n which [ c h a p t e r ] e i t h e r t h e curves which have t o be i n t e r s e c t e d a r e sought from t h e t r a j e c t o r i e s , o r t h e c a s e s a r e i n v e s t i g a t e d i n which t h e t r a j e c t o r y curves have a given r e l a t i o n t o t h e curves which a r e t o be i n t e r ~ e c t e d . " ~ ~ E u l e r ' s p l a n f o r book 4 covers a l l t h o s e i s s u e s which we have d i s c u s s e d i n t h e previous s e c t i o n s of t h i s c h a p t e r : t h e modular e q u a t i o n s from De i n f i n i t i s

curuis and t h e Additamentwn a r e mentioned i n ch. 1.3 and ch. 2.1; modular equat i o n s and d i f f e r e n t i a l e q u a t i o n s as p r e s e n t e d i n S o l u t i o problematwn and De

construetione a r e mentioned i n ch. 1.4; and t h e g e o m e t r i c a l a p p l i c a t i o n s i n De d i f f e r e n t i a t i o n e a r e mentioned i n ch. 2 . 1 and ch. 2 . 2 . However, t h e index of book 4 c o n t a i n s more than t h a t : t h e t o t a l d i f f e r e n t i a l e x p r e s s i o n s and t o t a l - o r P f a f f i a n - d i f f e r e n t i a l e q u a t i o n s of chs. 1 . 1 and 1 . 2 a r e new. Indeed, E u l e r had been informed of C l a i r a u t ' s work on t o t a l d i f f e r e n t i a l equat i o n s and i n t e g r a t i n g f a c t o r s o n l y e a r l y i n 1740. A s becomes c l e a r from t h e Euler-Clairaut

correspondence a s w e l l a s from a n o t e i n E u l e r ' s Adversaria

Mathematica f o r t h e y e a r s 1740-1744, E u l e r owed t o C l a i r i u t t h e c r i t e r i o n f o r a t o t a l d i f f e r e n t i a l e q u a t i o n t o be i n t e g r a b l e by means of an i n t e g r a t i n g f a c t o r 3 ' . Hence, t h e m a t e r i a l f o r t h e s e chs. 1 . 1 and 1.2 was brand new when Euler composed h i s Index capitum. Reviewing t h e c o n t e n t s planned f o r book 4 , i t i s c l e a r t h a t E u l e r i n 1740 regarded t o t a l o r P f a f f i a n d i f f e r e n t i a l e q u a t i o n s , such a s

(5.82)

P(x,y, z l dx+Qfx,y,z)dy+R(x, y, z l dz=O

t o be the obvious g e n e r a l i s a t i o n of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s t o t h r e e v a r i a b l e s . Indeed, w r i t i n g o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i n t h e form

P(x,yyldx+Qfx,yldy=O

s u g g e s t s t h a t i n c a s e of t h r e e v a r i a b l e s t h e d i f f e r e n t i a l

of t h e t h i r d v a r i a b l e must be added, and t h a t t h e c o e f f i c i e n t s should a l s o c o n t a i n t h i s t h i r d v a r i a b l e . By 1740, p a r t i a l d i f f e r e n t i a l e q u a t i o n s had n o t

159

160

Euler’s Theory of Modular Equations in the 1730s

y e t emerged as t h e o t h e r , e q u a l l y r e a s o n a b l e , g e n e r a l i s a t i o n of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s t o t h e c a s e of t h r e e v a r i a b l e s .

161

EPILOGUE

I n 1741, E u l e r moved t o B e r l i n , where he w a s t o e s t a b l i s h h i m s e l f a s a w r i t e r of g r e a t textbooks i n t h e n e x t decade. Within t e n y e a r s E u l e r had completed h i s Methodus inveniendi l i n e a s curuas

... , d e a l i n g w i t h

the c a l c u l u s

of v a r i a t i o n s , h i s I n t r o d u c t i o in analysin i n f i n i t o r w n , and t h e I n s t i t u t i o n e s

c a l c u l i d i f f e r e n t i a z i s . However, none of t h e s e books followed t h e l i n e s s e t o u t i n t h e Index c a p i t m ; i n p a r t i c u l a r , modular e q u a t i o n s and t h e i r a p p l i c a t i o n s t o o r d i n a r y d i f f e r e n t i a l e q u a t i o n s and t o t r a j e c t o r i e s i n f a m i l i e s of curves were n o t e l e v a t e d t o textbook s t a t u s i n those y e a r s , n o r were t h e y i n c o r p o r a t e d i n t o the l a s t of E u l e r ' s q u a r t e t of t e x t b o o k s , t h e I n s t i t u t i o n e s

calculi i n t e g r a l i s which E u l e r composed i n t h e 1760s. Modular e q u a t i o n s d i e d o u t i n t h e 1740s. Apart from a few s p o r a d i c r e f e r e n c e s t o t h e s e e q u a t i o n s i n t h e Adversaria Mathernatica they vanished c o m p l e t e l y , and p a r t i a l d i f f e r e n t i a l e q u a t i o n s no l o n g e r o c c u r r e d e i t h e r . Why was t h i s so? Why d i d E u l e r r e f r a i n from e x p l o i t i n g t h e s e l i n e s of thought any f u r t h e r i n h i s e a r l y y e a r s i n Berl i n ? I t h i n k t h e b a l a n c e was wrong. The techniques employed i n t h e 1730s t o f i n d modular e q u a t i o n s had been h i g h l y s o p h i s t i c a t e d and powerful. But t h e a p p l i c a t i o n s were r a t h e r d u l l and u n a t t r a c t i v e . E u l e r had shown how t o make l i s t s of d i f f e r e n t i a l e q u a t i o n s t h a t could be s o l v e d by means of a given f a m i l y of c u r v e s . The way t o more such r e s u l t s was c l e a r and u n a t t r a c t i v e f o r a mat h e m a t i c i a n a t the top of h i s c a p a c i t i e s . Hence t h e b a l a n c e was wrong: t h e a p p l i c a t i o n s d i d n o t i n v i t e f u r t h e r development of modular e q u a t i o n s o r t h e d e t e r m i n a t i o n of o t h e r c l a s s e s of f u n c t i o n s a s e l e g a n t a s t h e homogeneous ones. Then, i n 1746, t h e Academie des Sciences e t des B e l l e s L e t t r e s found d'Alembert's "Reflexions on the General Cause of t h e Winds" among t h e p i e c e s t h a t had been s e n t i n f o r t h e p r i z e . And o n l y s h o r t l y a f t e r w a r d s , d'Alembert s u b m i t t e d h i s paper on t h e v i b r a t i n g s t r i n g . I n both i n s t a n c e s , E u l e r , b e i n g d i r e c t o r of t h e mathematical s e c t i o n of t h e Academy, was t h e f i r s t t o r e f e r e e d'Alembert's work, and h e must have r e c o g n i s e d t h e e s s e n t i a l l y new type of problem

-

v i z . p a r t i a l d i f f e r e n t i a l e q u a t i o n s and t h e i r i n t e g r a t i o n

-

almost

immediately. d'ALembert r e f e r r e d t o E u l e r ' s a r t i c l e De i n f i n i t i s cumis exp l i c i t l y , a n d h e made use of t h e c o e f f i c i e n t lemma f o r t o t a l d i f f e r e n t i a l s i n almost t h e same way a s Euler had done when d e t e r m i n i n g c l a s s e s of g e n e r a l i s e d homogeneous f u n c t i o n s .

162

Epilogue

d'Alembert had opened up a new f i e l d of i n v e s t i g a t i o n : continuum mechanics, r e q u i r i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s a s i t s n a t u r a l language of d e s c r i p t i o n , and p r o v i d i n g more and more d i f f i c u l t problems than could be s o l v e d by t h e means then a v a i l a b l e . Now

-

with one s t r o k e

-

t h e balance went t h e o t h e r way:

t h i s c h a l l e n g i n g new f i e l d provided demanding problems and c a l l e d f o r f u r t h e r development of p a r t i a l d i f f e r e n t i a l c a l c u l u s i n a way f a r s t r o n g e r than the modulus had e v e r done.

163

FOOTNOTES CHAPTER I

c f . c h a p t e r 2 , f o o t n o t e 40. c f . d'Alembert 1747 and 1 7 4 9 . The q u e s t i o n s e t f o r t h e year 1746 was formulated t h u s : "DEterminer l ' o r d r e e t l a l o i que l e v e n t d e v r o i t s u i v r e s i l a terre S t o i t environnee de tous cBt6s p a r l'OcCan, de s o r t e qu'on p u t en t o u t t e m p s t r o u v e r l a d i r e c t i o n e t l a v i t e s s e du v e n t pour chaque e n d r o i t " ( c f . Harnack 1 9 0 0 , v o l . 2 . 1 , p. 305; o r Youshkevitch/Taton i n LEOO 4 A . 5 , p. 250, f n . 4 ) . Euler was chairman of t h e committee t h a t was a p p o i n t e d t o review t h e e s s a y s s e n t i n f o r t h e p r i z e ; t h e d e c i s i o n t o award t h e p r i z e t o d'Alembert was taken unanimously and i t was confirmed a t t h e s e s s i o n of 2/6/1746 ( c f . Winter 1 9 5 7 ) . E u l e r ' s r e p o r t on d'Alembert's e s s a y i s h i s 1746. On d'Alembert and t h e P r u s s i a n Academy see Hankins

1970. The most important and comprehensive account of d'Alembert's work on p a r t i a l d i f f e r e n t i a l e q u a t i o n s i s Demidov 1 9 7 4 . The d i s c u s s i o n between d'Alembert, E u l e r , Daniel B e r n o u l l i and Lagrange concerning t h e v i b r a t i n g s t r i n g has been d e a l t w i t h by Ravetz ( 1 9 6 1 ) , who most c l e a r l y i d e n t i f i e d p o i n t s of view and q u e s t i o n s under review. T r u e s d e l l 1960 and Burkhardt

1908 a l s o d e a l w i t h t h i s d i s c u s s i o n , and, more b r o a d l y , w i t h o t h e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s o c c u r r i n g i n 18th c e n t u r y mathematical p h y s i c s ; on E u l e r and f l u i d mechanics s e e T r u e s d e l l ' s i n t r o d u c t i o n s t o LEOO 2.22 and 2.13.

Other secondary s o u r c e s d e a l i n g w i t h t h e h i s t o r y of p a r t i a l d i f -

f e r e n t i a l e q u a t i o n s i n t h e 18th c e n t u r y from an i n n e r mathematical p o i n t of view a r e : Demidov 1975 and 1980 and Engelsman 1 9 8 0 . It

The most r e c e n t source i s Greenberg 1 9 8 1 , who d e a l t w i t h t h e o c c u r r e n c e of p a r t i a l d i f f e r e n t i a t i o n i n t h e e a r l y works of F o n t a i n e . Greenberg mentions t h e l i n k w i t h f a m i l i e s of c u r v e s , b u t h i s main argument s t a t e s t h a t p a r t i a l d i f f e r e n t i a t i o n a r o s e from formal m a n i p u l a t i o n s of e x p r e s s i o n s involving s e v e r al v ar i ab l es . An i l l u s t r a t i n g example o f t h e confusion a r i s i n g from d i f f e r e n t i d e a s on t h e term " p a r t i a l " i s provided by t h e polemic between Hoppe (1927) and C a j o r i ( 1 9 2 8 ) . While d i s c u s s i n g t h e same s o u r c e s t h e y came t o e n t i r e l y d i f f e r e n t c o n c l u s i o n s , a l l of which, i n f a c t , were overdone.

Footnotes Chapter 1

164

c f . Newton, MuthemuticaZ Papers I, esp. p. 289 f f . References t o o t h e r secondary s o u r c e s t h a t have adopted W h i t e s i d e ' s view a r e q:iven

i n Appendix 3 .

c f . G r a t t a n Guinness 1979.

'

c f . Haas 1 9 5 5 , pp. 253 f f . o r Edwards 1 9 7 9 , pp. 129-131.

a

c f . Bos 1 9 7 4 a , e s p . § I . 10 and 55.0. c f . Bos 1 9 7 4 0 , 1 2 . 4 . c f . f o r example JohB:GWL 1 4 / 8 / 1 6 9 7 , where Johann B e r n o u l l i wrote: "[.

.. ]

p r o t r a n s i t u a curva ad curvam v a r i a b i l i u m , de huiusmodi

differentiatione,

[...I".

i n : Johann B e r n o u l l i 1 6 9 7 b , problem 1. c f . Enestr6m 1 8 9 9 , who p r o v i d e s r e f e r e n c e s . Nicolaus I B e r n o u l l i t a c k l e d t h e problem by means of p a r t i a l d i f f e r e n t i a l c a l c u l u s i n an undated and a s y e t unpublished manuscript, k e p t as i t e m L I a 2 3 , 5139-142 a t t h e b f f e n t l i c h e Bibliothek der U n i v e r s i t a t Basel. l 3

c f . Greenberg 1 9 8 1 , esp. p. 253. c f . C a j o r i 1928.

l5

c f . Taylor 1974 f o r a d i s c u s s i o n of t h e t o t a l d i f f e r e n t i a l i n 19th and 20th century calculus.

l6

c f . Schl6milch 1 8 6 8 .

I have c o n s u l t e d t h e t h i r d e d i t i o n , w h i l e L i n d e l E f r e f e r r e d t o t h e second e d i t i o n of 1862. The p r o o f s however seem t o be i d e n t i c a l , s i n c e i n t h e t h i r d e d i t i o n Schlzmilch s t i l l used the lemma "Piz+h,y)-P(,,yl=h,P,fz+th,y)

t between 0 and I, and t independent of y". This l a t t e r assumption was r e f u t e d by LindelEf by means of t h e example y=fa2-z21'. for a certain

However, Schlzmilch d i d n o t i n t e n d t o prove t h e n a i v e form of t h e e q u a l i t y theorem, s i n c e h e had added c o n d i t i o n s which he found n e c e s s a r y f o r t h e lemma t o h o l d ; t h e s e c o n d i t i o n s amounted t o t h e requirement of c o n t i n u i t y of P,

P,, Py, Pyx. Hence, t h e s e c o n d i t i o n s were indeed s u f f i c i e n t f o r t h e

e q u a l i t y theorem t o h o l d . 135 f o r t h e lemma and pp.

l7

c f . B e r t r a n d 1 8 6 4 , p.

I9

E u l e r ' s 1 7 5 8 , f o r example, was e n t i t l e d "Exposition de quelques paradoxes

156-158 f o r t h e p r o o f .

c f . Higgins 1940 who drops names and p r o v i d e s r e f e r e n c e s .

dans l e c a l c u l i n t g g r a l " . One of t h e s e paradoxes was t h e f o l l o w i n g : Discussing t h e d i f f e r e n t i a l e q u a t i o n (Z) z&+ydy=dyJxr+yy-aa t h a t t h i s e q u a t i o n i s s o l v e d by (*a!)

E u l e r remarked

zz+yy-aa=O, which s o l u t i o n he s a i d

could not be found by i n t e g r a t i n g t h e d i f f e r e n t i a l e q u a t i o n p r o p e r l y . E u l e r i l l u s t r a t e d h i s p o i n t by i n t e g r a t i n g t h e e q u a t i o n ( z ~ Z ) fz&+ydy)/J~z+yy-aa=dy, which he regarded t o be e q u i v a l e n t t o (*). I n t e g r a t i o n o f (22Z) y i e l d s

165

Footnotes Chapter 1 (&&&e) Jmiyy-aa=y+c. Indeed, f o r none of t h e v a l u e s of c does

(St*&)

t r a n s f o r m i n t o (StSt). E u l e r d i d n o t go on t o "explain" t h e paradox, which obviously d e r i v e s from t h e f a c t t h a t ( S t ) and (&t&)a r e e q u i v a l e n t o n l y i f

xxiyy-aa#O. 2o

c f . L i t z e n 1978. c f . JohB:CWL 1 4 / 8 / 1 6 9 7 .

22

c f . Nicolaus I1 B e r n o u l l i 1 7 2 0 , esp. 926.

23

c f . Nicolaus I B e r n o u l l i Demonstratio.

'' c f .

E u l e r De i n f i n i t i s curvis, § 4 .

25

c f . f o r example Johann B e r n o u l l i 1697d, JohB:CWL 2 / 9 / 1 6 9 4 , GWL:JohB 7 / 6 / 1 6 9 4

26

Recently, t h e main r e s u l t s of Demidov's 1974, 1975 and 1980 have a l s o become a v a i l a b l e i n French o r E n g l i s h ; c f . h i s 1982a, 1 9 8 2 b .

166

FOOTNOTES CHAPTER 2

L e i b n i z ' s 1692 i s e n t i t l e d : "De l i n e a e x l i n e i s numero i n f i n i t i s o r d i n a t i m d u c t i s i n t e r s e c o n c u r r e n t i b u s formata easque omnes t a n g e n t e , a c de novo i n e a r e analysis infinitorum

usu". The follow-up a r t i c l e 1694 i s e n t i t l e d :

"Nova c a l c u l i d i f f e r e n t i a l i s a p p l i c a t i o e t usus ad m u l t i p l i c e m l i n e a r u m c o n s t r u c t ionem e x d a t a t a n g e n t ium cond it ione". T o r r i c e l l i 1644, pp. 183-184. c f . f o r example: Huygens 1690, p. 475.

Here t h e page number r e f e r s t o t h e

volume of t h e Oeuvres Completes c i t e d i n t h e b i b l i o g r a p h y . A s a r u l e , I a l ways r e f e r t o c o l l e c t e d works, opera omnia, o e u v r e s e t c . i n those c a s e s where more than one p l a c e of p u b l i c a t i o n i s l i s t e d i n t h e b i b l i o g r a p h y . The term "given by p o s i t i o n " had been d e f i n e d a l r e a d y by Euclid i n h i s Data ( c f . Thaer 1962), and i t was used t o denote one of t h e t h r e e p o s s i b l e ways i n which a geometric o b j e c t could be c o n s i d e r e d "given";

it contrasted with

"given b y s i z e " and "given by shape". I have found no e x p l i c i t d e f i n i t i o n o f t h e concept of a curve "given by p o s i t i o n " a s used i n 17th c e n t u r y

mathematical p r a c t i c e . I t s meaning, however, may be paraphrased t h u s : The p o s i t i o n o f t h e c u r v e w i t h r e s p e c t t o t h e coordinate-axes

o r w i t h res-

p e c t t o o t h e r c u r v e s i s known. Hence a l l a l g e b r a i c c u r v e s a r e given by p o s i t i o n , s i n c e they a r e d e f i n e d by an a l g e b r a i c e q u a t i o n between t h e coord i n a t e s , and s o t h e i r p o s i t i o n w i t h r e s p e c t t o t h e axes i s always known. I f t r a n s c e n d e n t a l curves, d e f i n e d by a n e q u a t i o n o f t h e form y=/p(z,a)& a r e given by p o s i t i o n , then one may assume t h a t t h e bounds of i n t e g r a t i o n a r e known, s i n c e t h e s e bounds d e f i n e t h e p r e c i s e p o s i t i o n of t h e c u r v e s w i t h r e s p e c t t o the axes. from: Leibniz 1692, p . 267. Here t h e page number r e f e r s t o t h e volume of L e i b n i z ' s Mathernatische S c h r i f t e n ( e d i t e d by C . I .

Gerhardt) mentioned i n

the bibliography; c f . footnote 3 . "Exempli causa, s i speculum a l i q u o d , v e l p o t i u s s e c t i o e j u s a plano per axem, cujuscunque f i g u r a e p o s i t i o n e d a t a e , r a d i o s s o l a r e s s i v e immediate s i v e p o s t a l i a m quandam r e f l e x i o n e m a u t r e f r a c t i o n e m a d v e n i e n t e s r e f l e c t a t ,

i s t i r a d i i r e f l e x i erunt i n f i n i t a e l i n e a e r e c t a e ordinatim ductae, e t d a t o quovis puncto s p e c u l i ( c a e t e r i s manentibus) d a b i t u r r a d i u s r e f l e x u s e i respondens."

Footnotes chapter 2

6

167

from: Leibniz 1692, p. 267. "Verum ego sub o r d i n a t i m d u c t i s non tantum r e c t a s , sed e t c u r v a s l i n e a s qualescunque a c c i p i o , mod0 l e x h a b e a t u r , secundum quam d a t o l i n e a e cujusdam d a t a e (tamquam o r d i n a t r i c i s ) puncto,respondens e i puncto l i n e a d u c i p o s s i t , quae una e r i t e x o r d i n a t i m ducendis s e u o r d i n a t i m p o s i t i o n e d a t i s . Ordine enim percurrendo puncta o r d i n a t r i c i s ( v e r b i g r a t i a l i n e a e , c u j u s r o t a t i o n e f i t speculum paulo a n t e dictum, seu s e c t i o n i s e j u s per axem), o r d i n e prodibunt l i n e a e i l l a e o r d i n a t i m d a t a e . " from:Leibniz 1694, pp. 301-302. " c o e f f i c i e n t e s a , b , c i n a e q u a t i o n e cum i p s i s z e t y u s u r p a t a e , s i g n i f i c a n t q u a n t i t a t e s i n eadem curva c o n s t a n t e s , a l i a s quidem i n s i t a s (nempe p a r a m e t r o s ) , a l i a s vero e z t r u n e a s , quae s i t u m curvae (adeoque v e r t i c i s a x i s q u e ) d e f i n i u n t . Sed comparando curvas s e r i e i i n t e r se seu t r a n s i t u m de curva i n curvam considerando, a l i a e c o e f f i c i e n t e s s u n t c o n s t a n t i s s i m u e s e u permanentes (quae manent non tantum i n una, sed i n omnibus s e r i e i curv i s ) , a l i a e s u n t v a r i a b i l e s . E t quidem u t s e r i e i curvarm l e x d a t a s i t , n e c e s s e e s t unicam tantum i n c o e f f i c i e n t i b u s s u p e r e s s e v a r i a b i l i t a t e m . " from: L e i b n i z 2 6 9 4 , p . 301. "manifestum e s t , c o n c u r r e n t e s quidem adeoque lineam e x concursu formutam t a n g e n t e s e s s e geminas, i n t e r s e c t i o n i s autem s e u concursus punctum esse unicum, adeoque e t ordinatam e i respondentem unicam e s s e , cum a l i o q u i i n i n v e s t i g a t i o n e s o l i t a linearum propositam tangentium, r e c t a r u m v e l curva-

rum ( v e l u t c i r c u l o r u m , parabolarum e t c . ) e x d a t a e curvae o r d i n a t i s quaerendarum, o r d i n a t a e geminae,tangentes unicae c o n c i p i a n t u r . I t a q u e quoad praesentem calculum, quo i p s a e e x t a n g e n t i b u s r e c t i s v e l c u r v i s p o s i t i o n e d a t i s i n v e s t i g a n t u r o r d i n a t a e (contraquam i n communi),manent c o o r d i n a t a e 3:

e t y i n hoc t r a n s i t u ( a proximo ad proximum) i n v a r i a t a e , adeoque s u n t

indifferentiabiles;

a t c o e f f i c i e n t e s (quae i n communi c a l c u l o i n d i f f e r e n -

t i a b i l e s c e n s e n t u r , q u i a c o n s t a n t e s ) quatenus h i c v a r i a b i l e s s u n t , d i f f e rentiantur.

"

l ' H 6 p i t a l posed t h e problem t o Johann B e r n o u l l i i n t h e l e t t e r Gl'H:JohB

8 / 1 2 / 1 6 9 2 , and Johann B e r n o u l l i provided t h e s o l u t i o n i n a l e t t e r of 18/12/1692, which, however, i s r e p o r t e d l o s t . H i s s o l u t i o n h a s been recons t r u c t e d by S p i e s s from l ' H 6 p i t a l ' s answer G2'H:JohB 2 / 1 / 1 6 9 3 ;

c f . Brief-

wechsel p. 162,fn. 4 . L ' H 8 p i t a l then communicated t h i s s o l u t i o n as h i s own t o L e i b n i z i n GZ'H:GWL 2 4 / 2 / 1 6 9 3 , and

-

i n a form more g e n e r a l

duced t h e s o l u t i o n a g a i n i n h i s 1 6 9 6 , 15146-147.

-

he r e p r o -

Footnotes chapter 2

168

Checking t h e s e r e s u l t s i s a n a s t y j o b , which may be c a r r i e d o u t i n t h e f o l l o w i n g way: E l i m i n a t i o n of y from (2.7) by means of (2.6) and e x p l i c i t s o l u t i o n of t h e r e s u l t i n g e q u a t i o n f o r J: y i e l d s

(a)

X = ~ SI s d t - t d s ) / l s d t - 2 t d s ) .

By means of (2.5) and t h e d i f f e r e n t i a l e q u a t i o n 2sds+$tdt-4adt=O,

resulting

from ( 2 . 5 ) by d i f f e r e n t i a t i o n w i t h r e s p e c t t o s and t, t h e d i f f e r e n t i a l s ds

dt can be e l i m i n a t e d from

and

(*x)

(k)

t o produce

x=as/t.

E l i m i n a t i n g t from (2.5) by means of (**) y i e l d s s=4a2x/(x2+4a2)and insertion

of t h i s v a l u e i n t o (%h) y i e l d s t = 4 a 3 / f x 2 + 4 a 2 ) . S u b s t i t u t i o n of

t h e s e v a l u e s f o r s and t i n (2.6) f i n a l l y y i e l d s t h e e q u a t i o n (2.9) of t h e envelope. Jakob B e r n o u l l i ' s 1694 i s h i s f i r s t r e a c t i o n t o L e i b n i z ' s 2694; he e x p l i c i t l y s e t o u t t o show t h a t L e i b n i z ' s r e s u l t s could a l l be found by c l a s s i c a l methods, s i n c e t h e c u r v e s L e i b n i z c o n s i d e r e d were a l l a l g e b r a i c . Jakob repeated h i s demonstrations i n h i s 1 6 9 5 , b e i n g a remark t o the 1695-edition of D e s c a r t e s ' s Geometria. Cf. a l s o Hofmann 1956,p.32 and f o o t n o t e s . from: L e i b n i z 1692, p. 269. "Hinc p a t e t , eandem aequationem posse h a b e r e d i v e r s a s a e q u a t i o n e s d i f f e r e n t i a l e s , s e u v a r i i s modis e s s e d i f f e r e n t i a b i l e m , p r o u t p o s t u l a t scopus i n q u i s i t i o n i s " . l 3

from: Joh. B e r n o u l l i 1 6 9 6 , p. 161. "Datis i n plano v e r t i c a l i duobus p u n c t i s A & B , a s s i g n a r e Mobili M viam

AMB, per quam g r a v i t a t e sua descendens, & moveri i n c i p i e n s a puncto A , brevissimo tempore p e r v e n i a t ad a l t e r u m punctum B". Compare S p i e s s ' s account of l ' H 8 p i t a l ' s s o l u t i o n i n Brieflechsez, pp. 148-149. I n t h i s f a s c i n a t i n g comedy of e r r o r s o n l y t h e f i n a l answer was c o r r e c t . l5

D i f f e r e n t i a l e q u a t i o n (2.13) can be solved t h u s :

y=Jx g

d t =

+

XX-a % arcsin a - ~az-rc2.

0

This s o l u t i o n indeed r e p r e s e n t s a c y c l o i d g e n e r a t e d by a c i r c l e w i t h d i a meter a and r o l l i n g along t h e y - a x i s , geometric c o n s i d e r a t i o n

a s can be seen from t h e f o l l o w i n g

(see f i g . ) :

L e t t h e g e n e r a t i n g c i r c l e be a t p o i n t G on t h e h o r i z o n t a l a x i s A G , and l e t t h e p o i n t L be on t h e c y c l o i d AL g e n e r a t e d by t h i s c i r c l e . Then arcGL=AG. Let A S = y be the o r d i n a t e and A T = x be t h e a b s c i s s a of t h e c y c l o i d . a a n Obviously, A S = A G - SG; s i n c e arcGL =-(LGUU + L U U L ) =-f+ arcsin-)2x-a and

2

2 2

a

169

Footnotes Chapter 2

I

X

S G = G 2 t h e s o l u t i o n given above indeed r e p r e s e n t s t h e c y c l o i d AL. 16

D e s p i t e a l l n i c e p r o p e r t i e s of t h e c y c l o i d s , n e i t h e r a r e t h e i r o r t h o g o n a l t r a j e c t o r i e s c y c l o i d s , nor d i d Johann B e r n o u l l i put any c l a i m t o t h a t ext e n t , c o n t r a r y t o what S t r u i k (1969, p. 396) and Hofmann (1956, p. 89 f n . 322) s t a t e ( c f . a l s o Goldstine 2980, p.43). However, i n h i s l e t t e r JohB:GWL

21/7/1696 Johann B e r n o u l l i n o t i c e d t h a t t h e f a m i l y of congruent c y c l o i d s , g e n e r a t e d by s h i f t i n g one f i x e d c y c l o i d a l o n g t h e h o r i z o n t a l a x i s , a g a i n has c y c l o i d a l o r t h o g o n a l t r a j e c t o r i e s . The f a m i l y of o r t h o g o n a l t r a j e c t o r i e s i s symmetrical w i t h the given f a m i l y and emerges through r e f l e c t i o n of t h e given f a m i l y i n t h e h o r i z o n t a l l i n e a t h a l f t h e amplitude of t h e given c y c l o i d . To my knowledge, t h i s was t h e f i r s t example of a f a m i l y of r e c i procal t r a j e c t o r i e s . 17

c f . Jakob B e r n o u l l i 1697. S t r u i k 1969 c o n t a i n s an English t r a n s l a t i o n .

18

c f . f o r example G o l d s t i n e 1980, o r D i e t z 1959.

19

from: Jakob B e r n o u l l i 1697, p. 774. "

...

quaenam ex i n f i n i t i s c y c l o i d i b u s ( a u t s a l t e m c i r c u l i s , p a r a b o l i s ,

a l i i s v e c u r v i s ) p e r A t r a n s e u n t i b u s , a c super eadem b a s ? AH c o n s t i t u t i s ,

i l l a s i t , per quam descendens grave minimo tempore ex A ad datum perpendiculum ZB a p p e l a t " . 20

c f . JohB:GWL 7/6/1697.

21

The l e t t e r t o Varignon was p u b l i s h e d a s Johann B e r n o u l l i ' s 1697~. The l e t t e r t o l ' H 8 p i t a l i s : JohB:GZ'H 15/10/1697.

22

i n : JohB:GWL

23

This r e c o n s t r u c t i o n c o i n c i d e s w i t h t h e one given by Cramer i n JBO 11, p . 8 0 4 , fn.h

,

14/8/1697.

and c i t e d by S p i e s s i n BriefwechseZ,p.345,

f n . 2 . Hofmann's recon-

s t r u c t i o n i n h i s 1956, fn.334 i s wrong, s i n c e t h e segments P& and PII are r e l a t e d t o each o t h e r by t h e p r o p o r t i o n PQ:PrL=OP:O?,

and hence P Q f P n .

Footnotes Chapter 2

170 24

LetARB'andACC'be members of t h e f a m i l y o f s i m i l a r c u r v e s , and l e t t h e a r c s A 2 and A?

and t h e i n f i n i t e s i m a l segments B%'

r e s p e c t t o A . Put A?=cxA%,

hence C?'=aB%',

and

6'be

similar with

o r , w i t h l%'=ds and E'=ds',

ds'=ads. The f a l l i n g - t i m e s d t and d t ' a l o n g ds and ds' r e s p e c t i v e l y ( f o r a p a r t i c l e s t a r t i n g i t s f a l l w i t h z e r o v e l o c i t y i n A ) can be expressed a s :

dt=ds/& and d t '=ds '/@=cids/&=&dt.

Hence, d t : d t = & = m :

m.

I n t e g r a t i o n of t h i s e q u a t i o n y i e l d s ( 2 . 1 8 ) .

A

S i m i l a r i t y of t h e synchrones i n a family of similar c u r v e s can b e proved i n t h e f o l l o w i n g way: consider two synchrones BB' and PP', chosen a r b i t r a r i l y ,

P' a r e taken such t h a t A , B and P and A , B' and P' a r e c o l l i n e a r . I and (**)tAlp'tA^p,. We have t o prove t h a t AB:AP=AB ' : A p t . Hence ( a )tA%=tA% Due t o (2.18) we f i n d : tA%: tAT=m: and tA%: t A F , = m :JAP'. Now by con-

where P and

sequence o f (*) and (**) one f i n d s :

AB:AP=AB ' : A f t , q . e . d.

m:&=m:m,and

hence:

Footnotes chapter 2

25

171

from: JohB:GWL 17/7/1697. " F a c i l e credam, quod t u m u l t u a r i a c o n s i d e r a t i o i n t e r scribendum T i b i s u g g e s s i t , Synchronas semper posse per q u a d r a t u r a s h a b e r i : primum enim hoc e s t quod s e s e o f f e r t i n contemplatione harum curvaram, quod s c i l i c e t d a t o tempore d e t e r m i n a r i queat punctum i n curva d a t a , ad

quod mobile p e r v e n i t ,

e t quod hoc f i e r i p o s s i t pro eodem tempore i n q u a l i b e t curva o r d i n a t i m p o s i t i o n e d a t a , e t s i c t o t a Synchrona c o n s t r u i . Sed hujusmodi c o n s t r u c t i o eo i p s 0 non e s t aestimanda, q u i a non per continuam

Quadraturam u n i u s

ejusdemque i n d e t e r m i n a t i s p a t i i p e r a g i t u r , e t q u i a per consequens e x i n d e non h a b e r i p o t e s t modus ducendi t a n g e n t e s ad Synchronam, q u i tamen h i c summe n e c e s s a r i u s e s t . Rogo i t a q u e u t paulo p e n i t i u s i n s p i c i a s negotium; f o r s a n r e v o c a b i s Tua v e r b a , quando d i c i s : A s s u m a t u r e x

S y n c h r o n i s ,

d a t a e

r e c t a e

p o t e s t,

e t

a d

earn

p a r a l l e l a ,

s a 1 t e m

d u c a t u r q u o d

a 1 i q u a t a n g e n s

u t i q u e

f i e r i

t r a n s c e n d e n t e r ; nam nondum v i d e o

quomodo v e l t r a n s c e n d e n t e r v e l algebra'ice d u c i p o s s i t tangens ope cons t r u c t i o n i s i l l i u s per q u a d r a t u r a s diversorum spatiorum. Ego quidem i n hoc puto l a t e r e maximum a r t i f i c i u m , u t d i v e r s a e i s t a e q u a d r a t u r a e r e ducantur ad quadraturam indeterminatam u n i u s s p a t i i c o n t i n u i , quod ego f e l i c i ter p r a e s t i t i . " 26

from: JohB:GWL 17/7/1697. "Neque hactenus p e r s p i c e r e p o t u i ullam viam p e r v e n i e n d i ad t a n g e n t e s : s i aliquam mihi m o n s t r a b i s , quamvis t r a n s c e n d e n t e r , habebo T i b i g r a t i a s haud mediocres."

27

from: GWL:JohB 3/8/1697. " L i t e r a s meas nuperrimas a c c e p e r i s . I n t e r e a Moscorum Monarcham e j u s q u e Legationem i n v i c i n i a vidimus, e t quidam ex c o m i t a t u i n s e r e c e p i t mihi p r o c u r a r e responsiones ad q u a e s i t a quaedam mea c i r c a r e s Moscorum s c r i p t o c o n s i g n a t a . Dum huc redeo, more me0 i n i t i n e r e m e d i t a t u s , d e s i d e r a t a m a Te Me thodum generalem i n v e n i . The Russian Monarch mentioned above was Tsar P e t e r t h e f i r s t , who was on h i s way from B e r l i n t o t h e Netherlands i n e a r l y August 1697. L e i b n i z t r i e d t o have an i n t e r v i e w with t h e Tsar

-

o u t of c u r i o s i t y t o s e e one of the m i g h t i e s t

men i n the world, as he confessed

-

but he f a i l e d i n t h i s purpose

and had t o watch t h e d e l e g a t i o n p a s s i n g by i n Minden; cp. Muller/KrSnert

1 9 6 9 , p. 147.

'*

c f . f o r example Bos 1974a, e s p e c i a l l y c h a p t e r 2 .

Footnotes chapter 2

172

29

from: GWL:JohB 3/8/1697.

R C v e l B F e t i a m h a b i t a e f u i s s e n t per quandam quadraturam, u b i a 1 1 1 1 f u i s s e t i n g r e s s a vinculum quadratorium, eodem mod0 f u i s s e t procedendum

"Si

p r o d i f f e r e n t i a i n t e r l B I C e t l B I F s e u p r o lFIC, u t processimus i n exhibenda d i f f e r e n t i a i n t e r V C e t V F, nempe d i f f e r e n t i a n d a f u i s s e t quan1 1 t i t a s sub v i n c u l o q u a d r a t o r i o c o n t e n t a , sed secundum a ; e t proveniens rursum summandum, sed secundum x. Nec v i d e o quid hunc processum impedire unquam pos s i t

.

"

I n t h e t r a n s l a t i o n I have r e p l a c e d L e i b n i z ' s conventions by my own.

''

The Beilage i s d a t e d " I n i t i o Augusti 1697" (Early August 1697) and i t was published by Gerhardt i n t h e LMS n e x t t o t h e l e t t e r GWL:JohB 3/8/1697. The t i t 1 e " B e i l a g e " i s G e r h a r d t ' s . The Beilage h a s n o t been mailed t o Johann B e r n o u l l i , a s i s q u i t e c e r t a i n from t h e f a c t t h a t no r e f e r e n c e a t a l l i s made t o i t i n t h e l a t e r l e t t e r s . The n o t a t i o n s d(secund.a) o r xFa do n o t occur e i t h e r i n t h e subsequent correspondence.

31

from: BeiZage, p . 4 5 I . "Hujus p r o b l e m a t i s sane d i f f i c i l i s e t n o s t r i s Methodis h a c t e n u s non par e n t i s similiumque a l i o r u m s o l u t i o n e m a me p e t i i t Dn. Johannes B e r n o u l l i u s mense J u l i o 1697. Re a l i q u a n d i u c o n s i d e r a t a mihi tandem v i d e o r quaesitum a s s e c u t u s . Quod sane magni e s t momenti e t insignem aliquem i n n o s t r o c a l c u l o d i f f e r e n t i a l i defectum s u p p l e t . Devenimus autem i n hujusmodi quaest i o n e s occasione eorum, quas Dn. Jacobus B e r n o u l l i u s , P r o f e s s o r B a s i l e e n s i s , Dn. F r a t r i suo Johanni, P r o f e s s o r i Groningano, p r o p o s u i t , quas i s t e quidem s o l v i t , q u i a tantum agebatur de c u r v i s ejusdem s p e c i e i s e u s i m i l i b u s e t similiter

32

pos i t i s

.

"

from: Beilage, pp. 453-454. " P a t e t e t i a m e x h i s , summari h i n c i p s a s d i f f e r e n t i a s arcuumper a r c u s , nempe: Summa

d i f f e r e n t i a r u m elementarium simul sumtae I F 1 ( C I , IFl((CI)

e t c . aequatur d i f f e r e n t i a e i n t e g r a l i seu d i f f e r e n t i a e i n t e r arcum ultimum e t primum, e t i t a habentur summationes d u p l i c a t a e a n t e a i g n o t a e , v e l u t i

hic J ( a d d ( c h : x ~ ) = j d x ~ : (secund. x prim

x et a)-/dxG:J:

( s e c m d . u z t i m x e t a ) . Nempe h a c t e n u s non n i s i secundum unius l i t e r a e v a r i a t i o n e m summare potuimus v e l d i f f e r e n t i a r e , velsecundum p l u r e s simul v a r i a t a s ubique, sed non s i p l u r e s pro p a r t e v a r i a t a e , pro p a r t e invar i a t a e concurran t

.

'I

L e i b n i z ' s remark i s n o t v e r y c l e a r , and a t some p o i n t s mistaken. E.g.

(secund. p r i m . x e t a) and (secund. ultim. x e t a) s h o u l d i n f a c t r e a d (secund. prim. a ) and ( s e c m d . ulytim. a).

Footnotes chapter 2

173

from: GldL: JohB 9/8/1697. "Binas meas a c c e p e r i s . P r i o r e s Tuis respondebant: s e q u e n t e s novam Methodum d i f f e r e n t i a t i o n i s a Te d e s i d e r a t a m c o n t i n e b a n t . Has nunc s c r i b o , u t a l i q u i d addam, quod nuperrimas s c r i b e n t i e f f l u x i t . S e n t e n t i a nimirum mea e s t , r e c t e nos f a c t u r o s , s i n o n n i h i l

adhuc novam hanc Methodum dissimu-

lemus, donec i p s i s a t i s u s i simus; nam m u l t a i b i l a t e n t m a j o r i s momenti, quam q u i s prima f r o n t e s u s p i c e t u r . I t a q u e optimum puto, u t neque proponanus a l i i s quaerendam hanc d i f f e r e n t i a n d i vel t a n g e n t e s ducendi r a t i o n e m ,

neque a n o b i s inventam dicamus, multo minus exponamus i n quo c o n s i s t a t a r t i f i c i u m , donec n o b i s i p s i s l i c u e r i t p r o s e q u i pro d i g n i t a t e . " 34

aIdxs=Jd adxs w i t h r e s p e c t t o x, L e i b n i z

By d i f f e r e n t i a t i o n of t h e e q u a l i t y d

would have found d d s=d d s immediately. However, he o n l y i n t e g r a t e d t h e

x a

a x

e q u a l i t y w i t h r e s p e c t t o a t o a r r i v e a t h i s double i n t e g r a t i o n s . 35

from: JohB:GWL 1 4 / 8 / 1 6 9 7 . "Sed pro t r a n s i t u a curva ad curvam v a r i a b i l i u m , d e hujusmodi d i f f e r e n t i a t i o n e , l i c e t jam o l i m e t i a m i n t e r nos actum f u e r i t , nunc tamen ingenue f a t e o r , non c o g i t a v i . Quam v e r o i n g e n i o s e , quam a c u t e i l l u r n h u i c n e g o t i o accommodaveris, s a t i s m i r a r i nequeo; p r o f e c t o n i h i l e l e g a n t i u s e s t neque e x c o g i t a r i p o t e s t , quam modus i l l e Tuus d i f f e r e n t i a n d i curvam p e r summam d i f f e r e n t i u n c u l a r u m numero i n f i n i t a r u m . "

36

from: JohB:GWL 14/8/1697. "Annon p o s s e n t depromi problemata, q u a l i a jam d e d i i n E l l i p s i b u s , quibus m i s e r i e e x e r c e r e possemus Geometras, i n t e r i o r i Geometria l i c e t maxime v e r s a t o s ? V i d e r e n t sane omnes suos c o n a t u s i r i t o s , quamdiu i n nostrum a r t i f i c i u m non p e n e t r a r e n t , suamque i n f i r m i t a t e m t a n t o magis m i r a r e n t u r , quod hujusmodi problemata v i d e a n t u r f a c i l i a e t e x d i r e c t a tantum methodo tangentium desumta."

37

from: Joh. B e r n o u l l i 1697b, p . 205.

"IV. Sur l ' a x e BA donnd de p o s i t i o n E t a n t d E c r i t e s t o u t e s l e s Courbes d ' u n e msme e s p s c e , par exemple, t o u t e s l e s p a r a b o l e s BC, BC, BC & c. E t e n a y a n t coupd d e s arcs Cgaux RC, BC, BC, & c . On demande l e p o i n t C l e p l u s proche du p o i n t B; c ' e s t - a - d i r e ,

q u ' i l f a u t d s t e r m i n e r l e q u e l de c e s

a r c s a l a p l u s c o u r t e s o u t e n d e n t e BC. V. LesmGmes choses supposzes, on demande l a n a t u r e & l e s t o u c h a n t e s de l a

Courbe CCC,I ' 3a

from: Gl 'H:JohB 18/11/1697. " J e vous avou? que l o r s q u e l e s courbes n e s o n t pas semblables il f a u t quelque chose de p l u s , mais j ' a i compris que p a r courbes d'une meme

Footnotes Chapter 2

174

espece vous e n t e n d i e z courbes semblables, d ' a u t a n t p l u s que l'exemple des p a r a b o l e s que vous donnez p a r o i s t confirmer dans c e t t e

o p i n i o n ; en t o u t

cas j e ne p r e t e n d s a v o i r r e s o l u vos d e r n i e r s problemes que dans c e s e n s e t j ' a t t e n d s de l ' a p p r e n d r e de v o w , l o r s q u e les courbes s o n t dissemblables!' 39

from: JohB:Gl 'H 24/12/1697. "Par courbes d'une m6me espece j ' a y entendu t o u t e s les courbes d ' u n meme nom ou q u i s o n t donn6es ordonn6ment (ordinatirn d a t a a ) , t e l l e s que s o n t p a r exemple t o u t e s l e s e l l i p s e s s u r un meme axe; s i j ' a v o i s entendu par

15 seulement l e s courbes semblables e t semblablement p o s s e s , j e l ' a u r o i s dit;

[...I

Vous avez r a i s o n de d i r e que l o r s q u e l e s courbes s o n t d i s -

semblables i l f a u t quelque chose de p l u s , mais c e quelque chose de p l u s que vous croyez p e u t e t r e de peu d'importance e s t s i c o n s i d e r a b l e , que j e r e g a r d e c o m e r i e n l a d i f f i c u l t s qu'on a quand les courbes s o n t semblables

2 l ' e g a r d de c e l l e q u i s e r e n c o n t r e quand e l l e s s o n t dissemblables.'' 4Q

from: JohB:Ci:'I! 24/12/1697. " C ' e s t i c y une maniere t o u t e n o u v e l l e de c a l c u l e r , l e s r e g l e s du c a l c u l d i f f e r e n t i e l connues jusqu'ii p r e s e n t s o n t i n u t i l e s , il f a l l o i t i n v e n t e r pour c e l a une n o u v e l l e s o r t e de d i f f e r e n c e s pour b Z t i r aussy de n o u v e l l e s r e g l e s ; Nous avons dLja p e n e t r s , M r . L e i b n i t s e t moy, f o r t avant dans ce monde inconnu, M r . L e i b n i t s e n a trouv6 l ' e n t r g e , e t moy l u y e n ayant donnE l ' o c c a s i o n e t montrL l e s premieres t r a c e s ; J ' e n ay aussy t i r 6 une methode f a c i l e e t g e n e r a l e pour t r o u v e r l a courbe q u i coupe t o u t e s l e s courbes donnses ordonnsment s o i t geometriques ou mechaniques, s o i t semb l a b l e s ou d i s s e m b l a b l e s , s e l o n une c o n d i t i o n p r e s c r i t e quelconque, p a r ex. 5 l ' a n g l e d r o i t . "

41

The Meditationes a r e k e p t a t t h e d f f e n t l i c h e B i b l i o t h e k d e r U n i v e r s i t z t Basel a s i t e m LIu3. Nicolaus I B e r n o u l l i ' s s c r i p t f o r the Varicr Posthuma i n t h e JBO i s k e p t under L I a l . U n f o r t u n a t e l y , no complete e d i t i o n of t h e

Meditationes has a s y e t been p u b l i s h e d . 42

from: Jakob B e r n o u l l i 1744b. "Tangens l i n e a e e x i n f i n i t i s genere iisdem c u r v i s a e q u a l e s a r c u s a b s c i n d e n t i s i t a r e p e r i t u r . Ductis p e r datum i n a b s c i n d e n t e punctum una e x i n f i n i t i s , eiusque t a n g e n t e e t a p p l i c a t a ; f i a t , u t e x c e s s u s h u i u s t a n g e n t i s s u p r a summam t e r t i o r u m p r o p o r t i o n a l i u m ad elementa a b s c i s s a e curvae e t elementa a p p l i c a t a e ad ipsam tangentem i t a subtangens ad quartam. Den o t a b i t haec portionem a x i s t a n g e n t i b u s u t r i u s q u e curvae, a b s c i n d e n t i s e t a b s c i s s a e i n t e r cep tarn. 'I

175

Footnotes Chapter 2

'

Jakob B e r n o u l l i ' s p r o p o r t i o n a l i t y can be checked as f o l l o w s :

.

K

G

/

In t h i s f i g u r e , AB and AB' a r e two curves of t h e given f a m i l y , EK i s t h e t a n g e n t t o AB i n B, BS i s t h e t a n g e n t t o t h e equal a r c s t r a j e c t o r y BE' i n

8 ; DE and BS meet a t r i g h t a n g l e s i n E , and Q i s t h e p o i n t of i n t e r s e c t i o n of t h e o r d i n a t e DE w i t h t h e curve AB'. S i m i l a r i t y of t h e t r i a n g l e s ABQB' and ABDE y i e l d s : BQ:QB'=BD:DE, and hence w i t h BQ=d y , QB'=d S , BD-y: a a B DE=ydas/day= J ( d g j 2 / d x s . Furthermore, s i m i l a r i t y of AXBS and ADES y i e l d s :

A DE=DS.XB/XS, and w i t h K B = t , KS=o+u, D S = ~ : D E = ~ t / ( o + ~ ~ . J a k o bp 'rso p o r t i o n a l i t y f o l l o w s from e q u a t i n g t h e s e two v a l u e s of

DE and some r e w r i t i n g .

"

c f . Hofmann 1 9 7 0 , e s p . p. 4 8 .

45

This quote and t h e subsequent argument a r e from: Jakob B e r n o u l l i 1 7 4 4 b . " D i f f e r e n t i e t u r a e q u a t i o harum curvarum, tam j u x t a c o o r d i n a t a s x & y, quam j u x t a parametrum p , & emergat fdx+gdy+hdp=O." A s u s u a l I have r e p l a c e d Jakob B e r n o u l l i ' s c o n v e n t i o n s by my own i n t h e

translation.

176

FOOTNOTES CHAPTER 3

The orthogonal t r a j e c t o r y problem h a s n o t y e t

a t t r a c t e d much a t t e n t i o n i n

secondary l i t e r a t u r e . Cantor, i n h i s 1880, provided t h e b a s i c f a c t s conc e r n i n g the c h a l l e n g e s i n 1697 and 1716. The e x t e n t of t h e c h a l l e n g e i n 1716 and t h e f o l l o w i n g y e a r s i s a p p a r e n t from Newton's Correspondence V J . This f i n e e d i t i o n c o n t a i n s most of t h e l e t t e r s (not o n l y Newton's) pert a i n i n g t o the problem d u r i n g t h e s e y e a r s and g i v e s a g r e a t d e a l of information i n t h e f o o t n o t e s . Recently Mejlbo has p u b l i s h e d a s h o r t a r t i c l e

( 1 9 8 0 ) about t h e t r a j e c t o r y problem, showing how p a r t i a l d i f f e r e n t i a t i o n emerged i n t h e s t u d y of t r a n s c e n d e n t a l c u r v e s . Greenberg 1981 a l s o touches upon the s u b j e c t . from: JohB:GWL 2/9/1694. "Datis i n f i n i t i s c u r v i s p o s i t i o n e i n v e n i r e curvam quae omnes ad angulos r e c t o s secat". F i g u r e 1 i s copied from Huygens 1690, OEuvres Comple'tes XIX.

'

c f . Joi2B:GJdL 2/9/1694. i n h i s l e t t e r GWL:JohB 16/12/1694. from: JohB:GWL 21/7/1696. "Memini me t i b i olim g e n e r a l i t e r p r o p o s u i s s e , i n v e n i r e curvam, quae a l i i s p o s i t i o n e d a t i s o c c u r r a t n o r m a l i t e r , quod ego i n p l u r i b u s solveram. Modum quidem Tuum generalem t r a d e b a s ; sed s i resumere p l a c e t , v i d e b i s i l l u m plerumque locum non h a b e r e , quando curvae p o s i t i o n e d a t a e s u n t t r a n s c e n d e n t e s , u t i n hoc i p s 0 exemplo a p p a r e b i t . A l t e r i u s exempli, quod i n

m e 0 schediasmate propono de L o g a r i t h m i c i s n o r m a 1i t e r

s e c a n d i s

, nondum

p e r

c u r v a m

quidem c o n s t r u c t i o n e m , n e c

aequationem d i f f e r e n t i a l e m p r i m i gradus i n v e n i , s e d tamen s e r i e m quamdam simplicissimam p r o i l l a e x h i b e r e possum"

'

from: Johann B e r n o u l l i 1697a,p. 193. "Si q u i s methodum suam i n a l i i s e x e r c e r e v e l i t , q u a e r a t lineam, quae ord i n a t i m p o s i t i o n e d a t a s c u r v a s (non quidem a l g e b r a i c a s , quod haud arduum f o r e t , sed) t r a n s c e n d e n t e s , ex. g r . Logarithmicas super communi a x e , &

p e r idem punctum d u c t a s , ad angulos r e c t o s s e c a t " .

Footnotes chapter 3

*

An equation l i k e

177

X

y= J p ( x , a ) d x could o n l y be s o l v e d e x p l i c i t l y f o r a by r e 20

t u r n i n g t o t h e d i f f e r e n t i a l e q u a t i o n dy=p(x, a)&.

However, combination of

t h i s d i f f e r e n t i a l e q u a t i o n w i t h t h e c o n d i t i o n &=-pix,a)dy

f o r t h e tra-

j e c t o r i e s would o n l y l e a d back t o c o n d i t i o n ( 3 . 1 ) . On t h e o t h e r hand, exp l i c i t s o l u t i o n of dx=-p(~,a)dyf o r a and i n s e r t i o n of t h e r e s u l t i n g exX

p r e s s i o n i n t o y= J p ( x , a ) a h was impossible f o r t h e r e a s o n s d i s c u s s e d i n 91.5. XO

This now f o r g o t t e n branch of mathematics, f o r long a s t e p - c h i l d i n t h e h i s t o r y of mathematics a s w e l l , h a s been d i s c u s s e d by Bos. Cf. e s p e c i a l l y h i s 1 9 7 4 b , which d e a l s w i t h t r a n s c e n d e n t a l c o n s t r u c t i o n s and d i s c u s s e s 17th c e n t u r y p r e f e r e n c e of r e c t i f i c a t i o n s over q u a d r a t u r e s . For o t h e r t y p e s of constructions, cf. h i s

1981a and 1981b.

from: GUL:JohB 23/8/1696. "Non memini amplius, quid T i b i s c r i p s e r i m o l i m de Methodo mea p r o invenienda p e r p e n d i c u l a r i ad curvas o r d i n a t i m p o s i t i o n e d a t a s , quam p r o t r a n s c e n d e n t i b u s v a l e r e negas, rogoque u t mihi i n d i c e s , i n quo c o n s t a t . Saltem accomodari p o t e r i t ad t r a n s c e n d e n t e s curvas a e q u a t i o n e exponential i t e r p e r c u r r e n t e d a t a s . E t sane has earum e x p r e s s i o n e s semper p r o perf e c t i s s i m i s habui. Expressiones p e r c u r r e n t e s u t genus; e x p o n e n t i a 1 e s species"

''

apud mihi s u n t

v e r o s u n t p e r f e c t i s s i m a earum

.

I t i s not e x a c t l y c l e a r what o t h e r t r a n s c e n d e n t a l e x p r e s s i o n s L e i b n i z might

have had i n mind; t h e quote of f o o t n o t e 10 s t r o n g l y s u g g e s t s t h a t he d i d t h i n k of t r a n s c e n d e n t a l s o t h e r than those which can be w r i t t e n as a X

y

X

, x",

and so f o r t h . He presumably thought of i n t e g r a l s t h a t could be e x p r e s s e d

n e i t h e r a l g e b r a i c a l l y nor e x p o n e n t i a l l y , such a s f o r i n s t a n c e i n

y= <@a,

p e r t a i n i n g t o t h e brachystochrone c y c l o i d s . To such t r a n s -

c e n d e n t a l curves L e i b n i z ' s method could c e r t a i n l y n o t be adapted. c f . JohB:GWL 27/10/1696. Nowadays one would r a t h e r d e s c r i b e Johann B e r n o u l l i ' s l o g a r i t h m i c curves ( a l l p a s s i n g through one given p o i n t and a l l having t h e same asymptote) by means of t h e e x p o n e n t i a l e q u a t i o n y=a

3:

. These

c u r v e s indeed s a t i s f y t h e

c o n d i t i o n t h a t t h e a b s c i s s a e a t corresponding o r d i n a t e s are i n t h e same p r o p o r t i o n a s t h e s u b t a n g e n t s : t h e s u b t a n g e n t of t h e curve y=a

X

i s given by

s ( a ) = l / l o g ( a ) (where 2og=elog). Consider two c u r v e s , given by y=al X

and

y=aZ3: r e s p e c t i v e l y , then t h e i r a b s c i s s a e x1 and x2 a t corresponding o r d i n a t e s a r e provided by alX1=a2x2, and hence x l l o g ( a sequence,

xl:x 2=logia2 ) : Z o g ( a1) = s ( a l ) : s ( a2) .

)-x l o g ( a 2 ) . By con-

1 - 2

178

l4

Footnotes chupter 3

c f . Jakob Hermann 1 7 1 7 .

JohB:GWL 14/8/1697. 2 3 ?Y x, CL x, ax,

l5

cf. the postscriptum t o

l6

Johann B e r n o u l l i introduced t h e symbols

...

t o denote any quan-

t i t i e s which a r e expressed i n terms of x and a . He then immediately ident i f i e d the l i n e segment BE i n f i g u r e 4 a s

ixdx; hence, daZJ=& xda and l7

dg=iX dx.

AXda, and

t h e l i n e segment BD as

from: Jakob B e r n o u l l i 1698b. " Q u a e s t i o h i c e s t de t a l i , quae d a t a s omnes ad angulos r e c t o s s e c e t . Dependet autem Problema a methodo tangentium i n v e r s a , u t nullam generalem solutionem a d m i t t a t , e s t q u e pro v a r i a datarum p o s i t i o n e mirae d i v e r s i t a t i s ; neque gradus, v e l s p e c i e s Curvarum, e s t c h a r a c t e r f a c i l i t a t i s v e l d i f f i c u l t a t i s P r o b l e m a t i s ; cum non nunquam i n a l g e b r a i c i s r e s d i f f i c u l t e r , i n transcendentibus contra f a c i l e succedat." from: Jakob B e r n o u l l i 1698b. "Atque horum omnium s o l u t i o f a c i l i s admodum f u i t : d a r i autem p o s s u n t a l i a e Curvarum p o s i t i o n e s , quae Problema magis arduum r e d d u n t , & v e l i n s i m p l i c i P a r a b o l a ad casus methodi tangentium i n v e r s a e nondum exp l o r a t o s deducunt; v e l u t i , si quaeratur Curva, quae omnes Parabolas super

eodem uxe e x h u c t a s , lateraque sua r e c t a r e s p e c t i v i s verticum a puncto

fix0 d i s t a n t i i s aequalia habentes, ad r e c t o s angulos t r u j i c i t . &c." I n h i s "grande l e t t r e " NicIB:PRM 32/3/2716 Nicoiaus I B e r n o u l l i d e a l t w i t h the c a s e y2=px+p2, which a l s o r e p r e s e n t s a f a m i l y of p a r a b o l a s w i t h l a t u s rectum equal t o t h e d i s t a n c e from v e r t e x t o t h e o r i g i n . In t h i s c a s e , t h e p a r a b o l a s a r e a l l d i s j o i n t . B e r n o u l l i provided t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n f o r the t r a j e c t o r i e s : ( - S ! d y + ( - k l p d y / ( f ~ l p 2 + y 2 1 ~ ~ ~ / f ( - k I p 2 + y 2 ) and he remarked: "La d i f f i c u l t 6 de s e p a r e r l e s indeterminges de c e t t e Gquation d i f f e r e n t i e l l e a f a i r d i r e mon o n c l e probZema hoe ad easus methodi tangentiurn i n -

versae nondum exploratos deduci. J ' e n s u i s p o u r t a n t venu a bout e t j ' a i t r o w 6 que l a courbe cherchee e s t une courbe a l g e b r a i q u e dont 1 ' B q u a t i o n e s t (2y2-x2+tjZ2+qy2)~)3/2=,y(t2*qy2)~-arcy." I have n o t been a b l e t o v e r i f y N i c o l a u s ' s d i f f e r e n t i a l e q u a t i o n . However,

i n h i s Tentmen Nicolaus a g a i n d e a l t w i t h t h e s e p a r a b o l a s by means of t h e v a r i a b l e parameter e q u a t i o n , and he a g a i n found t h i s r e s u l t ( c f . Ex.IX). 2o

c f . Johann B e r n o u l l i 1698 and 1702. I n t h i s l a t t e r a r t i c l e Johann B e r n o u l l i c o r r e c t e d a mistake he had made i n h i s 1698.

Footnotes Chapter 3

21

179

from: Johann B e r n o u l l i 1 6 9 8 , pp. 268-269. " A l l a t a s u f f i c i a n t ad ostendendum, methodum ad aequationem p e r v e n i e n d i , non i n uno a l t e r o v e tantum exemplo a l g e b r a i c a r u m , sed i n omnibus, dudum n o b i s f u i s s e familiarem: sed quoniam i n t r a n s c e n d e n t i b u s non s u c c e d i t , n i s i i n s i m i l i b u s , u t i n C y c l o i d i b u s ; v e l e t i a m i n i l l i s , quae ad unam constantem r e d u c i p o s s u n t , u t i n L o g a r i t h m i c i s ; i l l a m v e l u t i n s u f f i c i e n t e m n e g l e x i , nec quam excolerem d i g n a t u s sum: a l i a i t a q u e e x c o g i t a n d a e r a t , quae g e n e r a l i s e s s e t , & ad quascunque o r d i n a t i m p o s i t i o n e d a t a s , t r a n s cendentes aeque a c a l g e b r a i c a s , p o r r i g e r e t u r . Hanc

autem, postquam

a c u t i s s i m u s L E I B N I T I U S , occasione eorum, quae i p s i super hac a f f i n i q u e m a t e r i a communicaveram, i p s e novam d i f f e r e n t i a l i s c a l c u l i a p p l i c a t i o n e m p e r u t i l e m sane i n v e n i s s e t , mecumque v i c i s s e m communicasset, de qua h a c t e n u s n i h i l i n publicum c o n s t a t ; hanc, inquam, quam optaveram methodum generalem s e c a n d i o r d i n a t i m p o s i t i o n e d a t a s , s i v e a l g e b r a i c a s , s i v e t r a n s c e n d e n t e s , i n angulo r e c t o , s i v e o b l i q u o , i n v a r i a b i l i , seu d a t a l e g e var i a b i l i , tandem ex v o t o e r u i ; c u i L E I B N I T I O a p p r o b a t o r e ne ypd a d d i p o s s e t ad u l t e r i o r e m perfectionem; & v e l ideo tantum, quod p e r p e t u o ad aequationem deducat; i n qua s i interdum i n d e t e r m i n a t a e s u n t i n s e p a r a b i l e s , methodus non i d e o i m p e r f e c t i o r e s t ; non enim h u i u s , sed a l i u s e s t methodi i n d e t e r m i n a t a s s e p a r a r e . Rogamus i t a q u e Fratrem, u t v e l i t s u a s quoque v i r e s e x e r c e r e i n r e t a n t i momenti." 22

J.E.

Hofmann ( i n h i s 1 9 5 6 , p. 90, f o o t n o t e 328) claimed t h a t Jakob B e r n o u l l i

f i n a l l y found such a method and t h a t he formulated i t i n t h e Meditationes, a r t . 2 5 4 , The r e f e r e n c e t o t h e Meditationes c o i n c i d e s e x a c t l y w i t h Jakob B e r n o u l l i ' s 1744b, which, a s I have d i s c u s s e d i n 52.2.10,

deals only

w i t h t h e t a n g e n t problem f o r e q u a l a r c s t r a j e c t o r i e s . 23

from: GWL:JohB 30/12/1714. "Dabo e t i a m operam, u t quaedam edam, i n quibus Newtono aquam h a e r e r e s c i o . "

"

from: JohB:GWL 6 / 2 / 1 7 1 5 . "Suppetunt haud dubie multa eorum, quae o l i m i n t e r nos a g i t a t a f u e r e , e t quae p e r communem d i f f e r e n t i a l i u m methodum non f a c i l e o b v i a s u n t : q u a l i a s u n t quae de t r a n s i t u e x curva i n curvam habuimus, quae p e r a g u n t u r singul a r i quadam d i f f e r e n t i a t i o n e a d h i b i t a . I'

25

Along w i t h h i s l e t t e r JohB:GWL 23/11/1715.

L e i b n i z had asked f o r t h i s copy

i n h i s l e t t e r GWL:JohB 4 / 1 1 / 1 7 1 5 . Although t h e copy has n o t been p r i n t e d by G e r h a r d t , i t i s c l e a r from i n t e r n a l e v i d e n c e i n Johann B e r n o u l l i ' s l e t t e r t h a t B e r n o u l l i had s e n t a copy of the p o s t s c r i p t u m of h i s l e t t e r JohB:GWL 1 4 / 8 / 1 6 9 7 , which c o n t a i n e d t h e d e r i v a t i o n o f t h e v a r i a b l e parameter e q u a t i o n .

Footnotes Chapter 3

180

26

from: GWL:ASC 6 / 1 2 / 1 7 1 5 . " P o u r t a t e r un peu l e p o d s 'a nos A n a l y s t e s Anglois, ayEs l e bontE, Mons i e u r , de l e u r proposer ce probleme comme de vous mfme ou d ' u n amis: Trouver une l i g n e ECD q u i coupe 'a a n g l e s d r o i t s t o u t e s l e s courbes d'une s u i t e determinEe d'un mfme g e n r e , par exemple t o u t e s l e s Hyperboles AB, AC, AD, q u i o n t l e mcme sommet e t l e mcme c e n t r e , e t c e l a p a r une voy g e n e r a l e . 'I Cf. a l s o : GWL:JohE Decemb. 1 7 1 5 . The p r i n t e d v e r s i o n of t h e l e t t e r t o Conti c o n t i n u e s w i t h t h e f o l l o w i n g exp l a n a t i o n i n which L e i b n i z c l a r i f i e d t h e scope of h i s c h a l l e n g e : "Car on marque ce probleme p a r t i c u l i e r seulement pour se f a i r e e n t e n d r e , c a r dans l e s s e c t i o n s coniques i l a s e s f a c i l i t s s p a r t i c u l i e r e s , mais il s ' a g i t de donner une methode g e n e r a l e . E t ce probleme g e n e r a l p e u t S t r e

con@ a i n s i : E s t a n t donne'e l a courbure des rayons de lumiere dans l e m i l i e u diaphane, changeant c o n t i n u e l l e m e n t de r e f r a c t i v i t e , t r o u v e r l ' o n d e d e lumiere s e l o n l a maniere de p a r l e r de M. Hugens, ou s e l o n l a fa$on de

p a r l e r de M. B e r n o u l l i l a synchrone, 'a l a q u e l l e l e s rayons ou l e s mobiles,

p r i s convenablement, p a r v i e n n e n t en msme temps.'' However, from c i r c u m s t a n t i a l evidence H a l l and T i l l i n g ( i n : Correspondence V I , p. 2 5 4 , f o o t n o t e 6) argue t h a t t h i s a d d i t i o n t o t h e l e t t e r t o Conti must be of a l a t e r d a t e . Apparently, L e i b n i z d i d n o t r e a l i z e t h a t h e had n o t made himself c l e a r enough on t h e i s s u e of t h e g e n e r a l i t y which he r e q u i r e d f o r t h e s o l u t i o n of t h e t r a j e c t o r y problem u n t i l Johann B e r n o u l l i drew h i s a t t e n t i o n t o t h i s p o i n t (see f o o t n o t e 2 7 ) . The a d d i t i o n , t h e r e f o r e , must have reached Conti only i n February 1716, and presumably a f t e r he had informed Newton about t h e problem. Cf. a l s o : H a l l 1 9 8 0 , p . 216. 27

from: JohB:GWL 1 5 / 1 / 1 7 1 6 . "Fateor hoc problema g e n e r a l i t e r sumptum ab A n a l y s t i s i l l i s non f a c i l e solutum i r i , n i s i c o g i t e n t de mod0 n o s t r o d i f f e r e n t i a n d i p a r a m e t r o s , v e l a l i a s l i n e a s quae parametrorum l o c o s u n t , seu de t r a n s i t u d i f f e r e n t i a t i o n i s a curva i n curvam. Sed v e l i m animadvertas, non omnia h u j u s exempla p a r t i c u l a r i a aeque d i f f i c i l i a e s s e : s u n t enim quae p e c u l i a r i hac d i f f e r e n t i a n d i r a t i o n e non i n d i g e n t , u t s u n t e a curvarum, ad quarum t a n g e n t e s determinandas p a r a m e t r i v a r i a b i l e s i n considerationem non v e n i u n t ; e x horum numero e s t exemplum, quod p r o p o s u i s t i , de Hyperbolis, u t p o t e quae, s i c u t omnes S e c t i o n e s conicae commune centrum e t v e r t i c e m h a b e n t e s , h a b e n t eandem subtangentem pro communi a b s c i s s a , quascunque habeant parametros:

18 1

Footnotes Chapter 3 e s t enim p r o omnibus [ r e f e r e n c e t o f i g u r e ] OF:OA=OA:OE, s i c per i t a q u e

timendum e s t , ne Analystae Angli hoc exemplum s o l v a n t p e r communes method o s , e t p o s t e a u b i v i d e n t , rem tam f a c i l e s i b i s u c c e s s i s s e , i n d e evadant i n f l a t i o r e s e t i n opinione suae s u p e r i o r i t a t i s magis c o n f i r m e n t u r . " c f . Nicolaus I1 B e r n o u l l i 1716. 29

c f . Newton 1716. For d r a f t s of t h i s s h o r t a r t i c l e , see: Correspondence VI, pp. 290-293.

30

John K e i l l conveyed a s o l u t i o n t o Newton i n h i s l e t t e r JK:IN 5 / 3 / 1 7 1 6

(which,

i n f a c t , was d a t e d 23/2/1715 a c c o r d i n g t o t h e o l d s t y l e ) . K e i l l ' s s o l u t i o n

i s i d e n t i c a l with L e i b n i z ' s method d i s c u s s e d i n § 3 . 2 . References t o o t h e r B r i t i s h s o l u t i o n s - produced by James S t i r l i n g , Henry Pemberton and John Machin 31

-

a r e provided i n : Correspondence VI, p. 284, f o o t n o t e s 2, 5 , 6.

from: GWL:JohB 31/1/1716. "Addidi enim d i s e r t e , q u a e r i methodum generalem. Quod s i mihi s u p p e d i t a r e exemplum v o l e s , quod non p a r t i c u l a r i a l i q u e f a c i l i t a t e a d j u v a r i p u t e s , sed ad generalem a d i g e r e , rem gratam f a c i e s . " Compare a l s o : f o o t n o t e 26.

32

from: J0hB:GWL 1 1 / 3 / 1 7 1 6 . "Problema 1'.

Super r e c t a AG tanquam axe e x puncto A c o n s t r u e r e i n f i n i t a s

c u r v a s , q u a l i s e s t ABD, e j u s n a t u r a e , u t r a d i i o s c u l i e x s i n g u l i s singularum curvarum p u n c t i s B e d u c t i , s e c e n t u r ab axe AG i n C i n d a t a r a t i o n e , u t nempe s i t BO:BC=l:n. 2'.

Construendae s u n t t r a j e c t o r i a e , q u a l i s ENF, p r i o r e s c u r v a s ABD ad

angulos r e c t o s s e c a n t e s . " 33

This a u x i l i a r y curve does n o t , i n f a c t , p a s s through A, s i n c e x=O i m p l i e s

z=+m.

and hence t h e curve h a s a h o r i z o n t a l asvmDtote _ . f o r t h i s v a l u e of z.

Jz

does e x i s t f o r a l l 5 between 0 and a, n m 06n<1. These bounds f o r n r e s u l t from t h e d e f i n i t i o n

However, t h e i n t e g r a l

0

since n s a t i s f i e s

x

BO:BC=l:n, i n which BO, by d e f i n i t i o n , i s g r e a t e r than BC. For n=l t h e int e g r a l does n o t e x i s t . H a l l and T i l l i n g s t a t e t h a t Johann B e r n o u l l i ' s sol u t i o n was f a u l t y ( c f . Correspondence V I , p. 324, f o o t n o t e 4 ) ; t h i s remark i s o n l y v a l i d with r e s p e c t t o t h e diagram,

the solution i t s e l f i s perfectly

i n order. 34

from: JohB:GWL 1 1 / 3 / 1 7 1 6 . "Communicabo (quamvis s i n e A n a l y s i ) s o l u t i o n e m exempli, quod omnia quae d e s i d e r a s h a b e t r e q u i s i t a ; n u l l a quippe p a r t i c u l a r i f a c i l i t a t e r e s o l v i t u r , sed p o t i u s p a r t i c u l a r i quadam d e x t e r i t a t e opus e s t , quae non c u i v i s o b v i a

Footnotes Chapter 3

182

e r i t ; deinde recurrendum quoque e s t ad d i f f e r e n t i a s secundas, quae autem p e r Methodum nostram ad primam r e d u c u n t u r ; postremo t a l e e s t , u t f a c t i s e v o l u t i o n i b u s , a r t e quadam s i n g u l a r i ad q u a d r a t u r a s r e d u c a t u r , cum a l i o q u i n , n i s i r i t e t r a c t e t u r , a e q u a t i o p r o d e a t , i n qua i n d e t e r m i n a t a e cum s u i s d i f f e r e n t i a l i b u s i n t r i c a t a e adeo invicem permixtae r e p e r i a n t u r , u t i n s e p a r a b i l e s v i d e a n t u r . I' 35

from: JohB:GWL 2 2 / 8 / 1 7 1 6 . "Aliquid p r o f e c t o i n casu transcendentium deprehendo, quod e t mihi etiamnum remoram i n j i c i t , f a c i t q u e , u t nondum p l e n a r i a m s o l u t i o n e m talemque qualem optaverem, i n v e n i s s e me j a c t a r e possim, s a l t e m i n quibusdam t r a n s cendentium g e n e r i b u s , nam i n p e r m u l t i s a l i i s r e s mihi p e r p u l c h r e s u c c e d i t . "

36

i n : GWL:ASC 1 4 / 4 / 1 7 1 6 .

37

quote from: Correspondence V I , p. 323, f o o t n o t e 1.

38

c f . Jakob Hermann 1717, 1718, 1719.

34

c f . Taylor 1717.

''

Johann B e r n o u l l i ' s hand can b e r e c o g n i s e d , f o r i n s t a n c e , i n 528 of Nicolaus I1 B e r n o u l l i ' s 1720 where t h e f o l l o w i n g r e f e r e n c e i s made:

" A l t e r modus i n t e g r a n d i

[...I,

c o n t i n e t u r i n methodo mihi o l i m u s i t a t a i n

Actis 1697 p. 115 ad construendam aequationem sitam,

[...I

a F r a t r e mihi propo-

...'I

("The o t h e r method of i n t e g r a t i n g

[...I

have once used i n t h e Acta [ Eruditorum]

i s contained i n t h e method which I 1697 p . 115, t o c o n s t r u c t t h e

e q u a t i o n [ . . . I proposed t o me by my b r o t h e r " ) .

This r e f e r e n c e p e r t a i n s t o

the a r t i c l e " S o l u t i o A n a l y t i c a Aequationis i n A c t i s A'.

1695, pag 553 pro-

p o s i t a e " ( c f . J B O O I , pp. 174-179) i n which Johann B e r n o u l l i responded t o a problem s e t b y h i s b r o t h e r Jakob.

I n 1697 Nicolaus I1 B e r n o u l l i w a s about 2

y e a r s of age. The d i s t i n c t i v e d i f f e r e n c e i n type between t h e p u r e l y mathem a t i c a l and t h e polemic o r n a r r a t i v e s e c t i o n s of Nicolaus I1 B e r n o u l l i ' s a r t i c l e 1720 ( a t l e a s t i n t h e v e r s i o n p r i n t e d i n JBOU I I ) s u g g e s t s t h a t d i f f e r e n t a u t h o r s h i p i s r e f l e c t e d by d i f f e r e n t t y p e .

'' c f . '' c f .

f o r example: Correspondence V I , pp. 438-439, Nicolaus I1 B e r n o u l l i 1 7 2 0 , 934.

" i n t e l l i g i t u r per

X q u a n t i t a s qualiscunque composita e x z & c o n s t a n t i b u s ,

hoc e s t , f u n c t i o q u a e l i b e t i p s i u s 43

f o o t n o t e s 7 and 9.

2".

from: Nicolaus I1 B e r n o u l l i 2 7 2 0 , 535. "Nunc v e l i m comparet B . L e c t o r hanc u n i v e r s a l i s s i m a m s i m i l i u m solutionem ad extremum

p e r f e c t i o n i s gradum, nempe ad q u a d r a t u r a s , redactam, eamque

simplicem, planam a c f a c i l e m , cum s o l u t i o n i b u s p a r t i c u l a r i s exempli

Footnotes chapter 3

183

d x = y m d y : m a V i r i s C l a r i s s i m i s HERMANNO & TAYLOR0 d a t i s , a c d e i n j u d i c e t p e r s e ipsum, annon i l l a e ( s a l t e m r e s p e c t u ad nostram) s i n t int r i c a t a e , longae e t d i f f i c i l e s . " 44

c f . Nicolaus 11 B e r n o u l l i 1 7 2 0 ,

42-61.

§§

The d e f i n i t i o n s of t h e s e d i f f e r e n t t y p e s of s i m i l a r i t y a r e given i n 9 4 2 . They may be summarised i n t h e f o l l o w i n g way: Consider a f i x e d curve k , t h e p r i n c i p a l i s , having c o o r d i n a t e s

3:

and y ; suppose t h i s curve i s given by an

e q u a t i o n y=f(x). Now any o t h e r curve k ' , having c o o r d i n a t e s X and Y , i s called:

- " l a t e r a l i t e r s i r n i l i s " with r e s p e c t t o k i f t h e r e e x i s t c o n s t a n t s a and b such t h a t f o r a l l (x,y) on k , t h e p o i n t (X,Y) i s on k ' where X=(a/b)x and Y=(a/b)y. The curve k ' can then be r e p r e s e n t e d by an e q u a t i o n of t h e form Y=(a/b) f (bX/a).

-

"exponentialiter

seu p o t e n t i a l i t e r

s i m i l i s " with r e s p e c t t o k i f t h e r e

b , t o g e t h e r w i t h an exponent r , such t h a t f o r a l l (x,y) r r on k t h e p o i n t (X,Y) i s on k', where X=(a/b)x and Y=(a /b )y. The curve k ' e x i s t c o n s t a n t s a and

r

r

can then be r e p r e s e n t e d by an e q u a t i o n of t h e form Y=(a /b )f(bX/al.

-

" f u n c t i o n a l i t e r sirnilis" w i t h r e s p e c t t o k i f t h e r e e x i s t c o n s t a n t s a and

b, t o g e t h e r with a f u n c t i o n g of a s i n g l e argument, such t h a t f o r a l l (x,yl on k t h e p o i n t (X,Y) i s on k', where X=la/b)x and Y=lg(aI/g(bI)y. The curve k ' can then be r e p r e s e n t e d by an e q u a t i o n of t h e form: Y=(g(al/g(b))flbX/al. Having d e f i n e d t h e s e d i f f e r e n t t y p e s of s i m i l a r i t y , one can conceive of f a m i l i e s of s i m i l a r curves - s i m i l a r i t y taken i n any of t h e g e n e r a l ways defined here

-

by c o n s i d e r i n g t h e c o n s t a n t a t o be t h e parameter of such a

f a m i l y . Johann B e r n o u l l i provided t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n s f o r o r t h o g o n a l t r a j e c t o r i e s t o s u c h f a m i l i e s o f s i m i l a r c u r v e s : For a f a m i l y of l a t e r a l l y s i m i l a r curves (cf.

da

(I)

-

7-

dx2 + d y z xdx+ydy

§§

43-48):

'

For a family of e x p o n e n t i a l l y s i m i l a r c u r v e s ( c f . 5 5 4 9 - 5 5 ) :

For a family of f u n c t i o n a l l y s i m i l a r curves ( c f . (3)

da - d Z 2 + d y 2 7 - xdx + aa-'Aydy , where u=g(a)/g(b) and

§§

56-61):

A =

du da

.

(Z), ( 3 ) embody t h e r e l a t i o n s between t h e c o o r d i n a t e s z and y and t h e parameter a , a l l considered b e i n g v a r i a b l e a l o n g

These d i f f e r e n t i a l e q u a t i o n s ( I ) ,

t h e t r a j e c t o r i e s . For a p p l i c a t i o n s of t h e s e e q u a t i o n s , c f . 5 5 . 3 . 1 .

Footnotes Chupter 3

184

45

from: Nicolaus I1 B e r n o u l l i 1720, 565. " L i c e t i d d i r e c t e i n v e n e r i t , & haud dubie e x f o n t e a l i q u o i n s u p e r i o r i b u s indicato.

46

"

from: Nicolaus I1 B e r n o u l l i 1 7 2 0 , 520. "Solvere autem hoc l o c o v a r i o sensu a c c i p i p o t e s t : 1'.

enim, s o l v i t u r a l i -

quo mod0 Problema, perveniendo duntaxat ad aequationem d i f f e r e n t i a l e m qualemcunque, c u i p l u r e s i n s u n t quem duae i n d e t e r m i n a t a e earumve elementa; q u i s o l v e n d i modus d i c i p o s s e t p r i m i & i n f i m i gradus: 2'.

s o l v i t u r non-

n i h i l p e r f e c t i u s , s i r e d u c i p o t e s t ad aequationem quamvis d i f f e r e n t i o d i f f e r e n t i a l e m , sed i n qua n o n n i s i duae r e p e r i u n t u r i n d e t e r m i n a t a e cum s u i s e l e m e n t i s ; q u i secundus e s t p e r f e c t i o n i s gradus: 3'.

Perfecte solvi-

t u r inveniendo aequationem d i f f e r e n t i a l e m s i n e d i f f e r e n t i i s a l t i o r i b u s , & quae c o n s t a t duabus tantum i n d e t e r m i n a t i s ; t a l i o s o l u t i o o b t i n e t t e r t i u m p e r f e c t i o n i s gradum. 4'.

S o l v i t u r p e r f e c t i s s i m e , s i p r a e t e r e a integrando

ad aequationam t e r m i n i s f i n i t i s constantem, a u t s a l t e m p e r separationem indeterminatarum ad q u a d r a t u r a s r e d u c a t u r , ut c o n s t r u i p o s s i t ; q u i quart u s & s u m u s e s t gradus p e r f e c t i o n i s . " 47

from: Nicolaus I1 B e r n o u l l i 1 7 2 0 , 525. "Quamdiu i t a q u e f u g i t quid p r o q substituendum s i t , a e q u a t i o i n v e n t a n u l l i u s e r i t u t i l i t a t i s , u t p o t e quae ne quidem i n infimo p e r f e c t i o n i s gradu Problema s o l v i t ; proinde m i n o r i s aestimanda, quam quae p e r primam methodurn i n v e n t a e s t b i l e s y,z

&

dy=-p&;

e t s i h a e c , jam dictum, q u i a t r e s i n v o l v i t v a r i a -

a l a t e n t e m i n p , n o n n i s i imperfectissimam c o n s t i t u a t s o l u t i o -

nem. Sed s i , qua a r t e d e t e r m i n a r i p o t e s t q , s c i l i c e t i n t r a n s c e n d e n t i b u s , d e quibus h i c sermo e s t ; a e q u a t i o p r o v e n i e n s , e x s u b s t i t u t i o n e e j u s v a l o r i s

in

-da=iZ+ppldy:pq s o l v e t Problema i n longe e x c e l l e n t i o r i gradu quam a l -

t e r a dy=-pdz;

siquidem i n i l l a tunc n o n n i s i duae c o n t i n e b u n t u r v a r i a b i l e s ,

s u p p o s i t o p d a r i duntaxat p e r a & y ; nam s i e t i a m z i n g r e d e r e t u r , utraque p a r i passu ambularent. Quacirca operae p r e t i u m e r i t o s t e n d e r e quomodo i n transcendentibus v a l o r i p s i u s q inveniatur

.I1

from: Nicolaus I1 B e r n o u l l i 1 7 2 0 , §32. "Sed missa hac u n i v e r s a l i s s i m a T r a j e c t o r i a r u m formula, quae qua t a l k , quod f a t e o r , p l u s h a b e t c u r i o s i t a t i s quam u t i l i t a t i s i n a p p l i c a t i o n e ad exempla; p e r v e n i o tandem ad v a r i a s methodos, non quidem u n i v e r s a l i s s i m e s o l v e n t e s Problema, sed tamen g e n e r a l e s s i n g u l a s i n s u i s generibus."

185

FOOTNOTES CHAPTER 4

'

c f . S p i e s s 1940. This s k e t c h draws from t h e f o l l o w i n g secondary s o u r c e s : F l e c k e n s t e i n 1970 f o r d a t e s , Kohli 1975a,b and Henny 1975 f o r p r o b a b i l i t y t h e o r y and Montmort, S p i e s s Eriefwechse?,, e s p . pp. 21-24

f o r t h e h i s t o r y of t h e Nachlass, Schaf-

h e i t l i n 1922 f o r t h e Johann B e r n o u l l i l e c t u r e s t o 1 ' 8 8 p i t a l ; N i c o l a u s ' s correspondence w i t h L e i b n i z i s p u b l i s h e d i n LMS III/Z, and t h e E u l e r correspondence i n b o t h Fuss 1862 and Fuss 1843. The Montmort-N.

Bernoulli

correspondence concerning t h e problem of o r t h o g o n a l t r a j e c t o r i e s h a s n o t y e t been p u b l i s h e d ; I have c o n s u l t e d t h e o r i g i n a l s i n t h e O e f f e n t l i c h e B i b l i o t h e k d e r U n i v e r s i t z t B a s e l , k e p t i n volumes LIa21 and LIa48. from: P R M : J O ~ B5 / 3 / 1 7 1 3 .

"Mr.

v o s t r e neveu e s t furieusement s c a v a n t e t e s t tres i n f a t i g a b l e . J e n e

peux t r a v a i l l e r deux h e u r e s de s u i t t e , i l e n t r a v a i l l e s i x s a n s e s t r e las. i l m'a proposE p l u s i e u r s problemes f o r t j o l i s e t je m'en s u i s t i r e avec

a s s e z d'honneur e t de bonheur. ce s e r a des Augmentations pour mon l i v r e . " Ir

from: Henny 1 9 7 5 , p . 4 6 0 . "der Drang nach Verallgemeinerung sowie e i n e

zzhe, auf K l a r h e i t aus-

g e r i c h t e t e Beweiskraf t . " c f . f o r example NicIB:LE 2 4 / 1 0 / 1 7 4 2 . c f . NicIE:LE 2 9 / 1 1 / 1 7 4 3 i n which Nicolaus wrote: "Theorema i l l u d , c u j u s inventionem mihi

a s s e r u i s t i , nempe de a e q u a l i t a t e

d i f f e r e n t i a l i u m i p s i u s Pdx e t Q d y , p o t e s t quidem usum non exiguum h a b e r e i n i n t e g r a n d i s a e q u a t i o n i b u s d i f f e r e n t i a l i b u s , sed ego non ausim hanc u t i l i t a t e m eousque e x t e n d e r e , u t credam, omnem aequationem d i f f e r e n t i a l e m h u j u s formae PR&+QRdy=O f e r e n t i a l i i p s i u s PR&

integrationem admittere, q u o t i e s f a c t a d i f (ponendo z constantem) a e q u a l i d i f f e r e n t i a l i i p s i u s

QRdy (ponendo y constantem), q u a n t i t a s R d e t e r m i n a r i p o t e s t . Verum quidem e s t , s i q u a n t i t a s quaedam i n t e g r a l i s f i n i t a p r o d i f f e r e n t i a l i h a b e a t

PR&+QRdy,

tunc f o r e

diff. P R d x = d i f f . QRdy; sed d u b i t o , an h u j u s proposi-

t i o n i s conversa e t i a m s i t Vera." ("This theorem, t h e i n v e n t i o n of which you a s s i g n t o m e , namely about t h e e q u a l i t y of t h e d i f f e r e n t i a l s of t h o s e P d x and Qdy, can c e r t a i n l y have no l i t t l e use f o r t h e i n t e g r a t i o n of d i f f e r e n t i a l e q u a t i o n s ; b u t I d a r e n o t

Footnotes Chupter 4

186

extend t h i s u t i l i t y t h a t f a r t h a t I would b e l i e v e t h a t a l l d i f f e r e n t i a l e q u a t i o n s of t h e form PRdx+QRdy=O admit i n t e g r a t i o n , o r t h a t always when t h e d i f f e r e n t i a l of PI?&

( p u t t i n g z c o n s t a n t ) i s made e q u a l t o t h e d i f -

f e r e n t i a l of QRdy ( p u t t i n g y c o n s t a n t ) t h e q u a n t i t y R can be determined. Certainly it is t r u e that i f a c e r t a i n f i n i t e i n t e g r a l quantity h a s

PR&+&Rdy f o r i t s d i f f e r e n t i a l , then d i f f . P R d c c = d i f f . QRdy h o l d s ; b u t I doubt whether t h e converse of t h i s theorem i s a l s o t r u e . " ) from: NicIB:PRM 1 1 / 6 / 1 7 1 8 . "Vous me demandez pourquoi j e ne me mets pas a u s s i s u r l e s rangs avec ceux

q u i o n t r e s o l u l e fameux problzme de M r . L e i b n i t s , e t pourquoi j e r e f u s e

l a g l o i r e q u i p o u r r o i t m ' a p p a r t e n i r ; j e vous r e p o n d i s , parceque j e ne s u i s pas ambitieux, e t que j ' a i c r u , q u ' i l s u f f i s o i t de vous a v o i r

assure

dans

m a grande l e t t r e du 31 mars 1716 que j e l ' a i r e s o l u . "

from : N7IcIB: PRM 1 1/6/1718. " J e vous p r i e , Monsieur, de n e c o m u n i q u e r m a s o l u t i o n 'a personne s a n s l e consentement de mon o n c l e , 1 q u i j e donne a v i s de ce que j e v i e n s de vous e c r i r e . " from: NicIB:LE 6 / 4 / 1 7 4 3 . "communicabo h i c Tecum ejusdem c o n s t r u c t i o n i s demonstrationem a n a l y t i c a m , quam olim concinatam nondum p u b l i c i j u r i s f e c i . " lo

Both Poggendorff ( 1 8 6 3 , p.159) and F l e c k e n s t e i n ( 1 9 7 0 ) l i s t an a r t i c l e ent i t l e d "Modus i n v e n i e n d i aequationem d i f f e r e n t i a l e m completam e x d a t a aequatione d i f f e r e n t i a l i incompleta", published i n SuppZ. AE 7 (1719) i n t h e i r b i b l i o g r a p h y of Nicolaus I B e r n o u l l i . F l e c k e n s t e i n even provided page numbers: 310-859.

However, t h e volume S u p p l . AE 7 i s d a t e d 1721 r a t h e r than

1719, and i t does n o t c o n t a i n any a r t i c l e by t h e hand of Nicolaus I Bern o u l l i ; n e i t h e r do t h e volumes SuppZ. AE 6 (1717) o r SuppI. AE 8 (1724). Thus i t emerges t h a t no such a r t i c l e as l i s t e d by Poggendorff and Fleckens t e i n d i d e v e r e x i s t ! S t i l l t h e S u p p l . AE 7 (1721) c o n t a i n s some of t h e s e c t i o n s of Nicolaus I1 B e r n o u l l i ' s survey a r t i c l e 1 7 2 0 , and indeed on pp. 310-312 one f i n d s a quote from t h e e x c e r p t of t h e l e t t e r NicIB:JohB 2/1/1718. The h i s t o r y l e a d i n g up t o t h i s annoying d i s o r d e r presumably r u n s a s f o l l o w s : when going through t h e Acta Eruditorum and i t s supplements Poggendorff encountered t h e quote from Nicolaus I B e r n o u l l i ' s l e t t e r t o Johann B e r n o u l l i and denoted i t by t h e

-

indeed a p p r o p r i a t e

-

t i t l e "Modus i n v e n i e n d i e t c " ,

w h i l e making a mistake with r e s p e c t t o t h e d a t e of t h e supplement volume 7 and w i t h o u t i n d i c a t i n g t h e p r e c i s e b i b l i o g r a p h i c a l environment of t h i s quote i n h i s f i l e s . When making up t h e f i n a l s e c t i o n on Nicolaus I B e r n o u l l i

Footnotes Chapter 4

187

- quite some time later? - he must have forgotten about all niceties and just presented the item as a genuine article by the hand of Nicolaus I Bernoulli. Fleckenstein presumably copied the Poggendorff entries and by some incomprehensible divination came up with the page numbers. An almost identical story goes for the article "Calculus pro invenienda

linea curva, quam describit projectile in medio resistente"AE1719, listed by Poggendorff as nr. 2 and by Fleckenstein as nr. 5. Again this title does not pertain to an independent article of Nicolaus I Bernoulli, but to a quote from the letter NicIB:JohB 1 / 1 / 1 7 1 8 in Johann Bernoulli's art cle "De curva quam projectile describit in medio resistente" AE 1719 pp ff l1

=

216

JBOO I1 pp. 3 9 3 - 4 0 2 .

cf. Demonstratio, lines 32-34. from: Tentamen, p. 307. "Voco autem aequationem differentialem Curvarum secandarum completam illam, quae exprimit relationem, quam habent inter se differentialia, non tantum coordinatarum Curvarum secandarum, sed & parametri variabilis, sive ejus lineae, quam C 1 . HERMANNUS modulwn appellat. A e q u a t i o n e m v e r o d i f f e r e n t i a lem incompZetam voco illam, quae exprimit tantum relationem, quae est inter differentialia coordinatarum unius ex Curvis secandis, parametro sive modulo manente constante. Aequationem differentialem completam Curvarum secandarum generaliter design0 hoc mod0 drc-pdyiqda, incompletam hoc mod0 d3:=pdy, in quibus aequationibus litterae

3: &

y denotant coordinatas

Curvarum secandarum, a parametrum variabilem, p & q quantitates datas quomodocunque per 13

2,

y, a,

& constantes

.

"

cf. for example Youshkevitch 1 9 5 6 , Enestrsm 1901.

14

cf. Demonstratio, lines 11-14.

15

cf. footnote 12.

16

cf. Bos 1974a, ch. 3.

17

cf. Demonstratio, lines 24-26.

18

cf. Demonstratio, lines 26-28.

19

cf. Demonstratio, line 4 0 .

20

from: NicIB:LE 6 / 4 / 1 7 4 3 . "Ego quidem hac in re nullius inventionis gloriam mihi tribuo, utpote qui proprietatem illam formularum differentialium, quae huic controversiae ansam dedit, non instar theorematis proposui, sed instar axiomatis supposui, quod ex sola notione differentialium, etiam sine inspectione figurae, cuivis manifestum esse putabam."

21

cf. Demonstratio, lines 29-34.

188

Footnotes chapter 4

22

c f . 53.11.2.

23

from: Tentamen, pp. 307-308. "Logarithmus q u a n t i t a t i s & parametrum

J(l+ppi:p

a variabilem,

d i f f e r e n t i e t u r , sumendo y constantem

& substituendo pro

dz

e j u s valorem qdu; p o s t e a

i n v e n t a d i f f e r e n t i a l i s i t e r u m i n t e g r e t u r , sumendo e t i a m y v a r i a h i l e m , & s u b s t i t u e n d o , ubi opus f u e r i t , valorem harum aequationum, dx+ppdx=qda, &

dy+pppdy-pqda, i n t e g r a l i s ( s i qua h a b e r i p o s s i t ) h a b e a t u r p r o Logarithmo, c u j u s numerus v o c e t u r n, p e r hanc q u a n t i t a t e m n d i v i d a t u r q u a n t i t a s q , & e x q u o t i e n t e , postquam p r o J: s u b s t i t u t u s f u e r i t e j u s v a l o r e x p r e s s u s i n

y,a,

& c o n s t a n t i b u s , e j i c i a n t u r omnes illi t e r m i n i , i n quos i n g r e d i t u r

r e s i d u i per

y,

da m u l t i p l i c a t i surnatur i n t e g r a l i s , quae p o n a t u r =A. Quo f a c t o , a b s c i s s a e s i n t =y, & o r d i n a t a e , quas vocabo z=(Z+pp):pn:

s i i n Curva, c u j u s

a b s c i n d a t u r a r e a =C-A, ubi p e r C i n t e l l i g o q u a n t i t a t e m a r b i t r a r i a m cons t a n t e m pro s i n g u l i s ejusdem T r a j e c t o r i a e p u n c t i s ; a r e a e h u j u s a b s c i s s a y d a b i t ordinatam T r a j e c t o r i a e q u a e s i t a e i n puncto i n t e r s e c t i o n i s e j u s & Curvae secandae. A l t e r a r e g u l a , quae Corollarium e s t p r a e c e d e n t i s , l o c u m tantum h a h e t i n

i i s Curvis s e c a n d i s t r a n s c e n d e n t i b u s , quarum a e q u a t i o d i f f e r e n t i a l i s i n completa dx=pdy i t a comparata e s t , u t q u a n t i t a s /(l+ppi:p m u l t i p l i c a t i o n e componatur e x duobus f a c t o r i b u s , quos nominabo s i t per

a

& constantes, h i c per

B

&

Y , quorum i l l e d a t u s

y & constantes. In h i s casibus construenda

e s t Curva, i n qua a b s c i s s i s e x i s t e n t i b u s 3 ,a p p l i c a t a e s i n t =(Z+pp):pR, & r e l i q u a peragenda u t p r i u s , cum hoc tamen d i s c r i m i n e , quod loco f r a c t i o -

n i s qda:n, quae s e r v i t p r o invenienda

q u a n t i t a t e A , h i c adhiberi debeat

f r a c t i o i s t a dE:B, ubi p e r E i n t e l l i g i t u r d i s t a n t i a v a r i a b i l i s p u n c t i i n t e r s e c t i o n i s Curvarum secandarum cum axe a puncto quopiam d a t o , e x quo nempe sumitur i n i t i u m abscissarum T r a j e c t o r i a e q u a e s i t a e ; d a t u r autem i l l a d i s t a n t i a E p e r a & c o n s t a n t e s : quare h i c a r e a abscindenda semper e r i t =C-/(dE':B),

adeoque c o n s t a n s , s i Curvae secandae omnes t r a n s e a n t p e r

idem a x i s punctum." 24

I n example I X i n t h e Tentamen t h e d e f i n i t i o n of (q/n)* even l e d

to a serious

mistake. I n t h i s example Nicolaus c o n s i d e r e d t h e f a m i l y of p a r a b o l a s des-

(q/n)=-4y4/u3+3u-u3/y2. Thus i n c r i b e d by t h e e q u a t i o n y2=ax+a', y i e l d i n g f t h i s c a s e (q/nl =3a, whereas (q/n)y=o does n o t e x i s t (nor does any o t h e r y e x i s t such t h a t ( q / n )

Y=Yo

=3a). This value o f (q/n)* y i e l d s Afa)=3a2/2 and

consequently t h e a r e a t o be c u t o f f i n t h e a u x i l i a r y

C-3a2/2. However, t h e a u x i l i a r y

curve i s equal t o

curve i n t h i s case i s d e s c r i b e d by t h e

e q u a t i o n z=(Z6y2-22u2y4+a6)/2uy3and thus t h e i n t e g r a l

Y

0

zdy i s obviously

189

Footnotes chapter 4

divergent. from: Ll?:NicIB 1 4 / 5 / 1 7 4 3 . "Plurimum autem T i b i , V i r Celeberrime, me o b s t r i c t u m agnosco p r o demons t r a t i o n e e l e g a n t i s s i m a e Tuae c o n s t r u c t i o n i s t r a j e c t o r i a r u m orthogonalium, quam i n A c t i s L i p s . 1719 p u b l i c a v e r a s . Equidem jam pridem i n i l l i u s dem o n s t r a t i o n e eruenda desudaveram, neque tamen alium casum e l i c u i , p r a e t e r eum, quo p / -

f i t functio

i p s i u s p a r a m e t r i a tantum. Quanquam enim

q u a e s i v i q u a n t i t a t e m n , p e r quam a e q u a t i o O = d y c l + p P )

P

+ qda d i v i s a i n t e -

g r a b i l i s r e d d a t u r , tamen i n mentem mihi non v e n i t , i n i n v e s t i g a t i o n e i p s i u s n ipsam aequationem propositam i n subsidium v o c a r i posse. Hac i g i t u r methodo Tua plurimae a e q u a t i o n e s d i f f e r e n t i a l e s e x p e d i t e i n t e g r a r i possunt." 26

In h i s l e t t e r LE:NiclB 4 / 2 / 1 7 4 4 E u l e r could e a s i l y ward o f f B e r n o u l l i ' s c r i t i c i s m by p o i n t i n g o u t t h a t t h e e q u a l i t y theorem g u a r a n t e e s t h a t

~ o x m I x , y ~ +d x

sy

nf0,yldy

i s an i n t e g r a l e x p r e s s i o n o f t h e d i f f e r e n t i a l

m d x + ndy. With t h i s answer Nicolaus seemed t o be s a t i s f i e d , a s becomes c l e a r from h i s l e t t e r Nic1B:LE 20/4/2745. The f a c t t h a t N i c o l a u s ' s o b j e c t i o n could e a s i l y be e l i m i n a t e d by E u l e r does n o t , of c o u r s e , d e t r a c t from t h e v a l i d i t y of t h e o b j e c t i o n . Nicolaus I B e r n o u l l i i s one o f t h e very few i n t h e 18th c e n t u r y t o a p p r e c i a t e s u c h pedantry. I am i n d e b t e d t o Hans Duistermaat who urged me t o c l a r i f y t h i s point.

190

FOOTNOTES CHAPTER 5

c f . a l s o Hofmann 1959. c f . Montucla 1 8 0 2 , v o l . 111, p. 344. A f o o t n o t e composed by L a c r o i x s t a t e s : "Le c i t . Cousin a rappelE, dans l a p r 6 f a c e de son Astronomic phy s ique , q u ' E u l e r a v o i t , dSs 1734, intEgrE complettement une Equation de ce genre

( M h de Petersb. tom. V I I ) , mais q u ' i l o u b l i a son nouveau c a l c u l , j u s qu"a ce que d'Alembert en e u t f a i t l e s premieres a p p l i c a t i o n s aux s c i e n c e s physico-mathEmatiques. La g l o i r e de ces a p p l i c a t i o n s d o i t demeurer t o u t e e n t i s r e 'a d'Alembert, mais I ' i n v e n t i o n du c a l c u l a p p a r t i e n t i n c o n t e s t a blement 1 E u l e r . "

'

c f . L . Euler De d i f f e r e n t i a t i o n e ,

51.

c f . L. Euler De d i f f e r e n t i a t i o n e , 1 2 . c f . L. Euler De d i f f e r e n t i a t i o n e , 93. c f . L. Euler De d i f f e r e n t i a t i o n e , 54, E u l e r f i r s t published t h e proof i n De i n f i n i t i s curvis, and he r e p e a t e d i t , f o r i n s t a n c e , i n h i s l e t t e r LE:ACC 19/10/1740 and i n h i s I C D 15226-227. cf. L

E u l e r De d i f f e r e n t i a t i o n e , 5 10.

The term "homogeneous of degree n" appears n e i t h e r i n De differentiatione nor i n De i n f i n i t i s c u m i s , I n s t e a d E u l e r c a l l e d such f u n c t i o n s " f u n c t i o n s of dimension n i n x and a". lo

The assumptions u n d e r l y i n g E u l e r ' s i m p l i c a t i o n "Q homogeneous of d e g r e e

n - I , then P homogeneous of degree n" can be found i n t h e f o l l o w i n g way: Let Q ( x , a ) be homogeneous of degree n-1 i n x and a; t h i s i s e q u i v a l e n t t o n-1 X s a y i n g t h a t f o r a l l t : Q ( t x , t a ) = t Q ( x , a l . Now d e f i n e P(x,a)= J Q f u , a ) d u . Then: P(tx, t a l = J

tx

XO

Q(u,ta)du; s u b s t i t u t e v=u/t, and hence tdv=du; u=tx im-

Hence P ( t z , t a l = t n P ( z , a ) i f and only i f xO=O. c f . L . Euler De d i f f e r e n t i a t i o n e , 113. l2

c f . L . Euler De d i f f e r e n t i a t i o n e , marginal remark (m3).

l 3

The Adversaria Mathematica f o r t h e y e a r s 1749-1753 a r e given t h e number H5

i n Enestrsm 1913 and nr. 401 i n Kopelevic 1 9 6 2 ; they have n o t y e t been

Footnotes Chapter 5

191

published. The calculation of the orthogonal trajectories is on p. 2 6 2 :

*=*

sJ,:eAA,

"Casus pro trajectoriis orthogonalibus: sit y= erit aequatio + & - "dx existente X=funct.x et A funct.ips.a. (XX-A-4) JXX-Al.1 k ergo subnorrna:is pro curvis see. = unde

modularis dy=

AdY

{(XX-AA)

'=J(XX-AA) l 4

A&

+ &=&- A A d x

XX-AA

+A&.-%

+

xxdx (XX-AA)

JIXX-AA

xxdx

I sj(x-AAI.y

+ dx ergo pro secantibus

aeq. diff. completam undeh(xX-AA) xxdx =eonst. ''

cf. L. Euler 1 7 7 0 , 1 7 7 3 , 1 7 8 7 . Dates of composition are provided by Enestrzm 1 9 1 0 , pp. 2 4 6 , 2 4 8 , 2 5 5 .

l5

from: Euler 1 7 8 7 , 538. "Ex his abunde perspicitur, quam profundae indaginis fuerit investigatio huius aequationis: siquidem ex natura functionum binarum variabilium est deducta, quae ill0 tempore prorsus adhuc erat ignorata. Primus autem, qui hoc praeclarissimum specimen taLis Analyseos in medium attulit, iam ante sexaginta annos, fuit acutissimus NICOLAUS BERNOULLI, NICOLAI filius, Professor Iuris in Universitate Basiliensi; cui ergo maxima incrementa, quae hinc deinceps in Analysin sunt inducta, potissimum accepta sunt referenda."

l6

cf. LE:DB 1 8 / 2 / 2 7 3 4 . "Wann unendlich vie1 Ellipses auf einem Axe conjug. bgesetzet werden, und daraus eine neue curva formirt wird, davon die Abscissae den Axibus transversis r gleichgenomen, die Applicatae aber den Peripheriis dieser Ellipsium u gleichgesetzet werden, so wird die Natur dieser Curvae durch nach-

dudr folgende Aequation exprimiert werden; ddu = -+r falls nicht anderst als indirecte gekommen."

udr2

r2-b2

worauf ich gleich-

The differential equation is wrong in that the coefficient (b2+r2)/(b2-r2) o f dudr is missing. Euler provided the correct equation ( 5 . 4 7 ) in

Solutio

problematwn, 5 1 7 ; cp. 55.5.1. l7

from: LE:DB November(?) 1 7 3 4 . "Seit der Zeit habe ich nachfolgende Problemata solvirt worGber ich gerne Deroselben nebst Dero Herren Vatters und anderer Mathematicorum Urtheil vernehmen m6chte. Das erste ist eine Curvam zu finden, welche von unendlich vie1 Ellipsibus, welche auf einem Axe transverso stehen, gleiche Arcus abschneidet. Ingleichem von unendlich vielen Ellipsibus, welche einen Verticem und gleiche Axes conjugatos haben gleiche Arcus abzuschneiden. Die Construction dieser curvarum ist per rectificationem ellipsium leicht,

Footnotes Chupter 5

192

ich verlange aber eine aequationem vor diese curvas, welche so beschaffen seyn wird, dass man daraus zu keiner Construction gelangen kann, ohne meine Methode, dadurch ich auch die Aequationem RICCATIanam construirt." from: DB:LE 1 8 / 1 2 / 1 7 3 4 . "Ew. problema de abscindendis arcubus aequalis in serie ellipsium etc. ist sehr profundurn und, wie ich glaube, schwer anders als a posteriori, methodo serierum auf Ihre Weise, zu solviren." l9

Euler had announced this method of series in his 1 7 3 3 , in which he provided a construction for the differential equation (4t) dy+y2dx/x=xdx/(x2-l) and in which he listed the separable cases of the Riccati equation uzndx=dy+y2dy. The arguments themselves were published in his 1738a and 1 7 3 8 b respectively. The basic idea of this method may be illuminated by a sketchof Euler's treatment of the former equation (Sl:The perimeter s of a quarter ellipse with 3-

semi-axes I and in+ll" can be represented by the integral i&t) s=/" J(1+t2'n2t2ht. Expanding the integrand of ( m ) as a power series in c ll+t2)% n , integrating this series term by term, and substituting n=-x2, a power

series for s emerges in terms of x. Manipulating this series (differentiating with respect to x, multiplying by x again etc.) Euler found the folwhich lowing second order differential equation: dds=-~ds/x+sdx2/ix2-2/, he could reduce to the first order equation (a!) Hence, 2u

(3)can

by suitable substitutions.

be constructed by means of the rectifications

(sttt).

from: LE:DB 2 / 6 / 1 7 3 5 . "Problema de abscindendis arcubus aequalibus in serie ellipsium a posteriori methodo mea serierum resolvere non potueram, neque arbitror ideo modo, quo aequationis Riccatianae constructionem dedi, solvi posse. Incidi vero in aliam methodum directam latissime patentem, cujus ope non solum multo plura in problemate trajectoriarum orthogonalium praestare possum, quam ad hisce factum est, sed etiam eandem, quam ante inveneram aequationis Riccatianae constructionem sine seriebus elicui. Eadem methodo aequationem pro curva ab infinitis ellipsibus arcus aequales abscindentem sum adeptus, quae est differentialis secundi gradus, et praeterea ita complicata ut eam ad differentialem primi gradus reducere nullo modo potuerim. Novae hujus methodi principia jam praeterito anno in Conventibus nostris praelegi et solutionem problematis hujus ellipsium paratam habeo, quae proxime praelegantur. Pluribus autem dissertationibus opus erit ad hanc materiam ut decet pertractandam."

Footnotes Chapter 5

''

193

In his 1 7 1 9 , 9 5 , Jakob Hermann had applied the term "modular equation" in connection with the differential equation da/a=(dz2+dy describes the relation between the coordinates

2

)/(x&+ydy)

which

and 9 of the orthogonal

trajectoriesandthe parameter a of a given family of similar curves. The term occurs in Nicolaus I1 Bernoulli's 1720, e.g. 454, as well, with exactly the same meaning. However, in Euler's sense this equation is neither a modular equation of the given family of curves nor of the family of orthogonal trajectories, since the coordinates

I%

and y and the parameter a res-

pectively pertain to different families of curves. 22

from L. Euler De in:inztis

curvis, 4 2 .

"Infinitae igitur curvae eiusdem generis omnes unica aequatione exprimuntur, quam modulus qui nobis semper litteraa indicabitur, ingreditur. Huic enim modulo, si successive alii atque alii valores tribuantur, aequatio continuo alias dabit curvas, quae omnes in una aequatione continentur. Aequationem hanc modulum continentem cum HERMANNO modularem vocabimus; in qua igitur praeter alias constantes et eiusdem valoris in omnibus curvis quantitates insunt modulus a et duae variabiles ad curvam quamlibet pertinentes, cuiusmodi sunt vel abscissa et applicata, vel abscissa et arcus curvae, vel area curvae et abscissa etc., prout problema solvendum postulat." 23

cf. L. Euler De infinitis curtris, 9 9 . Discussing whether da=Pdx+da/Bdx may be taken as the modular equation equivalent to z = / P d z , Euler stated: "L..]aeque inutilis est haec equatio ac prima z=/P&,

utraque enim invol-

vit integrationem differentialis, in qua a ut constans debet considerari, id quod est contra naturam aequationis modularis, quippe in qua a aeque variabile esse debet ac z et 24

2.''

cf. L. Euler De i n f i n i t i s curvis, 9 4 . "Ad construendas quidem et cognoscendas curvas aequatio dz=/Pdx sufficit.

[...I

Sed si in his curvis certa puncta debeant assignari prout problema

aliquod postulat, talis aequatio z=/i'dx

non sufficit sed requiritur

aequatio a signis summatoriis libera." 25

from: L. Euler Additarnentwn,

55.

"Deprehendi vero in superiore dissertatione Q toties algebraicum habere

x functio, ut numerus dimenx constituunt, sit ubique idem atque -I, seu quoties Px vel Pa fuerit functio ipsarum a et x nullius dimensionis. Deinde etiam valorem, quoties P talis fuerit ipsarum a et sionum, quas a et

Footnotes Chapter 5

194

observavi, quoties in P litterae u et z eundem tanturn ubique constituant dimensionurn numerum, toties Q ab integratione ipsius Pdcc pendere. Ex quo cum tam eximia consequantur subsidia ad aequationes modulares inveniendas, maxime iuvabit investigare, num forte aliae dentur huiusmodi functiones ipsius P, quae iisdem praerogativis gaudeant. Has igitur a priore investigare constitui, quo simul methodus tales functiones inveniendi aperiatur . ' I 26

from: L. Euler S o l u t i o problematm,

§1

"Agitata iam superiori seculo inter Geometras sunt huiusmodi problemata, in quibus linea curva requirebatur, quae ab infinitis positione datis curvis arcus aequales abscinderet. Communicaverunt etiam ill0 tempore C 1 . C 1 . Geometrae elegantes solutiones pro casu, quo curvae positione datae

inter se sunt similes, uti cum ab infinitis circulis vel parabolis arcus aequales abscindendi essent. Nemo autem, quantum constat, ulterius est progressus neque quisquam pro curvis dissimilibus problemati satisfecit, etiamsi iam tum quaestio de infinitis ellipsibus proponeretur. Atque etiamnum, cum Insigni Geometrae per litteras significassern me aequationem pro curva, quae ab infinitis ellipsibus dissimilibus arcus aequales abscinderet, invenisse, ille mihi respondit huius problematis solutionem in sua non esse potestate meque simul rogavit, ut meam solutionem in non conternnendum analyseos augmentum communicarem." This preference of rectifications over quadratures has been discussed by Bos in his 1974b. Rectifications weremore acceptable intuitively, because it seemed

to be easier to measure the length of a curve than to measure

an area under a curve.

**

from: L. Euler S o l u t i o problematwn,

56.

"Quod enim ad curvam, quae ab infinitis ellipsibus arcus aequales abscindat, attinet, eius constructio eo ips0 est facilis, quod ope rectificationis curvarum, quae facillime describi hanc

possunt, perfici queat. Atque

ipsam constructionern longe anteferendam esse censeo aliis per qua-

draturas curvarum peractis constructionibus. Non igitur tam illius curvae constructio requiritur quam eius aequatio, quo, quales aequationes tam facile construi queant, cognoscatur. Hanc ob rem analysis non parum augmenti accipiet, si illae aequationes proferantur, quae ope rectificationis ellipsium constructionem admittunt." 29

from: L. Euler De construetione, § I , "QGoties in resolutione problematurn ad aequationes differentiales pervenitur, ante omnia inquirendum est, an istae aequationes integrationem ad-

Footnotes chapter 5

195

mittant; perfectissime enim problema resolvi censendum est, quod ad constructionem aequationis algebraicae deducitur. At si aequatio, quod saepissime evenit, in formam algebraicam nullo mod0 transmutari potest, tum vel quadraturis vel rectificationibus curvarum, quarum constructio habetur, ad problemata resolvenda uti oportet. Ad hoc vero efficiendum necesse est, ut aequatio solutionem problematis continens et primi tantum gradus sit differentialis et praeterea separationem variabilium admittat, si quidem regulis receptis atque iam satis cognitis uti velimus. Hoc enim istae regulae laborant

defectu, ut earum ope neque aequationes differentiales

altiorum graduum, neque differentiales primi gradus, quarum separatio non constat, construi queant. Hanc ob rem nisi aequatio ad differentialem primi gradus reduci, simulque separatio variabilium detegi potest, frustra per illas regulas constructio

aequationis investigatur."

30

cf. footnote 1 9 .

31

from: L. Euler De construct7~one, 52. "Dedi autem ego iam aliquoties specimina methodi cuiusdam peculiaris multo latius patentis, cuius ope non solum plures aequationes differentiales separationem variabilium non admittentes construxi, sed etiam aequationes differentiales secundi gradus, quae nequidem ad differentiales primi gradus reduci poterant. Initio quidem seriebus infinitis, in quas aequationes propositas transmutaveram, sum u s u s , earumque summas ad quadraturas reduxi. Tum vero hanc viam non satis genuinam iudicans, in methodum directam inquisivi, qua ad easdem constructiones pertingere possem. In quo etiam negotio operam non inutiliter collocavi; incidi enim in methodum aequationes modulares eruendi, quarum ope ad constructiones difficillimarum aequationum via paratur. Methodum quidem hanc fusius iam exposui, sed illius usum eximium in construendis aequationibus ill0 tempore monstrare non vacabat. Interim tamen nuperrime dedi specimen illarum aequationum, quae ope rectificationis ellipsis construi possunt. Nunc vero, quo u s u s huius methodi plenius perspiciatur, casus nonnullos pervolvam speciales, ex quibus plurimarum aequationum constructiones consequantur. Principia autem ex dissertatione de infinitis curvis eiusdem generis, quam praecedente anno praelegi,petam."

32

from: L. Euler De construetione, 53. "Huiusmodi ergo aequatio, quam cum HERMANNO modularem vocavi, tres continebit variabiles z, J: et a, quae autem in aequationem duarum variabilium abibit, si vel ipsi z vel J: determinatus vel ab a pendens valor tribuatur. Talis vero aequatio quamcunque habuerit formam, et cuiuscunque sit gradus differentialis, semper opeaequationis z=/P&

construi poterit. Nam

Footnotes chapter 5

196

si pro dato quoque ipsius a valore J P d x exhiheatur, quod per quadraturas fieri potest, et z vel x illi valori assignato aequale capiatur, determinabitur altera ipsarum z vel x per a, eiusque ideo quantitas innotescit. Quocirca hac ratione pro dato alterius indeterminatae valore alterius quantitas poterit reperiri, in quo ipsa aequationis cuiusvis constructio consistit .I1 33

The title is provided by Kopelevic 1962, p. 4 2 ; the Index c a p i t m has np.

92 in Kopelevic's catalogue and appears as item H26 in Enestr6m 1 9 1 3 . It has been published by Speiser in the preface to LEO0 1 . 2 9 , pp. XXXI-XXXVIII; since Speiser mixed up the photographs of the manuscript he regarded

it to

consist of three fragments. However, if the fragment on p. XXXVI be placed at the end, then there appears to exist no gap any longer between book 1.2.8

and the end, book 6 . 2 . 4 .

This reconstruction is confirmed by inspec-

tion of the photographs kept at the Euler Archives in Basel; moreover, it is

confirmed by Kopelevic, Zoc.cit., p . 4 3 . The index capitwn is undated; however, 1740 may reasonably be taken to be the year of composition for the following reasons: A terminus post quem is provided by the fact that Euler obviously knew about total differential equations and the conditions of integrability of such equations when he composed the Index c a p i t m . A s is clear from his letter LE:ACC 19/10/1740 a s well as from a remark in his Adversaria Mathernatica (cf. footnote 3 8 ) ,

Euler had learned about these matters from Clairaut's article 1 7 4 1 , which had been conveyed to Euler in manuscript by Daniel Bernoulli (cf. Taton

1976, p. 1 1 9 ; LEO0 4 A . 5 , p. 7 0 footnote 3 ) in January 1 7 4 0 . Hence the Index capitwn must have been written after January 1 7 4 0 . A terminus ante quem can be established in the following way: Book 5 . 2 of the Index capitwn was planned to contain variational calculus. The list of chapters, however, is meagre, and at the end of this list Euler stated: "Ubi notandum est singula hujus libri capita plures requirere subdivisiones, quae autem in ipsa tractatione se manifestabunt." ("It should be noted that the single chapters of this book require many subdivisions, which, however, will make themselves clear during writing"). From this remark I infer that Euler had not yet started writing his textbook

Methodus inveniendi Zineas curvas ( 1 7 4 4 ) concerning the variational calculus. The book appeared in 1 7 4 5 , but ina letter to Maupertuis (unpublished; cf. LEU0 4 A . 1 , R 1 5 0 4 ) Euler seems to have stated that he had completed that book

before he left Petersburg. Since Euler left St. Petersburg on June 19th, 1 7 4 1 , and allowing for at least six months for writing this book, the Index

197

Footnotes chapter 5 c a p i t m must have been written before 1 7 4 1 . 34

from: L. Euler Index capitwn. "LIBER SECUNDUSI De integratione formularum differentialium unicam variabilem involventium

[...I

LIBER TERTIUSI De aequationibus differentialibus

duas variabiles continentibus [ . . . ]

LIBER QUARTUS/ De aequationibus dif-

ferentialibus tres variabiles involventibus

[...I

LIBER QUINTUS/ Continens

aliquot methodos peculiares et ad plurima eximia problemata solvenda accomodatas [ . . . ] LIBER SEXTUSI De usu praecedentium tum in doctrina serierum, tum in naturis curvarum ex generatione earum indagandis." 35

from: L. Euler Index capiturn. "Haec ergo sunt capita, quibus tota Analysis infinitorurn eo, ad quod hoc tempore evecta est, perfectionis fastigio includi posse mihi videtur. Necque enim quicquam adhuc reperire potui, quod non ad quodpiam horum capitum referri posset. Quocirca opus foret summae utilitatis, si quis hanc matheseos praecipuam partem ordine hic praescripto perspicue explicaret, quo tandem tot et tam ingentia impedimenta, quae plurimos ad interiora Matheseos penetrare volentes absterruerunt et repulerunt, ad eximium reliquarum etiam scientiarum incrementum tollantur."

36

Euler used the Leibnizian concept of function here. A function in this sense is a variable - or a combination of variables

-

defined in points of

a curve. 37

from: L. Euler Index c a p i t m . "LIBER QUARTUS/ De aequationibus differentialibus tres variabiles involventibusl S e c t i o primal De inventione et resolutione aequationum tres variabiles involventiuml Capita! 1 . De formulis differentialibus tres variabiles involventibus. In quo capite discrimen investigatur inter formulas determinatas seu eas, quae integrationem admittunbet indeterminatas.1 2. De aequationibus differentialibus tres variabiles involventibus. Ubi iterum ostenditur alias harum aequationum esse determinatas, alias vero indeterminatas, quae integrationem sine facta hypothesi non admittant.1 3 . De aequationibus modularibus earumque inventione. In quo capite modus ostenditur ex aequatione duas variabiles continente aliam inveniendi, in qua praeter illas duas variabiles nova insit variabilis, quae ante constans erat posita.1 4 . De usu aequationum modularium in constructione aequationum differentialium altiorum graduum. In hoc capite ope aequationum modularium plurimarum aequationum differentialium tam primi quam altiorum graduum constructiones exhibentur, quae priores methodos superant.1 S e c t i o secundal De traiectoriis cuiusque generisl

Footnotes C%apter 5

198

C a p i t a l 1. De traiectoriis, quae ab innumeris curvis positione datis vel areas vel arcus vel alias functiones aequales abscindant. In quo capite methodus aperitur pro innumeris curvis datis aequationes modulares inveniendi earumque ope huiusmodi problemata resolvendi./ 2. De curvis traiectoriis, quae curvas innumerabiles datas ad datos secant angulos. In quo capite praeter aequationes p r o istis traiectoriis etiam earum constructio investigatur./ 3 . De solutione plurium problematum reciprocarum.

In quo vel ex traiectoriis curvae secandae quaeruntur vel casus investigantur, quibus curvae traiectoriae ad secandas datam teneant relationem." 38

On page 245 of his Adversaria Mzthematica for the period 1740-1744 (Kopelevic 1 9 6 2 , nr. 400; Enestram 1913, H4; unpublished), Euler investigated the problem: "Probl. Invenire utrum aequatio tres variabiles involvens Pdx+&dy+Kdz=O sit realis seu an integrationem admittat, an secus". dQ dR dR dP dp dQiTO He derived the integrability condition P(----)+Q(~-;~)+R(-J-d z dy z y & and finally made the following remark: "Quae aequatio nisi habeat locum, aequatio proposita non habebit integrale. Hocque est criterium a Viro Clar. Clairaut inventum mecumque comm mica t urn. ' I ("Prob. To find out whether the equation F&+&dy+Rdz=O involving three variables is real or admits integration, o r not.

[...I

Unless this equation holds the proposed equation will not have an integral. This is the criterion found by the very clever gentleman Clairaut and communicated to me").

199

APPENDIX 1 NICOLAUS I BERNOULLI'S "DEMONSTRATIO ANALYTICA CONSTRUCTIONIS CURVARUM, QUAE ALIAS POSITIONE DATAS AD ANGULOS RECTOS SECANT, TRADITAE IN ACTIS LIPS. 1719 PAG 295 ET SEQQ."

Introduction The autograph of this treatise is kept at the bffentliche Bibliothek der Universitat Base1 as item LIa48, 5.24-25. I am indebted to Dr. M. Steinmann, curator of the manuscript department of this library for his permission to publish this text. The manuscript is written in a fine, very clear hand which provides no difficulties in transcription. Abbreviations for "ae" and "mm" have been written out in full without further acknowledgement. The autograph is undated; from internal evidence, discussed in chapter 4 , its date of composition may be taken as 1718 or early 1719. In his letter NicIB:LE 6 / 4 / 1 7 4 3 Nicolaus I Bernoulli provided Euler with an offscript of the Demonstratio. This letter has been published by P.H. F u s s in his 1 8 4 3 b , pp. 701-707.

The text of the autograph and the text published

by F u s s differ only in a few minor details, such as punctuation and the use of the symbols & and 6 respectively to denote the partial differential d

I have reproduced the Demonstratio here in order to make this most

Y'

important source for Nicolaus I Bernoulli's partial differential calculus more easily accessible to the reader of this book. Line numbers have been added for reference.

Appendix 1

200

Text Demonstratio analytica constructionis Curvarum, quae alias positione datas ad angulos rectos secant, traditae in Actis Lips. 1719 pag 295 et seqq. 1 Sint Curvae secandae DEF,

GHI, Curva

has ad angulos rectos secans HF, ipsarumque coordinatae communes AB,

AC=y, BE vel BH, aut CF vel CI=x,

5 sitque aequatio Curvarum secandarum generalis &=pdyy+qda, ubi u significat parametrum variabilem, sive lineam ex cujus mutatione mutatur Curva secanda, p vero et q sunt 10 quantitates datae per

et con-

x,y,a

stantes. Sit porro cfnota differentialium quando a constans ponitur, et 6 nota differentialium quando y

constans ponitur. Quia Curva H F 15 secat Curvas DEF, GHI, ad angulos rectos, subtangens Curvarum DE’F,

CHI, eadem est ac subnormalis Cur-rcdx 2 vae secantis HF, id est, - = - sive dy=-pdx, quae est aequatio generalis P dY Curvarum HF, in qua si pro dx substituatur ejus valor p d y i q d a orietur 20 dy=-ppdy-pqdu,

vel

2 = *.

Eadem aequatio etiam sic invenitur: quia

triangulum EFH es t rectangulum erit HE’=EF’+HF’,

EF’=&’+dy

’=ppdy i d y

’

’, HF2=&’ ’

sed HE2=6x2=qqda2,

’ ’ ’, hinc ’, subtrahendo q q d a ’ et pos tea per

+dy ’=ppdy +2pqdyda+qqda i d y

qqda =ppdy i d y i p p d y +2pqdydu+qqdu +dy

2dy dividend0 orietur U=dy+ppdy+pqda, ut antea. Quia vero quantitas q in 25 Curvis secandis transcendentibus non data est, tentari debet ejus eliminatio per sequentem considerationem, in qua valor lineolae I F duplici mod0 invenitur. Nimirum IF=HE+&E=Gx+d$x,

sed est quoque I F = G C F = G B E i G ~ E = 6 x + 6 ~ ,

hinc ablato utrinque 6x, erit &x=6&,

& d u = 6 p d y , hinc 30 draturas, si

&=%,

id est (quia Gx=qda, et &=pdy)

cujus integralis haberi potest, saltem per qua-

x non ingrediatur quantitatem 6 ~ debet ; autem in integratione

addi talis quantitas ex a et aliis constantibus composita, ut in casu y=O,

HE sive qda evadat GD=GAD; datur autem recta AD ob datam positionem Curva33 rum secandarum in a et constantibus, quae si ponatur 3, erit in casu y=O,

201

Appendix 1

dE

34 q = -

da

.

Si modo inventa aequatio differentialis

GY

&=‘g

comparetur cum

= - ~4 reperietur SZ= , du l+pp ’ 4 I+PP quae aequatio inserviet ad inveniendam Curvam LMN pro qualibet Curva secan-

35 aequatione Curvae HF supra inventa

da G H I , ut abscindendo aream datae magnitudinis ALM3, ordinata M 3 producta secet Curvam G H I in puncto aliquo if t r a j e c t o r i a e q u a e s i t a e H F . Sit ordinata Curvae construendae B M z z , respondens abscissae AB-y. Appelletur Area ALMB=S, 40 sitque generaliter dS=zdy+uda, eritque ut supra dFx=6&,

ita hic d F S = 6 6 ,

id est, duda=bzdy; quia vero 6S=zidu, et in casu y=O, omnes areae ALMB evanescunt, evanescet quoque 6S, adeoque in casu y=O, erit u=O. Ponatur

z=.3PP , et area abscindenda ALMB=C-A, Pn

ubi C significet quantitatem con-

stantem, et A quantitatem inveniendam compositam ex a et constantibus,

a

u+b

45 sitque dA=bda; et erit &=zdy+uda=dC-dA=-bd, sive = -= hinc da -2 l+pp -___ et quia in casu z = u + b 1 +pp = I + P P , e t u + b = q , e t ~ u =

.

w;

n nn y = O est u = O , erit in hoc casu b ~4 = (si m ponatur = n in casu y = O ) dE . Sed supra inventa est aequatio duda = 6zdy sive mda , et bda = dA = m ?4

?n

* 50

Vfi-7= 6Log: 2 quod est illud ipsum, quod praecipit n I+PP P P 5 2 constructio tradita in Actis Lips. loco citato.

-dn _ -*-@

202

Appendix 1

T m n sl u t i o n Analytical

demonstration of t h e c o n s t r u c t i o n of c u r v e s , t h a t i n t e r s e c t

o t h e r c u r v e s , which a r e given by p o s i t i o n , a t r i g h t a n g l e s , t r e a t e d i n t h e Acta of L e i p z i g 1719 page 295 f f .

L e t the curves which are t o be

i n t e r s e c t e d be DEF, GHI, and t h e curve which i n t e r s e c t s them a t r i g h t a n g l e s be HF; l e t t h e common coordin a t e s A B , AC be y , BE o r B H , o r CF o r C I be

3:,

and l e t t h e g e n e r a l

e q u a t i o n of t h e c u r v e s which a r e t o be i n t e r s e c t e d be where

Q

dx = p d y + qda,

s i g n i f i e s t h e v a r i a b l e para-

meter, o r t h e linesegment, by t h e v a r i a t i o n of which t h e i n t e r s e c t e d curve v a r i e s ; p and q a r e q u a n t i t i e s given i n

x, y , a and c o n s t a n t s .

Furthermore, l e t & b e t h e symbol of t h e d i f f e r e n t i a l s , when a i s taken c o n s t a n t , and 6 be t h e symbol f o r t h e d i f f e r e n t i a l s when y i s taken c o n s t a n t . Because t h e curve HF i n t e r s e c t s t h e curves DEF, G H I a t r i g h t a n g l e s t h e subtangent of t h e c u r v e s DEF, GHI and t h e subnormal of t h e i n t e r 3:

s e c t i n g curve HF a r e i d e n t i c a l , i . e . - =

P

-xdcc -

dY '

o r dy = -p&.

g e n e r a l e q u a t i o n f o r t h e c u r v e s H F , from which d y = -ppdy

$=

emerges by s u b s t i t u t i n g f o r

dcc

i t s v a l u e pdy

This i s t h e

- pqda,

or

+ q d a . The same

e q u a t i o n can a l s o be found i n t h e f o l l o w i n g way: s i n c e t h e t r i a n g l e EFH

i s r e c t a n g u l a r , one w i l l have HE2 = E F 2 + HF';

now HE2

= 6x2 = qqda',

= cE2 + d y 2 = ppdy' + d y ' , H F 2 = dcc' + dy' = ppdy' + Zpqdyda + qqda' + + d y 2 , and t h u s qqda2 = ppdy' + dy' + p p d y z + Zpqdyda + qqda' + d y ' ; s u b t r a c t i n g qqda' and d i v i d i n g by Zdy a f t e r w a r d s y i e l d s 0 = dg + p p d y + pqda,

EF'

a s b e f o r e . Since f o r t r a n s c e n d e n t a l curves t h e q u a n t i t y q i s n o t g i v e n , one has t o attempt i t s e l i m i n a t i o n by means of t h e f o l l o w i n g c o n s i d e r a t i o n , i n which t h e v a l u e of t h e l i t t l e linesegment IF i s found i n two ways. Namely, I F = H E + &E

away 6x

= 6x + ddx, but a l s o I F = 6CF = 6BE

i n both c a s e s y i e l d s a x = 6&,

i.e.

f

6&E

= 63:

i6&;

taking

(because 6x = qda and & = p d y )

203

Appendix 1

&da = 6 p d y , and t h u s % =

9,

t h e i n t e g r a l of which can be found, a t

l e a s t i n terms o f q u a d r a t u r e s , i f

z does n o t e n t e r i n t o t h e q u a n t i t y 6 p .

In t h e i n t e g r a t i o n , i t i s n e c e s s s a r y t o add such a q u a n t i t y composed from

u and o t h e r c o n s t a n t s , t h a t f o r y = 0 HE o r qda change i n t o GD = S A D . Because t h e curves t h a t a r e t o be i n t e r s e c t e d a r e given by p o s i t i o n , t h e linesegment AD i s given i n terms of a and c o n s t a n t s ; c a l l i n g i t E, i t y i e l d s

= dE f o r y = 0. I f t h e d i f f e r e n t i a l e q u a t i o n c?q = 6 d i s compared w i t h t h e e q u a t i o n d?d = _7Ep of t h e curve HF found above, one w i l l f i n d l+pp da 2% = B, which e q u a t i o n s e r v e s t o f i n d a curve LMU f o r each of t h e (I

9

I+PP

i n t e r s e c t e d c u r v e s G H I , such t h a t i f an a r e a of given magnitude A L W i s c u t o f f underneath i t , then t h e prolonged o r d i n a t e MB w i l l i n t e r s e c t t h e c u r v e G H I i n a p o i n t H of the r e q u i r e d t r a j e c t o r y curve which we have t o c o n s t r u c t be BM =

HF. Let t h e o r d i n a t e of t h i s

z, corresponding t o t h e a b s c i s s a

AB = y . The a r e a ALME be c a l l e d S; l e t i n g e n e r a l dS = zdy + uda, t h e n

&x = 6&,

one has h e r e : &S

= 663, i . e . dztda = 6 z d y . Since 6S = uda, and moreover s i n c e i n t h e c a s e y = 0 t h e analogously t o t h e above

e n t i r e a r e a ALME v a n i s h e s , a l s o 6s v a n i s h e s , and t h u s f o r y = 0 one h a s

u = 0. P u t z =

-,

and t h e a r e a ALMB t o b e c u t o f f = C-A,

Pn

where C

s i g n i f i e s a c o n s t a n t q u a n t i t y , and A a q u a n t i t y which i s t o b e found and which i s composed from a and c o n s t a n t s ; l e t o r 5& =

dS = zdy +uda = dC-dA = -Ma,

z z (u+h)

’

(lipP)

Pn

P9

u i s e q u a l t o 0,

dE = n ) -, mda

z

o r u+b =

da

-z

and

dA = bda; one h a s = and t h u s l+pp

= ne-qdiz nn

.

Because f o r y = 0,

= ( i f i n c a s e y = 0 m be Put dA = dE Above t h e e q u a t i o n duda = 6zdy has been found, m

one h a s i n t h i s case b =

and bda =

-.

-h- n

which e q u a t i o n because of

hz

dn

dn + Sn y i e l d s - =

1

*I+PP

, L%! = 6 L o g :

P

JI+ P

This i s t h e very e q u a t i o n which s u p p l i e d t h e c o n s t r u c t i o n given i n t h e Acta of L e i p z i g a t t h e c i t e d p l a c e .

204 APPENDIX 2 LEONHARD EULER'S "DE DIFFERENTIATIONE FUNCTIONUM DUAS PLURESVE VARIABILES QUANTITATES INVOLVENTIUM"

I n tr o duct i on The autograph of this text is kept at the Archives of the Soviet Academy

P.236,op.I,nr.97,pp.2-9 (cf. Kopelevic et al. 2962 nr. 86, and EnestrEm 1913 H 2 0 ) . My edition is based on the set of photographs of Sciences as item

available at the archive of the Euler-Kommission of the Schweizer Naturforschende Gesellschaft in Basel. I am indebted to Dr. E.A. Fellmann of the Euler-Kommission for his kind permission to publish the text here. The manuscript De differentiatione consists of 15 written pages, bearing the pagenumbers 2-16, taken

KO

of a size approximately 2lcmxlbcm; page 17, usually

be part of the manuscript contains some unconnected calculations,

which have no relation with the argument on the preceding 15 pages. The handwriting is very legible, and apart from a few exceptions provides no difficulty of transcription. At those scant instances where a word is partly unreadable, the context provides sufficient information for reconstruction. Such reconstructions have been put between brackets

[ I ; when uncertain, a

question mark is added. Euler's abbreviations for the endings 'I-que", "-tur", "-um" and for the diphthong "ae" have been written out in full without further acknowledgement. References in the margin show that Euler intended to add four figures to his text, which, however, are missing. I have supplied them myself Apart from obvious additions or corrections to the main text, the margins contain five calculations or comments which were presumably not in ended to be put into the text. They have been edited separately at the end of the main text, and their location in the manuscript is indicated by page number and number of the section next to which they are scribbled. The page numbers of the original have been put between brackets

[ I into the main text.

The manuscript must have been written in the early 1730s (cf. ch. 5).

205

Appendiv 2

Text

[2 1 De Differentiatione Functionum duas pluresve variabiles quantitates involventium. §.I.

Ad solutionem problematum, quae infinitas curvas eiusdem generis conside-

rant, cujusmodi varia de trajectoriis orthogonalibus, de secanda curvarum serie juxta datam legem, proposita et soluta inveniuntur; definiri debet differentiale quantitatis ex duabus variabilibus compositae cognito differentiali ejusdem quantitatis sed altera variabili pro constante habita. Ut si P sit functio ex a et z composita, et Qdz ejus differentiale, si a pro constante habeatur; ex his datis quaeritur differentiale ipsius P si et a ut variabilis consideretur. Hoc quidem differentiale formam habebit hujusmodi &dz+Rda; at quomodo R ex P et Q inveniri possit, ostendi oportet. Si autem ipsa quantitas P aliter non sit data nisi per J&dx,ita ut praeter Q nihil aliud constet, difficile admodum videtur ex s o l o Q determinare R . i i . 2 . Quod autem ad duas variabiles, ex quibus P et Q consistit, attinet, totum negotium sequente conficietur principio. Si quantitas ex duabus variabilibus composita bis differentietur, altera variabilium in prima differentiatione altera in secunda tanquam constante tractata, id quod duobus modis fieri [ 3 ] potest: erunt, quae in utroque casu proveniunt inter se aequalia. Ut proposita sit haec quantitas z3y-y2z2,differentietur posita

3:

constante prodibit

z3dy-2x2ydy: hoc denuo differentiatum posita y constante, dabit 3z2dxdy-4ccy&dy.

Tam eadem quantitas iterum bis differentietur sed ordine in-

verso, ut primo y et postea z pro constante habeatur. Proveniet ergo prima differentiatione 3z2y&-2y2z&,

atque secunda 3 z 2 & d y - 4 ~ d x d y id quod prorsus

congruit cum priore.

5.3. Qui in pluribus exemplis has operationes instituere voluerit, non solum hanc convenientiam perpetuo observabit, sed etiam perspiciet hujus principii veritatem in omnibus casibus necessario locum habere debere. In eorum tamen gratiam, qui hanc necessitatem minus perspicere valent, sequens demonstratio geometrica afferri potest. Sit curva quaecunque AM ad axem AP relata, in qua posita abscissa AP=x sit applicata PM=P, quae quantitas P erit functio quaecunque ex z et parametro curvaeaaliisque constantibus constans. Augeatur parameter a differentiali &, et habebitur curva AN proxima ipsi AM. Iam cum applicata PM=P instituatur haec duplex operatio: differentietur scilicet P posita a constante, et prodibit rm elementum applicatae in curva AM. In expressione hujus elementi rm si

Appendix 2

206 [fig. I ]

a abeat in a+da, habebitur ejus loco elementum ns . Quare ipsius rm differentiale posita z constante et a variabili erit ns-mr. Instituatur nunc similis duplex ipsius P differentiatio constantibus inverso or-[4]-dine accipiendis: erit ergo MN differentiale ipsius PM posita a variabili at z constante. Hoc elementum MN porro, si in ejus expressione ponatur z+dx loco z,abibit in mn. Quamobrem ipsius MN differentiale posita

3:

variabili et a constante erit

mn-MN. Prima igitur operatione resultat n s - m , altera vero mn-MN. Quia autem est ns=nr-sr et rnr-nr-mn erit ns-rnr-m-sr-m-MN.

Consequenter utraque opera-

tione idem productum datur ex hocque principii veritas cognoscitur. 5 . 4 . Quia autem haec demonstratio ex alieno fonte est petita, aliam ex ipsius differentiationis natura derivabo. Sit igitur ut ante P quantitas ex a, et z, et constantibus composita; abeat ea in Q posito z+dz loco z. At si a+& pro a scribatur mutetur P in R. Sin autem et z i d z loco z et a+& loco a ponatur prodeat S loco P. Ex his perspicuum est si in & loco a ponatur aida provenire S, similique mod0 si in R ponatur x+dx loco x haberi quoque S. Iam differentiata P posita a constante prodibit &-P. Porro posita a variabili et z constante, quia Q mutatur in S et P in R, erit ipsius &-P differentiale S-&-R+P. Idem autem proveniet si in prima differentiatione z in altera vero a ponatur constans; habebitur enim ex prima R-P et ex altera S-R-QiP. Cujus cum ill0 convenientia clarissime perspicitur. [51 1.5.

Quo autem hoc principium ad nostrum usum accomodemus. Sit P functio ipsa-

rum z et y et constantium, ejusque differentiale =Q&+Rdy,

in quo Q et R quo-

que erunt functiones ipsarum z et y. Ponatur propterea d&=Kdz+Ldy et

dR=Mdx+Ndy. Atque ope nostri principii relationem quandam inter has quantitates K, L , M et N poterimus invenire. Nam differentietur primo P posita y constante prodibit &&.

Hujus autem differentiale posita z constante erit

Ldxdy. Deinde differentiato P posita

3:

constante habebitur R d y , cujus differen-

201

Appendix 2 tiale posita y constante erit Mdxdy. Quare cum vi principii debeat esse

Ldxdy=Mdxdy eri t M=L.

5.6. Plures determinationes inveniemus, si proprietas quaedam functionis P fuerit nota. Ponamus P esse functionem ipsarum x et y nullius dimensionis, cujus modi sunt e.g. 3:

X ,

( ~ + & ~ ~ ~ - dico ) / y ; fore Qx+Ry=O. Nam quia ipsarum Y et y dimensiones conjunctim se destruunt in P, provenietur posito y = z z , mera

functio ipsius z . Quare in differentiali ipsius P praeter dz aliud differentiale non reperietur. Substituto igitur in differentiali Q d z i R d y loco d y valor zdx+zdz habebitur Qdz+RzdxiRxdz. Sed quia & non [ 6 1 adesse potest erit Qydx- Q ~ d y Q+Rz=O seu Qx+Ry=O. Quocirca erit R=* et d P = Y Y I) g . 7 . Simili mod0 si fuerit P functio ipsarum x et y dimensionum n , erit -

Yn

functio nullius dimensionis. Quare ejus differentiale eam habebit proprirtatem loco dx et y loco d y ponatur, proveniat nihilo aequalis quantiP YdP-nPdY seu Q y d x i R y d y - n P d y tas. Est vero ipsius - differentiale Yn+l Yn yn+l nP - Qx Quamobrem fieri debet Q y ~ + ~ y ~ - n P y = seu O Qx+Ry=nP. Ex quo erit R=ita ut Qydx-Qxdy+nPdy Y futurum sit dP= ut, si in eo

3:

.a 5.8. Ex his ulterius ad determinationes functionum K , L , M y N progredi licebit.

Si enim P est functio dimensionum n erunt Q et R functiones dimensionum n-I. Quare cum sit dQ=Kdx+Ldy erit K x i L y = ( n - l ) Q , similique mod0 Mx+Ny=(n-1)R. nP- Qx Quia vero est R=erit Y nydP-yzdQ-Qydz-nPdy+Qxdy - (n-l)Qydx+(n2-nIPdy-Z (n-l)Qxdy-Kyxdx+Kx2dy &= Y2 Y2 (n-I)Q-Kx et N= (n2-n)P-2(n-1)Qx+Kx2 Hocque ipso fit L=M. [ 7 1 Consequenter M=

.

g.9.

Y Y2 Praetermittere hic non possum, quin bre[viter?l observem formulam

nP=Qx+Ry ad integrandas formulas differentiales duas variabiles involventes utilitatem habere posse. Nam si differentiale Qdx+Rdy ita fuerit comparatum, ut Q et R sint functiones ipsarum

3:

et y ejusdem dimensionis puta n-I fore

.

ejus integrale *='

Id quod semper verum est, quoties Qdx+Rdy habet inten grale. Deipde si fuerit n=O, integrale provenit infinite magnum, nihilque ex inde cognoscitur. His enim casibus integralia [aeque?] a logarithmis pendent, qui hoc mod0 exhiberi non possunt. Quoties ergo integratio succedit et n non

est =O, integrale semper algebraicum reperitur. Q=lt%et R= Quare integrale erit G25'J---2xL+y

.

+

ax2

3.10.

i ay2

=x

+

u

m

Ut in

dx+axh+aydy est n=I , J-2-2x +Y

.

Sit nunc P functio quaecunque ipsarum x et y , quam autem aliter non novi

nisi quod ejus differentiale sit &&,

si y tanquam constans consideretur. Ex

Appendix 2

208

hoc vero inveniri oporteat differentiale ipsius P si ex

[!I

x et y ut variabiles

tractentur. Sit hoc differentiale quaesitum Qdx+Rdy in quo, quod sit R definiendum est. Est autem dR=Mdx+Ndy, seu R=lMdx si y pro constante habeatur. At

dQ cum sit M=L [ 8 ] et L=-,

si in differentiatione ipsius & x pro constante ha-

" et d P = & d x + d yd& l-dx.

beatur; erit H=J-&*dQ

dY

Ex quo haec nascitur regula:

dx

dY

Differentietur Q posita x constante, hocque ductum in - iterum integretur y

dY

ut constante considerata. Integrale resultans erit Vera quantitas quae in d y differentiale quaesitum ipsius P exhibeat.

ducta una cum Q& 8.11.

Si Q pariter non detur nisi per integrationem sitque Q=ITdx, cujus

differentiale d& requiritur, quod provenit si x fit constans. Hoc ex priore facile invenietur, nam cum ipsius P seu IQdx differentiale posita x constante

dQ et y variabili sit dyi--.dx;

dY

dT

erit simile differentiale ipsius Q seu JTdx hoc

dT

dyl-. dx. Consequenter fiet dP=dxlTdx+dyldx/-. d x . dY dY

Hocque mod0 ulterius progredi licebit, si etiam T per quadraturam ut JVdx determinetur, fieret enim dP=dxldxIVdx+dyldxJdxJ$fx tiari debent, ut

3:

sit constans et y variabilis.

Y

ubi T et V ita differen-

d 8.12. Si P fuerit functio n dimensionum ipsarum x et y , erit dP=Qdx*(nP-Qx) quemadmodum ex 5.7 intelligitur. Est vero etiam [ 9 ] R = l L d x , ut L=(%Q-Kx 1 Y 7' Sed est K=-dQ posita y constante. Fiet igitur R=-i((n-l)QdexdQ), et conse-

&

d

Y

quenter dP=Qdxc4F/((n-l)Qdx-xdQ) , ubi tam in differentiatione ipsius Q etiam

Y

in integratione ipsius (n-l)Qdx-xdQ, y ponitur constans et sola x ut variab i l k consideratur. Hoc autem statim ex illa aequatione [invleniri potuisset,

d

quae erat dP=Qdx&(nP-Qx).

Y

Est enim nP=JnQdx habita y pro constante. Ergo

nP-Qx=l ((n-l)Qdx-xdQ). 5 . 1 3 . Quo usus horum theorematum melius percipiatur, adhibebimus ea ad s o l -

venda problemata quaedam. Et primum quidem sit maxime agitatum problema trajectoriarum orthogonalium. [fig. 21

209

Appendix 2 Curva igitur data sit BM ad axem AP relata, in qua dicta abscissa A P , x sit

applicata PM=P functio quaedam ex x parametro a aliisque constantibus composita. Si jam a variabilis fiat, ejusque loco alii atque alii valores substituantur, prodibunt innumerabiles curvae ejusdem generis seu eadem aequatione contentae, quarum una ipsique BM proxima sit brn orta, si a augeatur elemento

da. Has autem curvas omnes ad angulos rectos secare linea quadam invenienda EM id est quod problema postulat. [ l o ] 5 . 1 4 . Sit igitur EM curva quaesita, faciens angulos rectos in M cum curva d a t a BM. [Volcetur ejus applicata PM-y; eritque P=y. Sed quia P non habet determinatum valorem ob a variabilem, vi conditionis propositae littera a debet eliminari. Problema autem requirit, ut elementum curvae quaesitae MI perpendiculare sit in curvas BM, bm. Quare ducta Mr, quae erit =dx, fiet Mr2=mr.nr, seu ob mr=dy, habebitur ch2-72r.dy. Est vero rn differentiale ipsius P posita a

-dx Cum igitur P et Q per a et x sint datae, dY -&

constante seu Qdx, ex quo erit &=-.

poterit ex his aequationibus P-y et Q=-

dY

pro curva quaesita.

nova formari eliminanda a , quae erit

5 . 1 5 . Ad trajectorias igitur orthogonales inveniendas alio artificio non est

opus, si et P et Q algebraice dentur, seu si curvae secandae fuerint algebraicae. Sed si P per quadraturas determinari debeat, Q vero algebraice exprimatur, haec solutio non erit sufficiens; verum loco aequationis P - y ejus differentialem adhiberi oportet dP=dy, ubi P ita debet differentiari, ut et a et x sint variabiles. Quamobrem habebitur dy=Q&+daJ+ dQ

( § . l o ) hic existente

-dx a quod ibi erat y, seu dx2+dy2=dyJdQ&. [ I l l Ex aequatione igitur Q=dY definiatur a et da hique valores substituantur in integrali ipsius dQ&,

quod

prodit erit aequatio ab a vacua [inter] x et y pro curva quaesita. 9.16.

Si fuerit P functio n dimensionum ipsarum a et

(nP-Qx)=

-

da a

X& dx2 da (ny+-). dY dK& a

Hinc fiet -=

-+-

x erit d P = d y = Q d x i -d a

dx2 + d y 2 nydy +xdx

.

In qua aequatione

si ex altera Ij= - valor ipsius a substituatur prodibit aequatio pro curva

dY

quaesita. Simili mod0 solutio succedit si non P sed PX denotante X functionem quamcunque ipsius x et constantium excepta a , fuerit functio n dimensionum

da D.PX-ydX+Xdy=Qdx+,(nyX-Q.r). Posita igitur Q h y d X a constante erit ydX+Xdy=Q& et dy= . Quare cum ante erat Q=-& -, hoc dY dx X& casu debet esse &d;c-ydx=-& Unde fit Q = L - - - , ideoque xdx g dx dY daXfdx2+dy2) Est vero Qdx differentiale ipsius PX posita a cona -nXydy+XxdpyxdXdyXdy/dx ’ stante, s e u 1121 functionis illius n dimensionum ipsarum a et z. Cognita igitur Q in a et x aequatio QzYdx-* dabit valorem ipsius a , qui in alters ipsarum a et x. Fiet enim ob P-y,

dx

dY

aequatione substitutus producet aequationem trajectoriae quaesitam.

210 5.17.

Appendix 2

P r a e t e r hos casus a l i o s s p e c i a l e s c o n s i d e r a s s e j u v a b i t : S i i n Q p r o r s u s

non i n s i t a, i d quod a c c i d i t , s i f u e r i t P=A+X, d e n o t a n t e A functionem i p s i u s

a tantum e t constantium; X vero functionem i p s i u s z tantum e t constantium;

-dx

hoc casu a e q u a t i o &= - seu dXdy=-&' dY quaesitae. [See f i g . 31

jam exprimet naturam t r a j e c t o r i a e

Hic autem casus locum h a b e t , s i eadem curva AL motu s i b i p a r a l l e l 0 sursum moveatur j u x t a a p p l i c a t a m PM. Quia enim e s t PM=P=A+X, s i c o n s t i t u a t u r curva

AL, i n qua s i t PL=X, e r i t LM=A ideoque c o n s t a n s per totam curvam BM.

E t quo

A f u e r i t v e l major v e l minor eo l o n g i u s v e l p r o p i u s e r i t c u r v a AL p o s i t a . S i t ex. g r . AL p a r a b o l a p a r a m e t r i b , e r i t PL=X=&,

4 2x&

seu b i y

- ci

Quamobrem

= -dx c u j u s i n t e g r a l i s e s t

t r a j e c t o r i a exprimetur hac a e q u a t i o n e (c- y l 6 =

bdx dX=---. Z&

ideoque

2vG

=g z 3 9

,

[ f i g . 31

.

L P

A

9 e j u s q u e ramus i n f r a axem Quae e s t pro p a r a b o l a c u b i c a l i N e i l i a n a p a r a m e t r i 4 16 AP p o s i t u s , atque sursum deorsumque motus ob indeterrninatam c q u a e s i t o s a t i s f a c i e t . [I31 5.18. P o s i t i s A e t

X ejusmodi u t s u p r a , s i f u e r i t P=y=AX e r i t Qdx=AdX. Quare ydx .-_--&'. Pro t r a j e c t o r i a i g i t u r h a b e b i t u r i s t a a e q u a t i o x dY

ob A =g fret Q&= X ydydX+Xdx2=0. Sumto i g i t u r PL=X, cum s i t PM=AX habebunt a p p l i c a t a e curvae BM constantem rationem ad a p p l i c a t a s curvae AL. Quare h i s c a s i b u s i s t a s o l u t i o

v a l e b i t . S i n t ex. g r . omnes hae curvae e l l i p s e s eodem v e r t i c e A eodemque axe t r a n s v e r s o b d e s c r i p t a e e r i t y = a G ' i ideoque X = G 2 7 e t d X = b&- ZX& Unde f i t ydy+ 2dX(bx-x2 '=o s e u y 2 = b z - z 2 + + b bb-Zx n Z L [ Z = Z q I. b-2x 9.19.

--. dY

Znbx - ~

2

)

m-1 -h2 dX= A t v e r o e s t A+X=yl/m.ConseS i P=y=(A+Xlm, e r i t Qdx=(A+X)

quenter haec h a b e b i t u r a e q u a t i o p r o t r a j e c t o r i a , my (i"-1)/mdydX+&2=0.

s i s i t y=IAX+Y)

m d e n o t a n t e Y u t X functionem i p s i u s

ergo & d x = r n l A X + Y l m - l ( A d X + d Y ) =*

dY

. Sed

cum s i t

3:

S i m i l i modo

e t constantium. E r i t

AX+^=^^/^ e t

A -Y

l/m

X-*

21 1

Appendix 2

e r i t m y f m - l”mdy(yl’mdx-YdX+XdYI

+ X h 2 =O. H i casus omnes i n s e r v i u n t [ I 4 1

quando omnes curvae secandae e x una quadam curva p r o p o s i t a v e l p l u r i b u s c o n j u n c t i s formantur, i n quo n e g o t i o p e r i n d e e s t s i v e hae curvae d a t a e s i n t a l g e b r a i c a e s i v e t r a n s c e n d e n t e s . Datae autem curvae s u n t e a e , quae a p p l i f a t a s habent X e t Y a b s c i s s a e

5

r e s p o n d e n t e s . Quomodo enim e x h i s d a t i s , curvae

secandae formentur, i p s a e x p r e s s i o i p s i u s P v e l y d o c e t . 5 . 2 0 . N e autem d i u t i u s imnoremur p r o b l e m a t i jam p l e n i s s i m e p e r t r a c t a t o ad a l i a progrediamur.

[See f i g . 4 1 . S i n t i g i t u r i t e r u m innumerabiles

curvae u t AM, Am eadem a e q u a t i o n e , s i parameter a v a r i a b i l i s ponatur c o n t i n e n t a e , o p o r t e a t q u e curvam EM i n v e n i r e , quae ab omnibus c u r v i s AM a e q u a l i a s p a t i a u t APM a b s c i n d a t . P o s i t a AP=x s i t PM=P f u n c t i o n i cuicunque ipsarum a e t constantium, a r e a autem APM e r i t quoque hujusmodi f u n c t i o , quae s i t S.

et

Pro curva q u a e s i t a EM v e r o s i t a p p l i c a t a PM-y, i t a u t s i t y=P, ex qua aequat i o n e a e l i m i n a r i d e b e t . S i curvae p r o p o s i t a e f u e r i n t r e i p s a q u a d r a b i l e s ponatur S=C seu s p a t i o abscindendo d a t o , huicque q u a e r a t u r a, c u j u s v a l o r i n a e q u a t i o n e y=P s u b s t i t u t u s , d a b i t aequationem curvae q u a e s i t a e . 9.21.

A t s i curvae non f u e r i n t q u a d r a b i l e s e t S p r o p t e r e a non d a t u r , l o c o

S

ponatur e j u s v a l o r JP& p o s i t a a c o n s t a n t e . Quia autem h i c valor i n d i v e r s i s c u r v i s d e b e t e s s e idem, o p o r t e t e j u s d i f f e r e n t i a l e quod h a b e t u r s i [I5 1 e t a et

x ponatur v a r i a b i l e , f ? e r i

=O.

Hoc autem d i f f e r e n t i a l e e s t P&+da/

3,

i n quo dP e s t d i f f e r e n t i a l e i p s i u s P p o s i t a z c o n s t a n t e , e t i n i n t e g r a t i o n e

dP a p r o c o n s t a n t i e s t habenda. E r i t i t a q u e P&+daf&x=O. dP Postquam da i g i t u r @ i n t e g r a t u m f u e r i t , ponatur i n hac a e q u a t i o n e l o c o a v a l o r e x i p s i u s -dx

a e q u a t i o n e P=y i n v e n t u s , hocque modo o b t i n e b i t u r a e q u a t i o pro curva q u a e s i t a .

dP da P positis a 5.22.

S i -dx non p o t e s t i n t e g r a r i hoc mod0 o p e r a r i o p o r t e t . D i f f e r e n t i e t u r e t x v a r i a b i l i b u s prodeatque Qdx+Rda quo f a c t o cum i n t e g r a r i

212

Appendix 2

debeat Rdx sumamus ejus loco expressionem aequivalentem Rx-JxdR, in qua a itidem est constans. Habebitur ergo Pdrc+Rxda=&/xdR.

Hocque mod0 pergendum est,

donec post signum summatorium a non amplius reperiatur, vel conjungendis hujusmodi aequationibus eliminari possit.

5.23. Sit PzA+X, A functionem ipsius A [Sic] et X functionem ipsius x de-

dP

xdA

notante erit J&dx=--;i;~-,quare A d x + X d x + x d A = O seu A x + i X d x = C = y x - X ~ + J X d x ob A-y-X:

vel in differentialibus iterum ydx+xdy=xdX, quae est aequatio [ 1 6 ] pro

curva quaesita. Similiter si ,?=AX3 erit S=AJX&=C.

&! erit C =

X

Posito autem loco A valore

X / X d x seu CydX- CXdy = y 2 X d x . Exempli loco sumamus infinitas ellipses

eodem vertice A eodemque axe transverso b descriptas. Erit itaque X = G 2 - e t aequatio pro curva quaesita haec

cG2=J&,.G27 Y

Sit P functio ipsarum a et x dimensionurn n, erit S ejusmodi functio dida mensiones n+Z habens. Quare erit dS=Pdx+-((n+l)C-yz) =O. Ex aequatione autem 5.24.

a

P=y definitur a , et substituto ejus valore in illa aequatione prodibit aequa-

=e, 4cydY+4p&

tio pro curva quaesita. Sint ex. gr. curvae circuli ex eodem vertice A deseripti erit P = G 2 - y , unde n=l eta ergo l a = l ( y 2 + x 2 ) - 1 2 x atque da-2ydy+2xdcc &. Prodibit ergo sequens aequatio --2 a y2+x2 X Y2+X pro &va quaesita in qua C aequa[lis] est areae abscissae propositae.

-

Appendix 2

213

---

The symbol "1" indicates a new line.

-

Along the last two sentences of 512 on page 9 of the ms. (ml) dQ QdQ / Z Q - Z G ~ F ~

-

Q Q

Almost entire margin of page 10 of the ms: the argument in the margin pro-

ceeds from the equation dx2+dy2=dy/dQ.dx occurring in the regular text of 515.

(m2) ob dy=-dx -/ 'Q2+1)dx+c+/dQdT=0 / dQ=Rdx+Sda, /

Q

Q

7 + l )dT+rsdah=o (Q

( Q 2 + 1 ) A h + i S A d a d x = 0 / /Tdx=O, dT=Fdx+Gda / Tdx+/Gdadx=O

Q

/ G=SA, / T=c?(Q~+I)A

dA=Bda / dQ=Rdx+Sda / G=-----B+( Q 2 + 1 ) (Q2-*)A'' / Q&AS=Q2AS-AS+Q3B+QB/ AS=Q3B+QB / Q QQ [ (ml) is an auxilliary calculation for this Posito x constante / -=-

Qf$

step]/ ?=------A Q 4 2 7 7

Q=

-

/ -=A 2 Q2 / @I---A---Y2 Q2+1

Y(Y2-AZ)

[this is a mistake]/Y2=B2Z, A2=B /

I / r/Bz-I

Along sentences 2-4 of 516 on page 1 1 of the ms.

da (m3) Pro functionaliter similibus invenit / -= a institui, juvabit /

-

dx2+dy2

/ Hinc comparationem

kydy+xdx

Along sentences 8 and 9 of 916 on page 1 1 of the ms. x a bx (m4) -=-t=-E funct. t b a / funct. nullius dimensionis ipsius x - a / r = % y = b nullius dimensionis ipsius x et a,. /

-

Along sentence 4 of 5 1 7 on page 12 of the ms.

Appendix 2

214

On d i f f e r e n t i a t i o n of f u n c t i o n s i n v o l v i n g two o r more v a r i a b l e s

5.1.

What i s proposed and solved h e r e is found i n d e a l i n g w i t h problems which

c o n s i d e r i n f i n i t e l y many curves of t h e same s o r t , such as t h o s e on o r t h o g o n a l t r a j e c t o r i e s , o r on i n t e r s e c t i n g a s e r i e s of c u r v e s a c c o r d i n g t o a g i v e n law. Required i s to d e f i n e t h e d i f f e r e n t i a l of a q u a n t i t y , composed from two v a r i a b l e s , given t h e d i f f e r e n t i a l of t h a t same q u a n t i t y i n c a s e one of t h e v a r i a b l e s i s c o n s i d e r e d t o be c o n s t a n t . For example, l e t P be a f u n c t i o n c o n s i s t i n g of

3:

and a, and Qdx i t s d i f f e r e n t i a l , a being c o n s i d e r e d c o n s t a n t , then from t h e s e d a t a t h e d i f f e r e n t i a l of P i s s o u g h t , i n which a i s c o n s i d e r e d t o be v a r i a b l e a s w e l l . This very d i f f e r e n t i a l w i l l have such a form: & d x + R d a . The means of f i n d i n g R from

P and Q i s what h a s t o be e s t a b l i s h e d . Now i f t h i s q u a n t i t y P I&&, such t h a t a p a r t from Q n o t h i n g

i s not given o t h e r w i s e b u t by means of

e l s e i s known, i t appears t o b e very d i f f i c u l t t o determine R merely from &.

5 . 2 . However, a s t o t h e v a r i a b l e s from which P and

Q

a r e composed, t h e e n t i r e

t a s k can be d e a l t w i t h by means of t h e f o l l o w i n g p r i n c i p l e . When a q u a n t i t y composed of two v a r i a b l e s i s d i f f e r e n t i a t e d t w i c e , where i n t h e f i r s t d i f f e r e n t i a t i o n one v a r i a b l e , and i n t h e second d i f f e r e n t i a t i o n the o t h e r var i a b l e i s t r e a t e d a s a constant

-

a s can be done i n two ways - then t h e r e s u l t s

i n both c a s e s w i l l b e e q u a l . For example, i f t h e q u a n t i t y z 3 y - y 2 ~ . 2i s g i v e n , d i f f e r e n t i a t i n g i t w h i l e keeping

3:

c o n s t a n t y i e l d s x3dy - 23:c2ydy; d i f f e r e n t i a t i n g a g a i n w h i l e keeping

y c o n s t a n t w i l l g i v e : 33:'&dy

- 4;cydxdy. Then t h e same q u a n t i t y i s d i f f e r e n t i a -

t e d twice a g a i n , b u t i n i n v e r t e d o r d e r , such t h a t f i r s t y and then c o n s t a n t . From t h e f i r s t d i f f e r e n t i a t i o n 33:'ydX- 2y2z&

3:

i s kept

a r i s e s , and from t h e

second 3z2dxdy - 4 y ~ d 3 : d y , a g r e e i n g e n t i r e l y w i t h t h e former r e s u l t .

5 . 3 . One who wishes t o apply t h e s e o p e r a t i o n s i n s e v e r a l examples w i l l n o t only always n o t i c e t h i s agreement, b u t h e w i l l a l s o u n d e r s t a n d t h a t t h e t r u t h of

t h i s p r i n c i p l e must n e c e s s a r i l y h o l d i n a l l c a s e s . B u t f o r the sake o f

those who a r e l e s s a b l e t o understand t h i s n e c e s s i t y , t h e f o l l o w i n g g e o m e t r i c a l demonstration can be produced: Let an a r b i t r a r y curve AM be r e l a t e d t o the a x i s AP, i n which a t t h e a b s c i s s a

AP = z t h e o r d i n a t e PM w i l l be e q u a l t o P, which q u a n t i t y P w i l l be an a r b i t r a r y f u n c t i o n c o n s i s t i n g of z,of t h e parameter a of t h e c u r v e s and of o t h e r c o n s t a n t s . Augmenting t h e parameter a by t h e d i f f e r e n t i a l day one w i l l have the curve AN c l o s e t o AM. Now w i t h t h e o r d i n a t e PM = P t h e f o l l o w i n g twofold

Appendix 2

[fig.

215

11

o p e r a t i o n i s c a r r i e d o u t : namely, P be d i f f e r e n t i a t e d w h i l e keeping a c o n s t a n t producing t h e d i f f e r e n t i a l r m of t h e o r d i n a t e o f t h e curve A M . I f i n t h e exp r e s s i o n o f t h i s d i f f e r e n t i a l rm a transforms i n t o

a + & , t h e n one w i l l i n s t e a d have t h e d i f f e r e n t i a l n s . T h e r e f o r e , t h e d i f f e r e n t i a l of t h i s rm, f o r

constant

x

and v a r i a b l e a w i l l be n s - m r . Now a s i m i l a r twofold d i f f e r e n t i a -

t i o n of P i s a p p l i e d w h i l e t a k i n g t h e c o n s t a n t s i n i n v e r t e d o r d e r : hence, MU

w i l l be t h e d i f f e r e n t i a l of P M f o r v a r i a b l e a and c o n s t a n t 2. I f i n i t s exp r e s s i o n x+&

is substituted for

x,

t h i s d i f f e r e n t i a l MU w i l l furthermore

change i n t o mn. T h e r e f o r e , t h e d i f f e r e n t i a l o f MN f o r v a r i a b l e x and c o n s t a n t

a w i l l be m n - M U . Hence, from t h e f i r s t o p e r a t i o n ns-rnr r e s u l t s , and from t h e second one

mn-MN.

B u t s i n c e ns = n r - s r and

mr = n r - m n , one h a s ns-rnr = rnn-sr = m n - M N .

By consequence, the same r e s u l t i s produced by b o t h o p e r a t i o n s ; from t h i s t h e

t r u t h of t h e p r i n c i p l e w i l l be p e r c e i v e d .

1 . 4 . However, s i n c e t h i s demonstration i s drawn from an a l i e n s o u r c e , I w i l l

P again be a q u a n t i t y which i s made up from a, x and c o n s t a n t s ; by s u b s t i t u t i n g x+& i n t h e p l a c e of x i t w i l l t u r n i n t o Q. And i f a +da i s w r i t t e n i n s t e a d of a , P changes i n t o R. B u t i f both x + & C . s s u b s t i t u t e d f o r x and a + d a f o r a , t h e n S w i l l emerge i n s t e a d of' P. From t h i s i t i s c l e a r t h a t i f i n Q a tda i s subd e r i v e a n o t h e r one from t h e n a t u r e of such d i f f e r e n t i a t i o n i t s e l f . L e t

s t i t u t e d f o r a then S w i l l appear, and i n t h e same way one w i l l a l s o have S i f in R x i

d3: i s

substituted for

x.

Now d i f f e r e n t i a t i n g P while keeping a c o n s t a n t w i l l y i e l d Q - P . F u r t h e r more, t a k i n g a t o b e v a r i a b l e and

Q - P w i l l be S - Q - R + P , same w i l l appear i f

x is

x t o b e c o n s t a n t , t h e d i f f e r e n t i a l of t h i s

s i n c e Q transforms i n t o S and P i n t o R . And t h e v e r y k e p t c o n s t a n t i n t h e f i r s t d i f f e r e n t i a t i o n and a i n

Appendix 2

216

t h e second; from t h e f i r s t one w i l l have R - P ,

and from t h e second S - R - Q + P .

The i d e n t i t y of t h i s e x p r e s s i o n and t h e former one can be most c l e a r l y p e r c e i v e d .

5.5,

i n which we a d a p t t h i s p r i n c i p l e t o our purpose. Let

P be a f u n c t i o n of

z,y and c o n s t a n t s , and l e t i t s d i f f e r e n t i a l be e q u a l t o Q d x + R d y , where Q and R a r e a l s o f u n c t i o n s of x and y. T h e r e f o r e , t a k e d Q = K d x + L d y and dR = M d x i N d y . By means of our p r i n c i p l e we s h a l l b e a b l e t o f i n d a c e r t a i n rethose

l a t i o n between t h e s e q u a n t i t i e s K , L , M and N . Namely, d i f f e r e n t i a t i n g P f i r s t with c o n s t a n t y y i e l d s Q d x . I n t u r n , i t s d i f f e r e n t i a l w h i l e keeping

3:

constant

w i l l be L d x d y . Thereupon one w i l l have Rdy by d i f f e r e n t i a t i n g P w h i l e keeping

z c o n s t a n t , t h e d i f f e r e n t i a l of which w i t h c o n s t a n t y w i l l be Mdzdy. By f o r c e of t h e p r i n c i p l e , t h e r e f o r e : Ldxdy = Mdxdy, and hence: M = L . 9.6. W e w i l l f i n d more r e l a t i o n s if a c e r t a i n p r o p e r t y of t h e f u n c t i o n P i s known. I f we suppose P t o be a f u n c t i o n of x and y of dimension z e r o , such as qx-@+x I a s s e r t t h a t i n t h i s c a s e Q x + R y = 0 w i l l h o l d . Namely, beY' Y cause t h e dimensions of x and y - taken t o g e t h e r - a n n i h i l a t e each o t h e r i n P, a mere f u n c t i o n of z w i l l r e s u l t from p u t t i n g y = zx. T h e r e f o r e , i n t h e d i f f e r e n t i a l of P no o t h e r d i f f e r e n t i a l b u t d z w i l l t u r n up. By s u b s t i t u t i n g t h e

.

value z d x + z d z i n s t e a d of dy i n t h e d i f f e r e n t i a l Z d x + R d y one w i l l have

'

Qdx + R z & + Rxdz. B u t because dx v a n i s h e s , one w i l l have Q + Rz = 0 , o r -Qx d x - Qxdy Q x + R y = 0. Therefore, R = - and dP Y Y P is a 5 . 7 . S i m i l a r l y i f P i s a f u n c t i o n of x and y of dimension n , t h e n f u n c t i o n of dimension zero. T h e r e f o r e , i t s d i f f e r e n t i a l w i l l have Vhne same p r o p e r t y t h a t when

dx i s r e p l a c e d by x and dy by y , a q u a n t i t y w i l l then P dP - nPd

emerge which is e q u a l t o zero. The d i f f e r e n t i a l of

- is

n i l ' 9 0' yn Y Q y d x + R y d y - n P d y . Therefore n e c e s s a r i l y : 6 & x + R y 2 - n P y = 0 , o r Q x + R y = nP. Yn + nP - Qx , and by consequence dp = 6& d x - Qxdy + nPdy This y i e l d s R = Y Y

5.8. With t h e s e p r o p e r t i e s one may advance f u r t h e r a t d e t e r m i n a t i o n s of t h e

f u n c t i o n s K, L , M , N . Namely, i f P i s a f u n c t i o n of n dimensions then Q and R

w i l l b e f u n c t i o n s of n - I

dimensions. Because of dQ

= K&+Ldy

one f i n d s :

K x + L y = I n - I I Q , and i n t h e same way: Mx+Ny = ( n - l ) R . Now s i n c e R =one h a s :

dR=

nydP - y x d Q - Q y d z - nPdy + Qxdy -

Y2 - In - I I Q y d x + In'- nIPdy

- 2In - I I Q x d y - K y x d x + Kx2dy

Y2 ( n l i Q - K z and By consequence: M = Y

,

= (n2- n/P - 2(n - I I Qx+ Kx2 Y2

n P - Qx Y

217

Appendix 2 And p r e c i s e l y t h e r e f o r e : L = M.

S . 9 . I cannot r e f r a i n h e r e from s w i f t l y o b s e r v i n g t h a t t h e formula nP = Qx+Ry can be of use i n t h e

i n t e g r a t i o n of d i f f e r e n t i a l formulas i n v o l v i n g two va-

r i a b l e s . Namely, i f t h e d i f f e r e n t i a l QdxiRdy i s made up s u c h , t h a t Q and R a r e f u n c t i o n s of x and y of t h e same dimension, say ( n - l ) , then i t s i n t e g r a l

.

w i l l be =

This i s always t r u e a s soon a s Qdx+Rdy h a s an i n t e g r a l .

Furthermore, i f n = 0 , then t h e i n t e g r a l w i l l b e i n f i n i t e l y l a r g e , and n o t h i n g can then be l e a r n t from i t . I n t h i s c a s e t h e i n t e g r a l s depend on l o g a r i t h m s , which can n o t be e x p r e s s e d i n t h i s manner. So when t h e i n t e g r a t i o n succeeds and

n i s unequal t o z e r o , then the i n t e g r a l w i l l always be found t o be a l g e b r a i c a l . For i n s t a n c e , i n the c a s e d x i R =

. Therefore,

5 . 10. Now l e t

GX&

i-aydy

n i s equal t o one, Q = I

t h e i n t e g r a l w i l l be x i ax2 ay2 = x + a+/-.

izand mxp

+

@T-j7-

P be a n a r b i t r a r y f u n c t i o n of x and y , of which t h e only t h i n g

known i s t h a t i t s d i f f e r e n t i a l f o r c o n s t a n t y i s Qdx. From t h a t t h e d i f f e r e n t i a l of P h a s t o be found when both x and y a r e t r e a t e d a s v a r i a b l e s . Let t h i s d i f -

dR = MdxiNdy, dQ o r R = /Mdx i f y i s regarded a s c o n s t a n t . Since M = L and L = -, i f i n t h i s dy dQ and d i f f e r e n t i a t i o n of Q x i s regarded a s c o n s t a n t , one w i l l have R = J-dx dY dQ dP = Q d x i d y J - d x . The f o l l o w i n g r u l e emerges from t h i s r e s u l t : dY dx Q be d i f f e r e n t i a t e d w i t h c o n s t a n t x , t h e r e s u l t be m u l t i p l i e d by - and i n t e dY g r a t e d a g a i n while y i s regarded as c o n s t a n t . The r e s u l t i n g i n t e g r a l w i l l then f e r e n t i a l be QdxiRdy, where R denotes what has t o be found. Now

be t h e very q u a n t i t y which, m u l t i p l i e d by dy and added t o Qdx, forms t h e req u i r e d d i f f e r e n t i a l of 5.11.

P.

Likewise, i f Q i s n o t given b u t by means of i n t e g r a t i o n , say Q = JTdx,

then i t s d i f f e r e n t i a l dQ i s demanded which emerges i f

x i s c o n s t a n t . I t can P o r JQdx

e a s i l y b e found from the p r e l i m i n a r i e s ; s i n c e t h e d i f f e r e n t i a l of

dQ f o r c o n s t a n t x and v a r i a b l e y i s dyJ-;tdx,

dT

IT& i s dy/-dx.

dY

Y

By consequence:

t h e s i m i l a r d i f f e r e n t i a l of Q o r

dP = d x / T d x + d y / d x / - ddTx . dY

ceed i n t h i s manner i f T i s a l s o d e f i n e d by a q u a d r a t u r e /V&;

One may prothen w i l l h o l d

dv dxJdx/Vdx+dyJdxidxJ-dx, where b o t h V and T a r e t o b e d i f f e r e n t i a dY t e d such t h a t x i s c o n s t a n t and y i s v a r i a b l e . d 5 . 1 2 . I f P i s a f u n c t i o n of n dimensions of x and y , then dP = Q&+J (nP-Qx), ( n - l I Q Kx a s can b e s e e n from 5.7. B u t a l s o : R = JLdx, hence L = Now f o r Y Y dQ T h e r e f o r e , R = -IJ ( i n - 1)Qdx-xdQ), where c o n s t a n t y K = -. both i n t h e d i f dx Y f e r e n t i a t i o n of Q and i n t h e i n t e g r a t i o n of (n-1)Qdx-xdQ y i s k e p t c o n s t a n t

dP

---.

and only x i s c o n s i d e r e d t o be v a r i a b l e . However, t h i s could a l s o have been found from t h e e q u a t i o n dp = Qdxia (nP-Qx) i t s e l f . Namely, f o r c o n s t a n t y:

Y

218

Appendix 2

nP = J’nQdx. Hence n P - Q x = J - ( ( n - 1)Qd;C-xdQ). 8 . 1 3 . I n o r d e r t o p e r c e i v e t h e use of t h e s e theorems more c l e a r l y , we s h a l l apply them t o t h e s o l u t i o n of c e r t a i n problems. F i r s t of a l l , l e t t h i s be t h e thoroughly s t u d i e d problem of o r t h o g o n a l t r a j e c t o r i e s .

[ f i g . 21

Let a curve BM r e l a t e d t o an a x i s A P be g i v e n ; i n which a b s c i s s a AP, c a l l e d t h e o r d i n a t e PM = P i s an a r b i t r a r y f u n c t i o n made up from

5,

2,

t h e parameter a

and o t h e r c o n s t a n t s . I f a i s a l s o made v a r i a b l e , and i f d i f f e r e n t v a l u e s a r e s u b s t i t u t e d f o r i t , then i n f i n i t e l y many c u r v e s of t h e same s o r t , o r c o n t a i n e d i n t h e same e q u a t i o n emerge, the one of which c l o s e t o BM i s bm, which a r i s e s i f a is augmented by t h e d i f f e r e n t i a l da. Now t h e problem demands t o f i n d a c e r t a i n curve EM, i n t e r s e c t i n g a l l t h e s e c u r v e s a t r i g h t a n g l e s .

5 . 1 4 . L e t EM be t h e curve r e q u i r e d , making r i g h t a n g l e s i n M with t h e g i v e n curve BM. C a l l i t s o r d i n a t e

PM = y , and hence P = y . But s i n c e P does n o t have

a f i x e d value owing t o t h e v a r i a b i l i t y of a , t h e type a should be e l i m i n a t e d , i n v i r t u e of t h e proposed c o n d i t i o n . Now t h e problem r e q u i r e s t h a t t h e element

Mm of t h e curve i n q u e s t i o n i s p e r p e n d i c u l a r t o t h e c u r v e s BM, bm. T h e r e f o r e , , r because having drawn M r , which w i l l be = &, one w i l l have Mr2 = mr. n ~ o rnr = dy, dx2 = n r . dy. Now rn i s t h e d i f f e r e n t i a l of P keeping a c o n s t a n t , o r

Q&;

t h i s yields Q =

-dX .

dY

Since P and

Q a r e g i v e n i n terms of a and

2,

one may

form a new e q u a t i o n , p e r t a i n i n g t o t h e r e q u i r e d c u r v e , by e l i m i n a t i n g a from the e q u a t i o n s P = y and Q

-.-dx dY

9.15. A s long a s P and Q a r e given a l g e b r a i c a l l y o r the c u r v e s which a r e t o be i n t e r s e c t e d a r e a l g e b r a i c t h e r e i s no need f o r any o t h e r a r t i f i c e f o r f i n d i n g t h e orthogonal t r a j e c t o r i e s . However, a s soon a s P must be determined by quad r a t u r e s , while Q i s s t i l l e x p r e s s e d a l g e b r a i c a l l y , t h i s s o l u t i o n i s no l o n g e r s u f f i c i e n t ; then i n s t e a d of t h e e q u a t i o n

P = y i t s d i f f e r e n t i a l dP = dy must be

Appendix 2

219

t a k e n i n t o a c c o u n t , where P i s t o b e d i f f e r e n t i a t e d s u c h , t h a t b o t h a r e v a r i a b l e . T h e r e f o r e one w i l l h a v e dy = Q c k + d a . fdQ -&,

da

p o s i t i o n of y i n § . l o . ,

2

and a

where a t a k e s up t h e

o r d x z + d y * = dy I d Q d x . From t h e e q u a t i o n Q =

-dx -

dY

a

and du must t h e n b e f o u n d , and t h e i r v a l u e s must be s u b s t i t u t e d i n t h e i n t e g r a l of d Q d z ; t h e r e s u l t w i l l be a n e q u a t i o n between

and y f o r t h e r e q u i r e d c u r v e

3:

t h a t d o e s n o t i n v o l v e a.

5.16.

I f P i s a f u n c t i o n of n d i m e n s i o n s of a and

da dP = dy = Qdx+- ( n P - Q x ) = I.&?-& (ny +&). dY a dY

2,

then

This y i e l d s

-d3: I f t h e v a l u e of a , t a k e n from t h e o t h e r e q u a t i o n Q = --, dv

$ = n y y +xdx .

is substituted i n

t h i s e q u a t i o n t h e n a n e q u a t i o n f o r t h e r e q u i r e d c u r v e r e s u l t s . The s o l u t i o n s u c c e e d s i n t h e same way i f PX i n s t e a d o f P i s a f u n c t i o n of d i m e n s i o n n , X d e n o t i n g an a r b i t r a r y f u n c t i o n made up from Namely, b e c a u s e of P = y:

z and c o n s t a n t s o t h e r t h a n a. da ( n y X - Qx). T a k i n g a con-

D . PX = ydX + Xdy = Q d x

s t a n t y i e l d s y d X + X d y = Qd;C and d y =

+a

Qd33iydX. Therefore, such as previously

Qdx - YdX - -dx Q must h o l d . Hence Xdx - d4 i n t h i s c a s e t h e e q u a t i o n dq dX Xdx da XIdx2 + dy') Q = L--, dx dy and therefore 7= nXydy + Xxdx - yzdXdy/dx' Now Qdx i s t h e d i f f e r e n t i a l o f PX, k e e . p i n g- a c o n s t a n t , o r r a t h e r of t h i s f u n c t i o n o f 3: and a o f dimension n . Once Q i s known i n terms o f a and x, t h e e q u a t i o n Q = ydX X d x

dx

will

dY

p r o d u c e t h e v a l u e of a which, s u b s t i t u t e d i n t o t h e o t h e r e q u a t i o n , w i l l

produce t h e r e q u i r e d e q u a t i o n of t h e t r a j e c t o r y .

5.17.

I t w i l l h e l p t o c o n s i d e r some o t h e r s p e c i a l c a s e s b e s i d e s t h e s e : i f a

i s e n t i r e l y a b s e n t from

Q - as happens i f P = A + X , A d e n o t i n g a f u n c t i o n o n l y

o f a and c o n s t a n t s , X a f u n c t i o n o n l y of z and c o n s t a n t s - t h e n i n t h i s case -& t h e e q u a t i o n Q = __ or dXdy = -dx2 a l r e a d y e x p r e s s e s t h e n a t u r e o f t h e r e q u i r e d trajectory.

dY

T h i s v e r y case t a k e s p l a c e , i f a c u r v e

[ f i g . 31

M

B

I

L

A

P

AL i s , by a move p a r a l l e l t o

220

Appendix 2

i t s e l f , s h i f t e d upward along a n o r d i n a t e PM. Since PM

= P = A+X,

granted t h a t

PL = X f o r t h e curve A L , LM w i l l be e q u a l t o A , and w i l l t h e r e f o r e be c o n s t a n t along t h e e n t i r e curve BM. And t h e l a r g e r o r s m a l l e r A becomes, t h e f u r t h e r o r c l o s e r w i l l t h e curve AL be d i s p l a c e d . For example, l e t A L be a p a r a b o l a w i t h parameter b , then PL = X = expressed by t h e e q u a t i o n

6, hence

bd

dX = bdx

Therefore, the t r a j e c t o r y i s

7 7 .

2&x

-dx, t h e i n t e g r a l of which i s

Z&

4 ( c - y l A= - x&,

or b(y-c)’ = x3. This i s t h e e q u a t i o n of t h e c u b i c 3 99 , t h e branch of which i s placed underneath N e i l l i a n p a r a b o l a w i t h parameter -b 16 t h e a x i s AP; an up- o r downward motion w i l l s a t i s f y t h e r e q u i r e m e n t s , because of t h e undetermined c. §.

*

18. A and X d e f i n e d a s above; i f P = y = AX t h e n &&

=

has &&

=

dY

= A&.

Since A =

$ one

Thus one w i l l have t h e f o l l o w i n g e q u a t i o n f o r t h e t r a -

a

j e c t o r y : ydydX+Xdx* = 0 . I f one t a k e s PL = X, then t h e o r d i n a t e s of t h e curve

BM w i l l have a c o n s t a n t r a t i o w i t h t h e o r d i n a t e s of t h e curve A L , because of PM = AX. Thus i n t h e s e c a s e s t h e same s o l u t i o n w i l l apply. Let f o r i n s t a n c e t h e s e curves a l l be e l l i p s e s w i t h t h e same v e r t e x A and t h e same t r a n s v e r s e and dX = b d x - Z x d x . This y i e l d s = 2V5F-F x‘) 1 0 , o r y 2 = bx -x2+ %bn Zog b-ZX’

a x i s b ; then: y = a

ydy

t

§.19.

m

, hence X

m

If P = y = i A + X ) , t h e n Q d x = fA+X)m-ldX,andA+X=yl’m.

quence one w i l l have t h i s e q u a t i o n f o r t h e t r a j e c t o r y : my ( m -

Byconse-

dY& + &’

0.

S i m i l a r l y , i f P = I A X + Y l m , b o t h X and Y d e n o t i n g f u n c t i o n s of x and c o n s t a n t s , l/m - Y then Qdx = m ( A X + Y l m - l ( A d X + d Y ) = --dX’. Since A X + Y = yl/m and A = one has my ( m - li/m

dY dy ly”mdX- YdX + XdY) + Xdx’ = 0, A l l t h e s e c a s e s w i l l be of

use whenever the c u r v e s t h a t a r e t o be i n t e r s e c t e d a r e a l l moulded o u t of one curve o r s e v e r a l connected c u r v e s . The given c u r v e s a r e t h o s e t h a t have t h e o r d i n a t e s X and Y corresponding t o t h e a b s c i s s a x. How t o form t h e c u r v e s t h a t a r e t o be i n t e r s e c t e d from t h e s e given c u r v e s , emerges from t h e e x p r e s s i o n f o r

P or y. 5 . 2 0 . We w i l l n o t l i n g e r any l o n g e r w i t h t h i s problem t h a t has a l r e a d y been t r e a t e d e x t e n s i v e l y , b u t w i l l proceed t o o t h e r s now. T h e r e f o r e , l e t a g a i n inf i n i t e l y many c u r v e s l i k e AM, Am a l l be c o n t a i n e d i n t h e same e q u a t i o n , i n which t h e parameter a i s c o n s i d e r e d t o be v a r i a b l e ( s e e f i g u r e 4 ) ; r e q u i r e d i s t o f i n d a curve EM, t h a t c u t s o f f e q u a l a r e a s such a s APM from a l l c u r v e s AM. Putting AP

= x,

then PM =

P w i l l be some f u n c t i o n of t h o s e x, a and c o n s t a n t s

and t h e a r e a APM w i l l a l s o be a f u n c t i o n of t h i s t y p e ; l e t i t be S. The o r d i n a t e of t h e r e q u i r e d curve EM w i l l a l s o be PM

= y,

such t h a t y = P ; from t h i s

e q u a t i o n a has t o b e e l i m i n a t e d . I f t h e a r e a underneath t h e given c u r v e s can be

22 1

Appendix 2

[ f i g . 41

A

c a l c u l a t e d d i r e c t l y , t h e n one p u t s S = C, where C i s t h e g i v e n a r e a which i s t o be c u t o f f ; a being determined from t h i s e q u a t i o n , and i t s v a l u e b e i n g subs t i t u t e d i n t o t h e e q u a t i o n y = P w i l l y i e l d t h e e q u a t i o n of t h e r e q u i r e d curve. §.21.

However, i f t h e a r e a underneath t h e c u r v e s can n o t b e c a l c u l a t e d and,

t h e r e f o r e , S i s n o t g i v e n , then S must be r e p l a c e d by i t s v a l u e IPdx, t a k i n g u c o n s t a n t . Since t h i s v a l u e must be t h e same i n a l l c u r v e s , i t s d i f f e r e n t i a l f o r v a r i a b l e a and

P d x + d a J dP -dx, da

3:

must n e c e s s a r i l y be e q u a l t o zero. This d i f f e r e n t i a l i s

where dP i s t h e d i f f e r e n t i a l of P f o r c o n s t a n t x , and where

dP dP 0. a, then i n t h i s e q u a t i o n a i s t o be r e p l a c e d by i t s v a l u e which i s found from t h e e q u a t i o n P = y , and i n t h i s way t h e e q u a t i o n f o r t h e i n t h e i n t e g r a t i o n of z d x a must be taken c o n s t a n t . Thus: P d x + d a l & d x =

dP Having i n t e g r a t e d -dx da

r e q u i r e d curve w i l l b e found.

dF

9 . 2 2 . Whenever -dx

da

can n o t be i n t e g r a t e d , one must proceed i n t h e f o l l o w i n g

way: D i f f e r e n t i a t i n g P, taking, a and x v a r i a b l e y i e l d s Qdx+Bd.a.

Since Rdx

is t o be i n t e g r a t e d , we take t h e e q u i v a l e n t e x p r e s s i o n Rx-JxcdR i n s t e a d , i n which a i s c o n s t a n t a g a i n . One w i l l thus have Pdx+Rxda = d a l x a . One must go on i n t h i s way u n t i l a does no l o n g e r occur underneath t h e i n t e g r a t i o n s i g n , or u n t i l i t can be e l i m i n a t e d by combining e q u a t i o n s of t h i s t y p e .

P = A + X , where A denotes a f u n c t i o n of a , and X a f u n c t i o n of x ; xdA hence A d x + X d x + x d A = 0 o r , because of A = y - X : -; du Ax + IXdx = C = y x - Xx + JXdx. O r , i n d i f f e r e n t i a l s a g a i n : y d x + x d y = X d x , which i s t h e e q u a t i o n f o r t h e r e q u i r e d curve. S i m i l a r l y , i f P = A X = y, t h e n S = AIXdx = C. P u t t i n g $- i n s t e a d o f A , then C = $ JXdx, o r C y d x - Cxdy = y’xdx. For example, l e t us t a k e i n f i n i t e l y many e l l i p s e s w i t h t h e same v e r t e x A and and a l l d e s c r i b e d above t h e same t r a n s v e r s e a x i s b . Then X = and t h e 9 . 2 3 . Let then

dP I-&= da

equation f o r the r e q u i r e d c u r v e w i l l be

c=

Y

=/ & A X *

222

Appendix 2

x o f n d i m e n s i o n s , t h e n s w i l l be such a da f u n c t i o n h a v i n g n + l d i m e n s i o n s . T h e r e f o r e dS = Pdx+- ( ( n + l l C - y x l = 0. a S u b s t i t u t i o n of t h e v a l u e of a , d e f i n e d by t h e e q u a t i o n P = y , i n t h i s e q u a t i o n 5.24.

I f P i s a f u n c t i o n o f a and

y i e l d s t h e e q u a t i o n of t h e r e q u i r e d c u r v e . L e t f o r example t h e c u r v e s b e c i r c l e s w i t h t h e same v e r t e x A , t h e n P

=

-

= y ; hence n = 1 and a =

t h e r e f o r e log a = log (y2 +x2) Log 2x, and

da = a

e q u a t i o n t h e n emerges f o r t h e r e q u i r e d c u r v e :

4cydy+4cxh y2

+d

--2cdx -X

has t o be c u t o f f .

2y2xdy-2y3h, y2 +x2

+ , and $&-X2

.

2 y d y + 2 x d x - ~ The f o l l o w i n g y2+.2*

where C i s e q u a l t o t h e g i v e n area t h a t

223 APPENDIX 3 NEWTON’S RULE FOR THE RADIUS OF CURVATURE OF MAY ZlST, 1665

On May 21st, 1665, Newton formulated the following rule for the radius of curvature of an algebraic curve: “But from these

such like consideracons may bee pronounced a general1

&

Theoreme whereby ye crookedness of any line may be readily determined. To w

ch

purpose let mee suppose X to signifie all ye Algebraicall termes (expressing ye nature of ye given line) when they are considered as equal1 to nothing & not m some of y to others. Let ZC signifie ye same termes ordered according to ye n dimensions of x, & y multiplyed by any Arithmeticall progression. let X signifie the same termes ordered according to ye dimensions of y

&

then multi-

plyed by any Arithmeticall progression. let X signifie those termes ordered according to x

&

multiplyed by any two Arithmeticall progressions[,] one of

them being greater then the other by a terme. Let X signifie those termes n ordered according to ye dimensions of y, & y multiplyed by any two Arithmeticall progressions differing by a terme. Let ZE signifie those termes ordered according to x & multiplyed by ye greater of ye progressions wch multiplyeds, then ordered according to y & multiplyed by ye greater o f ye progressions ch w multiplyed J3 &

.“

The radius of curvature is then given by:

This rule is contained in Newton’s manuscripts of the period 1665-P666, grouped together by Whiteside under the title: “Normals, curvature and the resolution of the general problem of tangents”, and published in Mathematical

Papers vol. 1 (cf. esp: pp. 289-290). Obviously Newton considered algebraic curves, given by a polynomial equation in the variables x and y . If we denote this equation by (2)

f (x,y ) =o,

where f(x,y)

is of the form

Appendix 3

224

then Newton’s symbol X corresponds to our f ( x , y ) . According to their verbal definition, the side-dotted variants of X can be identified as the following set of polynomial expressions:

where all the sums have the same limits as the original sum given in (3). The arithmetic progressions introduced by Newton occur in ( 4 ) as (5)

a , a+h, a+Zh,

.. .

(6)

b, b+u, b i b ,

...

In

and

terms o f f ( x , y ) and its partial derivatives yet another transcription

evolves, of the following form: (7)

X =Xxfx+af x =uyf +bf ix: =h2x f + 2 A a x f x + a i a - l ) f xx

Y

3 Z = p 2 y 2 f +2ubyf +bib-1)$ Y YY =Xuxyf +payf +hbxfz+abf X Y Y Now under the following two conditions:

x

(a) the arithmetical progressions given in (5) and (6) have the same difference, hence 131, and ( b ) this difference is equal to one, hence h = u = I ,

only the first of which is mentioned explicitly by Newton when calculating through some,examples,the values of the side-dotted X ’ s as defined by Newton and as transcribed in (7) produce the correct expression (8)

R= i f x 2 + f 2y, 3/2/(-f~2fyy+2f~yfxy-fy2f5CC)

for the radius of curvature. This rule for the radius of curvature, and these side-dotted expressions form the source for certain claims that Newton possessed a concept o f , and a notation corresponding to partial derivatives as early as 1665. All these claims (e.g. in Cohen 1 9 7 4 , Grattan Guinness 1 9 7 9 , Baron 1969 to mention just a few) derive from Whiteside’s article 1960 (esp. p.362) and Whiteside’s monumental edition of Newton’s M a t h e m a t i c a l Papers Vol. 1 (esp. pp. 146-147 and 289-290). Let me quote Whiteside’s remarks about the side-dotted quantities

225

Appendix 3 in his 1960:

"Clearly side-dots are equivalent to partial differential operators x%X, ax 9 yx-ZE, and Newton saw as much. Further he states - ( X ) [ Z EC ]=x2 axc2 , aY 1.(=) [: (X)* [E X]=zj'9 and ('x)~. aY axay (for the elementary functions, at least, considered in Newton's day), using them to derive a formula for the

=]=w-

'

radius of curvature

..."

Now the definition of the side-dotted quantities as given here is both wrong and a deviation from Newton's practice. Apart from the obvious lack of the arbitrariness allowed by the arithmetical progressions (5) and (6) - to which we shall turn later

-

Whiteside's definition differs from Newton's in that

Newton did not apply repeated application of the single dot in order to define the multiple dot. Hence it is not Newton's mistake that these definitions are wrong: for instance, repeated application of the single left-dot yields:

*('X)=xfx+x2fxx rather than x2fxx. In his comments upon Newton's definition in the Mathematical Papers, Whiteside was far more modest: "In an extension of Hudde's scheme of multipliers, Newton defines operationally the homogenized first- and second order partial derivatives of X:f(x,y)=O. and Z=zj2f " (cf.p. 289 fn.75). Specifically, ~ = x f x ,= Y f y , X = x 2 f , X = x y f xx XY' YY In these transcriptions, however, the arbitrariness which Newton allowed to exist in the side-dotted expressions is still left out. In fact, the arbitrariness was not neglected by Newton, who used it to facilitate computations (e.g. in Example 3, op. c i t . p. 293). Apparently, Whiteside's claims of 1960 - and hence all subsequent claims that derive from the article 1960

-

are based on the assumption that one can sensibly distinguish between

a conceptual aspect of the side-dotted quantities

-

which could fairly be

transcribed as the homogenized partial derivatives xf2 etc.

-

and a

computational aspect, of minor importance only and not necessary to account for in the transcription. I have found no evidence in Newton's work

that would justify such a

dichotomy, the subsequent stress on a conceptual aspect, and the neglect of the arbitrariness in Newton's definitions of the side-dotted expressions. On the contrary, all available evidence points towards the conclusion that Newton by his side-dot notation sought to distinguish and codify exactly certain subroutines which occur in his computationofthe radius of curvature - hence, the computational aspect! This evidence comes from the chronology of Newton's research. The day before he hit upon the rule for the radius of curvature, May 20th, 1665, Newton had formulated the algorithm for the subtangent of an algebraic curve:

226

Appendix 3

"Multiply t h e termes of ye e q u a t i o n ordered a c c o r d i n g t o ye dimensions of y, by any A r i t h m e t i c a l l p r o g r e s s i o n , wch s h a l l b e e a Numerator; Againe change ye s i g n e s of ye e q u a t i o n & o r d e r i n g i t according t [ o ]

2, m u l t i p l y

ye termes by any

A r i t h m e t i c a l l p r o g r e s s i o n & t h e product d i v i d e d by 2 s h a l l bee ye denominator of ye v a l o r of t" ( c f . op. c i t . p. 280). No s i d e - d o t s h e r e . On t h e f o l l o w i n g day, May 2 1 s t , 1665, Newton came t o f o r m u l a t e t h e r u l e f o r t h e r a d i u s of c u r v a t u r e , which he had found by i n d u c t i o n from t h e p a r t i c u l a r c a s e s which h e had c a l c u l a t e d through. But now, i t seems, t h e complexity of t h e a l g o r i t h m r u l e d o u t a c i r c u m l o c u t i o n a s had been employed i n t h e c a s e o f t h e subtangent r u l e , and Newton was f o r c e d t o a c c e p t a n o t a t i o n f o r t h e d i f f e r e n t e x p r e s s i o n s which had t o be c a l c u l a t e d from t h e e q u a t i o n of t h e curve. Thus, on t h e 2 1 s t of May 1665 t h e s i d e - d o t s appeared i n o r d e r t o s i m p l i f y t h e f o r m u l a t i o n of an a l g o r i t h m . Now, of c o u r s e , i t i s c o n c e i v a b l e t h a t Newton immediately r e a l i s e d t h e c o n c e p t u a l

importance of t h e s e ex-

p r e s s i o n s , i n which c a s e one would expect him t o e x p l o i t t h e s e concepts as soon a s t h e r e was o c c a s i o n t o do s o . But Newton d i d n o t . Over a y e a r l a t e r , i n October 1666, Newton composed h i s famous "October 1666

t r a c t on f l u x i o n s "

(op. c i t . pp. 4 0 0 - 4 4 8 ) , i n which t h e r u l e s f o r t h e s u b t a n g e n t and t h e r a d i u s of c u r v a t u r e a r e formulated a g a i n . And h e r e a g a i n t h e s u b t a n g e n t r u l e was o n l y formulated v e r b a l l y , and t h e s i d e - d o t s

-

i n a n improved v e r s i o n w i t h double

i n s t e a d of t r i p l e d o t s - were r e s e r v e d e x c l u s i v e l y f o r t h e r a d i u s of c u r v a t u r e . I f Newton d i d indeed possess a concept c o r r e s p o n d i n g t o p a r t i a l d i f f e r e n t i a t i o n , and a n o t a t i o n f o r i t , why then d i d he n o t u s e h i s concept t o d e r i v e h i s r u l e s , and why d i d he n o t u s e t h e n o t a t i o n t o f o r m u l a t e t h e s u b t a n g e n t r u l e ? Newton had no such concept; i n s t e a d , h e had a m a g n i f i c e n t r u l e f o r t h e r a d i u s of c u r v a t u r e , and a f i n e n o t a t i o n t o p r e s e n t t h i s r u l e as e l e g a n t l y as p o s s i b l e .

227 BIBLIOGRAPHY

Abbreviations used i n r e f e r e n c e s t o l e t t e r s : ACC

=

Alexis-Claude C l a i r a u t (1713-1765).

ASC

=

Antonio-Schinella Conti (1677-1749).

DB

= Daniel B e r n o u l l i (1700-1782).

G1'H

= Guillaume F r a n c o i s , Marquis de l ' H 6 p i t a l e t c .

GWL

= G o t t f r i e d Wilhelm L e i b n i z (1646-1716).

IN

= I s a a c Newton (1643-1727).

JacH

= Jacob Hermann (1678-1733).

JK

= John

JohB

=

Johann B e r n o u l l i (1667-1748).

(1661-1704).

Keill (1671-1721).

LE

=

Leonhard E u l e r (1707-1783).

NicIB

=

Nicolaus I B e r n o u l l i (1687-1759).

PRM

= P i e r r e R6mond de Montmort (1678-1719).

A b b r e v i a t i o n s used i n t h e b i b l i o g r a p h y :

AE = Acta Eruditorum; Brieflechsel Correspondence

c f . b i b l i o g r a p h y Johann B e r n o u l l i ; c f . b i b l i o g r a p h y I s a a c Newton;

LEO0 c f . b i b l i o g r a p h y Leonhard E u l e r ;

LMS c f . b i b l i o g r a p h y G o t t f r i e d Wilhelm Leibniz. L e t t e r s t o and from E u l e r , and books, a r t i c l e s and m a n u s c r i p t s from E u l e r a r e always i d e n t i f i e d by t h e i r r e s p e c t i v e numbers i n t h e f o l l o w i n g indexes: l e t t e r s (e.g. R 9 7 ) a r e indexed i n LEO0 4.4.1,

a r t i c l e s and books a r e indexed i n

EnestrSm 1920 (e.g. E & ) , manuscripts are indexed i n EnestrSm 1913 (e.g. HZO) and i n Kopelevic 1962 (e.g. Kop. 86).

Bibliography

228

List o f l e t t e r s GZ ‘ H : JohB

8/12/1692

In:

BrieJbechseZ, pp. 159-160.

G?, ‘H:JohB

2/ 1 / 1 6 9 3

In:

Driefwechse?,, pp. 161-164.

G I ’H:GWL

24/ 2 / 1 6 9 3

In:

LMS 11, pp. 223-227.

GWL: JohB

7/ 6/1694

In:

LMS I I I / 1 , pp. 141-143.

Jo hB :GWL

2/ 9 / 1 6 9 4

In:

LMS I I T / 1 , pp. 143-152.

GWL: JohB

16/12/1694

In:

LMS I I I / 1 , pp. 152-157.

JohB:GWL

21/ 7 / 1 6 9 6

In:

LMS I I I / 1 , pp. 295-302.

GWL: JohB

23/ 8/1696

In:

LMS I I I / l , p p . 318-323.

JohB: GWL

27/10/2696

In: LMS I I I / 1 , pp. 332-334.

JohB:GWL

7/ 6 / 1 6 9 7

In:

LMS I I I / 1 , pp. 413-420.

JohB: GWL

17/ 7 / 1 6 9 7

In:

LMS I I I / 2 , pp. 439-444.

GWL:JohB

3/ 8 / 1 6 9 7

In:

LMS I I I / 2 , pp. 449-451,

GWL:JohB

9/ 8/1697

In:

LMS I I I / 2 , pp. 454-455.

JohB: GWL

14/ 8/1697

In:

LMS I I I / 2 , pp. 455-466.

JohB:G?,’ H

1 5 / 1 0/169 7

In:

BrieflechseZ, pp. 353-358.

GI ’ H : JohB JohB:GZ ‘H

18/11/1697

In:

BriefweckseZ, pp. 359-361.

In:

BrieflechseZ, pp. 361-364.

24/12/1697

U n p u b l i s h e d , e x c e p t f o r q u o t e i n Henny 1 9 7 5 , p . 461.

PRM: JohB

5/ 3 / 1 7 1 3

GWL:JohB

30/12/1714

In:

LMS I I . l / 2 , pp. 933-935.

JohB:GWL

6/ 2 / 1 7 1 5

In:

LMS I I I / 2 , pp. 935-938.

GWL:JohB

4/11/1715

In:

LMS I I I / 2 , pp. 948-949.

Jo hB :GWL

23/11 /1715

In:

LMS I I I / 2 , pp. 949-951.

GWL: ASC

6/12/1725

In:

Correspondence V I , pp. 250-255. LMS I I I / 2 , pp. 951-953.

GWL: JohB

December 1715

In:

JohB: GWL

1 5 / 1/1716

In:

LMS I I I / 2 , pp. 953-955.

GWL: JohB

31,’

1/1716

In:

LMS I I I / 2 , pp. 956-957.

JK:IN

5 / 3/1716

(Dated 23/2/1716 Old S t y l e ) . I n : Correspondence VI, pp. 282-284. LMS I I I / 2 , pp. 957-959.

Jo hB :GWL

11/ 3/1716

In:

NicIB: PRM

31/ 3/1716

Autograph a t o f f e n t l i c h e B i b l i o t h e k d e r Univers i t a t Basel, L I a 2 1 , f.232-240.

GWL:ASC

14/ 4 / 1 7 1 6

In:

Correspondence V I , pp. 322-324.

GWL:JohB

26/ 7 / 1 7 1 6

In:

LMS I I I / 2 , pp. 967-968.

JohB :G WL

22/ 8/1716

In:

LMS I I I / 2 , pp. 968-970.

NicIB: JohB

I / 1/1718

U n p u b l i s h e d e x c e r p t a t o f f . B i b l . Univ. Basel, LIa21, f.13-14.

229

Bibliography

NicIB:PRM

11/ 6 / 1 7 1 8

Autograph a t o f f . B i b l . Univ. Basel, L I a 2 1 , f.253-255.

LE:DB

28/ 2/1734

(R97). I n : B i b Z i o t h e c a M a t h e m a t i c a ( 3 ) 7 ( 1 9 0 6 / 7 ) , pp. 134-137.

LE:DB

IVovember I ? ) 1734

(R98). I n : B i b l i o t h e c a Mathematica ( 3 ) 7 ( 1 9 0 6 / 7 ) , pp. 137-141.

DB:LE

18/12/1734

LE':DB

2/ 6 / 1 7 3 5

LE:ACC

19/10/1740

(R387). I n :

LEO0 4A.5 , pp. 71-78.

NicIB: LE

24/10/1742

(R237). I n :

F u s s 2 8 4 3 b , pp. 690-700.

NicIB: LE

6/ 4 / 1 7 4 3

(R239). I n :

Fuss 1 8 4 3 b , pp. 701-707.

(R99). I n :

F u s s 1 8 4 3 b , pp. 415-418.

(R102). U n p u b l i s h e d ; t r a n s c r i p t a t E u l e r A r c h i v e Basel.

LE :Ni elB

14/ 5/1743

(R240). I n :

FUSS

1 8 6 2 a , pp. 536-538.

NieIB: LE

29/11/1743

(R241). I n :

FUSS

1 8 4 3 b , pp. 708-713.

LE: MicIB

4 / 2/1744

(R242). I n :

F u s s 1 8 6 2 a , p p - 538-541.

NicIB: LE

20/ 4 / 1 7 4 5

(R243). I n :

Fuss 1862a, pp. 542-545.

Bibliography

230

Books, a r t i c l e s , manuscripts d'Alembert, J. l e Rond

1747.

R 6 f l e x k m s sur l a cause ge'n6rale des vents. P a r i s , 1747.

1749.

"Recherches s u r l a courbe que forme une c o r d e tendue m i s e e n v i b r a t i o n " . Histoire Acad. B e r l i n 3(1747; p u b l . 1749), pp. 214-219.

Baron, M.E.

1969.

The Origins o f t h e I n f i n i t e s i m a l Calculus. Oxford, 1969.

B e r n o u l l i , Jakob

JBO I , I I .

Jacobi Bernoulli E a s i l e e n s i s Opera. Gengve, 1744. 2 Vols.

R e p r i n t : B r u x e l l e s , 1967.

Meditationes.

Unpublished m a n u s c r i p t : k e p t a t t h e O f f e n t l i c h e B i b l i o t h e k d e r U n i v e r s i t a t B a s e l as i t e m LIa3 ( c f . f o o t n o t e 41, c h a p t e r 2 ) .

AE (1694; O c t . ) , pp. 391 f f .

2694.

"De methodo tangentium i n v e r s a = JBO I , pp. 618-623.

2695.

"Nota VII: Ifethodus t a n g e n t i u n C a r t e s i i proruota".

1697.

...'I.

" S o l u t i o problematum f r a t e r n o r u m I T , pp. 7cQ-778.

,..". AE

JBO II, pp. 680-684.

(1697; May), pp. 21 1-217,

= JRO

. ..".

3698a.

"Demonstratio s y n t h e t i c a p r o b l e m a t i s de i n f i n i t i s c y c l o i d i b u s AE (1698; May), pp. 223-226. = JBO 11, pp. 785-794. = JBOO I , pp. 253-256.

2698b.

" S o l u t i o p r o b l e m a t i s f r a t e r n i i n A c t i s mensis Maii 1697 p r o p o s i t i de c u r v a i n f i n i t a s l o g a r i t m i c a s ad a n g u l o s r e c t o s s e c a n t e " . AE (1698; May), pp. 230-232. = JEO 11, pp. 806-813. = JBOO I , pp. 259-261.

2698c.

" S o l u t i o s e x problematum f r a t e r n o r u m ..'I. AE (1698; May), pp. 226230. = JBO I I , pp. 796-806. = JBOO I , pp. 256-259.

1698d.

" E x t r a i t d ' u n e l e t t r e d e Monsieur B e r n o u l l i de B Z l e , du 26 j u i n 1698, c o n t e n a n t l'examen de l a s o l u t i o n de s e s problsmes . . . ' I . Journal des Savans (1698; Aug.), pp. 355 f f . = JBO 11, pp. 829-839. = JEOO I , pp. 222-229.

1744a.

" A r t i c u l a I V : Demons t r a t i o anagrammatis Ephemerid. P a r i s . 1 1. a u g u s t . 1698 i n s e r t i " . JBO 11, pp. 1017-1020.

1744b.

" A r t i c u l a V: D e m o n s t r a t i o p o s t e r i o r i s anagrammatis Ephemer. P a r i s . 1 1 . aug. 1698 i n s e r t i " . JBO .TI, pp. 1021-1023.

.

B e r n o u l l i , Johann I

JBOO I,11,111, I V .

Johannis Bernoulli @era Omnia, tam m t e a sparsim e d i t a , quam hactenus i n e d i t a . Lausanne/GenSve, 1742. 4 V o l s . R e p r i n t : J.E.

Briepilechsel.

Hofmann ( e d . ) , H i l d e s h e i m , 1968.

Der EriefwechseZ von Johann Bernoulli.

B a s e l , 1955-.

Vol.

0. S p i e s s (ed.),

1.

1696.

"Supplemen tum d e f e c t u s geome t r i a e Cartes i a n a e c i r c a i n v e n tionem locorum". AE (1696; J u n e ) , pp. 264-269. = JBOO I , pp. 155-161.

1697a.

" C u r v a t u r a r a d i i i n d i a p h a n i s non u n i f o r m i b u s pp. 206-211. = JEOO I , pp. 187-193.

. . . ' I .

AE (1697; May),

23 1

Bibliography

1697b.

"Problemes a r e s o u d r e " . = JBOO I, pp. 204-205.

JournaZ des Savans (1697; Aug.), pp. 394 f f .

1 6 9 7 ~ . " L e t t r e de M r . B e r n o u l l i , p r o f e s s e u r d e Groningue, B M r . Varignon du 15 o c t o b r e 1697, s u r l e probleme d e s i s o p e r i m e t r e s " . Journal! des Savans (1697; D e c . ) , pp. 458 f f . = JEOO I, 206-213.

AE (1697;

169 7d.

" P r i n c i p i a c a l c u l i exponentialium, s e u percurrentium". March), pp. 125 f f . = JBOO I, pp. 179-187.

1698.

"Annotata i n s o l u t i o n e s f r a t e r n a s problematum quorumdam suorum AE (1698; O c t . ) , pp. 466-474. = JBOO I, pp. 262-271.

1702.

" S o l u t i o n d ' u n probleme c o n c e r n a n t l e c a l c u l i n t s g r a l

..

.'I.

Acad. S c i . Paris (1702; p u b l . 2nd. e d . 1737), pp. 289 f f .

..

.I1.

Me'moires I,

= JEOO

pp. 393-400. B e r n o u l l i , N i c o l a u s I (1687-1759)

Tentamen.

"Tentamen s o l u t i o n i s g e n e r a l i s p r o b l e m a t i s de c o n s t r u e n d a c u r v a , quae a l i a s o r d i n a t i m p o s i t i o n e d a t a s ad a n g u l o s r e c t o s s e c a t " . AE (1719; J u n e ) , pp. 295-304. = JBOO 11, pp. 305-314.

Demonstratio.

"Demonstratio a n a l y t i c a c o n s t r u c t i o n i s curvarum, quae a l i a s p o s i t i o n e d a t a s ad a n g u l o s r e c t o s s e c a n t , t r a d i t a e i n Actis L i p s . 1719 pag 295 e t seqq.". I n : a p p e n d i x 1 of t h i s book; a l s o i n : NicIB:LE 6 / 4 / 1 7 4 3 .

Modus inveniendi.

[Modus i n v e n i e n d i aequationem d i f f e r e n t i a l e m comple tam e x d a t a aequatione d i f f e r e n t i a l i incompleta] I n : N i c o l a u s I1 B e r n o u l l i 1 7 2 0 , 530. F o r more d e t a i l s on t h i s b i b l i o g r a p h i c a l m o n s t e r see f o o t n o t e 10, c h a p t e r 2.

.

B e r n o u l l i , N i c o l a u s I1 (1695-1726)

1716.

"Problema: d a t a s e r i e l i n e a r u m p e r r e c t a e i n eadem l i n e a c o n s t a n t i s v a r i a t i o n e m p r o d e u n t e ; i n v e n i r e a l i a m seriem l i n e a r u m , quarum q u a e v i s p r i o r e s omnes ad a n g u l o s r e c t o s s e c e t " . AE (1716; May), pp. 226-230. = JBOO 11,pp. 270-272.

1718.

"De t r a j e c t o r i i s c u r v a s o r d i n a t i m p o s i t i o n e d a t a s ad a n g u l o s r e c t o s v e l a l i a d a t a l e g e s e c a n t i b u s " . P J (1718; J u n e ) , pp. 248-262. = JBOO 11,pp. 286-298.

1720.

" E x e r c i t a t i o geometrica de t r a j e c t o r i i s o r t h o g o n a l i b u s .I1. Section I : AE (1720; May), pp. 223-237. S e c t i o n 11: AE Supplement 7 ( 1 7 2 1 ) , pp. 303-326. S e c t i o n 111: AE Supplement 7 ( 1 7 2 1 ) , pp. 337-353. = JEOO 11,pp. 423-472.

..

Bert r a n d , J 1864.

.

Trait6 de caZcuZ d i f f e ' r e n t i e l e t de caZcuZ intSgraZ. P a r i s , 1864. R e f e r e n c e t o Vol. 1 : CaZcuZ d i f f g r e n t i e z .

Bos, H . J . M . 1974a. 1974b.

" D i f f e r e n t i a l s , Higher-Order D i f f e r e n t i a l s , and t h e D e r i v a t i v e i n t h e L e i b n i z i a n Calculus". Archive H i s t . Ex. S c i . l 4 ( 1 9 7 4 / 5 ) , pp. 1-90. "The Lemniscate of B e r n o u l l i " .

In:

R.S. Cohen e t a l . ( e d s . ) , For

Dirk S t r u i k . D o r d r e c h t , 1974. pp. 3-14. 1981a.

"De ' c o n s t r u c t i e van v e r g e l i j k i n g e n ' , e e n v e r g e t e n v o o r o u d e r v a n de a l g e b r a i s c h e meetkunde". Nieuw T i j d s c h r i f t VOOT Wiskunde 6 8 ( 1 9 8 1 ) , pp. 154-165.

Bibliography

232

1981b.

"On t h e R e p r e s e n t a t i o n of Curves i n D e s c a r t e s ' GGomGtrie".

Hist. EX. S c i . 24(1981), pp. 295-338.

Archive

B u r k h a r d t , H.

1908.

fintwicklungen nach o s c i l l i e r e n d e n Functionen und I n t e g r a t i o n der DiSferentialgleichungen der mathematischen Physik. L e i p z i g , 1908. 2 Vols. = Jahresbericht DMV 10.

C a j o r i , F. 2928.

"The E a r l y H i s t o r y of P a r t i a l D i f f e r e n t i a l E q u a t i o n s and of P a r t i a l D i f f e r e n t i a t i o n and I n t e g r a t i o n " . Amer-ican Mathematical Monthly 35( l 9 2 8 ) , pp. 459-467.

1929.

A History of Mathematical Notations. Chicago, 1928-1929. Reference t o Vol. 2.

2 Vols.

C a n t o r , M. 1880.

Vorlesungen iiber Geschichte der Mathematik. L e i p z i g , 1880- 1908. 4 Vols. Reference t o Vol. 3.

C l a i r a u t , A . C.

1741.

(Taton 1976, n r . 25) "Recherches gEnErales s u r l e c a l c u l i n t E g r a 1 " . Me'moires Acad. Sci. Paris (1739; p u b l . 1741), pp. 425-436.

Cohen, I . B .

1974.

" I s a a c Newton". I n : C.C. G i l l i s p i e ( e d . ) , &ctionary of S c i e n t i f i c Biography. New York, 1970-1980. Vol. 10, pp. 42-103.

Demidov, S.S. 1974.

" D i f f e r e n t s i a l n y e u r a v n e n i y a s chastnymi proizvodnymi v r a b o t a k h zh. dalambera". Istoriko-Mutematicheskie Issledovuniya 19 ( 1974) , pp. 94-124.

1975.

"Vozniknovenie t e o r i i d i f f e r e n t s i a l n y k h u r a v n e n i i s chastnymi proizvodnymi". Ibidem 20( 19751, pp. 204-220.

1980.

" R a z v i t i e i s s l e d o v a n i i PO uravneniyam s chastnymi proizvodnymi pervogo p o r j a d k a v X V I I I - X I X w.". Ibidem 2 5 ( 1 9 8 0 ) , pp. 71-103.

1 9 8 2 ~ . "CrEation e t developpement de l a t h E o r i e d e s g q u a t i o n s d i f f g r e n t i e l l e s aux d E r i v e e s p a r t i e l l e s dans l e s t r a v a u x d e J. d ' b l e m b e r t " . Revue d ' f f i s t o i r e des Sciences 35 ( 1982), pp. 3-42. 1982b.

"The Study of P a r t i a l D i f f e r e n t i a l E q u a t i o n s of t h e F i r s t Order i n t h e 1 8 t h and 1 9 t h C e n t u r i e s " . Archive Hist. E's. S c i . 2 6 ( 1 9 8 2 ) , pp. 325-350.

D e s c a r t e s , R.

.. ..

"La GGomEtrie". I n : R. D e s c a r t e s , Discours de la me'thode L e i d e n , 1637. pp. 297-413. R e p r i n t e d and t r a n s l a t e d i n t o E n g l i s h i n : D.E. Smith & M.L. Latham, The Geometry of Rene Descartes w i t h a FucsimiZe of t h e First Edition. New York, 1954. L a t i n t r a n s a t i o n i n : F. van Schooten, Geometm'a 6 Renuto Des Cartes. Amsterdam, 659-166 1 . 2 Vols. bound i n I .

Geome'trie.

D i e t z , P.

1959.

"Die Ursprcnge d e r V a r i a t i o n s r e c h n u n g b e i Jakob B e r n o u l l i " . VerhandZungen der naturforschenden Gesellschaft i n Basel 7 0 ( 1959), PP 8 1- 146.

-

Bibliography

233

Edwards, C . H .

1979.

The H i s t o r i c a l Development o f t h e Calculus. New York e t c . , 1979.

EnestrEm, G.

2899.

"Sur l a d P c o u v e r t e de 1 ' P q u a t i o n gPnErale d e s l i g n e s ggodesiques".

Bibliotheca Mathematica (nouv. s e r . ) 1 3 ( 1 8 9 9 ) , pp. 19-24.

I. Bibliotheca Mathemutica

1901.

[ M i t t e i l u n g 3:88l

1970.

" V e r z e i c h n i s d e r S c h r i f t e n Leonhard E u l e r s " . Jahresbericht DMV Erganzungsband 4(1910).

1913.

" B e r i c h t a n d i e Eulerkommission d e r S c h w e i z e r i s c h e n n a t u r f o r s c h e n d e n G e s e l l s c h a f t Gber d i e E u l e r s c h e n Manuskripte d e r P e t e r s b u r g e r Akademie". Jahresbericht DMV 2 2 ( 1 9 1 3 ) , pp. 191-205 ( i t a l i c i s e d pagination).

(3) 2 ( 1 9 0 1 ) , p . 443.

Engelsman, S.B.

7980.

"Lagrange's E a r l y C o n t r i b u t i o n s t o t h e Theory of F i r s t - O r d e r P a r t i a l D i f f e r e n t i a l Equations". H i s t o r i a Mathematica 7 ( 1 9 8 0 ) , pp. 7-23.

Euclid

Data.

c f . Thaer 1962.

E u l e r , L.

LEOO 1.YY. Leonhardi Euleri Opera Omnia; s e r i e s prima: opera mathernatica. Various e d i t o r s , v a r i o u s p l a c e s of p u b l i c a t i o n , 1911-1956. 29 V o l s . R e f e r e n c e t o Vol. YY.

Leonhardi Euleri @era Ornnia; s e r i e s secunda: opera mechanica e t astronomica. V a r i o u s e d i t o r s , v a r i o u s p l a c e s of p u b l i c a t i o n ,

LEOO 2. Y Y .

1912-. R e f e r e n c e t o Vol. YY.

Leonhardi Euleri Opera h i a ; s e r i e s quarta A : cornercium epistolicwn. E.A. Fellmann e t a l . ( e d s . ) , B a s e l , 1975-. R e f e r e n c e

LEOO 4A.Y.

to

VOl.

Opera postuma.

Y.

See Fuss

2862a,b.

Adversaria Mathematica. 397-408.

Unpublished n o t e b o o k s ; c f . K o p e l e v i c 7 9 6 2 , n r s . A l s o : EnestrEm 2 9 1 3 , H I - 1 3 .

De d i f f e r e n t i a t i o n e .

(H20; Kop. 86) "De d i f f e r e n t i a t i o n e Functionum d u a s p l u r e s v e v a r i a b i l e s q u a n t i t a t e s i n v o l v e n tium". I n : a p p e n d i x 2 of t h i s book.

Indez capitum.

(H26; Kop. 92) " ( I n d e x c a p i t u m t r a c t a t u s cujusdam m a j o r i s de a n a l y s i i n f i n i t o r u m ) " . I n : LEOO 1 . 2 9 , pp. XXXI-XXXVIII.

De infinitis curvis.

(E44) "De i n f i n i t i s c u r v i s eiusdem g e n e r i s s e u methodus i n v e n i e n d i a e q u a t i o n e s p r o i n f i n i t i s c u r v i s eiusdem g e n e r i s " . Corn. Acad. S c i . Petrop. 7(1734/5; p u b l . 1740), pp. 174-189 & pp. 180-183. = LEOO 1 . 2 2 , pp. 36-56.

Additarnentzm.

(E45) "Additamentum ad d i s s e r t a t i o n e m de i n f i n i t i s c u r v i s eiusdem g e n e r i s " . Corn. Acad. S c i . Petrop. 7(1734/5; p u b l . l 7 4 0 ) , pp. 184-200. = LEOO 1 . 2 2 , pp. 57-75.

SoZutio problematwn.

( E 5 1 ) " S o l u t i o problematum r e c t i f i c a t i o n e m e l l i p s i s r e q u i r e n t i u m " . Corn. Acad. S c i . Petrop. 8(1736; p u b l . 1741), pp. 86-98. = LEOO 2 . 2 0 , pp. 8-20.

234

Bibliography

constructione. (E70) "De Petrop. 9(1737; publ. ICD. (E212) I n s t i t u t i o n e s finitorwn uc doctrina

~e

constructione aequationum". Corn. &ad. S c i . 1744), pp. 85-97. = LEOO 1 . 2 2 , pp. 150-161. caZculi d i f f e r e n t i a l i s cwn e i u s usu in analysi serierwn. Berlin, 1755. = LEOO 1 . 1 0 .

(E342, E366, E385) I n s t i t u t i o n w n calculi i n t e g r a l i s volwnen prirnwn/secwzdwn/tertim Petersburg, 1768-1770. 3 Vols. = LEO0 1 . 1 1 / 1 . 1 2 / 1 . 1 3 .

ICI I , 11, I I I .

....

1733.

(Ell) "Constructio aequationum quarundam differentialium quae indeterminatarum separationem non admittunt". Nova A E (1733), pp. 369-373. = LEO0 1 . 2 2 , pp. 15-18.

1738a.

(E28) "Specimen de constructione aequationum differentialium sine indeterminatarum separatione". Comm. Acad. S c i . Petrop. 6( 1732/3; publ. 1738), pp. 168-174. = LEOO 1.20, pp. 1-7.

1738b.

(E31 ) "Constructio aequationis differentialis a x &=dy+y*&". Corn. Acad. S c i . Petrop. 6(1732/3; publ. 1738), pp. 124-137. = LEO0 1 . 2 2 , pp. 19-35.

1744.

(E65) Methodus inveniendi l i n e a s cumas maximi rninimive p r o p r i e t a t i s gaudentes Lausanne/GenSve, 1744. = LEOO 1.24.

1746.

(E851) "Recensio dissertationis de ventis". In: pp. 793-797.

n

....

Fuss 1 8 6 2 b ,

(E101/2) Introductio i n analysin i n f i n i t o r w n . Lausanne, 1748. 2 Vols.

1748.

= LEOO 1 . 8 / 9 .

1758.

(E236) "Exposition de quelques paradoxes dans le calcul int6gral". Memoires Acad. S c i . B e r l i n 12(1756; publ. 1758), pp. 300-321. = LEOO 1.22, pp. 214-236.

1770.

(E390) "Considerationes de traiectoriis orthogonalibus". Nevi Corn. Acad. S c i . Petrop. 14(1769; publ. 1770), pp. 46-71. = LEOO 1.28, pp. 99-119.

1773.

(E433) "Digressio de traiectoriis tam orthogonalibus quam obliquangulis". Novi Corn. Acad. S c i . Petrop. 17(1772; publ. 1773), pp. 205-248. = LEO0 2.28, pp. 215-247.

1787.

(E609) "Considerationes super traiectoriis tam rectangulis quam obliquangulis". Nova Acta Acad. S c i . Petrop. 1 ( 1 783; publ. 1787) , pp. 3-46. = LEOO 1 . 2 9 , pp. 28-71.

Fleckenstein, J.O. "Nikolaus I Bernoulli". In: C.C. Gillispie (ed.) , Dictionary of S c i e n t i f i c Biography. New York, 1970-1980. Vol. 2 , pp. 56-57. Fuss, Paulus H. 1970.

1843a, b.

Correspondunce mathgrnatique e t physique de quelques cdlDbres g9ome'tres du XVIII?e sie'cle. Petersburg, 1843. 2 Vols.

FUSS, Paulus H. & N. Fuss (eds.)

1862a,b. Leonhardi Euleri Opera Postwna. Petersburg, 1862. 2 Vols. Goldstine, H.H. 1980.

A History of the CalcuZus of Variations from the 17th through the 29th Century. New York etc., 1980.

235

Bibliography

G r a t t a n Guinness, I .

1979.

B r i t i s h Journal

"Essay review 'Babbage's Mathematics i n i t s Time'".

f o r the History of Science 12(1979), pp. 82-88. Greenberg, J. L. "Alexis F o n t a i n e ' s ' F l u x i o - D i f f e r e n t i a l Method' and t h e O r i g i n s o f t h e C a l c u l u s of S e v e r a l Variables". Annals of Science 38(1981), pp. 251-290. Haas, K.

1955.

"Die mathematische A r b e i t e n von Johann Hudde (1628-1704) B i r g e m e i s t e r von Amsterdam". Centaurus 4(1955/6), pp. 235-284.

H a l l , A.R.

1980.

Philosophers a t War; the Quarrel between Newton and Leibniz. Cambridge, 1980.

Hankins, T.L.

1970.

Jean d'Alembert. Science and the EnZightenment. Oxford, 1970.

Harnack, A.

1900.

Geschichte der Kliniglich Preussischen Akademie der Wissenschaften zu Berlin. B e r l i n , 1900. 3 Vols. Reference t o Vol.

2.1.

Henny, J.

1975.

"Niklaus und Johann B e r n o u l l i ' s Forschungen auf dem G e b i e t d e r Wahrscheinlichkeitsrechnung i n ihrem B r i e f w e c h s e l m i t P i e r r e RBmond de Montmort". I n : B.L. van d e r Waerden ( e d . ) , Die Werke don Jakob Bernoulli. Vol. 3, B a s e l , 1975. pp. 457-507.

Hermann, J.

1717.

"Schediasma de t r a j e c t o r i i s d a t a e s e r i e i c u r v i s ad angulos r e c t o s o c c u r r e n t i b u s . . . ' I . AE ( 1 7 1 7 ; Aug.), pp. 348-352. = JBOO 11, pp. 275-279.

1718.

"Supplementum s o l u t i o n i s suae p r o b l e m a t i s de t r a j e c t o r i i s curvarum i n v e n i e n d i s ...'I. AE (1718; J u l y ) , pp. 335-336. = JBOO II, pp. 279-281.

1719.

"Additamentum ad schedas s u p e r problema t r a j e c t o r i a r u m (1719; F e b r . ) , pp. 68-77. JBOO 11, pp. 299-305.

.,

.'I.

AE

Higgins, Th. J.

1940.

"A Note on t h e H i s t o r y of Mixed P a r t i a l D e r i v a t i v e s " .

Scripta Mathe-

matica 7(1940), pp. 59-62. Ho fmann , J. E.

1956,

Ueber Jakob Bernoullis Beitriige z u r Infinitesimalmathematik. Gensve,

1959.

"Um E u l e r s e r s t e Reihenstudien". I n : K. SchrEder ( e d . ) , SameZband der zu Ehren des 250. Gebwtstages Leonhard EuZers der Deutschen

1956.

Akademie der Wissenschaften zu Berlin vorgelegten AbhandZungen. B e r l i n , 1959. pp. 139-208.

1970.

"Jakob B e r n o u l l i " . I n : C.C. G i l l i s p i e ( e d . ) , f i c t i o n a r y of Scient i f i c Biography. New York, 1970-1980. Vol. 2 , pp. 46-51.

236

Bibliography

1 1 H 8 p i t a l , G.F.A.

1696.

de

AnaZyse des i n f i n i m e n t p e t i t s , p o w 2. ' i n t e l l i g e n c e des Zignes courbes. 1 s t E d i t i o n : P a r i s , 1696. 2nd E d i t i o n : P a r i s , 1715.

Hoppe, E.

I 9 2 7,

.

"Die En t deckung d e r p a r t i e 1l e n D i f f e r e n t i a 1g l e i chungen" Archiv Geschichte der Mathematik, der Natumissensch. und der Technik 10(1927), p p .

158-165.

Huygens, C. OEuvres Corriple'tes. OEuvres compldtes de C h r i s t i o m Huugms. Den Haag, 18881950. 2 2 V o l s .

1690.

Trait6 de Za Zumie're

....

Leiden, 1690. = OEuvres CompZBtes 19,

pp. 453-548. Kohli, K. 1 9 7 5 ~ . "Zur P u b l i k a t i o n s g e s c h i c h t e dor A r s Conjectandi". I n : B.L. van d e r Waerden ( e d . ) , Die Werke von Jakob Bernoulli. Vol. 3, B a s e l , 1975. pp. 391-401.

1975b.

"Komentar z u r D i s s e r t a t i o n von Niklaus B e r n o u l l i : De Usu A r t i s Conjectandi i n Jure". I n : I b i d e m , pp. 541-556.

Kopelevic, Yu.Kh. e t a l .

1962.

Rukopisnye materiaZy L. EiZera w arkhiwe Akademii Nauk SSSR. Moscow1 Leningrad,

1962.

Lakatos, I.

1976.

Proofs and Refutations; The Logic of MathematicaZ Discovery. John Worrall and E l i e Zahar ( e d s . ) , London e t c . ,

1976.

L e i b n i z , G.W.

LMS Y .

Leibnizens Muthematische S c h r i f t e n . C . I . Gerhardt ( e d . ) ,

Halle, 1855. 7 Vols. Reference t o Vol. Y. R e p r i n t : Hildesheim e t c . , 1971.

BeiZage. 1684.

1686.

[Beilage t o t h e l e t t e r GWL:JohB 3 / 8 / 1 6 9 7 . ]

LMS I I I / Z , pp. 451-454.

"Nova methodus p r o maximis e t minimis, itemque t a n g e n t i b u s quae nec f r a c t a s nec i r r a t i o n a l e s q u a n t i t a t e s moratur, e t s i n g u l a r e p r o i l l i s c a l c u l i genere". AE (1684; O c t . ) , pp. 467-473. = LMS V , pp. 220-226. "De geometria r e c o n d i t a e t a n a l y s i i n d i v i s i b i l i u m a t q u e i n f initorum". = LMS V , pp. 226-233.

AE (1686; J u n e ) , pp. 292-300.

1692.

"De l i n e a ex l i n e i s numero i n f i n i t i s o r d i n a t i m d u c t i s i n t e r se c o n c u r r e n t i b u s formata easque omnes t a n g e n t e , ac de novo i n e a re a n a l y s i s i n f i n i t o r u r n usu". AE (1692; A p r . ) , pp. 168-171.

1694.

"Nova c a l c u l i d i f f e r e n t i a l i s a p p l i c a t i o e t usus ad m u l t i p l i c e m linearum constructionem ex d a t a tangentium c o n d i t i o n e " . AE (1694; J u l y ) , pp. 311-316.

LindelGf, L. 1867.

"Remarques s u r l e s d i f f 6 r e n t e s maniPres d ' 6 t a b l i r l a formule

d2z/dxdy=d2z/dydx1'. Acta Soc. S c i . Fennicae 8 ( 1867), pp. 206-213. Lctzen, J.

1.978.

" ' P r o o f s and R e f u t a t i o n s ' , 104(1978).

A Reunion". B e p r i n t Math. I n s t . Utrecht

237

Bibliography

Mejlbo, L.C. 1980.

"Ortogonaltrajektorier 1694-1720. Et kapitel i analysens tidlige historie". Nordisk Matematisk T i d s k r i f t 2 8 ( 19801, pp. 41-51.

Montucla, J.F. 1802.

Histoire des Mathe'matiques. Paris, 1799-1802. Paris, 1968.

4 Vols. Reprint:

Muller, K. and G. KrEnert

Leben und Werk von G o t t f r i e d WiZheIm Leibniz. Eine Chronik. Frankfurt am Main, 1969. Newton, I. 1969.

Correspondence. The Correspondence of Isaac Newton. H.W. Turnbull et a l . (eds.), Cambridge, 1959-1977. 7 Vols. Mathematical Papers. The Mathematical Papers of Isaac Newton. D.T. Whiteside (ed.), Cambridge, 1967-1981. 8 Vols. 1716.

"Problematis olim in Actis Eruditorum Lipsiae propositi solutio generalis". PhiZosophical Transactions 2 9 ( 1 7 1 6 ) , pp. 399-400. = JBOO 11, pp. 273-274.

Poggendorff, J. C. 186 3.

Biographisch- Ziterarische s Handwijrterbuch zur Geschichte der exacten Vissenschaften e t c . Vol. 1 (A-L) : Leipzig, 1863.

Ravetz, J. R. 1961.

"Vibrating Strings and Arbitrary Functions". In: Logic of Personal Knowledge: Essays Presented t o Michael Polanyi on h i s S e v e n t i e t h Birthday, 11th March 1961. London, [ 1 9 6 1 ] . pp. 71-88.

Schafheitlin, P. 1922.

"Johannis (I) Bernoullii Lectiones de calculo differentialium". VerhandZungen der Naturforschenden Gesellschaft in Basel 3 4 ( 1 9 2 2 ) , pp. 1-32.

Schlgmilch, 0. 1868.

Compendium der Hijheren Analysis. Braunschweig, 1868. Vol. 1 .

Schwarz, H . - A . 1873.

[Comunications]. Archives des Sciences physiques e t naturezles (nouv. per.) 4 8 ( 1 8 7 3 ) , pp. 33-44.

Spiess, 0 .

Briefwechsel. 1940.

Cf. Johann Bernoulli Briefwechsel.

"Johannes Bernoulli und seine SEhne. Zur Geschichte der grossen Basler Mathematiker-Familie". A t l a n t i s ( l 9 4 0 ) , pp. 6 6 3 ff.

Stolz, 0 . 1893.

Grmdziige der D i f f e r e n t i a l - und IntegraZrechnung. Leipzig, 1893.

Struik, D . J . 1969.

A Source Book in Mathematics, 1200-1800. Cambridge Mass., 1969.

238

Bibliography

Taton, R. 2976.

" I n v e n t a i r e chronologique de l ' o e u v r e d'Alexis-Claude C l a i r a u t (1713-1765)". Revue Hist. S c i . 29(1976), pp. 97-122.

Taylor, Angus 1974.

"The D i f f e r e n t i a l : N i n e t e e n t h and Twentieth Century Developments". Archive Hist. Ex. S c i . 12(1974), pp. 355-383.

T a y l o r , Brook

1727.

" S o l u t i o p r o b l e m a t i s a Domino G. G. L e i b n i t i o g e o m e t r i s Anglis nuper p r o p o s i t i " . PhiZosophical Transactions 30( 17171, pp. 695-701. = JBOO 11, pp. 281-285.

Thaer, C. 1962.

Die Data von E u k l i d . B e r l i n e t c . ,

1962.

T o r r i c e l l i , E.

1644.

De motu gravium n a t u r a l i t e r descendentium e t proiectorwn Zibri duo. I n : @era Geometrica. F l o r e n c e , 1644. pp. 95-243.

T r u e s d e l l , C.A.

1960.

I11

The Rational Mechanics o f Flexible o r E l a s t i c Bodies 1638-1788. I n : LEO0 2 . 1 1 / 2 .

Whiteside, D.T. 1960.

" P a t t e r n s of Mathematical Thought i n t h e L a t e r S e v e n t e e n t h Century". Archive Hist. Ez. S c i . 1(1960/2), pp. 179-388.

Winter, E. 1957.

Die Registres der Berliner Akademie der Wissenschaften 1746-1 766. Dokumente fiir das Wirken Leonhard Euler's i n Berlin. B e r l i n , 1957.

Youshkevitch, A.P.

1956.

" H i s t o r i s c h e r Abriss". I n : W.W. Stepanow, Lehrbuch d e r D i f f e r e n t i a l gleichungen. B e r l i n , 1956. pp. 415-449.