Numerical notation: A comparative history

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Numerical Notation This book is a cross-cultural reference volume of all attested numerical notation systems (graphic, nonphonetic systems for representing numbers), encompassing more than 100 such systems used over the past 5,500 years. Using a typology that defies progressive, unilinear evolutionary models of change, Stephen Chrisomalis identifies five basic types of numerical notation systems, using a cultural phylogenetic framework to show relationships between systems and to create a general theory of change in numerical systems. Numerical notation systems are primarily representational systems, not computational technologies. Cognitive factors that help explain how numerical systems change relate to general principles, such as conciseness and avoidance of ambiguity, which also apply to writing systems. The transformation and replacement of numerical notation systems relate to specific social, economic, and technological changes, such as the development of the printing press and the expansion of the global world-system. Stephen Chrisomalis is an assistant professor of anthropology at Wayne State University in Detroit, Michigan. He completed his Ph.D. at McGill University in Montreal, Quebec, where he studied under the late Bruce Trigger. Chrisomalis’s work has appeared in journals including Antiquity, Cambridge Archaeological Journal, and Cross-Cultural Research. He is the editor of the Stop: Toutes Directions project and the author of the academic weblog Glossographia.

Numerical Notation A Comparative History

Stephen Chrisomalis Wayne State University

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521878180 © Stephen Chrisomalis 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13

978-0-511-67934-6

eBook (EBL)

ISBN-13

978-0-521-87818-0

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Acknowledgments

page vii

1

Introduction

1

2

Hieroglyphic Systems

34

3

Levantine Systems

68

4

Italic Systems

93

5

Alphabetic Systems

133

6 South Asian Systems

188

7

Mesopotamian Systems

228

8

East Asian Systems

259

9

Mesoamerican Systems

284

10

Miscellaneous Systems

309

11

Cognitive and Structural Analysis

360

v

Contents

vi 12

Social and Historical Analysis

401

13

Conclusion

430

Glossary

435

Bibliography

439

Index

471

Acknowledgments

Although the history of scholarship on numeration is lengthy and includes such illustrious figures as Alexander von Humboldt, Alfred Kroeber, and Oswald Spengler, its temporal and spatial breadth inevitably means that its practitioners frequently operate in a seeming near-vacuum. For this reason I am doubly grateful for the assistance I have received over the decade since this work’s inception. This book had its genesis during my time at McGill University. The late Bruce Trigger was the shepherd and guiding hand behind this book, beginning in its formative stages and continuing almost to the final draft. The central premise of this book stems from Bruce’s conviction that comparative research is not only possible but indeed necessary in order for anthropology to be theoretically meaningful. Without Bruce’s mentorship and support for me throughout this decidedly unorthodox anthropological pursuit, this book would not exist. Bruce’s death in 2006 was a momentous loss for the discipline and for comparativism. At McGill, in addition to Bruce, Michael Bisson, Andre Costopoulos, Jim Lambek, and Jerome Rousseau read the manuscript and provided useful suggestions for improvement at various stages, as well as providing invaluable moral support to me. Funding at this stage of the research was provided by a Social Sciences and Humanities Research Council (Canada) doctoral fellowship. I wish to thank particularly the interlibrary loan staff at McGill’s McLennan Library, who went well beyond the call of duty in tracking down obscure material.

vii

viii

Acknowledgments

Further refinements and a new draft of the book were produced under a SSHRC postdoctoral fellowship at the University of Toronto. While I was in Toronto, Richard Lee, Trueman MacHenry, and David Olson were particularly helpful to me and provided useful insights on the theories and concepts underlying my work, forcing me to clarify my own positions in ways that I had not previously done. Bob Bunker, John Gilks, Heather Hatch, Andy Pope, and Shana Worthen read portions of the manuscript at this stage and provided very useful editorial advice. A work of this scope inevitably relies upon the individual and collective experience of regional specialists in the writing systems and mathematical practices of various regions and periods, and of theorists working in cognitive and psychologically oriented anthropology and linguistics. I have benefited tremendously from the specialized expertise of Priskin Gyula, Christopher Hallpike, Jim Hurford, Joel Kalvesmaki, Eleanor Robson, Nicholas Sims-Williams, Matthew Stolper, and Konrad Tuchscherer. A School of Advanced Research Advanced Seminar entitled The Shape of Script was the key to moving my work into its final completed form, and introduced me to many additional regional specialists whose advice has been of assistance: John Baines, John Bodel, Beatrice Gruendler, Stephen Houston, David Lurie, Kyle McCarter, John Monaghan, Richard Salomon, Kyle Steinke, and Niek Veldhuis. Scholars of numeration include historians, anthropologists, archaeologists, linguists, mathematicians, and psychologists, to name only a few, and it is all too easy in such a disparate crowd of research traditions to lack a sense of disciplinary cohesion and of one’s scholarly influences. I therefore acknowledge my intellectual forebears in the comparative study of numerals, most notably Florian Cajori, Genevieve Guitel, Karl Menninger, Antoine Pihan, and David Eugene Smith. Although I disagree with his conclusions in many places, I thank Georges Ifrah, whose gargantuan and important Histoire universelle des chiffres (1998) inspired me to produce this volume. Eric Crahan and Frank Smith at Cambridge University Press deserve great credit for their skillful guidance of my work through the editorial process at all stages. Russell Hahn guided the complex copyediting masterfully, and Leah Shapardanis prepared the index and read proofs. Many thanks to the fourteen anonymous reviewers who read and commented on one or more chapters on behalf of the Press, and to the entirety of the production staff for their handling of dozens of specialized typefaces. To my family, all my love and thanks. Arthur Chrisomalis provided useful firsthand insights into the childhood acquisition of lexical and graphic numeration, and rekindled his father’s wonderment at the magic of numbers. Finally, this work is dedicated with love to my wife, Julia Pope, for her patience with me over

Acknowledgments

ix

the past decade, her keen editorial eye, her endless willingness to reread manuscript chapters, and her ongoing conviction that this work is worthwhile. Despite the advice and assistance of the abovementioned, and any others I have forgotten, I have doubtless made many errors of fact and interpretation, and I eagerly anticipate the opportunity to broaden my knowledge of numerical notation systems in the future.

Notes on Style Throughout the book I have used the conventions “bc” and “ad” to refer to chronological periods. Where no era indicator is associated, ad dates are assumed; I do so only when the interpretation of a date is obvious.

chapter 1

Introduction

The Western world is a world of written numbers. One can hardly imagine an industrial civilization functioning without the digits 0 through 9 or a similar system. Yet while these digits have pervasive social and cognitive effects, many unanswered questions remain concerning how humans use numerals. Why do societies enumerate? How does the representation of numbers today differ from their representation in the past? Why does the visual representation of number figure so prominently in complex states? What cognitive and social functions are served by numerical notation systems? How do numeral systems spread from society to society, and how do they change when they do so? And, despite their present ubiquity, why have the vast majority of human societies not possessed them at all? If you look up from this page and examine your surroundings, I am certain that you will encounter at least one instance of numerical notation, probably more. Moreover, unless you have a Roman numeral clock nearby, I am nearly certain that all of the numerals you encounter are those of the Hindu-Arabic or Western1 system. Numerals serve a wide variety of functions: denotation – “Call George, 1

The conventional term used in popular literature, “Arabic numerals,” and the term used in most scholarly literature, “Hindu-Arabic numerals,” can lead to considerable confusion because the scripts used to write the Hindi and Arabic languages use numerical notation systems that differ from those of the West in the shape of the signs. I use the term “Western numerals” to refer to this system because it developed in Western Europe in the late Middle Ages, while fully acknowledging its Indian and Arabic ancestry.

1

2

Numerical Notation

876–5000”; computation – “21.00 × 1.15 = 24.15”; valuation – “25 cents”; ordination – “1. Wash dishes, 2. Sweep floor, 3. Finish manuscript”; and so on. Most of the thousands of numerals we see each day barely register on our conscious minds; regardless, we encounter far more written numbers in our lifetime than we do sunsets, songs, or smiles. Until the past few centuries, the opposite was true for most people. These ten digits are so prevalent that it is easy to equate our numeral-signs with the set of abstract numbers. In this view, 62 does not merely signify the abstract concept “sixty-two” – it is the raw form of the number itself, the stuff of pure mathematics (or perhaps pure numerology). That these signs are frequently encountered and used in mathematical contexts contributes to the prevalence of such attitudes. According to this view, our numeral-signs constitute abstract number, and other systems (when recognized as such) are simply archaic deviations from the abstract entity comprised by these signs. This view is erroneous, and rests on the confusion of a mental concept (signified) with its symbolic representation (signifier). Our numerical notation system has an extensive history, as do the more than one hundred systems that have existed over the past five thousand years. Still, the worldwide prevalence of Western numerical notation is undeniable. Most literate individuals worldwide, as well as a sizable number of illiterates, understand them. Nor does any competing system have any reasonable chance of supplanting our system in the near future. This has led many scholars to assert its supremacy solely on the evidence of its near-universality (Zhang and Norman 1995; Dehaene 1997; Ifrah 1998). Nevertheless, this situation does not imply that our system will dominate the whole world forever. The study of numerical notation remains mired in a theoretical framework that has much more in common with late nineteenth-century unilinear evolutionism in anthropology than it does with early twenty-firstcentury critiques of unfettered scientific progress. Despite this theoretical weakness, numerical notation as a topic of academic study is a relatively common pursuit, with linguists, epigraphers, archaeologists, anthropologists, historians, psychologists, and mathematicians all making significant contributions to the literature. These studies are mostly restricted to the analysis of one or a few numerical notation systems, although a small number of synthetic and comparative works dealing with numerical notation exist (Cajori 1928; Menninger 1969; Guitel 1975; Ifrah 1998). However, such works rarely consider more obscure numerical notation systems, such as those of subSaharan Africa, North America, and Central Asia. Similarly, social scientists such as the anthropologist Thomas Crump (1990), the psychologist David Lancy (1983), and the ethnomathematicians Marcia Ascher (1991) and Claudia Zaslavsky (1973) have undertaken major comparative research on numeracy and mathematics in

Introduction

3

non-Western societies. Yet numerical notation has not been a primary focus of this body of research. This study is a comparative analysis of all numerical notation systems known to have existed throughout history – approximately one hundred distinct systems, most of which can be grouped into eight distinct subgroups. By presenting a universal study of such systems and examining the historical connections and contexts in which they are encountered, I will develop a framework that accounts for cultural universals, identifies evolutionary regularities, and yet remains cognizant of idiosyncratic features, seeking to determine, rather than to assume, the amount of intercultural variability among them. I will distinguish several major types of numerical notation, evaluate their efficiency for performing specific functions, link their features to human cognitive capacities, and relate systems to their sociopolitical contexts.

Definitions A numerical notation system is a visual, relatively permanent, and primarily nonphonetic structured system for representing numbers. Signs such as 9 and 68, IX and LXVIII, are part of numerical notation systems, but numeral words such as nine and achtundsechzig are not. Though there are ties between numeral words and numerical notation, a lexical numeral system, or the sequence of numeral words in a language (whether written or spoken), has a language-specific phonetic component. Every language has a lexical numeral system of some sort, while numerical notation is an invented technology that may or may not be present in a society.2 Some numerical notation systems contain a small phonetic component, as in acrophonic systems whose signs are derived from the first letters of the appropriate number-words in a language. However, since such systems are still comprehensible without having to understand a specific language, they are numerical notation systems. Numerical notation systems must be structured. Simple and relatively unstructured techniques, such as marking lines on a jailhouse cell to count one’s days or piling pebbles in a basket, are largely or entirely unstructured. They rely on oneto-one correspondence, in which things are counted by associating them with an equal number of marks or other identical objects. A numerical notation system, by contrast, is a system of different discrete numeral-signs: single elementary symbols, or, in the terminology used in writing systems, graphemes, which are then 2

I will leave aside for the moment discussions of counterevidence questioning the assumption of the universality of lexical numeral systems (Hurford 1987: 68–78; Gordon 2004; Everett 2005).

4

Numerical Notation

used in combination to represent numbers.3 A numeral-phrase is a group of one or more numeral-signs used to express a specific number (e.g., MMDXXV); numeral-phrases such as 8 or Roman L are nonetheless complete even though they only use one sign apiece. All numerical notation systems (and most lexical numeral systems) are structured by means of powers of one or more bases. A power is a number X multiplied by itself some number of times (its power); 101 = 10, 102 = 100, 103 = 1000, etc. By mathematical definition, a number raised to the power 0 equals 1. A base is a natural number B in which powers of B are specially designated. While mathematicians normally require that a base be extendable to an infinite number of powers of B (e.g., 10, 100, 1000, 10,000, ... ad infinitum), most numerical notation systems are not infinitely extendable. It is sufficient that some powers of B are specially designated within a numerical notation system. Western numerals and many other systems use a base of 10, but this is not universal. In addition to its base, a numerical notation system may have one or more sub-bases that structure it. The Roman numeral system has a primary base of 10 with a sub-base of 5. Unlike bases, the powers of sub-bases are not specially designated; there are no special Roman numerals for 25 or 125. It is, rather, the products of a sub-base and the powers of the primary base that are specially designated – for the Roman numerals, 50 (5 × 10) and 500 (5 × 100). Two topics that I will present only peripherally are number and mathematics. Number is an abstract concept used to designate quantity. For the purposes of my study, a simple (if philosophically naïve) definition will suffice. Questions such as whether numbers are “real” or Platonic entities, or the connection of the set of natural numbers to formal logic, are beyond its scope. The distinction between cardinal numbers – denoting quantity but not order – and ordinal numbers – designating ordered sequences – is extremely important for lexical numerals, where many languages use different series of words (e.g., two versus second ) for the two concepts. This distinction also has implications for our understanding of the origin of numerals and numerical concepts in humans (Crump 1990: 6–10), but has little influence on numerical notation. In defining mathematics as the science that deals with the logic of quantity, shape, and arrangement, I am consciously employing a simple definition for a complex term. In order to understand numerical notation, one needs no mathematical ability save some knowledge of basic arithmetic. While some parts of mathematics make frequent 3

A few numeral-signs are more complex in that they graphically combine two or more signs into one in order to represent multiplication, but they are treated as elementary numeral-signs because their use is identical to that of all other simple signs in the systems in question.

Introduction

5

use of numbers (number theory being the most obvious example), large parts of the discipline have only infrequent or peripheral encounters with numerical notation. Numerical notation systems are not necessarily designed with mathematical purposes in mind. Even in contemporary industrial societies, where mathematical ability is more extensive than in any other historical or modern society, most numerical notation is nonmathematical.

Universal Comparison The present study is, as far as possible, a universal one. I have not excluded any numerical notation system intentionally save where data are not plentiful enough to undertake a reasonable analysis. Most comparative research in anthropology aims to discover generalizations and patterns in human behavior, but using the universe of cases is neither possible nor desirable in most cross-cultural studies. In order to use most analytical statistics on cross-cultural data, each case must be independent of the others, which requires that each case may not be historically derived or diffused from any other case. This issue, known as Galton’s problem, is the thorniest methodological issue in statistical cross-cultural research (Naroll 1968: 258–262). The establishment of correlations between traits among historically independent societies is enormously useful, and is the basis for most cross-cultural research in modern anthropology. Yet to do so in a study such as this one, in which there are perhaps only seven independently invented numerical notation systems, would be pointless. Firstly, seven cases would be too small a sample to analyze statistically. Secondly, by studying all cases, I am able to show that the total observable variability among numerical notation systems is far greater than has previously been believed. This variability cannot be understood by studying only a fraction of numerical notation systems. To paraphrase the old fable, if we study only the elephant’s trunk or tail, we ignore most of the animal. Thirdly, I wish to explain structural variation among historically related systems, which frequently differ considerably from their relations. This would be impossible using a sampling technique that omitted related cases. Finally, were I to omit related cases, I could not analyze how systems change over time or how new systems develop out of existing ones. By taking events of change, rather than static systems, as the units of analysis in my comparisons, I am able to elucidate both synchronic and diachronic patterns among numerical notation systems. It is worth noting that Galton’s problem does not apply to events of change of the sort I am analyzing, since every event is essentially independent of every other, and can thus be analyzed statistically, where relevant. I reject as false the dichotomy in anthropology between universalism (Tylor 1958 [1871], White 1949, 1959; Steward 1955; Harris 1968) and relativism (Lowie 1920,

6

Numerical Notation

Boas 1940, Sahlins 1976, Geertz 1984), both of which presume rather than evaluate the degree of regularity that social phenomena display. While numerical notation systems display remarkable regularities and even universals, historical contingencies also played a major role in shaping the cultural history of numerical notation. Yet the only way to determine which features of numeration are cross-culturally regular and which are idiosyncratic is to undertake cross-cultural comparison. The best way to deal with the messiness of the world – less universal than universalists would like, less relative than relativists prefer – is through a body of theory that deals with constraints. Most anthropological theory is predicated on the existence of very strong constraints on the forms possible within human societies. Some of these constraints are so strong as to produce cross-cultural universals (Brown 1991). Most cultural relativists dismiss these universals as minimally true, but facile, irrelevant, and useless for understanding humanity (cf. Geertz 1965, 1984). The denial of comparativism on this basis is an overly negative position, given that those who criticize comparativism most harshly are very often those who have not undertaken it. One of the most crucial theoretical contributions of anthropology should be to indicate the degree to which human societies are alike and the degree to which they differ. While some aspects of human existence are truly universal, and others are almost infinitely variable, most of the really intriguing domains of activity fall somewhere in the middle. In the early 1900s, Alexander Goldenweiser developed his “principle of limited possibilities,” which stated that for any social or cultural phenomenon, there are a limited number of possible forms that can be expressed in human societies (Goldenweiser 1913). Goldenweiser was particularly interested in the limitations imposed by human psychology on the expression of cultural traits, although, given the inchoate nature of psychological theory at the time, he was unable to describe these mechanisms precisely. Bruce Trigger (1991) has rejuvenated the idea of constraints, proposing that anthropologists should use the concept of constraint to describe the limitations on human sociocultural variation – whether those constraints are biological, ecological, technological, informational, psychological, or historical – in order to analyze statistical regularities among cultures without implying determinism. We must be cautious, with both the “limited possibilities” and the “constraint” approaches, not to restrict our formulations and assume the restricting influence of various factors to be more important than positive (enabling) effects. A very strong propensity in favor of some trait is not the same thing as a very strong constraint against all other possibilities. Constraints and inclinations can and do coexist, and the negative limitations of one variable must be weighed against the positive inclinations of another. Despite this caveat, I find a constraint-based approach to be the most

Introduction

7

promising theoretical perspective for explaining the regularities found in numerical notation systems, something to which I will return in Chapters 11 and 12. In much of my analysis, I follow Joseph Greenberg (1978), whose analysis of significant regularities in lexical numeral systems presents a list of fifty-four generalizations. Unlike much of his later work, Greenberg’s study of numerals is universal and cognitive in orientation rather than phylogenetic. It is synthetic, based on the detailed empirical work of earlier scholars, such as the German linguist Theodor Kluge, who spent years compiling sets of numeral terms in languages throughout the world (Kluge 1937–42). While many of Greenberg’s regularities are extremely complex4 or have some exceptions, others reveal truly universal and nontrivial features of every natural language; for instance, every numeral system contains a complete set of integers between one and some upper limit – each system is finite5 and has no gaps (Greenberg 1978: 253–255). Similarly, no natural language expresses “two” as “ten minus eight” or “twenty” as “one-fifth of one hundred.” While every language has a set of lexical numerals, most pre-modern societies functioned quite well without numerical notation. It is possible to conceive of a world in which there are many regularities in lexical numerals, but in which numerical notation systems are highly specific and unique responses to local needs. We do not live in such a world. There is considerable uniformity among the world’s numerical notation systems, and they display many synchronic and diachronic regularities. In fact, the number and variety of conceivable numerical notation systems is far greater than what is attested historically. To take only a very limited example, a numerical notation system can very easily be imagined that is just like the Western system but – instead of being a decimal system – having a base of any natural6 number of 2 or higher. Yet most numerical notation systems have a base-10 structure (and those that do not use multiples of 10). This does not preclude the existence of binary and hexadecimal numerical notation for specialized computing purposes. Similarly, while there are only five basic principles of numerical notation systems found historically (as described earlier), it is easy to imagine other types that could have existed: a system where the size of a numeral-sign is relevant to its 4

5

6

For instance: “37. If a numeral expression contains a complex constituent, then the numerical value of the complex constituent itself in isolation receives either simple lexical expression or is expressed by the same function and in the same phonological shape, except for possible automatic phonological alternations, stress shifts, or overt expressions of coordination” (Greenberg 1978: 279–280). This is not true of numerical notation systems, some of which (like our own) are truly infinite. Or even, as discussed in some aspects of number theory, having a fractional or negative base!

8

Numerical Notation

value, or where all nonprime numbers are expressed multiplicatively using prime number numeral-signs. Several modern writers, abandoning traditional principles of numerical notation, have created new systems ex nihilo that rely on rather different principles than do the systems discussed in this study (Harris 1905; Pohl 1966; Dwornik 1980–81). Explaining regularities from a constraint-based perspective allows us to speculate about why certain numerical notation systems flourish while others do not. Instead of denying the existence of exceptions, I use general rules to explain why special cases are special, and why some imaginable systems are unattested in the ethnographic and historical records. Yet one might wish to contend that comparison of any sort, much less the universal type of this study, is misleading because each culture, and hence each numerical notation system, is a product of unique historical circumstances. If so, comparing Egyptian hieroglyphic numerals to Shang oracle-bone numerals and Inka khipus might be misleading. At best, even if there is a core of features common to all numerical notation systems, I would be labeling oranges apples in order to compare them to other apples. At worst, if these systems are entirely different phenomena, I am trying to make apples out of abaci. Yet the relative ease of intercultural communication refutes the claim that all cultures are incommensurable. The intercultural transmission of ideas relating to numerical notation systems is frequent and poses a serious challenge to this degree of relativism. Prior to comparing phenomena among multiple societies, one cannot assume either that the phenomenon is cross-culturally regular or that it is not. Having compared numerical notation systems on a worldwide basis, I regard the systems as being sufficiently similar to warrant their theoretical analysis as variations on a single theme. I regard numerical notation as translatable cross-culturally without significant loss of information or change of meaning. The number 1138 is practically identical in referent to MCXXXVIII or t\s\rrr\qqq\qqq\qq or any other representation. These systems have very different structures, but, in Saussurean terms, the various signifiers refer to the same signified (Saussure 1959). Although the linguistic and symbolic signifiers for numbers may differ greatly (23, dreiundzwanzig, XXIII, viginti tres, etc.), the correlation of both numeral-phrases and lexical numerals with natural numbers is not culturally relative. Yet, while seemingly uncontroversial in the exact sciences, the cross-cultural universality of number concepts has been criticized recently by relativistic anthropologists and sociologists. In his recent work on Quechua number and arithmetic, Urton (1997) asserts that Western concepts such as “odd/even” are not appropriate to the Quechua arithmetical experience, and that the Quechua use a fundamentally different ontology of numbers than the Western one. Yet Quechua numbers can be understood in the same way as any others, and the Inka numerical notation used by Quechua speakers (Chapter 10) can be compared to others without any particular difficulty. Relativist philosophers such

Introduction

9

as Restivo (1992) claim that 1 + 1 cannot equal 2 in any absolute manner, because if one were to take a cup of popcorn and add a cup of milk to it, the result would not be two of anything, but somewhat more than a cup of pulpy mush. Resisting the temptation to describe such casuistry as pulpy mush, I simply point out that addition is an arithmetical function that can only represent adding discrete objects of a like nature. Such evidence does not convince me that the number concepts of non-Western societies are incommensurable with our own. On the contrary, my own research suggests that these differences are relatively inconsequential in comparison to the commonalities observed in all societies. I acknowledge that, by treating all numerical notation systems purely as systems for representing number, I do not do justice to the complex symbolism that complements many of them or to the scholarship on numerology (Hopper 1938, Crump 1990). The arrival of the year 2000 was not simply another cause for celebration (or trepidation); rather, the nature of our numerical notation system and the “rolling over” of the calendrical odometer on 2000/01/01 held great symbolic and even mystical significance for much of the world’s population. My decision to underemphasize numerology is based partly on space limitations, but also on my theoretical interest in the comparable core of features underlying all lexical numeral systems and numerical notation systems. These interesting differences do not affect the validity of cross-cultural comparisons, but merely highlight the need to establish, rather than assume, the level of regularity in sociocultural phenomena. It may be true, as Geertz (1984: 276) famously asserted, that “[i]f we wanted home truths, we should have stayed at home,” but if we want human truths, we must compare.

Structural Typology of Numerical Notation The systematic classification of numerical notation systems helps to identify their relevant features, distinguish independent inventions from cultural borrowings, and determine how their features relate to their uses. The goal of typology is not simply to develop a scheme into which every case fits, but to do so in a way that allows us to ask and answer questions that could not otherwise be considered. When poorly done, typology is descriptive but nonanalytical, and thus largely useless; when well done, it organizes knowledge in a way that answers inquiries. Any classificatory scheme is inherently theory-laden, and answers only some of the questions that might be asked of a set of data. The typology presented here represents all the major principles by which numbers are represented and emphasizes the features of numerical notation that are cognitively most important. It removes each system from its temporal, geographic, and spatial contexts and examines how numeral-signs are combined to represent numbers.

10

Numerical Notation

Any natural number can be expressed as the sum of multiples of powers of some base. In Western numerals, 4637 is 4 × 1000 + 6 × 100 + 3 × 10 + 7 × 1 – or, to use exponential notation, 4 × 103 + 6 × 102 + 3 × 101 + 7 × 100. Because the Western numerals use the principle of place-value, the value of any numeral-sign in the phrase is determined by its position – position dictates the power of the base that is to be multiplied by the sign in question. If the order changes, the value changes, so that 6437, 3674, and so on mean different things than 4637. We could also write the number out lexically as four thousand six hundred and thirty seven. Instead of using place-value, the powers (except for 1) are expressed explicitly – thousand, hundred, -ty. Because each multiplier corresponds to a word for a power, we could in theory move each power and its multiplier to a different spot without introducing ambiguity; German lexical numerals, among others, do exactly that – viertausend sechshundert sieben und dreizig “four thousand six hundred seven and thirty.” Some numerical systems, however, do not use multiplication at all. To use the Roman numerals, one simply adds up the values of all the signs: MMMMDCXXXVII – 1000 + 1000 + 1000 + 1000 + 500 + 100 + 10 + 10 + 10 + 5 + 1 + 1. Although there is no logical requirement that systems like the Roman numerals list their powers in order, they almost universally do so. The Roman numeral CCLXXVIII could be unambiguously read even if it were written as VIIICCXXL, or even as XLIVIXCIC, if we omit the slight complexity of the occasional use of subtraction. The fact that such disordered phrases are not valid tells us something about systems that lack place-value, however – they too are structured as the sum of multiples of powers. We can thus interpret the Roman numeral MMMMDCXXXVII as (1000 + 1000 + 1000 + 1000) + (500 + 100) + (10 + 10 + 10) + (5 + 1 + 1). I will return in a moment to the issue of the signs for 500 and 5, and how they affect our understanding of such systems. The only major systematic attempt to date to classify numerical notation systems is Geneviève Guitel’s Histoire comparée des numérations écrites (1975).7 Guitel classifies approximately twenty-five systems (drawn from about a dozen societies) according to whether they use addition alone to form numeral-phrases (Type I, like Roman numerals), addition and explicit multiplication (Type II, like English lexical numerals), or implicit multiplication with place-value (Type III, like Western numerals) – just as I have done here. Each type is further subdivided according to the systems’ base(s) and other features. Despite an admirable attempt, Guitel’s analysis fails the most basic test of classification, which is that it must classify similar systems together and separate dissimilar ones. It is problematic because its primary division is made only on the basis of the degree to which multiplication 7

See also Zhang and Norman (1995). Ifrah’s (1985, 1998) popular studies on the subject follow Guitel’s typology.

Introduction

11

is used in forming numeral-phrases. Her typology assumes that, since positional notation is very important to our own numerical notation system, positionality is the primary criterion by which all numerical notation systems should be judged. It is not factually incorrect in any significant way; systems that are identically structured do end up in the same category. Yet because it is tied to the misleading question “To what extent does system X use multiplication to form numeral-phrases?,” it fails to represent fully the similarities and differences among numerical notation systems.8 There is another way to approach the subject, however. Consider a different system, namely, the Babylonian cuneiform positional system, which expresses numbers as combinations of signs for 1 (1) and 10 (a). Like the Roman numerals, the Babylonian system relies on the repetition of like symbols – the number 37 is written as three signs for 10 followed by seven for 1. However, the system also has a base of 60, and multiples of 60 and powers of 60 are written using the principle of place-value. It shares the use of implied multiplication through place-value with the Western numerals but shares the use of repeated added signs with the Roman system. Yet according to Guitel’s typology there is nothing in common between the Babylonian and Roman systems. The resolution to this difficulty is that when one writes a numeral-phrase in any system, one is actually doing two things: expressing the value associated with each power of the base, and then organizing all the powers in a numeral-phrase into a single total value. Thus, I distinguish two separate dimensions of numerical notation systems, which I call intraexponential and interexponential, in order to analyze them adequately. Intraexponential organization determines how numeral-signs are constituted and combined within each power of the base. The major types of intraexponential organization are cumulative, ciphered, and multiplicative. Cumulative systems are those in which the value of any power of the base is represented through the repetition of numeral-signs, each of which represents one times the power value of the sign, and which are then added. For instance, XXX = 30 in the Roman system because the sign for 10 (X) is repeated three times. Ciphered systems, on the other hand, use at most a single numeral-sign for each power represented, with different signs being used to represent different multiples of the power. The Western numerals are ciphered: a number that has as 103 (thousands) as its highest power (e.g., 1984) will require at most four symbols, one for each power of the base. Multiplicative systems have two components for each power represented: a unit-sign (or sometimes multiple signs), which represents the quantity of that power needed to represent the number, and a power-sign, which represents a power of the base. 8

See Chrisomalis (2004) for a more detailed critique of Guitel’s typology.

12

Numerical Notation

The product of the two signs determines the value of that power. The multiplicative principle is often used only to structure higher powers of the base. Interexponential organization determines how the values of the signs for each power of the base are combined to symbolize the value of each entire numeralphrase. It is subdivided into additive and positional subtypes. Additive systems are those in which the sum of the intraexponential values in a numeral-phrase produces its total value. For instance, the Roman numeral CCLXXVIII consists of two 100s (102), one 50 (5 × 101), two 10s (101), one 5 (5 × 100), and three 1s (100), for a total of 278. Positional or place-value systems, of which the Western system is the best known, are those in which the value of a numeral-phrase is determined not only by its constituent numeral-signs but also by the place of each sign within the phrase. The intraexponential values within a numeral-phrase must all be multiplied by the appropriate power-values before the sum of the phrase can be taken. All numerical notation systems are structured both intra- and interexponentially, creating six theoretically possible pairings of principles. However, it is logically impossible for a multiplicative-positional system to exist because multiplicative systems represent the required positional value (10, 100, 1000, etc.) intraexponentially, leaving only five possibilities, as detailed in Table 1.1. Cumulative-additive systems, such as Roman numerals, have one sign for each power of the base; the signs within each power are repeated and their values added, and then the total value of the phrase is the sum of the signs. Cumulative-positional systems likewise use repeated signs to indicate the value of each power, but this value is then multiplied by the place-values (in the Babylonian example used earlier, 60 and 1) before summing the phrase. In order to be entirely unambiguous, some sort of placeholder or zero sign is required. Ciphered-additive systems have a unique sign for each multiple of each power of the base (1–9, 10–90, 100–900, etc., in a base-10 system like the Greek alphabetic system); the values of these signs are added to obtain the value of the numeral-phrase. Ciphered-positional systems like Western numerals have unique unit signs from one up to but not including the base (e.g., 1, 2, 3 ... 9) and a zero sign; the unit-value is multiplied by the power-value indicated by its position, and then the sum of these products gives the total value. Finally, multiplicativeadditive systems (like the traditional Chinese system shown earlier, but also for that matter spoken English lexical numerals, e.g., three thousand six hundred and twenty four) juxtapose a unit-sign (or signs) and a power-sign, which are multiplied together, and then the sum of those products gives the total value of the phrase. Most numerical notation systems use only one of these five combinations throughout the entire system. However, some additive systems use one intraexponential principle (either cumulative or ciphered) for lower powers of the base, and then use the multiplicative principle thereafter. These systems, which I call hybrids, comprise about 30 percent of those I examine in this study. Systems that

Introduction

13

Table 1.1. Typology of numerical notation systems Additive The sum of the values of each power is taken to obtain the total value of the numeral-phrase.

Positional The value of each power must be multiplied by a value dependent on its position before taking the sum of the numeral-phrase.

Cumulative Many signs per power of the base, which are added to obtain the total value of that power.

Classical Roman

Babylonian cuneiform

1000 + (100 + 100 + 100 + 100) + (10 + 10 + 10) + (1 + 1 + 1 + 1)

(10 + 10 + 1 + 1 + 1) × 60 + (10 + 10 + 10 + 10 + 10 + 1 + 1 + 1 + 1) × 1

Ciphered Only one sign per power of the base, which alone represents the total value of that power.

Greek alphabetic

Khmer

1434 = /auld

1434 =

((1 × )1000 + 400 + 30 + 4)

(1 × 1000 + 4 × 100 + 3 × 10 + 4 × 1)

Multiplicative Two components per power, unitsign(s) and a powersign, multiplied together, give that power’s total value.

Chinese (traditional)

LOGICALLY EXCLUDED

1434 =

MCCCCXXXIIII

1434 = ₏◒⥪䤍ₘ◐⥪

1434 = b3

e4

(1 × 1000 + 4 × 100 + 3 × 10 + 4)

use two principles are not exceptions to my typology. They simply need to be analyzed in two parts, with each part of the hybrid being assigned the appropriate principle. For instance, the version of the Greek alphabetic system shown in Table 1.1 is ciphered-additive for powers below 1000 and multiplicative-additive for those above 1000.9 No numerical notation system employs more than two of the five basic types, and no positional system uses more than one type. Systems that have a sub-base as well as a base require further typological clarification because they may use two intraexponential principles: one for units up to the sub-base, and another for multiples of the sub-base up to the base. For instance, the cumulative-positional Babylonian system shown in Table 1.1 has a base 9

This feature is the one that leads Guitel to place this version of the Greek alphabetic numerals in her Type II as opposed to Type I.

14

Numerical Notation

of 60 and a sub-base of 10. In this case, we must know both how units from 1 to 9 are expressed and how tens from 10 to 50 are expressed in order to fully describe its intraexponential structure. In this case, both the sub-base and the base use the cumulative principle, so we might more properly describe this system as a (cumulativecumulative)-positional system. However, no system uses a different intraexponential principle for its sub-base than for its base, so this elaboration is mostly unnecessary. Again, none of this affects the interexponential structure of these systems. Because it reflects both intra- and interexponential principles, this typology shifts the focus of analysis from systems to the structural principles that build systems, and thus allows a more nuanced comparison of systems’ structures. Moreover, it allows us to ask fruitful questions regarding the cognitive effects and historical development of numerical notation.

Cognition and Number Cognitive psychology examines how the brain processes information, including the study of sensation and perception, concept formation, attention, learning, and memory. Its methodologies are primarily experimental: because neuroscience cannot yet fully observe the workings of the brain directly, cognitive psychologists study the brain by its observable outputs – the behavior of humans under controlled conditions. Information processing is crucial for human survival, and the ability to form concepts is a major part of our species’ evolutionary adaptation. At the same time, however, these concepts are not perfect representations of reality, because the act of conceptualization requires some information to be emphasized, and because errors in information processing reflect the imperfect conceptual abilities of the brain. In this regard, cognitive psychologists would agree with the archaeologist Gordon Childe’s argument that humans do not adapt to the world as it really is, but rather to the world that they perceive as mediated through culture (Childe 1956: 65–68). Still, Childe insisted, human perceptions must correspond reasonably well to reality or else we would not survive. Two cognitive questions inform the study of numerical notation. Firstly, there is the question of origins – how did numerical abilities originate, and how do they relate to the origins of numerical notation? Secondly, there is the question of what cognitive effects, if any, numerical notation has on its users. George Miller’s seminal paper on the “magic number 7 ± 2” remains an essential work for understanding how the brain processes number (Miller 1956). Miller asserts that in several related aspects of human cognition, our capacity for processing information lies between five and nine “units.” Two aspects of his research are particularly relevant to the study of number. Firstly, using research conducted by Kaufman et al. (1949), Miller discusses subitizing, in which small quantities of figures or objects are perceived directly, while larger quantities must be encoded

Introduction

15

by counting, a more time-consuming process. This experiment involved showing groups of dots to subjects for one-fifth of a second, after which they would indicate how many were present; up to five or six dots, few errors were made (subjects were subitizing), while above that number subjects had to estimate and hence made more errors. In more recent studies, the limit of subitizing has been found to be somewhat lower than six, ranging around three or four for most experimental subjects under typical conditions (Mandler and Shebo 1982). Closely related to subitizing is chunking, which organizes large quantities of objects into smaller groups, thereby enabling the brain to process the larger number as a certain number of the smaller sets rather than requiring each object to be cognized independently. North American telephone numbers of ten digits are divided into three “chunks” such as 212-555-2629 rather than written 2125552629, in part to distinguish the area code, local exchange, and individual phone line but also to facilitate memorization and recall. Chunking normally involves the division of a collection of objects into groups of three or four units each, which speeds up the process of perception and accurate quantification by the brain. The perception of larger units as gestalts thus maximizes the brain’s efficiency within the limits of its biological evolution. A third element to be considered, related to the first two, is the principle of one-to-one correspondence defined earlier. This capacity has been studied primarily through research on infants and children (Piaget 1952, Lancy 1983, Wynn 1992). Adults use one-to-one correspondence when they hold up eight fingers to represent eight coconuts, put aside twenty-seven pebbles to count their flock of sheep, or mark twelve lines on a sheet of paper to indicate the number of pints of beer consumed before staggering out of the local pub. Counting (as opposed to subitizing) cannot take place without one-to-one correspondence. One-to-one correspondence can be used in combination with chunking to increase the ease of representation and cognition. After my fifth pint, I might place a horizontal stroke through the four existing strokes to indicate a group of five; my twelve pints would thereby be rendered as two groups of five strokes followed by a group of two (rather erratic) strokes. By extension, numerical notation systems, particularly cumulative ones, rely on one-to-one correspondence. Much of the debate on cognitive domains relating to mathematics and its origins takes place in the realm of comparative ethology, specifically studying number concepts in animals in order to create meaningful analogies with the abilities of human infants and adults (Fuson 1988, Gallistel 1990, Dehaene 1997, Butterworth 1999).10 There is much skepticism about the ability of animals to count, after it 10

These authors go into far more detail on the various research programs undertaken to study animal and human infant perception of numerosity than is warranted here.

16

Numerical Notation

was shown that the mathematical abilities of Clever Hans and other animal calculators were the result of subconscious cues passed from human trainers to these purported prodigies (Fernald 1984). Yet, following in the footsteps of Koehler’s (1951) work on counting among birds, and the enormous literature studying primate numeracy (Matsuzawa 1985; Boysen and Berntson 1989, 1996), we now know that many animal species are able to perceive quantity at least accurately enough to perform tasks involving small quantities, mostly up to three to five units. It is not yet known whether animal quantification is a homology inherited from an ancestral species, a specific convergent adaptation in many species to the requirements of similar physical environments, or a general cognitive response of animals of a certain level of neurological complexity. Many experiments involving many different species have confirmed that something more than a Clever Hans phenomenon is being observed. The same is true in the case of human infants, who are able to distinguish small numerical quantities (Gelman and Gallistel 1978, Wynn 1992). While our hominid ancestors did not need numerical notation, the ability to distinguish between two gazelles and three gazelles would have been cognitively important and evolutionarily adaptive. As the survival of early hominids was strongly predicated upon the ability to function in groups, the number concept likely developed relatively early in human prehistory, although direct evidence is limited. It is highly probable that by the time of the Upper Paleolithic (40,000 to 10,000 years ago), Homo sapiens sapiens possessed languages including two or more numeral words and the ability to conceptually distinguish cardinal and ordinal quantities (Marshack 1972; Wiese 2003, 2007). Any human being (save those suffering from certain types of brain damage or other serious mental deficiencies) has the capacity to learn how to use numerical notation. As a technology invented in particular historical contexts, however, its use is limited to those who have encountered it. Anatomically and cognitively modern humans survived for millennia without any need for numerical notation, and the variability among numerical notation systems cannot be explained fully by the universal human mathematical ability. Even so, this does not prevent us from considering the possible effects of human cognitive capacities on the types of numerical notation system that have been developed historically. It is very likely that the evolved capacity of some primates to distinguish five from six bananas is related to the human visual capacity to distinguish five from six strokes on a tally or knots on a cord. Three biological characteristics of humans pertain to the development of the concept of number, which in turn is necessary for the development of numerical notation. 1. Perception of discrete external objects. The ability, common at least to all animals, to distinguish foreground from background, to perceive the borders of external objects, is necessary to the creation of the concept of “oneness.”

Introduction

17

2. Perception and cognition of concrete quantity. The ability to distinguish the quantity of sets of objects is present in human infants and some animals, but is generally restricted to small quantities. 3. Possession of language. The ability to identify numbers using linguistic symbols, as opposed to the pre-linguistic quantitative abilities possessed by infants and animals, permits the conceptualization of number through a series of lexical numerals, each greater than the previous by one unit.

While numerical notation systems are useful because they enable the human brain to conceptualize quantities efficiently, we must not assume that their structure and evolution can be derived entirely from the principles of cognitive psychology. Some neuropsychologists examine the development of numerical notation from a cognitive perspective (Dehaene 1997, Butterworth 1999). Dehaene (1997: 115–117) uses a stage-based unilinear scheme to describe the development of numerical notation from its beginnings in one-to-one correspondence, through chunked groupings of notches and ciphered numerals, to the ultimate stage of positional notation with a zero. However, I am very suspicious of such schemes in the absence of significant historical documentation. The contention that the history of technology can be understood as a sequence of ever-superior inventions “the better to fit the human mind and improve the usability of numbers” is untested at best, ethnocentric at worst (Dehaene 1997: 117). There are three sociocultural features that are likely prerequisites for the development of numerical notation. These are nonuniversal and derive from contingent historical circumstances, so it is possible to establish whether they are necessary conditions for the development of numerical notation using my universal cross-cultural methodology. 1. Presence of organizing principles that structure the number line. This refers to the ability to structure the natural numbers in a manner most convenient to thought, usually taking the form of a numerical base. No known numerical notation system has ever been developed by speakers of any of the world’s many languages whose lexical numerals have no base. 2. Presence of a nonstructured tally-marking system based on one-to-one correspondence. Often claimed as the earliest stage of numerical notation through which all societies must pass, tallying is a very intuitive way to represent number visually. There is evidence for this form of representation as early as the Upper Paleolithic (Marshack 1972). 3. Social need for long-term recording and communication of number. The social need for a relatively permanent record of numbers is essential to the development of numerical notation. One of its main functions is to assist memory, so the social need to preserve

18

Numerical Notation numbers beyond the ordinary limits of memory – for whatever specific purpose – is probably necessary to its development. Related to this is the need to communicate number outside a local community. While verbal numbers suffice for local communication, the ability of numerical notation to communicate numbers across barriers of geography and language is an important feature that would make its development likely in such circumstances.

Because numerical notation is a human invention, it must be subject to the constraints imposed by our cognitive abilities. Yet, because it is an invention deriving from specific historical contexts, I study its historical development inductively before turning to cognitive approaches. A full explanation of the origin of numerical notation must consider both cognitive and sociohistorical factors. Turning from causes to consequences, I believe that numerical notation has important cognitive effects on its users. These consequences, I suspect, are of a similar nature to Goody’s (1977) suggestions regarding the cognitive consequences of literacy. Goody himself believes this to be the case, as seen from his observations regarding the process of counting cowrie shells among the LoDagaa (1977: 12–13). The LoDagaa separate large groups of cowries into smaller groupings of five and twenty cowries to facilitate the counting of the larger group. While this is not numerical notation, since it does not represent large numbers using new signs for a base and its powers, it is an efficient way of counting a large group of objects. Yet, while LoDagaa boys were expert cowrie counters, they had little ability to multiply, a skill they had only begun to acquire recently in school. The very existence of multiplication tables, a technique used by almost all Western children to learn to multiply, links literacy and the use of numerical notation. While Goody is careful not to overextend this distinction into a rigid dichotomy, he rightly insists that the formalization of numerical knowledge that accompanies written numeration is a more abstract way of using numbers. The comparison of the cognitive abilities of groups who lack numerical notation and those who possess it would best be done through the ethnographic study of a group before and after its members learned such a system, or in a group where some but not all members use numerical notation. To date, no such study exists, although Saxe (1981) has done so for the “body counting” system used by the Oksapmin of Papua New Guinea. I discuss only societies that possess numerical notation, and even then, there is rarely specific contextual information about how the numerals were used. However, it may be possible to determine whether different types of numerical notation have different cognitive effects on their users. It is often assumed that cumulative systems such as the Roman numerals represent “concreteness” in numeration because of their iconicity, while positional systems represent “abstraction” because of their infinite extendability (Hallpike 1986: 121–122; Damerow 1996). The existence of cumulative-positional systems is highly problematic for this

Introduction

19

dichotomy. All associations of numerical structure with cognitive ability are untested, and rely on the equally untested assumption that numerical notation develops from concreteness to abstraction over time. By examining the diachronic patterns that actually occurred in the evolution of numerical notation, I will show that these patterns are multilinear, not unilinear. By comparing the structure of systems to the functions for which they were used, I will examine the cognitive framework within which different groups used numerical notation, keeping in mind that it is only one part of a cluster of techniques that includes mental calculation, lexical numerals, finger numbering, and computational artifacts. Rather than assigning labels such as “concrete” and “abstract” to numerical notation systems, or identifying any other single factor on which the utility of a system should be judged, I focus on a constellation of features of numerical notation systems that have cognitive consequences. This approach is similar to that adopted by Nickerson (1988), who lists the relevant criteria as being ease of interpretation, ease of writing, ease of learning, extensibility, compactness of notation, and ease of computation. A set of nonhierarchical criteria for evaluating systems from a cognitive perspective is a very valuable tool. Nickerson notes usefully: If one accepts the idea that the Arabic system is in general the best way of representing numbers that has yet been developed, one need not believe that it is clearly superior with respect to all the design goals that one might establish for an ideal system. It may be, in fact, that simultaneous realization of all such goals is not possible. (Nickerson 1988: 198)

There is no ideal numerical notation system; rather, each system is shaped by a set of goals that its users and inventors seek to attain, and that they can achieve only by compromising on other factors. There may be patterns of change among systems, but the burden of proof lies with those who wish to maintain that numerical notation evolves in a unilinear sequence.

Numerals and Writing The scholarly analysis of numerical notation has often been pursued by scholars interested in writing systems. Therefore, numerical notation systems are usually regarded as a subcategory of writing systems (Diringer 1949, Harris 1995, Daniels and Bright 1996, Houston 2004). Most numerical notation systems are associated with one or more scripts, and conversely, most scripts have some special form of numerical notation. Numeral-signs are graphemes that undergo paleographic change over time, just as phonographic signs do. The process of recovering instances of numerical notation archaeologically and interpreting them thus

20

Numerical Notation

inevitably involves epigraphers, paleographers, and other scholars of writing. However, the uncritical acceptance of a close connection between numerical notation and writing can lead to unfounded assumptions. There are three basic ways that number is expressed by human beings: a set of spoken lexical numerals, the written expression of those words in scripts, and the graphic expression of number through numerical notation systems. We can divide these three types into auditory systems (verbal lexical numerals) and visual ones (written lexical numerals and numerical notation). Alternately, we might distinguish lexical (verbal and written numerals) from nonlexical (numerical notation) means of expressing number. If the similarities between the two visual representations are more significant than the similarities between the two lexical representations, then the connection between numerical notation and writing is strong. However, four differences between lexical and nonlexical representations of number suggest that this distinction is the more important one. Firstly, lexical numerals are linguistic, while numerical notation represents number translinguistically. Numerical notations followed the evolution of language chronologically, and could not have occurred in a nonlinguistic species, but they are not inherently linked to any language structurally or semantically. The distinction between “writing” and “not-writing” is an issue of great debate among modern scholars, particularly in Mesoamerican (Marcus 1992, Boone 2000) and Andean (Urton 1997, 1998) studies. The most restrictive approach holds that only phonographic scripts – those whose signs can represent phonemes – constitute writing. Accordingly, the Maya glyph system is a “true” script, while the Aztec system is a pictographic system that requires a great deal of context in order to be interpreted, and the Inka khipu notation is a numerical notation system with some undeciphered non-numerical component. A broader approach holds that phoneticism is not an essential feature of scripts, and classifies pictographic representational systems, numerical notation, and even pictorial art under the rubric “writing.” Gelb’s classic definition of writing as “a system of intercommunication by means of conventional visible marks” (Gelb 1963: 253) would suggest that a numerical notation system is a script, although I do not believe that Gelb meant to imply this. I am sympathetic to the argument that because classifying societies as illiterate can be used to denigrate them, a broad definition of writing helps to counteract ethnocentrism, but there is enormous theoretical value in distinguishing phonographic from nonphonographic representation systems. I do not consider numerical notation to be “writing” in this narrow sense. Because numerical notation is nonphonetic, it transcends language and can traverse linguistic boundaries more easily than scripts. It is also learned much more readily than scripts. To use the terms proposed by Houston (2004b), numerical notation systems are “open” and can thus be employed by many groups, as opposed to “closed” notations that are accessible to one or few linguistic or cultural

Introduction

21

communities. Once an individual learns a numerical notation system, he or she can communicate numerically with any other individual familiar with the system, regardless of their linguistic differences. This does not imply that they exist completely outside of culture. As Houston (2004a: 226) and others have noted, quantification systems exist within cultural contexts, which is why, even though we can identify Inka khipu (Chapter 10) as encoding specific numbers, we know little about the communicative acts or information systems underlying them. Yet the fact that we can read Linear A, or Indus Valley, or Inka numerals even though the rest of those representation systems have eluded decipherment is telling. This suggests that numbering is separate from writing, more decipherable and less bound to culturally conventional encoding than other forms of notation. Secondly, numerical notation systems are not limited to societies possessing scripts, nor do societies with scripts necessarily possess numerical notation systems. Unfortunately, while scholars such as Ifrah (1998) and Guitel (1975) mention the existence of tallies, knotted strings, and other such technologies, they are considered solely as peripheral and/or ancestral to numerical notation proper. However, the khipu and several other tallying systems lie within the scope of this study because they are structured by a numerical base and its powers. One problem with studying such systems is that they are notched on wood, drawn in sand, or knotted on ropes or strings, all of which are unlikely to survive archaeologically, while written numerical notation is often found on durable metal, stone, or clay. Far more numerical notation once existed in nonwritten contexts than has survived. Moreover, just as numerical notation is not necessarily encountered in conjunction with writing, many scripts have no corresponding numerical notation system. For instance, the Ogham script of Ireland, the Canaanite script, the early alphabets of Asia Minor such as Carian and Phrygian, and the indigenous scripts of the Philippines all lack numerical notation and instead express numbers lexically. In societies that possess both scripts and numerical notation systems, there are often strong norms prescribing the means of representing number depending on social context. Throughout the Western world, lexical numerals are preferred in literary or religious contexts, while numerical notation is preferred in commercial transactions and accounting. In cases where both systems are found in a single text, there is often a functional division between the two. For instance, the text of the Bible is written using lexical numerals, but chapters and verses are numbered using numerical notation. In writing checks, numerical notation predominates, but dollar amounts are written out in full to prevent forgery. Such contrasts suggest that lexical and nonlexical representations should be treated separately. Thirdly, numerical notation systems and scripts exhibit very different patterns of geographical distribution and historical change. In part, this may be because scripts are largely phonographic, so their diffusion can be constrained by patterns of

22

Numerical Notation

language use. Numerical notation, by contrast, is largely nonphonetic and translinguistic, and may diffuse more readily than scripts. The Western numerals diffused initially from India and passed through the Arab world before reaching Europe, while the Roman alphabet is of Greek and Phoenician ancestry. This historical differentiation is not uncommon; the path of diffusion of numerical notation is often radically different from that of the diffusion of scripts. Yet there may be a connection between the indigenous development of writing and numerical notation. In several historically unrelated cases (Mesopotamia, Egypt, China, and Mesoamerica), the independent invention of numerical notation immediately preceded or coincided with the development of a full-fledged script. Perhaps the social need for numerical notation and a phonetic script tends to arise under similar circumstances (i.e., during the formative phases of early civilizations). Alternatively, the idea of numerical notation, once developed, might naturally suggest to its users that other domains might also be represented visually. I return to this subject in Chapter 12. Finally, the structures by which written lexical numerals and numerical notation express number are quite different. The simple fact of being denoted visually is not as important as the different principles used in the two symbol systems. Lexical numerals (whether written or verbal) share a common structure that is very different from that of numerical notation.11 For instance, while the cumulative principle is commonly employed in numerical notation, it is nearly absent from lexical numeration. No known language expresses thirty as “ten ten ten,” even though cumulative numeral-phrases (like the Roman XXX = 30) are quite common. In lexical numeral systems that have a base, multiplicative-additive structuring is overwhelmingly prevalent, whereas numerical notation systems are only occasionally multiplicativeadditive. To take a familiar example, let us compare Western numerical notation with English lexical numerals. The English lexical numerals eleven and twelve do not follow the regular pattern for numbers between thirteen and nineteen, and words like dozen and score add further complexity. Our numerical notation system is base10 and ciphered-positional, while English lexical numerals use a mixed base of 10 and 1000 (one million = 1000 × 1000; one billion = 1000 × 1000 × 1000), a situation that becomes even more complex if we include British English, in which one billion normally means one million millions (1012). Finally, while Western numerals are infinitely extendable – one can add zeroes to the right of a number ad infinitum – English lexical numerals are only potentially infinite, since one needs to develop new words to express higher and higher values. The highest number in many English dictionaries is decillion (1033 in American English, 1060 in British English). 11

The major exception to this disjunction is the classical Chinese numerals, which due to the somewhat logographic nature of the Chinese script serve both as lexical numerals and as numerical notation. I will return to this definitional issue in Chapter 8.

Introduction

23

The relationship between the origins of writing systems and numerical notation systems is similarly complex. Visual number marks clearly precede phonetic writing by many millennia. A wealth of evidence from Upper Paleolithic portable artifacts (e.g., notched bones and stones) suggests that one-to-one marking of numbers for calendrical or other mnemonic purposes has roots extending back at least 30,000 years (Absolon 1957, Marshack 1972, d’Errico 1998, d’Errico et al. 2003). This may in turn have been related to the early use of the fingers and hands as a visual, though nonpermanent, numerical system around the same time (Rouillon 2006).12 Schmandt-Besserat (1992), on the basis of controversial interpretations of Mesopotamian evidence from the proto-literate period, has been the strongest advocate for an evolutionary sequence from numeration to writing. Houston (2004: 237) argues that most writing systems emerged as “word signs bundled with systems of numeration that probably had a different and far-more-ancient origin,” and this may be correct. However, numerical notation (as opposed to non-basestructured tallies) does not greatly precede, if at all, the earliest writing. As I shall show, in all the independent cases of the development of numerical notation, written numerical systems with bases emerge alongside other conventionalized signs, not as a unilinear predecessor to them. For these reasons, to analyze numerical notation systems as adjunct components of scripts does not do them justice. Nevertheless, throughout this study I will sometimes refer to numeral-signs and numeral-phrases as being “written.” When I do so, it is mere conventionality, and this usage does not indicate any specific relationship between numerals and scripts.

Diffusion and Invention Numerical notation systems develop out of purposeful human efforts to perform tasks related to the visual representation of number. In a handful of instances, they developed independently of influence from other numerical notation systems, while in the vast majority of cases systems were borrowed wholesale or with modification from one society to another. I want to explain the origin, transformation, transmission, and decline of systems, not merely to describe a sequence of historical events. This does not mean that technical and functional aspects should be given priority over social factors; rather, social context and historical contingencies must be incorporated into analyses of the histories of systems. It does require, however, that I distinguish analogies – similarities that derive from independent operation of cause and effect – from homologies – similarities that derive from the 12

It is absolutely clear that manual counting and numerical notation are connected in later societies known through written and oral evidence (see Chapter 12).

24

Numerical Notation

descent of cultural features from a common ancestor or the borrowing of features from one society to another. In anthropology, analogies and homologies are normally seen as dichotomous, with materialists (Steward 1955, Harris 1968) preferring analogical explanations and regarding independent invention as common and idealists (Elliot Smith 1923, Driver 1966, Rouse 1986) assigning priority to homologies and using borrowing to explain most cross-cultural similarities. There is no serious scholar who denies that some features are developed independently multiple times, and similarly no one doubts that societies borrow extensively from one another. Despite this, the anthropological effort to distinguish cultural analogies and homologies has not been especially fruitful (Steward 1955, Kroeber 1948, White 1959, Driver 1966, Tolstoy 1972, Jorgensen 1979, Maisels 1987, Burton et al. 1996). Harris (1968) has attempted to circumvent the debate by noting that regardless of how a trait was exposed to a society, it must still be accepted and integrated into that society, even if it is borrowed from elsewhere. He thus asserts that diffusion is a sterile “nonprinciple” that is “not only superfluous, but the very incarnation of antiscience” (1968: 378). Harris is right that simply classifying an innovation as representing either diffusion or independent invention is insufficient, but he is wrong in implying that it does not really matter whether a trait was of internal or external origin. In practice, Harris’s rejection of diffusion leads him to assume that cultural adaptation is a unitary process and that analogical explanations are the only ones worthy of scientific consideration. Yet if the social consequences of, and motivations for, adopting diffused numerical notation systems and adopting independent invention are different – as I believe them to be – then we must instead compare the two different circumstances while keeping an open mind as to potential differences. “Diffusion” is often implicitly taken to represent a largely benign transfer of features from one group to another, followed by a period in which the recipient society evaluates the innovation, followed by its acceptance or rejection. This extremely naïve view of processes of cultural contact denies entirely the role of imperialism, peer-polity networks, and power structures. For instance, many numerical notation systems developed in societies just as they began to enter into longdistance trading relationships with larger, more politically complex state societies that already possessed numerical notation. Numerical notation, in this instance, is not simply something that happens to be transmitted due to cultural contact; it is a medium through which contact takes place and a feature that becomes important just as societies are becoming integrated into larger intersocietal networks. Power relations are always involved in such cases, and we need to understand how the social statuses of individuals and groups within given contexts affect the transmission process and the eventual outcome. Yet the existence of historical

Introduction

25

contingencies need not be fatal to the development of a cultural-evolutionary theory of numerical notation. In order to demonstrate empirically the cultural evolution of numerical notation, we must examine how systems change in a patterned way, comparing analytically the conditions under which numerical notation systems are invented, transmitted, and adopted. In this study, I address three basic contextual questions regarding each numerical notation system: 1. What antecedent(s) does the system have, if any? I establish whether each system is descended from antecedent numerical notation systems. Numerical notation was independently invented six or seven times, and these “pristine” systems stand at the head of cultural phylogenies, but are certainly not the norm. Independent invention should not be the null hypothesis for any account of the origins of a system, but neither should it be restricted only to very ancient systems. Most systems have one antecedent only, while a few systems blend features of two antecedents. 2. Does the new system supplant one or more older systems? I establish what happens when a new system is introduced into a society that already uses numerical notation. Four outcomes are possible: a) the newly introduced system replaces the existing one; b) the new system is used in conjunction with the older one, normally with some sort of functional division between the two; c) elements of the original and new systems are commingled to create a third system; d) the new system is rejected entirely, while the older system is retained. All these outcomes are attested multiple times. 3. Does the new system use the graphic symbols and/or the structural principles of its antecedent(s)? I establish how specifically the new system resembles its antecedent(s), either in the form of its numeral-signs or in its structure (base, interexponential and intraexponential principle[s], and additional signs). Resemblances among closely related systems, in conjunction with other historical evidence, help to specify the exact connection between them.

To answer these questions, criteria must be adopted to distinguish endogenously invented systems from ones introduced from outside a society and to specify connections between ancestral and descendant systems. Discerning historical relations among cultural phenomena can be extremely contentious, particularly when only archaeological data are available. Rowe (1966) would permit diffusionary explanations only when abundant evidence of colonies, trading posts, or traded objects independently confirms contact between two regions, while Tolstoy (1972) deemed it sufficient to show that a particular combination of features is probabilistically unlikely to have occurred independently. This question is unresolvable in the abstract, because the ease of demonstrating cultural transmission depends on the nature of the specific trait or phenomenon being studied.

26

Numerical Notation

Few inventors of numerical notation systems have ever provided detailed information about the contexts of their inventions. Thus, I must build a circumstantial case for the origins of most systems. In order to demonstrate cultural affiliations between numerical notation systems, I use both internal (structural and graphic) resemblances between systems and external (contextual and historical) considerations. The main criteria I use are as follows: 1. Use of the two systems at the same point in time. This criterion is nearly unavoidable; some chronological overlap in the periods during which two systems are used is needed to sustain a hypothesis of cultural transmission. An extinct system might conceivably be revived and modified by a later society (for instance, on the basis of old inscriptions), but this is hardly a sufficient basis for a hypothesis of cultural transmission. Alternately, a system that is not attested to have survived may in fact have done so; this is the basis of the controversial theory that the Mycenean Linear B numerals (Chapter 2) gave rise to the Etruscan numerals (Chapter 4). Such hypotheses cannot be dismissed immediately, if other factors suggest that they could be true, but they require much more evidence. 2. Similarity in structural features. Because there are only three intraexponential principles (cumulative, ciphered, multiplicative), two interexponential principles (additive, positional), three common bases (10, 20, 60), and two sub-bases (5, 10), no one aspect that is similar in two systems is sufficient to prove a connection. However, when two systems are alike in all or most of these respects, cultural contact becomes a much more likely explanation for the resemblance. Many of the cultural phylogenies of systems that I discuss share a common structure; for instance, all the Italic systems (Chapter 4) are cumulative-additive with a base of 10 and a sub-base of 5. This does not mean that all identically structured systems must be placed in that phylogeny – the Ryukyu sho-chu-ma numerals (Chapter 10) and modern Berber numerals (Chapter 10) do not fit because they were used much later and have different numeral-signs. The use of structural features as evidence of contact suffers from the weakness that, if many systems in a phylogeny are identical or similar, it is often impossible to choose between several equally likely candidate ancestors. 3. Similarity of forms and values of numeral-signs. Because many graphic symbols are very complex, they are unlikely to have developed independently. If the forms of numeral-signs used in two systems are identical or very similar, and if those signs represent the same numerical values in the two systems, it is likely that cultural contact resulted in the invention of the later system based on the earlier one. The more signs that are shared between two systems, the more likely it is that there is a historical connection between them. However, when two systems use similar signs for different numerical values, this is not good evidence of such a connection. For instance, à and J represent 10 and 20 in the Kharoṣṭhī

Introduction

27

numerals (Chapter 3) but mean 7 and 9 in the Brāhmī numerals (Chapter 6). In this instance, even though the two systems were used in the same region at the same time (the Indian subcontinent in the fourth century bc) and have two similar numeral-signs, the dissimilarity of their values reduces the likelihood of a historical connection. Caution must be exercised when invoking this criterion for very simple symbols – vertical and horizontal lines, dots, circles, crosses, and the like – because such designs are cross-culturally common. This is especially true in the case of the use of lines and dots with the value of one, since these signs may have been part of tallying systems before being used in numerical notation systems. Cases where signs are similar but not identical must also be treated with caution. There is no general paleographic principle for identifying relations among graphically similar signs; hence, such efforts usually proceed on an intuitive basis. 4. Known cultural contact between the regions where the two systems are used. In general, where one cultural trait is transmitted from one region to another, multiple traits are likely to have been transmitted. Thus, where there is a known pattern of shared non-numerical features in two societies, or where there is substantial evidence of interregional trade, migration, or colonization, such evidence supports a postulated ancestor-descendant relationship between two numerical notation systems. Determining whether known cultural contact is sufficient to postulate the diffusion of a numerical notation system is always a tricky matter and involves an evaluation of various lines of evidence. For instance, one of the difficulties in postulating that the Brāhmī numerals (Chapter 6) are descended from the Egyptian demotic ones is that, despite structural and graphic resemblances between the two systems, Egypt is well down on the list of areas with which ancient India had contact. In no case do I postulate a connection between two systems solely on the basis that they were used at approximately the same time and in a single region. There must always be some structural or graphic resemblance between postulated ancestor and descendant systems. This problem is made more complex by stimulus diffusion, a complex blend of inventive and diffusionary processes in which awareness of an invention is transmitted, but, because of some obstacle to transmission or acceptance, the actual invention does not take hold in the adopting society (Kroeber 1948: 368–370). However, because the general principle is seen as useful, some members of the adopting society, stimulated by the original idea, invent their own version of the innovation. The most widely cited example of stimulus diffusion is the development of the Cherokee syllabic writing system by Sequoyah in the nineteenth century, based on his rudimentary knowledge of the Western alphabet. While several numerical notation systems resulted from stimulus diffusion (e.g., the abortive

28

Numerical Notation

Cherokee numerals, never used in the syllabary), no principles exist to help identify stimulus diffusion. It is tempting to postulate stimulus diffusion even when the basic fact of incomplete transmission cannot be established. However, I use stimulus diffusion as an explanation only when it can be established that the form of cultural contact that occurred between two regions fits Kroeber’s model. 5. Use of ancestor and descendant systems in similar contexts. If two systems serve similar purposes, on similar media, or among similar social groups in their respective societies, this can serve as further confirmatory evidence that the two systems are related historically. This factor, while useful, is never sufficient on its own to demonstrate such a connection, but it may provide further support. For instance, the spread of the Greek alphabetic numerals into Armenia and Georgia (Chapter 5), though poorly documented, is confirmed not only by the striking similarities in the systems but also by the systems’ use in Bibles and other liturgical texts. The similarities among some of the cuneiform systems of Mesopotamia (Chapter 7) rest on their common use of a wedge-shaped stylus on clay media as much as on specific resemblances in the numeral-signs or their organization. 6. Geographic proximity of the regions where two systems were used. All other factors being equal, a system is more likely to have been modeled on one that is used by neighboring groups than on one used more distantly. This is a particularly dangerous criterion to invoke, especially where there is less cultural contact between neighboring regions than with more distant regions. Many times, two very different and unrelated systems are used in proximity to one another, and other times, closely related systems are used at considerable distances from one another. Geographical proximity is such a weak measure that I will use it only as a last resort, and never as the sole factor for hypothesizing transmission. Establishing links between ancestor and descendant systems, within the limits of the available data, allows me to describe phylogenies of related systems. These are, however, analytical descriptions, which allow the explanation of evolutionary patterns of change in numerical notation systems. These explanations are analogical, because they describe independent recurrences of cause and effect. However, they are also explaining homological processes resulting from cultural contact and the transmission of knowledge among many societies. This is a paradox only if we accept the notion that these two concepts stand in opposition to one another. A phylogenetic perspective is both homological and analogical, seeking to describe particular historical contexts, but also to derive general processes by which numerical notation systems are related to one another. Diffusion may be, as Harris contends, a nonprinciple, but it is not a nonprocess. Comparing ancestor and descendant systems, and understanding the nature of the process of borrowing and adoption of cultural features, is absolutely essential to an evolutionary perspective on cultural change.

Introduction

29

Technology, Function, and Efficiency Modern historians of science such as Thomas Kuhn (1962) have effectively demolished the myth of linear progressivism in science. While in some fields, the accumulation of knowledge leads to a better understanding of reality, and technical innovations likewise have antecedents, the burden of proof has now rightly shifted to those who wish to demonstrate that progress occurs. Progressivist schemes that assume rather than demonstrate the superiority of new technologies, imply that where such progress exists it implies moral superiority, or argue teleologically that present achievements can never be exceeded, have no scientific credibility. Yet these preconceptions abound among scholars of numerical notation; this shift in our conception of progress has not yet taken hold. Consider the following collection of recent laudatory statements regarding Western numerals: Sa perfection va bien au-delà de la civilisation indienne puisqu’aucune autre numération de Type III n’a jamais été en mesure de l’égaler. (Guitel 1975: 758) If the evolution of written numeration converges, it is mainly because place-value coding is the best available notation. So many of its characteristics can be praised: its compactness, the few symbols it requires, the ease with which it can be learned, the speed with which it can be read or written, the simplicity of the calculation algorithms it supports. All justify its universal adoption. Indeed, it is hard to see what new invention could ever improve on it. (Dehaene 1997: 101) Our positional number-system is perfect and complete, because it is as economical in symbols as can be and can represent any number, however large. Also, as we have seen, it is the most efficacious in that it allows everyone to do arithmetic. . . . In short, the invention of our current number-system is the final stage in the development of numerical notation: once it was achieved, no further discoveries remained to be made in this domain. (Ifrah 1998: 592)

Such perspectives accept without proof that the Western numerals are the most efficient ever developed, and are not only the “best” in existence but also “perfect” – the best that could ever be conceived. Their adoption by the vast majority of human societies today is perceived as a natural and inevitable consequence of this superiority, only minimally mediated by social factors. Other, more cumbersome systems are to be evaluated in relation to the Western system, and in particular to their utility for arithmetical calculations and higher mathematics. Since so many modern technologies require mathematics, Western numerical notation is a partial cause of these evolutionary developments. The corollary of this proposition, often left unstated, is that those societies that did not develop or adopt Western numerals failed to compete politically with the West in part because of this.

30

Numerical Notation

I consider the decimal, ciphered-positional system of numerical notation developed in India in the sixth century ad and transmitted by Arab scholars to Western Europe to be a very remarkable invention. Its brevity, unambiguity, and ease of learning make it conducive to the practice of written arithmetic and mathematics. How well numerical notation systems represent number strongly affects the development of new systems, their acceptance after being transmitted, their modification over time, and their eventual abandonment. This pattern of long-term sociocultural change can meaningfully be called evolutionary. The primary difficulty with the assumption of the evolutionary progress of numerical notation is not the notion of evolution itself. The problem is that the efficiency of numerical notation systems cannot be evaluated in the abstract, but only by considering the purposes for which they were developed and used. It is often assumed that the function of numerical notation is to perform written computations. For instance, Ifrah, whose work is the most popular and influential study of the history of numerical notation, writes: To see why place-value systems are superior to all others, we can begin by considering the Greek alphabetic numeration. It has very short notations for the commonly used numbers: no more than four signs are needed for any number below 10,000. But that is not the main criterion for judging a written numeration. What matters most is the ease with which it lends itself to arithmetical operations. (Ifrah 1985: 431)

This view is entirely erroneous. Numerical notation was a necessary condition for the development of modern mathematics, but it is ethnocentric to argue from this that its purpose was to facilitate the development of mathematics. The efficiency of any technology can be evaluated only in terms of the purposes for which it was developed and/or used. There is thus no eternal abstract standard of efficiency for any technology. It smacks of teleology to argue that Western numerical notation is wonderful because it enabled modern mathematics to develop. The origin of Western numerals had little to do with mathematical computation and much to do with writing dates on ancient and medieval southern Asian inscriptions. The primary function of numerical notation is always the simple visual representation of numbers. Most numerical notation systems were never used for arithmetic or mathematics, but only for representation. Even when they are used in mathematical contexts, they frequently simply record the results of computations performed in the head, on the fingers, or with an abacus. Even in industrialized societies, computation remains a secondary function of numerical notation. I am looking at a not-so-crisp Canadian five-dollar bill, on which numerals indicate a monetary value (5), the date the bill was designed (1986), a serial number with some letters to render it unique (GPA6537377), and the number 64 penciled in

Introduction

31

one corner (probably to record the number of five-dollar bills received at some event). None of these numeral-phrases was actually ever used to compute.13 Numbers denote far more often than they reckon, even in our highly numerical society. This was doubly true in pre-industrial contexts. In defining a numerical notation system as a system for representing numbers, I am explicitly making a functional statement. At minimum, whatever else numerical notation may mean in a particular society, it must always express number as one of its functions. I am not saying that a numerical notation system must be fully integrated with other sociocultural phenomena, that it must be perfectly adapted to serve social needs, or that the purpose for which it is used must be that for which it was developed. The representational function of numerical notation is general enough that it can be stimulated by a variety of social or political needs. While trade is the most obvious one – making transactions possible over long distances, enabling monetary calculations, or recording results to facilitate accurate bookkeeping – it is not the only one. For instance, the main impetus behind the origin of the Mesoamerican numerical notation systems was probably astronomical and calendrical, while the Shang numerals were first used in the context of Chinese divination. We should not expect to find a single specific domain of activity correlated with the origin of numerical notation, and we should be skeptical of universal or unilinear schemes. If we wish to compare the efficiency of various numerical notation systems, we must compare systems that served a common purpose in terms of how well they served that purpose. Because all systems represent number visually, some general criteria can be used. A system that represents numbers using few number-signs is more efficient than one that requires many signs. One could then argue that the Roman numerals are not as efficient for representing number as Western numerals are because 1492 is much shorter than MCCCCLXXXXII (or MCDXCII). While the situation is slightly more complex – MMI is shorter than 2001, for instance – the Roman numerals are more concise for only a small fraction of all natural numbers. Two other criteria that are relatively easily definable are a system’s sign-count (how many signs it uses in total) and extendability (the highest number expressible).14 I return to these criteria in Chapter 11 and show how they can be used to ask

13

14

One might protest that the numeral on the bill is used in doing arithmetical computations such as providing change for purchases. To refute this, one need only go into a bank and ask for $100 in five-dollar bills, and see whether the teller looks at the number on each bill, or whether in fact he or she merely counts out twenty bills while doing mental arithmetic. The numeral on the bill denotes its value, but is not used in calculation. See Nickerson (1988: 189–197) for a different, but related, list of criteria used in comparing numerical notation systems.

32

Numerical Notation

fruitful questions that help explain synchronic and diachronic patterns among attested systems. By contrast, efficiency for computation is a Western-centered and historically inaccurate benchmark for comparing numerical notation systems. Zhang and Norman’s (1995) paper on the visual representation of numbers through numerical notation is a major step forward in our understanding of how numerical notation systems work. They analyze how specific systems visually represent (or fail to represent) numerical information, describe three general means by which numerical notation systems are structured (shape, quantity, and position), and then examine how these features are combined in numerical notation systems. Yet their analysis falls apart because they compare and evaluate different numerical notation systems based on their ability to aid in multiplication. Even if Western numerals are the best system for doing arithmetic (which would best be resolved through the use of the systems rather than abstract theorization), most other systems were never designed or used for such a purpose. The situation is analogous to denigrating screwdrivers for being inefficient hammers. The fact that one can use a screwdriver handle to drive in nails does not justify that comparison, just as the fact that one might use Egyptian hieroglyphic numerals to multiply does not justify comparing them to systems such as the Western numerals. To add insult to injury, even though Zhang and Norman recognize that calculation technologies such as the abacus are frequently better than numerical notation for doing arithmetic, they suggest that part of Western numerals’ superiority is that they are used for both calculation and representation, while other societies employed two separate systems (Zhang and Norman 1995: 293). They thus blame the carpenter for using both a hammer and a screwdriver where just the screwdriver would do. Such arguments are little more than elaborate rationalizations for a historical fact (the near-universality of Western numerals) that eludes simple explanation. The only way to compare numerical notation systems fairly is to use functional criteria that apply to all systems (namely, those related to simple representation). Yet, even where there are definite answers to these efficiency-related questions, this does not mean that individuals testing out a new system will immediately perceive its advantages and disadvantages. A familiar but in some respects inefficient system, so long as it is not entirely unworkable, may be retained, despite its “obvious” inferiority. There may be a steep learning curve preventing the easy adoption of the alternative system, or there may be cultural or political reasons for retaining one’s present system. Moreover, numerical notation, as a system for communicating information to others, requires not only that specific individuals adopt it, as would be the case with a more efficient plough or a better mousetrap, but also that an entire social group learn it before its usefulness will be evident. A system with many users is functional for that reason alone, because it can be

Introduction

33

used to communicate with more people than one with few users. Thus, it is inappropriate to evaluate numerical notation systems only in terms of their structural features. Rather, these features must be considered in the broader social context in which systems develop and are used. I examine these social factors and show how, far from negating structural factors, structural and social explanations of regularities combine to produce a more complete understanding of numerical notation than has previously been possible. I turn in the following chapters to the body of data itself. I endeavor to highlight the ways in which the general theoretical principles just discussed relate to the data. I have organized these data according to cultural phylogenies of related systems, presenting the earliest systems first, leading forward to systems developed more recently. The first five phylogenies are probably related to one another historically, so I treat them together, but no other principle has been used in the ordering of chapters. The eight major phylogenies, each of which merits a full chapter, are as follows: Chapter 2: Hieroglyphic – systems historically descended from the Egyptian hieroglyphic numerals; Chapter 3: Levantine – systems used in the Levant, descended from the Aramaic and Phoenician numerals; Chapter 4: Italic – systems used in the circum-Mediterranean region, descended from the Etruscan numerals; Chapter 5: Alphabetic – systems whose signs are mainly phonetic script-signs, descended from the Greek alphabetic numerals; Chapter 6: South Asian – systems originating on the Indian subcontinent and descended from the Brāhmī numerals; Chapter 7: Mesopotamian – systems used in Mesopotamia, descended from the protocuneiform numerals; Chapter 8: East Asian – systems descended from the Shang numerals; Chapter 9: Mesoamerican – systems descended from the Mesoamerican bar-and-dot numerals.

Chapter 10 is devoted to miscellaneous systems and cultural isolates that do not fit into any of these phylogenies, and also to the numerous systems invented in colonial contexts over the past two hundred years. Chapter 11 analyzes synchronic and diachronic regularities among numerical notation systems in a structural and cognitive framework, while Chapter 12 tempers these findings with considerations relating to social context.

chapter 2

Hieroglyphic Systems

A recognizable phylogeny of numerical notation systems was used in conjunction with a group of related scripts and their descendants, beginning with the Egyptian hieroglyphic numerals as early as 3250 bc, which thus rivals the Mesopotamian family (Chapter 7) as the oldest attested numerical notation anywhere. Among these, I include the Egyptian hieroglyphic system, obviously, but also the Hittite hieroglyphic, Cretan hieroglyphic, Minoan Linear A, Mycenean Linear B, and Cypriote numerals. In addition, I include the Egyptian hieratic and demotic systems, which are cursive reductions of the Egyptian hieroglyphic numerals, even though they are structurally closer to the alphabetic systems (Chapter 5), to which they are ancestral. I use the term “hieroglyphic” simply because several systems discussed in this chapter are associated with “hieroglyphic” scripts, rather than to imply anything about the systems’ structure. The hieroglyphic systems are summarized in Table 2.1.1 Of these systems, the Egyptian hieroglyphic has been discussed most extensively, though it has often been misinterpreted, while others, such as the Cypriote system, are severely understudied and poorly known. Identifying these systems and distinguishing them from other, superficially similar ones helps explain the diffusion of numerical notation throughout the ancient Mediterranean region. 1

The hieratic and demotic systems are too complex to be included on this chart; consult their individual entries for their numeral-signs.

34

Hieroglyphic Systems

35

Table 2.1. Hieroglyphic numerical notation systems System

1

10

100

1000

10,000 100,000 1,000,000

Egyptian hieroglyphic

q ù\= Å Å Å q

r • É\• É É É

s 0 æ æ

t ÿ Æ Æ

u\‹

(

)

Cretan hieroglyphic Minoan Linear A Mycenean Linear B Cypriote syllabic Hittite hieroglyphic

v

w

ô

The hieroglyphic phylogeny of numerical notation systems is ancestral to the Levantine (Chapter 3), Italic (Chapter 4), and Alphabetic (Chapter 5) phylogenies, but its systems differ sharply from those of its descendants.

Egyptian Hieroglyphic The hieroglyphic script is the best-known ancient Egyptian script. It was used between about 3250 bc and 400 ad, making it the longest surviving of all scripts (Loprieno 1995). However, its use was restricted geographically to the Nile Valley and nearby areas under Egyptian control. While the hieroglyphic script may well have arisen because of stimulus diffusion and trade with Mesopotamia, the scripts in these two areas emerged essentially simultaneously and show no substantial resemblances. Hieroglyphic inscriptions are written from top to bottom, left to right, or right to left, with the last of these three options being the most common (Ritner 1996: 80). The script is mixed in principle, with both phonograms (consisting of one, two, or three consonants) and logograms indicating words nonphonetically (Ritner 1996: 74). The later hieratic and demotic scripts used to write the ancient Egyptian language, as well as the Meroitic hieroglyphic script, are directly derived from the hieroglyphic, while the early scripts of the Levant and the Aegean are probably its less direct descendants. Numbers other than ‘one’ are very rarely expressed through lexical numerals in Egyptian, making it difficult to determine their structure, although evidence from some Old Kingdom Pyramid Texts and later Coptic writings establishes that they had a purely decimal structure with words for each power of 10 up to 1,000,000 (Loprieno 1995: 71). Most hieroglyphic inscriptions express numbers using numeralsigns rather than words, however, with separate signs corresponding to each power. These signs are shown in Table 2.2 (cf. Gardiner 1927: 191; Allen 2000: 97).

Numerical Notation

36

Table 2.2. Egyptian hieroglyphic numerals 1

10

100

1000

10,000

100,000

1,000,000

R-L

q q

r r

s „

t …

u †

v\ į

w İ

Lex.

w̒

mḏ

št

ḫ

ḏb̒

ḥfn

ḥḥ

L-R

68,257 =

qqqq rrr „„\\\…………\\\\††† qqq\ \\rr\\\\\\\ \\\\\\\…………\\\\†††

The system is purely decimal and cumulative-additive, with each sign repeated up to nine times as necessary, and ordered from highest to lowest rank. The direction in which a numeral is read is always the same as the direction of writing, but varies depending on the inscription in question. The set of signs in the top row of Table 2.2 are those used when the direction of writing is from left to right; when right-to-left writing is used, the signs are mirrored (i.e., q\r\„\…\†\‡). Occasionally, when days of the month are being expressed, the signs for 1 and 10 were placed on their side: ‘ or ^ instead of r or q (Gardiner 1927: 191). Numeral-signs could be used either cardinally or ordinally, with ordinals from ‘second’ through ‘ninth’ adding the ending nw (masculine) or nwt (feminine) to the numeral-phrase, and those from ‘tenth’ upward adding m (masculine) or m t (feminine). To aid in reading long numeral-phrases, five or more identical signs were usually grouped in sets of three or four rather than placed on a single line. Thus, 5 is written as a row of three signs above a row of two signs, 6 as a row of three above a row of three, 7 as a row of four above a row of three, 8 as a row of four above a row of four, and 9 either as a row of five above a row of four or as three rows of three.2 The sign for 1 is a simple vertical stroke. Gunn (1916: 280) suggests that in early well-executed inscriptions, the sides of the vertical bar are curved inward slightly, thus making a biconcave bar, and postulates that it may represent “a small object of bone or wood used in some kind of tally or aid to reckoning,” but I tend to think that it is simply an abstract stroke. The sign for 10 has been described as a heel bone (Kavett and Kavett 1975: 390), a tie made by bending a leaf (McLeish 1991: 42), or even, anachronistically, as a croquet wicket (Boyer 1959: 127). In fact, it corresponds to the phonetic value m (masc. m w, fem. m t) ‘hook, handle’, 2

However, other groupings were sometimes used when it was more convenient for the scribe to do so.

Hieroglyphic Systems

37

and is a rebus for the Egyptian lexical numeral for 10 (m ) (Sethe 1916: 2). The higher power signs also have specific representational qualities and can also represent phonetic values in Egyptian apart from their use as numerical symbols. The sign for 100 ( t) is probably a coiled length of rope; that for 1000 ( ) is a lotus plant; the sign for 10,000 ( b ) is an extended finger; and that for 100,000 ( fn) is a tadpole. These numeral-signs, as well as the overall structure of the system, remained remarkably stable throughout its history. In some older instances in which the sign for 1000 occurs, rather than grouping the signs in clusters of three to five separated signs (as in the numeral-phrase mentioned earlier), multiple “lotus plants” were depicted as emerging from a single bush (e.g., 3000 = ‰). The sign for one million ( ) could also mean “multitude” or “a countless quantity,” just as the Greek word ‘myriad’ can mean a group of ten thousand or, more generally, a large quantity (Loprieno 1995). After the Early Dynastic period, this nonspecific lexical sense predominated over the specific numerical value. In most other respects, Predynastic hieroglyphic numerals would have been completely intelligible to Late Period scribes. The earliest known Egyptian hieroglyphic numerals are those from Tomb U-j at Abydos, which dates to around 3250-3200 bc (late Naqada II or early Naqada III period), and also contains the earliest examples of Egyptian writing (Dreyer 1998). Numeral-signs occur on many drilled bone and ivory tags found in this royal tomb, which were probably once attached to containers of goods. Other tags have other signs that resemble later Egyptian hieroglyphs, but only a very few contain both numerals and hieroglyphs (Baines 2004: 154–157). Some tags have six to twelve vertical or horizontal strokes, others the sign for 100, and one has both a sign for 100 and a sign for 1 (Dreyer 1998: 113–118). This system has three distinctive features as compared to the mature hieroglyphic system: it uses both horizontal and vertical strokes for units; there is no attested numeral-sign for 10; and there are tags with more than nine unit-strokes. Dreyer (1998: 140) explains the first two of these differences simultaneously by noting that on some Old Kingdom linen-lists, horizontal strokes stand for 10. The Tomb U-j tags are very similar to others found at Naqada and Abydos that date from the Naqada III and Early Dynastic periods, which contain the sign for 10 and use only vertical strokes for 1 (Dreyer 1998: 139). The very early date of the tags suggests that the system was developed independently of Mesopotamian influence, although the U-j tags are essentially contemporaneous with the Uruk IV tablets. The margin of error and discrepancies in the different radiocarbon dates from Tomb U-j are large enough that no definite conclusion regarding priority can be reached (Baines 2004: 154). Even though the U-j tags are apparently administrative or commercial, the context of the discovery (a royal tomb) suggests instead that the

38

Numerical Notation

Figure 2.1. A scene from The Narmer mace-head, a late Predynastic ceremonial artifact bearing early but recognizable Egyptian hieroglyphic numerals. At the far right, a quantity of 120,000 prisoners is indicated, while the lower register indicates quantities of 1,422,000 goats and 400,000 cattle. Source: Quibell 1900: Plate XXVI B.

signs were part of a nascent “visual high culture” of interest to elites seeking to legitimate their authority (Baines 2004: 170–171). While we have no evidence for numeral-signs higher than 100 from the Tomb U-j tags, by the Early Dynastic period the system was fully developed. Figure 2.1 depicts the Narmer mace-head found at Hierakonpolis, which may describe the unification of Upper and Lower Egypt by around 3100 bc, and which demonstrates that even the very highest signs were being used at that time (Arnett 1982: 42). The mace-head indicates an exaggerated tally of 400,000 oxen, 1,422,000 goats, and 120,000 humans (Quibell 1900: 8–9, Pl. XXVI). This is traditionally interpreted as an exaggerated and propagandistic tally of booty and prisoners acquired through Narmer’s military victories. Millet (1990), however, provides an alternate interpretation of the mace-head inscription as a year-identifier and suggests that the numbers are purely artificial, meant only to signify the taking of a census. Another early example of hieroglyphic numerals is found on the Second Dynasty statue of Khasekhem indicating the slaughter of 47,209 of the pharaoh’s enemies (Guitel 1958: 692). These large numerical values figure prominently in early hieroglyphic inscriptions, further supporting the idea that early Egyptian monumental writing was primarily oriented toward display purposes relating to the ideological justification of the kings’ authority (cf. Baines 2004). Other numerical inscriptions from the Early Dynastic include tags for commodities from Naqada like the

Hieroglyphic Systems

39

earlier ones found at Abydos, but using ordinarily structured hieroglyphic numerals (Imhausen 2007: 14). In the Old Kingdom (2575–2134 bc), variants on the basic hieroglyphs were not uncommon. Clagett (1989: 56–57) discusses a variant of the hieroglyphic numerals used on the Palermo Stone (a Fifth Dynasty / 2400 bc pharaonic annal), in which certain notations of the aroura measure of land are represented with unit-strokes in a quasi-positional manner. However, this system is not used regularly throughout the Palermo Stone and is not found in any other inscriptions; hence its value for understanding the hieroglyphic numerals is somewhat limited. Another unusual Old Kingdom system has been proposed by Posener-Kriéger (1977) to have been used in papyrus documents from Gebelein indicating area measures of fabric on so-called linen-lists (mentioned earlier). In this system, a single cubit sign meant one square cubit, horizontal strokes ten, and long vertical strokes meant one hundred – each of which were followed by short vertical strokes indicating how many of the requisite units were denoted. Both of these variant systems were employed solely for enumerating particular types of goods, and were never used more generally for denoting numbers. In the Ptolemaic era (332–30 bc), the hieroglyphic numerals became more complex. The sign for 1,000,000 was reintroduced into the numerical sequence, though it is unclear whether its numerical meaning was truly understood. In a few inscriptions from this period, a “ring” sign – ß – is found in the sequence between v and w. While Sethe (1916) believed that the ring sign was a meaningless addition, Gunn (1916: 280) protested that perhaps, in order to lengthen the series of numerals without assigning the god w a subordinate place, ß was assigned the value of 1,000,000, while w either shifted upward in value to ten million or else retained its lexical meaning of “an uncountable number.” Curiously, on the stela of Ptolemy Philadelphos (r. 282–246 bc) at Pithom, the sign used for 100,000 is not v but rather ˆ, with the ring sign placed underneath the ordinary tadpole sign (Sethe 1916: 9). Another curious change in the late hieroglyphic numerals is the occasional use of cryptographic ciphered numeral-signs for many numbers, as shown in Table 2.3. These signs replaced the standard cumulative sets of signs with single signs whose association with the number was homophonic, pictorial, religious, or related to the corresponding hieratic numeral-sign. They were used as early as 950 bc on a wooden votive cubit rod of Sheshonk I, but are found on no artifacts between that point and the Ptolemaic era (Priskin 2003). The most common of these signs is that for 5, a five-pointed star, which often combines with unit-strokes in the same way as V = 5 in Roman numerals (Sethe 1916: 25). However, unlike the Roman numerals and related systems, no signs were developed for 50, 500, or other half-powers. The origin of this sign is almost certainly pictorial, from the five

Numerical Notation

40

Table 2.3. Ptolemaic-era cryptographic hieroglyphic numerals 5 7 9 60 80

Q U Z A B

points of the star. Other common signs were a human head for 7, from the Egyptian understanding of the head as having seven orifices, and a scythe for 9, from the resemblance between that sign and the hieratic numeral-sign for 9 (Sethe 1916: 25). In addition to the signs for the units 1 through 9, there were cryptographic hieroglyphs for 60 and 80, both of which were derived from resemblances to hieratic numerals (Fairman 1963). These new signs never led to a fully ciphered-additive set of hieroglyphic numeral-signs, and were often included in otherwise perfectly ordinary cumulative numeral-phrases. Hieroglyphic numerals are largely written on monumental inscriptions, but not exclusively so. Texts including hieroglyphic numerals include seals, funerary stelae and tomb inscriptions, annals, lists relating to conquest and plundered goods, and certain administrative texts. An often-overlooked source of hieroglyphic numerals is the wide variety of stone balance-weights bearing inscriptions indicative of their weight (Petrie 1926, Petruso 1981). Numerals indicated dates, weights and measures, and a wide variety of quantities of goods, animals, and people. In all of these texts, the numerals are formed in the ordinary fashion just described. The hieroglyphic numerals were rarely if ever used for mathematics and calculation. The vast majority of Egyptian literary texts, and all Egyptian mathematical texts, are written in the hieratic or later demotic scripts (cf. Gillings 1978: 704–705). Nor are there hieroglyphic numerals marked on potsherds, tallies, or other such media that would suggest their use as an intermediate step in performing calculations. Some hieroglyphic numerals are used in an inscription from the tomb of Methen (Fourth Dynasty, twenty-sixth century bc), which indicates the calculation of the area of a rectangle, but this inscription indicates only that the calculation was done; the numerals were not actually used in the calculation process (Peet 1923: 9). Yet, because Egyptologists regularly transliterate documents in the hieratic script into regularized hieroglyphs, historians of mathematics have sometimes

Hieroglyphic Systems

41

inferred wrongly that the Egyptians calculated using hieroglyphic numerals. The hieroglyphic system is cumulative-additive, while the hieratic system is cipheredadditive, but since this difference in structure was underemphasized by Egyptologists of earlier generations (e.g., Gardiner 1927: 191; Peet 1931: 411), historians of mathematics frequently presume that the hieroglyphic numerals were the only ones available to Egyptian scribes (cf. McLeish 1991: 42; Guedj 1996: 34–35; Palter 1996: 228–229; Dehaene 1997: 97). It is to be hoped that the presence of new Egyptological literature may remedy this deficiency (Ritter 2002, Imhausen 2007). It is necessary to treat the hieroglyphic and hieratic systems separately, not despite their very strong historical connection, but because of that connection, inasmuch as the two systems were different in structure and used in entirely different functional contexts. A very few hieroglyphic inscriptions express large numbers (particularly multiples of 100,000) through multiplicative formations instead of purely additive ones. In one Ptolemaic-era text, the number 27,000,000 is expressed by placing a single sign for 100,000 above the ordinary additive hieroglyphic phrase for 270 (Brugsch 1968 [1883]: III, 604).3 The only other way to write 27,000,000 would have been to use 270 signs for 100,000 or 27 signs for 1,000,000, neither of which is an attractive option. Such phrases would be unappealing from the perspective of Egyptian aesthetic canons, in addition to the clear economy of symbols enjoyed through multiplicative phrases. In a second instance (from the time of Amenhotep III, around 1400 bc), 100,000 is expressed multiplicatively using the tadpole-sign v placed above a vertical stroke – 100,000 × 1 = 100,000 (Sethe 1916: 9; Loprieno, personal communication), thus, curiously, requiring more signs than the standard numeral-phrase. Finally, multiplicative hieroglyphs are found on a number of votive cubits from the New Kingdom and later, cubit-long polygonal stone objects inscribed with metrological and religious information, with clear multiplicative phrases for multiples of 100,000 and possibly also for 100 and 1000 (Ritter 2002: 308–309). Yet there is no evidence that this multiplicative-additive structure was widespread. The ciphered-additive hieratic numerals use multiplicative forms far more frequently, and earlier, than do the cumulative-additive hieroglyphs (Sethe 1916: 8–10; Möller 1936: I, 59). Yet because Egyptian grammars mention the hieratic examples (e.g. Gardiner 1927: 191; Allen 2000: 97) but transcribe the numerals as hieroglyphic numerals, it is easy to conclude that multiplicative expressions are common in the hieroglyphic numerals, when in fact almost all such expressions come from hieratic texts. 3

While Brugsch interprets this figure as 100,270, the figure being represented is the amount (in arouras) of land in Egypt, for which 27,000,000 is the only reasonable interpretation (cf. Kraus 2004: 225).

42

Numerical Notation

The question of when this borrowing took place remains open; the first hieratic documents to use this structure date to the Middle Kingdom (2040 to 1652 bc), while the first hieroglyphic example (mentioned earlier) dates to about 1400 bc. Because the hieratic multiplicative numeral-phrases are more common and earlier than the hieroglyphic, I think it likely that the ancestral system (hieroglyphic) borrowed the feature from its descendant (hieratic). Because hieroglyphic numerals were used only for monumental purposes at that time, numbers higher than 100,000 would have been expressed only infrequently. It is entirely possible, given the small number of hieroglyphic inscriptions using multiplication, that it was an exceptional response to the occasional requirement for expressing high numbers in hieroglyphic numerals. There is no evidence supporting Guitel’s assertion that this occasional use of multiplication, which is paralleled in certain Aztec texts (Chapter 9), represents even an abortive step toward a fully positional notation (Guitel 1958: 692–695; 1975: 44). Rather, it represents an alternative means of increasing the conciseness of some (but not all) numeral-phrases and extending a system’s capacity to write numbers while retaining its basic structure. The Egyptian hieroglyphic script possessed two distinct systems for representing fractional values, both of which normally expressed only unit-fractions – those in which the numerator is 1. The first such system, the standard system for expressing fractional quantities, simply required the scribe to place the sign r, which could also mean “part” or “mouth,” above any hieroglyphic numeral-phrase to indicate the corresponding unit-fraction (Loprieno 1986: 1307). If the mouth sign was too small to place over the entire phrase, it was simply placed over the signs for the highest power of 10. This system also used special symbols for some of the most commonly used fractions: 1/2 (gs), 1/4 (r-4), 2/3 (rwj), and 3/4 (hmt-rw) (Sethe 1916: Table II; Allen 2000: 101). The last two of these are not unit-fractions, and are thus exceptions to the general rule. This system was not used in the Predynastic era, but is found in abundance during the Old Kingdom and thereafter. While, in theory, this system could express any fraction, most have denominators smaller than 20. The second system was used primarily for measurements of volume of grain, fruit, and liquids by indicating fractions of the heqat (ḥḳ t), a measure probably equal to 292.24 cubic inches, or roughly 4.8 liters (Chace et al. 1929: 31). This notation is sometimes known as “Horus-eye fractions” because the six hieroglyphic symbols for fractional values can be combined to form the glyph of the wḏ t or eye of Horus (#), a symbol of health, fertility, and abundance. The sum of these signs is only 63/64; symbolically, the remaining 1/64 would be supplied magically by the god Thoth when he healed the Eye of Horus, thus producing unity (Gardiner 1927: 197). Ritter (2002) makes a persuasive case, however, that these signs were originally nonpictographic, hieratic submultiples of metrological units

Hieroglyphic Systems

43

Table 2.4. Egyptian hieroglyphic fractional ideograms 1/2

1/4

x

5

2/3

3/4

2

3

“Horus-eye fractions” 1/2

1/4

1/8

1/16

1/32

1/64

$

%

& *

(

)

rather than pure numerals, and that they were not originally associated with the Eye of Horus at all, and thus he renames them “capacity system submultiples.” Despite the early caution of Peet (1923), the “Horus-eye” interpretation has misleadingly become standard among historians of mathematics. The hieroglyphic forms of these signs are shown in Table 2.4, presuming a left-right direction of writing (Sethe 1916: Table II; Gardiner 1927: 197). This system’s binary structure was probably most useful for dividing and multiplying by two, a standard operation needed when manipulating volumes of goods. The system probably originated in an earlier hieratic series of fractional signs, of which the earliest example is from the Abusir Papyri of the Fifth Dynasty, and only later did the signs assimilate to resemble the parts of the Horus-eye symbol (Reineke 1992: 204). Other than one possible sign for 1/2 from the Fifth Dynasty, the earliest hieroglyphic Horus-eye fractions are from the Nineteenth Dynasty or later (Priskin 2002: 76; Ritter 2002: 304). The Egyptian hieroglyphic numerical notation system has several direct descendants, the most direct of which is the ciphered-additive Egyptian hieratic system (to be discussed later), which developed as early as the First Dynasty as a scribal shorthand for the hieroglyphs (Peet 1923: 11). Egyptian scribes would have learned both the hieroglyphic and hieratic numerals during their education, and used both systems in the appropriate contexts – the hieroglyphs on stone monuments, and the hieratic numerals written in ink on papyrus and ostraca (inscribed potsherds). It is also very likely that the civilizations of the Aegean used the Egyptian hieroglyphic numerals as the model for their own indigenous numerals – the Cretan hieroglyphic system, the Linear A and B numerals, and the Hittite-Luwian hieroglyphic numerals. There was considerable commercial and political interaction between Egypt and the Aegean in the second millennium bc, when the Aegean numerical notation systems began to emerge (Cline 1994). Despite the dissimilarity in the numeral-signs of the two systems, they are identically structured, and thus a hypothesis of diffusion is likely correct. A less direct descendant of the Egyptian hieroglyphs is the Phoenician-Aramaic system, which

44

Numerical Notation

was developed around 750 bc, blending the numeral-signs and script tradition of the Egyptian hieroglyphs with the structure of the Assyro-Babylonian cuneiform numerals (Chapter 7). This development marks the formation of the Levantine systems (Chapter 3), reflecting the intermediary position of the Levantine civilizations between the larger polities of the eastern Mediterranean. By the Greco-Roman period, the use of the hieroglyphic script and numerals had declined greatly, and both writing and numerals had increased in the number of signs used and the complexity thereof, to the point where it was considered to be a purely symbolic or cryptographic script by outsiders (Ritner 1996: 81). By the third century ad, Egypt was becoming increasingly Christian in its religion, and its language was being written in the Greek and Coptic scripts. The latest dated hieroglyphic inscription is on the temple of Isis at Philae, and dates from August 24, ad 394 (Griffith 1937: I, 126–127). By the fifth century, knowledge of how to read and write hieroglyphs had disappeared. The hieroglyphic numerals, as well as their immediate descendants, were replaced by the Coptic alphabetic numerals (Chapter 5).

Egyptian Hieratic The hieratic script was developed around 2600 bc by Egyptian scribes as a sort of cursive shorthand for the earlier hieroglyphic script (Loprieno 1995) and, like its forerunner, used a mixture of logographic and phonographic components. However, unlike the hieroglyphs, hieratic writing was designed for cursive writing on papyrus and on ostraca, making it suitable for administrative and literary purposes. Furthermore, while the hieroglyphs could be written in a variety of directions, hieratic texts are always linear and written from right to left. While the form of the hieroglyphs was very regular and formalized, hieratic writing varied greatly by period, location, and the idiosyncrasies of the scribe’s handwriting. The Old Kingdom divergence of Egyptian scripts into monumental (hieroglyphic) and cursive (hieratic and demotic) variants continued throughout the remainder of ancient Egyptian history. A base-10 ciphered-additive numerical notation system accompanied the hieratic script. The hieratic numeral-signs, like the script itself, changed considerably over the system’s extensive history. The paleographic development of hieratic numerals is traced in the charts provided by Möller (1936). In Tables 2.5, 2.6, and 2.7, I present three distinct sets of numerals, the first and earliest from the Kahun papyrus (Twelfth Dynasty / 2000–1800 bc), the second from Pap. Louvre 3226 (fifteenth century bc), and the third from the Harris papyri (twelfth century bc) (Möller 1936, vol 1: 59–63, vol. 2: 55–59). These three texts contain mostly complete sets of numeral-signs at least as high as 1000, and are thus very useful for

Hieroglyphic Systems

45

Table 2.5. Hieratic numerals (Kahun papyri, Twelfth Dynasty) 1

2

a b 10s j k 100s s t 1000s 1 2 10,000s : :: 100,000s > 4367 = gou4 1s

3

4

5

6

7

8

c l u 3 :::

d m v 4 =

e n w 5

f g h o p q x y z 6 8

9

i r 0 9

Table 2.6. Hieratic numerals (Pap. Louvre 3226, Eighteenth Dynasty) 1

2

3

4

5

6

7

8

9

A B C D E F G H I 10s J K L M N O P Q R 100s S T U V W X Y Z [ 1000s \ ] 657 = GNX 1s

Table 2.7. Hieratic numerals (Pap. Harris, Twentieth Dynasty) 1

2

a b 10s j k 100s s t 1000s | } 10,000s 8 9 100,000s , 56,207 = gt4; 1s

3

4

5

6

7

8

9

c l u 1 0

d m v 2 :

e n w 3 ;

f o x 4 <

g p y 5 =

h q z 6 >

i r { 7 ?

46

Numerical Notation

comparative purposes. The Harris papyri numerals, from Table 2.7, include all of the numbers up to 100,000; this is the only text to do so. Looking only at the signs for 5, 6, 7, and 9, the three series appear remarkably distinct. At the same time, however, most of the hieratic numeral-signs show remarkable continuity. Many of the hieratic signs used in the Old Kingdom would have been perfectly comprehensible to a scribe in the Late Period or even the Ptolemaic era. Many of the numeral-signs are very similar to others from the same period; for instance, it is very difficult to distinguish 400 from 600 or 3000 from 5000 in Table 2.7. When used to express days of the month, hieratic numerals, like hieroglyphic numerals, were often rotated ninety degrees counterclockwise to reflect their function. Given the nature of the Egyptian calendar, these forms exist only for numerals less than 30. To write fractional values, a small dot was placed above the numeral-phrase for an integer to indicate the appropriate unit fraction (1/x). The hieratic system is primarily ciphered-additive, and its signs each represent a multiple of a power of 10. Many of the hieratic numeral-signs bear a clear relationship to their cumulative-additive hieroglyphic forerunners, seen particularly in the signs for 1 through 4, 10, 10,000 through 40,000, and 100,000. Other hieratic numerals show no clear correspondence with their hieroglyphic ancestors except in very early periods. The ciphered-additive hieratic system thus shows traces of its cumulative-additive ancestry. For this reason, I include the hieratic system in this chapter even though it is structurally different from its hieroglyphic ancestor. In some hieratic texts, irregular numerical systems were used in conjunction with grain measures (Allen 2000: 102). One early Middle Kingdom system notated sacks of grain using regular numerals, and heqats (1/10 sack) using one to nine dots in a cumulative fashion. Later in the Middle Kingdom, a notation developed whereby ordinary numerals placed before a heqat unit indicated multiples of 100, those after the numeral multiples of 10, and then one to nine dots for the units. These systems were not used outside of this metrological context. As discussed earlier, hieratic fractions were frequently written in unit-fraction form or through capacity system submultiples (Ritter 2002). For writing many numbers above 10,000, multiplicative notation was used in the hieratic numerals; for instance, the sign for 60,000 is written by placing the sign for 6 below the sign for 10,000. This principle is not used for 10,000 through 30,000, but was used occasionally for 40,000, and normally for 50,000 through 90,000 and for values above 100,000. While the multiplicative principle is seemingly used for certain values of the hundreds and the thousands, paleographic analysis of the numeral-signs shows that the sign for 300 represents the abbreviation of the first two of three cumulative 100-signs and the extension of the third rather than the juxtaposition of 3 and 100. Imhausen (2006: 25–26) discusses an

Hieroglyphic Systems

47

Table 2.8. Evolution of cursive from linear Egyptian numerals

6

9

300

Hieroglyphic

Old Kingdom

Kahun papyrus (Dyn. 12)

P. Louvre 3226 (Dyn. 18)

P. Harris (Dyn. 20)

qqq qqq qqq qqq qqq

F

f

F

f

I

i

I

i

„„„

D

u

U

u

ostracon from the New Kingdom workers’ village of Deir el Medina, a scribal exercise in which 600,000, 700,000, 800,000, and five, six, and seven million are expressed multiplicatively using the hieratic signs for 100,000 and one million, respectively. The regular use of multiplicative-additive structuring allowed very high numbers above 100,000 to be expressed easily in hieratic numerals by placing the appropriate multiplier below the “tadpole” sign. The earliest multiplicative hieratic numerals are from the twentieth century bc Kahun papyrus fragments, which are arithmetical problems and accounting documents, suggesting that this technique originated in the context of mathematical or arithmetical practice (Griffith 1898: 16). The development of hieratic numerals was thus a highly creative process involving both the shift to ciphered notation and the use of multiplication where it was deemed useful. The strong similarities between the hieratic numerals and the earlier hieroglyphic numerals, coupled with the indisputable historical connections between the two scripts, prove the historical relationship of the two systems. Whereas the hieroglyphic numerals are found in Predynastic inscriptions, hieratic numerals first appear in the First Dynasty (Peet 1923: 11). Their use became widespread from the Old Kingdom onward, with the two systems (hieroglyphic and hieratic) being used for parallel purposes. The earliest hieratic numerals were little more than cumulative-additive cursive forms of the appropriate hieroglyphic numerals. Over time, the numerals became increasingly removed from their hieroglyphic ancestors as multiple strokes were condensed into single strokes, probably for greater ease of writing. By the Fifth Dynasty, the numerals written in the Abusir papyri (archives of royal funerary cults) had acquired a strongly cursive character that had moved away from the original cumulative signs (Goedicke 1988: xvi–xvii). Table 2.8

48

Numerical Notation

compares how the numbers 6, 9, and 300 were written in Old Kingdom hieratic to the numeral-signs from the three sets of numerals presented earlier. While ciphered signs were the ordinary ones, the system’s origins were not completely forgotten; cumulative hieratic numeral-signs were occasionally employed even into the New Kingdom. No single individual invented ciphered notation in Egypt; rather, its development was a process of abbreviating and combining cumulative signs by scribes over many centuries until, by the Late period, very few hieratic signs bore any resemblance to their hieroglyphic counterparts. It is even possible that the scribes making these changes were not really aware of the importance of the new structural principle they were using. Hence the origin of ciphered notation may, in some sense, have been accidental. Strikingly, in some hieratic documents from the Ptolemaic era, there is a reversion in the numeral-signs away from the ciphered signs used in older hieratic texts and back to the common use of the cumulative principle. In several texts (Leinwand, P. Bremner, Isis-N., Leiden J. 32, and P. Rhind),4 hieratic units were expressed with repeated vertical strokes, tens with horseshoe-shaped curves, and hundreds with coils, in an exact imitation of the hieroglyphic numeral-phrases of the same value (Möller 1936: vol. III, 59–60). While some of these documents retained the ciphered signs for some values, there is a trend over time toward the use of cumulative numeral-signs in these late hieratic documents. Some scribes may have forgotten the ciphered signs; more likely, however, the reversion to cumulative-additive numerals was a deliberate archaism, resulting from the desire to emulate hieroglyphs more exactly. Egyptian scribes would have learned both hieroglyphic and hieratic writing and numerals during their education, and used whichever was appropriate according to the context. Accordingly, while the functions of the hieratic numerals are quite distinct from those of the hieroglyphic numerals, the users of the two systems would have been the same individuals. For the hieratic numerals, two functions stand out above all others: administration and mathematics. Almost all extant Egyptian legal, commercial, educational, and literary texts from 2600 to 600 bc are written in hieratic, and numerals abound on such documents. While hieroglyphic numeral-phrases were very lengthy, requiring an enormous number of symbols to express many small values, hieratic numerals were highly concise, facilitating their use in accounting, commerce, and law, as well as for expressing dates and cardinal quantities. Because they would have been learned and used by only a small and well-educated segment of the populace (i.e., the scribes), their main disadvantage – the large number of signs one needed to learn in order to use the system – would not have been a serious problem. 4

P. Rhind does not refer here to the famous Rhind Mathematical Papyrus, but to a different text dating to 9 bc and having nothing to do with mathematics.

Hieroglyphic Systems

49

A limited but interesting set of hieratic texts directly concern mathematical subjects. The hieratic numerals were first used in Egypt for arithmetic and mathematics in the late Middle Kingdom and the early Second Intermediate Period (Twelfth and Thirteenth Dynasties). The well-known Reisner, Berlin, Kahun, and Moscow mathematical papyri all date from the nineteenth century bc (Gillings 1978: 704–705). Later, around 1650 bc, during the period of Hyksos domination, the Egyptian Mathematical Leather Roll and the famed Rhind Mathematical Papyrus were written using hieratic numerals, though the latter may be a copy of an earlier document. Egyptian mathematics was never perceived as a separate field of activity, but was thoroughly enmeshed and embodied within daily scribal practice, so to search for “pure” mathematics divorced from administrative activities is futile and ethnocentric (Imhausen 2003: 386). While the most thoroughly mathematical texts from ancient Egypt date from roughly 1900 to 1650 bc, these texts are but a minuscule fraction of the total number of hieratic texts containing numerals, and cannot represent the full scope of mathematical practice over four millennia of Egyptian history. A full discussion of the mathematics of ancient Egypt is well beyond the scope of this work (cf. Peet 1923; Neugebauer 1957; van der Waerden 1963; Gillings 1972, 1978; Rossi 2004; Imhausen 2003, 2006, 2007). Tracing the diffusion of the hieratic numerals is quite difficult. As the system was primarily used for administrative purposes, it spread wherever Egyptian domination extended – for instance, into Canaan in the Nineteenth and Twentieth Dynasties (Millard 1995: 189–190). The early Israelites used a minor variant of hieratic numerals (to be described later) starting in the tenth century bc. In addition, the hieratic numerals gave rise to two distinct descendant systems. By the eighth century bc, the hieratic of Upper Egypt (“abnormal hieratic”) was no longer mutually legible with that of Lower Egypt, which is now known as “demotic,” and which eventually replaced its ancestor. In addition, the Meroitic cursive script, found on ostraca in the Sudan starting in the third century bc, contains numeral-signs to which Griffith (1916: 23) assigns ancestry from the hieratic numerals. While the hieratic numerals have relatively few direct descendants, through their demotic descendant they are ancestral to a great number of systems. In the Twenty-sixth Dynasty (664 to 525 bc), the demotic script and numerals, which had only begun to diverge from hieratic a century or so earlier, were accorded royal preference for most purposes. After that point, demotic began to replace hieratic for more and more functions throughout Egypt. By the early Christian era, when hieratic was encountered by the Greeks, it was used only in religious texts – by which means it got its name, hieratikos ‘sacred’. The name that we now give to this script and numerical notation system is, ironically, taken from a purpose for which it was rarely used throughout over two millennia of its history. By around 200 ad, even these religious functions had ceased.

50

Numerical Notation

Hebrew Hieratic In the ninth century bc, the Egyptian scribal tradition, including the use of the hieratic script and numerals, was adopted by the ancient Israelites, who incorporated a great deal of Egyptian learning into their own thought (Millard 1995, Rollston 2006). Prior to this point, there is no evidence that the Israelites used any numerical notation whatsoever, although some may have become familiar with Egyptian notations while in Egypt. From around the ninth5 through the sixth centuries, Hebrew scribes combined their own (Paleo-Hebrew) script with a variant form of hieratic numerals, which I will call “Hebrew hieratic” for simplicity, although they differ only paleographically from ordinary Egyptian hieratic numerals (Kletter 1998: 142). The earliest Hebrew inscriptions containing hieratic numerals are the Samaria ostraca from the late ninth or early eighth century bc. The most notable and complete example of Hebrew hieratic numerals is a large (30 × 22 cm) ostracon, KBar6, excavated from Tell el-Qudeirat (Kadesh-barnea) in 1979, depicted in Figure 2.2, with the numerals transcribed in Table 2.9. The ostracon contains a very complete series of hieratic numerals; only the signs for the units 1 through 9 and 60 were missing, blurred, or unreadable (Lemaire and Vernus 1980; Cohen 1981: 105–107). It was probably a scribal exercise or practice text in writing numerals and measures (Dobbs-Allsopp et al. 2005: 251–260). The number 10,000 is expressed by combining the Egyptian hieratic sign for 10 with the lexical numeral ‘thousand’ in Paleo-Hebrew script, and there are also Hebrew units of measurement (heqat, shekel, gerah) on the ostracon, so it cannot have been written by an Egyptian scribe. The numeral-signs are paleographically very similar and structurally identical to the late hieratic ones, indicating that these numerals were directly borrowed under conditions of economic domination by and cultural contact with Egypt. Ostraca found at Arad and Lachish and mostly dating to the late seventh and early sixth centuries record weights and measures in letters and accounts, and served an administrative function (Levine 2004: 433).6 While Gandz (1933: 61) argued that the numerals from the Samaria ostraca had an Aramaic origin, the numeral-signs are in fact hieratic in origin (Lemaire 1977: 281). The Samaria ostraca signs for 5, 10, and 20 are , , and , respectively, which closely resemble the late hieratic e, j, and k, but not Aramaic aaa\aa, A, and B.

5

6

Shea’s claim (1978) that this system is attested on a Late Bronze Age jar handle from the thirteenth century bc has not been addressed by Semitic epigraphers, but both the paleographic interpretation and the date are questionable. Some of the Arad ostraca record quantities in a West Semitic (Aramaic-Phoenician) notation rather than in hieratic numerals (see Chapter 3).

Hieroglyphic Systems

51

Figure 2.2. The KBar6 ostracon from Kadesh-barnea, most likely the work of a student practicing writing numerals in Hebrew hieratic numerals. Source: Cohen 1981: 106. Reprinted with the permission of the American Schools of Oriental Research.

The Hebrew hieratic numerals were also used on Judaean inscribed limestone weights, a metrological system that reflected the growing commercial importance of Judah within the context of Near Eastern commerce between the late eighth and early sixth centuries bc. Kletter’s (1998) study of 434 of these Judaean weights represents the most thorough examination of these artifacts to date. Although it was once argued that these notations were cumulative-additive and indigenous in origin (Allrik 1954, Yadin 1961), apart from the paleographic variability one would expect when transferring a cursive notation onto stone, they are clearly hieratic, Table 2.9. Hebrew hieratic numerals from KBar6 ostracon from Kadesh-barnea 1 10s 100s 1000s

2

3

4

5

6

7

8

9

52

Numerical Notation

ciphered-additive numerals like those on the ostraca (Aharoni 1966). The numerals on these weights range from 1 to 50 to indicate multiples of the shekel (approx. 11.3 grams), but curiously, other than the 1- and 2-shekel weights, the numerals are in a 5:4 ratio to the expected masses of the weights. This puzzle has been the source of considerable recent debate among metrologists and remains incompletely resolved, but in any case the numerical interpretations of the signs are paleographically secure (Ronen 1996, Kletter 1998). The ostraca and shekel weights are of a relatively early date in the history of Hebrew writing, and must be understood in the context of growing administrative needs in the Iron Age states of the Levant. This period marks the first in which literacy was relatively widespread in Judah (Kletter 1998: 144). Rollston (2006) rightly sees the presence of hieratic numerals and artifacts such as KBar6 as positive (if not conclusive) evidence for the introduction of formal schooling in late Iron Age Israel. After Judah lost its independence in 586 bc, the system appears to have become obsolescent. The Hebrew hieratic numerals are not directly ancestral to the cumulative-additive Levantine systems that emerged in the eighth century bc, among Phoenicians and Aramaeans. These systems were used contemporaneously with the hieratic numerals and, for instance, occur in the same contexts on ostraca from Arad. There is no evidence of an indigenous Hebrew numerical notation system until about 125 bc, when the use of the familiar alphabetic numerals (Chapter 5) began.

Meroitic The kingdom of Meroë, which flourished from roughly 300 bc to 350 ad, made use of two distinct scripts. The first, Meroitic hieroglyphs, were based on Egyptian hieroglyphs and were used on some stone monuments. The attested Meroitic hieroglyphic inscriptions contain no numerals. The other script, the Meroitic cursive, was written from right to left on ostraca as well as on stone, and was accompanied by a set of numerals. Almost all of our information on the Meroitic numerals rests on the work of F. Ll. Griffith, the original decipherer of the Meroitic scripts. Unfortunately, because the Meroitic language has no known relatives, we are largely unable to read Meroitic inscriptions, even though the values for the signs of the cursive script are more or less fully deciphered. Griffith (1916: 22) first presented the interpretation of the Meroitic numeral-signs shown in Table 2.10. On structural and paleographic grounds, the values for the units, 10, and all of the hundreds are unquestionable, and the remainder of the numeral-signs are fairly certain. This system is ciphered-additive and decimal, and written from right to left (like the Meroitic cursive script). The number 2348 as shown in Table 2.10 appears on the stela of Akinidad, which dates to the late first century bc (Griffith 1916: 22).

Hieroglyphic Systems

53

Table 2.10. Meroitic numerals 1

2

a b 10s j k 100s s t 1000s A B 2348 = hmuB

1s

3

4

5

6

7

c d e f g l m n p u v w x C

8

9

h

i

z

As with the hieratic numerals, the signs for the units, low hundreds, and possibly 1000 through 3000 are somewhat cumulative. There is only one case (again, from the Akinidad stela) where a number greater than 10,000 is expressed; interestingly, where hieratic uses a single sign for 10,000 (8), Meroitic appears to use a multiplicative formation (10 × 1000). However, this evidence is far too limited to conclude that the Meroites regularly used multiplicative-additive structuring to express higher powers. In addition, cumulative sets of one to nine dots apparently indicated tenths from 1/10 to 9/10, while a dot in a semicircle (0) represented 1/20 (Griffith 1916, 1925). Griffith (1916: 22–23) believed this system to be purely metrological, representing tenths and twentieths of some larger unit of measure rather than abstract numbers. By the time of the development of the Meroitic scripts, the hieratic script and numerical notation system had largely been replaced by demotic throughout Egypt. Nevertheless, on paleographic grounds (citing especially the signs for 6, 10, and 20, but also the cumulative unit-signs), Griffith (1916: 23) argued that the Meroitic numerals resemble the late hieratic numerals (eighth to third centuries bc) more closely than the demotic forms, even though the characters of the Meroitic cursive script are almost certainly derived from a demotic rather than a hieratic prototype (Millet 1996: 85). More paleographic analysis is desirable to settle this question. The Meroitic numerals were used for administrative purposes such as tax records and mensuration, as well as in funerary and monumental contexts indicating yeardates and quantities of individuals. Griffith suggests that something akin to the Egyptian heqat or artaba measures, used to indicate volumes of produce such as corn or dates, was probably indicated on some ostraca (1916: 23). There is no evidence that the Meroitic numerals were ever used for arithmetic or mathematics. Even on ostraca upon which multiple numerals have been written, Griffith was unable, except in one instance, to establish any arithmetical correspondence

54

Numerical Notation

between the numerals that would indicate that a tally or sum had been taken (1916: 24). The Meroitic numerals were used until the fourth century ad, but did not outlast the kingdom of Meroë. Millet (1996: 84) suggests that the script may have continued in use until the introduction of Coptic Christianity in the sixth century, but there is no textual evidence to establish whether the Meroitic numerical notation system existed during this late period. The Coptic and Ethiopic numerals, both of which are derived from the Greek alphabetic numerals, were used widely in the region from the sixth century onward.

Egyptian Demotic The demotic script developed in the late eighth century bc (Twenty-fifth Dynasty) and began to replace the hieratic script about a century later. It was a cursive script consisting largely of consonantal characters, derived from the “business hand” hieratic used in the Nile Delta (Ritner 1996: 82). During the Late period and the Ptolemaic era, demotic writing was used very widely for administrative and literary purposes, and more sporadically throughout the Roman period. A set of ciphered-additive, base-10 numerals accompanied this script throughout its history. As with the hieratic numerals, there is a great deal of variation in the demotic numeral-signs; the ones presented in Table 2.11 (after Sethe 1916: Table I) are typical of those found in papyri of the Late and Ptolemaic periods. Griffith (1909: 415–417) provides an interesting paleographic comparison of the demotic numeral-signs found on a selection of papyri dating from the Twenty-sixth dynasty to the Roman period. The demotic numerals are a base-10, ciphered-additive system, written from right to left. They are less reliant on the cumulative principle than their hieratic ancestor (compare hieratic c and demotic C for 3). Some of the signs for the thousands may be vaguely multiplicative, as there is a general resemblance between the signs for the hundreds and the corresponding signs for the thousands, but it is more likely that they are simply further reductions of the nonmultiplicative hieratic signs. Sethe (1916: Table I) suggests that additive phrases incorporating two lower signs (3000 + 2000, 4000 + 3000) were used for the missing 5000 and 7000 signs. Above 10,000, the demotic numerals, like the hieratic ones, are multiplicative (though such large expressions are fairly rare); for instance, Parker has found multiplicative expressions for 90,000 (7, = 9 × 10,000) and 100,000 (8, = 10 × 10,000) in his study of demotic mathematical papyri (Parker 1972: 86). As in the hieratic numerals, a small dot placed above a numeral-sign indicated the corresponding (1/x) unit fraction. The demotic numerals are directly derived from the hieratic forms used in the eighth century bc; as the hieratic numerals were used as late as 200 ad, the two

Hieroglyphic Systems

55

Table 2.11. Demotic numerals 1

2

3

A B C 10s J K L 100s S T U 1000s @ # $ 6268 = HOT&

1s

4

5

D E M N V W %

6

7

8

9

F G H I O p Q R X Y Z ! & * (

systems were used side by side in Egypt for nearly a millennium. This long coexistence can be explained in part by regional variations, with Upper Egypt retaining the “abnormal” hieratic numerals and Lower Egypt using demotic. Unlike the corresponding writing systems, the hieratic and demotic numerals would have been largely mutually intelligible until the Ptolemaic period at least, which may have facilitated communication between different parts of Egypt. The demotic script and numerals were accorded royal preference in the Twenty-sixth Dynasty, and thus they were used for most royal functions thereafter, while the hieratic system was retained primarily for calligraphic religious texts (Ritner 1996: 81–82). Unlike the hieratic script and numerals, which were rarely written on stone except at the very end of their history, demotic inscriptions are found on stone as well as ceramics and papyrus. Like their predecessor, demotic numerals were used for a wide variety of commercial, legal, and other administrative functions, as well as for indicating dates. A number of demotic mathematical papyri have survived from the Ptolemaic period, confirming the suitability of the system for arithmetical and mathematical purposes (Parker 1972, Gillings 1978). However, as with the hieratic numerals, most demotic texts that contain numerals serve no mathematical function. Much of our paleographical knowledge of the demotic numerals comes from administrative texts, such as dowry records and educational papyri (Griffith 1909). An extensive set of demotic numerals is found in P. Tsenhor, the private archive of a sixth-century woman (Pestman 1994). The importance of the demotic numerical notation system lies not in any structural feature or unusual function, but rather in its historical role as the immediate ancestor of several other numerical notation systems. The demotic numerals are almost certainly ancestral to the Greek alphabetic numerals (Chapter 5). These numerals, which are structurally identical to the demotic numerals, first appear in the sixth century bc in Ionia and Caria, at which time Greek trade with Egypt was beginning in earnest, and when the Ionian trading city of Naukratis in the

56

Numerical Notation

Nile Delta was the major center for trade between Egypt and Greece (Chrisomalis 2003). Furthermore, the alphabetic numerals became common in the late fourth century bc, at which time Egypt came under Ptolemaic control. Remarkably, the similarities between the demotic and Greek alphabetic numerals have been substantially ignored over the past century, with most scholars inclined to treat the latter system as a case of independent invention (but cf. Boyer 1944: 159). Secondly, there are strong similarities between the demotic numerals and the Brāhmī numerals (Chapter 6), which began to be used in India around 300 bc. In this case, the historical connection between the two regions is not as clear, but the structural similarities between the two systems suggest some connection. While trade between Egypt and India became common only in the Roman period, there are strong indications of overseas trade dating from the Ptolemaic period and perhaps even somewhat earlier. Again, few historians of mathematics have proposed this connection, although it has held some popularity among Indologists for over a century (Bühler 1896, Salomon 1998). By the Roman period (30 bc–ad 364), the demotic numerals were used increasingly rarely, as the general decline of Egyptian cultural institutions continued apace. However, even though Roman imperialism was the immediate circumstance surrounding the decline of the demotic numerals, they were not replaced with Roman numerals, but rather with the Coptic numerals, which were themselves descended from the demotic through the Greek alphabetic numerals. As Christianity began to take hold in Egypt, and the Coptic script and numerals became more widespread, demotic suffered a fatal decline. The last text with demotic numerals is a graffito on the temple of Isis at Philae (the same temple that contains the last evidence for hieroglyphs), which dates to December 2, ad 452 (Griffith 1937: I, 102–103). The last known demotic text of any sort (dated in Greek) was written nine days later.

Linear A (Minoan) The Linear A script was the standard script used in the Minoan civilization of Crete between 1800 and 1450 bc (Bennett 1996: 132). It is perhaps the most famous of all undeciphered scripts, having foiled decades of effort to interpret it. Only the numerals and a few other ideograms for commodities can be deciphered. Linear A is written from left to right and is almost certainly a mixture of syllabograms and logograms. Its well-attested numeral-signs are shown in Table 2.12 (Sarton 1936b: 378; Ventris and Chadwick 1973: 36). The Linear A numerical notation system is decimal and cumulative-additive, and is written from left to right with the powers in descending order. Where appropriate, signs are grouped in two rows of up to five signs each rather than placing them in an uninterrupted row. The variant dot symbol for 10 is found only in

Hieroglyphic Systems

57

Table 2.12. Linear A numerals 1

Å

10

100

É\ • æ 7659 = ÆÆÆÆ æææ ÆÆÆ æææ

1000

Æ ÉÉÉ ÉÉ

qqqqq qqqq

early Linear A documents and is probably related to the identical numeral-sign for 10 in the contemporaneous Cretan hieroglyphs (see the following discussion). Other than this, however, the system remained unchanged throughout its history. While Evans (1935: 693) suggested that there may have been a sign R or ¹ that stood for zero, this was later shown to be a sort of check-mark or sign for completion of an item, or perhaps served some other bookkeeping function (Bennett 1950: 205). Using statistical methods, Daniel Was (1971) has postulated the existence of a complex base-24 system for representing fractions in Linear A. If correctly deciphered, this system is likely to have been metrological in function. While Struik (1982: 56) suggests that this system is related to the Egyptian unitfractions, no real resemblances exist between the two fractional systems. The Linear A script and numerals were probably borrowed in some manner from the identically structured Egyptian hieroglyphic system (cf. Sarton 1936b: 378). Trade between Egypt and Crete was extensive in the Middle Minoan II period (ca. 1800–1700 bc), when Linear A developed (Cline 1994). Admittedly, there is no real similarity between the numeral-signs of the two scripts, except in the use of vertical strokes for the units, which is common to almost all systems used in the Mediterranean region. Whereas Egyptian hieroglyphic numerals are pictorial representations with phonetic values mostly originating as homonyms of lexical numerals, Linear A numerals are abstract and simplified. However, we would not expect the Minoans to adopt the Egyptian signs, because the signs would have no such phonetic associations for them. I am unconvinced by the isolationist position with regard to Minoan literacy (e.g., Dow and Chadwick 1971: 3–5). The link suggested between Linear A and the Proto-Elamite numerals (Chapter 7) of fourth millennium bc Iran, however, requires implausibly great chronological and geographical gaps (Brice 1963). Egypt is the only plausible ancestral region for the Minoan numerals. The abstract and geometric character of the numeral-signs makes it impossible, however, to exclude an independent origin for the system. Branigan (1969) speculates that concentric circles on sealings from Phaistos may have represented tens, hundreds, and thousands, and may be a geometric precursor to the Linear A

58

Numerical Notation

numerals. A similar system of small circles and large circles inscribed on cylindrical stone weights from the palace at Knossos may have indicated one and ten units of some metrological value (Evans 1906). While either of these systems could be related to Linear A, at present the hypothesis of borrowing from Egypt best explains the structure of Linear A numerals, with the numeral-signs developed indigenously. Numeral-signs are the only known means of representing numbers in Linear A; although it remains possible that lexical numerals were written using syllabic signs, the closely related (and deciphered) Linear B script does not do so, suggesting that this is unlikely. The vast majority of Linear A documents are clay tablets having an accounting or bookkeeping function, and thus we have many examples of the use of numerals. Vertical strokes that probably represented numbers have been found in other contexts – for example, on Minoan balance weights; these marks, however, do not show any clear relation to the Linear A signs found on the clay tablets and are probably simply unstructured unit-marks or tallies (Petruso 1978). What are likely Linear A numerals occur on a number of pieces of pottery from Bronze Age Cyprus (Grace 1940). Stieglitz proposes that a numerical graffito found at Hagia Triada and containing the sequence of numbers (1, 1 1/2, 2 1/4, 3 3/8), in which each number is 1.5 times the previous one, represents a series of musical notes or tunings for a stringed instrument (Stieglitz 1978). I think it equally likely that the series served an economic function such as calculating interest. Since we do not have significant literary or monumental texts in Linear A, we do not know if the numerals were ever used in other contexts. While the Cretan hieroglyphic numerals were formerly thought to be ancestral to Linear A, it now appears that Linear A predates the Cretan hieroglyphs, perhaps by as much as a century. The exact historical relationship between the two numerical notation systems is unclear, but I believe it most likely that the Cretan hieroglyphic numerals were a local variant of the Linear A system. The Linear B Mycenean script used on Crete and the Greek mainland definitely derived from Linear A. Its numerals (to be discussed later) are nearly identical to those of Linear A. The precise relation between the peoples using the Linear A and B scripts is still unclear, as is the question of the cause of the collapse of the Minoan civilization in the fifteenth century bc. Presumably, during this period, the Greek-speaking Myceneans adapted Linear A for their own language, resulting in Linear B. The two scripts coexisted in Crete from about 1550 to 1450 bc, after which time Linear B replaced Linear A completely.

Cretan Hieroglyphic The Cretan Hieroglyphic or Pictographic script was first identified by Sir Arthur Evans (1909) based on his work at Knossos. While it was once considered ancestral to the other Aegean scripts, it probably developed about the same time as, or

Hieroglyphic Systems

59

Table 2.13. Cretan hieroglyphic numerals 1

10

100

1000

ù\\=

• 0 ÿ 8357 = ÿÿÿÿ\000\•••••\\==== ÿÿÿÿ === slightly later than, the Linear A script. Its use is generally thought to have lasted from 1750 to 1600 bc (Bennett 1996: 132). It is found on around 300 attested seal-stones and clay documents (Olivier et al. 1996). While the script is still undeciphered, it is probably of a mixed syllabic and logographic structure, like other Aegean scripts. Among the few Cretan hieroglyphic signs that can be interpreted securely are the numerals, which are shown in Table 2.13 (Evans 1909: 258; Sarton 1936b: 378; Ventris and Chadwick 1973: 30–31). The system is cumulative-additive and decimal, and most often written from left to right, although right-to-left numeral-phrases are also attested. Groups of multiple repeated signs were sometimes organized using two rows, one above the other, each with no more than five signs, but this rule was not strictly applied, and in other cases the organization of signs was more haphazard. Figure 2.3 depicts a Cretan clay rectangular bar on which numerals are written on three of the four long sides plus the base (Evans 1909: 177). While the number 483 is written at the bottom of side (b) according to this principle, for instance, many of the other numerals are oriented irregularly, grouping signs in clusters of six or more. Evans (1909: 257) assigns the uncommon sign P the value 1/4 because it is repeated not more than three times at the end of a few numeral-phrases, while Dow and Chadwick (1971: 12) suggest a quite different fractional system with signs for 1/2, 1/4, and 1/8. Since the Cretan hieroglyphs are largely undeciphered, it is difficult to speculate on the history of their numerals. As with other Aegean scripts, an Egyptian origin for the system has been proposed (Sarton 1936: 378), though this cannot be demonstrated conclusively. There is limited similarity between the numeral-signs for the Cretan hieroglyphs and any other system, except that the use of the dot for 10 is common to some early Linear A inscriptions. Dow and Chadwick (1971: 14) suggest that the differences between the Cretan hieroglyphs and Linear A are attributable to the fact that the former were designed for chiseled inscriptions while the latter were intended to be written with ink. The Cretan hieroglyphic numerals are probably a local variation of the Linear A numerals or, less plausibly, a direct borrowing from the Egyptian hieroglyphic numerals. The contexts in which the

60

Figure 2.3. Cretan hieroglyphic inscriptions on a clay bar containing numerals. For instance, at the bottom of face b, the numeral 483 is represented with four diagonal strokes, eight dots, and three curved strokes. Source: Evans 1909: 177.

Hieroglyphic Systems

61

numerals are found are similar to those for Linear A. The Cretan hieroglyphic inscriptions include information on commodities such as wheat, oil, and olives and thus are probably records of transactions, inventories of goods, and similar administrative documents (Ventris and Chadwick 1973: 31). By around 1600 bc, Cretan hieroglyphs had been entirely replaced by Linear A.

Linear B (Mycenean) The Linear B script was used on Crete and the Greek mainland in the middle to late second millennium bc to write an archaic Greek dialect on clay administrative tablets. It is written from left to right, and consists of a syllabary with a large repertory of logograms and taxograms (classifiers), including a numerical notation system. The Linear B numerals are shown in Table 2.14. The Linear B signs are mostly identical with the Linear A signs, except that the sign for 10 is always a horizontal stroke (never a dot), and there is a sign for 10,000 that is not found in the earlier system. The 10,000 sign is probably a multiplicative combination of the signs for 10 and 1000. The structure of the system is cumulative-additive and decimal, with the highest powers on the left, written in descending order and with five or more identical signs divided into two rows. Unlike the Linear A numerals, Linear B lacks a separate system for expressing fractions; instead, specific logograms express divisions of metrological units and then combine with numeral-signs as appropriate (just as one might say 10 cm instead of 0.1 m). Ventris and Chadwick (1973: 54–55) note that some of the Mycenean logograms for metrological units resemble the Minoan signs for fractions, and may have originally indicated specific ratios of two types of units, which further shows the indebtedness of Linear B to its Minoan forerunner. The Linear B system definitely originated through direct contact with the Minoan civilization and the Linear A numerals. The earliest Linear B inscriptions date from the sixteenth century bc, so the two scripts coexisted on Crete for about a century. Their numerical notation systems are so similar that some authors do not distinguish between the two (Ventris and Chadwick 1973: 53; Struik 1982). The distinction between the two is not nearly as great as between the two scripts, which record different languages. Throughout the history of the Linear B numerical notation, there is no observable change in the form of the numeral-signs or in the structure of the system. Linear B numerals are found almost solely on clay tablets serving accounting and financial purposes (Olivier 1986: 384–386). Numerals are used both for counting discrete objects (men, chariots, etc.) and for measures of dry and liquid volume and weight. Almost all Linear B documents relate to administrative and bookkeeping functions, suggesting a very limited level of literacy and numeracy

Numerical Notation

62

Table 2.14. Linear B numerals 1

10

100

1000

10,000

Å

É æ Æ ô 68,357 = ôôô\ÆÆÆÆ\æææ\ÉÉÉ ÅÅÅÅ ôôô\ÆÆÆÆ\\\\\\\\\\\\\\\ÉÉ \\\\ÅÅÅ throughout Mycenean society. Even so, the consistency of the numerals throughout several centuries and across a substantial geographic area suggests that some sort of scribal education system was in place to transmit knowledge of both the Linear B script and its numerals. We do not know if Linear B numerals were written on papyrus or other materials, though such uses are certainly possible. We also do not know whether the Myceneans used their numerals for arithmetical purposes. Anderson’s (1958) theory on the means by which such calculations could be undertaken suffers from the defect that it involves aligning and manipulating numbers as one would in Western arithmetic, although there is no evidence that such a procedure was ever undertaken. Dow (1958: 32) and Anderson (1958: 368) both point to a clay tablet found at Pylos (designated Eq03) in which tallying in groups of five units is used to reach 137. Other tablets from Pylos discussed by Ventris and Chadwick (1973: 118–119) show that the Myceneans could successfully compute complex ratios in order to determine the contributions of goods required from towns of different sizes. Rather than proving that the Myceneans used numerical notation for arithmetic, however, these examples indicate that tallying by units and in groups of five, rather than the purely decimal-structured numerical notation, was the method used for computation. None of this denies that clay tablets recorded the results of rather complex computations done mentally, through tallying, or perhaps by some other method. There is no relationship between the Mycenean numerals and either of the later Greek numerical notation systems (the acrophonic and alphabetic systems). It is conceivable, however, that there is some relationship between the Mycenean and Etruscan numerals (Chapter 4). Both Haarmann (1996) and Keyser (1988) have raised this claim, which will be discussed in detail when considering the origins of the Etruscan system. Mycenean settlements have been found in Sicily and southern Italy, providing one possible locus for cultural contact. However, this theory is controversial, not least because of the time elapsed between the latest known Linear B documents (twelfth century bc) and the first Etruscan ones (seventh century bc). A more likely descendant of Linear B numerical notation is the Hittite hieroglyphic system, which was invented around 1400 bc and used by Hittite and

Hieroglyphic Systems

63

Luwian speakers in Anatolia. The Hittite signs for 1 and 10 are identical to the Linear B ones, and at the time when the Hittite numerals were developed, there were Mycenean settlements in western Anatolia (such as at Miletus) and on Cyprus that were engaged in trade throughout the eastern Mediterranean. The contemporaneity of the two systems makes this scenario plausible, if not proven. The perplexing and apparently violent end of the Mycenean civilization in the twelfth century bc, and the repeated razing of major sites such as Mycenae and Pylos, marks the end of the Linear B inscriptions and the start of the “Dark Age” of Greek civilization. No writing or numerical notation of any kind is attested from the Aegean region between 1100 bc and the introduction of the Greek alphabet a few centuries later.

Hittite Hieroglyphic The Hittites lived in central Asia Minor from about the end of the third millennium bc. The Hittite and closely related Luwian languages are the first Indo-European languages for which we have solid textual evidence. By the middle of the second millennium bc, two distinct scripts were in use in the Hittite Empire. Firstly, a cuneiform script (borrowed from Mesopotamia) was used to write the Hittite language. Its numerals are closely related to the Assyro-Babylonian cuneiform system, and so will be treated in Chapter 7. Additionally, an indigenous hieroglyphic script was used to represent the Luwian language on monumental inscriptions, on a few lead tablets, and probably also on wooden tablets that have not survived (Melchert 1996: 120). This script was used from about 1500 to 1200 bc, during the apogee of the classical Hittite Empire, and then is found only sporadically until the rise of the Neo-Hittite kingdoms between around 1000 and 700 bc, during which time it was again common (Hawkins 1986: 368). This script is known as Hieroglyphic Hittite or Hieroglyphic Luwian, and has a mixed syllabic and logographic structure. Among the purely ideographic signs, the Hittites used a set of written numerals as shown in Table 2.15 (cf. Laroche 1960: 380–400). The system is purely cumulative-additive and uses a base of 10. Numeral-phrases were written from left to right, right to left, or top to bottom, depending on the overall direction of the inscription. As in the Egyptian and Aegean systems, Hittite numeral-signs were sometimes but not always grouped in clusters of three to five unit-signs. Laroche (1960: 395) indicates that 9 was variously written using three rows of three strokes, a row of five above a row of four, or simply with nine strokes in sequence on a single line. The Hittite hieroglyphic numerals were most likely based on one of the Aegean numerical notation systems. Both the Linear A and Linear B scripts were in use around 1500 bc, when the first Hittite hieroglyphic inscriptions are found, but

Numerical Notation

64 Table 2.15. Hittite numerals 1

10

100

1000

q

^ ( ) 3635 = )))((((((*qqqqq Linear A was almost extinct by that time. Like the hieroglyphs, the three Aegean scripts use a combination of syllabograms and logograms. The Linear A, Linear B, and Hittite hieroglyphic numerical notation systems are all decimal and cumulative-additive, and use a horizontal stroke for the units and a vertical stroke for the tens. There was a significant degree of intercultural contact between the Aegean and Asia Minor during this period. The Myceneans had settlements in western Anatolia and traded throughout the eastern Mediterranean, and were possibly the “Ahhijawa” (Achaeans) mentioned in the Hittite archive from Bogazkoy. Because the Luwian language was spoken primarily in western Asia Minor and only later was used in the Hittite Empire, the transmission of the numerals from the Aegean to western and then central Anatolia is plausible (Hawkins 1986: 374). An alternate hypothesis is that the Hittite system was based directly on the Egyptian hieroglyphic numerals, since the Hittites were in contact with Egypt at that time. Due to the paucity of extant examples, little can be said about the function and use of the system. The numerals are found on a variety of stone inscriptions and lead tablets. Most notable among these are the Kululu lead strips (mid to late eighth century bc), which record village census data using an abundance of numerical signs (Hawkins 2000: 503–505). The Hittite numerical notation is used far more frequently than lexical numerals, which is also true of the Egyptian hieroglyphs and Aegean scripts. There is no discernable change in the structure or sign-forms of the system throughout its history, even though there is little evidence for its use between 1200 and 1000 bc, following the invasion of Phrygians and others who ended the classical Hittite kingdom. During these two centuries, the hieroglyphs were likely used only on perishable materials, such as wooden tablets (Hawkins 1986: 374). A few inscriptions on clay jars found at the Urartian site of Altintepe (in eastern Asia Minor) use a syllabary closely related to the Hittite hieroglyphs to write single words in the Urartian language, starting in the early eighth century bc (Laroche 1971, Klein 1974). Many of these inscriptions contain numeral-signs for small numbers using either “pitted” dots or vertical strokes to represent units (i.e., 5 = 554 or 11111), but never to express numbers larger than eight, making this system an unstructured tally system having no base. Klein (1974: 93) accurately states that this usage “should thus be viewed as an isolated and short-lived

Hieroglyphic Systems

65

phenomenon, possibly not outlasting the career of a single (foreign?) scribe.” The numerals that accompany the Cypriote syllabary, which was invented around 800 bc, are also potentially derived from the Hittite hieroglyphic numerals. The proximity of the Neo-Hittite kingdoms to Cyprus, the extensive trade relations between the regions, and the identical structure of the two systems all suggest that such a derivation is likely. However, there are too few Cypriote syllabic inscriptions containing numerals to establish an accurate chronology or even to secure values for certain numeral-signs. Less plausible descendants of the Hittite hieroglyphic system are the earliest Levantine systems, Phoenician and Aramaic (Chapter 3). However, these systems developed around 750 bc, at the very end of the Hittite system’s history, and are structurally distinct from it, since they have a sign for 20 and are multiplicative-additive above 100. The subjugation of the Neo-Hittite kingdoms under the Assyrian empire ended the use of Hittite hieroglyphic numerals around 700 bc, and the system was replaced for all functions by the Assyro-Babylonian common numerals. Later numerical notation systems developed for related peoples of Asia Minor, such as the Lycians, were based on a Greek model and display no obvious relation to the Hittite hieroglyphs.

Cypriote Syllabary As its name suggests, the Cypriote syllabary was a syllabic script used only on the island of Cyprus. It was used between about 800 and 200 bc for writing the Greek language, and thus coexisted with the much more prominent and long-lasting Greek alphabetic script (Bennett 1996: 130). Cypriote is always written from right to left. None of the synthetic works concerning numerical notation have dealt with the (admittedly small) evidence for a distinct Cypriote numerical notation system. However, Masson (1983: 80), whose discussion of the Cypriote syllabary is the most detailed currently available, presents about a dozen inscriptions in which the system shown in Table 2.16 was used. This rudimentary system was decimal and cumulative-additive and, like the syllabary itself, was written from right to left. The numbers expressed using the system are very small; unless certain undeciphered signs are in fact numeral-signs (as discussed later), the largest number expressed in any Cypriote inscription is 22. This system parallels the Aegean Linear systems from which the Cypriote numerals are probably derived. This is strongly suggested by the use of the Cypro-Minoan script, which was very probably borrowed from Linear A, on Cyprus as early as 1500 bc. However, eastern Cyprus was under Phoenician domination well into the period of the use of the syllabary, and the Phoenician numerical notation system is also written from right to left, and uses vertical strokes for units and horizontal

Numerical Notation

66

Table 2.16. Cypriote numerals 1

10

q

^

strokes for tens. Furthermore, Masson (1983: 80) notes the use of two unusual symbols: æ, found in but a single inscription but possibly indicating 100 on the model of the Aegean systems, and Ö, also in only a single document, but possibly signifying 20. It is notable that the Phoenician system used @\and D at various times as the sign for 20. Because Cypriote inscriptions do not contain dates, it is often difficult to place them in chronological context, but it is possible that the Cypriote system is either ancestral to or descended from the Phoenician system. A final complexity is that the Hittite hieroglyphic numerals, which were still in use in the Neo-Hittite kingdoms in 800 bc, also use a vertical stroke for 1 and a horizontal one for 10. Trade between Cyprus and Asia Minor was substantial, and it would have been an extremely short sea voyage between the two regions. None of this material categorically excludes the possibility that the aberrant signs found by Masson are non-numerical and that the Phoenician, Hittite, and Cypriote numerals are unconnected except by their temporal and geographic proximity on the island of Cyprus. The corpus of inscriptions containing numerical signs is simply too limited, and the numbers expressed too small, to resolve the issue of their origin.

Summary Despite the enormous amount of work being done in the archaeology of the eastern Mediterranean, the genetic relations among the systems of this phylogeny have not been analyzed adequately in the past. The connections between the Egyptian hieroglyphic, hieratic, and demotic systems are well established, but more data are needed to establish the specific links between the Egyptian and Aegean systems. Nevertheless, on the basis of a shared set of features that distinguish it from other, superficially similar phylogenies such as the Levantine (Chapter 3) and Italic (Chapter 4), the inclusion of all the hieroglyphic systems in a single group is warranted. First, all the hieroglyphic systems have a base of 10, but they do not use a sub-base of 5 or any additional structuring signs. Second, they mostly have a cumulative-additive structure, although the hieratic, demotic, and Meroitic systems are ciphered-additive reductions of the original structure. Third, large numbers of cumulative signs in a numeral-phrase are grouped in sets of three to five. Fourth, their direction of writing can be quite variable (left-right, right-left, top-bottom,

Hieroglyphic Systems

67

or boustrophedon). Finally, hieroglyphic numerical notation systems are used far more frequently than lexical numerals for expressing numbers. While no hieroglyphic systems survived past 400 ad, its less direct descendants include the Roman numerals and probably even Western numerals (though greatly transformed). In the following four chapters, I will discuss a) the Levantine systems (Chapter 3), the Phoenician-Aramaic numerals and related systems; b) the Italic systems (Chapter 4), the Etruscan and Roman numerals and their descendants; c) the Alphabetic systems (Chapter 5), the Greek alphabetic numerals and related systems; and d) the South Asian systems (Chapter 6), the Brāhmī system and its descendants. While they are distinct enough to warrant placing them in separate families, all originate ultimately from hieroglyphic systems.

chapter 3

Levantine Systems

The first millennium bc was an era of considerable interregional commerce, warfare, and colonization in the Levant. This region was peripheral to both Egypt and Mesopotamia and thus exposed to multiple cultural influences. The various Levantine numerical notation systems that developed in the first millennium bc share several common features that reflect their debt to both Mesopotamia and Egypt, while demonstrating their indigenous creators’ considerable inventive energy. While this phylogeny of numerical notation systems was developed and most widely used in the Levant, it would eventually be adopted in various script traditions in Asia Minor, Arabia, Iran, the Indian subcontinent, and Central Asia. The Aramaic notation is the most important of the Levantine family of systems, which also includes the Phoenician, Palmyrene, Nabataean, Kharohī, Hatran, Old Syriac, Middle Persian, Sogdian, Manichaean, and Pahlavi systems. The most commonly used signs of these systems are shown in Table 3.1. Unfortunately, despite their widespread use over a large geographical area, these systems remain poorly analyzed in recent scholarship, so we must turn to the earlier work of epigraphers and paleographers such as Schroder (1869), Duval (1881), Lidzbarski (1898), Cooke (1903), and Cantineau (1930, 1935) for analyzing Levantine numerical notation. Despite the age of these works, there is no reason to question the data presented. However, this tradition of scholarship was primarily oriented toward the study of the texts of specific societies. Issues of diffusion and 68

69

Pahlavi

Manichaean

Sogdian

Middle Persian

Old Syriac

Hatran

Kharoṣṭhī

Nabataean

Palmyrene

Phoenician

Aramaic

ë

a a a a a a a

10

a b d

f

C D

H

G A

A A H A N K à > A > A

H

5

2

R g

4

1

BA

2

1

Table 3.1. Levantine numerical notation systems

g

E

I

4

C B @ D J J J ê J @

20

ä

s

F

J

5

F\ Ç A I L ü è

100

â

å

² ¶

–

E

ï

500

u

7

7

ç

G μ

1000

—

±

10,000

Numerical Notation

70 Table 3.2. Aramaic numerals 1

10

20

100

1000

a

A C F G 2894 = \0\aaa\A\CCCC\F\aa\aaa\aaa\G\aa cross-cultural comparison have not previously been addressed, and much work remains to be done.

Aramaic The Aramaeans, who originally inhabited a large portion of modern-day Syria, are first recognizable in the archaeological and written records around the end of the second millennium bc. During the ninth and eighth centuries bc, Aramaeans ruled a number of small states in the Levant, until these came under the domination of the Assyrian empire. Around this time, they developed a consonantal script on the model of the pre-existing Phoenician consonantary. By the eighth century bc, Aramaic inscriptions began to include numerical signs, shown in Table 3.2. The system is purely cumulative-additive for numbers up to 99, written (like the script itself ) from right to left, using signs for 20, 10, and 1. The unit-signs are grouped in threes, since up to nine such signs could be required. Occasionally, when an ungrouped unit-stroke was present in a numeral-phrase, it was written at a slight angle (so that 7 would be 0\aaa\aaa). Because there was a sign for 100, no more than four 20-signs and one 10-sign would ever be required, obviating the need for such groupings for higher values. The 10-sign appears to have originally been a simple horizontal stroke, with a tail added cursively. The 20-sign is almost certainly a ligatured combination of two 10-signs, as shown by the occasional use of a variant form B. There is a gradual trend over time toward the use of a special sign for 5 (H), which Lidzbarski (1898: 199) notes appearing on an Assyrian clay tablet as early as 680 bc. However, the majority of Aramaic numeral-phrases do not use a symbol for 5. Above 100, the Aramaic numerical notation system is multiplicative-additive rather than cumulative-additive, and it is thus a hybrid system. To form 800, for instance, eight unit-signs (appropriately grouped) were placed in front of the sign for 100 in order to indicate that the values should be multiplied. The same principle was followed for the thousands. There were apparently two signs for 1000; the first, G, is actually no more than the final two letters of the Aramaic lexical

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71

numeral ‘LP ‘thousand’ (Gandz 1933: 69–70), while the second, μ, is the same as the corresponding Phoenician numeral-sign (Lidzbarski 1898: 201–202). While there is no distinct sign for 10,000 in the Aramaic system used in the Levant (though see the following discussion for Egyptian variants), rarely numbers greater than 9,999 were written using 10- and 20-signs in conjunction with the sign for 1000. Fractions are apparently found in a handful of inscriptions in which ungrouped unit-strokes aaaa and aaaaa mean 1/4 and 1/5, and one inscription contains a special sign for 2/3 (¼) (Lidzbarski 1898: 202), but they were normally written out lexically. The first Aramaic inscription with numerical notation is an eighth-century bc ostracon from Tell Qasile, in which 30 is expressed as three horizontal strokes (*) rather than the normal form (Lemaire 1977: 280). However, it may be a Hittite hieroglyphic numeral-phrase (Chapter 2), since that system was still in use in the eighth century bc in the Neo-Hittite kingdoms to the north. The earliest uncontestable examples are from the Assyrian bronze lion-weights found at Nimrud by Layard in the nineteenth century. These eighth-century inscribed weights have texts in Aramaic and Akkadian; on the largest (BM 91220; CIS II/1, 1), dating to the reign of Shalmaneser V (726–722 bc), the number 15 indicates the object’s weight of fifteen minas in three different ways on its three lines of text: in Aramaic lexical numerals, as fifteen ungrouped single strokes, and according to the structure detailed above (aa\aaa\A) (Fales 1995: 35). This threefold repetition using different methods of representation suggests that the system was unfamiliar, either because of its novelty or because it was intended for speakers of several languages. Structural similarities between the Aramaic system and the Assyro-Babylonian common system (Chapter 7), with which it shares a decimal base and the use of multiplicative-additive structuring for the hundreds and thousands, suggest a historical connection (Gandz 1933: 69; Ifrah 1985: 356). The conquest of the Aramaeans in 732 bc by the Assyrian empire establishes a clear historical context in which this transmission could have taken place. The lion-weights from Nimrud may well have been taken from the Levant as war booty shortly after this time (Fales 1995: 54). Yet the Aramaic system is also similar to the Egyptian hieroglyphic system. Aramaic speakers would certainly have had considerable contact with Egypt in the eighth century bc, and by the sixth century bc the Aramaic script was being used by settlers in Egypt at Elephantine and Saqqara. There are a number of similarities in the forms for signs. Like the Egyptian hieroglyphs but unlike the Assyro-Babylonian system, Aramaic uses vertical unit-strokes grouped in threes to express the units. A relationship between Aramaic A and hieroglyphic r (both signifying 10) has also been postulated (Schroder 1869: 186), although it is more likely that the hooked Aramaic sign is simply a cursive alteration of a horizontal stroke. Regardless, both signs are very different from the cuneiform Assyro-Babylonian

Numerical Notation

72

Table 3.3. Aramaic, Egyptian hieroglyphic, and Assyro-Babylonian numerals

Aramaic

424 =

\aaaa\C\F\aaaa 4

Hieroglyphic

424 =

qqqqrr„„„„ 4

Assyro-Babylonian

424 = Æ

20 100 4 Å

20

400 Å

4i\b4 4 100 20 4

system. The Aramaic use of unit fractions along the Egyptian hieroglyphic model, including the exception of having a special sign for 2/3, further suggests Egyptian borrowing. West Semitic accounts, like those in the Egyptian hieroglyphic and Aegean scripts, are written with the item being enumerated placed before the numeral, in contrast to Mesopotamian texts, which follow a “quantity + item” order (Levine 2004: 435). Finally, Egyptian hieroglyphic numeral-phrases are primarily written from right to left, as in Aramaic, whereas the Assyro-Babylonian system runs in a left-right direction, although of course the Aramaic script is also written right to left, so this cannot be taken as positive evidence in its own right. These differences are compared in Table 3.3. To muddy the waters even further, two other Hieroglyphic numerical notation systems were used in the eastern Mediterranean around 750 bc and could potentially have been known to the early users of Aramaic numerals. The Neo-Hittite kingdoms, although on the wane by that time, were still present in southeastern Anatolia, immediately abutting the Aramaeans. Moreover, the Cypriote numerals were invented just before that time, and there was enormous trade between Cyprus and the Levantine coast. But while the Cypriote and Hittite systems are cumulativeadditive and decimal and use vertical strokes for 1 and horizontal strokes for 10, they lack the other characteristics that might identify them as potential ancestor systems. The Aramaic numerals were likely developed under a dual cultural influence from Egypt and Mesopotamia. The system’s basic structure is very similar to the Assyro-Babylonian common system, but many paleographic and contextual similarities are far more similar to the Egyptian hieroglyphs. Geographically and historically, the Aramaeans and other Levantine peoples were peripheral to both civilizations in the mid first millennium bc, at the time of the system’s invention.

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73

Although cultural phylogenies for scripts and numerical notation systems are usually arranged in accordance with a biological taxonomic scheme, cultural phenomena may have multiple origins, each making a contribution to the descendant, much as biological parents contribute to a child’s genetic makeup. If this explanation is correct, we must ask why the Egyptian hieroglyphic numerals, rather than the hieratic, would be chosen as a model for the Aramaic numerals. As I discussed in Chapter 2, the hieratic numerals were widely used in the Kingdom of Judah in the first half of the first millennium bc. Ostraca from Arad in the Negev dating to the late seventh and early sixth centuries attest both hieratic and Aramaic numerals (Levine 2004: 433). Like Millard (1995: 190–191), I find the failure of the Aramaeans to adopt the hieratic numerals to be rather curious. The existence of a distinct sign for 20 in Aramaic, and the recombination of features of two quite different systems, demonstrates that the Aramaeans were numerically inventive. In most of the Semitic languages, the word for ‘twenty’ is etymologically the plural of ‘ten’ – for example, Hebrew eser ‘ten’ versus esrim ‘twenty’ (Menninger 1969: 14). This may explain why the graphic “etymology” of the Aramaic numeral-sign for 20 is two ligatured 10-signs. This development of a special sign for 20 outside the regular decimal base of the numerical notation system is a unique development of the Levantine numerical notation systems; neither the Assyro-Babylonian system nor the systems of the Hieroglyphic phylogeny have this feature. Like the script to which it was attached, the Aramaic numerical notation was used in the Levant, Egypt, Mesopotamia, and farther afield throughout the second half of the first millennium bc. Segal (1983) gives ample evidence for the use of the system among Aramaic texts from Saqqara in Lower Egypt throughout the fifth and fourth centuries bc, and Aramaic papyri found at the fifth-century bc military colony at Elephantine demonstrate the use of the system in numerous administrative documents. While the system as used in the Levant had no special sign for 10,000, the Aramaic papyri found at Saqqara and Elephantine do use such a sign (±), which obeys the multiplicative principle in the same way as detailed earlier for 100 and 1000 (Segal 1983: 131; Ifrah 1985: 335). An alternate sign for 100 (²) was also used in Egyptian Aramaic, but it resembles none of the signs used in the Levant and is not similar to any of the Egyptian demotic or hieratic signs used at that time. The Aramaic script was widely used throughout the Achaemenid Empire from the sixth to fourth centuries bc on clay administrative and legal tablets, stone monuments, and leather and papyrus. While the scripts used in official royal proclamations and dedications were Old Persian, Babylonian, and Elamite, Aramaic was the lingua franca of the empire and served most administrative functions. As such, it was used as widely as Lower Egypt, Asia Minor, and the Transcaucasus and

74

Numerical Notation

even as far east as the Indus River. Throughout its history, the Aramaic numerical notation was used extensively on monumental inscriptions, ostraca, and administrative papyri. In literary and religious texts, however, numbers were more often written using lexical numerals only. Aramaic numerals were used to record the results of calculations used in commerce and administration, but none of the extant inscriptions demonstrate the use of written arithmetic. The end of the Achaemenid Empire did not spell the end of Aramaic influence over the Middle East; however, it did result in the fragmentation of what previously had been a unified script and numerical notation into several regional variants. During this period, Greek alphabetic numerals were often used administratively, although Aramaic numerals continued to be used in a variety of contexts. By the second century bce, political and ethnic divisions in the Near East had led to the emergence of variant numerical notation systems. The Hellenized Palmyrene, Nabataean, Hatran, and Edessan Syrian populations of the Levant each possessed its own variant numerical notations based on Aramaic. In these variants, the use of a distinct sign for 5 was far more prominent than in Aramaic numerals. The Kharoṣṭhī numerical notation used in parts of modern Afghanistan and Pakistan, and the Middle Persian and Pahlavi systems used in Iran, are also variant forms of Aramaic.

Phoenician The Phoenicians, who inhabited various cities (Tyre and Sidon foremostly) along the Levantine coast in the first millennium bc, were perhaps the greatest mercantile people of the ancient Mediterranean. The Phoenician consonantal script was a descendant of the earlier Canaanite consonantary that diverged from its ancestor late in the second millennium bce. However, none of the earliest Phoenician inscriptions contain numerical notation. While the Aramaic writing system developed from the earlier Phoenician, the Aramaic numerals appear to be slightly prior to the Phoenician, and there is no reason to assume that script and numerical notation must be borrowed jointly. The Phoenician numerical notation system is similar in structure to the Aramaic, with distinct signs for 1, 10, 20, 100, and 1000. These signs (including some paleographic variants) are shown in Table 3.4 (cf. Schroder 1869; Lidzbarski 1898; Gandz 1933; van den Branden 1969: 42–43). Like Aramaic, this system is purely decimal with the exception of the 20-sign, cumulative-additive below 100 and multiplicative-additive thereafter. Unit-signs are simple vertical strokes, although a left-slanting stroke is often used for ungrouped single strokes, and are grouped in threes, as in the Egyptian hieroglyphic and Aramaic systems. Like the Phoenician script itself, numeral-phrases are nearly always read from right to left, although van den Branden (1969: 43) notes at least

Levantine Systems

75

Table 3.4. Phoenician numerals 1

a

10

20

A B @ D 697 = 0\aaa\aaa\ADDDD\â\aaa\aaa

100

ä

Ç

1000

â

¶

E

μ

one exception (CIS.87,ph) in which left-to-right ordering is used, probably in error. The most notable feature of Phoenician numerals is the wide variety of forms for number-signs, particularly for 20 and 100. Schroder (1869: 188–189 and Table C) lists over twenty variants each for these two numbers, some of which can be attributed to differing individual scribal styles, while others may reflect regional or diachronic variation. I list only the more common forms for the sake of brevity. The 1000-sign is extremely rare; Lidzbarski (1898: 201) reports only a single instance from Tyre. There is no evidence whatsoever of the use of a distinct sign for 5, in contrast to later Levantine systems, nor is there any evidence of numeralsigns for fractions. For numbers greater than 100, a multiplicative-additive structure is employed as in Aramaic; a group of cumulative unit-signs preceding a single 100-sign indicates multiples from 100 to 900, with any additional signs to the left indicating the component of the number less than 100. It is likely that the rare sign for 1000 also combined multiplicatively with sets of grouped unit-signs. It is sometimes claimed that the Phoenicians used an alphabetic (presumably ciphered-additive) numerical notation system as early as 900 bc (Dantzig 1954: 295; Zabilka 1968: 117–119). The myth of Phoenician alphabetic numeration has been repeated for more than a century, but there is no foundation for this assertion. The first alphabetic numerals were developed by the Greeks in the late sixth century bc (Chapter 5). Zabilka (1968: 118) claims that the first ten letters of the Phoenician alphabet were used on coins minted at Sidon to represent the numbers 1 through 10, based on Harris (1936:19), who is, however, referring only to Alexandrine coins. By this time, the Greek alphabetic numeral system was used throughout the Levant by speakers of both Indo-European and Semitic languages. Even this does not prove the existence of a true alphabetic numerical notation system among the Phoenicians in the fourth century bc; it could, rather, indicate a system of letter labeling as used by the Greeks (Tod 1979: 98–105), which is not really different from the practice of modern writers who label points of discussion A, B, C, and so on. The first example of numerical notation in a Phoenician inscription is the Karatepe inscription of around 750 bc, which contains a single stroke for 1 (Millard 1995: 191). If this is a true example of the system just described, then its appearance

76

Numerical Notation

is virtually simultaneous with that of the Aramaic system. However, one unit-stroke is scant evidence for this. From the seventh century onward, however, Phoenician texts containing numerals are relatively common, including inscriptions on stone, ink writings on clay, administrative documents on papyrus, and, at a somewhat later period, inscriptions on coins. Phoenician numerals were often used to enumerate regnal years and for record keeping of commodities. However, between the seventh and first centuries bc, Phoenicia was politically dominated in turn by the Assyrian, Neo-Babylonian, Achaemenid, and Alexandrine Greek empires. The Phoenician numerical notation thus predominated in the Levant only during its early history. However, in the Phoenician colonies in North Africa and Spain (including, most importantly, Carthage), the Phoenicians continued to use the system detailed here, without significant regional variation, until Roman and Greek conquests in the second century bc effectively ended its use. Coins from Akko, Tyre, and Sidon used Greek alphabetic numerals as early as 265 bc, though at Arvad and Marathus, Phoenician numerals were used on coins until about 110 bc (Millard 1995: 193). The fruitful transmission of the Phoenician consonantal script throughout the Aegean and the Middle East has led some to speculate as to the transmission of its numerical notation system. Millard argues that Phoenician may have been the model for the Greek acrophonic (base-10, sub-base 5, cumulative-additive) numerical notation system (Millard 1995: 192). However, the acrophonic system’s sub-base of 5, coupled with the more obvious derivation of acrophonic numerals from the very similar Etruscan system, makes such an origin unlikely. Schroder (1869: 187f ) suggests that the Lycian numerals (Chapter 4) are a variant of Phoenician, but that system much more closely resembles the Greek acrophonic numerals than the Phoenician. It is entirely possible that one or more of the later Levantine systems have a Phoenician as opposed to Aramaic origin, but there is no good way to demonstrate this in most circumstances. The lack of a symbol for 5 in Phoenician numerals suggests that most of these later systems were Aramaic-based. Phoenician numeration, then, unlike the Phoenician writing system, is essentially a side branch of the broader Levantine family.

Palmyrene Palmyra was an important mercantile city located in modern Syria around 200 km northeast of Damascus, and whose inhabitants, Aramaic-speaking Semites, managed to retain considerable control over their own affairs despite Greek and Roman influence in the area. Palmyrene inscriptions are found dating from the first century bc to the mid third century ad, continuing the tradition of the earlier Aramaic script. Palmyrene numerical notation retained much of the structure of

Levantine Systems

77

Table 3.5. Palmyrene numerals 1

a 178 =

5

10

20

H A J aaa\\H\\\A\\J\J\J\\A\a

the older Aramaic system, while introducing new numeral-signs. Despite their relative obscurity, the Palmyrene numerals were first analyzed over a quartermillennium ago by Swinton (1753–54). The Palmyrene system had distinct signs for 1, 5, 10, and 20, as shown in Table 3.5. These four symbols express any number less than 100. While in earlier Aramaic scripts the sign for 5 appeared only sporadically, it was a fundamental part of the Palmyrene system. Because of this, only four unit-signs were required at most, so there was no need to group sets of unit-signs into threes. Like its Aramaic ancestor, Palmyrene numerical notation is base-10 and cumulative-additive below 100. For numbers greater than 100, Palmyrene, like Aramaic, is multiplicative-additive, with the complexity that the sign for 100 is identical to that for 10. The possibility of confusion is avoided by the requirement of having one or more unit-signs before the 100-sign, whereas no such signs could precede a 10-sign. While this feature resembles the use of the positional principle, such phrases are multiplicative, not positional. To represent 100, the sign A had to be combined with unit-signs; alone, it always meant 10, not 100. Cantineau (1935: 36) contends that the original Palmyrene sign for 100 was a horizontal stroke placed above a dot, but that it was later reduced until it was identical to the 10-sign. If so, the identity of the two signs may be largely coincidental. In monumental inscriptions on stone, Palmyrene numerals are among the clearest and most unambiguous of all the Levantine systems. Figure 3.1 is a memorial inscription dating to the year 492 (ad 181); the phrase is clearly visible on the last three lines of the inscription (Arnold 1905: Plate IV). Palmyrene numerical notation was restricted geographically and temporally to the city of Palmyra during the period from about 100 bc to 275 ad. During that time, however, it was used widely on inscriptions and records of commercial transactions, though not normally in literary contexts. We do not know the extent to which it may have been used in a broader range of genres due to the poor survival of evidence. The importance of Palmyra as a commercial center rested on its strategic location on the Roman frontier and its trade ties with peoples outside the empire. Despite considerable Hellenisation and Latinisation, Palmyra retained its script and numerical notation through the third century ad, though Greek alphabetic

78

Numerical Notation

Figure 3.1. A Palmyrene memorial inscription dating to ad 181; the year-date (492) is clearly visible on the final three lines of the inscription, including multiplicative use of the sign for 100. Source: Arnold 1905: Plate IV.

and Roman numerals came to be used more frequently for administrative and mercantile purposes. In 273 ad, following the short-lived independent rule of Queen Zenobia over the province (266–272 ad), Palmyra was destroyed by the Roman emperor Aurelian, abruptly ending its importance as a commercial center. Thus, political factors, rather than criteria of function and efficiency, led to the complete replacement of the Palmyrene numerical notation system by those of Greek and Roman colonizers. It has sometimes been argued that Palmyrene is ancestral to the Syriac numerical notation, though I will show below that this is only one of many possible scenarios of transmission.

Nabataean The Nabataeans were a South Semitic people of Arabian ancestry who inhabited the area between Syria and Arabia in the southeastern Levant in the late first millennium bc and into the Christian era. Though not Aramaeans, they came under considerable Aramaean influence and adapted the Aramaic script for their South

Levantine Systems

79

Table 3.6. Nabataean numerals 1

A

4

5

R N 178 = aaa\\NA\\J\J\J\\I\a

10

20

100

K

J

I

Semitic language, including a variant of its numerical notation. This system was used from approximately 100 bc to 350 ad in inland areas of the Levant (modern southern Syria and Jordan) including the cities of Damascus and Petra, and even as far south as the port of Aqaba on the Red Sea. Its signs are indicated in Table 3.6. As with all the Levantine systems, Nabataean is decimal, cumulative-additive below 100 and multiplicative-additive above, with additional signs for 4, 5, and 20. Unit-strokes are grouped in threes where necessary and are sometimes joined together at the base in cursive writing. The sign for 4 is used only in some inscriptions, and then only in numeral-phrases for 4 (never 5 through 9); 8 is expressed as 111N (5 + 3) or 11\111\111, but never (to my knowledge) as RR. Lidzbarski (1898: 199) argues that its shape represents four unit-strokes placed in a cross, strictly on graphic principles, but this is unproven. Its historical connection with the identical Kharoṣṭhī sign for 4 is still unclear, but some link seems probable, as the Nabataeans were frequently engaged in commerce with peoples to the east. However, Gibson (1971: 13) notes that the eighth-century bc Samaria ostraca, in which the Hebrew variant of the Egyptian hieratic numerals (Chapter 2) predominates, contain a “+” or “X”-shaped sign for 4, which would antedate either the Nabataean or Kharoṣṭhī symbol by several centuries. Finally, Cantineau (1930: 36) and Lidzbarski (1898: 199) believe the signs for 4 and 5 to be quite late inventions, possibly independent of any other system. The Nabataean sign for 10 is a more arched version of the hooked horizontal stroke used in most Levantine systems, while the 20-sign can easily be shown to derive from the Aramaic form. In one inscription from Egypt, the year-number 160 (ad 266) is written irregularly as “100 20 10 20 10” instead of the expected “100 20 20 10 10”; this seems unlikely to be a scribal error but is otherwise unexplained (Littmann and Meredith 1953: 16). The sign for 100 is not obviously related to that of any other notation, though Cantineau (1930: 36) argues for its possible derivation from Phoenician Ç. The sign for 100 combines with signs for 1, 4, and 5 multiplicatively. Accordingly, the 4-sign is used to express 400, as in an inscription from Dumêr (near Damascus) from 94 ad in which the number 405 is expressed as NIR (4 × 100 + 5) (Cooke 1903: 249). Such numeral-phrases make the system look less cumulative than it actually was. No Nabataean writings contain numbers higher than 1000.

80

Numerical Notation

The Nabataean numerical notation system is found on inscriptions dating from around 100 bc to the late fourth century ad, primarily in the inland Levant from Damascus south to Petra. Throughout its history, it was used in inscriptions on edifices, on ostraca, and on coins. I do not know of any attested Nabataean numerals in the poorly attested cursive script tradition, which varies from the monumental script in several paleographic respects. The Nabataean legal papyri from the Cave of Letters dating to the early to mid second century ad express all numerals lexically (Yadin et al. 2002). The Greek alphabetic numerals (Chapter 5) and Roman numerals (Chapter 4) were also well known and frequently used in otherwise Nabataean inscriptions. Though the Nabataeans were politically subordinate to Rome throughout most of the period under consideration, they held a monopoly over the caravan trade that passed from inland Arabia to the Levantine coast. Nabataean numerical notation has been found on economic documents and inscriptions from Greece, Italy, and Egypt. In the fourth century ad, the Nabataean numerals began to be replaced by the Greek alphabetic and, to a far lesser extent, Roman numerals. While the Nabataean script is ancestral to the earliest Arabic script, there is no connection between the Nabataean numerals and the systems used by the early Arabs. Millard (1995: 193–194) reports the use of Nabataean numerals on the pre-Islamic Arabic inscriptions from En-Namara (dated 328 ad), and possibly on the sixth-century ad Zabad and Harran inscriptions. Such late occurrences become increasingly rare, however, as Greek alphabetic numerals and systems based thereupon predominated throughout the Middle East, and by the time of the introduction of Islam, no trace of the Nabataean system remained.

Hatran A variant Aramaic script was used in the region around the city of Hatra (modern Al-Hadr, in northern Iraq), an outpost of the Parthian Empire and later the capital of the small autonomous state of Araba. The Hatran script, for which inscriptions have been found dating from about 50 bc to 275 ad, possessed a distinct numerical notation system with signs for 1, 5, 20, and 100, as shown in Table 3.7. As with all Levantine systems, the Hatran numerical notation is decimal, cumulativeadditive for numbers less than 100, multiplicative-additive above 100, and written from right to left. The precise relation of the Hatran system to the other Levantine systems is unclear, but it is descended in some way from the Aramaic system used around Hatra in the centuries prior to the development of the Hatran script, given the similarity of signs for 1, 10, and 20 to earlier Aramaic forms. The sign for 5 is identical to that of the Old Syriac script used around Edessa at that time. Finally, the 100-sign is of entirely mysterious origin, though a case could be made that it is related to the Phoenician â.

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81

Table 3.7. Hatran numerals 1

5

10

20

a

>

A

ê

100

J

ü

697 = 11>AJJJJü1>

The Hatran numerals were probably ancestral in some way to the Middle Persian numerals, at least giving rise to some of the Middle Persian numeralsigns. Hatran numerical notation was used widely on ostraca, on inscriptions on stone, and in economic documents. Unlike the Palmyrene and Nabataean states, which were subjected to Roman political and economic domination for most of their history, Hatra remained independent from both Roman and Parthian control until 272 ad, when the Middle Persian king Shapur I conquered the region. After this time, Hatran inscriptions are more rarely encountered, and the Middle Persian script and numerals (and its successors) replaced them.

Old Syriac A consonantal script was used at the ancient city of Edessa (modern Urfa, in southeast Turkey) in the early years of the Christian era. Based on an Aramaic model, this script, which resembles the estrangela Syriac script that emerged in the later manuscript tradition, was used to write the Old Syriac language, a close relative of Aramaic. A large number of Old Syriac inscriptions on stone, mosaics, and parchment have survived, dating from the beginning of the Christian era to 500 ad, and largely found in northern Syria and southern Turkey.1 The Old Syriac numerical notation system used signs for 1, 5, 10, 20, and 100, and possibly also 2 and 500, as shown in Table 3.8 (cf. Rödiger 1862, Duval 1881, Segal 1954). Numeral-phrases, like the script itself, are always written from right to left. The system is decimal (with a special sign for 20), cumulative-additive for numbers less than 100, and multiplicative-additive for higher values, all of which points to its membership in this phylogeny and its close relationship to Aramaic. However, it has some curious features. The sign for 2, as defined by Duval (1881: 14–15), is simply a ligatured form of two unit-strokes, a paleographic convenience that was never used consistently or regarded as a structural feature of the system. The sign for 5 is identical to that of the Hatran system, and the sign for 20 to one variant form used in Phoenician. The sign for 5 was not consistently used in numeral 1

The earliest dated Syriac inscription is from ad 6, although Drijvers and Healey (1999: 17) argue that it may have originated earlier.

Numerical Notation

82

Table 3.8. Old Syriac numerals 1

2

5

10

A

ë > A 697 = ë>A@@@@èï

20

@

100

‚

è

500

’

ï

phrases for 5 through 9 (Segal 1954: 35). For instance, in three separate inscriptions dating to ad 165, the date 476 of the Seleucid era is written in three different ways, as indicated in Table 3.9 (Drijvers and Healey 1999). The earliest dated Old Syriac inscription (As55, dating to ad 6) expresses the year-date (317) using seven unitstrokes (Drijvers and Healey 1999: 141). This suggests that the sign for 5 may have been a later development, and at best an occasional one. Duval (1881: 14) argues that the 100-sign is a slightly modified form of the 10sign, resembling the Palmyrene numerical notation in this respect. The Old Syriac symbol for 500 is rare, partly because numbers of this magnitude are infrequent in Syriac writings. Duval (1881: 14) insists that it ought to be understood in many numeral-phrases where it is not written, such as in year-dates. However, in the Old Syriac slave sale contract, P. Dura 28, written at Edessa in ad 243, 700 is written multiplicatively as seven unit-strokes (some ligatured) followed by the powersign for 100, rather than with a sign for 500. The notion of a sign for 500, if not its form, may have been borrowed from the Roman numerals, given the cultural and political dominance of Rome in Syria throughout the period. Old Syriac numerical notation was used on numerous inscriptions on stone around Edessa, but not in the short-lived tradition of mosaic inscriptions from the third century ad. It evidently originated as a variant of the Aramaic system, but its specific relationship to the other Levantine numerical systems remains unclear. Three Old Syriac legal texts written on parchment survive, dating from ad 240–243; of these, only P. Dura 28 (just discussed) contains Syriac numerals (Drijvers and Healey 1999: 232–235; Goldstein 1966). In many inscriptions and manuscripts, lexical numerals were used instead of numerical notation. Although the Edessan Table 3.9. Old Syriac year-dates for 476 / ad 165 Inscription

Date

As29

a>A\‚‚‚\’\aaaa

As36

aaaaaaA\‚‚‚\’\aaaa

As37

ëaëaA\‚‚‚\’\aaaa

Transliteration (1 + 1 + 1 + 1) × 100 + 20 + 20 + 20 + 10 + 5 + 1 (1 + 1 + 1 + 1) × 100 + 20 + 20 + 20 + 10 + 1 + 1 + 1 + 1 + 1 + 1 (1 + 1 + 1 + 1) × 100 + 20 + 20 + 20 + 10 + 1 + 2 + 1 + 2

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83

Christians were subject to Roman imperial authority throughout most of the history of their script and numerical notation, and although both Greek alphabetic and Roman numerals were widely used in Syria, the Old Syriac numerals were not displaced by either system. By the fifth and sixth centuries ad, however, the older numerals began to be replaced by the ciphered-additive Syriac alphabetic system (Chapter 5), which assigned numerical values to the twenty-two letters of the Syriac consonantary. This gradual obsolescence corresponds to the development of the indigenous Syriac Christian manuscript tradition, particularly oriented toward liturgical subjects. Many texts use the two systems side by side. A seventh-century Syriac religious commentary (BM Add. 14,603) contains several lines of Old Syriac numerals that are incomprehensible until the numeral values are converted into their corresponding values in the Syriac alphabetic numerals; the resulting alphabetic signs can then be read as the author’s epigraph (Wright 1870: II, 586–587). While this demonstrates that the numerals were still in use, the cryptographic nature of the note suggests that they may not have been well known. The very latest evidence of the Old Syriac system is from the eighth century, after which only alphabetic numerals were used (Duval 1881: 15).

Kharos.t.hī The Kharoṣṭhī script was used in the region of Gandhara in eastern Afghanistan and northern Pakistan from around 325 bc to 300 ad and, from the second century ad onward, in parts of Central Asia. Given that this region was under the control of the Achaemenid Empire (for which Aramaic was a lingua franca) from 559 to 336 bc, the similarity in form and value of many of the signs in the two scripts, and their common right-to-left directionality, Kharoṣṭhī is clearly descended from Aramaic. During the earliest periods of its use (before about 100 bc), Kharoṣṭhī inscriptions containing numerals are quite rare, being found in only a few royal inscriptions of the Mauryan King Aśoka, who reigned from about 273 to 232 bc. Only the numbers 1, 2, 4, and 5 are represented, and they are always formed using simple unit-strokes. In the later Saka, Parthian, and Kusana inscriptions (dating from about 100 bc onward), a more complex system was used, and much larger numbers were represented. This system possessed unique signs for the numbers 1, 4, 10, 20, 100, and 1000, as shown in Table 3.10 (cf. Das Gupta 1958: Table XIV; Salomon 1998: Table 2.6; Glass 2000: 139–143). In common with all the Levantine systems, Kharoṣṭhī is purely cumulativeadditive up to 100 and multiplicative-additive thereafter. As in the script as a whole (and in other Aramaic-derived scripts), the direction of writing is always from right to left. Unlike other Levantine systems used at the time, Kharoṣṭhī has no special sign for 5; numbers from 4 through 9 were always expressed through combinations of units and 4-signs. Unit-signs in the cursively written texts of Central Asia are

Numerical Notation

84

Table 3.10. Kharoṣṭhī numerals 1

20

100

g à J 697 = aaa\R\à\JJJJ\å\aaR

L

a

4

10

1000

å

–

ç

—

usually ligatured together in groups of two and three. The sign for 1000 is found only in the late (perhaps fifth century ad) texts from Inner Asia (Das Gupta 1958: 259; Mangalam 1990: 48; Glass 2000: 143). It is most likely a variant of the similar Aramaic sign, which was a conventionalized version of the lexical numeral for 1000, ‘LP (Salomon 1998: 64). The signs for 100 and 1000 combine multiplicatively with signs for units less than 10, with the units to the right (before) the power-sign. Figure 3.2 is the obverse of a leather text written in cursive script found at Niya by Sir Aurel Stein; the numerals 3 and 25 (20 + 4 + 1) are readily visible at the bottom of the text (Boyer, Rapson, and Senart 1920: 120, Plate V). The Aśokan-period system of vertical strokes may or may not be of Aramaic origin, though the geographical proximity of its users, coupled with the obvious relation of the Kharoṣṭhī alphasyllabary to the Aramaic consonantary, suggests that it was. In its fully developed form, however, it is definitely part of the Levantine phylogeny, and not related to the Brāhmī numerals (Chapter 6) used in India. Kharoṣṭhī shares with the other systems the right-to-left direction of writing, the use of vertical strokes for units, similar forms for the numeral-signs for 10 and 20, and the use of the multiplicative principle for 100. The use of X\ for 4 is common to both Kharoṣṭhī and Nabataean, and this is unlikely to be coincidental, since they share a common sign for 20 as well, and both systems developed around 100 bc. These are the only two cumulative systems worldwide ever to use a special sign for 4. As mentioned earlier, the Hebrew hieratic ciphered sign for 4 was + or X, suggesting transmission from west to east. However, Datta and Singh (1962: 23) argue that the sign may have developed by rotating the Brāhmī sign for 4 (+) by forty-five degrees, and may then have been transmitted westward to the Nabataeans. Buhler (1896: 73), in turn, contends that the Nabataean and Kharoṣṭhī signs were invented independently of one another. The Kharoṣṭhī numerical notation system was used primarily on inscriptions on stone and on copper, but there are also surviving documents from Inner Asia written on wood, palm leaf, birch bark, and leather (Salomon 1996: 378). Throughout its history, it was in competition with its rival, Brāhmī numerals (Chapter 6), the system used on the Mauryan inscriptions of the Indian heartland. The use of Kharoṣṭhī was tied to the political independence of the Greek, Scythian, and Parthian kingdoms, which looked to Bactrian and Iranian traditions rather than

85

Figure 3.2. A Kharoṣṭhī leather text found by Sir Aurel Stein at Niya. The numeral 25 is clearly visible at the bottom left of the page (20 + 4 + 1). Source: Boyer, Rapson, and Senart 1920: Plate V.

86

Numerical Notation

to Indian ones. By the late third century ad, the Bactrian and Indo-Scythian polities of the Kharoṣṭh heartland were seriously weakened, and the advent of the Gupta Empire in the fourth century ad heralded the predominance of Brāhmī throughout the Indian subcontinent. However, Kharoṣṭhī survived longer in the small states of Inner Asia. Inscriptions on wooden documents from the city of Niya date to as late as the seventh century ad, and contain a variant of the Kharoṣṭhī script and numerals.

Middle Persian The Persians of the Achaemenid, Seleucid, and Parthian Empires used the Old Persian numerals (Chapter 7), the Greek alphabetic numerals (Chapter 5), or the common Aramaic numerals already described. The Sassanian dynasty began in ad 228 when Ardashir I destroyed the Parthian Empire, which had ruled much of the territory of modern Iraq and Iran for several centuries prior. For several centuries, the Sassanian empire was a rival of Rome and later Byzantium to the west, and of the Gupta Empire in India to the east, and was the dominant power in Mesopotamia and Persia until the Islamic conquest. The Middle Persian language (the ancestor of modern Farsi), the language of Sassanian administration and commerce, was written in an Aramaic-derived consonantary, reflecting the legacy of Achaemenid, Seleucid, and Parthian rule in the region. The numerical notation system associated with the Middle Persian script is shown in Table 3.11 (Frye 1973). Like its ancestor, the Aramaic system, the Middle Persian system was cumulativeadditive and decimal for numbers below 100, and written from right to left with the highest powers at the right. The signs for 1, 10, and 20 resemble closely the signs used in the other contemporary Levantine systems, while the sign for 100 resembles only that of the Hatran system. Unlike most of the later variants of Aramaic, however, there was no sign for either 4 or 5, and units from 5 to 9 were written using grouped sets of three or four unit-strokes. The sign for 1000, as in Aramaic and Kharoṣṭhī, is a reduced version of the Aramaic lexical numeral ‘LP ‘thousand’. For numbers above 100, the signs for 100 and 1000 combined multiplicatively with cumulative numeral-phrases, as in the other Levantine systems. Thus, in the Qa’ba inscription of Kartir, the number 6798 is written as shown in Table 3.11 (Frye 1973: 4). It is impossible to determine the precise historical affiliation of the Middle Persian system to the other Levantine systems, other than to note that it is most definitely descended from the Aramaic system in some way. The lack of a sign either for 4 or 5 is quite unusual for such a late descendant of Aramaic, as all of the other contemporary Levantine systems have some such sign. The Middle Persian script is most closely affiliated with the Hatran script, and the two systems share similar signs for 100, suggesting a historical connection. Middle Persian numerals

Levantine Systems

87

Table 3.11. Middle Persian numerals 1

10

20

100

1000

1

2 3 4 5 6 6798 = 1111\1111\24444\5\111\1111\7\111\111

7

were employed on silver bowls and plates to indicate weights, inscribed on stone texts, and written in ink on ostraca. As the Middle Persian period progressed, the script and numerals tended to be written increasingly cursively, with signs ligatured together. The Middle Persian Empire came to an abrupt end in ad 637 after the child-king Yezdigird III was overthrown by the Islamic Umayyad caliphs. By that time, the Middle Persian script had diverged into several variants, one of the more important of which was Book Pahlavi. The Book Pahlavi numerals are sufficiently different from their Middle Persian ancestor to warrant separate treatment.

Sogdian The Sogdian language was an Iranian language closely related to Middle Persian but spoken further to the north, in modern Uzbekistan and Tajikistan. Sogdian was written using three separate scripts: the Sogdian script descended from Middle Persian, the Manichaean script used by Sogdian followers of that religion and descended from the Estrangelo Syriac script, and the Christian Sogdian script used by Nestorian Christians and descended from Nestorian Syriac (Skjaervø 1996). This religious and scriptal pluralism greatly complicates the history of Central Asian Iranian scripts and numerals. The Sogdian script is first attested from the “Ancient Letters” dating to ad 312–313 found by Stein in Chinese Turkestan, but may have originated in the third century. The Sogdian script proper and the Manichaean script had distinct numerical notation systems of the basic Levantine structure, which I will treat in turn, while the Christian Sogdian script used lexical numerals or Syriac alphabetic numerals (Sims-Williams, personal communication). There has been no systematic comparative treatment of Sogdian numerals to date, and minimal paleographic work. The Sogdian numeral-signs are shown in Table 3.12 (cf. Sundermann and Zieme 1981). Table 3.12. Sogdian numerals 1

10

20

100

1000

G

H

I

J

7

697=

G\GGG\GGGHIIII\JGGG\GGG

88

Numerical Notation

The Sogdian system has numeral-signs for 1, 10, 20, and 100. It is cumulative-additive below 100 and multiplicative for the hundreds and thousands, with numeral-phrases always written in descending order from right to left. The sign for 1 is never used to write 1 alone but always phonetically as ‘yw (Sims-Williams, personal communication). Units from 2 to 9 are usually ligatured, although they can be arranged in groupings of two to four units, or sometimes ungrouped. There is no special sign for 5, in contrast to many of the Levantine systems (including Manichaean), but in common with Middle Persian, Kharoṣṭhī, and Pahlavi. While the sign for 10 is similar to the Sogdian letter δ that begins the word δs ‘ten’, the similarity is probably the result of later paleographic assimilation rather than being indicative of an alphabetic origin for the sign. Paleographically, the sign for 20 was originally two superimposed signs for 10, and in the “Ancient Letters” 30 was occasionally expressed using three such signs (Sims-Williams, personal communication). By the seventh and eighth centuries the Sogdian letters dāleth (d) and ‘ain (‘) had become assimilated to the forms of the numerals 20 and 100, respectively (Livshitz 1970: 259). The sign for 1000 is not a numerical-sign per se, but, as in Aramaic and Middle Persian, an abbreviated ideographic form of the Aramaic word ‘LP ‘thousand’; similarly, 10,000 is written using an ideogram RYPW (Aramaic ribbō) (Sims-Williams, personal communication). Fractions are poorly understood, although Grenet, Sims-Williams, and de la Vaissière (1998: 96) suggest that there is a sign for 1/2 that had previously been interpreted as a variant for 100. The majority of the texts in which this system was used are religious in nature (the so-called Sogdian sutra script), although the “Ancient Letters” are personal correspondence, and there are a few inscriptions on stone from Pakistan (Skjaervø 1996: 517). Numerals are used ordinally and cardinally in texts in various ways. Sundermann and Zieme (1981) discuss some fragmentary lists of sequential numbers in the “Sogdian-Turkish word lists” used as translation glossaries, one of which simply lists numerals from 88 to 100, and another of which contains undeciphered (non-Sogdian) numerical symbols associated with the Turkish numeral words ‘one’ through ‘five’. Numeral-signs could be combined with lexical numerals, as in the phrases 100 ‘št ‘108’ and δwy 100 20 ‘220’ in the Padmacintāmaņi-dhāraņīsūtra dating to around ad 700 (Mackenzie 1976: 12–17). Multiplicative powerideograms could also be combined together, as in the phrase 100 1LPW RYPW ‘100,000 myriads’ (= billions) in the Dhyāna text (Mackenzie 1976: 72–73). Following the conversion of most of the Central Asian peoples to Islam, the Sogdian script was used increasingly infrequently. In the eighth and ninth centuries Sogdian writing was adopted by Buddhist Uygurs, around which time the script was rotated ninety degrees counterclockwise, resulting in vertical columns (Coulmas 1996: 472–473). By this time, however, it appears that no numerical

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89

symbols were used (numbers were written lexically). It had largely fallen out of use by the tenth century ad.

Manichaean The development of a specifically Manichaean writing system is usually attributed to Mani himself in the third century ad, although this tale is likely mythical, and, the script may indeed be older than the religion (Skjaervø 1996). Manichaean writing is derived from the Estrangelo variety of Syriac, and its numerical system owes much to its ancestor. It was used to write a wide variety of languages, including Middle Persian, Sogdian, and Uygur. The best-known Manichaean texts are those from the oasis of Turfan along the Silk Road, which date from the eighth and ninth centuries ad. As with many of the Central Asian scripts, Manichaean numerals remain understudied paleographically and comparatively; the signs shown in Table 3.13 derive from the manuscript published by Müller (1912). Manichaean numerals are always written from right to left with the highest powers written first. The system is cumulative-additive below 100, with signs for 1, 5, 10, 20, and 100, while the hundreds are expressed multiplicatively. Units are generally ligatured together, with long flourishes at the end (left) of the phrase. Figure 3.3 depicts a section of a hymn book (Mahrnâmag) in Manichaean script dating to ad 761–62; the year-number 546 (dated from Mani’s birth) is found on lines 1–2 (5 100 20 20 5 1), while the last line contains the year-number 162 (100 20 20 20 1 1), reckoning from the death of Mari Schad Ormizd (Müller 1912: 36, Taf. II). The presence of a distinct sign for 5 very similar to that of Syriac, as well as similarities in the sign for 20, suggest that Iranian and Central Asian scripts such as Middle Persian and Sogdian played little role in shaping the Manichaean numeral-signs. However, all of the Manichaean numeral-signs, with the exception of the upright stroke for 1, are assimilated to letters of the Manichaean consonantal script (5 = ‘; 10 = h; 20 = p; 100 = m), so reconstructing the diachronic paleographic history of the system is complex (Sims-Williams, personal communication). It is unclear whether Manichaean had any signs for 1000 or higher powers. While the Manichaean religion flourished for several centuries after its peak, texts in the Manichaean script became less numerous after the tenth century, by which time it seems to have acquired a dignified and prestigious but also arcane quality (Sundermann 1997). The Manichaean numerals do not appear to have left any descendant systems.

Pahlavi Following the Islamic conquest, the Arabic script was normally used for writing the Middle Persian language. The Zoroastrian Persians, however, continued to use

Numerical Notation

90

Table 3.13. Manichaean numerals 1

10

20

100

BA C D 697 = BACDEEEEFAC

E

F

A

2

5

their own Aramaic-derived script for their religious texts and for other purposes. No Persian texts on papyrus survive from the early Middle Persian period, but late in the Middle Persian period, and following the Islamic conquest, Persian began to be written using a cursive, highly ligatured version of the earlier Pahlavi script, known as Book Pahlavi. Alongside this script, a set of numerals was employed (which I will call simply “Pahlavi”), shown in Table 3.14 (Abramian 1965: 285; Mackenzie 1971: 145). The Pahlavi system is decimal and written from right to left with the highest powers at the right. Frye (1973: 4–6) established conclusively that the Pahlavi numeral-signs are cursively derived from those of the earlier Middle Persian system. The signs for 1 through 9 are ligatured and cursive reductions of unit-strokes, and the Middle Persian practice of grouping unit-signs in groups of three and four

Figure 3.3. A portion of the Manichaean Mahrnâmag of ad 761–62, with the numeral 546 spanning the first two lines and 162 at the beginning (reading from right to left) of the last line shown. Source: Müller 1912: Taf. II.

Levantine Systems

91

Table 3.14. Pahlavi numerals 1

2

3

4

5

6

7

8

9

a b c d bc cc cd dd ccc 10s f j l m n o p q r 100s s 1000s u 4697 = cdrsccud

1s

strokes can also be seen in the phrases for 5 through 9. Like the Middle Persian system, the Pahlavi system is multiplicative-additive for the hundreds and thousands. Yet the Pahlavi numerals are structurally quite divergent from their ancestor. The signs for the tens, in particular, show almost no trace of their cumulative ancestry, and the unit-signs have largely become ligatured into single signs or, in the case of 5 through 9, into combinations of two or three signs. Combinations of tens and units were usually ligatured together. The Pahlavi system is thus, for all intents, ciphered-additive rather than cumulative-additive below 100. This transformation from cumulation to ciphering occurred when the epigraphic Middle Persian script and numerals, written mostly on stone and metal, were transferred to papyrus, which is more amenable to cursive and ligatured writing. This transformation is directly analogous to the derivation of Egyptian hieratic numerals from their hieroglyphic ancestor (Chapter 2), corresponding to the switch in medium from stone to papyrus. These two instances, in fact, are the only two known cases where a cumulative system directly gave rise to a ciphered one. The Zoroastrian Persians continued to use the Book Pahlavi script for their religious writings and for new pieces of literature into the tenth century ad. Most surviving Pahlavi numerals are found in these papyrus texts, although there are also epigraphic texts on stone and metal. After about ad 1000, the Middle Persian language underwent a set of changes that led to its transformation into Modern Persian (Farsi). By this time, the abjad numerals (Chapter 5) and Arabic positional numerals (Chapter 6) had completely replaced the Pahlavi system.

Summary The Levantine phylogeny is descended from the Aramaic and Phoenician systems developed around 750 bc, based on the dual model of the Egyptian hieroglyphic system and the Assyro-Babylonian common system. Over the second half of the first millennium bc, the Aramaic system and its descendants spread throughout

92

Numerical Notation

Assyria, Persia, Egypt, Asia Minor, and even into India and Central Asia. While these systems were used for over a millennium, they ceased to be used once the polities in which they predominated (most significantly, Achaemenid and Seleucid Persia) declined in importance. Levantine numerical notation systems steeply declined in frequency of use after the rise of Islam, as the systems were replaced by those of the alphabetic (Chapter 5) and South Asian (Chapter 6) phylogenies. They persisted the longest in Central Asia, where Islam was somewhat slower to take hold. The central features of Levantine numerical notation systems are as follows: a) a decimal base; b) a special sign for 20 (sometimes a combination of two 10signs); c) the use of vertical strokes for units and horizontal strokes (usually with some degree of curvature) for tens; d) a cumulative-additive structure for numbers smaller than 100; and e) the use of multiplicative-additive notation for expressing multiples of 100 (and also of 1000 and 10,000, where appropriate). Signs for 4 are found in Nabataean and Kharoṣṭhī. The presence of a sign for 5 used in late Aramaic, Palmyrene, Nabataean, Hatran, Old Syriac, and Manichaean helps to clarify some of the relationships among the systems of the family. The late Pahlavi system, which is heavily ligatured, is essentially ciphered-additive and thus somewhat anomalous, but it shares all the other structural features of this phylogeny, and is clearly derivative of a cumulative-additive ancestor. While these systems were used extensively for administrative and mercantile purposes, as well as on inscriptions, there is no direct evidence that any Levantine numerical notation system was ever used as a computational aid. We simply do not know by what means the users of these script traditions performed arithmetic, but there is no reason to assume that it was done with pen and paper. There are issues relating to the survival of perishable materials such as wooden tablets, papyrus sheets, and leather scrolls, none of which survive well in the archaeological record. Yet, even in surviving texts, numbers were often written out lexically in religious and literary contexts and even occasionally in economic documents. As such, numerical notation occupied a less significant role in the script traditions of these societies than would otherwise have been the case. Moreover, in comparison to the incredibly wide diffusion of Aramaic-derived scripts throughout Europe, the Middle East, and South Asia, Levantine numerical notations spread only sporadically, and their imprint was impermanent. The reason for this deserves careful attention, and I will return to the question in Chapter 12, after looking at the history of these systems’ competitors.

chapter 4

Italic Systems

The Roman numerals are undoubtedly one of the better-known numerical notation systems, and have received a tremendous amount of scholarly attention. Nevertheless, they constitute only a part of a larger phylogeny of numerical notation systems that originated, not among Romans, but among Etruscans and Greeks on the Italian peninsula around 600–500 bc. The name “Italic” refers only to this geographical origin, and thus does not reflect any shared linguistic or cultural affiliation. Italic systems flourished between 500 bc and 500 ad throughout the Mediterranean region, Western Europe, and North Africa, under conditions of Greek and Roman cultural hegemony and political domination. Ironically enough, however, the collapse of the Roman Empire brought about the greatest expansion of one particular system – the Roman numerals – in medieval Europe, and ultimately throughout the modern Western world. The most common variants of the Italic numeral-signs are shown in Table 4.1.

Etruscan The Etruscans were a non-Indo-European people whose civilization had its center in north central Italy, in the region of modern Tuscany (whose name is taken from the Latin Tusci, meaning Etruscan). The origins and civilization of the Etruscans are poorly understood, and large parts of their language remain undeciphered. Yet Etruscan civilization was the most potent political force on the Italian peninsula between around 800 and 300 bc, and significantly influenced Roman culture 93

Numerical Notation

94

Table 4.1. Italic numerical notation systems System

1

5

10

50

100 500 1000 5000 10K 50K 100K

Etruscan

1 Q R ÷ ; \¬ \ì ¡ Ä

Greek – archaic

1 Â È Á

Greek acrophonic

1 b c d e f g h

Greek –Argos/Nemea

•

ó d\ Ü

g

Greek – Epidaurus

•

2

Ü

g

Greek – Olynthus

1

g

á

¥

Lycian

1 < ó Ð 1Ò

Roman

1 P R S\ U W Y\ ,

. ½ ~

Roman multiplicative

1 P R S U W a

d e g

Arabico-Hispanic

Ø Ù Ú Û Ý

Calendar numerals

A E J

South Arabian

1 ! @ # $

À

b

i j

%

throughout the Republic and even later. The Etruscan alphabet, developed in the early seventh century bc on the model of the archaic Euboean Greek alphabet, usually runs from right to left (Bonfante 1996). The Etruscan lexical numerals were probably base-10 with a special term for 20, zathrum (but not for 40, 60, 80 ...), and it appears that subtractive structures formed the words for 17 through 19 (Lejeune 1981; Bonfante 1990: 22). However, these irregularities are not reproduced in the Etruscan numerical notation system, shown in Table 4.2. Table 4.2. Etruscan numerals 1

5

10

50

100

500

1000

5000 10,000

1 Q R :\ ÷\ ;\ VaU U î ¬ ì /\ ¡ 1378 = aaaQRR÷;;;ì

Ä ?\

Italic Systems

95

This system is cumulative-additive with a base of 10 and a sub-base of 5, very much like the Roman numerals, but is most often written from right to left, with the highest values at the right side of the numeral-phrase. Each power-sign of the primary base (10) may be repeated up to four times, but the half-decade values may occur only once in any numeral-phrase. The numeral-signs for 1, 5, 10, and 50 are quite regular throughout the system’s history. The sign ; for 100, though rarer, is found in many inscriptions from a relatively early date; the use of C is seen by Keyser (1988: 542) as a development occurring between 250 and 200 bc. It may be that C = 100 arose first in Latin inscriptions and found its way into Etruscan only when the Etruscan system was already declining. The signs for 1000 and 10,000 are encountered only rarely (Keyser 1988, Bonfante 1990). The signs for 500 and 5000 are unattested, but because the numeral-signs 5 and 50 are the bottom halves of the signs for 10 and 100, we would expect this same graphic principle to be followed (Buonamici 1932: 244; Keyser 1988: 544–545). This theory is given some support in that the earliest Roman sign for 500 is X (later to become W). Furthermore, two Etruscan inscriptions contain an unidentified sign ¬ that might have had such a value (Keyser 1988: 545). A special sign, í, is used primarily on coins to indicate ½ (Bonfante 1990: 48). While the Etruscan script is attested from around 700 bc, there is no evidence of Etruscan numerals until the late sixth century bc. Their invention may have been entirely independent of other base-10, cumulative-additive systems used around the Mediterranean in the early first millennium bc. No earlier system used a mixed base of 5 and 10, and most of the numeral-signs can be derived from successive crossings and circlings of tally marks for 1, 5, and 10. The most common sign for 100 is simply 10 with a vertical line through it, while 50 is made by drawing a straight line from the apex of the “upside-down V” 5-sign (Keyser 1988: 533). This theory is closely allied to Zangemeister’s (1887) theory regarding the origin of Roman numerals. Tallying practices in which numbers were marked sequentially, then crossed off as appropriate, could thus have led to a numerical notation system. However, because tally sticks are normally wooden, no evidence survives that the Etruscans ever used tallies in such a manner. Alternatively, the Etruscan numerals may be descended from the Mycenaean Linear B system (Chapter 2). Peruzzi (1980) argues that Mycenaean Greece influenced some aspects of Etruscan culture but does not discuss the numerical evidence. Mycenaean settlements have been found in southern Italy and Sicily, though these are too early to have had much direct cultural contact with the Etruscans. Keyser (1988: 542–543) notes that the Aegean systems, like Etruscan, use strokes to represent the lower powers of the base and strokes in conjunction with circles for higher powers. Unfortunately for this theory, 100 is ; in Etruscan but æ in Linear B. The similarity between Etruscan Ä and

96

Numerical Notation

Linear B ô for 10,000 is notable, but it is the only numeral-sign to be relatively close in form and value in both systems, other than their historically meaningless use of a vertical stroke for 1. I think it likely that the Etruscan system arose relatively independently of other systems, but with some continuity or influence from Linear B numerals. Base-10, cumulative-additive systems abounded in the Mediterranean between 1100 and 650 bc – the Egyptian hieroglyphic system, the Aramaic and Phoenician systems, the Hittite hieroglyphic numerals, the Cypriote numerals, and any remnants of the Linear B numerals. Regardless, the invention of a mixed base of 5 and 10 is an important development, and the use of halved signs for the sub-base of 5 is an ingenious means of deriving sign-values, suggesting that whatever system(s) the Etruscans knew, their numeral-signs are of their own invention. The early history of the Etruscan numerals is, in essence, a shared history with that of the Greek acrophonic numerals to be described later. While the numeralsigns of the mature systems are quite different, they are structurally identical. The ancestral role of the Greek scripts with respect to Etruscan is now very widely accepted, and many other aspects of Etruscan culture owed much to contact with Greek traders. Many early Greek numerals are found in the late sixth century bc in south Italy in the context of contact with the Etruscans (Johnston 1975: 362–364; Johnston 1979: 31). Yet it is impossible to assign chronological priority to one or the other. Instead, we might recognize a single ancestral system in the late sixth century bc, which later diverged into Greek and Etruscan variants. The Etruscan system is, however, the direct ancestor of the Roman numerals (Rix 1969, Keyser 1988). Etruria was politically dominant over Rome throughout its early history and remained a potent force in Roman culture well into the Republican period. Similarly, several Indo-European languages of the Italian peninsula, including Oscan, Umbrian, and Faliscan, adopted scripts and numerical notation systems based on an Etruscan model; their numerals are essentially identical to the Etruscan numerals. Etruscan numerals were used in a wide variety of contexts. Out of nearly 10,000 Etruscan inscriptions known to Kharsekin (1967), about 200 contain numerical notation, while only 40 contain the (as yet poorly understood) Etruscan lexical numerals. Numeral-signs were often used on funerary inscriptions to indicate the age of the deceased; other inscriptions on stone make use of numerical notation, but more rarely. From the fifth century bc onward, Etruscan coins were stamped with the numeral-signs for ½, 1, 5, 10, 50, and 100 in various combinations. As well, many graffiti inscribed on potsherds contain Etruscan numerals. These graffiti often recorded the quantity or value of goods in containers. A lead tablet whose purpose has not been reliably established contains the numeral-signs for 1000 and 10,000 (Keyser 1988: 544, Fig. 9).

Italic Systems

97

Figure 4.1. The Etruscan “abacus-gem” (CII 2578 ter) showing a figure seated at a board working with Etruscan numerals. Source: Fabretti 1867: 224.

Figure 4.1 depicts the so-called Etruscan cameo or abacus-gem (CII 2578 ter), a small gem (1.5 cm high) dating from the fifth century bc that depicts a seated individual working at a large board upon which rows of Etruscan numerals have been inscribed, including the elusive signs for 1000 and 10,000, but not 500 or 5000 (Fabretti 1867: 224; Keyser 1988: 545). This demonstrates the association of the numerals with pebble-board computation at an early date. Similarly, Etruscan numerals may have been used on wooden tallies and similar perishable materials, despite the lack of evidence for such a function. There is no evidence, however, for the use of the Etruscan system for performing arithmetical calculations as we would (on papyrus or slate). Computations would have been done in the head, with the fingers, or on a counting board. Large numbers and long numeral-phrases are very rarely encountered; even the sign for 100 is relatively uncommon. The demise of the Etruscan numerical notation system was a direct consequence of the rising fortunes of the Roman republic. The lack of Etruscan political unity in the third century bc, coupled with the political advantages of association with Rome, led to the slow but steady assimilation of the cities of Etruria into the Roman political and cultural milieu. While the Etruscans remained a culturally

Numerical Notation

98

Table 4.3. Post-Etruscan tally numerals 1 5

10

50

100

500

1000

1 Q P\U R O A ÷ Õ õ ; « ì ¾ ø Z û ð distinct people at least until the beginning of the Roman Empire, by 100 bc they were entirely within the Roman political sphere. This inevitable trend was accompanied by the slow replacement of the Etruscan language, script, and numerical notation with those of the Romans. Given the similarity of the two numerical notation systems, there would have been little difficulty in making the change to the new system. The last certainly dated examples of Etruscan numerical notation are from the second century bc. However, A. P. Ninni (1888–89) first presented the theory that the Etruscan numerals survived into the nineteenth century. While studying the tally marks used by fishers along the coast of the Adriatic Sea, near the town of Chioggia (near Venice), Ninni discovered a numerical notation system that he called cifre chioggiotte (numerals of Chioggia). This potential vestige is cumulative-additive, with a mixed base of 5 and 10, like both the Roman and Etruscan systems. Its numeralsigns are shown in Table 4.3 (Ninni 1888–89: 680). Ninni noted that the signs of the cifre chioggiotte more closely resemble the Etruscan numerals than the Roman numerals, and on this basis proposed that they were of ancient origin. Furthermore, in both the Etruscan system and the cifre chioggiotte, several of the sub-base numeral-signs are halved versions of the signs for powers of ten (Ninni 1888–89: 680–681). Could this system in fact be a survival, over 2,000 years, of an Etruscan tradition among modern fishers? At present, there is simply not enough surviving data to speculate on the possibility of such longterm cultural survivals, particularly in a region, such as Italy, that has experienced immense social change over two millennia. On the one hand, certain signs of the cifre chioggiotte (e.g., the first signs in Table 4.3 for 50 and 100) are identical to the Etruscan numeral-signs for those numbers, but are quite dissimilar to the intervening Roman numerals. On the other hand, Chioggia is not in modern Tuscany, and there is no evidence that the systems’ users believed the cifre chioggiotte to be ancient. Since no information has come to light for over a century, perhaps we have lost our opportunity to learn more about this system.

Greek Acrophonic Between 750 and 500 bc, what we now call archaic Greece was a conglomeration of small city-states in mainland Greece, the Aegean islands (including Crete), the southern half of the Italic peninsula (known as Magna Graecia), and western Asia

Italic Systems

99

Table 4.4. Greek acrophonic numerals 1 5

10

50 100

500 1000

5000 10,000

50,000

1 b

c

d e

f g

h

j

ΠΕΝΤΕ ΔΕΚΑ

ΗΕΚΑΤΟΝ

ΧΙΛΙΟ∑

i ΜΥΡΙΟ∑

36,849 = iiihgfeeeccccbaaaa

Minor, sharing in common only the use of Greek dialects. A tremendous number of local scripts, known as epichoric scripts (from Greek epi-, upon, over, and chora, place, country), were used during this period, all of which were based on the model of the Phoenician consonantary around 800 bc. In their earliest phases, some of these alphabets were written from right to left or in alternating directions (boustrophedon), although by around 500 bc all the epichoric scripts were written from left to right. Adjoining these scripts were two very distinct types of numerical notation: the acrophonic, to be described here, and the cipheredadditive alphabetic numerals (Chapter 5). For our present state of knowledge of these two systems, we are greatly indebted to the tireless and unparalleled work of Marcus Niebuhr Tod.1 The acrophonic system as used in classical Athens is shown in Table 4.4 (Tod 1911–12: 100–101). The system is cumulative-additive, uses vertical strokes for units, has a base of 10 with a sub-base of 5, and is always written from left to right, with numeralphrases in descending order of numeral-sign value. The acrophonic system is so named because the signs for many numbers are taken from the first letter (akros = highest, outermost; phone = sound) of the corresponding (classical) Greek word. Other names for this system, now largely rejected, include “Herodianic” and “decimal” (Tod 1911–12: 125–127). The signs for 50, 500, 5000, and 50,000 combine the sign for 5 with the sign for the appropriate power of 10. Whether we choose to see these sub-base signs as single signs or as two ligatured multiplicative ones is largely a matter of definition, and does not substantially affect how we classify the entire system. Similar acrophonic signs were used in large portions of the Hellenic world, the only difference being that the appropriate letters from each epichoric script were used in place of the letters used in the Attic inscriptions. Dow (1952) notes that the variety of acrophonic Greek numerical notation systems stands in sharp contrast to the Greek alphabetic system (Chapter 5), which is remarkably consistent 1

Tod’s six papers on Greek numerical notation (Tod 1911–12, 1913, 1926–27, 1936–37, 1950, 1954) have been reprinted in one volume (Tod 1979). My citations are taken from the original papers.

Numerical Notation

100

Table 4.5. Non-acrophonic archaic Greek numerals 1

5

10

50

100

a

Â

È

Á

À

throughout its geographic and temporal range. This degree of variation among local systems is far greater than the variety of lexical numerals used in the Greek dialects. However, the differences in sign-forms were probably not great enough to affect their comprehensibility (Tod 1936–37: 246). For expressing monetary values, the acrophonic numerals were often modified to reflect the forms of currency being expressed; for example, in Attica, £ (talanton = 1 talent = 6,000 drachmas), Z (mna = 1 mina = 100 drachmas), Σ (1 stater), ¢ (1 drachma), I (1 obol), Ã (1/2 obol), » or £ (1/4 obol), and R (1/8 obol) (Threatte 1980: 111). These could sometimes be ligatured to the sign for 5, just as the ordinary acrophonic powers of 10, to express multiples of units of currency. While there is some potential for confusion (£ can mean 1 talent or ¼ obol; Z can mean 1 mina or the numeral 10,000, etc.), numeral-signs are always listed in descending order, which averts most ambiguities. In some regions, special signs were used to indicate monetary values that did not fit easily into the standard system. For instance, a system found in inscriptions from Thespiae (in Boeotia) uses numeral-signs for 30 and 300, which consist of a sign £ (for triobole, or 3 obols) ligatured to the appropriate sign for 10 or 100 (Tod 1911–12: 109; Feyel 1937). Other acrophonic subsystems used cumulative signs related to systems of weight or volume, such as those described by Lawall (2000) on graffiti from the Athenian Agora from the last quarter of the fifth century bc; for example, EEEE = 4 hemichoes. Despite the name of the system, not all numeral-signs used in the Greek epichoric scripts are acrophonic, and in fact the earliest ones are nonacrophonic. Johnston (1975, 1979, 1982) has found several instances of a very early Greek cumulative-additive but nonacrophonic system with a mixed base of 5 and 10 dating from the sixth and fifth centuries bc. The signs of the system are shown in Table 4.5 (cf. Johnston 1979: 29–30; Johnston 1982: 208). Johnston argues that this system was built up systematically by cumulatively adding oblique lines to a vertical stroke to obtain higher numeral-signs. Curiously, he does not note that the signs for the sub-base (5 and 50) are the right halves of the appropriate primary bases (10 and 100). This structure parallels the halving of Etruscan numeral-signs, which is notable because many of the examples of this “pre-acrophonic” system are of South Italian provenance. Johnston (2006: 17) notes several sixth-century Greek inscribed vases where X = 10, paralleling the Etruscan practice but in contrast with later Greek acrophonic practice. A very unusual numerical notation system used only to express monetary values is found in five fourth-century bc inscriptions from the Greek colony of Cyrene

Italic Systems

101

Table 4.6. Cyrenaic numerals 20,000 10,000 5000 1000 500 100

¿

ƒ

Z

í

20

b 2 R

4

2

1

1/5 1/10 1/50

¿ ƒ Z í 2 c

(in modern Libya). These numeral-signs are nonacrophonic, and their interpretation is controversial (Tod 1926–27, Oliverio 1933, Tod 1936–37, Gasperini 1986). Our best evidence comes from the temple of Demeter at Cyrene, where inscriptions list the prices of various goods and the temple’s revenues and expenditures (Tod 1936–37: 255). They present a dual series of figures in which each numeralsign has both a higher and lower value; the specific amount must be inferred from the context within the numeral-phrase. Normally the higher is 5000 times the value of the lower, but this breaks down for some of the lower signs. The relative values of different units of currency used in Cyrene during this period (drachmas, staters, minas, and talents, where 1 talent = 50 minas = 1250 staters = 5000 drachmas) help explain its unusual structure. The interpretation presented by Oliverio, Tod, and Gasperini is derived from an analysis of the maximum number of times each sign is repeated (and is thus open to question if more inscriptions are found). This system is shown in Table 4.6.2 Still another aberrant acrophonic system is found in fourth-century bc inscriptions from Olynthus (in the northern Chalcidice region). There, a system was used that is nonacrophonic and lacks a sub-base of 5 (Tod 1936–37: 248–249; Graham 1969). The signs for 10, 100, and 1000 (R, á, and ¥, respectively) are the last three letters of the western Greek alphabet used in the region. Of course, R = 10 is common to the Roman and Etruscan systems as well.3 On this basis, Graham (1969: 356) argues that the Roman/Etruscan system was borrowed from the Chalcidian colony at Cumae (in southern Italy). This theory, while attractive, has several flaws, many of which derive from Mommsen’s (1965 [1909]) flawed “lost-letter” theory of the Roman numerals discussed later. Moreover, the fourth-century bc numeral-signs of Olynthus cannot have spread to the sixth-century bc Etruscans by means of a colony at Cumae that never used the numeral-signs in question. I suspect that the Greek letters were borrowed for the higher powers, just as the Romans began with nonalphabetic numeral-signs, but later modified their signs into alphabetic ones for mnemonic purposes. 2

3

The numbers listed are amounts in drachmas, based on the assumption that the lower Z sign represents one drachma, without which the absolute value of each sign would be indeterminate. No significance should be attributed to the fact that the sign _, a common Roman numeral-sign for 1000, is rotated ninety degrees from the Olynthian sign á for 100.

Numerical Notation

102

Table 4.7. Epichoric Greek numerals System

1

5

10

Standard Acrophonic

1 1 • •

b

c d None g 2 None ó\\« d\\b

Olynthus Epidaurus Argos and Nemea

None None None

50

100

500

1000

e á Ü Ü

f

g ¥ g g

None None None

A similar system was used in Epidaurus, on the southern Greek mainland (Tod 1911–12: 103–104). It is acrophonic for 100 and 1000 but not for the lower powers. Nearby, in Argos and Nemea, a closely related system was used that apparently had a sign for 50, but not for 5 (Tod 1911–12: 102–103; Ifrah 1985: 235). The systems of Epidaurus and Argos, alone among the Italic numerical notation systems, use a dot rather than a vertical stroke for 1. Table 4.7 compares the numeral-signs of these irregular systems to the standard acrophonic system. Most scholars explicitly or implicitly assume that the acrophonic system was invented independently of the Roman, Phoenician, and other systems used at the time (e.g., Ste. Croix 1956: 52). Because of the use of the acrophonic principle, the numeral-signs are often Greek letters, which makes reconstructing the history of the system rather difficult. It could be argued that the acrophonic nature of the system suggests that it could only have been invented in Greece. Yet like the Roman numerals, the earliest Greek acrophonic numerals are not phonetic signs at all, which provides crucial evidence allowing us to reconstruct their origin. While the traditional and widely quoted dates given for the use of the acrophonic system in Athens are 454 to 95 bc (Heath 1921: 30), there is indisputable evidence of an earlier origin for the system. Tod argues, solely on logical grounds, that a seventh-century bc origin is not unreasonable, as the system was fully developed by the middle of the fifth century bc (Tod 1911–12: 128). Mabel Lang mentions a seventh-century bc decorated Greek amphora inscribed with three vertical strokes, but this does not prove that the numeral was part of the acrophonic system; it might have been part of an unstructured tallying system or almost any numerical notation system in use in the Aegean at the time (Lang 1956: 3). For the second half of the sixth century bc, however, there is more promising evidence of the acrophonic system. Johnston (1979: 27–29) discusses three different variations of the “pre-acrophonic” system mentioned earlier, used in the sixth century bc in southern Italy, Sicily, western Asia Minor, the Aegean islands, and various parts of mainland Greece – almost the entirety of Greek civilization during that period. Several vases from southern Italy and Sicily, which Johnston dates to the last quarter of the sixth century bc, bear marks used in commercial transactions

Italic Systems

103

( Johnston 1975, 1979, 1982). It is telling that so many variations of the acrophonic system were used in the fifth and fourth centuries bc, suggesting an initial period of experimentation followed by consolidation and agreement on a single form of the numerals. The Greek acrophonic numerals likely originated on the Italian peninsula around 575–550 bc, around the same time and in similar contexts as the Etruscan system. As I accept Keyser’s (1988) contention that the Etruscan numerals developed relatively independently as an outgrowth of tally marks, the obvious conclusion is that the Greek system developed on the model of the Etruscan numerals in southern Italy and Sicily, an area of considerable Greco-Etruscan commercial and cultural contact. It is difficult to believe that two cumulative-additive, quinary/ decimal numerical notation systems developed on the Italian peninsula in the second half of the sixth century bc independently of one another. Yet because the Etruscans owe their script to contact with the Greeks, it is counterintuitive to think of the transmission of numerals moving in the opposite direction. In any event, separating out questions of chronological priority of the two systems is virtually impossible. There may also have been some influence from the Phoenicians, who were in contact with both the Greeks and the Etruscans in the sixth century bc. In at least one document, tablet V from Entella in west central Sicily, the early acrophonic numerals for 10, 50, and 100 were written with the smallest numbers on the left and ascending to the right – perhaps in emulation of the right-to-left direction of the Phoenician system (Nenci 1995). The Phoenician system, however, has a special sign for 20, is a hybrid multiplicative system above 100, and does not have a sign for 50 at all. In the early classical period, acrophonic numerals were used in Asia Minor, the Aegean islands, North Africa, southern Italy, and Sicily, in addition to mainland Greece, but their spread to the non-Greek world was relatively limited. The Lycians of southern Asia Minor used a nonacrophonic numerical notation system in the late fifth and fourth centuries bc that is probably an epichoric variant of the acrophonic system, although their language was not Greek (see the following discussion). The enormous cultural debt of Lycia to classical Greece is beyond doubt, and its geographic and temporal proximity strengthens this hypothesis. Also likely is the possibility that the South Arabian numerals, which arose in the fifth century bc, derive from the acrophonic system. The South Arabian numerals are cumulativeadditive, base-10 with a sub-base of 5, and use acrophonic numeral-signs. However, more evidence of cultural contact is desirable before this hypothesis can be proven. Acrophonic numerals are found on inscriptions on stone, lead, and silver as well as on potsherds; they may also have been used on wood or other perishable materials, though evidence is lacking. Of the thousands of Greek papyri from

104

Numerical Notation

the fourth century bc onward, only a handful from Saqqara contain acrophonic numerals (Turner 1975). Inscriptions on stone use acrophonic numerals far more frequently, including accounts, inventories, lists, regulations, treaties, and boundary markers. As well, graffiti or other marks on pottery often indicate quantities for commercial purposes. The acrophonic numerals expressed measures of volume or distance, quantities of goods, or monetary values. What is notable is the wide range of purposes for which acrophonic numerals were not used, even compared to other cumulative-additive systems used in the Mediterranean in antiquity. Firstly, the numerals could only be used to express cardinal numbers; ordinal numbers were expressed using lexical numerals or, when available, alphabetic numerals (Tod 1911: 128). Similarly, with the exception of monetary amounts, there was no acrophonic numeral expression for fractions. The Greeks never expressed dates in acrophonic numerals. The practice of expressing the age of the deceased at death on funerary inscriptions, a source of much information on other numerical notation systems, was not customary in Greece. The custom of dating using regnal years did not arise until the Alexandrine period. Documents in connected prose (decrees, for instance) rarely contain acrophonic numerals, except to indicate the price of executing the inscription (Threatte 1980: 112). There is no evidence that the acrophonic numerals were used direcly for arithmetic or accounting. For these purposes, as with the Roman and Etruscan systems, the Greek acrophonic system was complemented by the use of the pebble-board abacus, in which several grooves were labeled with the appropriate acrophonic numerals. Lang (1957) has established that many of the mathematical errors made by Herodotos demonstrate his use of the abacus to perform calculations, with which certain types of errors (especially in multiplication and division) can occur easily. All of the thirteen examples of abaci (and fragments thereof ) known from classical Greece have the row values inscribed with acrophonic numerals (Lang 1957: 275–276). Most notable among these abaci is the remarkably well-preserved “Salamis tablet,” which probably dates from the fifth century bc (Menninger 1969: 299–303). The numerals on it range from T (one talent) to X (1/8 obol); the monetary values of the numeral-signs suggest that it was used for practical commercial computations. The decline of the acrophonic system is thoroughly entwined with the fate of the Athenian state as a Greek power. As Athens ceased to be a dominant power in Mediterranean affairs, acrophonic numerals were used less often; by the third century bc, they had been supplanted by the alphabetic numerals for most purposes throughout large parts of the Hellenistic world, including Ptolemaic Egypt and Seleucid Persia. Only in Athens and the surrounding areas did the acrophonic system continue to flourish. There are only a handful of known first-century bc

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examples from Athens (Threatte 1980: 113). By this time, Greece was firmly under Roman control. Yet there is no evidence that the acrophonic system was replaced by Roman numerals except, as one might expect, in southern Italy, where Latinspeaking populations dominated. However, the use of acrophonic numerals did continue in one very limited domain – stichometry, or the enumeration of lines of verse in classical texts (Tod 1911–12: 129–130). This practice continued as late as the third century ad with the writings of the Neoplatonist philosopher Iamblichus. Such late examples are analogous to the use of Roman numerals in the modern West, in contexts in which it is very useful to have two separate numerical notation systems – for paginating introductory sections versus the body of a work, or for distinguishing volume numbers from page numbers in certain texts.

Lycian Lycia was a small state of southern Asia Minor in the middle of the first millennium bc, centered around the city of Xanthus. The Lycians spoke an incompletely understood Indo-European language related to the earlier Luwian language, which was spoken in the Neo-Hittite kingdoms of Asia Minor. Lycia occupied an intermediate position between the Greek and Persian spheres of influence, and was intimately involved in interregional commerce and conflict. The Lycian alphabet, which was developed around 500 bc, is an epichoric variant of the Greek script, like many others used in the Greek peninsula and western Asia Minor, except that the language of the inscriptions was not a Greek dialect. A few hundred instances of the Lycian script have survived, mostly from inscriptions on stone and on coins; they are written almost exclusively from left to right and date from the fifth and fourth centuries bc. The Lycian numerical notation system is still very poorly understood. The Lycian system, like the Greek acrophonic, Etruscan, and Roman systems, is cumulative-additive with a base of 10 and a sub-base of 5. However, the exact values of the numeral-signs are still in debate. The signs of this system are shown in Table 4.8 (cf. Shafer 1950, Bryce 1976). There is also a sign, 2, that probably represents ½, although Shafer (1950: 260) argues that it may represent an additional one-half of any numeral-sign that precedes it; ó2 would be 15 and <2 would be 7½ according to this theory. The numeral-signs for 50 and 100 are found only on a few inscriptions. The value 50 is assigned to Ð primarily by default; its value is certainly between 10 and 100 (it is found after 1Ò but before ó). I follow Frei (1976: 15) in assigning it the value of 50. We can be fairly certain about the value of the sign for 100, because it is found in the Lycian portion of a trilingual Greek-Lycian-Aramaic inscription found at Letoon and dating from 358 bc (Frei 1976: 13–15). Shafer (1950: 258–259) suggests,

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106 Table 4.8. Lycian numerals 1

1

5

< 127 = 1Òóó<11

10

50

100

ó

Ð

1Ò

based on one inscription, that the Lycians may have used the subtractive principle to express the number 4 as a<. However, in other inscriptions, 4 is expressed as aaaa. The Lycian numerals are very likely a previously unidentified variant of the Greek cumulative-additive systems. Lycian numerals arise in the early fifth century bc at the time of the peak use of the Greek acrophonic numerals in Athens and throughout the Hellenic world. Both systems were cumulative-additive, had a base of 10 with a sub-base of 5, and were used in the fifth century bc in the Aegean region. While the Lycian numerals are not acrophonic, this is true of many of the epichoric numerical notation systems of the classical Greek world. Alternatively, the Lycian system may have been based on an Aramaic model, with the sign for 100 (1Ò) being in fact multiplicative (1 × 100) rather than constituting a single numeral-sign (Frei 1976, 1977). No numbers higher than 120 are expressed in any Lycian inscriptions, so we do not know whether, for instance, the Lycian numeral-phrase for 200 was additive (1Ò\ 1Ò) or multiplicative (11Ò). Because none of the Semitic systems had separate signs for 50, but all of them had signs for 20, we would need to modify the value of the Lycian Ð to 20, which is consistent with the numeral-phrases known from inscriptions. There is little similarity, however, between the numeral-signs of Lycian and either Phoenician or Aramaic, and the Lycian system uses a sign for 5, which is very rare in Aramaic. As well, Lycian, like the acrophonic numerals, is written from left to right, whereas the Levantine systems are all written from right to left. The Lycian numerical notation system apparently did not diffuse outside Lycia. Although Shafer (1950) argues that the similarities between the Roman and Lycian numerals are sufficient to indicate the derivation of the former from the latter, this likeness is no greater than that between Lycian and the Greek acrophonic system. Furthermore, while there is some similarity between the Lycian numeral-signs and other systems of the Italic phylogeny (especially South Arabian), these similarities do not correspond to any plausible circumstances of cultural contact. In Asia Minor, scripts such as Phrygian and Lydian, both of which are closely related to Lycian and were used in the fifth and fourth centuries bc, used numerical notation based on the Phoenician-Aramaic model rather than on the Greek or Lycian. Lycian numerals are found primarily in a single context – sepulchral epitaphs indicating monetary amounts, normally including a numeral-phrase preceded by the word ada, now considered to be a monetary unit (Bryce 1976: 175). The

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monetary values may have stipulated a penalty to be paid should the tomb in question be violated (Shafer 1950), or they may indicate fees paid in advance by the family for a tomb site (Bryce 1976). The only nonfunereal context in which Lycian numerals are used is the trilingual inscription found at Letoon, a public legal regulation (Frei 1976). As Lycian numerals are not found on coins or on financial inscriptions, they are quite distinct from the numerals of the rest of Asia Minor and the Aegean. As the Lycians became increasingly caught up in imperial conflicts between the Persians and the Greeks, their script was used increasingly infrequently. By 300 bc, the Lycian script had assimilated to the Greek, and its numerical notation ceased to be used, replaced by the Greek alphabetic numerals.

South Arabian The Old South Arabian scripts are of a very ancient origin, first appearing around the turn of the first millennium bc in the southern part of the Arabian peninsula (modern Yemen). They are consonantal and are characterized by large, well-formed letters and by their extremely varied direction of writing (left-to-right, right-to-left, or boustrophedon, depending on the inscription). They were used to write South Semitic languages such as Minaean, Sabaean, Qatabanian, and Hadramauti. During their early history, these scripts did not possess any numerical notation system, but following the rise of the kingdoms of Minaea and Saba in the fifth century bc, numerical notation began to be used in monumental inscriptions. The numeralsigns used are shown in Table 4.9, including both left-to-right and right-to-left sign forms, where appropriate (Halévy 1873: 511–512; Hommel 1893: 8). The system is cumulative-additive, with a base of 10 and a sub-base of 5, and is written in whichever script direction is used in the inscription as a whole. The sign for 1 is, as in all Italic systems, purely iconic. The signs for 5, 10, 100, and 1000, however, are acrophonic; each is simply the first letter of the appropriate South Arabian lexical numeral (Beeston 1984: 8). The sign for 50 is non-acrophonic, but is simply a halved version of the sign for 100. In one inscription (Biella 1982: 531), the sign X is apparently used with the numerical value 4, possibly in imitation of the Nabataean system (Chapter 3). There are no signs for 500 or 5000 known from any South Arabian inscriptions; these numbers were written with five signs for 100 and 1000, respectively (Biella 1982: 1, 265). Normally, numeral-phrases were placed between large hatched bars to avoid confusing numeral-phrases with words, given the use of the acrophonic principle (Halévy 1875: 78). Large sets of unit-signs were not divided into smaller groups, which creates a legibility issue because the sub-base of 5 is not used throughout the system; one inscription from Sirwah denotes 12,000 using twelve signs for 1000 (Ifrah 1998: 187).

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Table 4.9. South Arabian numerals 1

5

10

50

1 ! @ # Right to left 1 Ê @ Ë 3697 = Î\%%%$$$$$$#@@@@!11\Î (L-R) Î\11Ê@@@@ËÌÌÌÌÌÌÍÍÍ\Î (R-L)

Left to right

100

1000

$ Ì

% Í

In some inscriptions, the South Arabian numerals used an unusual technique of implied multiplication that resembles positional notation. Most often, this was done when expressing values greater than 10,000, by placing signs to the left of a sign for 1000 (when reading from left to right), which were implicitly taken to represent multiples of 1000. For instance, one inscription (R3943/2) has Î\@@@%\Î for 31,000, in which the three @ signs have the value of 10,000 instead of 10, while % retains its ordinary value of 1000 (Biella 1982: 349; Ifrah 1998: 187). Apparently this technique was also sometimes used for multiples of 100; Halévy (1875: 79) notes an inscription that has Î\111\Î instead of Î\$$$\Î for 300. We know that the multiplied value is correct because of contextual information and because South Arabian inscriptions commonly list the appropriate lexical numeral beside the numeral-phrase. Ifrah sees in this use of implied multiplication “what might be called the germ of our place-value notation” (Ifrah 1985: 232). However, without contextual information, such numeral-phrases would simply be confusing and ambiguous, as there is no sign for zero. These formations are very rarely attested throughout the system’s history. The South Arabian system is derived from the Greek acrophonic system (Ifrah 1998: 186; Fevrier 1948: 579). Both systems are decimal and cumulative-additive, and both have a sub-base of 5. Additionally, both systems use the acrophonic principle, a feature that is otherwise uncommon during this period. The numeral-signs are not similar to those of any other system, but this is unsurprising, since the system is acrophonic. In the fifth century bc, when the South Arabian numerals were first used, the Minaeans and Sabaeans were actively engaged in trade with the Greeks at a time when the acrophonic numerals were the only ones the Greeks were using for monetary and metrological purposes. While one would expect the South Arabian scripts to have numerical notation systems similar to those used in North Semitic scripts at the time (Aramaic or Phoenician), this is not borne out by comparing the systems. There is no sign for 20 in the South Arabian system, while the Aramaic and Phoenician systems did not normally use signs for 5 and 50. There is no evidence that the South Arabian numerals ever diffused outside the Arabian peninsula. The Ge’ez script used for the Ethiopic languages, which is

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derived from a South Arabian model, used numerals based on the Greek alphabetic system (Chapter 5). Although some South Arabian cursive inscriptions on wood have been found, these contain no numerals. The system just described is documented only in monumental contexts. The functions of the numerals included recording details of sacrifices or offerings to gods, quantities of booty obtained, numbers of military troops, and information on construction projects such as monuments and irrigation systems. The South Arabians did not use an enumerated dating system, nor do South Arabian coins contain numerical notation of any kind. By the second century bc, instances of the South Arabian numerals were normally preceded by the appropriate lexical numeral written out in full. While this aids modern scholars in their interpretation, doing so also removed any incentive to continue to use the system. By the first century bc, although the South Arabian scripts continued to be used, the numerical notation system had become extinct, and it was not replaced until the seventh century ad, when the Islamic conquest brought alphabetic and later positional numerals to southern Arabia.

Roman Despite the importance and continued use of Roman numerals, the early history of the system was very poorly understood until Keyser’s (1988) study. There are several different classical variants of the Roman numerals, while the sign forms and structure of the Roman numerals used today are medieval in origin. The Roman alphabet was developed on an Etruscan model around 600 bc at a time when much of Italy was under Etruscan political domination; it was written from left to right, as it is today. Like its Etruscan precursor, the Roman numerical notation system has a base of 10, with a sub-base of 5, and is essentially cumulativeadditive; unlike it, the Roman numerals are written from left to right and are sometimes used subtractively. The great variety of numeral-signs used throughout two millennia of its history contrasts strongly with the highly static quality of the equally long-lived Babylonian cuneiform and Egyptian hieroglyphic numerals. Table 4.10 presents the numeral-signs used during the republican period (Ifrah 1985: 132; Keyser 1988). These numerals were cumulative-additive in structure in most inscriptions. Only the signs for 1 and 10 remain unchanged throughout the entire history of the system. The “inverted V” sign for 5, Q, is found only in early contexts, and is evidence of the system’s indebtedness to its Etruscan ancestor, but P is used exclusively thereafter. The signs for 50 in Table 4.10 are roughly in chronological order; the “inverted arrow” forms are earliest, with the “inverted T” forms prevalent until about ad 25, and L most common thereafter (Gordon and Gordon 1957: 181).

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110

Table 4.10. Roman numerals (republican period) 1

5

10

50

100

1

Q P\

R

õ\ V § U T \¸ S

500

X W

1000 5000 10,000 50,000 100,000

¤ , Y\\ ® _ \© ¦ | 19,494 = ¯®YYYYUUUUTRRRRaaaa 19,494 = ¯Y¯UXRUaP

.\ ¯

½

~\ °

There is no sign for 100 in any early Roman inscription although there surely must have been one to complete the series. The V form for 100 is extremely rare; Ifrah (1998: 188) lists only a single inscription where it is found. Keyser indicates that the first Roman C = 100 whose date is secure is from 186 bc, but he postulates a third-century bc origin for the symbol as a reduction of the Etruscan ;, even though there is no example of any sign for 100 at this early date (Keyser 1988: 542). The number 500 is expressed using X in all early contexts, with assimilation to the alphabetic D occurring around the transition to empire. The familiar M = 1000 used from the Middle Ages to the present occurs only in one classical Latin inscription as part of a numeral-phrase, although it is also found in various places where M is simply an abbreviation for mille or milia (Gordon and Gordon 1957: 181–182; Gordon 1983: 45). The signs for 5000, 10,000, 50,000, and 100,000 are rarely encountered, though they are all attested as early as the third century bc. Adding arcs on either side of the most common sign for 1000, Y, indicates successive powers of 10, while the right half of the appropriate base-10 sign represents the quinary component. In a very limited set of texts, ´ is used to represent 500,000 (Mommsen 1965 [1909]: 788–791; Gordon 1983: 45). The sign is probably derived from alphabetic Q and is thus an abbreviation of quingenta milia. Its use was limited to the later Republic, and it was certainly not familiar a century later to Pliny the Elder, who, in his Natural History, wrote “Non erat apud antiquos numerus ultra centum milia” or “Among the ancients there was no numeral larger than 100,000” (Natural History 33.47.133). Alongside the Roman numerals, the Romans had a duodecimal fractional system based on the as of twelve unciae (Menninger 1969: 158–162; Cagnat 1964: 33).

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There was very little regional variation in the signs or the structure of the classical Roman numerals (due, no doubt, to the centralization of the empire). In North African inscriptions, however, the signs for the sub-base were sometimes not used, for example, IIIII for 5 and XXXXXX for 60 (Cagnat 1964: 30–31). Similarly, the signs for C and L were often written cursively, even in inscriptions on stone, to distinguish them from the corresponding letters (Salama 1999). This practice of cursively writing Roman numerals on stone was also attested in Spanish inscriptions of the fifth and sixth centuries ad, borrowing the numeral forms used on much earlier Roman paypri written in Egypt (Mallon 1948). Starting in the late republican period, the subtractive principle was occasionally used for writing multiples of 4 or 9 (or rarely 8) of powers of 10. As would later become the rule with modern Roman numerals, placing a lower-valued power of 10 to the left of a higher numeral-sign indicated subtraction of the former from the latter (IX for 9, XL for 40, but never using sub-base signs as the subtrahend – i.e., VC is unacceptable for 95). This reduced the length of numeral-phrases – four or five numeral-signs were replaced by two. Similarly, in the Augustan period and into the early empire, the use of the subtractive XIIX and XXIIX for 18 and 28 were common, and IIX and XXC for 8 and 80 are also attested (Gordon and Gordon 1957: 176–181). Addition was used almost exclusively in the early republican period, and is the more usual form even in later classical inscriptions (Sandys 1919: 55–56).4 Subtractive numerals are more common where a numeral is at the end of a line of an inscription, allowing the engraver to avoid crowding many numeral-signs into a limited space (Cajori 1928: 31). They may be more common in informal texts than in formal inscriptions (Gordon and Gordon 1957: 180–181; Cagnat 1964: 30–31). Despite Guitel’s (1975: 202–203) denigration of subtractive notation because it lacks the simplicity of the pure additive principle, it is a very economical way of structuring numeral-signs. The Latin lexical numerals use the subtractive principle for 18 and 19 (duodeviginti, undeviginti), perhaps explaining the origin of this practice. Note, however, that while duodeviginti and undeviginti are subtractive, novem (VIIII/IX), quatuordecim (XIIII/XIV), nonaginta et novem (LXXXXVIIII/XCIX), and all other Latin lexical numerals are not. Around the same time, the multiplicative principle began to be employed to write very large numbers. Even as early as the third century bc, the Roman republic had become a large centralized state, and the need to express large numbers was acute, yet the highest numeral-sign was 100,000. At times, this led to extremely cumbersome numeral-phrases, such as the inscription on the famed Columna 4

In the tradition of modern Roman numeral hour-numbers on clocks, 4 is normally denoted additively (IIII), while 9 is denoted subtractively (IX), possibly because IIII aesthetically balances the left and right sides of the clock face (Hering 1939: 319).

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Numerical Notation

rostrata of the consul Gaius Duilius, originally erected in Rome in 260 bc, which celebrated a naval victory over Carthage in which over two million aes worth of loot was plundered. The column is inscribed with at least twenty-two signs for 100,000, and possibly as many as thirty-two, as the inscription is fragmentary (Menninger 1969: 43–44). Although it was recut, probably in the Augustan period when other means of expressing this number were available, the original structure of the expression was retained (Stenhouse 2005: 59–60). One is struck, in looking at this inscription, at the sheer enormity of the numeral-phrase, and thus by the impressive amount of booty obtained, and in fact this may have had something to do with the reason it was written at all. Gordon and Gordon (1957: 180–181) suggest that one of the factors working against the widespread acceptance of subtractive notation was the desire of public officials to impress and indulge. This “conspicuous consumption of numerals” is cross-culturally frequent, for example, in the “Narmer macehead” (Chapter 2) and other royal commemorative inscriptions. Starting in the late republican period, a horizontal bar (vinculum or virgula) placed above a numeral-phrase or some portion thereof indicated that the number under the bar should be multiplied by 1000 (Smith 1926: 76–78; Gordon and Gordon 1957; Cagnat 1964: 31–32; Gordon 1983). This principle was first used in the Lex de Gallia Cisalpina written between 49 and 42 bc (Gordon 1983: 47). For most numbers, doing so did not improve conciseness; both gedddda\SROOO and °½....YSROOO for 191,063 require twelve symbols. The main advantage is that one need no longer remember so many numeral-signs or invent new ones; the signs for 1, 5, 10, 50, and 100 are sufficient to express any number up to 500,000, whereas eleven different signs would be needed under the purely additive system. One slight complexity is that starting around the same time, barred numerals were often used to distinguish ordinal from cardinal numbers, as in abbreviations such as aaVIR for duumvir (Gordon and Gordon 1957: 166–176). In one irregular inscription from Pompeii from the first century ad, the numeral-phrase LXXXX¦ apparently indicated 90,000 using a regular sign for 1000 multiplicatively, a practice not encountered again until the Middle Ages (Smith 1926: 7). Starting around the early Imperial period, three vertical bars enclosing a numeral-phrase on the top and sides signified multiplication by 100,000. Thus, instead of the thirty-two signs for 100,000 found on the Columna rostrata, one would need only to write mdddaan. This technique was first used in the late first century bc, according to Gordon (1983: 47), but was employed rather sparsely until the second century ad.5 In this way, any number less than 500 million could 5

According to Suetonius (Galba, 5), the emperor Tiberius willfully read the will of Livia in such a way as to read a three-barred numeral min as i in order to reduce by a factor of 100 the amount of inheritance owed to the future emperor, Galba (Cagnat 1964: 32).

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Table 4.11. Roman numerals (multiplicative) 1

Regular signs Multiplicative (1000)

5

10

50

100

500

1000

a

P

R

S

U

W

Y

1000

5000

10,000

50,000

100,000

500,000

1,000,000

a

b

d

e

g

j

k

100,000 500,000 1,000,000 5,000,000 10,000,000 50,000,000 100,000,000

Multiplicative (100,000)

man mbn mdn men 35,863,120 = mgggebaaan edaaa URR

mgn

mjn

mkn

be expressed with just the lowest numeral-signs plus two types of bar to express multiplication. This revised system is still a decimal system with a sub-base of 5; however, instead of being purely cumulative-additive, it is a hybrid system using cumulative-additive structuring for numbers up to 1000 and multiplicativeadditive thereafter. The entire system (up to 100,000,000) as used in the Imperial period is shown in Table 4.11. Gordon (1983: 47) claims that the largest number expressed using this hybrid cumulative and multiplicative system is 35,863,120, though an inscription at Ostia from ad 36 apparently indicates 100 million as mkn (Menninger 1969: 245). Most of the higher signs are attested only rarely. Curiously, Guitel (1975: 215) regards Roman multiplicative notation as an evolutionary dead end, because, she argues, they no longer needed to develop a more efficient positional system. The teleology of this argument is immediately apparent, as it regards this development only with respect to its failure to lead to a “superior” system. The Romans themselves likely perceived it as a means of improving conciseness, while reducing the number of signs one needed to memorize. Although the use of the 1000 “bar” continued among some post-Roman scribes, the use of the 100,000 “box” did not outlast the empire. While the origin of the Roman numerals is a common topic of inquiry, unfortunately, as Cajori (1928: 31) noted, “the imagination of historians has been unusually active in this field.” Fortunately, Keyser’s (1988) panoptic essay on the origin of the Roman numerals, which examines a variety of theories, ranging from the sixth-century theories of the grammarian Priscian to the twentieth-century theories of modern classicists, has firmly settled the issue. The popular belief that the Roman numerals originated as alphabetic signs is false. While the modern Roman numeral-signs are also letters, the signs I, V, and X mean 1, 5, and 10, rather than U, Q, and D, for unus, quinque, and decem. C is the first letter of centum, but this is coincidental, since C is a reduction of the older Etruscan sign ; or VaU (Keyser

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114

Table 4.12. Etruscan and Roman numerals

Etruscan Roman

1

5

10

50

100

1 1

Q Q\P\

R R

÷\: U\\; õ\§\T U

500

1000

\¬ \X

ì\\\/ Y\\\¤

1988: 542). The signs for 50 and 500 were not associated with the letters L and D until the late Republic, and M was not used for 1000 until the Middle Ages. It is fortuitous that the older sign for 1000 (Y) could be easily transformed into an M. Similarly, theories employing the pictographic principle (for instance, I = 1 from a single finger; V = 5 from an outstretched hand, and X = 10 from two hands together) were proffered by many early modern antiquarians, and later by classicists such as Sandys (1919), but, while they are imaginative, there is no evidence to support them. Until recently, the most widely accepted theory was that of Theodor Mommsen (1965 [1909]), who argued that the signs for 50 (õ), 100 (U), and 1000 (Y) were taken from letters of the Chalcidic Greek alphabet that were not needed to transliterate the Latin language: chi, theta, and phi, respectively, which somewhat resemble the numeral-signs. Unfortunately, this attractive theory has several flaws: the sign for 100 does not really resemble the Chalcidic theta; these “lost letters” were sometimes used in Etruscan and Roman inscriptions; and special pleading is required to derive the origin of X = 500. While it was once plausible, it is no longer a parsimonious theory. In fact, the Roman numerals up to 1000 developed through direct diffusion from the Etruscans. The astonishing similarity between the Etruscan and archaic Roman numeral-signs, as shown in Table 4.12, ought to be enough to prove a relationship between the two systems. The Etruscan numerals have temporal priority over the Roman numerals, which do not appear until well into the fifth century bc, and are not frequently encountered until the third century bc. In fact, the similarities between the two systems are so great that one could treat them as a single numerical notation system; they are identically structured, and many of their numeral-signs are similar or identical. I treat them separately because the two systems are written in opposite directions and because the Roman system used signs for much larger powers at an early date. The Roman numerals were used in a broader range of contexts than any other cumulative-additive system. In its earliest forms, they were used on coins, on pottery, and on inscriptions on stone. Dates, monetary values, and measures were all frequently expressed in the Roman numerals. The Roman numerals could be employed to express both cardinal and ordinal values. Their use in administration and literature was widespread from the republican period onward. In texts, Roman numerals were used to enumerate page and line numbers. Accounts, inventories,

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and legal documents also occasionally provide us evidence of their use in commercial and institutional contexts. While Roman numerals certainly were used in the contexts of arithmetic and calculation, there is minimal evidence that they were ever used for calculation. Glautier (1972) discusses the Roman account records, which he characterizes as primitive from an accounting standpoint because of the lack of positionality. A similar point is raised by Meuret (1996) in his discussion of the Lamasba tablet, an irrigation regulation from North Africa during the reign of Elagabalus (218–222 ad), which contains a multiplication table to enable quick calculation of water supplies, thus overcoming the computational deficiencies of the system. Maher and Makowski (2001) demonstrate persuasively, however, that the assumption that Roman mathematics was poor because of Roman numerals cannot be correct, given the complexity of the arithmetical calculations attested in Latin texts. Nevertheless, their stronger claim – that Roman mathematicians actually used written numerals to do arithmetic, in particular for calculations involving fractions – remains unproven. Roman numerals were used for computational functions – if not directly for computation, then certainly to mark the rows on the abacus. While few Roman abaci survive, Taisbak claims that the Romans did all their calculations with them, and even that “the notation of Roman numerals originates from the abacus reckoning” (Taisbak 1965: 158). This finding is contradicted by the derivation of the Roman numerals from the Etruscan system and ultimately from an older tallying system. Because cumulative systems use one-to-one correspondence intraexponentially, just as one counter equals one multiple of a power on the abacus, the Roman techniques of numeration (Roman numerals) and computation (the abacus) complement one another. This correlation is confirmed by the quinary (base-5) component of the abacus (there are rows not only for the powers of 10 but also for their halvings). No row on the abacus ever would have contained more than four counters, which would have facilitated reading and working with them. However, because the original Etruscan system probably emerged from a system of tallying, it is more likely that the structure of the abacus emerged out of the structure of the numerals than vice versa. Despite the enormous influence of Roman civilization on Europe, North Africa, and the Middle East, and despite the extraordinary chronological duration of the Roman numerals (almost 2,500 years), they produced relatively few descendants. While other systems in use over similar periods, such as the Brāhmī numerals and Greek alphabetic numerals, changed their form greatly as they spread across time and space, the Roman numerals of antiquity spread largely unmodified throughout Western Europe and other areas where the Roman alphabet was used. While the Indian and Greek systems spread throughout many different scripts, changing

116

Numerical Notation

the forms of signs as they were transmitted, Roman numerals were infrequently adopted by users of other scripts. Of the few descendants of the Roman numerals, I have already discussed the hybrid multiplicative-additive system used occasionally from the first century bc onward. In the medieval period, Roman numerals were essentially the same as classical ones, though with slight differences in form and structure. In Arabinfluenced Spain, certain variant Roman numeral systems were used starting in the tenth century ad. Around the same time in northern Europe, certain types of medieval calendars contained unusual Roman numerals. Finally, as the Roman numerals came increasingly under assault from the rival Western system, certain positional variants of the numerals were occasionally used, combining features of both systems.

Computation with Roman Numerals The computational efficiency of the Roman numerals is a common subject in the history of mathematics. The consensus of these arguments, with which I am in general agreement, is that the Roman numerals are poorly suited to performing arithmetical calculations. This accords with the historical finding that people used abaci or finger computation to actually manipulate numbers, with the Roman numerals being used only to express the result. Yet the rejection of the efficiency of the Roman numerals for computation has been used to draw two unwarranted implications. Firstly, it is unlikely that the use of the Roman or other cumulative-additive numerals has any simple or unilinear correlation with developmental stages of psychology, as Murray (1978), Hallpike (1979), and Dehaene (1997) have suggested, either as the cause or effect of a less abstract way of thinking about number. Although many ancient numerical notation systems are cumulative-additive, this is not evidence that the Roman number concept is less abstract than the modern one. Other evidence, such as the use of abaci and finger reckoning, refutes any simple correlation between the cultural evolution of numerical notation and cognition. There may be some correlation, but it must be demonstrated, not assumed from the inefficiency of the Roman numerals for a task for which they were never intended. Secondly, some scholars claim that the Roman numerals prevented the Romans from developing other useful institutions or techniques. Glautier (1972) asserts that the Romans did not develop an efficient accounting system due to the lack of a suitable numerical notation system, while Williams (1995) bemoans the limitations of Roman numerals for doing the complex calculations with fractions needed for alloying coinage. Yet the Romans evidently had sufficient accounting techniques

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to administer their empire, and while double-entry bookkeeping is made easier by a ciphered-positional numerical notation system, it must have functional equivalents, or else no society lacking such numerals could administer a large political entity. The inferiority of Roman numerals is used by Guitel (1975) to explain why the Romans were poor mathematicians as compared to their Greek subjects. However, both Greek and Roman mathematicians relied considerably on Babylonian knowledge, and educated Romans knew the Greek alphabetic numerals. Maher and Makowski (2001) show that the Romans were, in fact, quite good at arithmetical computation. If, in fact, the Romans were poorer mathematicians than the Greeks, evidence other than their numerals must be sought. In contrast, a small body of research within the subdiscipline of the history of mathematics holds that the Roman numerals are not less efficient for computation than the Western numerals. At least four modern scholars claim to show how the Roman numerals could have been used in written calculations without the aid of an abacus or similar technology (Anderson 1956, Krenkel 1969, Detlefsen et al. 1975, Kennedy 1981). These analyses, apparently derived independently of one another, differ in the exact technique used in performing calculations, but all conclude that even if the Romans never used their numerals in such a fashion, the Roman numerals are in fact amenable to computational functions. While this is superficially true, I regard this argument as highly spurious. The proposed techniques are often more complicated than they are presented to be, and certainly more complicated than the standard arithmetical techniques using Western numerals. For instance, the technique proposed by Detlefsen and colleagues (1975) involves a complex “transformational grammar” that is far removed from the knowledge systems of Roman or medieval scholars. Additionally, none of these studies establishes that the Roman numerals are equally or more efficient for computation than Western or other numerals. A system’s minimal utility for a function is hardly proof that it was or should have been used for it. These studies presume that it is natural for numerals to be manipulated for computation, even in the absence of historical evidence. Anderson (1956: 145) suggests that “any reader, once he discovers how simple the operations are, will be inclined to imagine that some Roman engineers and surveyors, in building their great projects, did occasionally do their computations very much in the way described below, even though they left no records of their work.” Detlefsen and colleagues (1975: 147) go so far as to blame the Romans for not recognizing the computational potential of their numerals. While the possibility that the Roman numerals were used in this way cannot be ruled out, the argument ex silentio is highly implausible. We have persuasive evidence that ancient and medieval scholars used the abacus, finger computation, and other techniques (Lang 1957, Taisbak 1965, Murray 1978). The best way to counteract the denigration of the Roman numerals is not to show that they are

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mildly (or even greatly) useful for a function for which they were never known to have been used. The issue is not simply a mathematical game to see whether a system can serve some arithmetical function. By understanding the functions for which the Roman numerals were not used, we may better understand the circumstances of their eventual replacement.

Medieval – Additive As early as the second century ad, but most prominently in the fifth and sixth centuries, Roman numerals inscribed on stone inscriptions frequently took on a cursive quality, and some signs became ligatured together (Gordon 1983: 46). A special sign for 6, ³, which is nothing more than a cursively written and ligatured vi or ui, occurs on many inscriptions from late antiquity (Lassère 2005: 57–59). On sixth-century Byzantine imperial coins, ³ was as common as VI (Wroth 1966: cx). The use of this sign probably ceased in the eighth century (Bischoff 1990: 176). Similarly, in late antique and early medieval inscriptions, 40, 60, and 500 could be written cursively, even in inscriptions on stone. The Visigothic script used between the seventh and thirteenth centuries was particularly characterized by these and other ligatured Roman numerals, particularly for 40 (XL) (Schapiro 1942, Bischoff 1990). These sporadic shifts toward ligatured notation would later play some role in the paleographic modification of Arabic signs into what would become the Western numerals (Lemay 1982: 393; see also Chapter 6, of this volume). Yet these ligatures did not represent an inexorable diachronic trend toward ciphering in Roman numerals, and the majority of early medieval Roman numerals remained cumulative-additive. After the fall of the western Roman Empire, literate knowledge was distributed rather sparsely – for instance, among Western monks, Byzantine bureaucrats, and Middle Eastern scholars. The great early medieval Mediterranean polities – the Byzantine Empire and the early Muslim caliphates – mainly used cipheredadditive numerical notation systems such as the Greek and Arabic alphabetic numerals (Chapter 5). In Western Europe during the Middle Ages, the Roman numerals were the only ones in common use. Knowledge of other systems was restricted to peripheral regions such as Spain and southern Italy, and to a tiny well-educated elite. Even at the height of the Carolingian Renaissance (around 800 ad), arithmetic was the province of a learned few, and was acquired late in the scholar’s education (Murray 1978). Still, while the need for large-scale bureaucracy and the corresponding need to express large numbers had declined since the height of the Roman Empire, Roman numerals were still frequently encountered, and even expanded in the range of functions they served. The medieval Roman numeral-signs differ from the classical ones in several aspects. Rather than being written solely in majuscule characters, Roman numerals

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were frequently written in the lowercase script on perishable materials. When written in minuscule form, placing a stroke through the last numeral-sign of a numeral-phrase indicated that one-half was to be subtracted from the represented value. As a measure against fraud, the last i in cursive numeral-phrases was often extended into a j, preventing anyone from adding additional signs to the end of the phrase (Menninger 1969: 285). This practice originated around ad 900 in South Italian manuscripts, although it may have had an antecedent in the upward rather than downward extension of phrase-final cursive i into I in a few classical texts (Smith 1926: 74). Alphabetic forms of the numerals for 500 (D) and 1000 (M) replaced the earlier X and _, marking the end point of a long process of alphabetization. This would have had the advantage of some mnemonic convenience; even if most of the numeral letters did not correspond to the numbers they represented, at least literate learners of the system would not need to learn an entirely new set of signs. Alphabetization of the numeral-signs also enabled one to use them for numerical riddles, particularly chronograms, in which the total value of the Roman numerals in a line of verse expressed the date of an event described in that verse (Menninger 1969: 281). The multiplicative vinculum bar for 1000 used in classical antiquity continued to be used under a new name, the titulus, but the three-sided box symbol for multiplying by 100,000 was no longer used (Menninger 1969: 281). Large numbers were very rarely needed, and even the need for the titulus was limited. Subtractive forms were used more frequently in the Middle Ages, though purely additive forms (e.g., IIII) were still common. Finally, starting in the tenth century, numeral-phrases for ordinal numbers were often adverbialized using superscript endings such as -o and -mo – for instance, Mmo CCmo Lmo IVto, incorporating aspects of the medieval Latin lexical numerals into numeral-phrases (Smith 1926: 75). Shipley (1902) suggests that the changing use of Roman numerals between classical antiquity and the ninth century ad led to transcription errors in medieval manuscripts. Comparing the fifth-century ad Codex Puteanus, containing sections of the works of Livy, and the ninth-century ad Codex Reginensis, a copy of the former, the analysis of copying errors reveals much about the Roman numerals used at the time of copying. Where the classical Roman form for 1000 was _, the medieval scribe was more accustomed to using Z, and thus _ was often transcribed as R. Where the classical manuscripts contained X for 500, medieval scribes used W, and thus often omitted the X symbols entirely on the theory that the horizontal stroke indicated that the scribe had crossed out an error. Finally, because the subtractive form XL for 40 was used in medieval times as opposed to RRRR, instances of RRRR were abbreviated to RRR to correspond to correct medieval numeral-phrases. Shipley’s analysis confirms the increased use of both subtractive structuring and alphabetic Roman numeral-signs.

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The range of functions for which Roman numerals were used expanded considerably in the Middle Ages. Astronomical texts, which in antiquity were almost exclusively written using Greek numerals, often employed Roman numerals in the medieval era. As mentioned earlier, the alphabetizing of the signs for 500 and 1000 to D and M allowed the creation of number-riddles such as chronograms in which the numerical value of the Roman numerals in a phrase expressed the date of an event. Evidence for their use in legal documents and account records increases greatly, though this may be a function of the differential survival of perishable materials from later periods.

Medieval Modifications The replacement of the Roman numerals by the Western system was neither easy nor uncontested. Instead, between the twelfth and seventeenth centuries, there was great controversy throughout Western Europe regarding which system to use, with cultural, sociopolitical, and practical considerations being invoked in favor of one system or the other. Nor was the situation simply a choice between two options. While the general trend was toward the Western system in the great majority of contexts, as knowledge of the ciphered-positional system spread into Western Europe, a few individual writers made idiosyncratic modifications to the Roman numerals in response to the interloping newcomer. While none of these modifications was adopted on a wider scale, they can enlighten us about the circumstances under which the Roman numerals were replaced. The most complete positionalized version of the Roman numerals is one of the earliest. Around 1130, the mathematician “H. Ocreatus,” a student of Adelard of Bath, invented a positional numerical system using the Roman numeral-phrases for 1 through 9 (I, II, ... IX) along with a special sign (O or t, called cifra) to indicate an empty position (Smith and Karpinski 1911: 55; Burnett 1996, 2002c, 2006). This system is attested only in one thirteenth-century manuscript, a collection of arithmetical texts (Cashel, G.P.A. Bolton Library, Medieval MS 1), of which Ocreatus’s Helcep Sarracenicum (Saracen Calculation) is only one part. Positions were separated using a dot to avoid confusion; thus, 1089 was expressed as I.O.VIII.IX.6 This system, which blends the Roman cumulative-additive and Arabic ciphered-positional systems, is cumulative-positional and base-10 with a sub-base of 5. While Murray (1978: 167) characterizes Ocreatus’s system as clumsy, it should be noted that it is fully positional, far more so than later compromises 6

Smith (1926: 72) argues that this practice was attested in a few classical inscriptions as well, such as XII.L.D for 1,250,500 (!), but I know of no inscriptions where this was actually done.

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Table 4.13. Medieval multiplicative Roman numerals Year

Numeral-Phrase

Roman with Multiplication

Western Source

1220–25

II•DCCC•XIIII

Menninger 1969: 285

IX.XX.XVI

aaWUUUROP URUPO ZUUSPOOO ZZSROOO SRRRPOOO

2814

1231

196

Smith 1926: 6

1258

Steele 1922: xvii

2073

Crosby 1997: 208

88

Guitel 1975: 225

aaaUSRRROOO

3183

Cajori 1928: 33

USOR ZWP UU UWSR ZWROP ggg baWUUROOO

159

Preston 1994

1505

Menninger 1969: 285

200 460

Cajori 1928: 34

1514

Smith 1926: 7

300000

Menninger 1969: 285

6713

Menninger 1969: 283

gddaaaUWSPO

123456

Cajori 1928: 33

c

1258

Mij lviii

1340

IILXXIII

1388

1437–38

IIIIxx et huit M XX III C IIII III VIIXXXIX

1505

I•Vc•V

1392

C

1514

II IIIIC.LX

1514

XVCXIV

1550

CCCM

1554 1771

M

C

vi vii xiii c m c i xxiij iiij lvj

made between the Roman and Arabic systems. Nevertheless, this system was not taken up more widely in medieval scribal or mathematical texts. The speculation of Busch (2004) that a difficulty with seafaring travel times in the fifteenthcentury Icelandic Landnámabók can be resolved by interpreting VI (6) as “V I” (e.g., 51) is highly dubious. While the use of barred numerals to indicate multiplication by 1000 was of early origin and continued throughout the Middle Ages, new multiplicative forms began to be used starting in the twelfth and thirteenth centuries. Some examples of such numeral-phrases are listed in Table 4.13, along with the transcriptions of the appropriate number both in the classical Roman system (including the use of subtractive and multiplicative forms, where appropriate) and in the Western system. These numeral-phrases primarily express multiplication by 100 or 1000 by juxtaposing C or M either immediately beside the appropriate multiplier, above it, or in superscript, and sometimes interposed with a dot. Often, the number 20 occupies a special role as a multiplier; such phrases are almost all from France and result from assimilation to the partly vigesimal structure of the French lexical numerals and/or to the monetary system (Preston 1994). These numeral-phrases have a multiplicative component, but they are not positional – the value of the

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Table 4.14. Partially positional Roman numerals M•CCCC•8II CC2 ICC00 1•5•IIII 15X5 MDZ4 MCCCC4XVII IV0II

1482 202 1200 1504 1515 1624 1447 1502

numeral-signs does not change due to their position, but rather due only to their juxtaposition with another sign. It might be thought that multiplicative forms were adopted in order to write numerals more concisely or with a smaller set of numeral-signs – as was the case with the initial use of multiplicative forms in classical Rome. However, as seen by comparing the numeral-phrases in Table 4.7 to their equivalents in standard Roman numerals, there is no such benefit. In addition to these multiplicative forms, we find many cases where the signs of the Roman system were intermingled with the positional principle of the newer Western system, as well as its actual numeral-signs. The earliest known example is from the late twelfth-century Microcosmographia, in which the only known copy (from the thirteenth or fourteenth century) is dated mclxxviii (1178) but in another place as mc87 (an error for 1178) (Williams 1934: 107). Preston (1994) describes a number of such mixed uses in Durandus of Saint-Pourçain’s Commentary on the Sentences of Peter Lombard of 1336, such as xxx3 for 33 and xl7 for 47. In one example from an English almanac from 1386, Western numerals both precede and follow Roman numerals in a numeral phrase for 52,220, written as 52mcc20 (Halliwell 1839: 116). Menninger (1969: 287–288) provides examples of such admixtures starting in the late fifteenth century, as indicated in Table 4.14. In these partly ciphered-positional numeral-phrases, conciseness is greatly increased over their Roman counterparts. It is unknown whether this was being done consciously as a compromise between the two systems, as a misunderstanding of the Western system, or as numerical playfulness. In at least one case, the blending is obviously erroneous; in an astronomical table, the scribe wrote MCC6 for 1269, then crossed out the entire phrase and wrote 1269 in Western numerals (Steele 1922: xvii). Such combinations are not necessarily advantageous; the idiosyncratic nature of these numeral formations almost certainly decreased their comprehensibility. None were used frequently or consistently enough to create a true variant system. These hybrid formations no longer appear after about 1650. The roughly contemporaneous expansion of Roman multiplicative notation and the introduction of hybrid forms employing ciphering and positionality in

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late medieval European scribal traditions meant that any number could be written in several ways. The wide range of textual genres containing such variants, and their geographical and chronological breadth, confirm that this variety was not isolated, but neither was it standardized. In any event, a parallel process was ongoing from the twelfth century onward that would eventually lead to the nearly complete replacement of Roman numerals by Western ones throughout Europe and eventually throughout the world.

Replacement and Persistence Although positional numeration was first introduced into Western Europe in the late tenth century by Gerbert of Aurillac (later Pope Sylvester II), it was rarely used before the publication in 1202 of Liber Abaci by Leonardo of Pisa, also known as Fibonacci. This mathematical text sparked an important debate between two camps: the abacists, those who preferred computation with the medieval abacus, and the algorithmists, who preferred pen-and-paper calculations using the Western ciphered-positional numerals. The history of this debate is well documented, as it involved many important commercial families, renowned mathematicians and clergymen, and even state authorities (Menninger 1969: 422–445; Evans 1977; Murray 1978: 163–175). Issues of computational efficiency were often addressed. In his dictionary of 1530, the English lexicographer-priest John Palsgrave included the sentence, “I shall reken it syxe tymes by aulgorisme or you can caste it ones by counters” as a sample sentence for the verb to reckon (Palsgrave 1530: 337).7 However, these debates did not compare the Roman and Western numerals, but rather two techniques of computation: pen-and-paper arithmetic versus abacus calculation. The use of these techniques was correlated with specific numerical notation systems, but evaluations of computational efficiency require more information about how numbers were manipulated. Yet the debate was not only about efficiency. In 1299, the Arte del Cambio or moneychangers’ guild of Florence prohibited the use of Western numerals in its registers, a prohibition that was maintained for at least twenty years (Struik 1968). The reason given in the document is to prevent fraud, which was encouraged by the numerals’ ease of falsification, but Struik rightly notes that socioeconomic explanations for this prohibition are just as persuasive, taking into account conflict between conservative (aristocratic) and progressive (mercantile) factions. In 1348, booksellers at the University of Padua were required to list prices in their 7

This statement must be considered in light of the fact that a) it is a sample sentence rather than part of an evaluation of the two methods and b) Palsgrave’s book was paginated throughout using Roman numerals.

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inventories “not by ciphers but by plain letters” (Berggren 2002: 361). As late as 1494, the Frankfurt Bürgermeisterbuch (a mayoral book of edicts) ordered reckoners not to compute using Western numerals (Menninger 1969: 426–427). One can certainly imagine the consternation of merchants and bookkeepers upon learning that one can simply add zeroes to the end of a numeral ad infinitum to multiply its value by ten each time, but the role of xenophobia and traditionalism in all of these cases should not be underestimated. The Western numerals may have been rejected in part because of practical considerations, but they were, after all, a foreign and newfangled invention that involved the unusual principle of positionality and that (because of their novelty) could be used to conceal information or deceive others. The rarity of paper in the earlier Middle Ages may also have contributed to the continued use of the Roman numerals (Smith and Ginsburg 1937: 29). Until penand-paper calculation became feasible, the “Roman numeral–friendly” abacus was the computational technology of choice, whereas the switch to pen and paper made the conciseness of Western numerals more attractive. This attractiveness increased with the introduction of printing presses, where long strings of movabletype Roman numerals would have been unwieldy in comparison to Western numerals. Yet despite Eisenstein’s contention that, “[t]he use of Arabic numbers for pagination suggests how the most inconspicuous innovation could have weighty consequences – in this case, more accurate indexing, anotation, and crossreferencing resulted” (1979: 105–6), most incunabula contained no Western numerals. Even where the utility of Western numerals for computation was recognized, the Roman numerals were not always abandoned entirely, as in Italian metrological documents studied by Travaini (1998), in which computations were undertaken using Western numerals, but totals were then written in Roman numerals. William Cecil (Lord Burghley), England’s Lord High Treasurer from 1572 to 1598 and a chief advisor of Elizabeth I, would frequently transcribe economic documents from Western back into Roman numerals, and was decidedly uncomfortable with the new system (Stone 1949: 31). Despite such resistance, the Western numerals had taken hold by around 1300 in the Italian city-states. Elsewhere in Western Europe, particularly in Germany and England, the Roman numerals predominated until the late fifteenth century or even later. Jenkinson (1926) finds limited evidence for the use of Western numerals in English archives before 1500, and notes that Roman numerals were forbidden from use in state accounts only in the nineteenth century. Barradas de Carvalho’s (1957) study of fifteenth- and sixteenth-century Portuguese texts showed that the Roman numerals were replaced there only around 1500. Several important sixteenth-century arithmetical texts continued to use Roman numerals in preference to Western ones. Jacob Köbel’s (1470–1533) Rechenbiechlin, published in

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Figure 4.2. Tables of variant Roman numerals from the Mysticae numerorum significationis (1583–84) of Petrus Bungus. Source: Smith 1908: 380–383.

Augsburg in 1514, an important commercial arithmetic book intended for mercantile instruction, uses Roman numerals throughout and strongly advocates the use of the counting board, although Köbel frequently used multiplicative phrasing for hundreds and thousands, and his text is the source of the unorthodox “260/400” Roman numeral fraction in Table 4.13 (Smith 1908: 100–107). However, by the seventeenth century, the battle was essentially finished, and the Roman numerals ceased to be used in most contexts. Figure 4.2, from the Mysticae numerorum significationis of Petrus Bungus (d. 1601), first published in 1583–84, depicts a wide variety of archaic and curious Roman numeral phrases, but is essentially a text on number mysticism and symbolism rather than a manual for practical use (Smith 1908: 380–383). Why, after over a millennium of essentially unchallenged use in Western Europe, should the Roman numerals have ceased to be used for most purposes? The traditional answer given – that Roman numerals were less efficient for computation – is correct but incomplete. The Roman numerals were used alongside the Greek

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alphabetic numerals for well over a thousand years, despite the great increase in conciseness that could have been achieved by abandoning the Roman system. In medieval Western Europe, debates over the use of Roman numerals lasted into the seventeenth century, with the relative usefulness of the Roman numerals for calculation being greatly increased by the accompanying use of counting boards. However, two developments were fundamental in rendering the Roman system obsolete in the West. Firstly, the development of the printing press and the consequent rise in literacy after 1450 correlates very well with the rapid adoption of Western numerals throughout Europe (particularly in northern regions). The newly literate middle classes of Western Europe were learning to read and calculate, unconstrained by centuries of traditional use of the Roman numerals. At the same time, the rise of mercantile capitalism in the Renaissance changed how numbers were viewed and used (cf. Swetz 1987). The functional needs of Western society to represent number changed dramatically between 1300 and 1700, and the computational functions that are better served by Western than by Roman numerals increased in importance. Nevertheless, Crosby’s (1997: 41) insistence that Roman numerals were “adequate for the weekly market and for local tax collection, but not for anything grander” is misplaced if intended as an explanation for their replacement. In fact, as Crosby himself notes, Roman numerals persisted for centuries in the record books of institutions as grand as the British Exchequer (Crosby 1997: 115–116). Double-entry bookkeeping was facilitated to some degree by the transition to Western numerals, but it was by no means a requirement. The introduction of a more fully mercantile and urban economy in Western Europe did, however, expose more people, particularly members of the newly wealthy and increasingly literate middle class, to a new form of numeration. Rather than examining this change in notation simply by comparing the two notations, it is more productive to examine the question from a social perspective. If Roman numerals were truly so inferior, how did they persist for two millennia without being replaced? Why did it take six centuries even after the introduction of Western numerals for this shift to take place? The most likely scenario is that the new users of writing and written numeration, unconstrained by past practices, provided a critical mass of users for the Western numerals that facilitated certain mercantile functions. Even today, the Roman numerals have not disappeared. The use of the Roman system on clock faces, in the enumeration of kings and popes, on many dated inscriptions, and to mark the copyright dates on films is rather archaic, but the cultural importance of its symbolic connotations of antiquity, tradition, and prestige likely guarantee its future survival. In media where archaism is desired, Roman numerals are often preferred; this is the case among the Cornish church sundials from 1670–1850 studied by Burge (1994), for instance. An inscription on a gateway at Harvard uses archaic Roman numeral-signs, noting the college’s 264 years of

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existence in 1900 as “ANN. DOM. _WUUUU. COLL. HARV. UUTROOOO” (McPharlin 1942: 18). Pot (1999) notes the existence of several twentieth-century postage stamps in which additive Roman numerals rather than the ordinary subtractive ones are employed – for instance, XXXXI instead of XLI on a 1990 German issue. Presumably these variants are not errors per se, but are mobilized for aesthetic reasons, just as on Roman numeral clock faces, 4 is normally written IIII instead of IV, even though 9 is IX (Hering 1939). The fifteenth-century accountant Luca Pacioli advocated the use of Roman numerals for the year-dates in accounting books for aesthetic reasons (although Western everywhere else), recommending “[u]se the ancient letters in making this entry, if only for the sake of beauty” (Crosby 1997: 222). At the other end of the class and prestige spectrum, Roman numerals were often retained in peasant or rural traditions and trades long after they had ceased to be used in formal texts and administration. Gmür (1917) describes a wide variety of wooden tally sticks of Swiss provenience, dating from the sixteenth to the nineteenth centuries, containing Roman numeral-phrases. The same principle applies to the much earlier “calendar-numerals” used on Northern European calendrical documents (see the following discussion). With the inexorable spread of industrialization and capitalist commercial practices throughout Europe, the use of wooden tallies containing Roman numerals has essentially disappeared. When used to paginate the introductions of books, or to enumerate items in sublists or subsections, Roman numerals continue to serve the very practical function of providing an alternate form of enumeration wherever two different series of things must be listed. I would not expect that any industrial society could function solely with a cumulative-additive system such as the Roman numerals. Conversely, however, I would not expect them to be replaced entirely. The failure of over five hundred years’ worth of predictions of the Roman numerals’ imminent demise, and complaints regarding their utility, suggests that functional considerations, while important, do not tell the entire tale with respect to the history of numerical notation systems.

Arabico-Hispanic Variants Arabic and Western European knowledge systems interacted intimately on the Iberian peninsula between the tenth and sixteenth centuries – the period of the Reconquista and somewhat beyond. At this time, the Roman numerals were well known in Spain and Portugal, as in the rest of Europe. Arab mathematicians and astronomers had, since around 800 ad, used a ciphered-positional system much like the one used today (Chapter 6), along with older cipheredadditive abjad numerals related to those of the Greeks (Chapter 5). Medieval

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Spain thus presents us with a remarkable case where three coexistent systems each had a different structuring principle (cf. Labarta and Barceló 1988). But rather than a simple case of the two Arabic systems replacing the Roman numerals, medieval Spanish astronomers, bookkeepers, and scribes experimented and mixed principles, incorporating ciphering and positionality into the basic structure of the Roman numerals in innovative ways. It was, as described by Berggren (2002: 358), a “promiscuous blend of systems,” a mélange that persisted for several hundred years. In the tenth and eleventh centuries ad, Spanish astronomers wrote extensive astronomical texts using primarily the Roman numerals. Even the use of the subtractive principle was very infrequent in Spain at this time. Despite the use of Arabic positional numerals in the region for some time previous and familiarity with works such as the Arithmetic of Al-Khwarizimi, Spanish astronomers did not adopt the system directly (Lemay 1977: 458). Instead, the Roman numerals were modified to increase their conciseness. Instead of using the cumulative principle to express the numbers 3 through 9 (III, IIII, V, etc.), they used new acrophonic symbols based on the first letters of the Latin numeral words: 3 = t = tres; 4 = q = quatuor; 5 = Q = quinque; 6 = s = sex; 7 = S = septem; 8 = o = octo; 9 = N = novem. Of these, only the symbols for 4, 8, and 9 were common, probably because they are the longest numeral-phrases below 10 in Roman numerals (IIII, VIII, VIIII). In addition, a special sign for 40 was employed: Ó, a cursive ligatured version of XL, and the only exception to the abandonment of subtractive forms (Lemay 1977: 459). Bischoff (1990: 176) reports that this sign for 40 is also found in some Visigothic manuscripts, suggesting considerable antiquity for the sign on the Iberian peninsula. These changes altered the classical Roman numerals into a partly ciphered-additive system. However, these modifications were not accompanied by the adoption of positional notation, and the system still had a mixed base of 5 and 10 and was purely additive. These developments increased the conciseness of numeral-phrases considerably. The number 99, which would have been LXXXXVIIII, could now be expressed as LÓN. This system was used in mathematical and astronomical texts until about the mid twelfth century, at which time the Western numerals took hold in astronomy, as in the rest of Western Europe. Yet while the Arabic positional numerals had firmly established themselves and were later transformed into the Western numeral-signs familiar to us, the Roman numerals did not cease to be used on the Iberian peninsula, though their use became increasingly limited. Labarta and Barceló discuss two curious offshoot Roman numeral systems found in early modern Spanish documents, as shown in Table 4.15 (Labarta and Barceló 1988: 32–34). Both systems are cumulative-additive, decimal systems with a sub-base of 5, unlike those of the earlier astronomical texts. Unlike most Roman numerals,

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Table 4.15. Arabico-Hispanic numerals 1

5

10

50

100

Ø 1

Ù V

Ú ó

Û

Ý

numeral-phrases are written from right to left (highest values at the right), which is curious because even in the Arabic script, which has a right-to-left direction, the numerical notation system has a left-to-right direction. The first system is found in a few late sixteenth- and early seventeenth-century documents to indicate monetary quantities, and its numeral-signs are somewhat similar to letters of the Arabic script. The second system is found only in a single Inquisition document from 1576 (Labarta and Barceló 1988: 34).8 From the sixteenth century onward, Western numerals became increasingly common throughout the Iberian peninsula, but certain anomalies remained in the Roman numerals of the early modern period. Subtractive numerals remained rare in comparison to the rest of Europe. A symbol known as the caldéron was sometimes used; it meant 1000 but was used multiplicatively, preceded by unit-signs, not additively (Cajori 1922, Lowe 1943). The caldéron was most often shaped like a U, sometimes with one or two diagonal bars across it (™). Thus, a manuscript from the Ponce de Leon papers from 1501 has the year dated as J™DJ (1 1000 500 1), reminiscent of the use of M for that purpose in other parts of medieval Western Europe (Lowe 1943: 9). Cajori (1922) hypothesizes that the caldéron originated in Italy in the fifteenth century, possibly as a modification of the older Y for 1000 that was not fully replaced by M until the modern period. Curiously, even after the Roman numerals had been abandoned entirely, it continued to be used multiplicatively with Western numerals, as in a contract written in Mexico City in 1649 in which a sum of 7291 pesos is expressed both as 7U291 and VIIUCCXCI (Cajori 1922: 201).

Calendar Numerals In the late Middle Ages, unusual numerals were used in some Northern European documents and inscriptions pertaining to calendrical calculations. Known as “calendar numerals,” “runic numerals,” or “peasant numerals,” they are a variant of the Roman numerals. I reject the term “runic” because few runic inscriptions contain 8

However, see Chapter 10 for a discussion of a potential descendant of this system in the form of Berber numerals used in North Africa.

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Table 4.16. Calendar numerals 1

2

3

4

5

6

7

a b c d e f g A B C D E F G 1 2 3 4 5 5 5 1 2

8

9

10 11 12 13 14 15 16 17 18 19 20

h H 5 3

i j k l m I J K L M 5 6 6 6 6 4 1 2 3

n o p q r N O P Q R 6 7 7 7 7 4 1 2 3

s t S 8 7 8 4

them.9 Likewise, the term “peasant numerals” tells us who may (or may not) have been using them, but lacks the precise functional association of “calendar numerals.” Some examples of these signs are shown in Table 4.16 (cf. Kroman 1974: 121; Ifrah 1985: 146–147). These systems are all cumulative-additive and have a base of 10 with a sub-base of 5. Units are marked by strokes or dots; fives are marked by angled or curved lines or loops to create “U” or “V” shapes, and tens are marked by transecting the vertical line perpendicularly, creating a cross or X. Although these numeralsigns are often joined together into single figures resembling digits by using a vertical stroke, the system is not ciphered. There are no signs for 50, 100, or higher values, and I am not aware of calendar numerals being used for numbers higher than 31 (the number of days in the longest months). Other than their unusual numeral-signs, the calendar numerals are identical to ordinary Roman numerals. While they were used in rural traditions that suggest to some that they were part of the large cultural substratum of tallying and notching in medieval and early modern Europe (Menninger 1969: 249–251; Ifrah 1985: 146–147), the calendar numerals are not tallies in this sense. A tally is used ordinally – one places marks as necessary on some material (wood, paper, stone, etc.) in sequence in order to keep a running total, rather than, as with numerical notation systems, marking an already totaled value. The calendar numerals are simply a variant of Roman numerals in which numeral-phrases are written vertically and attached to a line. By the late fifteenth century, the Western numerals were fairly well known in Germany and were becoming much more common in England and Scandinavia. Calendar numerals were used in a very delimited set of contexts, namely

9

The medieval antiquarian tradition of manuscripts discussing runic writing contains minimal evidence of any specifically “Runic” numerical system. The clophruna (Old Norse klapprúnir) system of notating numbers by dots (a = 1 = .; b = 2 = ..; c = 3 = … ) discussed by Derolez (1954: 134) is of obscure origin and is not clearly related to any other numerical notation.

Italic Systems

131

on documents of wood, stone, or horn designed to assist the largely nonliterate populace in determining the dates of festivals, especially Easter. Throughout the medieval period, the Metonic cycle of nineteen years, after which the moon’s phases recur on the same day of the year, was used as a rough-and-ready guide to calculate the date of Easter. The function of the calendar numerals was solely to denote the various years of this cycle on stylized perpetual calendars, known as “runic calendars” in German- and Norse-speaking areas and as “clog almanacs” in England. Because of this specialized function, there was almost never any reason to express numbers higher than 19. While these texts were used for computation, the numerals were not used directly for arithmetic. One merely needed to line up the appropriate days and years to get the correct value. Calendar numerals were, however, reasonably compact and easily understandable by anyone who knew the Roman numerals. The calendar numerals lie at the heart of one of the major pseudoscientific controversies of New World archaeology – the so-called Kensington Rune Stone of Minnesota, which purportedly contains a Viking inscription left in 1362 by Norse explorers who had traveled westward from Vinland. At issue are the calendar or runic numerals used in the inscription, particularly the numbers 14, 22, and 1362, expressed as ad, bb, and acfb, respectively (Hagen 1950; Nielsen 1986: 51). These numeral-phrases are ciphered-positional, like our Western numerals but unlike the cumulative-additive calendar numerals (in which 14 would be written as n). It is improbable that a Norse explorer in “Vinland” would have been familiar with the Western numerals in 1362, which were known only by the very educated in northern Europe at the time (Struik 1964: 167). Even if this knowledge is presumed, we need to explain why the savant substituted the cumulative-additive runic numeral-signs for the appropriate Western figures. While some fourteenth-century runic inscriptions used Western numerals (Nielsen 1986), calendar numerals were never used in such a positional fashion. There is no evidence for the use of calendar numerals except in calendrical texts, which the Kensington stone is not. Ohman, the Swedish-American “discoverer” of the stone, was likely familiar with the calendar numerals (easily found in any book on runology) but not with the proper structure of the system. Similar inscriptions with runic numerals, such as that on No Man’s Land, an island south of Martha’s Vineyard, Massachusetts, are undeniable forgeries (Marstrand 1949). The calendar numerals were a brief and local phenomenon in northern Europe. As the Western numerals came increasingly to be used for calendrical purposes, their use waned. By 1643, when Ole Worm wrote his Fasti Danici describing the runic numerals used in calendar tables, it was primarily as a curiosity and to aid the transition to newer methods of calculation (Worm 1643).

132

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Summary The Italic numerical notation systems developed in the early sixth century bc with the invention of the Etruscan and Greek acrophonic numerals, based on a previously existing but still poorly understood system of tallies, and spread eastward from Italy during the period of classical Greek preeminence in the Mediterranean. However, of all the Italic systems, only the Roman numerals had any extensive use in the Christian era, as the Greeks had switched to the ciphered-additive alphabetic numerals (Chapter 5) by the Hellenistic period. The remarkable persistence of the Roman system and the swift decline of other systems are best explained by the changing political fortunes of their users. The use of the earlier Italic systems was mainly limited to inscriptions and commercial marks, though the Roman numerals were later used for an enormous variety of functions in different social contexts. All the Italic numerical notation systems are cumulative-additive and decimal, with a sub-base of 5, and virtually all use a single vertical stroke for the units. Some unusual structural features occasionally emerge in the higher powers of 10, such as the use of implied multiplication in South Arabian and the hybrid multiplicative structure of later Roman numerals. Although the epichoric numerals of Argos, Nemea, and Epidaurus use dots rather than vertical strokes for units, they are related to the Greek acrophonic numerals and must be considered part of this phylogeny. The cultural history of some Italic systems is intermingled with those of the Hieroglyphic (Chapter 2) and Levantine (Chapter 3) phylogenies, making the construction of accurate cultural phylogenies more difficult, particularly because the systems of all three are cumulative-additive, decimal, and used in the eastern Mediterranean. Often the three families can be distinguished on structural grounds: the Hieroglyphic systems all lack a quinary component, while the Levantine systems all have special signs for 20 and are multiplicative-additive above 100. This structural distinction can be confirmed independently by examining known patterns of historical contact. Despite the jumbled state of our present knowledge, we can identify phylogenies, not only because of similarities in their systems’ structure, but also as a result of the attested cultural connections among the societies in which they were developed. Nevertheless, to insist too strongly on “pure” cultural phylogenies following a model of speciation would be erroneous. In comparison to either the Alphabetic (Chapter 5) or South Asian (Chapter 6) systems, however, the diversity of Italic systems is far less than might be expected, given the preeminence and impact of the Roman Empire on European social life. The Roman numerals gave rise to relatively few descendant systems – when they were borrowed, they were copied rather than modified. We would do well to remember that the Roman numerals are simply one numerical notation system among the world’s diverse systems, despite their importance within the Western consciousness.

chapter 5

Alphabetic Systems

The systems discussed in the previous three chapters are primarily cumulative, repeating signs within each power of the base to indicate addition. In contrast, the next two families – the Alphabetic and South Asian systems – consist mainly of ciphered systems, which use, at most, a single sign for any power to indicate its different multiples: 1 through 9, 10 through 90, 100 through 900, and so on, in the case of decimal systems. Ciphered numeral-phrases are thus much shorter than cumulative ones, but require their users to be familiar with many more signs. Alphabetic numerical notation systems generally use phonetic script-signs, in a specified order, to express numerical values, and thus mitigate the effort needed to memorize both script-signs and numeral-signs. Despite the name, the scripts in question are not always alphabets; some, such as the Hebrew and early Arabic, are abjads or consonantaries, expressing primarily consonantal phonemes, and one, the Ethiopic Ge’ez script, is an alphasyllabary or abugida, expressing consonant + vowel clusters.1 Alphabetic systems were used as far north as England, Germany, and Russia and as far south as Ethiopia, and throughout Africa and the Middle East from Morocco eastward to Iran. Their history spans over two thousand years, from the development of the Greek numerals around 600 bc to the present, but in some 1

For more extensive discussion of script typology, which is not warranted here, see Daniels and Bright (1996).

133

134

Numerical Notation

cases important historical questions remain unresolved. While they are mostly ciphered-additive, they are not structurally identical. We can learn much more from these structural differences than from the paleographic curiosities of the signs of various systems. I hope in this chapter to illuminate areas of study where our knowledge is less than perfect in order to draw attention to the need for further specialized research. Table 5.1 shows the most common variants of the major alphabetic systems.

Greek Alphabetic In Chapter 4, I discussed the cumulative-additive Greek acrophonic numerals, which were given their name because the letters used are the first letters of the appropriate Greek numeral words. Another Greek system, much more common, is sometimes called the “Ionic” or “Milesian” system due to its origin in western Asia Minor, but is most commonly called the alphabetic system. While the Greek alphabet was based on a Phoenician model, probably in the ninth or eighth century bc, none of the earliest Greek inscriptions contains numerical notation; thus, the debates on the time of the origin of the alphabet are not relevant to the origin of alphabetic numerals (cf. McCarter 1975, Swiggers 1996). The first examples of the alphabetic numerals date to the sixth century bc and are written using the letters of the archaic Greek script used in Ionia and the Ionian cities of Caria, such as Miletus. Table 5.2 has representative examples of these early signs. The system was purely ciphered-additive and decimal, and was usually written from left to right, though right-to-left and boustrophedon (alternating direction) inscriptions are not unknown. The numeral-signs are archaic variants of the twenty-four familiar Greek letters, plus three special signs called episemons: vau or digamma (6), qoppa (90), and san or sampi (900), which were added to reach a full complement of twenty-seven signs for all the values from 1 through 900, allowing one to write any natural number less than 1000.2 Vau and qoppa were occasionally used phonetically in the Ionic script, with the values of [v] and [k], while san appears to have been borrowed from Phoenician sādē [ts], and was possibly used in archaic Greek with a similar phonetic value (Swiggers 1996: 265–266). There are very few examples of alphabetic numerals from this early period, and all of them express values under 1000, so we do not know whether higher values could be represented. 2

There was no general name for the three signs in classical Greek; the term episemon originally referred only to the sign for 6, but following its etymology, some classicists use the term in a more generic sense (J. Kalvesmaki, personal communication; Foat 1905).

135

70

60

50

40

30

20

10

9

8

7

6

5

4

3

2

1

a b g d e v z\ h q i k l m n x o

Greek

Coptic

Ethiopic

.a. .b. .c. .d. .e. .f. .g. .h. .i. .j. .k. .l. .m. .n. .o. .p.

Gothic Hebrew

Table 5.1. Alphabetic numerical notation systems

g h i j k l m n o p

Syriac

a b c d e f g h i j k l m n o p

Arabic

a b c d e f g h i j k l m n o p

Fez

a b c d e f g h i j k l m n o p

a b c d e f g h i j k l m n o p a b c d e f g h i j k l m n o p

a b c d e f g h i j k l m n o p

(continued )

a b c d e f g h i k l m n o p q

Armenian Georgian Glagolitic Cyrillic Latin

136

6000

5000

4000

3000

2000

1000

900

800

700

600

500

400

300

200

100

90

80

p , r s t u f c y w . /a /b /g /d /e /v

Greek

Coptic

Ethiopic

.q. .r. .s. .t. .u. .v. .w. .x. .y. .z. .{.

Gothic Hebrew

Table 5.1. Alphabetic numerical notation systems (continued )

q r s t u v sv tv uv vv svv /a /b /c /d /e /f

Syriac

q r s t u v w x y z : ;

Arabic

q r s t u v w x y z ,

Fez

q r s t u v w x y z ! @ # $ % ^ &

q r s t u v w x y z 1 2 3 4 5 6 7 q r s t u v w x y z / 1 2

q r s t u v w x y z {

r s t u x y z θ φ

Armenian Georgian Glagolitic Cyrillic Latin

137

1,000,000

100,000

90,000

80,000

70,000

60,000

50,000

40,000

30,000

20,000

10,000

9000

8000

7000

z h q i p

/z /h /q a b g d e V

Greek

Coptic

Ethiopic

Gothic Hebrew

-f -g -h -i -j -s

/g /h /i -a -b -c -d -e

Syriac

Arabic

Fez

* ( )

8 9 0 -

Armenian Georgian Glagolitic Cyrillic Latin

Numerical Notation

138

Table 5.2. Greek alphabetic numerals (archaic) 1

a 10s j 100s s 562 = wob

1s

2

3

4

5

6

7

8

9

b k t

c l u

d m v

e n w

f o x

g p y

h q z

i r ,

In the classical and Hellenistic periods, the familiar Greek alphabet supplanted the archaic regional (or “epichoric”) variants, and the alphabetic numerals developed along with them, retaining their order and numerical values but assuming their modern (majuscule) forms. In addition, starting in the middle of the fifth century bc, two new techniques were used to express higher values. For multiples of 1000, a small slanting mark (known as a hasta) was placed to the left and below a sign for 1 to 9 to indicate that its value should be multiplied by 1000; thus, G means 3, but /G means 3000 (Threatte 1980: 115). Values above 10,000 are rarely encountered except in mathematical works, and individual writers used different methods to do so. The most common method, used by Aristarchus, involved placing a small alphabetic numeral-phrase (less than 10,000) above a large M character (= myriades) to indicate multiplication by 10,000 (Heath 1921: 39–41).3 Thus, 3,000,000 would be expressed with only two signs, as y. This allowed any number less than 100 million to be easily expressed.4 The system thus appeared as shown in Table 5.3, using the Athenian letters. The system is thus ciphered-additive for values under 1000, and thereafter is multiplicative-additive at two different levels: firstly, through the use of a hasta to indicate multiplication by 1000, and then through the use of an M to indicate multiplication by 10,000. The alphabetic numerals were generally written in descending order, with the highest values on the left. Numbers between 11 and 19 were often written with the 10-sign (I) following the unit-sign, however, to correspond with the way in which the ancient Greek lexical numerals were formed: 3

4

Heath also discusses techniques such as that of Heron’s Geometrica, where two dots placed over a sign indicate multiplication by 10,000; that of Apollonius, using “tetrads,” turning the system into a mixed base-10/10,000 ciphered-additive system; and that of Nicholas Rhabdas, a fourteenth-century scholar who used Heron’s technique, except that additional pairs of dots above a number indicated successive powers of 10,000. None of these systems was ever widely used. For the ancient Greeks, ‘ten thousand times ten thousand’ – 100 million, the first uncountable number in the alphabetic system – was of symbolic significance, as in Revelation 5:11: “and the number of them was ten thousand times ten thousand.”

Alphabetic Systems

139

Table 5.3. Greek alphabetic numerals (classical) 1

2

3

4

5

6

7

8

9

Aa Ii

Bb Kk

Gg Ll

Dd Mm

Ee Nn

Vv Xx

Zz Oo

Hh Pp

Qq <,

1000s

Rr /A

Ss /B

Tt /G

Uu /D

Cc /E

Ff /V

Yy /Z

Ww /H

>. /Q

10,000s

A

B

G

D

E

V

Z

H

Q

1s 10s 100s

562 = CXB

hendeka, dodeka, treis kai deka, tettares kai deka, and so on. For instance, Threatte (1980: 114) provides a number of examples from Attica where GI, ZI, HI, and QI appear for 13, 17, 18, and 19. Starting in the Roman period, the signs became more rigidly fixed in highest-to-lowest order. There was a mild taboo against the use of theta (Q) for 9 in some texts because it is the first letter of θανατοσ ‘death’; circumlocutions were used instead, such as writing the number lexically, or as an additive combination of alphabetic numerals such as EΔ (5 + 4) or AH (1 + 8) (Smith 1926: 69; Thomas 1977). Because alphabetic numerals could easily be confused with written words, classical Greek alphabetic numerals were sometimes distinguished from the rest of the text with special signs, most commonly a horizontal stroke above the numeralphrase, but occasionally with dots placed to either side of it. One of the problems in identifying earlier Greek alphabetic numerals is the lack of such marks, meaning that any single letter could be an alphabetic numeral or a non-numerical label. Even in later periods, numerals frequently appear without any indicator mark whatsoever (Threatte 1980: 115). In most Greek monumental inscriptions, the only fractions used are acrophonic signs for fractions of different monetary units (Tod 1950: 134). In mathematical and literary texts, an entirely different system was used in which small accents or strokes placed above and to the right of a numeral indicated unit-fractions (Thomas 1962: 43). Special signs existed for 1/2 (<’ and U’) and 2/3 (ω’), in addition to standard unit-fractions (Thomas 1962: 45). From the second century ad onward, the requirement of using only unit-fractions was lifted, and fractions were expressed with both numerators and denominators using alphabetic numerals. Finally, a special system for fractions was used in astronomy, combining the alphabetic numerals with sexagesimal structures borrowed from the Babylonians. There is no evidence for an early (eighth century bc or earlier) origin of the numerals, although prominent classicists such as Larfeld (1902–07) endorsed this

140

Numerical Notation

theory, arguing for descent from the Phoenician alphabetic system, because the Greek alphabet was borrowed from a Phoenician ancestor and because many Semitic scripts have alphabetic numerals (Brunschwig and Lloyd 2000: 388). Yet no Semitic consonantal numerical notation systems existed before the second century bc, and they were based on Greek rather than the other way around (Gow 1883). None of the very earliest Greek alphabetic inscriptions contains numerals. Conversely, however, the once-popular theory of a very late origin (late fourth or even third century bc) cannot be sustained in light of evidence from earlier periods (Gow 1883). The first epigraphic evidence for alphabetic numerals comes from a vase, dating to around 575 bc, found at Corinth, which contains the inscription “SYM g,” which Johnston reads as “mixed batch of 7” (Johnston 1973: 186). There is good evidence from Attica and Corinth for the system’s use on mercantile vases in the late sixth and early fifth centuries bc (Hackl 1909). Yet it is unlikely that the numerals actually developed in either of these localities. Rather, the numerals probably developed in western Asia Minor, in the regions of Ionia and Caria, especially in the cities of Miletus5 and Halicarnassus, where several early instances of the numerals have been found (Heath 1921: 32–33). Another early example is found in the “tunnel of Eupalinos” on the Ionian island of Samos, constructed around 550 bce, in which distances are noted using letters that are probably alphabetic numerals (as opposed to non-numerical or ordinal labels) (Verdan 2007: 12–13). All the early examples of the alphabetic numerals, even those found outside Asia Minor, are written using the Ionic script, which was used in Ionia, Caria, and various Ionic colonies throughout the Mediterranean. This reflects the predominance of Ionia in regional and international commerce during the sixth and early fifth centuries bc. While vau and qoppa were used phonetically in the early Ionic script, san was not, and so had lost its place between pi and qoppa in the numerical order (Jeffery 1990: 327). Later, it was reincorporated into the script for the purpose of providing a twenty-seventh sign for the numerals to function, but was placed at the end of the system, with the numerical value 900. The principle of the Greek alphabetic numerals was borrowed directly from the Egyptian demotic numerals in the early sixth century bc, only using alphabetic signs as numeral-signs (Chrisomalis 2003; see also Zaslavsky 2003). Boyer (1944: 159) regarded the similarity between the two systems as indicative of a historical connection, but his paper was not primarily oriented toward such an argument. Because the alphabetic numerals were the first to use phonetic signs as numeralsigns, there will be no paleographic similarity between the Greek numeral-signs 5

Miletus, from whence the adjective “Milesian,” was the most important Ionian city in Caria, the region of Asia Minor immediately to the south of Ionia proper.

Alphabetic Systems

141

and any other system. Yet the alphabetic numerals are structurally similar to the demotic numerals. They are both ciphered-additive, base-10 systems. While it has yet to be established whether the alphabetic numerals used multiplicative notation at an early date, both systems are multiplicative-additive above 10,000. Unlike the demotic numerals, the alphabetic system is also multiplicative for the thousands. The Greeks could simply have continued the series above 1000 using 10 through 90 and 100 through 900 (/I = 10,000; /K = 20,000; etc.), so the use of two-stage multiplication may be a clue to the alphabetic numerals’ history. It would be reasonable for the Greeks to adopt the multiplicative principle at the same level as in the demotic numerals, namely 10,000. However, it would not have been feasible to find nine extra signs for the values 1000 through 9000. Consequently, the inventor(s) of the alphabetic numerals may have had the idea of using multiplication for the thousands values as well as the ten thousands. The only remaining problem is to explain why the Greeks, recognizing this irregularity, did not then abandon the higher multiplicative series. Furthermore, Greek arithmetical techniques for dealing with fractions strongly resemble the Egyptian unit-fraction (1/x) tradition of computation (Knorr 1982, Fowler 1999a). Both systems used unit-fractions formed by placing a small mark above numeral-signs to indicate the appropriate unit-fraction. As well, both used non-unit-fractions for specific values such as 1/2 and 2/3. Historians of mathematics are unanimous that the Greeks borrowed the unit-fraction technique from the Egyptians, and I see no reason to doubt that the Greek use of special signs for 1/2 and 2/3 is also a result of Egyptian influence. Turning to the historical evidence, the demotic numerals were the predominant ones in use in Egypt (especially Lower Egypt) in the early sixth century bc, when Greeks were just starting to encounter Egyptians in large numbers for the purposes of international trade. Most notable among the Greek traders in Egypt were Ionian colonists from Miletus, who set up an important emporion (port of trade) at Naukratis in the western Nile delta in the seventh century bc. Naukratis quickly became the central locus for cultural contact between Greece and Egypt, a position that it held until the Ptolemaic era. Inscriptions in the Ionic Greek script dating as early as 650 bc have been found at Naukratis (Heath 1921: 33). It should be noted, however, that no known inscriptions from Naukratis contain alphabetic numerals, and there are later (fourth-century bc) inscriptions with acrophonic numerals (Gardner 1888). Since the earliest examples of the alphabetic numerals are from containers for commercial goods, the context of the system’s development was probably in mercantile activity. The only alternative to the hypothesis of Egyptian origin is that the Ionians independently developed a ciphered-additive, decimal numerical notation system within a few decades of coming into contact with Egyptians in large numbers,

142

Numerical Notation

founding a colony at Naukratis, and being exposed to the demotic numerals used throughout Lower Egypt. These connections are too significant to be coincidental. This should not be taken as a denial of the Greeks’ inventiveness, however, because the alphabetic numerals have several distinctive properties. Firstly, while some of the demotic numeral-signs use the cumulative principle, the alphabetic numerals use purely ciphered signs. Secondly, as mentioned earlier, the alphabetic numerals use the multiplicative principle for 1000 through 9000, obviating the need for nine more signs for those values, as one could simply write a hasta before a unitsign. Finally, correlating numeral-signs with the ordered set of alphabetic signs meant that anyone who knew the order of the alphabet could determine the signs’ values as long as the episemons were taken into account. The often-mentioned “weakness” of the alphabetic numerals, that too many signs needed to be learned, is thus illusory. In learning to read and write, Western pupils must learn twentysix alphabetic signs (in their proper order) plus ten digits in order, making thirtysix total signs in two separate series, while the ancient Greeks needed to learn only twenty-seven alphabetic signs and two auxiliary signs (/ and :), and needed only twenty-nine total signs in one series. After a period of Ionian cultural dominance between 575 and 475 bc, when alphabetic numerals were commonly found, alphabetic numerals are found only rarely in a period starting in 475 bc and lasting around 150 years (Johnston 1979: 27). During this period, the height of Greek civilization, Athens came to the forefront as an Aegean power, while Ionia’s power waned after the Milesian-led Ionian revolt of 499 to 494 bc against Achaemenid Persia. The system did not disappear entirely; at Halicarnassus, there is evidence of its continued use (Keil 1894; Heath 1921: 31–33). A curious inscription from Athens (IG I2 760) from the middle of the fifth century bc contains a long series of alphabetic numerals, written with Ionic letters (Tod 1950: 137). There is, as well, good reason to believe that some of the graffiti on amphorae from the Athenian Agora had weight and volume marks notated in alphabetic numerals (Lawall 2000). In general, however, acrophonic numerals were used in most Greek texts during this period. The resurgence of the alphabetic numerals in Greece around 325 bc corresponds precisely with the rise of the Ptolemies in Egypt. In this renewed period, some of the earliest instances of the numerals come from Egypt. Figure 5.1 depicts a portion of a Greco-Egyptian astronomical text including a calendar for the Saite nome (Hibeh Papyrus i 27) dating to around 300 bc, one of these very early instances (Grenfell and Hunt 1906, Fowler and Turner 1983). Ordinary (unmarked) Greek letters stand for whole numbers, while 1/x “unit-fractions” are represented by signs modified with long oblique strokes. This is the earliest attested example of alphabetic numerals used in a scientific context. Coins dating to 266 bc indicating the regnal year of Ptolemy II Soter are, to my knowledge, the first coins bearing

Alphabetic Systems

143

Figure 5.1. Hibeh papyrus i 27, with Greek alphabetic numerals indistinguishable from letters, but with unit-fractions clearly distinguished through the addition of long upward slanting strokes. Source: Grenfell and Hunt 1906: Plate VIII.

any ciphered-additive numerals (Tod 1950: 138). It is interesting that while these examples come from Egypt, we have no record of the alphabetic numerals’ use in Egypt during the interlude of the fifth and fourth centuries bc. The evidence, at present, is simply too scanty to conclude what stimulus caused the rejuvenation of the alphabetic numerals. From the third century bc onward, the alphabetic numerals were preferred over the acrophonic numerals throughout most of the Greek-speaking world, with only Athens retaining the acrophonic system until around 50 bc (Threatte 1980: 117). While the acrophonic numerals used in different city-states varied quite widely, the alphabetic numerals had no regional variants. As such, they could be used as an effective instrument of cross-cultural communication and trade among diverse regions (Dow 1952: 23). Whereas Greece before Alexander was highly fragmented, rendering the development of a universal Greek numerical notation system unlikely, Alexandrine and especially Roman Greece provided a suitable environment for the development of a single pan-Hellenic notation. That the system was a very concise way to represent numbers, and that it relied on alphabetic symbols that had become invariant throughout Greece by this period in history, cannot have hurt this process.

144

Numerical Notation

In the sixth and fifth centuries bc, the alphabetic numerals were no more than a system for labeling mercantile containers. All the early instances of the system’s use are from marked vases and potsherds. Even then, most numerals on vases are acrophonic or other cumulative-additive Greek numerals, not alphabetic ones. In these very early contexts, the alphabetic numerals, like the acrophonic ones, were used for cardinal quantities, particularly of money, weights and measures, and discrete quantities of commodities, the sorts of numerical expressions likely to be found in inventories and decrees. From the third century bc onward, though, when the alphabetic system became the predominant one throughout the Greek world, the numerals were used in a much wider range of contexts. In contrast to the acrophonic numerals, which are found solely on ceramic vessels and stone, alphabetic numerals are found, in addition, in manuscripts of various sorts as well as on coins. As described by Tod (1950: 130–134) and Threatte (1980: 115–116), the functions for the alphabetic numerals include: a. b. c. d. e. f.

cardinal quantities of commodities, persons; phrases indicating lengths of time in days, months, and years; monetary values (denarii, drachmas, and obols); weights, measures, and distances; ordinal numerical adjectives and adverbs; ordinal dates, for example, to indicate a specific year in the tenure of an archon.

Before the Hellenistic period, we have no evidence that the alphabetic numerals were used for arithmetic or mathematics. Early writers such as Herodotus instead used the acrophonic numerals to record the results of computations performed with a pebble-board or abacus (Lang 1957). This situation changed once the alphabetic numerals began to be used more widely. Most surviving Greek mathematical manuscripts use the alphabetic numerals, which were used by all Greek mathematicians beginning with Archimedes and Apollinius (third century bc). Users of alphabetic numerals had a wide variety of computational techniques available. Schärlig (2001) suggests that while the earliest abaci are most conducive to working with acrophonic numerals, certain types of abaci were well suited to working with alphabetic numerals also. Furthermore, the Greeks developed a complex system of computing on the fingers up to 9999, reminiscent of the alphabetic system’s structural shift above 10,000 (Williams and Williams 1995). To facilitate multiplication and division, which were difficult to undertake using the abacus or finger computing, the Greeks used multiplication tables written with alphabetic numerals (Dilke 1987: 16; Sesiano 2002). Finally, Greek arithmetic was sometimes done in a manner analogous to modern Western practice, by writing down numbers and manipulating numeral-signs directly; the primary difference between the

Alphabetic Systems

145

two techniques is that in the Greek case, because the numerals are nonpositional, there was no need to line up numbers in columns (Smyly 1905). One would think, as Boyer (1944: 160) comments, that the adoption of the alphabetic numerals by such prominent Greek mathematicians would curb the criticism of modern scholars that the system was dysfunctional for mathematics. Yet many modern scholars denigrate the alphabetic numerals in comparison not only to ciphered-positional systems but also to cumulative-additive systems such as the acrophonic numerals (cf. Boyer 1944: 160–166). In particular, the need to learn many signs and the lack of resemblance between numeral-signs for common multiples of different powers (e.g., 5, 50, and 500) are perceived as serious weaknesses. In the only instance where a modern scholar actually attempted to learn and use the numerals, however, the system fared very well. The classicist Paul Tannery found that calculation with alphabetic numerals took little more effort than with Western numerals, with which Tannery, despite all his efforts, surely had far greater experience and familiarity (Boyer 1944: 160–161). In any event, only a small fraction of the Greek texts containing alphabetic numerals serve mathematical functions. Like virtually all numerical notation systems, numerals are primarily for representation and only secondarily for computation. Throughout their history, the Greek alphabetic numerals were used primarily in Greek-speaking areas or in regions under the control of Greek speakers. During its early history, the system was used in the eastern Mediterranean (particularly the Aegean), Ptolemaic Egypt, and Seleucid Persia. Cursive varieties of the alphabetic numerals on Greco-Egyptian papyri show a great deal of paleographic variability (Foat 1902, 1905). A large number of variant multiplicative signs for 1000 and 10,000 were used in these papyri, most notably Ô for 10,000, which was the normal form starting in the second century ad (Brashear 1985). While alphabetic numerals were not used for administration during the height of Roman power, from the fourth century ad onward, they were used as the primary numerals of administration, law, literature, and mathematics in the Eastern Roman Empire. Whenever and wherever the Greek alphabet was used in the Middle Ages, the alphabetic numerals followed. Additionally, the Greek alphabetic numerals were the most common system used in Arabic papyri for several centuries after the Islamic conquest for recording the results of financial transactions (Grohmann 1952: 89). Most of the descendants of the Greek alphabetic numerals substituted the letters of their own scripts for the Greek signs. In the late second century bc, the Israelites developed the Hebrew alphabetic numerals along a Greek model. In the mid fourth century ad, the Goths developed alphabetic numerals along with their Greek-influenced script, while in regions of Africa under Greek influence, the Coptic script of Egypt and the Ge’ez script used in Ethiopia both developed alphabetic numerical notation systems based on a Greek model. In Armenia and

146

Numerical Notation

Georgia, right on the border of the Eastern Roman Empire, scripts and alphabetic numeral systems developed in the fifth century ad at around the time they were Christianized. In the sixth century ad, the Syriac script abandoned an earlier cumulative-additive numerical notation system (Chapter 3) in favor of one based on the Greek. Around the same time, the Arabic abjad numerals used following the Islamic conquest of the Middle East were derived at least in part from the Greek alphabetic system. The Glagolitic and Cyrillic numerals, developed in Slavic regions in the late ninth century ad, under the auspices of the missionary-related script development of Cyril and Methodius, closely resemble their Greek ancestor. Finally, knowledge of the Greek alphabetic numerals in Western Europe led to the development of a set of Latin alphabetic numerals in the twelfth century ad. It is remotely possible that the Greek numerals are ancestral to the Brāhmī numerals (Chapter 6), which were used from the late fourth century bc onward in India, and which eventually gave rise to Western numerals. The Brāhmī numerical notation system is ciphered-additive and decimal, and used a variety of the multiplicative principle. The chronology of its invention, corresponding almost exactly with the Alexandrine conquests and journeys in India, is also suggestive. However, the Brāhmī system is more similar in structure and numeral-signs to the Egyptian demotic numerals than to the Greek alphabetic numerals. The large number of descendants of the Greek alphabetic numerals is due in part to their longevity. This cannot be a full explanation, however; other systems, such as the Egyptian hieroglyphs, were used over a much longer period, yet generated few direct descendants. Other long-lived systems, such as the Roman numerals, spread very widely over large parts of the world due to Roman imperial power, but they were often accepted by colonized or subordinate societies unchanged, and did not replace indigenous systems entirely. Because the Greek system was alphabetic, cultures borrowing the principle of alphabetic numeration tended to modify the signs to fit their own scripts (whether alphabets or consonantaries) rather than adopting the Greek alphabetic numerals directly, and also made minor structural changes to the system. In the early Middle Ages, when the Eastern Roman Empire’s fortunes were prosperous, the numerals were widely used throughout Greece, the Balkans, Egypt, the Levant, and Asia Minor, and were incorporated into the learning of all European mathematicians. For instance, they were known to the English scholar Bede, who described them in his De temporum ratione (The Reckoning of Time) in the early eighth century ad (Wallis 1999). Many Western European manuscripts include numerical glossaries, for instance, a tenth- to eleventh-century codex (Vatican Library, Codex Urbinas Latinus 290) that lists (among other information concerning non-Latin alphabets) the Greek lexical numerals, letter names, and alphabetic numerals along with their translations into Roman numerals

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147

(Derolez 1954: 106–109). Such tables would have aided medieval scribes in understanding texts, but the Greek alphabetic numerals were not actively used except in rare instances. For instance, in one ninth-century Latin manuscript, the Greek numerals 1 (A) to 21 (KA) stand for the letters A through X in a cryptogram to indicate that it was written by Irish writers at the Welsh court of Merfyn Frych (824–844) (Derolez 1954: 97–99). Such cryptographic uses confirm rather than refute the obscurity of the system, however. In the Western European debate between the “abacists” and “algorithmicists,” the Greek alphabetic numerals, used by all the great mathematical minds of antiquity, did not rate a mention. The eventual fate of the Greek numerals was directly tied to the fortunes of the Byzantine Empire. Byzantine mathematicians and astronomers used alphabetic numerals exclusively until the twelfth century, and numerals such as regnal years were often stamped on Byzantine coins. By 1300, however, the geographical extent of the numerals’ use had become very limited, and mathematicians were using Western numerals under the influence of Arab learning in Spain and Italy. In the Byzantine Empire, mathematicians used Arabic positional numerals in marginal notes on Euclid’s Elements in the twelfth century (Wilson 1981). The first major Byzantine mathematician to recommend the switch to the Arabic numerals was Maximus Planudes (c. 1260–1310) (Schub 1932). In some Byzantine astronomical texts, Arabic numerals were used to write year numbers while Greek alphabetic numerals were used for all other functions; even then, the Greek equivalent of the Arabic numeral-phrase was often written in the margin of the page (Neugebauer 1960: 5). In a few late Byzantine mathematical texts, combinations of Greek alphabetic numerals and Arabic positional numerals with a zero occurred, such as υνο (400 + 50 + 0) instead of υν for 450 (Neugebauer 1960: 5). The ultimate extension of this principle was the use by some writers of only the first nine Greek letters (A through θ) to indicate the units, then adding a dot, a circle, or a special sign, Ч, to indicate a zero position (Schub 1932: 59; Menninger 1969: 273–274). In 1453, with the fall of Constantinople, the Greek numerals ceased to be used administratively. Nevertheless, they continued to be used thereafter for restricted purposes, such as paginating the introductions to scholarly texts and enumerating ordinal lists, just as the Roman numerals were used in Western Europe. Starting in the seventeenth century, the sign for 90 (qoppa), little used in alphabetic contexts, underwent a palaeographic alteration, so that a new, roughly Z-shaped sign served only as a numeral, while the older Q-shaped sign could serve both alphabetic and numerical functions, but was regarded as a different sign (Everson 1998). Alphabetic numerals are still used today in many Greek Bibles to enumerate chapter and verse numbers, in certain legal contexts, and in pagination. Given the prestige associated with these functions, it is unlikely that the alphabetic numerals will cease to be used entirely.

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Coptic The Coptic language is the last descendant of the Egyptian languages. It was written using the Greek alphabet until the fourth century ad, when a distinctive Coptic alphabet originated based on a Greek model, but using six additional characters taken from the demotic script to express uniquely Egyptian phonemes. Unlike the earlier Egyptian scripts, it is written from left to right and has signs for vowels. The adoption of the Coptic script was accompanied by the introduction of a numerical notation system based on the model of the Greek alphabetic numerals (Megally 1991, Messiha 1994). The Coptic numeral-signs are shown in Table 5.4. The numerals, like the script itself, were written from left to right, and were ciphered-additive and decimal. The numeral-signs closely resemble the Greek uncial signs used between the fourth and ninth centuries ad. In addition, as in the Greek alphabetic numerals, a horizontal stroke above the numeral-phrase indicates that it is a numeral rather than a word, and a slanted subscript stroke under a unitsign (the Greek hasta) indicates multiplication by 1000 (Megally 1991: 1821). There is no known sign or multiplier for 10,000 or higher values for these numerals. The classical age of Coptic lasted from the fourth to the tenth centuries ad, during which time the script and numerals were used extensively, surviving the seventh-century ad Muslim conquest of Egypt. There may have been a geographical division in the frequency of their use, with northern Egyptian scribes using them frequently, while southern writers tended to write out numbers using lexical numerals (Till 1961: 80). While the Coptic numerals were generally used in formal manuscripts, the ordinary cursive Greek alphabetic numerals were used for calculation and administration, possibly because the Coptic numerals, being uncials without tails, were less practical for rapid writing (Megally 1991: 1821). It is unclear whether the Ethiopic numerals (used to write the Ge’ez language from the fourth century ad onward) were based directly on the Greek alphabetic numerals or derived through a Coptic intermediary. While the Ethiopic system is generally said to derive directly from Greek, the Coptic uncial letter-signs are similar enough to the Greek to render such a determination premature, and given the geographic proximity of Egypt and Ethiopia, this possibility cannot be dismissed outright. While the primary function of Coptic numerals has always been religious, in the context of the Coptic Church, their administrative and arithmetical functions should not be discounted. Despite the control of the population of Egypt by a succession of foreign powers, and despite the decline of Coptic as a spoken language from the twelfth century onward, the use of Coptic numerals continued as late as the fourteenth century. They are still used today in Coptic Christian liturgical texts for pagination and stichometry, although for most ordinary purposes, either Western or Arabic positional numerals are preferred by those familiar with Coptic.

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Table 5.4. Coptic numerals 1

2

3

4

5

6

7

8

9

1s 10s 100s 1000s 6085 =

ZIMĀM Numerals In Egypt in the tenth century ad, when the Coptic script was being replaced by Arabic for most administrative purposes, a different Coptic cursive numerical notation system developed, known as “numerals of the Epakt” (named after the computation used to resolve the discrepancy between the solar and lunar calendars) (Messiha 1994: 26) or, more generally, as ḥurūf al-zimām ‘account book; register’ numerals. Often simply labeled “Coptic numerals” in modern scholarship, it seems prudent to distinguish them from their uncial antecedents, particularly since the zimām numerals are often found alongside Arabic or Hebrew texts, not ones in the Coptic language or the Coptic alphabet. Their long-standing use in a wide variety of contexts renders their importance far greater than the restrictedfunction Coptic uncials. While doubtless many users of the zimām numerals were Coptic Christians from Egypt, this was certainly not an ironclad rule, and Egyptian Jews and Muslims writing in Arabic script frequently used Coptic numerals instead of the Arabic abjad or Hebrew alphabetic numerals. The zimām system is shown in Table 5.5. This system is structurally identical to the classical Coptic system, but the numeral-signs are cursive minuscule letters rather than uncial ones. Many of these signs bear little or no resemblance to the classical signs. Some of them are probably taken from the signs of the Arabic abjad numerals, which I will describe later, while others are of indigenous development. This system also uses two stages of multiplication (at 1000 and 10,000), like the Greek alphabetic system. A horizontal stroke and two dots placed below a sign indicated multiplication by 10,000 (Sesiano 1989: 64). A fifteenth-century multiplication table includes instructions on writing “numerals of the Epakt,” and indicates that this sign for 10,000 could be used in conjunction with any of the twenty-seven letters, thus allowing any number less than ten million to be expressed (Sesiano 1989: 54–55). It may be that the two-stage multiplicative principle at 1000 and 10,000 existed even in

Numerical Notation

150 Table 5.5. Zimām numerals 1

2

3

4

5

6

7

8

9

1s 10s 100s 1000s 7104 =

the earlier uncial numerals, and that we simply have no paleographic evidence to confirm this. The paleographic relation between the first and second Coptic systems remains unclear; although some of the signs are related, more are not, and in fact many are modifications of Greek minuscule letters rather than Coptic uncials. These numerals were often used in early bilingual Arabic-Coptic documents, suggesting that the Arabs were making concessions to local administrators. This situation is quite extraordinary, because many Egyptian Arabs by this time were employing the ciphered-positional Arabic numerals used today. This suggests that the advantages of ciphered-positional systems over ciphered-additive ones may not have been perceived to be important. In addition to Arabs, many users of the zimām system were Egyptian Jews who used the system alongside their own alphabetic numerals (Goitein 1967–88: II, 178). Their primary functions were commercial, including contracts, bills of sale, payments, and accounts. There is no evidence for or against the position that numerals were manipulated for arithmetic; this surely would not have been done on permanent media, even perishable ones like papyrus. Although zimām numerals originated in commercial contexts and are rare in attested astronomical and mathematical texts, Lemay (1982: 384) has found one early text dating to ad 938 (MS Garullah Efendi 1508, Süleymanie Library, Istanbul) in which the numbers in the text are written lexically, the Arabic abjad numerals used in astronomical tables, and the zimām numerals used in pagination. King (1999: 76–78) discusses an astrolabe made in Cairo in ad 1282–83 with an Arabic inscription but zimām numerals. At this period, particularly in Egypt, the Coptic calendar was used for various purposes by Muslims, potentially explaining this artifact’s unusual blend of representational systems. Zimām numerals were also used very widely for pagination and foliation of Arabic texts written by adherents of all three Abrahamic faiths, throughout Egypt and more broadly throughout the Maghreb (Troupeau 1974). Mingana (1936: II, 183–184) reports that an Arabic

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151

letter of bequest from 1586 relating to a Syriac Christian manuscript (Mingana Syriac 617) is dated in Coptic numerals (presumably zimām rather than uncial), suggesting that even at this late date, their use was widespread and not restricted to Coptic Christians and Egyptian Muslims. Although Coptic began to become obsolescent as a language in the twelfth century, and Coptic Christians made the transition to the Arabic language and script, the zimām numerals continued to be used widely in economic documents otherwise written in Arabic, and for divinatory purposes. Their obscurity lent itself to cryptographic uses; a letter sent in 1139 from the Egyptian Jewish trader Halfōn ben Nethanel to the poet Isaac ibn Ezra uses zimām numerals as a code for Hebrew characters that in turn denote Arabic words (Goitein 1971: II, 303). Their opacity and rarity could also be useful in avoiding fraud, for instance, on a thirteenth-century check in which the amount was written in Arabic words and zimām numerals, presumably because their obscurity made it less likely that they would be altered fraudulently (Reif 2002: 26).6 In the fourteenth-century Muqadimmah of Ibn Khaldūn, he mentions the use of “zimâm ciphers, the same as those used for numerals by government officials and accountants in the contemporary Maghrib” alongside the Arabic abjad numerals and the ghubar positional numerals (Chapter 6) in the zā’irajah technique of numerical divination (1958: I, 239). The zimām numerals played a considerable role in the development of the socalled Fez numerals used in Spain and parts of North Africa from the twelfth century onward (see the following discussion). Both the Greek alphabetic numerals and the Arabic abjad numerals were also involved in the development of the Fez numerals, however, and the paleographic lines of descent among these systems are complex. The problem is made more complex by the fact that the later users of zimām numerals were otherwise writing Arabic texts, not Coptic ones. Moreover, the Fez numerals are first attested in Spain, not in Africa. However, the similarities of some of the sign-forms for particular numerals suggest some relationship. The zimām numerals were used for several centuries, and survived the conquest of Egypt by the Ottoman Empire in 1517. Egyptian Coptic children in the sixteenth century learned Arabic before Coptic in kuttāb schools, and their practical education emphasized their assimilation to Ottoman practices, but they continued to learn and use zimām numerals for bookkeeping and accounting, thereby acquiring a reputation for competent and secure financial practices associated with 6

The use of highly ligatured Roman numerals in late medieval Europe, and of special complex numeral-signs in Chinese banking, are additional examples of the practice of using unusual numerical systems to avoid fraud, and are somewhat paralleled by the modern requirement of writing out numerals lexically in addition to using Western numerals on checks.

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what was by this time a highly cryptic numerical code (Hamilton 2006: 38). They ceased to be used only in the seventeenth century, after which time Arabic positional numerals were used for these functions (Ritter 1936; Messiha 1994: 26).

Ethiopic The Ethiopic script developed in the late third century ad, primarily on the model of the Minaeo-Sabaean consonantary used in South Arabia, but also influenced by the Greek and Coptic alphabets used to the north. It was used (and continues to be used) for writing various languages of Ethiopia, especially Ge’ez, the liturgical language of the Ethiopian church, and modern Amharic. While the earliest Ethiopic script has no signs for vowels, by the fourth century ad, the script was an alphasyllabary, in which each individual sign represents a consonant + vowel. The direction of writing is always left to right. While the Ethiopic script is based on South Arabian writing, its numerical notation is strictly Greek and Coptic in origin. The earliest Ethiopic numerical notation, such as that found on third- and fourth-century inscriptions from the kingdom of Aksum (on which the signs for 70 and 90 are not found) is shown in Table 5.6 (Drewes 1962; Ifrah 1998: 247). The earliest inscription with Ethiopic numerals is on the late third-century Anza stela erected by Bazat, negus (local king) of Agabo, in which the highest number expressed is 20,620 (loaves of bread) (Drewes 1962: 65–67). The modern Ethiopic script uses a system derived from these early inscriptions, structurally unchanged but slightly modified, as shown in Table 5.7 (Fossey 1948: 99; Haile 1996: 574). The Ethiopic numeral-signs are not letters of the Ethiopic script, but rather are derived from the Greek or Coptic letters. Even though the signs have no nonnumerical meaning in Ethiopic, the signs normally have marks both above and below them to indicate that their value is numerical. This practice was universal only from the fifteenth century onward, and it is not found at all in the Aksum inscriptions (Ifrah 1998: 246–247). In addition to these signs, modern Amharic texts use a sign for 1000 ( ), which lacks marks above and below it (Bender et al. 1976: Table I). It may be that the minimal demand for writing very large numbers led to the abandonment of the higher multiplicative formations and the introduction of an indigenous 1000-sign. The system has a hybrid structure: it is ciphered-additive for the units and tens but multiplicative-additive for numbers over 100. For 10,000, a multiplicative sign consisting of two ligatured 100-signs was used, as shown in Table 5.7. Thus, because there are special signs for 100 and 10,000, the Ethiopic system has a mixed base of 10 and 100, with the base-10 formations governed by eighteen ciphered characters for 1 through 9 and 10 through 90 and the hundreds and ten thousands governed by multiplicative power-signs that combine with the ciphered signs.

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Table 5.6. Ethiopic numerals (Aksumite period) 1

2

3

4

5

6

7

8

9

1s 10s 100s 1000s

etc.

10,000s

etc.

20,620 =

This flexibility allowed the Ethiopic system to express very large numbers with a very limited set of numeral-signs. In theory, it could be extended infinitely by adding additional signs for 10,000 as needed, although such large numbers rarely occurred in practice (Guitel 1975: 272–273). The development of the Ethiopic system occurred under cultural contact and Christianization by Egyptian and Syrian missionaries. While some scholars argue that borrowing of the signs must have been from the Greek uncial script (Bender et al. 1976: 124; Ifrah 1998: 246), Haile (1996: 574) raises the possibility that this transmission might have taken place by means of a Coptic intermediary. Egyptian missionaries were active in Ethiopia throughout the fourth century ad, which might tip the scales slightly in favor of a Coptic origin. Saint Frumentius, generally held to be the first major converter of the Aksumites, was a Syrian by birth, trained by Greeks, but his missionary work was based in Alexandria and focused on establishing connections between the Aksumites and the Egyptian Copts. The Table 5.7. Ethiopic numerals (modern) 1

2

3

4

5

6

7

1s 10s 100s 1000s 10,000s 647,035 = 100,000,000 =

(60 + 4) × 10,000 + 70 × 100 + 30 + 5 (10,000 × 10,000)

8

9

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Numerical Notation

South Arabian numerals were already extinct by this time and had no influence on the Ethiopic system. The use of base-100 for the multiplicative component of the system is the Ethiopic system’s most notable feature. No other alphabetic system begins using the multiplicative principle until 1000 or 10,000. This innovation had a clear antecedent in the Greek and Coptic use of multiplication at 10,000, but it eschews the extra nine signs for 100 through 900 needed for both Greek and Coptic numerals. Instead, it employs the entire set of eighteen ciphered signs, then the same set again beside the 100-sign, then the same set beside the 10,000-sign. While the Ethiopic numerals were based on the Greek or Coptic systems, the Ethiopic script was not. Of all the descendants of the Greek alphabetic numerals, the Ethiopic system is the only one to use non-phonetic signs as numeral-signs. While this makes it more cumbersome, since one needs to learn all the script-signs as well as twenty numeral-signs, there is also no impetus to use all the borrowed numeral-signs. In systems that assign numerical values to an ordered series of script-signs, it is natural that one would assign values to all the signs, rather than stopping at some arbitrary point. In this case, however, the Aksumites borrowed the first eighteen symbols of the alphabet and then, rather than adopting another nine signs for 100 through 900, which were meaningless to them, took only the sign for 100 and used the multiplicative principle thereafter, thereby reducing the number of new signs they needed to learn. Many Aksumite inscriptions on coins (indicating regnal years) and stone monuments (indicating cardinal and ordinal quantities of various kinds) contain Ethiopic numerals. After the fall of the kingdom of Aksum and the Islamic conquest, the Ethiopic script was used only rarely, and over time became an esoteric script known only to scholars and Ethiopian Orthodox priests. It was used in the mathematical, numerological, and astronomical texts of medieval Ethiopia, both for writing whole numbers and for writing fractions in a special sexagesimal notation based on Greek traditions (Neugebauer 1979; see the following discussion). In Amharic texts, Ethiopic numerals were used for a wider variety of functions from the fifteenth century to the present day; for instance, they were used in the personal correspondence of Amharic elites in the nineteenth century (Pankhurst 1985). The New Testament printed in Amharic in 1852 uses the Ethiopic numerals throughout for page, chapter, and verse numbers (Novum Testamentum in linguam amharicam 1852). Today, the numerals are still occasionally used for writing dates, but have largely been supplanted by the Western numerals (Bender et al. 1976: 124).

Gothic The Gothic alphabet was developed around 350 ad by Wulfila, a bishop who translated the Bible into his native language. Gothic was an East Germanic language

Alphabetic Systems

155

Table 5.8. Gothic numerals 1

2

a b 10s j k 100s s t 665 = .xoe.

1s

3

4

5

6

7

8

9

c l u

d m v

e n w

f o x

g p y

h q z

i r {

spoken by the Germanic tribes who migrated throughout Europe in the latter years of the Western Roman Empire (Ebbinghaus 1996: 290). The script was alphabetic and written from left to right. Along with the script, an alphabetic numerical notation system was employed, as indicated in Table 5.8 (Braune and Ebbinghaus 1966: 10). Like the Greek alphabetic numerals, the Gothic numerals were usually distinguished from the rest of the text either through dots to either side of the numeralphrase or by placing a horizontal stroke above the phrase (Braune and Ebbinghaus 1966: 10). The Gothic numerals never expressed quantities higher than 1000, and thus there is no evidence of the use of the multiplicative principle. Larger numbers were always written out in full (Menninger 1969: 260). The system is therefore ciphered-additive and decimal throughout. The numeral-signs are related to the Greek uncial letters. Of the episemons, f (6) was the sixth letter of the Gothic alphabet, and had the phonetic value [kw]. The other two episemons, qoppa (r) and san ({), had no phonetic value and were simply used to fill out the full complement of twenty-seven signs using Greek models (Fairbanks and Magoun 1940: 319). The possibility has been considered, but now largely rejected, that the Gothic script owes its ancestry at least in part to either the Latin alphabet or the Germanic runes (Ebbinghaus 1996: 290–291). However, since neither of these scripts uses alphabetic numerals, the Gothic numerals must be of Greek origin. The Gothic alphabet is attested in only a limited set of documents, mostly translations of parts of the New Testament, but also in a small number of secular texts. Most numerals in Gothic texts indicate chapter and verse numbers in Bibles. Additionally, they were used within the text to indicate numerical values, while in Greek Bibles such numbers were always written out in full (Menninger 1969: 260). There is no evidence that the Goths ever did arithmetic or mathematics using these numerals. The sign for 900 is only attested in the Codex Vindobonensis 795, an antiquarian text of the ninth century, by which time the script had already fallen out of use (Fairbanks and Magoun 1940, Ebbinghaus 1978).

156

Numerical Notation

Very little of what surely must have been written in Gothic has survived. Most surviving texts date to the sixth century ad, although the assignment of a fourth- century ad origin to the numerals is undisputed. It is unclear exactly when in the seventh or early eighth century the Gothic language died out, but around that time the script and numerals ceased to be used, and were replaced by the Roman numerals that were coming to be used throughout Western Europe.

Hebrew Alphabetic The earliest Hebrew scripts began to diverge from the earlier Phoenician consonantary in the ninth or tenth century bc. Then as now, Hebrew consisted of twenty-two consonantal signs, written from right to left, and placed in a customary order. Early Hebrew inscriptions used a variant of the Egyptian hieratic numerals (Chapter 2). Somewhat later, particularly in the fifth and fourth centuries bc, some Hebrew speakers used the Aramaic numerals (Chapter 3) for administrative and commercial purposes. Only at a much later date, probably in the second century bc, did a uniquely Hebrew set of numerals develop. This system is indicated in Table 5.9. The first twenty-two signs, indicating 1 through 400, are the letters of the Hasmonean Hebrew script circa 125 bc, at the time of the writing of the Dead Sea Scrolls and also when the numerals were first used (Goerwitz 1996: 488). The system is decimal and ciphered-additive, and written from right to left. Values between 500 and 900 were represented using the sign for 400 in conjunction with one or more signs for the lower hundreds (i.e., 500 = 400 + 100, 600 = 400 + 200 ... 900 = 400 + 400 + 100). This structural irregularity exists because there are too few letters in the Hebrew consonantary to fill out the twenty-seven signs needed to extend the system to 900. The number 400 occupies a special structural role, but is not a base of the system, as its powers (16,000, 6,400,000, etc.) do not receive any special treatment. While the very earliest Hebrew inscriptions contain no signs for numbers above 1000, the need to do so quickly arose, as the numerals began to be used for dating on grave inscriptions using the Hebrew calendar. For multiples of 1000, a mark – either a small curved stroke to the left of a numeral-sign or two dots placed above it – could be used to indicate multiplication by 1000; thus, would signify 9000 and 90,000. This feature is similar to, but distinct from, the Greek alphabetic system, which adds a stroke above or below a numeral to indicate multiplication by 1000, but begins again at 10,000 by placing the multiplicand above the sign Z. Thus, while the Greek system could express any numeral up to ten million, as opposed to one million for the Hebrew numerals, the Hebrew system is arguably easier to use because it has only one value at which the multiplicative principle is employed.

Alphabetic Systems

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Table 5.9. Hebrew alphabetic numerals (Hasmonean) 1

2

3

4

5

6

7

8

9

1s 10s 100s 369 =

The Hebrew alphabetic numerals were derived from the Greek alphabetic numerals between 125 and 100 bc for use on coins inscribed with the Hasmonean script (Gandz 1933: 76; Millard 1995: 192). Theories of a much earlier (ninth- to seventh-century bc) independent origin are no longer credible (Smith and Karpinski 1911: 33; Gandz 1933: 75–76; Schanzlin 1934; Zabilka 1968: 176–178). A bilingual ostracon from the Palestinian village of Khirbet el-Qôm and dating to 277 bc uses both Aramaic and Greek alphabetic numerals to denote a financial transaction (Geraty 1975). Although it is not a Hebrew text, the village was in Judah and hundreds of contemporary Hebrew inscriptions have been found there. If the Hebrew alphabetic numerals had been in use at that time, they likely would have been used instead of the Aramaic numerals in a situation where the Greek inscription a few lines below used a similar system. The first safely dated instance in which the use of Hebrew alphabetic numerals is certain is on coins from the reign of the Hasmonean king Alexander Janneus (103 to 76 bc), some of which were stamped with Greek script and alphabetic numerals, others with the Hebrew script and alphabetic numerals, in both cases using alphabetic signs (Naveh 1968). Avigad (1975) suggests that a clay seal reading “Jonathan high priest Jerusalem M)” might refer to the fortieth (m = 40) year of the Hasmonean kings, and thus dates it to 103 bc, early in the reign of Alexander Janneus, whose Hebrew name was Jonathan. That these early Hebrew alphabetic numerals would both be found in the context of the same man is strong circumstantial evidence. Despite the special formation of Hebrew numerals from 500 to 900, the similarities between the Greek and Hebrew numerical notation systems are striking. The two systems share not only a similar structure (decimal and ciphered-additive) but also the alphabetic principle for forming the numeral-signs. The notion that the Hebrew numerals were independently developed can no longer seriously be sustained, despite the agnostic attitude of some scholars, including Ifrah (1998: 239). That Hasmonean coins were struck in both languages and using both systems provides specific contextual evidence that the Hasmonean kings adopted the technique

158

Numerical Notation

from the Greeks. The cultural influence of the Ptolemaic and Seleucid kingdoms in the Levant at this time was enormous; Greek alphabetic numerals were used on coins from the Phoenician cities of Sidon, Tyre, Byblos, and Akon from the mid third century bc onward (Millard 1995: 193). At the same time, the Hebrew use of 400 as a “stepping-stone” for representing the higher hundreds is an important innovation, as it did not require that Hebrew speakers learn and adopt additional nonphonetic signs. Despite the development of the Hebrew numerals, most Semitic peoples of the ancient Levant (Nabataeans, Aramaeans, Palmyrans) never used alphabetic numerals, but continued to use their own hybrid cumulative-additive/multiplicativeadditive systems (Chapter 3), which coexisted with the Hebrew alphabetic system for several centuries, and were replaced only over a long period. Until the seventh century ad, inscriptions on Jewish graves throughout the Mediterranean region were often written, not with the Hebrew numerals, but rather in the Greek alphabetic numerals (Ifrah 1998: 238–239). Gradually, the Hebrew numerals replaced the Greek and Levantine systems, until they were firmly established as a distinctive system peculiar to the Jewish populations of Europe, North Africa, and the Levant. Also, in the sixth century ad, the Hebrew numerals were partly or wholly used by the creators of the Syriac alphabetic numerals (to be discussed), which are also ciphered-additive and decimal and have the same break at 400 as the Hebrew system. The modern Hebrew alphabetic numerical notation system has the same structure as the ancient system, but uses modern script-signs, as shown in Table 5.10. Around the beginning of the tenth century, five additional Hebrew characters, the signs used for kof, mem, nun, pe, and tsade in word-final position, were sometimes used to complete the sign set for 500 through 900. These signs occur in some Masoretic commentaries on the Old Testament, but were never the regular forms used (Gandz 1933: 96–102). The older formations using additive combinations of hundreds-signs are the usual means of representation. The newer forms probably were not widely accepted because the word-finality of these signs is inconsistent with the principle of the numerical notation system that the highest values should come first in a numeral-phrase (Gandz 1933: 98). To put a word-final letter for 500 through 900 at the head of a numeral-phrase would be inconsistent with its original purpose; to put it at the end of the numeral-phrase would be inconsistent with the rule of decreasing ordering of the powers. Also in the tenth century, Hebrew scholars became aware of positional numerals, and occasionally experimented using combinations of the alphabetic numerals and the positional principle. In a Masoretic poem by Saadia Gaon (ad 882–942), numbers are written quasi-positionally in two columns; the rightmost represents the thousands, while the leftmost column represents the ones, so that 42,377 was

Alphabetic Systems

159

Table 5.10. Hebrew alphabetic numerals (modern) 1

2

3

4

5

6

7

8

9

1s 10s 100s OR

, i.e., (7 + 70 + 300) + ((2 + 40) × 1000) (Gandz 1970: 487–488). written as Some medieval Hebrew writers simply used the first nine alphabetic symbols in place of the ordinary Western or Arabic signs, supplementing them with a circle for zero, which made the system ciphered-positional (Gandz 1933: 110). This practice is first attested in the mid twelfth-century Sefer ha-Mispar of Abraham ibn Ezra (Burnett 2000c: 14). The ordinary Hebrew numerals were always used in the regular text of such works, with the positional variants used only for mathematical expressions (Schub 1932). This technique was not commonly used, however, and later Hebrew mathematicians and astronomers simply used Western or Arabic numerals. One of the more important functions for which the Hebrew numerals have been used historically is gematria, the art of number-magic (Ifrah 1998: 250–256). Because every letter of the Hebrew script has a numerical value, every Hebrew word has a numerical value equal to the sum of its letters’ values. Among medieval and early modern scholars, this practice was commonly employed for interpreting passages from the Talmud and the Midrash and for finding symbolic associations among words that share the same numerical value. For instance, two of the terms associated with the Messiah, shema ‘seed’ and menakhem ‘consoler’, have the same numerical value (8 + 40 + 90 = 40 + 8 +50 + 40 = 138). A related practice is the construction of chronograms, in which a specified set of words has a numerical value equal to the date of an event (e.g., a person’s death) to which the words refer. Because these practices (also used with the Arabic abjad, described below) can only be done with a system for correlating phonetic signs with numerical values, they probably contributed to the retention of the alphabetic numerical notation systems long after ciphered-positional systems had been adopted. While the Western numerals are used in modern Israel for most purposes, the alphabetic system is regularly used for dates using the traditional Jewish calendar, especially in religious texts and on graves. In 1999, the Israeli Supreme Court ruled that gravestones in orthodox Jewish cemeteries could henceforth record numbers using Western numerals rather than the Hebrew alphabetic system, whose use had previously been mandatory in that context (Copans 1999). It is unclear whether this ruling will have any long-term effect on the use of the Hebrew numerals for

160

Numerical Notation

dating. However, the fact that such a ruling needed to be made at all shows the continued health and vibrancy of Hebrew alphabetic numerals, albeit in a limited set of religious contexts. The Hebrew alphabetic numerical notation system is now over two millennia old, and is one of the oldest systems in continuous and regular use.

Syriac Alphabetic In Chapter 3, I described the numerals used alongside the Old Syriac script between about 50 and 500 ad. Around the fifth century ad, this script diverged into two forms: an eastern variety, Nestorian, used by the Christians of Persia, and a western variety, the Serto script used by the Jacobite Christians of Syria (Daniels 1996). This split was precipitated by the expulsion of the Nestorian Christians from the Byzantine city of Edessa and their subsequent migration into the Sassanian Empire (Duval 1881: vii). Soon thereafter, both the Nestorian and Serto scripts began to use an alphabetic numeral system akin to those used elsewhere in the Middle East. The basic signs of this system (sign-forms are those of the Serto script) are shown in Table 5.11. The system is decimal and ciphered-additive, and written from right to left. As in many alphabetic numeral systems, numerals were sometimes distinguished from letters by placing a horizontal stroke above a numeral-phrase, but often no special mark was present. Like the Hebrew alphabetic numerals, values from 500 to 900 were usually written using the signs for the lower hundreds with the sign for 400 in various additive combinations. Alternately, the upper hundreds were occasionally expressed multiplicatively, by placing a small dot above the signs for 50 through 90 to indicate multiplication by 10 (Duval 1881: 15; Noldeke 1904: 316–317). For values above 1000, a slanted stroke placed beneath a unit-sign indicates multiplication by 1000, while a horizontal stroke placed beneath a sign indicates multiplication by 10,000 (Hoffman 1858: 6–7). In this way, any number below ten million could be expressed. Even higher values could be expressed by placing two small strokes beneath a sign to indicate multiplication by ten million, and placing one small stroke above and one small stroke beneath a sign indicated multiplication by ten billion; however, these techniques were used extremely rarely and not followed uniformly (Duval 1881: 15–16). As in several other alphabetic systems, placing an oblique stroke above a numeral-sign indicated the appropriate (1/x) unit fraction (Duval 1881: 16). The Syriac alphabetic numerical notation system was probably invented in the sixth century ad, and gradually replaced the Old Syriac system over the next two hundred years (Duval 1881: 15). Independent invention of this system can be ruled out, given its strong similarity to others used in the region. Two likely possibilities

Alphabetic Systems

161

Table 5.11. Syriac alphabetic numerals

1s 10s 100s

1

2

3

4

5

6

7

8

9

a j s

b k t

c l u

d m v

e n sv w\ /e -e

f o tv x /f -f

g p uv y /g -g

h q vv \z /h -h

i r svv , /i -i

OR

/a 10,000s -a 369 = iou 1000s

/b -b

/c -c

/d -d

are that it was modeled on the Greek alphabetic numerals prevalent in the Byzantine Empire or else on the Hebrew alphabetic numerals. The Hebrew and Syriac numerals are the only two systems in which 400 occupies a special structuring role, in that the higher hundreds are expressed using additive combinations of 400 and the lower hundreds. The ordering of the Syriac numerals follows the letter-order shared by the Syriac and Hebrew scripts, not that of Greek. Finally, the Syriac system was written from right to left like the Hebrew system, but unlike the Greek. On the other hand, the Syrians were closely affiliated with Eastern (Greek) Christianity, and many Syrians lived under Byzantine rule. The hypothesis of Greek ancestry is supported by the shared feature of the two systems that the multiplicative principle was used at two different powers of the base, 1000 and 10,000, whereas the Hebrew numerals did so only at 1000. The inventor(s) of the Syriac numerals probably would have been familiar with both the Greek and Hebrew numerals, so it is possible that features of both systems were blended in the Syriac system. The Syriac scripts were never used as the official script of any polity, and thus Syriac numerals are rarely found on stone monuments or coins. However, their use in religious texts is extremely prevalent from the time of their invention to the present day, a situation afforded by the relative separation of the Jacobites and Nestorians from both Roman and Greek Christianity. Most Syriac liturgical texts are dated and paginated using the numerals, making it easy to examine paleographic changes. There is no evidence that the Syriac numerals were ever used in mathematical texts, nor, in contrast to the Hebrew numerals, were they used for letter-magic or numerology. Although the heyday of the Syriac scripts came and went before ad 1000, both the Nestorian and Serto scripts survive to this day, the former in Iraq, Turkey, and Iran among a small number of Nestorian Christians, the latter in Lebanon

162

Numerical Notation

among the Maronite Christians of that country. Both scripts retain their distinctive numerical notation systems, albeit restricted in the contexts of their use to the same liturgical functions for which they have been employed for nearly 1,500 years. Arabic or Western numerals serve most other purposes. Nevertheless, there is no reason to think that the Syriac numerals are about to disappear, particularly given the special status accorded to Maronite Christianity in Lebanon’s 1990 constitution.

Arabic Abjad In pre-Islamic times, Arabic speakers used a variety of the Nabataean script and numerals (Chapter 3). While the classical Arabic script is directly descended from this ancestor, the Nabataean hybrid cumulative-additive/multiplicative-additive numerals were abandoned in favor of a ciphered-additive system based on the Arabic script-signs. The basic signs of this system are shown in Table 5.12 (Saidan 1996: 332). The system is decimal and ciphered-additive below 1000 and, like the Arabic script, is written from right to left with the signs in descending order. The numeralsigns shown are the unligatured signs of the Arabic consonantary; in numeralphrases, signs were ligatured to one another as appropriate for the letters in question. Often, a horizontal stroke was placed above a numeral-phrase to distinguish it from an ordinary word. Curiously, the signs are not valued according to the normal Arabic letter-order, but rather according to the letter-order of the Hebrew and Syriac scripts, which was also the order used early in the Arabic script’s history. The first three signs of this order (‘alif, ba, jim) gave the system its most common name, h.isa-b abjad.7 Because the Arabic script has twenty-eight basic consonantal signs, the remaining sign, ghayn (;), was assigned the numerical value of 1000. Ghayn was not used as a single unit-sign, but as a multiplier in numeral-phrases by placing another sign before it. In this way, any number up to and including one million could be written. The system is thus multiplicative-additive above 1000. The values assigned to six of the abjad numerals were different among users of the Arabic script in the Maghreb (North Africa and Spain). This ordering developed somewhat later than that used elsewhere, probably in the ninth century ad. Other than the different values assigned to the six signs in Table 5.13, the system is structurally identical to the regular abjad numerals. It is possible that the abjad numerical system is of pre-Islamic origin, and that it spread from the north. However, it is more likely to have originated around 650 ad, at or shortly after the time of the early Islamic conquests in Syria, Egypt, 7

The first three letters in the modern Arabic script are ‘alif, ba, and ta.

Alphabetic Systems

163

Table 5.12. Arabic abjad numerals 1

2

3

4

5

6

7

8

9

b k t

c l u

d m v

e n w

f o x

g p y

h q z

i r :

1000

a j s ;

7642 =

b m x;g (7 × 1000 + 600 + 40 + 2)

1s 10s 100s

and Mesopotamia. Under Byzantine rule, this region used the Greek alphabetic numerals for administrative and commercial functions; furthermore, both the Syriac and Hebrew alphabetic numerals were used in their respective scripts. The abjad numeral-signs have exactly the same order as the corresponding Syriac and Hebrew characters up to 400 (above which point the other two systems are structurally irregular). The Arabic script adopted a different letter-order starting in the eighth century ad, but the older order was retained for the numerals. The Greek system follows an order similar to that of the Syriac and Hebrew numerals up to 80, but diverges thereafter by putting sampi (equivalent to the Hebrew tsade) at the end of the system, rather than in the middle. Thus, either the Hebrew or Syriac systems inspired the development of the Arabic abjad numerals (Guitel 1975: 276–278; Ifrah 1998: 243). Table 5.14 illustrates how the Arabic order is directly parallel to the Hebrew and Syriac, while the Greek numerals diverge from them starting at 90. Table 5.13. Arabic abjad numerals (Eastern vs. Maghreb) Sign

o r u z : ;

Letter-name

Eastern Value

Maghreb Value

sin

60

300

sad

90

60

shin

300

1000

dad

800

90

ẓa

900

800

ghayn

1000

900

Numerical Notation

164

Table 5.14. Arabic, Greek, Hebrew, and Syriac numeral-signs and letters Arabic 60 70 80 90 100 200 300 400 500 600 700 800 900

o p q r s t u v w x y z :

Greek sin ayin fa sad qaf ra shin ta tha kha dhal dad ẓa

X O P < R S T U F C Y W >\

Hebrew

Syriac

xi

samekh

omicron

ayin

pi

pe

qoppa

tsade

rho

quf

sigma

resh

tau

shin

upsilon

tav

phi chi psi omega sampi

o p q r s t u v w x y z ,

semkat ‘e pe sode quf rish shin taw

However, the Arabic abjad was rarely used in the same texts as either the Hebrew or the Syriac numerals. In contrast, Greek alphabetic numerals are found in Arabic documents from the seventh to the ninth centuries ad. In ad 706, Caliph Walid I dictated that, although his Greek financial administrators in Damascus should no longer use the Greek alphabet, they could continue to use the Greek alphabetical numerals (Menninger 1969: 410). In an eighth-century ad Arabic tax record, numbers are expressed in both Greek alphabetic numerals and abjad numerals (Cajori 1928: 29). Because many of the regions conquered by the Arabs – even those such as Syria and Palestine in which the Syriac or Hebrew numerals were found – were under Greek rule, Greek numerals were the normal system used for administration, on coins, and in inscriptions. It is unlikely that the Greek system played no role in the development of the abjad numerals. The phonetic values of the final six Arabic characters do not correspond to the Greek, however. Furthermore, the Greek system has only twenty-seven rather than twenty-eight signs (lacking a sign for 1000). While the Arabs were aware that the Greek system had signs for the higher hundreds, and may thus have attached

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165

numerical values to the remaining signs in their own script, the use of a special sign for 1000 is unique to the Arabic system among all four systems under consideration. More likely, the Arabic system was based on the Semitic letter-order but employed the structural advantages of a system, such as the Greek, with a full complement of numeral-signs. This feature would have been particularly important, since the administrative needs of the new Islamic caliphate were growing rapidly. By the late eighth century ad, the Arabic abjad numerals had spread throughout the Middle East and into the Maghreb, including southern Spain. They were used on administrative documents, in literary and scientific texts, on dated coins, and on monuments, though not for the most part on ephemeral media such as ostraca. In areas in which an existing administrative apparatus was retained from the Byzantine Empire (such as Egypt), the Greek and Coptic alphabetic numerals were used much more frequently on administrative and financial documents than were the abjad numerals (Grohmann 1952: 89). While the systems are structurally similar to the abjad numerals, it would have taken some effort for Arabic writers to learn an entirely new set of twenty-seven signs, so its failure to be used more widely is somewhat surprising. Issues of identity and ethnicity may have played a significant role in determining the scope of their use. For instance, Ifrah describes a ninth-century ad Christian manuscript written in Arabic but in which the verses are numbered in Greek (Ifrah 1998: 243). In this case, the writer was a Christian who associated himself with Greek Christianity through the alphabetic numerals even though he wrote in the Arabic language and script. Abjad numerals were rarely if ever used directly for arithmetical calculation. Instead, a system of finger computing borrowed from the Byzantine Greeks, and known as h.isab al-’uqūd (finger-joint arithmetic), was employed (Saidan 1974: 367–368), as well as computation on dust-covered boards. Despite their use over a wide area, the abjad numerals did not give rise to a large number of descendants. In part, this must be due to the remarkable stability of the Arabic script itself. The Coptic zimām numerals used in Egypt from the tenth century ad onward are an interesting blend of Greek and Arabic influences, although relatively few of the numeral-signs are evidently based on those of the abjad, and it is unclear to what extent the Arabic abjad affected the structure of the system. A similar situation probably arose in Morocco, where “Fez numerals,” incorporating elements of the Arabic abjad as well as Greco-Coptic alphabetic numerals, were used until very recently (Colin 1933). Finally, like Greek and Hebrew scholars, Arabic astronomers used a very unusual system for writing fractions, which combines a ciphered-additive system with a base-60 (sexagesimal) positional notation. Shortly after they were invented, the abjad numerals began to be supplanted by the well-known ciphered-positional Arabic system (Chapter 6) borrowed from

166

Numerical Notation

the Hindus, and known in Arabic as al-h.isāb al-hindī (Indian arithmetic). The Islamic conquest of enormous territories to the east brought the Arabic and Indian spheres of influence into close contact by the mid seventh century. As Islam spread eastward throughout the eighth century ad as far as the Indus River, the Indian style of numeration began to diffuse westward and supplant the Arabic abjad, which itself was still a novelty in western regions such as North Africa. This replacement was hastened by the arrival in 773 ad of Hindu astronomers and astronomical knowledge at the court of Caliph al-Mansur in Baghdad, which formed the basis for the early ninth-century ad writings of the famed Arab mathematician al-Khwārizmī, who popularized the Arabic positional system (Menninger 1969: 410–411). By the late ninth century, the positional system was being used in administrative and financial documents, and by the late tenth century, on inscriptions (Grohmann 1952: 89). The latest Arabic papyrus in which abjad numerals express a date is from 517 ah, or 1123/4 ad (Destombes 1987: 131). Despite the introduction of positional notation, abjad numerals survived in a variety of contexts. Astronomers continued to use ordinary abjad numerals much later than other writers, probably because they had also adapted them for use in the quasi-positional sexagesimal fractions to be described later. Until the thirteenth century all astronomical and astrological texts used abjad numerals alone, and for several centuries thereafter, abjad and positional numerals were often used side by side in any given text (Lemay 1982: 385–386). Abjad numerals were commonly used on most Arabic astrolabes (both for marking gradations on the instrument and for recording dates of construction) until the sixteenth century (Destombes 1987). Mauritanian mathematicians did not adopt Arabic positional numerals until the end of the eighteenth century (Rebstock 1990). That positional numerals would be adopted more rapidly in nontechnical manuscripts than in scientific ones is seemingly paradoxical, but only because it is customarily assumed that positional numerals would be accepted owing to their efficiency for arithmetic. In fact, however, Arabic arithmetic was largely conducted using dust-boards, flat surfaces covered with a layer of fine dust on which calculations could be made using any numerical system, but of course no record of this would survive; in contrast, permanent records were often made in more traditional and authoritative numerical systems like the abjad numerals (Lemay 1982: 383–384). We do not know to what extent pre-modern Arabic astronomers may have performed computations using Arabic positional numerals. The abjad numerals survive to the present for very limited cryptographic, literary, and magical functions, such as chronograms (described earlier for Hebrew numerals), known in Arabic as h.isāb al-djummal (Colin 1971: 468). Chronograms using the abjad numerals were common throughout the Middle Ages, particularly in Persia and Islamized parts of India (Ahmad 1973). Babajanov and Szuppe

Alphabetic Systems

167

(2002) have recently published an unusual Persian cryptographic inscription in Arabic script dating to 1790 from the site of Chār Bakr near Bukhara, Uzbekistan. It is notable not only for its late date, but also because it records the computational work performed in translating the name whose numerical value is being taken (into abjad numerals, then into Arabic figures, then performing arithmetical calculations [using Arabic positional numerals] to double the entire abjad value to give the number associated with the unnamed writer). Finally, the abjad numerals were retained for pagination of prefaces and tables of contents of books, in a manner similar to the Western conventional use of the Roman numerals (Colin 1960: 97). Abjad numerals survived particularly well in the Maghreb, and continue to be used there for chronograms, cryptographic correspondence, and functions related to magic and divination. Several abjad-derived systems were used by nineteenth-century Ottoman administrators for cryptographic purposes (Chapter 10) (Decourdemanche 1899). Similarly, Monteil (1951) describes a text from Mali that discusses many cryptographic systems derived from the abjad numerals that were used in North Africa in the mid twentieth century.8 Most modern Arabic grammars mention the existence of abjad numerals and note the numerical values of the various letters.

Astronomical Fractions In addition to the ciphered-additive decimal systems peculiar to their civilizations, ancient and medieval astronomers from various linguistic backgrounds and script traditions used distinct systems for fractions, combining their ordinary alphabetic numerals with a base of 60 and the positional principle, borrowed from Babylonian astronomy. This ingenious system of astronomical fractions represents a curious digression in the history of numerical notation. From about 2100 bc to 0 ad, Babylonian astronomers and mathematicians used a cumulative-positional numerical notation system with a base of 60 and a sub-base of 10 (see Chapter 7). Numbers smaller than 60 were expressed through cumulative combinations of cuneiform signs for 10 (l) and 1 (k), while the positional principle was used to express multiples of powers of 60 (60, 3600, 216,000, ...). By the third century bc Greeks firmly controlled most of the lands formerly under Babylonian rule, under the potent Seleucid kingdom that came into existence after the Alexandrine conquests. At some point between the third and first centuries bc, the Babylonian positional notation and the sexagesimal base were married to the Greek ciphered-additive numerals and used thereafter by Greek astronomers (Jones 1999: 9). The first Greek 8

Curiously, one of these systems is known as el-Yunani (Ionian!), suggesting that its users were aware of the Greek origin of such notations.

Numerical Notation

168

Table 5.15. Astronomical numerals (Greek) 1

2

a b 10s i k /afie k ie 1s

1515

3

4

5

6

7

8

9

g l

d m

e n

v

z

h

q

+ 20 (× 1/60) + 15 (× 1/3600) = 1515.3375

mathematical astronomer to use sexagesimal notation was Hypsicles in the second century bc (Fowler 1999). The first major text in which this new system appeared was the Syntaxis of Ptolemy, written in the second century ad (Heath 1921: 44–45). In place of the cumulative Babylonian signs in each position, fourteen of the Greek alphabetic numerals were used (the units 1 through 9 and the decades 10 through 50) to write any number from 1 through 59, as shown in Table 5.15. Unlike the Babylonian system, however, the Greek sexagesimal system was never used for expressing integers. Numbers greater than 60 were always written with the ordinary alphabetic numerals. The sexagesimal numerals correspond to Greek astronomy, in which, as today, the circle is divided into 360 degrees, each degree into 60 minutes, each minute into 60 seconds, and so on. Thus, in Theon of Alexandria’s (fourth-century ad) commentary on Ptolemy’s Syntaxis, the numeral-phrase /afie k ie expresses 1515 (/afie) degrees, 20 (k) minutes, and 15 (ie) seconds (Thomas 1962: 50–51). The degrees value uses the decimal alphabetic numerals,9 while the latter two positions are sexagesimal. Sexagesimal fractions could be used to express any fractional value; successive positions represent 1/60, 1/602, 1/603, and so on, to express as small a value as desired. Within each sexagesimal position, decimal ciphered-additive numeral-phrases represent the value from 1 to 59. However, the primary base of the system – the one involved in the positional aspect of the system – is 60; thus, 10 (the base of the ordinary alphabetic numerals) becomes the sub-base of the sexagesimal fractions. The system’s base and sub-base thus follow the Babylonian system, but the intraexponential principle used in forming the signs of each position is ciphered rather than cumulative. This system is thus both ciphered and positional, but it is not identical to ordinary ciphered-positional systems such as the Western numerals. Rather, because it has a sub-base, it is intraexponentially cipheredadditive rather than simply ciphered. It is thus a (ciphered-additive)-positional system. 9

If this value were expressed sexagesimally, we would expect it to be written as ke ie, or (25 × 60) + 15.

Alphabetic Systems

169

The astronomical numerals used a special sign as a placeholder to indicate an empty position in order to avoid ambiguity. In some late (fourth- to first-century bc) Babylonian texts, a placeholder sign that served some of the functions of zero was used (see Chapter 7). Greek astronomers from the first century ad onward took up this practice, originally using a sign (, which in later manuscripts was written as ) (Irani 1955; Jones 1999: 61–62). The latter sign is sometimes held to represent omicron, the first letter of the Greek word ouden, ‘nothing’, with a stroke added above to distinguish it from the appropriate letter (Ifrah 1998: 549). Yet it is unlikely that the Greeks would have chosen a sign that already had a numerical value (o\= omicron = 70) in the alphabetic system (Neugebauer 1957: 14; Jones 1999: 61). The zero-sign was probably a paleographic outgrowth of the earlier form, which was, like the Babylonian placeholder sign, purely ideographic. As far as can be discerned, however, it is completely unrelated to the later Hindu zero, which emerged in the sixth century ad in India. The principle of sexagesimal fractions could, in theory, be combined with any alphabetic numeral system, replacing the Greek signs for 1 through 59 with those of any other system. The Arabs, who inherited the bulk of Greek astronomical knowledge when they took control of Mesopotamia in the mid seventh century ad, used the signs of the Arabic abjad rather than the Greek alphabetic numeral-signs. Unlike the abjad numerals, the Arabic sexagesimal fractions are written from left to right, following the Greek practice, and the Arabic symbol for zero (), or later () is derived from the Greek symbol (Irani 1955). Similarly, medieval Hebrew astronomers adopted sexagesimal fractions, as in the fourteenth-century astronomical writings of Levi Ben Gerson (Ifrah 1998: 158). From the Arabic translations of Greek astronomical texts, sexagesimal positional numeration reached the Byzantine Greeks and, by ad 1000, Western Europe (Neugebauer 1960, Berggren 2002). Medieval European astronomers used sexagesimal notation for fractions thereafter, and in the Alphonsine tables edited in Paris in 1327, even whole numbers were written sexagesimally, the only case I know of where this was done (Berggren 2002: 364). Medieval Ethiopian astronomers used sexagesimal fractions in their own computations and in translations of Greek and Arabic documents (Neugebauer 1979). The use of the Greek variant waned after the end of the Byzantine Empire in the mid fifteenth century, after which Western numerals were used for most purposes. The Arabic version of the system survived even longer; Irani (1955: 3) lists many texts from the sixteenth and seventeenth centuries and one from as late as 1788, although I suspect that this latter text is deliberately archaic. Astronomical numerals often occur in the same texts as pure decimal systems (either ciphered-additive ones such as alphabetic numerals, or cipheredpositional ones such as Arabic or Western positional numerals). In effect, sexagesimal notation is not so much a distinct numerical notation system as it is a technique

170

Numerical Notation

that can be used with any numerical notation system as a means of combining numeral-phrases for 1 through 59 into a positional notation system for fractions. The sexagesimal fractions were used only by astronomers and mathematicians working with astronomical problems and writing in manuscripts. Even so, when writing non-astronomical material, or when paginating and dating astronomical texts, they used other numerical notation systems. Yet, on astronomical instruments and in astronomical manuscripts, sexagesimal notation alongside alphabetic numerals was employed continuously for well over a millennium. That sexagesimal notation was used solely for astronomy suggests that the demands of the discipline led to its retention. The division of the circle into 360 degrees (with subdivisions of 60 minutes per degree and 60 seconds per minute) is very useful, since 60 has a large number of divisors.10 Faddegon (1932) showed that this feature enables quick and easy multiplication and division using sexagesimal fractions. The utility of sexagesimal fractions must therefore be evaluated in relation to computations involving this specific metrological system. The tenth-century Persian mathematician al-Biruni reported, however, that because it was inconvenient to multiply using a 60 × 60 multiplication table, Islamic astronomers would convert sexagesimal fractions into decimal numbers, multiply them, and then convert them back into sexagesimal notation (Berggren 2002: 362). Although sexagesimal fractions are no longer used, modern astronomers still use the sexagesimal division of the circle, and anyone who can read a digital clock uses a kind of sexagesimal numeration. While we no longer mix additive and positional principles in notating time and angles, astronomers continue to restrict themselves to values under 60 for the division of the sky into segments, just as everyone is able to realize that thirty minutes pass between 1:50 and 2:20. These vestiges come to us, via Greek and Arabic sexagesimal fractions, from the Babylonian custom of numbering by 60. In this way, a peculiar custom of numeration, useful for astronomy but not much else, influenced how humans perceive and structure time throughout most of the world today. However, these do not represent a sexagesimal numerical notation system, but simply a sexagesimal division of various metrological units (for angles and for time) that are then represented with decimal numerals. The notation “11:05” does not mean 665 minutes (11 × 60 + 5), but simply 11 hours and 5 minutes. That we continue to measure time and angles in this way is a fascinating issue in the history of astronomy and of timekeeping.

10

In number theory, 60 is a “highly composite” number, defined as a natural number that has more divisors than all the numbers below it (Wells 1986: 127–128). The number 60 has twelve divisors: {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}.

Alphabetic Systems

171

Table 5.16. Fez numerals 1

a A 10s j J 100s s 766 = foy 1s

2

3

4

5

6

b B c d D e f k l L m n N o t T u v V w W x X

7

8

9

g G h H i I p q Q r y z , <

Fez Numerals In the western extremity of the Muslim world, first briefly in medieval Spain, then around the city of Fez in modern Morocco starting in about the sixteenth century, an alphabetic numeral system was used, distinct from both the Arabic and Greek systems. This system was known as ḥisāb al-qalam al-Fāsī (‘Fez signs’) or rūmī signs (‘Roman’, the name given to the Byzantine Greeks) (Guergour 1997: 68). The numeral-signs, including paleographic variants where appropriate, are indicated in Table 5.16 (Colin 1933: 199–201).11 The system is decimal and ciphered-additive, and is written from right to left with the highest values on the right. The twenty-seven signs are thus sufficient to express any number smaller than 1000. The resemblance between the signs for 6 and 7 and the modern Western numerals is likely a coincidence. For higher numbers, a stroke placed to the left of or beneath any of the twenty-seven signs indicates that its value should be multiplied by 1000. In some documents, two strokes placed underneath a figure indicated multiplication by one million (1000 × 1000) (Guergour 1997: 69). The Fez system is thus a hybrid multiplicative-additive system for values above 1000. At least in later periods, there were signs for specific fractions, and for specific monetary values (Sanchez Perez 1935). The earliest version of the Fez numerals was used among the Mozarabs (Arabic Christians) of Toledo, Spain, in the twelfth and thirteenth centuries (Colin 1933: 204). Levi Della Vida’s (1934) study of these documents, which includes a table of these numeral-signs, confirms that they are essentially identical to the later ones except that they are written from left to right. The question then arises how these numerals reached Spain in the twelfth century. Three possible ancestors – the Arabic abjad (Maghreb variant), the Greek minuscule alphabetic numerals, and the Coptic zimām numerals – are depicted alongside the Fez numerals in Table 5.17.

11

See also Sanchez Perez (1935) and Guergour (1997) for different paleographic variants of this system.

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172

Table 5.17. Arabic abjad, Greek, zimām, and Fez numerals Arabic Greek Zimām Fez Numerals 1 2 3 4 5 6 7 8 9 10 20 30 40 50

a b c d e f g h i j k l m n

A b\ G D E V z\ E Q I k l m n

a A b B c d D e f g G h H i I j J k l L m n N

Arabic Greek Zimām Fez Numerals 60 70 80 90 100 200 300 400 500 600 700 800 900

r p q z s t o v w x y : ;

x o p < r s t u f c y w >\

o p q Q r s t T u v V w W x X y z , <

All four systems are written cursively and have an enormous amount of variation. It is possible that the Mozarabs’ numerals are paleographic variants of the Greek alphabetic numerals, and thus came to the Arabs of Spain via direct diffusion from the Byzantines (Levi Della Vida 1934: 283). The attribution of these signs as rūmī (Byzantine) would tend to confirm this origin. However, many of the numeral-signs bear no resemblance to the Greek alphabet. Colin suggests, rather, that the Fez numerals (and their Spanish antecedent) were borrowed, not directly from the Greek alphabetic numerals used in the Byzantine Empire, but by way of the Egypto-Coptic zimām numerals (Colin 1933: 213). The paleographic similarities between several of the Coptic and Fez numeral-signs (e.g., 8, 80, and 500) suggest that some connection must exist between the two. Yet while the Fez numerals are multiplicative at only one level (1000), both of these candidates for its origin are multiplicative at both 1000 and 10,000. An alternate ancestor is the Arabic abjad system used for number-magic and astronomy in the Maghreb at that time. Some of the paleographic resemblances between the Fez numerals and the zimām numerals (e.g., the signs for 7, 30, and 90) can be explained by both systems’ connection to the appropriate letters of the Arabic abjad. Furthermore, unlike the other two

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173

systems, the Arabic abjad is multiplicative only in combination with 1000. Most likely, the Fez numerals are an unusual blend of the Greek, zimām, and Arabic alphabetic systems adopted among a very unusual group of users, highly educated Arabized Christians living in Muslim-dominated southern Spain. The Fez numerals did not last long in Spain; I know of no texts from the fourteenth century or later in which they are used. They were described by the Moroccan mathematician Ibn al-Banna in the early fourteenth century, and later by the great historian Ibn Khaldun in the late fourteenth century, both of whom lived and worked at Fez (Colin 1933: 206; Guergour 1997: 68). Their use in Morocco began in earnest only in the sixteenth century, however, after the expulsion of the Moors from Spain in 1492. They were used frequently throughout the sixteenth and seventeenth centuries in accounting and other commercial contexts, after which time they began to be replaced by the Arabic positional numerals. While they were known to scholars, their use was always mercantile and legal, never mathematical or astronomical (Guergour 1997: 74). The Fez numerals were still used in the early twentieth century for indicating monetary values in wills and in related legal documents (Colin 1933, Sanchez Perez 1935). Because the meaning of the numerals was known to only a few learned notaries by this time, the system had become a cryptographic notation to prevent fraudulent modifications and forgeries, and generally to restrict access to information (Colin 1933: 195). Political changes in post-colonial Morocco have ended this system’s use.

Armenian Before the introduction of Christianity, there was no native Armenian script, and the Babylonian, Greek, and Old Persian scripts were used for literary purposes. The Armenian adoption of Christianity in the early fourth century ad was followed by enormous influence from the Greek-speaking world. In the early fifth century ad (probably in 406 or 407), the Armenian scholar-monk Mesrop Mashtots (c. 360–440) developed the first uniquely Armenian script, an alphabet of thirtysix letters, in order to translate the Bible from Greek into Armenian (Sanjian 1996: 356).12 At the same time, the letters of the alphabet were assigned numerical values as shown in Table 5.18. These signs are the erkat’agir ‘iron-forged letters’ preferred from the fifth through thirteenth centuries ad, and still used for epigraphic inscriptions (Thomson 1989; Sanjian 1996: 357). In the tenth century, cursive letters known as bolorgir began to be used, and are the standard forms used in modern Armenian writing. The Armenian system is ciphered-additive and decimal, and is written from left to right. 12

The modern Armenian script has thirty-eight letters, the last two of which (o and fé) were introduced in the medieval period and have no numerical value.

Numerical Notation

174

Table 5.18. Armenian erkat’agir numerals 1

A 10s J 100s S 1000s 2 346 = UMF

1s

2

3

4

5

6

7

8

9

B K T 3

C L U 4

D M V 5

E N W 6

F O X 7

G P Y 8

H Q Z 9

I R 1 0

Because the ancient Armenian alphabet had thirty-six letters, it had enough signs to express the complete series from 1000 to 9000 as well as all the units, tens, and hundreds. The system could thus denote any number less than 10,000. However, unlike many ciphered-additive alphabetic systems, the Armenian system does not use multiplication to express higher values, which were written in full using lexical numerals. Very little epigraphic or paleographic evidence survives from the earliest centuries of the system’s use. The Armenian numerals were probably developed on the model of the Greek alphabetic numerals, just as the Armenian script itself was derived from the Greek. Many other scripts have been suggested as possible ancestors of the Armenian script, based on resemblances in the shapes of certain characters (Gamkrelidze 1994: 37), while there are few resemblances to the Greek alphabet. However, of these likely ancestors, only the Greek alphabet used appropriate alphabetic numerals. Thus, regardless of the origins of the script-signs, the principle of alphabetic numeration was certainly borrowed from Greece. It is unclear whether the Armenian alphabetic numerals were developed by Mesrop Mashtots himself (or his assistants) in the early fifth century ad, or whether they were produced later in the century. Figure 5.2 is a monumental grave inscription from the temple of Garni east of Yerevan, which commemorates a ninth-century Armenian Catholicos, also named Mashtots; the first three signs are numerals (300 + 40 + 6), indicating his death-year to be 346 according to the Armenian calendar, or 897 ad. Although a connection is sometimes asserted to exist between the Armenian and Georgian alphabetic numerals, the evidence for this is too tenuous to suggest any definite link. The primary similarities between the two are that they were used in the same region and had distinct signs for 1000 through 9000. The only system that is derived from the Armenian alphabetic numerals is the variant Armenian system developed in the seventh century ad by Anania Shirakatsi. The Armenian numerals did not spread beyond the limited area around Lake Van where Armenian was spoken, nor do they appear to have inspired the creation of any foreign

Alphabetic Systems

175

Figure 5.2. Grave inscription of the Armenian Catholicos Mashtots (897 ad); the first three visible signs of the inscription are 300 + 40 + 6, the year of his death. Courtesy Gabriel Kepeklian.

systems. After the development of the minuscule Armenian script, these signs were also used numerically in the same manner. Ciphered-positional numerals – the Arabic system used by the neighboring Seljuk Turks – were first used in Armenia in the twelfth century (Shaw 1938–39: 368). Yet Armenian writers retained the alphabetic numerals for most ordinary purposes long afterward. Only in the mid seventeenth century, when Armenia had been firmly under Ottoman control for some time, did ciphered-positional numerals (Arabic, then later Western) replace the alphabetic system. Wingate (1930) has published an unusual, undated, and unsolved “magic square” arithmetical puzzle using both Arabic positional and Armenian alphabetic numerals, part of a scroll contained within a Armenian “family amulet” designed to be worn upon the person. The Armenian alphabetic system is still sometimes used for numbering chapters of the New Testament, although page and verse numbers are most often written using Western numerals. Otherwise, the numerals used in modern Armenia are the standard Western numerals.

Shirakatsi’s Notation The Armenian astronomer, geographer, and mathematician Anania Shirakatsi13 was born ca. ad 595–600 and was most likely a monk in the Armenian Church (Hewsen 1968: 34). While little-known today outside his native country, Shirakatsi’s contribution to Armenian learning is unparalleled, particularly his synthesis 13

Also known as Ananiah Shiragooni, or Ananias of Shirak.

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176

Table 5.19. Armenian numerals: Shirakatsi’s notation

1s 10

1

2

3

4

5

6

7

8

9

A j

B

C

D s

E

F

G 2

H

I

100

1000

of Persian, Arabic, Greek, and other scientific knowledge. In addition to these accomplishments, Shirakatsi developed a very interesting numerical notation system in his “Book of Arithmetic” (T’uabant’iwn), which is a collection of arithmetical tables designed for the instruction of pupils. The basic form of this system uses twelve signs, as shown in Table 5.19 (Shaw 1938–39: 270). The individual signs are identical to those used for the appropriate numbers in the traditional Armenian system. However, Shirakatsi showed how these signs could be combined to express numbers through multiplication as well as addition. In this system, a unit-sign followed by one of the three power-signs (for 10, 100, or 1000) indicates that the values of the two should be multiplied; these pairs of signs were combined into numeral-phrases through addition. Instead of writing 9642 as 0XMB (9000 + 600 + 40 + 2), as in the traditional Armenian alphabetic numerals, Shirakatsi would write the same number as I2FSDjB (9 × 1000 + 6 × 100 + 4 × 10 + 2). Thus, where the traditional Armenian system is ciphered-additive, Shirakatsi’s system is multiplicative-additive. Any numeral-phrase can be written more compactly with the traditional alphabetic numerals than with Shirakatsi’s variant – so why would Shirakatsi advocate its use? Firstly, it requires knowing fewer symbols (twelve versus thirty-six) in order to express any number less than 10,000. More importantly, numbers greater than 10,000 could be expressed using multiplicative combinations of two or three signs. To do so, however, one needs the entire repertoire of Armenian numerals from 1 through 9000, as described earlier. For numbers from 10,000 through 90,000, Shirakatsi juxtaposed the signs for 10 through 90 with the sign for 1000. Similarly, the numeral-phrases for 100,000 through 900,000 combined the signs for 100 through 900 with the sign for 1000. Alternatively, the hundred thousands could be expressed using unit-signs followed by a 100-sign and then a 1000-sign. Thus, one could write 460,000 as VO2 – (400 + 60) × 1000 – or DSO2 – ((4 × 100) + 60) × 1000. This system is no longer a purely decimal system, but has a mixed base of 10 and 1000. For values below 1000, it is purely multiplicativeadditive, but above 1000, the multiplicand that is juxtaposed with the sign for 1000 (2) is not a single sign, but rather a ciphered-additive numeral-phrase. In the “Book of Arithmetic,” numbers up to the ten millions are expressed relatively compactly (Abgarian 1962: 46; Hewsen 1968: 42).

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177

Shaw (1938–39: 369) believes that this system was not developed by Shirakatsi in the seventh century, but was a commonly used variant system, of which Shirakatsi’s writings are the only surviving remnant. I do not believe there is any reason to regard the system as anything other than the creation of Shirakatsi himself, since its structure is never found in Greek, Syriac, Hebrew, or any other alphabetic system. Shirakatsi may have borrowed the notion of multiplicative structuring from one of two foreign sources. The numerals developed by the fifth century ad Indian mathematician Âryabhata (Chapter 6) were multiplicative-additive; it is possible that Shirakatsi, a mathematician with extensive knowledge of foreign writers, knew of Âryabhata’s numerals and emulated them. Similarly, it is vaguely plausible that Shirakatsi knew of the classical Chinese multiplicative-additive numerical notation system (Chapter 8). Neither hypothesis has any direct evidentiary support. Shirakatsi’s system is thus a structurally innovative local variant of the Armenian numerals designed to facilitate the representation of large numbers of the sort that would be needed for his astronomical and mathematical calculations. There is no evidence that his system was adopted by any later writers, or that it had any effect on the development of other numerical notation systems throughout the world. Instead, we should view this system as the creative invention of a single individual, used only within his lifetime.

Georgian Like the Armenians, the Georgians developed a script and numerical notation system modeled after the Greek alphabet shortly after they converted to Christianity. While the creation of this first Georgian alphabet is often attributed in folklore to King Parnavaz in the third century bc, there is no direct evidence of Georgian writing until the fifth century ad, at which time the asomtavruli or majuscule script began to be used (Holisky 1996). More familiar to modern scholars, however, are the mxedruli characters developed in the eleventh century ad, which continue to be used to write the Georgian language today. The numerals associated with this script are shown in Table 5.20 (Holisky 1996: 366). The system is decimal and ciphered-additive and, like the Georgian script, is written from left to right. Like the Armenian script, the Georgian script had enough letters to serve for all numerical values up to 9000. Some later inscriptions even include a special sign for 10,000 (-). There is no evidence that the Georgian alphabetic numerals were ever used to express larger numbers than this, either through multiplication or through additional signs. Presumably, such numbers were always written out in full using lexical numerals. There is an undeniable structural similarity between the Georgian and Armenian systems, which both, unlike the Greek alphabetic numerals, have enough

Numerical Notation

178

Table 5.20. Georgian numerals 1

a 10s j 100s s 1000s 2 4808 = 5zh 1s

2

3

4

5

6

7

8

9

b k t 3

c l u 4

d m v 5

e n w 6

f o x 7

g p y 8

h q z 9

i r 1 0

additional letters to represent the values from 1000 through 9000. However, while the Georgian and Armenian scripts both use thirty-six signs for 1 through 9000, the letter-order of the two scripts is vastly different. The Georgian letter-order was modeled very closely on the Greek, with additional signs added as necessary at the end of the series, while the uniquely Armenian signs in that script were interspersed randomly within the original Greek letter-order. It is unlikely that the Georgian numerals would be modeled on the Armenian numerals but retain the Greek letter-order for their values (Gamkrelidze 1994: 77). There may have been some mutual influence between the two numerical notation systems, given certain similarities in the sign-forms, but the direction of this influence remains unclear (Gamkrelidze 1994: 81–82). For now, the hypothesis of direct diffusion from the Greek alphabetic numerals, while using distinctly Georgian signs and using distinct signs for the thousands, is most strongly supported. The Georgian numerals were used in literary and religious texts throughout the medieval period, particularly for pagination, dating, and stichometry, as well as in monumental inscriptions. Their regular use ended in the sixteenth century, when Georgia came under Ottoman control, after which Arabic positional numerals were used for administrative and commercial purposes, although the alphabetic numerals may have been retained for religious functions. However, Paolini and Irbach’s 1629 Georgian-Italian dictionary, the first book printed in Georgian, does not contain any mention of the alphabetic numerals alongside its list of Georgian letters. Since the eighteenth century, when Georgia fell under the Russian sphere of influence, the Western numerals have been those normally used for all purposes in written Georgian.

Glagolitic The Glagolitic script was probably developed between 860 and 870 by the brothers Cyril and Methodius, who, while on a mission to the Moravian Slavs of what is today modern Serbia, Croatia, and Macedonia, created an alphabet for liturgical

Alphabetic Systems

179

Table 5.21. Glagolitic numerals 1

a j 100s s 708 = yh

1s

10s

2

3

4

5

6

7

8

9

b k t

c l u

d m v

e n W

f o x

g p y

h q z

i r /

writings in the language now known as Old Church Slavonic (Schenker 1996: 166–167). There may have been a pre-Christian script in the region, which might explain why many of the Glagolitic letters have no correlation with the Greek alphabet, but no pre-Christian numerical notation existed (Cubberley 1996). The numeral-signs of the Glagolitic numerical notation system are shown in Table 5.21 (Vaillant 1948, Gardiner 1984). As with the Greek and many other systems, Glagolitic numerals were frequently distinguished from words in texts by placing dots to either side of a numeralphrase or by placing a mark of some sort above it (Vaillant 1948: 24; Schenker 1996: 182). In addition to these twenty-seven signs, additional signs for 1000, 1, and 2000, 2, were purportedly used in some texts. The system is ciphered-additive and decimal, and is always written from left to right. However, for the numbers 11 through 19, the ordinary sign order is reversed (e.g., bj instead of jb for 12), which reflects the Slavic lexical numerals for the teens (Schenker 1996: 182). Yet none of the surviving Glagolitic manuscripts apparently use numerals higher than 1000 (Gardiner 1984: 15; Lunt 2001: 28). Schenker (1996: 182) contends that the Glagolitic thousands were expressed by placing a small diagonal or curved stroke (like the Greek hasta) to the left of a numeral-sign to indicate that its value should be multiplied by 1000; if so, Glagolitic is a hybrid multiplicative-additive system above 1000. Gamkrelidze (1994: 39–40) and others contend that, because the earliest Glagolitic script had thirty-six characters, the last nine letters of the alphabet (most of which were later dropped from the script) originally had the values 1000 through 9000. The issue remains unresolved and contentious. The Greek alphabetic numerals were the sole and direct ancestor of the Glagolitic numerals. The similarities in structure between the Greek and Glagolitic systems are considerable, and Cyril, Methodius, and their followers were Greeks. Among other possible ancestors, the Gothic numerals were long defunct by the ninth century ad, and the Cyrillic numerals were not invented until later in the century. Nickel’s (1973) suggestion that the Glagolitic numerals may have originated from the tamgas (clan identifiers) used by Turks and Iranians in southern Russia as early as the first century is problematic. While a variety of scripts, such as the Latin,

180

Numerical Notation

Greek, Samaritan, and Hebrew, may have been used as the model for one or more of the letters of the alphabet, many other signs have no obvious correlates in other scripts (Schenker 1996: 168–172). Regardless, the Glagolitic letters must have been assigned their numerical values under the influence of Greek Christianity. Manuscripts were written in Glagolitic throughout the medieval period in the region of modern Croatia, Serbia, Slovakia, and even into the Czech Republic and Poland. Yet, even during the Middle Ages, Catholic or Western-influenced areas began to prefer the Roman numerals to the Glagolitic, while areas under Bulgarian or Serbian control tended to adopt the Cyrillic numerals and script. By the fifteenth century, almost all the Slavs had adopted either Roman or Cyrillic numerals. Only in Croatia, particularly along the Adriatic coast (Dalmatia), where they were retained for the Croatian Roman Catholic liturgy (Cubberley 1996: 350), did the Glagolitic script and numerals flourish. They were also used in monumental inscriptions in Croatia from the eleventh century onward, a context not seen elsewhere. Yet, even in Croatia, the Glagolitic script and numerals declined greatly in use after the Ottoman conquests of the sixteenth century, and were used only rarely from the seventeenth century onward (mostly in religious texts). It is not clear whether the Glagolitic numerals survived as long as the Glagolitic script, which persisted until the beginning of the twentieth century in the Quarner archipelago in northwestern Croatia.

Cyrillic Like Glagolitic, the Cyrillic script was developed under the guidance of the missionaries Cyril and Methodius. It is quite likely that Cyrillic was developed in ad 890– 900, after the deaths of Cyril and Methodius, by Cyril’s followers and disciples in Bulgaria, who then named the script after their deceased mentor. Cyrillic was originally used for writing the Old Church Slavonic language, but later was adopted for writing a variety of Slavic languages, most notably Russian. An alphabetic numerical notation system14 was developed around the same time. The Cyrillic numeral-signs are shown in Table 5.22 (Gardiner 1984: 16–17; Cubberley 1996: 348). The system is ciphered-additive and decimal, and is normally written from left to right. For the numbers 11 through 19, the ordinary sign order was often reversed (e.g., bj instead of jb for 12), which reflects the structure of Slavic lexical numerals (Vaillant 1948: 24). Numeral-phrases were often distinguished from ordinary letters by placing a bar or other mark (titlo) above the phrase, and sometimes also by 14

While some scholars call this system “Slavonic,” I use the term “Cyrillic” to prevent confusion with the Glagolitic system.

Alphabetic Systems

181

Table 5.22. Cyrillic numerals 1

A 10s J 100s S 708 = yh

1s

2

3

4

5

6

7

8

9

b k t

c l u

d m /

e n w

f o x

g p y

h q z

i r 0

placing dots on either side of the signs (Lunt 2001: 28). Placing a small stroke to the left of a number indicated that its value should be multiplied by 1000 (Vaillant 1948: 24; Schenker 1996: 182). The Cyrillic numerical notation system is thus a hybrid: purely ciphered-additive below 1000 and multiplicative-additive for higher powers. In some cases, higher Cyrillic numerals were expressed by using the signs preceded by an unusual sign, [, to indicate multiplication by 1000 (Gardiner 1984: 15). Apparently in some cases the multiplier-stroke could be repeated two or three times to indicate multiples of 1,000,000 and 1,000,000,000, respectively (Berdnikov and Lapko 1999: 16). In the earliest phase of the script’s history, there were a number of ideograms for powers of 10 from 10,000 up to 1,000,000,000, but these rare notations are poorly studied, and the range of their use is unknown (Berdnikov and Lapko 1999: 17). While there are only twenty-seven signs listed in Table 5.22, there are more than twenty-seven signs in all varieties of the Cyrillic script; modern Russian Cyrillic uses thirty-two letters, and earlier Cyrillic scripts used a number of older signs that have now fallen into disuse. The signs that are assigned numerical values in Cyrillic are those that are directly derived from Greek, including the otherwise rarely used signs for xi (o), psi (y), and theta (i). Yet numerical values were never assigned to the commonly used but non-Greek characters (Gardiner 1984: 14–15). Thus, the Cyrillic numerical values do not correspond to the customary order of letters, remaining faithful to the Greek order instead. The Cyrillic script and numerals originated around 890 ad, at which time Slavs and Greeks who had been influenced by Cyril and Methodius were extremely active in the Christianization of the Slavs in the region of modern Bulgaria and Serbia. That this missionary work was undertaken under the auspices of the Byzantine Empire confirms what is clear from the paleographic evidence – that the sole external influence on the Cyrillic script and numerals was the Greek uncial alphabet used at the time. The non-Greek signs for additional consonantal Slavic phonemes were never assigned numerical values, further confirming the Greek origin of the Cyrillic numerical notation system.

182

Numerical Notation

The Cyrillic numerical notation system spread to Kievan Rus in the tenth century. The earliest printed books in Cyrillic script, those printed in Kraków from 1491 onward, were paginated using Cyrillic rather than Roman or Western numerals (Zimmer et al. 1983). The Balkans fell under Ottoman influence in the fifteenth century, after the fall of Constantinople, and the alphabetic numerals largely ceased to be used there by around 1500. In Russia, the Cyrillic numerals were used much longer. The late sixteenth-century English Slavist, Christopher Borough, authored a Russian copy of the pseudo-Aristotelian Secret of Secrets (Bodleian MS Laud misc. 45 (SC 500)), which was paginated in Western numerals but used Cyrillic numerals and Arabic positional numerals elsewhere in the text (Pennington 1967: 681–682). Western numerals were known in Russia in the seventeenth century; a seventeenth-century sundial from Mangazeia (in Siberia, near the Arctic coast) is numbered using Western numerals, showing that they were known even in the Russian hinterland (Ryan 1991: 375). Yet the popular Schitanie udobnoe ‘ready reckoner’ arithmetic text of 1682, intended for merchants, contained only Cyrillic numerals (Okenfuss 1973: 329). For most purposes, the schety or Russian beadabacus, which had been used in Moscow since the eleventh century, was perfectly adequate for the computational needs of pre-Petrine Russia (Ryan 1991: 373–374; Simonov 1993). Not until the reforms of Peter the Great around 1700, and the introduction of technical training in mathematics by scholars from Western Europe, were the Western positional numerals introduced into Russia on a widespread basis (Hans 1959–60, Fedosov 1995). Magnitskii’s Arifmetika of 1703 used both systems side by side (Ryan 1991: 373). In the same year, logarithm tables published for the use of students at the Moscow School of Mathematics and Navigation were printed in Western numerals only. In 1710, however, Peter deferred to the clergy in decreeing that church texts were to be printed using the traditional Cyrillic numerals (Cracraft 2003: 103). Throughout the eighteenth century, aristocrats and literate officials would have needed to be familiar with Roman, Western, and Cyrillic numerals in order to read the full range of textual genres used in Russia (Billington 1968: 209). As late as 1918, Tsaritsa Alexandra (Alix of Hesse) was learning Cyrillic numerals and using them to paginate her final diary, demonstrating that their use was still relevant, if increasingly formal, in the late tsarist period (Kozlov and Khrustalev 1997: 2–3). Many texts were paginated using the older system even as Western numerals increasingly were employed for a wider variety of functions. Unlike the transition in Western Europe, where the Western numerals took centuries to be adopted fully, the change from the Cyrillic to the Western numerals took place relatively smoothly and rapidly, however. Today, the Cyrillic numerals are occasionally used in modern Church Slavonic texts (especially for numbering chapters and verses in Bibles), but never in ordinary Cyrillic writing (Gasparov 2001: 17–18).

Alphabetic Systems

183

Latin From the fifth through the twelfth centuries, knowledge of the Greek exact sciences in Western Europe was relatively limited. Starting in the eleventh century, as Arabic translations of Greek astronomical works began to be retranslated into Latin, Western European scholars became increasingly aware of the Greek numerals. In a handful of texts, an attempt was made to convert the Greek alphabetic numerals or Arabic abjad numerals for use with the Latin alphabet, as shown in Table 5.23 (cf. Lemay 2000, Burnett 2000c). The system is ciphered-additive and written from left to right, and simply employs twenty-two letters of the Latin alphabet as it existed at the time to represent 1 through 9, 10 through 90, and 100 through 400 (e.g., xlh = 328). Additionally, in the Dresden Almagest of 1121, z, θ, and ϕ represent 500 through 700 (Burnett 2000c: 61). There is, however, no known way to write 800 and 900, and there is no known multiplicative technique to express numbers higher than 1000. In one of the texts containing such an alphabetic system (MS Cambrai Bibliothèque Municipale 930, a copy of Hermann of Carinthia’s Astronomia), even the sign for 400 is not used (Lemay 2000: 378–379). Yet, because the texts containing the Latin alphabetical numerals are astronomical treatises, in which numbers higher than the 360 degrees of the circle are rarely needed, this was not a serious weakness of the system, which was perfectly adequate for such values. While minuscule letters were used most of the time, in one text, majuscule letters for K, M, and N, and occasionally for G, R, Q, and L were used, in order to prevent confusion with similar-appearing Western numerals (Burnett 2000a: 82). Burnett (2000c: 61–62) briefly discusses a few tenth- to twelfth-century astrolabes and manuscripts with Latin letters serving as numerals, but these are direct transliterations of Arabic or Hebrew numerals that follow the Semitic alphabetic order rather than the Latin one. The earliest documents containing the Latin alphabetic numerals in Table 5.23 date to 1121 and 1127 and were copied in Antioch in the Crusader States, probably in association with the work of Stephen of Pisa, an early twelfth-century translator of Arabic scientific texts. These contain no key, suggesting that the system would already have been understood by its intended audience (Burnett 2000a: 76). The copy of Hermann of Carinthia’s Astronomia similarly contains no key, and simply uses Latin alphabetic numerals to annotate astronomical diagrams (Lemay 2000). A later manuscript (MS London, British Library, Harley 5402), dated 1160 and written in a mixture of Latin and Italian, does, however, have a key for translating Roman numerals into the Latin alphabetic system (Burnett 2000a: 76). All of these manuscripts use Roman numerals in the text and for nonastronomical purposes such as column labels, and some of them use positional numerals as well. The Liber Mamonis uses Roman numerals

Numerical Notation

184

Table 5.23. Latin alphabetic numerals

1s

1 a

2 b

3 c

4 d

5 e

6 f

7 g

8 h

9 i

10s

k

l

m

n

o

100s

t

u

x

y

z

p

q

r

s

θ

ϕ

for single digits and some low compounds, Latin alphabetic numerals for many two- and three-digit compound numbers, and represents larger numbers using Arabic positional numerals (not Western numerals) (Burnett 2000c: 64–65). The Latin alphabetic numerals represent a brief and abortive attempt to adapt the Greek and Arabic systems to the Latin alphabet for the sole purpose of translating astronomical documents efficiently. They did not, however, lead to a consistent or long-lived tradition of Latin alphabetic numeration. By the time the Latin alphabetic system was developed, ciphered-positional numeration was already being widely adopted by Western European mathematicians and scholars, rendering the alphabetic system obsolete. I know of no thirteenth-century or later documents that use the system described here. However, two late thirteenthcentury manuscripts probably written in Flanders use the letters A through I in place of 1 through 9, in conjunction with 0, in a ciphered-positional alphabetic system written from right to left (e.g., B0G = 702), parallel to but probably independent from ibn Ezra’s use of Hebrew alphabetic numerals in the same way, discussed earlier (Burnett 2000b: 200; 2000c: 62–63). In some medieval Latin texts from Western Europe, a set of unusual lettersymbols were associated with numerical values in a sporadic and nonsystematic way, of which one example is shown in Table 5.24 (Cappelli 1901: 413–421). In some respects, this notation resembles the ciphered-additive Latin notation. These letter-symbols occur in very different contexts, however – in the same texts as Roman numerals rather than in translations of Greek or Arabic astronomical texts. The only letters that were not assigned values in this system were I/J, U/V, X, L, C, D, and M, for obvious reasons – these already had numerical values in the Roman numeral system. Like the corresponding Roman numerals, placing a bar above any of these abbreviations indicated multiplication by 1000; e.g., Ō = 11,000. Unlike the Latin alphabetic numerals or the Roman numerals, however, these abbreviations could not be combined to form numeral-phrases – that is, one Table 5.24. Medieval Latin numeral abbreviations A B E F G H K N O P Q R S T Y Z 500 300 250 40 400 200 151 90 11 400 500 80 70 160 150 2000

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could not write HN for 290 and be understood. Some numbers are represented twice (G and P both equal 400); many multiples of powers of 10 (30, 60, 600) are not represented; and strange numbers such as 151 are assigned letters. In other manuscripts (e.g., the tenth-century De inventione litterarum in Strasbourg, Bibliothèque Nationale et Universitaire MS. 326), the numerical associations of the letters are completely different than the ones in Table 5.24 (Derolez 1954: 332–335). In fact, this form of notation is not related to either the Greek or Latin alphabetic numerals, but instead is a complex but unstructured quasi-numerical abbreviatory system.

Summary Alphabetic numerical notation systems originated with the Greeks in the sixth century bc, who combined the structure of the Egyptian demotic system with the idea of using phonetic signs as numeral-signs. The political and ecclesiastical authority of Greek speakers, coupled with the brevity and adaptability of ciphered-additive numeration, led to the development of other alphabetic systems using numeral-signs specific to each script. This phylogeny expanded tremendously between the fourth and seventh centuries ad (the time of greatest Eastern Roman/ Byzantine power), with eight new systems arising during this period. Yet most systems of this phylogeny had died out, or had at least been greatly reduced in the contexts of their use, by the sixteenth century ad, during which time the Arabic and Western positional numerals replaced them. Many alphabetic systems are still used today, but only in limited contexts such as liturgical texts, numbered lists, and divinatory magic. There is no one feature common to all the alphabetic systems. Most are cipheredadditive and decimal, with or without the use of multiplicative structuring for the higher powers. However, the Armenian notation of Shirakatsi is multiplicativeadditive and sometimes uses a base of 1000, while the astronomical fractions are positional and involve a sexagesimal base. The number of signs used, the degree to which multiplication is used, and the correspondence of numeral-signs with script-signs are all variable. One of the great advantages of alphabetic systems is that, if the signs are ordered using a local script, one need not learn both a set of script-signs and a set of numeral-signs; one merely superimposes the decimal structure of the numerals onto the script, thereby lessening the mnemonic burden on both new learners and experienced users. The Greek alphabetic, Coptic, Gothic, Hebrew, Syriac, Armenian, Georgian, Glagolitic, and Latin systems all take advantage of this feature. However, the values assigned to Arabic and Cyrillic letters do not correspond to the customary letter-order, thus reducing this benefit. The Fez numerals and the

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zimām numerals are blended alphabetic systems, combining the numeral-signs of two or more existing alphabetic systems to create a new system that does not correspond with the sign-order of any script. Finally, in the Ethiopic system, the users of one script adopted the ordered numeral-signs of another (in this case, the Greek alphabet) rather than adopting both the script and numerals. Despite the advantage of combining phonetic and numerical representation systems, alphabetic numerical notation systems require many more signs than the cumulative-additive systems of Chapters 2 through 4. Even the Ethiopic system, which is multiplicative above 100, uses nineteen separate signs, more than any cumulative-additive system, and the Armenian and Georgian systems require thirty-six signs. Decimal ciphered-additive systems require nine signs for each power: twenty-seven signs to express all numbers up to 1000. In the case of the Hebrew and Syriac systems, whose scripts had only twenty-two signs, numerals above 400 were expressed through cumulative combinations of hundred-signs. While this solves the problem of having only twenty-two phonetic signs, it makes numeral-phrases longer and more complex. As it is inconvenient to develop nine new signs for each higher power of 10, many alphabetic systems are ciphered-additive for lower powers but use multiplicative-additive structuring above some specific point, a feature originally borrowed by the Greeks from the Egyptian demotic numerals. The Gothic, Armenian, Georgian, and Latin systems do not use multiplication at all, and express only numbers below 1000 (Gothic) or 10,000 (Armenian and Georgian). The Ethiopic system is multiplicative above 100, a feature that can exist only because the signs of the system are not Ethiopic script-signs. A large plurality of systems – Cyrillic, Hebrew, Fez numerals, Coptic, Arabic abjad, and possibly Glagolitic – use multiplication above 1000, a natural way to proceed in systems with twentyseven ordinary signs. Three other systems – the Greek, the Syriac, and zimām numerals – are multiplicative at both 1000 and 10,000; that is, after 8000 (8 × 1000) and 9000 (9 × 1000), one uses a new sign for 10,000 (1 × 10,000) rather than (10 × 1000) as in systems that are only multiplicative at 1000. Interestingly, only the obscure Armenian system developed by Shirakatsi takes the step of making the entire system multiplicative-additive. Because multiplicative-additive numeral-phrases are usually longer than ciphered-additive ones, doing so may not have been appealing. While there were occasional efforts to “positionalize” Greek, Hebrew, and Latin alphabetic numerals by simply using the first nine letters along with a zero, these attempts were sporadic and short-lived. The longevity of the alphabetic systems is remarkable. Seven alphabetic numerical notation systems were regularly used for a millennium or longer (Greek, Coptic, Ethiopic, Arabic, Hebrew, Syriac, Armenian, and Georgian). In cases such as the Greek system, this can be explained by the political importance of the

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system’s users, but in others, such as Hebrew and Armenian, the systems’ users have largely been politically and culturally marginalized. That these systems could survive in such circumstances and where, in many cases, ciphered-positional systems were available, requires explanation. Structural features may partly explain it: alphabetic systems’ “alphabeticity” means that one need not learn a set of numerals in addition to a script, and ciphered-additive systems are more concise than ciphered-positional systems – for any Western numeral-phrase containing zeroes, the corresponding ciphered-additive numeral-phrase will be shorter. A more satisfying explanation, however, is that alphabetic numerical notation systems, like scripts, can be important markers of cultural identity. In many cases (e.g., Coptic, Gothic, Armenian, Georgian, Glagolitic, and Cyrillic), a group of people developed a unique set of alphabetic numerals and developed their own script at the same time. The point of alphabetic numerals is not to be comprehensible translinguistically, but rather for each system to serve for one script alone. Under these circumstances, an alphabetic numeral system becomes an integral part of a script, and thus marks ethnic identity. Even when these systems ceased to be used regularly, many of them continued to be used in restricted functions, particularly in the domain of religion (e.g., Hebrew, Syriac, Coptic, Greek, Cyrillic). Of course, it must not be forgotten that ultimately, in the face of massive globalization over the past five centuries, Western and Arabic positional numerals have become the earliest and standard systems learned and used by almost everyone. The future of alphabetic numerals seems likely to be one of increasing vestigiality and obsolescence.

chapter 6

South Asian Systems

The South Asian numerical notation systems include all systems that derive from the Brāhmī numerals used on the Indian subcontinent, including Western numerals. With the possible exception of China, South Asian numerical notation systems are predominant throughout the entire world today. While most of the modern South Asian systems are ciphered-positional, the earlier systems were cipheredadditive or multiplicative-additive. An important evolutionary development was the shift from ciphered-additive systems, such as the early Brāhmī numerals, to ciphered-positional systems. Despite attempts to postulate the origin of the important ciphered-positional structure elsewhere (Greece, China, or Mesopotamia), this development came out of South Asia.

Brāhmī The Brāhmī script came to prominence in the mid third century bc, during the reign of the Mauryan emperor Aśoka, although inscribed potsherds from the site of Anuradhapura in Sri Lanka have been dated (controversially) to 400 bc (Coningham et al. 1996). Brāhmī script was probably derived from a Semitic prototype (Aramaic, South Semitic, or Phoenician), although many South Asian scholars still support the theory that the script was indigenously developed (Salomon 1996: 378–379). Along with the slightly earlier Kharoṣṭhī script used in the northwestern regions of India, it was the first script used in India after the collapse of the 188

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Table 6.1. Brāhmī numerals

1s 10s 100s 1000s

1

2

3

4

5

6

7

8

9

A J S 1

B K T 2

C L U 3

D M V 4

e N w 5

F O

G P

H Q

I R

Harappan civilization.1 Both scripts are alphasyllabaries (scripts in which each sign has a consonantal base that is modified to indicate which vowel sound is associated with it), but structural differences between the two suggest that their origins are different. This is supported by the fact that the Kharoṣṭhī numerals (Chapter 3) are a hybrid cumulative-additive/multiplicative-additive system very much like the Aramaic system, while the Brāhmī numerals are quite different in principle. The basic Brāhmī numerals are shown in Table 6.1 (Bühler 1896: Plate IX; Datta and Singh 1962 [1935]: Tables III-IX; Salomon 1998: 58). The signs shown in Table 6.1 are those found on Kṣatrapa coins (second to fourth century ad); however, there are enormous paleographic variations among Brāhmī inscriptions. Numeral-phrases are written from left to right, proceeding from higher to lower powers; thus, 289 might be written as TQI. The signs for 1 through 3 are cumulative, with horizontal strokes indicating units, but otherwise, the Brāhmī system is ciphered-additive up to 100, using separate signs for each of the units and the tens. In later inscriptions, even the cumulative signs become ligatured or distorted; for instance, the seventh/eighth-century ad grants of the Gangâ dynasty contain + and = for 2 and 3 (Datta and Singh 1962 [1935]: Table V]. Above 100, the system’s structure becomes more complex. The numeral-signs in Table 6.1 are not, as they appear, simple multiplicative formations that juxtapose a unit-sign with a power-sign for either the hundreds or the thousands; otherwise, one would expect 3 for 2000 rather than 2. Instead, these numeral-signs for 100 through 300 and 1000 through 3000 are quasi-multiplicative. Still, the graphic similarity of the various signs for the hundreds and thousands to the corresponding units is significant. On an inscription from Nana Ghat (first century bc), the numbers 400, 700, 1000, 4000, 6000, 10,000, and 20,000 are written in a straightforward multiplicative fashion as @, #, $, %, ^, &, and *, and thus combine unit-signs with signs for 100 (!) and 1000 ($) (Indraji 1876, 1877). Slightly different but equally straightforward signs are found on slightly later inscriptions 1

Most Western epigraphers and archaeologists accept that, aside from the Harappan script, there was no pre-Mauryan writing in India (Salomon 1995, 1998).

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Figure 6.1. Rubbing of an inscription from Nāsik, Maharashtra state, dating to 120 ad, bearing a variety of Brahmi numerals. Source: Senart 1905–6: 82.

from Nāsik, as in Figure 6.1, dated 42 Śaka (ad 120) (Senart 1905–6: 82; Sircar 1964: 164–167). The Vâkâtaka grants (fifth century ad), one of the latest texts containing signs for the thousands, denote 8000 as x, a ligature of the signs for 1000 (y) and 8 (z) that occur on the same grants (Datta and Singh 1962 [1935]: Tables IV, IX). Despite paleographic changes, the basic structure of the Brāhmī numerals was always ciphered-additive below 100 and multiplicative-additive at both 100 and 1000. No special sign for 10,000 was used; 10,000 and 20,000 are written as 1000 × 10 and 1000 × 20 rather than 10,000 and 10,000 × 2. In the Nana Ghat inscriptions, 24,400 is written as *%@ (1000 × 20 + 1000 × 4 + 100 × 4). The Brāhmī numerals appear on some of the earliest Aśokan inscriptions, dating to the middle of the third century bc, but not in the early Sri Lankan writings. These early inscriptions contain only a few signs (for 1, 2, 4, 6, 50, and 200), but already the hybrid cumulative-additive/multiplicative-additive structure of the system was in place.2 Most of the signs are recognizably ancestral to later ones, such as the more complete sets of numerals found at Nana Ghat and at Nāsik Cave. While there is no paleographic evidence of Brāhmī numerals prior to 300 bc, Datta and Singh (1962 [1935]: 37) claim that, because the Aśokan 2

This opinion contradicts that of Guitel (1975: 605), who sees the Aśoka numerals as being non-multiplicative on the basis that the sign for 200 does not sufficiently resemble a ligatured multiplicative 100 × 2.

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inscriptions are found all over India, the Brāhmī system must have been developed much earlier than the paleographic evidence would indicate, perhaps between 1000 and 600 bc. This is a spurious use of the discredited “age-area” method (determining the age of features by their geographical distribution). In the early Mauryan Empire, an enormous region was quickly encapsulated within a single polity, so it is unsurprising that Mauryan administrative inscriptions are widely distributed. While it was certainly plausible for nineteenth-century Indologists to hope to find earlier paleographic evidence for the numerals, such hopes now seem remote. I agree with Salomon (1996, 1998) and many other Indologists that a midthird-century origin for the Brāhmī numerals and script is probable. The question of the ultimate origin of the Brāhmī numerals – specifically, whether or not they constitute a case of independent invention, and if not, on which ancestor(s) they were modeled – is unresolved, and is made more complex by the politicization of the matter. Previous scholars have emphasized the paleographic comparison of individual signs. I believe that the consideration of the system’s structural features and historical context of origin – supplemented by paleography, where appropriate – will be a more fruitful approach. One set of theories regarding the origin of the Brāhmī numerals derives them from existing representational systems used in South Asia. Borrowing from the letters of the Brāhmī script to create an alphabetic numeral-system, while once a popular theory, is not really sustainable (Prinsep 1838, Woepcke 1863, Indraji 1876, Datta and Singh 1962 [1935], Gokhale 1966, Verma 1971). While a few Brāhmī numeral-signs resemble phonetic signs, if one accepts certain paleographic transformations, these correspond neither to the standard Brāhmī letter-order nor acrophonically to any language’s lexical numerals. Renou and Filliozat (1953) note that in texts containing both purported “letter-numerals” and the corresponding signs used phonetically, the two varieties are quite different. The derivation of the Brāhmī numerals from the Kharoṣṭhī letters is even more improbable, and has not been seriously proposed for over a century (Bayley 1882). Ifrah (1998) proposes but discards the theory that the Brāhmī numerals derive from the Kharoṣṭhī numerals, which can easily be dismissed by noting the temporal priority of the former. Finally, a more recent set of theories derives the Brāhmī numerals from those of the Indus Valley civilization (Sen 1971, Kak 1994), but there are no examples of any writing from India between the latest Harappan inscriptions (around 1700 bc) and the first Brāhmī inscriptions (around 250 bc), and only limited and conflicting evidence for the nature of the Harappan numerical notation system (Parpola 1994; cf. Chapter 10). If not derived from any South Asian system, the Brāhmī numerals could have developed independently. Woodruff (1994 [1909]: 53–60) speculated that both the Chinese and Brāhmī numerals derived from a hypothetical ancient set of cumulative tally signs for 1 through 9, which would then have spread to both China and

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India. Kaye (1919) argued that the Brāhmī numerals developed independently during Aśoka’s time, with their structural features representing three different stages of development, but inexplicably then argues against Indian creativity. Ifrah (1998: 390), arguing that there are “universal constants caused by the fundamental rules of history and paleography,” postulates nonattested cumulative signs for 1 through 9 which later became abbreviated and ligatured into the Brāhmī system. Salomon’s position is more sensibly agnostic; recognizing the problems involved with many other theories, he simply notes that numerical signs are sometimes “cursive reductions of collocations of counting strokes,” citing the hieratic and demotic systems as examples (Salomon 1998: 60). Finally, a number of theories argue for a foreign origin of the Brāhmī numerals. Falk (1993: 175–176), noting structural and paleographic resemblances between the Brāhmī and the earliest Chinese (Chapter 8) numerical notations, argues for a Chinese origin. However, there is little evidence of contact between the two regions at this period, and the only paleographic similarity between the systems is the common use of horizontal strokes for 1, 2, and 3. It has occasionally been proposed that the Greek alphabetic numerals inspired the Brāhmī numerals, given their appearance following the Alexandrine period, the strong trade ties with the Greco-Iranian kingdoms of Parthia and Bactria, and the structural similarities between the two systems. However, the evidence for the “alphabeticity” of the Brāhmī numerals is weak at best (see the previous discussion), and there is no paleographic correspondence between the Greek and Brāhmī numerals. It is most plausible that the Brāhmī numerals are derived from the Egyptian hieratic or demotic numerals. Burnell (1968 [1874]) argued for a demotic origin, while Bühler’s (1963 [1895]) much more prominent analysis argued for a hieratic origin. The three systems are structurally similar: they are all decimal, hybrid cipheredadditive/multiplicative-additive systems, and represent 200, 300, 2000, and 3000 by adding quasi-multiplicative strokes to the signs for 100 or 1000. There are resemblances in around one-third of the sign-forms, and very close resemblances for a few, such as 9 (hieratic = i; demotic = i; early Brāhmī = i) (Bühler 1963 [1895]: 115–119; Salomon 1995, 1998). While there was not tremendous Egypto-Indic cultural contact, Ptolemaic traders reached as far as the city of Muziris (modern Cranganore) on the Malabar Coast, and Aśoka is known to have sent Buddhist missionaries to Alexandria (Basham 1980: 187). Of the two Egyptian systems, I believe the demotic to be a more likely ancestor, because in the Ptolemaic period the use of hieratic numerals was very limited. Thus, although the demotic and Brāhmī systems differ in both the power at which multiplication is used and the direction of writing, I believe that a demotic origin should be adopted as a working hypothesis. The Brāhmī numerals spread throughout the Indian subcontinent during the Mauryan period. Only in the northwest, where Kharoṣṭhī numerals predominated,

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did the Brāhmī numerals fail to penetrate until around the fourth century ad. They were used primarily for writing dates on stone inscriptions and copper land grants. Thus, a full set of numeral-signs up to at least 1000 is attested, and the numeral-signs can be assigned exact dates. Other functions for which Brāhmī numerals were used include stichometry and the recording of financial transactions. While it is interesting to speculate on the use of Brāhmī numerals on other materials than stone and copper, the Indian climate and geography are unsuitable for the survival of perishable materials. In Central Asia, manuscripts in the Tocharian language were written using a variant of Brāhmī script and numerals from the sixth to ninth centuries, and in this drier climate, Brāhmī numerals are attested to have been used on wood tablets, palm leaves, and paper (Sander 1968). There is no surviving evidence that the Brāhmī numerals were used for arithmetic or accounting; the rather substantial body of medieval Indian mathematical works, sometimes attributed to very early dates, use either lexical numerals only, or employ one of the alphasyllabic notations to be described here. After the Kharoṣṭhī script died out in the fourth century ad, variants of the Brāhmī numerals were the only ones used in India until the late sixth or early seventh century. They spread not only throughout the Indian subcontinent, but also into Central and Southeast Asia, regions that were heavily influenced by India during this period. In some Central Asian manuscripts, numeral-phrases were written from top to bottom rather than from left to right (Renou and Filliozat 1953: 702). There was enormous variation in the shapes of the numeral-signs from location to location, suggesting that, as with the Indian scripts, no pressure existed toward the formation of interregional standards. The primary regional division, between northern and southern systems, began as early as the second century ad, and these two basic variants diverged further in later centuries. The end of the traditional Brāhmī numerals and the later local additive variants was a gradual process, instigated not by external influences but by the invention of ciphered-positional notation beginning in the late sixth or early seventh century ad. Over the next couple of centuries, the older ciphered-additive forms became increasingly rare, and by the ninth century ad, the Brāhmī numerals had been replaced by the modern ciphered-positional system throughout India and Southeast Asia. Only on the southern tip of India and in Sri Lanka were additive systems retained (though in an altered form) until significantly later.

From Addition to Position No numerical notation systems have been as widely studied and discussed as the ciphered-positional systems that originated in India, primarily because, through the intermediary of the Arabs, these systems are ancestral to Western numerals.

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While it would be teleological to portray the history of numerals in a linear fashion leading to our own system, the present near-universality of the cipheredpositional, decimal structure originating in India and spreading westward throughout the Islamic and Western spheres requires explanation. Around ad 600, a change began in the writing of dates in the Brāhmīderived scripts of India and Southeast Asia. Instead of writing smaller numbers with ciphered-additive notation and larger numbers with multiplicative-additive notation, all numbers were written using paleographic variations of the nine Brāhmī numeral-signs and a dot to indicate zero – a purely ciphered-positional system. The spread of the older additive systems throughout South and Southeast Asia between the third century bc and the seventh century ad was followed by a second wave of diffusion of the positional principle and zero (seventh century ad onward), wherein the additive systems changed into positional ones. The change is actually a very simple one. The unit-signs were retained, but the power-signs were replaced with a single sign for zero. This process is confirmed by comparing the signs for 1 through 9 in the nonpositional (Brāhmī-derived) systems with the very similar unit-signs used with a zero in later periods. This was not the first zero; the Babylonians (Chapter 7) and the Maya (Chapter 9) had already invented the positional principle and zero-signs well before this time. Some scholars have claimed that the Babylonian zero diffused eastward to India just as it diffused westward to Greece (Février 1948: 585; Menninger 1969: 398–9). Yet the Babylonian and Maya systems were both cumulative-positional, and used a sub-base in addition to a base (Babylonian 10 and 60, Maya 5 and 20). Similarly, while Greek and Arab astronomers used positional fractions with a zero, these had a sexagesimal (Babylonian-derived) base and never used positional notation for whole numbers. Thus, there is no apparent historical relation between these systems and the later Indian one. The Indian positional system first combined ciphering, a zero, and a single, decimal base. It is frequently claimed that the earliest example of ciphered-positional numerals is found on the Sankheda or Mankani copper plate bearing the date 346 in the Kalacuri era, which translates to 595 ad (Bühler 1896: 78; Smith and Karpinski 1911: 46; Das 1927a: 118; Kak 1990: 199). This plate is a donation charter of Dadda III, used to certify a land grant. When discussing any land grant, the issue of a later forgery always arises, as attempts to claim land by producing such evidence were common in India (as elsewhere). Since much of the paleographic evidence for early positional numerals comes from such land grants, we must be cautious to avoid dating inscriptions simply by the date as inscribed, and also take paleography and historical context into account. We must also remember that texts containing positional notation that are transcriptions or translations of earlier works must not be assigned an early date based simply on their putative earlier authors.

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While we need not go so far as Kaye (1919: 346), who claimed that all positional numerals in India prior to the ninth century ad were forgeries, most Indologists are very wary of the Sankheda plate (Salomon 1998: 61). The fact that it is ninety years older than any other positional numeral-phrase suggests that we need to question carefully any sixth-century ad evidence for ciphered-positional numerals in India. The earliest surviving and unquestionable examples of ciphered-positional numerals with a zero derive, not from India itself, but from Southeast Asia, in Khmer, Old Malay, and Cham inscriptions from the late seventh century ad. A calendrical inscription found at Sambor (in Cambodia) and written in a mixture of Old Khmer and Sanskrit is dated 605 in the Śaka dating system, or 683 ad; the zero appears as a small dot (Coedès 1931: 327). As the Sambor inscription is a calendrical passage rather than a land grant or financial document, it is unlikely to be a forgery (Diller 1996: 126). Similar inscriptions from the Old Malay kingdom of Sriwijaya have been found at Palembang and at Kotakapur on the nearby island of Bangka dating to 683, 684, and 686 ad, or 605, 606, and 608 Śaka, respectively; in these cases, the zero was written as a circle rather than as a closed dot (Diller 1995). It is intriguing that the Old Khmer and Old Malay inscriptions appear in the same year (Diller 1995: 66). Nevertheless, there is no reason to believe, as Kaye (1907) did, that the existence of these inscriptions must mean that ciphered-positional numerals actually originated in Southeast Asia and diffused from there to India. The existence of intermediate additive-positional forms from the sixth and early seventh centuries ad, which I will discuss later, coupled with the probability that some of the earlier copper grant plates are authentic, make it likely that the invention of ciphered-positional numerals occurred around ad 600. However, there is compelling evidence for something akin to the positional principle and zero in certain earlier literary and religious texts. In texts using the bhutasa khya or word-numeral system, special cryptic words for one through nine could be combined in a sort of positional fashion with a word for zero in order to represent dates lexically in a fashion quite different from that of ordinary Sanskrit number-words (Datta and Singh 1962 [1935]: 53–63; Mukherjee 1977). The earliest attested text to use this system is the Yavanajātaka of Sphujidhava (ad 269), a Sanskrit version of a Greek astronomical text (Yano 2006: 15). In it, the words ‘moon’ and ‘earth’ mean one, ‘eye’ and ‘twin’ two, and so on, with ‘sky’ and ‘dot’ meaning ‘zero’, suggesting a sort of positionality of words if not of graphemes, while also allowing multiple words to represent the number, a key to producing numeral-phrases that are unambiguous and yet flexible enough to fit into strict Sanskrit verse (Plofker 2007: 395). The bhutasa khya system is thus suggestive of positionality but does not constitute a system of graphic numerical signs, nor should its use be taken to imply the widespread use of decimal positional numerals

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in Indian manuscripts. Notably, the regular name for Indian numeral symbols, anka ‘mark’, is also a numerical word for ‘nine’ in this system, which suggests that there were originally only nine numerical graphemes – that is, that the zero, and place-value, were latecomers to Indian numerical notation even though they existed conceptually in the word-numerals (Clark 1929: 229–230). The earliest Sanksrit word for the zero-sign, śûnya-bindu (literally, ‘void-dot’), is first used in Subhandu’s Vasavadatta, written around 600 ad (Sen 1971: 175; Salomon 1998: 63). This evidence suggests a correspondence between the early use of numeral-words and the structurally identical later use of the ten numeral-signs. Thus, the literary concept of a “zero-space” in Hindu thought, the use of chronograms, and the term śûnya-bindu in the fifth and sixth centuries ad may have prefigured the eventual development of ciphered-positional numerals. Within the interlinked tradition of Indian religious and mathematical thought, the invention of the zero is as much a metaphysical concept as it is an arithmetical tool, if not more so. Almost all the attested early ciphered-positional numerals are nonarithmetical, and are simply used to register dates and other numbers on inscriptions and copper plates. It would be erroneous to assume that the positional numerals originally had an arithmetical function, and then to use this assumption to hypothesize an ancient mathematical tradition of ciphered-positional numerals. There is no evidence that true ciphered-positional numerals were used prior to the middle of the sixth century ad, and any claim prior to the middle of the seventh century ad requires careful examination. The Indian climate and topography are not particularly suitable for the survival of materials other than stone and metal, and we certainly do not have as much evidence as we would like. Nevertheless, there is plenty of inscriptional evidence for continued use of the old additive numerals from the sixth through the eighth centuries ad, they decline significantly in frequency only in the ninth century. It is unlikely that all the evidence for ciphered-positional numerals was lost where so much survives for the additive system – though not impossible, if positional numerals were used only on perishable media at first. Further evidence for the chronology of the shift comes from inscriptions dating from the late sixth to the middle of the eighth century ad from the Orissa region, which are written with unusual mixed structures combining the features of the older additive and newer positional notations (Datta and Singh 1962 [1935]; Acharya 1993; Salomon 1998: 62–63). The earliest of these is from the Urlam copper plates of the Eastern Gangâ king Hastivarman, dated to 578 ad, in which the Gangâ era year 80 is written as the additive sign for 80 followed by a zero, but this date is questionable (Salomon 1998: 62). Acharya (1993) describes many Orissan inscriptions dating from 635 to 690 ad in which dates such as 137 are written as 100 3 7 rather than 137. This series of dates leads directly into the first fully

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197

positional date found in India, on the Siddhantam grant of Devendravarman (195 Gangâ = 693 ad), just ten years after the Southeast Asian examples mentioned earlier (Salomon 1998: 62).3 Datta and Singh (1962 [1935]: 52) mention additional eighth-century ad examples combining additive and positional notation, which they characterize as representing the gradual forgetting of the older system. Their argument rests on the claim that these are quite late examples of additive notation and that the positional principle was well established by this time. In fact, it is far more likely that they represent incomplete attempts to incorporate the novel positional principle into inscriptions. These mixed numeral-phrases confirm the hypothesis of an early seventh-century origin of positional numerical notation in India. In the eighth century, the positional system gained significant ground, and was preferred by 800 ad. Around this time, the spread of scientific knowledge from India to China (primarily through the medium of Buddhist scholarship) led to awareness of the ciphered-positional numerals in China. In the Kaiyuan period (713 to 741 ad), the Indian astronomer Qutan Xida (Gautama Siddhartha) translated an Indian calendar into Chinese, using positional numerals (with a dot for zero), and commented on their ease of use (Gupta 1983: 24). In the ninth century ad, the additive system became much scarcer. Salomon (1998: 62) notes a striking late example from the Ahar stone inscription in north central India, which is a composite record of documents of different dates; those up to 865 ad are all dated using additive numerals, and those from 867 ad using positional numerals, providing precise information on the date of replacement. The plate of Vināyakapalā (931 ad) is an extremely late northern (Nagari) inscription containing the additive system (Singh 1991: 170). By the tenth century ad, only the far south of India (Tamil and Malayalam-speaking areas) consistently used additive systems, but even there, the old system was replaced by a purely multiplicative-additive structure. Of all the descendants of the Brāhmī system, only the Sinhalese numerical notation system preserved the old structure until comparatively recently.

Modern South Asian By the end of the ninth century ad, the transformation of Brāhmī numerals into modern ciphered-positional forms with a zero was complete. The structural evolution of the Indian systems ended at this point, although paleographic developments in the numeral-signs continued. It is well beyond the scope of this work 3

I am uncertain what to make of Mukherjee’s (1993) assertion that the copper-plate inscription of Devakhadga expresses the date 73 in the Harsha era (starting 606 ad) using positional numerals, which would thus be dated to 678 ad.

Numerical Notation

198

Table 6.2. North Indian numerical notation systems Script

0

1

2

3

4

5

6

7

8

9

Bengali

ؐ

ؑ

ؒ

ؓ

ؔ

ؕ

‫ؖ‬

‫ؗ‬

‫ؘ‬

‫ؙ‬

Devanagari Gujarati Marathi Oriya Punjabi Nepali

to present the paleographic data concerning the development of Indian numeralsigns from 800 ad to the present day (cf. Salomon 1998; Ifrah 1998: 367–385 for more complete analyses of this issue). Nevertheless, a look at some of the more important variations on this common pattern of ciphered-positional decimal systems is warranted. Many of these systems (or very close descendants thereof ) have been employed for well over one thousand years and continue to be used. Most major South Asian languages have their own alphasyllabaries and numeral-signs. Their numerical notation systems are structurally identical to one another and to Western numerals. Today, all these indigenous systems are in competition with Western numerals, especially for commercial and scientific purposes. In religious and formal contexts, the traditional numerals are still frequently preferred.

North India The ancestor of the northern Indian numerical notation systems is the Brāhmī system used in the Gupta Empire, which ruled most of northern India from the Indus to the Ganges from the fourth to sixth centuries ad, and also significantly influenced central India. These earliest Gupta Brāhmī numerals were nonpositional, but the idea of positionality and the zero sign spread quickly through the systems of the region. The most common modern varieties of this subgroup are the Bengali, Devanagari, Gujarati, Marathi, Oriya, Nepali, and Punjabi; they are thus used in a swath across Pakistan, northern and central India, Nepal, and Bangladesh. The northern Indian systems are also directly ancestral to both the modern Arabic and “Maghribi” numerals associated with the Arabic script, and thus, indirectly, to Western numerals. The similarities between Western numeral-signs and many of the north Indian numerals, especially those for 0, 2, and 3, are quite evident in Table 6.2.

South Asian Systems

199

Table 6.3. Central Asian numerical notation systems Script

0

1

2

3

4

5

6

7

8

9

Tibetan Mongolian

Central Asia The Gupta script also gave rise to a small number of scripts in the Himalayas and Central Asia, of which the most important are the Mongolian and Tibetan. The Tocharian script had used a variant of the Brāhmī additive numerals from the sixth to eighth centuries ad, but the Tocharian language and script died out before the introduction of positionality. Tibetan writing and numeration developed in the ninth century, and Mongolian numerals developed from Tibetan in the thirteenth century.4 These systems are related to the northern Indian systems. The classical Mongolian numerals were usually written from top to bottom in vertical columns, but the forms listed here are those used when they were written from left to right. These systems are shown in Table 6.3.

South India The scripts of the southern half of the Indian peninsula diverged from those of the north as early as the second century ad. There are five modern scripts in this subgroup: Telugu and Kannada, two closely related scripts of east central India, along with Tamil, Malayalam, and Sinhalese. Of these five, the Telugu and Kannada numerals shifted from addition to position in the seventh and eighth centuries ad, and are thus structurally identical to those of the northern systems (Syamalamma 1992: 51). The numerals of the Grantha script (sixth to twelfth centuries), which is ancestral to modern Tamil, Malayalam, and Sinhalese, did not switch principle. The Telugu and Kannada numerals are shown in Table 6.4, while the other three systems are described later in this chapter.

Southeast Asia Far from being a cultural backwater or simple recipient of positional notation, South Asian societies used ciphered-positional numerals very early. Scripts such 4

Despite Ifrah’s assertion (1998: 382) that each of the Agnean, Kutchean, and Khotanese scripts of Chinese Turkestan would have used a set of ten positional numerals, I know of no evidence that this was the case.

Numerical Notation

200

Table 6.4. South Indian numerical notation systems Script

0

1

2

3

4

5

6

7

8

9

Telugu Kannada

as Kawi (the ancient script of Java) and Cham (used in Vietnam until the thirteenth century) originally used hybrid additive numerical notation systems on the Brāhmī model, but these transformed into ciphered-additive positional systems in the seventh century. The modern descendants of these systems include Khmer, Thai, Burmese, Lao, Balinese, and Javanese. Of these, Balinese and Javanese are closely related to one another but paleographically distant from any other South Asian systems. They use Javanese letters to represent certain numbers, while retaining older signs derived from Kawi for the others (0, 4, 5, and 6). The Southeast Asian systems are shown in Table 6.5.

Tamil The Tamil script used in the far southeast of India and parts of Sri Lanka is derived ultimately from Bhattiprolu, the southern variety of the Brāhmī script that developed in the first or second century ad, and immediately from the Grantha script of the sixth through twelfth centuries, which is ancestral to Tamil, Malayalam, and Sinhalese. The Tamil script is alphasyllabic and similar to other Brāhmī-based scripts, but has unique features, such as the ability to represent consonant clusters as a sequence of individual consonant signs. Similarly, the Tamil numerical notation system is rather different from those of other Brāhmī-derived scripts. The Tamil numeral-signs are shown in Table 6.6 (Pihan 1860: 113–119; Guitel 1975: 614–615). Table 6.5. Southeast Asian numerical notation systems Script Khmer Thai Burmese Lao Balinese Javanese

0

1

2

3

4

5

6

7

8

9

South Asian Systems

201

Table 6.6. Tamil numerals 1

2

3

A B C 10 100 J 6408 = FLDKH

Units

4

5

6

7

8

9

D K

E

F

G L

H

I

1000

The numeral-signs are ultimately derived from Brāhmī, and are thus related to all the systems of India and Southeast Asia. Following the Indian pattern, numeral-phrases are written and read from left to right, but are structurally neither ciphered-additive, like those of the Brāhmī numerals, nor ciphered-positional, like those of most of the Indian systems. Rather, the traditional Tamil system is multiplicative-additive and decimal. There is no power multiplier for the ones. Tamil has no signs for 10,000 or higher powers of 10; large numbers were expressed by placing an appropriate numeral-phrase before the sign for 1000, then multiplying. Thus, 800,000 would be written as HKL (8 × 100 × 1000). There is no ambiguity in this phrase’s meaning, because phrases are always read strictly from left to right.5 This is the only instance where a lower power sign may precede a higher one. The Tamil numerals acquired their distinct structure in the medieval era, although it is not clear exactly when the divergence arose. The change from a hybrid cipheredadditive/multiplicative-additive to a purely multiplicative-additive structure is easily accomplished; because Brāhmī numerals are multiplicative above 100, all that is required is that the nine individual signs for the decades 10 through 90 be replaced by a single sign for 10. At this early period, we know them largely from inscriptions on stone, although we cannot exclude the possibility that they were used in other contexts. The numeral-signs are derived from those of the Grantha script, and are closely related to others of southern India. It has sometimes been claimed that the Tamil numerals are a uniquely Dravidian invention using letters of the alphabet, and, indeed, there are resemblances between the numeral-signs for 1 through 9 and nine Tamil phonetic signs (Burnell 1968 [1874]: 68). Nevertheless, since these resemblances can be found only by comparing the modern paleographic forms of the numbers and letters, this argument cannot be offered as a theory of their origin. Rather, the similarity is probably due to a later assimilation of the numeral-signs to the phonetic signs. Only Tamil and Malayalam, of all the South Asian systems, altered the Brāhmī ciphered-additive/multiplicative-additive system to a purely multiplicative-additive 5

Curiously, this system is structurally identical to the Armenian alphabetic notation of Anania Shirakatsi (Chapter 5), but it would be an error to make too much of this resemblance.

202

Numerical Notation

one. (Sinhalese retained the Brāhmī structure, while all other Brāhmī-derived systems became ciphered-positional.) The Chinese classical numerals (Chapter 8) are multiplicative-additive, so contact with Chinese Buddhists might have stimulated the development of the unique notations characteristic of areas of southern India that were also Buddhist, or, more likely, made their retention more appealing. However, there is no paleographic similarity between the Chinese and Tamil numerals, making diffusion from China improbable. Moreover, none of this evidence explains why the Tamils retained their system even after they ceased to practice Buddhism. Another possibility is that the users of the Tamil and Malayalam systems did not switch to positional notation as an effort to maintain their cultural distinctness from the north. Late in the system’s history, an abbreviated form of the Tamil numerals developed that, for some numbers, adds an element of positional notation by omitting the power-signs for 10, 100, and 1000. For instance, Pihan notes that while 21 was traditionally written BJA (2 × 10 + 1), it could also be written BA, abbreviating the phrase without any loss of information (Pihan 1860: 117). Such numeral-phrases are purely ciphered-positional. While this presents no problem for numerals that lack any empty positions, a zero sign is needed in other cases; however, no zero appears in any pre–twentieth-century Tamil numeral-phrases. Sometimes, rather than using a sign for zero, Tamil writers used the power-signs for 10, 100, and 1000 to eliminate ambiguity. In one case, 2205 is written as BBKE (2, 2, 100, 5), which indicates that the second 2 is to be understood as a hundreds value rather than as a tens value, and that therefore the first 2 must be understood as 2000 (Guitel 1975: 614–615). Such phrases combine multiplicative-additive and ciphered-positional notation. These mixed multiplicative and positional phrases are no longer used, and appear to be a transitional product of the colonial period, when contact with the West began in earnest. Apparently, in the nineteenth century some Tamil astronomers were computing using a mixed decimal and sexagesimal computation system by manipulating pebbles or shells on a flat surface in a manner reminiscent of Greek or Babylonian techniques, but there was no corresponding numerical notation (Neugebauer 1952). Today, some formal Tamil writings use the traditional numerals, while for most commercial and informal purposes an ordinary zero sign is used, making the system ciphered-positional. Most literate Tamils today are familiar with and use the Western numerals.

Malayalam The Malayalam script, like Tamil, is derived from the Grantha script of southern India. It is used to write the Dravidian language of the same name used in Kerala at India’s southwestern tip. It first emerged as a distinct script around 700 ad,

South Asian Systems

203

Table 6.7. Malayalam numerals 1

2

M N 10 W 6408 = RYPXT

Units

3

4

5

6

7

8

9

O

P X

Q

R

S Y

T

U

100

1000

although its letters and numeral-signs are closely related to those of the other Brāhmī-derived scripts. The Malayalam numerical notation system, like Tamil and Sinhalese, was unaffected by the cultural and political influence that rendered the northern Indian numerical notation systems structurally identical and paleographically similar. The traditional Malayalam numeral-signs are indicated in Table 6.7 (Pihan 1860: 122–125; Ifrah 1998: 335). The similarities between the Tamil and Malayalam systems are striking. Both systems are decimal and multiplicative-additive, and written from left to right. There are many paleographic similarities between the numeral-signs of the two systems, thus refuting the claim that the Tamil numerals are phonetic in origin. As in Tamil, there is no power-sign for the units, and the ‘1’ is understood in any numeral-phrase with a units value. Furthermore, Malayalam numbers above 10,000 are also expressed using multiplicative combinations of the sign for 1000 with those for 10 and 100, as necessary. None of these similarities is particularly surprising, given the close cultural and geographic proximity of these two Dravidian peoples. The only structural difference between Tamil and Malayalam numeration is that Malayalam numeral-phrases were never expressed using the hybrid additive and positional notation that was occasionally used later in the Tamil system’s history. There are few distinctly Malayalam inscriptions that date before ad 1000, by which time it had already acquired its multiplicative-additive structure. The numeral-signs are derived from those of the Grantha script, showing that the Malayalam numerals are native to South Asia, although, as with Tamil, we cannot exclude the possibility of some influence from Buddhist China. The fact that Buddhism was maintained longer in the south than in northern India may be a partial explanation for the difference in structure. A millennium of trade with and domination by other peoples of South Asia, most of whom used ciphered-positional notation, did not affect the structure of the Malayalam system. Malayalam writers employed this system regularly until the middle of the nineteenth century, at which time European contact introduced the zero and the idea of positionality. A new sign for zero was introduced (V), which, when combined with the nine regular unitsigns, produced a regular ciphered-positional system. Today, the older Malayalam

Numerical Notation

204

Table 6.8. Sinhalese numerals 1

2

3

a b c 10s j k l 100 s 1000 t 3684 = ctfsqd

1s

4

5

6

7

8

9

d m

e n

f o

g p

h q

i r

system is used rarely if at all (largely by those who need to understand old texts), and is quickly becoming a historical curiosity.

Sinhalese The Sinhalese (or Singhalese) script developed from the model of the southern Brāhmī scripts for use among the speakers of the Sinhalese language, and was influenced by the Grantha script that is ancestral to the Tamil and Malayalam scripts used for writing the Dravidian languages of southern India and northern Sri Lanka (although Sinhalese is an Indo-European language). It is an alphasyllabary, written from left to right, and is used today in Sri Lanka and the Maldives. The traditional Sinhalese numeral-signs are indicated in Table 6.8 (Pihan 1860: 140–141; Ifrah 1998: 332). Sinhalese has distinct signs for each of the units, each of the decades, 100, and 1000. The numeral-signs for the units resemble many of those used in other South Asian numerical notation systems, and many of the signs also resemble, but are not derived from, the phonetic signs of the Sinhalese script. Numeral-phrases are written from left to right. The system is ciphered-additive below 100, but it is multiplicative for the hundreds and thousands, combining the unit-signs with the appropriate power-signs. It is unknown how 10,000 and higher numbers were written, though Pihan (1860: 141) speculates that it may have been through multiplicative forms such as those used in Tamil and Malayalam. The Sinhalese numerals are thus structurally identical to the old Brāhmī system. The Sinhalese did not adopt the positional system when the peoples of India (except Tamil and Malayalam speakers) and Southeast Asia did, between the seventh and ninth centuries ad. Sinhalese inscriptions and texts used this system throughout the medieval period, seemingly unaffected by the shift to positionality occurring in other Indian numerical notation systems. Pihan (1860) shows no awareness of any structural changes in the Sinhalese numerals in use at the time

South Asian Systems

205

he was writing; although his knowledge of the numerals was limited, there is no reason to believe that they were in significant decline in the mid nineteenth century. Modern Sinhalese writings normally use the Western numerals, although the traditional numerals are retained for certain formal purposes.

Indian Alphasyllabic The primary numerical notation systems of India were ciphered-additive before the seventh century ad and ciphered-positional afterward, with only a few systems (Tamil, Sinhalese, Malayalam) remaining additive after that point. The numeral-signs of these systems are abstract and do not resemble closely the letters of the Brāhmī script or its descendants. However, starting around 500 ad, Indian astronomers and astrologers began to use a very different principle for representing numbers: assigning numerical values to the phonetic signs of various Indian alphasyllabic scripts. The basic principle of the Indian alphasyllabaries is that a set of consonant-signs are combined with a set of diacritic marks that indicate vowels to produce a set of signs for CV syllables; unmarked consonant-signs denote the syllable with the inherent vowel a.6 These numerical notation systems, known collectively as varnasankhya systems, were considered to be distinct from the normal Indian systems that had abstract numeral-signs (Ifrah 1998: 483). The three systems that I will now discuss – Âryabhata’s numerals, katapayâdi numerals, and aksharapallî numerals, represent an important side branch of the South Asian systems. Although used only by a limited group of initiates, they are very important for understanding Indian astronomy, astrology, poetry, and numerology, as well as serving important mnemonic functions by linking words and numbers.

Âryabhata’s Numerals While it is sometimes claimed that the Indian grammarian Pânini used alphasyllabic numerals in the seventh century bc (Datta and Singh 1962 [1935]), this is a highly dubious proposition given the lack of attested writing in India at that time. The first attested alphasyllabic numerals appear about 510 ad, near the end of the dominance of the Guptas over India. The system was very probably invented by the mathematician and astronomer Âryabhata, in whose works (later named the Âryabhatiya by his disciples) it first appears. Âryabhata, who lived in Kusumapura in modern Bihar, was renowned among Indian scholars of the Gupta empire and later centuries, was known to Muslim scholars as Arjabhad, and was likely the figure 6

The number of consonant-signs and vowel diacritics varies from script to script, and there are also signs for V syllables (isolated vowels) and CCV syllables.

Numerical Notation

206

Table 6.9. Âryabhata’s numerals



%

&

G



%



&

G

ka 1

kha 2

ga 3

gha 4

a 5

ki 100

khi 200

gi 300

ghi 400

i 500

<

E

;

D

A

<

E

;

D

A

ca 6

cha 7

ja 8

jha 9

ña 10

ci 600

chi 700

ji 800

jhi 900

ñi 1000

?

@

4

6

B

?

@

4

6

B

wa 11

wha 12

sa 13

sha 14

ta 15

wi 1100

whi 1200

si 1300

shi 1400

ti 1500

8

5

K

C

3

8

5

K

C

3

ta 16

tha 17

da 18

dha 19

na 20

ti 1600

thi 1700

di 1800

dhi 1900

ni 2000





!

F







!

F



pa 21

pha 22

ba 23

bha 24

ma 25

pi 2100

phi 2200

bi 2300

bhi 2400

mi 2500

'

7

#

'

7

#



ya 30

ra 40

la 50

va 60

yi 3000

ri 4000

li 5000

vi 6000

(



H



(



H



śa 70

va 80

sa 90

ha 100

śi 7000

vi 8000

si 9000

hi 10,000

named Ardubarius in the seventh-century Byzantine text, Chronicon Paschale. His work focused on astronomy (he is regarded as the first great Indian astronomer) but also contains much pure mathematics, in addition to the cosmological hypotheses from which the exact sciences in ancient India cannot be divorced. His numerals are concise and readable with little training, and while not infinitely extendable, are capable of expressing very high numbers. Table 6.9 shows the signs of Âryabhata’s numerals, using modern Nagari signs for convenience (Fleet 1911a; Guitel 1975: 582–583). Due to the structure of Indian alphasyllabaries, while there are many hundreds of possible syllables, to learn the signs of the system one need only learn the two sets of signs, which then can be combined with one another. The thirty-three unmarked signs, in their assigned order and divided into groups on the basis of similar phonetics, assume the numerical values 1–25, 30–90, and 100, as shown in the leftmost five columns of Table 6.9. The ingenious principle involved in this system is that changing

South Asian Systems

207

Table 6.10. Order of power diacritics



















ka 100 (1)

ki 102

ku 104

kri 106

kli 108

ke 1010

kai 1012

ko 1014

kau 1016

the vowel attached to one of the basic signs alters its numerical value. When combined with the vowel i, the signs take on the numerical values 100–2500, 3000–9000, and 10,000, as shown in the rightmost five columns. While this means that there are two signs with the value 100 – ha () and ki () – this has little potential to cause confusion. Each successive vowel diacritic multiplies the value of the sign by 100 with respect to its predecessor, as shown in Table 6.10 (indicating only the combinations of k + vowels). Using these signs in combination, any number up to 1018 could be expressed, and Âryabhata’s system does not exhaust the available diacritics. Numeral-phrases were written with the lowest powers on the left, which reflects the order of powers of the Sanskrit lexical numerals, but which is opposed to the Indian numerical notation systems, in which the highest power was on the left. No sign for zero was needed, and none used. The signs for 11 through 19 and 21 through 25 were not strictly necessary; 15 could be written as GA instead of B without any ambiguity, but the latter was more concise. These extraneous signs normally were not used in numbers such as 85, which was written as G (5 + 80) rather than B( (15 + 70). In some cases, these rules were violated (we do not know why), so that in one astronomical table, 106 is written as 16 + 90 and 37 as 16 + 21 (Guitel 1975: 587). Table 6.11 indicates several numeral-phrases written alphasyllabically. Table 6.11. Alphasyllabic numeral-phrases Alphasyllabic Representation Transcription and Sign-values kha va 62 % 2 60 ta ha ta OR 116 8or8 16 100 16 ra chu 70,040 7E 40 70,000 kha ya ghi li 765,432 %'&#<( 2 30 400 5000 ma vi nu jri 98,206,025  3;H 25 6000 200,000 8,000,000 phu ghe 40,000,220,000 & 220,000 40,000,000,000 Value

ki 100

cu śu 60,000 700,000 sri 90,000,000

208

Numerical Notation

The best way to conceive of this system is as a base-100, or centesimal, multiplicative-additive system with a decimal and ciphered sub-base. Unlike most multiplicative-additive systems, however, there can be up to two unit-signs within each power of 100, each of which combines with its own power-sign. For instance, in the representation of 9800 in Table 6.11, the signs for 8 (;) and 90 (H) each combine separately with the diacritic sign for 100 ( ). The system is slightly irregular below 100 in that the basic thirty-three signs include signs for 11 through 25. Âryabhata recognized that the system was centesimal rather than decimal, as he distinguished the set of centesimal powers, or varga ‘classes’, from the intermediate powers, or avarga (Das 1927a: 110; Jha 1988: 80). Âryabhata’s system is not positional, since placing an unmodified consonant-sign in the middle of a numeral-phrase would render it meaningless. If Âryabhata was unfamiliar with positional numeration, then he may have developed alphasyllabic numerals because of the insufficiency of the Brāhmī cipheredadditive system for writing large numbers, a task that could be done very concisely using his own system. However, there is no evidence that the calculations that Âryabhata undertook were done directly with these numerals. Instead, their function was primarily to efficiently express very large numbers in verse, in the manner in which traditional Indian mathematics was written (Clark 1929: 232–233). The versification of mathematical expressions, while seemingly constraining and complicating the expression of numbers, assisted the user mnemonically while retaining the brevity of other numerical systems. It is possible that Âryabhata was familiar with the Greek alphabetic numerals in addition to the Brāhmī numerals. Âryabhata’s work was inspired in part by Greek astronomical writings, and Fleet (1911a), among others, has argued that both Âryabhata’s astronomy and his numerals are derived from Greek sources. However, even if he borrowed the general idea of using script-signs as numerals from the Greeks – and there is no definite evidence either way – this does not tell us very much, because the two systems are radically different (Das 1927a: 111–114). Even if he did know the Greek numerals, they played little role in the invention of his own system, whose idiosyncratic features, such as a base of 100, are not found in other systems. Another possibility is that even though his system was not positional and lacked a sign for zero, Âryabhata had a complete knowledge of cipheredpositional numeration when inventing his alphasyllabic system (Ganguli 1927). As I discussed earlier, the concept of śûnya or “emptiness” existed in the fifth century ad and may have prefigured the use of positional numerals in India, but no good evidence survives for an actual ciphered-positional numeration system prior to the seventh century ad, long after Âryabhata’s death. Ifrah’s (1998: 450–451) statement that the use and adoption of Âryabhata’s system “caused the Indian discoveries of the place-value system and zero, which took place before Âryabhata’s time, to be

South Asian Systems

209

irretrievably lost to history” is typical of the confused anti-empiricism of recent research on this system. The nonpositionality and relative complexity of Âryabhata’s system argue against his having been familiar with positionality. While Âryabhata’s numerals were known to Indian astronomers and mathematicians long after his death, they were used solely in the context of commentaries on his work. The system’s lack of facility for arithmetic, coupled with the difficulty in pronouncing the resulting numeral-phrases in Sanskrit, led to its abandonment even among most of Âryabhata’s proponents (Jha 1988: 85–86; Yano 2006: 149). It was replaced, in part, by the regular numerical notation systems of India, but it also gave rise to a variety of successor systems for correlating phonetic signs with numerical values, most notably the katapayâdi system. While these successors were not as unusual as Âryabhata’s system, they were far more successful.

Katapayâdi Numerals When later scholars experimented with alphasyllabic numeration starting in the ninth century ad, they immediately saw that an alphasyllabary could also be turned into a ciphered-positional system. Known as katapayâdi, the signs of this system are shown in Table 6.12 (Fleet 1911b; Datta and Singh 1962 [1935]: 70). In this system, each V and CV syllable is given a value from 0 to 9. Unlike Âryabhata’s system, changing the vowel of the syllable does not change its numerical value, so that ka = ki = ku = 1. Two of the signs (ña and na) take on the value of zero, as did isolated vowel-signs (those representing a V syllable alone, without any consonantal component), which did not have a numerical value in Âryabhata’s system. CCV syllables do not have their own numerical values, but are considered to have the value of the consonant to the left of the vowel, so that tva = va = 4 and ntya = ya = 1. As a result, any sequence of syllables can be assigned a numerical value, and any number has a wide variety of possible phonetic transcriptions. Katapayâdi numerals were read with the lowest power on the left, as in Âryabhata’s numerals. Thus, the word bhavati or F 8 ‘is’ had the numerical value 644. The name katapayâdi itself is taken from the four syllables (ka, ta, pa, ya) that are assigned the value 1 in this system. Although it is unusual in that each digit from 0 to 9 has several alphasyllabic values that represent it, structurally this system is an ordinary ciphered-positional and decimal system. The earliest example of the katapayâdi numerals is from the Grahacāranibandhana by the astronomer Haridatta, written in ad 683 (Sarma 1999: 274). Datta and Singh (1962 [1935]: 71) place its invention around ad 500 and claim that it must have been known to Âryabhata himself, but there is no textual evidence to support this assertion. Haridatta was a direct intellectual descendant of Âryabhata, and used his predecessor’s system as the basis for his own. Nevertheless, as the katapayâdi

Numerical Notation

210

Table 6.12. Katapayâdi numerals



%

&

G

ka 1

kha 2

ga 3

gha 4

ua 5

<

E

;

D

A

ca 6

cha 7

ja 8

jha 9

ña 0

?

@

4

6

B

wa 1

wha 2

sa 3

sha 4

ta 5

8

5

K

C

3

ta 6

tha 7

da 8

dha 9

na 0





!

F



pa 1

pha 2

ba 3

bha 4

ma 5

'

7

#

ya 1

ra 2

la 3

va 4

(



H



śa 5

va 6

sa 7

ha 8

is ciphered-positional, like the general Indian positional numerals in ascendance at the time, the existing positional numerals must certainly have influenced him. Thus, this system is very likely a blend of Âryabhata’s numerals and the ordinary Indian positional numerals. Aside from being able to express any number, it gave every word a numerical value, and gave every number many corresponding words. This would have allowed for the construction of various mnemonic devices to aid scholars and students, and would have served a prosodic function (for astronomical texts were written in Sanskrit verse, which had strict metrical rules). The katapayâdi numerals were also important in the Hindu traditions of number-magic, divination, and chronograms in which the sum of the numerical values of the signs of a word or verse produced a meaningful date. The katapayâdi numerals, as well as related systems that are identical except for the use of local script-signs and the assigning of different digit-values to various signs, were used continuously throughout much of India for many centuries.

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Several variants of the katapayâdi developed, most of which changed a few numerical values or eliminated the values of certain categories of signs, such as the isolated vowels (Datta and Singh 1962 [1935]: 71–72). Some of these systems were unique to one writer, while others were used in specific regions over a longer period. Renou and Filliozat (1953: 708) claim that their use in paginating loose manuscripts served a cryptographic function in that the pages of the text, once jumbled, could be placed back in order only by initiates of the system. Katapayâdi numeration survived much more extensively in southern India, particularly in the province of Kerala – curiously, where the additive Tamil and Malayalam numerals were only recently and incompletely replaced by positional systems. Northern Indian katapayâdi numerals are rare, although Sarma (1999) discusses a sixteenthor seventeenth-century astrolabe labeled in the northern Devanagari script that uses them, perhaps in imitation of the Arabic abjad numerals. They were still used in astrological manuscripts and horoscopes in South India even in the late nineteenth century (Burnell 1968 [1874]: 79–80).

Aksharapallî A third variety of alphasyllabic numerals, sometimes confused with the cipheredpositional katapayâdi in the scholarly literature, is known as aksharapallî numeration (after akshara, the word for the CV syllable-clusters that comprise the basic unit of the Indian alphasyllabaries). Whereas Âryabhata’s system was multiplicative-additive, and the katapayâdi system was ciphered-positional, the aksharapallî systems are ciphered-additive and decimal, assigning the numerical values 1–9, 10–90, and sometimes also the low hundreds (but never as high as 1000, to my knowledge) to a set of phonetic signs. It was used very widely for paginating books, and was written in the margins from top to bottom with the highest power at the top. Unlike the first two alphasyllabic numerical notation systems, there was never a single regular system for correlating signs with numerical values in the aksharapallî. Datta and Singh’s (1962 [1935]: 73) search through old manuscripts revealed no fewer than three signs for 1, twelve different signs for 4, and nine signs for 60. This is less complex than it seems, because within specific traditions, there were set sequences of signs that would be understood by anyone working within them. In some instances, parts of these sequences may be comprehensible; for instance, in Nepali manuscripts from the eleventh to the fourteenth centuries, the numbers 1 through 3 were represented by the syllables e, dvi, and tri, which correspond to the Nepali lexical numerals (Burnell 1968 [1874]: 66). In other cases, the signs used appear to have been assigned almost randomly. Datta and Singh (1962 [1935]: 73) list many signs for which they cannot even attach a plausible syllabic value. Dialect

212

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differences in pronunciation or paleographic variation among scripts may account for the irregularity of the aksharapallî systems. The aksharapallî numerals are ciphered-additive, but the lack of a regular correlation between the signs and numerical values suggests no obvious origin. They may have developed directly from the Brāhmī ciphered-additive numerals, with only the use of phonetic rather than abstract symbols to distinguish them. This matter is made even more complex by the fact that many modern scholars still maintain that the origin of the Brāhmī numeral-signs was as a modification of phonetic signs (as mentioned earlier). Indian scholars considered the aksharapallî to be part of the varnasankhya tradition of alphasyllabic numeration, so I believe they are related to the other alphasyllabic Indian systems, although possibly with some influence from the Brāhmī system. The primary use of aksharapallî numerals was for the pagination of manuscripts. This may explain why there was apparently no need for any such system to express numbers in the high hundreds and thousands. Aksharapallî numerals had the greatest and most consistent level of use of any of the alphasyllabic numerals of India. They were used with great frequency in the manuscripts of the Jains until the sixteenth century, although it is not clear why this system would appeal specifically to Jains (Datta and Singh 1962 [1935]: 74). They also survived for a very long time in Nepal (Burnell 1968 [1874]: 65). Temple (1891) goes into great detail concerning a ciphered-additive numerical notation system used for arithmetic by Hindu astrologers in Burma in the late nineteenth century. While he does not indicate whether the signs were alphasyllabic, nor does he use the name aksharapallî (or any other name) to describe this notation, no other ciphered-additive notation is likely to have been used in Burma at that time. Aksharapallî numerals appear to have thrived along the Malabar Coast; they were still common enough in Malayalam-speaking regions in the middle of the nineteenth century to be included in some grammars (Bendall 1896). These groups are all relatively distant from the central political and religious movements of India, so their survival may reflect the marginal status of such places in Indian history. Aksharapallî systems continue to be used occasionally throughout India, Bangladesh, Nepal, Tibet, Burma, Cambodia, Thailand, and Java (Ifrah 1998: 484). The tradition of Indian alphasyllabic numeration, while lasting well over a millennium and playing a significant role in Indian astronomy, astrology, and mathematics, did not influence numeration practices outside of South and Southeast Asia. The different varnasankhya systems have in common a suitability for flexibly representing numerical phrases in verse, but in mathematical traditions that did not involve versification, alphasyllabic numeration was undesirable. Moreover, the Arabic and Western scripts were unsuitable for modification to suit the unusual structure of the alphasyllabic numerals, and both the Arabic abjad and the Greek

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213

alphabet had their own alphabetic numerical traditions. Thus, while the Indian tradition of ciphered-positional numerals spread fairly readily into the Arab world, alphasyllabic numeration remained a strictly South Asian phenomenon.

Arabic Positional The Arabic script is written from right to left, and is basically consonantal, though with some representation of vowel sounds. The earliest Arabic speakers used the hybrid cumulative-additive/multiplicative-additive Nabataean numerals (Chapter 3); this was replaced after the Islamic conquest of Greek-speaking regions by an alphabetic system (the abjad numerals) akin to the Greek, Hebrew, and Syriac systems (Chapter 5). Shortly after the introduction of abjad numerals, however, users of the Arabic script became aware of Indian ciphered-positional numerals, and developed their own system on this basis, whose modern numeral-signs are shown in Table 6.13. This system is written with the higher powers on the left, rather than from right to left following the direction of the Arabic script. Thus, 26,049 would be written . Burnett (2000b) discusses this phenomenon in light of the fact that Western numerals are read from left to right but in this case in accordance with the direction of the script, noting that this may have led to some confusion when the Arabic numerals were borrowed into the West as to the direction in which the numerals should be read. In the modern Arabic numeral-signs, there are alternate signs used for 4 ( ) and 5 ( ), and the modern zero sign is written with a dot instead of with a circle because the circle was already assigned the value of five. There are undeniable paleographic resemblances between the Arabic positional numeral-signs and those used in medieval north India. Table 6.14 compares the Arabic positional numerals found in eleventh-century mathematical and astronomical treatises with the inscription found at Gwalior, India, dated to 876 ad, containing the Nagari numerals used in medieval India. These signs are very similar, and it is thus safe to assume that the Arabic numerals have an Indian origin. In some cases, as for 2, 3, 7, 8, and 9, the Nagari numeral-sign became rotated or inverted, which may have resulted from the scribal practice of writing from top to bottom, then rotating the manuscript to read it (Ifrah 1998: 532–533). The fact that medieval and modern Arabic scholars are unanimous in attributing an Indian origin to these signs, and call them ḥisāb al-hindi (Indian numerals), abundantly confirms the paleographic evidence. The social context of the transmission appears to have been limited to the exact sciences initially, specifically to astronomy. In 662 ad, a Syrian Christian bishop, Severus Sebokht, noted the Hindu proficiency in astronomy, commenting that “as for their skilful methods of calculation and their computing which belies

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214

Table 6.13. Arabic positional numerals 0

1

2

3

4

5

6

7

8

9

description, they use only nine figures” (Nau 1910). The meaning of this statement is unclear, as he does not mention the zero, but it is likely that Sebokht was referring to ciphered-positional numerals; if so, the Arab-speaking world probably would have had some such knowledge as well, possibly through Persian intermediaries (Kunitzsch 2003: 3). In ad 773, an Indian embassy visited the court of the Abbasid caliph al-Mansur in Baghdad, among whose members was an astronomer who brought with him a copy of a Hindu work of astronomy, which was translated into Arabic (Folkerts 2001: 15). Within fifty years of this episode, the mathematician Muhammad ibn Mūsā al-Khwārizmī wrote his Arithmetic (c. 825 ad) using ciphered-positional numerals extensively, prompting later mathematicians and astronomers to follow his lead in replacing the old ciphered-additive abjad numerals with the new positional system. While al-Khwārizmī’s work does not survive in its original Arabic (the earliest surviving manuscript is a twelfth-century Latin translation), al-Khwārizmī knew of positional numeration and advocated its simplicity and functionality. We do not know specifically, however, what numeral-signs he used, and, indeed, almost no contemporary texts containing positional numerals survive. The earliest direct paleographic evidence for positional Arabic numerals comes from an Egyptian papyrus (PERF 789) with a numeral 260 at the bottom, but only if that numeral is the date 260 a.h., or 873/4 ad, a point that Kunitzsch (2003: 5) disputes. The Tārīkh of alYa’qūbī of 889 mentions the sign for zero as being a small circle, without describing the system further (Kunitzsch 2003: 4). The manuscript MS Paris, BNF ar. 2457 by the astronomer al-Sijzi, written between 969 and 972 ad, provides secure evidence for the mid tenth century, but this is a considerable gap in our direct evidence from the numeral-forms (Folkerts 2001: 14; Kunitzsch 2003: 5–6). Nevertheless, the textual non-paleographic evidence demonstrates that some Arabs were surely using them at least by al-Khwārizmī’s time, and possibly as early as ad 775. Table 6.14. Early Arabic and Nagari positional numerals 0

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

Arabic Nagari

South Asian Systems

215

From their origins in the late eighth and early ninth centuries ad, the numerals spread throughout the Islamic world, though not without resistance or confusion. Many conservative scribes and bookkeepers resisted the new numerals in favor of older calculation on the fingers and with numeral-words. In his Kitāb al-mu’allimīn, the ninth-century scholar al-Jāhiz recommended finger reckoning above the ḥisāb al-hindi because it needed neither speech nor writing, a position echoed a century later by the historian al-Sūlī in his Adab al-kuttāb (Kunitzsch 2003: 4–5). Whether this functional explanation is complete, or whether other, ideological considerations came into play, is unknown. Lemay (1977: 440–444) questions the extent to which the Indian numerals were known to the Arabs before the tenth century, and shows that there was confusion among some Arabic thinkers over how they worked. While positional numerals began to dominate in both mathematical and nonmathematical contexts starting in the eleventh century, astrological texts remained far more conservative, retaining the abjad numerals solely until the fourteenth century (Lemay 1982: 385–386). Contrary to the diffusion of most numerical notation systems, scientific functions rather than commerce or religion provided a significant impetus for the transmission of the positional numerals from India westward to the Arabs. It is nonetheless generally the case that medieval Arabic arithmetic did not distinguish commercial, astronomical, divinatory, and other arithmetical practices unambiguously. The Arabs borrowed not only the Indian numerals, but also a host of computational techniques and devices, including the dust-board, a flat tablet strewn with sand into which figures could be written for undertaking computations (Bag 1990: 290–293; Folkerts 2001: 14). Other techniques available included a complex Greek-derived system of finger reckoning and the use of counters or shells; accordingly, the use of written pen-and-paper arithmetic was apparently not part of the initial practice of Indian-derived numeration. The earliest major Arabic arithmetical text to advocate Indian numeration instead was the Kitāb al-fusūl fī al-hisāb al-hindī of al-Uqlīdisī ‘the Euclidean’, written in Damascus in ad 952/3, the earliest extant copy of which dates to ad 1186 (Saidan 1966; Burnett 2006: 16). Yet al-Uqlīdisī recommended concealing the Indian origin of the technique, using the first nine letters of the Greek alphabet or the Arabic abjad instead of the numeral-signs for 1 through 9 (Saidan 1966: 478–479). Many of the earliest Arabic arithmetic texts had names such as kitāb al-takht ‘book of the board’, confirming the association of the new numerals with dust-board computation (Smith and Mourad 1927). Yet once the ḥisāb al-hindi had been firmly established by the eleventh and twelfth centuries, Arabic mathematical texts began to advocate doing computations with paper and ink instead of the dust-board, which of necessity involved rubbing out intermediate steps of

216

Numerical Notation

computations and thus could lead to error. Clearly, Arab mathematicians’ sentiments toward the innovation were complex – both recognizing it as highly useful in comparison to earlier techniques, yet desiring to improve it to suit their own purposes and prejudices. Despite their prominence, the Arabic positional numerals have given rise to few structurally distinct descendants. Some of the cryptographic systems of the Ottoman Empire (Chapter 10) may have been inspired by the Arabic positional system, although most of these systems are additive and may be more closely related to the abjad numerals. A number of sub-Saharan African ciphered-positional systems developed in the twentieth century (Chapter 10) were created by inventors who knew the Arabic and/or Western positional numerals. While the debt of Western numerals to Arabic positional notation is undeniable, they are directly related to the ciphered-positional Maghribi numerals (known sometimes as ghubar numerals, and in medieval Latin as figure toletane ‘Toledan figures’) described in the following section, and are thus at best phylogenetic “cousins” of the signs commonly used in the modern Islamic world. Burnett (2000c) notes, however, that routes of transmission were complex, and the standard Arabic numerals were described in several Western European texts, through the medium of the Crusader state at Antioch in the twelfth-century Latin translation Liber Mamonis, and through the city of Pisa in the Latin translation of the Hebrew works of Abraham ibn Ezra (see Chapter 5). These “Eastern figures” (known in Latin as figure indice ‘Indian figures’) were, if never widespread, at least known to Western mathematicians, but by the early thirteenth century, the migration of Toledan translators to northern Italy marked the decline of “Indian figures” in Western Europe (Burnett 2002a). The Arabic numerals enjoy a degree of currency and use in the modern world second only to the Western numerals. They are used regularly in most contexts throughout all regions that employ the Arabic script, and are thus found regularly from Morocco to Indonesia. While global commerce and the effects of mass media have introduced Western numerals into the Arabic-speaking world, and most literate users of the Arabic script know them, it is unlikely that this will have any long-term effect on the use of Arabic numerals.

Maghribi (“GHUBAR”) Numerals A set of ciphered-positional numerals quite distinct from the regular Arabic system was used in North Africa and southern Spain during the medieval era and sporadically thereafter. These numerals, sometimes known in Arabic as ḥisāb al-ghubar ‘dust-reckoning’ and commonly known in the scholarly literature as ghubar ‘dust, sand’ numerals, are particularly important for this study because they are the

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217

Table 6.15. Maghribi numerals 0

1

2

3

4

5

6

^

U

V

Wc

X

Y e Z

7

8

9

[g

\ h

]

immediate ancestor of the Western numerals.7 While the paleographic forms of the Maghribi numeral-signs vary, representative examples are indicated in Table 6.15 (Gandz 1931, Souissi 1971, Labarta and Barceló 1988). In Chapter 5, I showed that North Africa and Spain were quite distinct from the rest of the Arabic world, both in their use of a different ordering of the abjad numerals and in their use of special “Fez numerals.” Likewise, comparing the Maghribi numerals to the standard Arabic positional numerals (either the medieval or modern forms), while they are both decimal, ciphered-positional numerical notation systems written with the highest powers on the left, the two systems differ paleographically. Potential early examples of the Maghribi numerals come from two documents dated from 874 and 888 ad, respectively, in texts from the Maghreb (Gandz 1931: 394). It is perhaps notable that the first textual example of the Maghribi numerals comes only one year after the appearance of the first regular Arabic positional numerals known in Egypt, but this may simply reflect accidents of survival and discovery. If these ninth-century texts actually contain Maghribi numerals, this would accord well with their tenth-century diffusion into the Latin manuscript tradition in Spain. However, Kunitzsch (2003: 11) argues, contrarily, that the earliest unambiguous Western Arabic/Maghribi numeral-signs in Arabic texts are from MS Florence, Or. 152, dating to the middle of the thirteenth century (!). A good part of this discrepancy arises from the fact that it is not a simple matter to distinguish the Maghribi numerals from the Eastern forms in early texts; clarifying the origin of the divergence will help to resolve the problem. Smith and Karpinski (1911: 98), Das (1927b: 359), and Datta and Singh (1962 [1935]) argue that the Maghribi numerals are closer to the original Indian forms, and thus represent an earlier transmission, than the later Eastern Arab forms. They claim from this that the Maghribi numerals were the ones used by al-Khwārizmī and other early mathematicians; however, such a conclusion is overly speculative. Another theory, popular from about 1915 until 1935, holds that the Maghribi numerals (and hence Western numerals) came from India to Spain via NeoPythagoreans in Byzantium, while the standard Arabic numerals came from India via the caliphate of Baghdad (Carra de Vaux 1917; Cajori 1919; Gandz 1931: 7

Following Kunitzsch (2003: 10), who argues persuasively that the term “ghubar” has been misapplied, I use the label Maghribi, reflecting the system’s geographical origin.

218

Numerical Notation

395; Miller 1933; Lattin 1933: 184–185). Yet there is no evidence for ciphered-positional numerals in Byzantine Greece prior to the twelfth century (Wilson 1981). The “rejectionist” view is further refuted by the fact that medieval Arabs, Western Europeans, and Byzantines were in accord that the numerals were of Indian origin (Lemay 1982: 382). It is best now regarded as an ethnocentric relic of the misguided notion that Indians and Arabs were uncreative. The term ghubar, with its unusual meaning of “dust” or “sand,” has prompted some comment as to the function of the number. Das (1927b: 358) and Gandz (1931) assert that the ghubar tradition represented a sort of Arabic abacus, but Kunitzsch (2003) shows that texts discussing the system refer to a takht ‘board’ on which writings were made and from which items could be erased – that is, boards covered with dust or sand that were used as calculating boards by drawing figures on them. Some of the variation between the Arabic positional and Maghribi numerals may be explained by their use on differing media, the former in permanent media and the latter for arithmetical calculations on sand-boards. Their forms, thus fixed by the separation of contexts, might have become entrenched through centuries of use in disparate parts of the Islamic world. However, no actual “dust-numerals” in that medium survive, of course. Kunitzsch (2003: 9–10) argues that despite the terms ḥisāb al-hindi and ḥisāb al-ghubar being references to different media and computational techniques, they do not necessarily imply two distinct sets of graphemes associated with each. While the term ghubar seems to have originated in Tunisia and is associated with Maghribi scholars such as ibn Khaldun (whose fourteenthcentury Muqaddimah mentions only ghubar, abjad, and zimām [Coptic] numerals, not hindi) (Lemay 1982: 384–387), the distinction in sign-forms is thus better understood as a geographical rather than a functional one. In fact, the ordinary Arabic numerals and the Maghribi numerals were quite similar until the twelfth century; their numeral-signs for almost all values are similar enough to be explained as graphic variations of a common system of Indian derivation (the medieval Nagari ciphered-positional system). In a tenth-century manuscript written by the Persian astronomer Sijzi, the form of numerals used is intermediate between the Arabic and Maghribi forms (Mazaheri 1974). Maghribi numerals are thus a subset of the larger class of Indian-derived positional, decimal numerals (Lemay 1977: 437), both of which stand in contrast to the abjad numerals described in Chapter 5. This does not mean that the transmission of positional numeration was a singular event, but it does suggest that it was not a matter of two distinct “waves” of diffusion, one into the Maghreb and the other into the rest of the Arab world, but rather a significantly more complex series of episodes that resulted in two parallel systems. While the Maghribi numerals began as a paleographic variant of the Indian numerals, they eventually took on a distinct cultural meaning among the scribes,

South Asian Systems

219

astronomers, and mathematicians of the western Islamic world. This is partly because of the relative independence of polities such as the caliphate of Cordoba from the Baghdad-based Abbasid caliphate. The traditionalism of the Maghrebi and Andalusians may partly explain why the numerals persisted even after the rest of the Islamic world had adopted the signs now used throughout the modern world (Ifrah 1998: 539). They were still regularly used in Spain and North Africa in the fifteenth and sixteenth centuries, and sporadically thereafter (Labarta and Barceló 1988). The Maghribi numerals would be little more than a paleographic curiosity, merely one of many ways of writing Arabic numerals, if not for the fact that through them, Western Europe adopted ciphered-positional numerals. For the past seventy-five years, all major scholars have agreed that the resemblances between Maghribi and Western numerals, coupled with the first appearance of the latter system in medieval Spain, demonstrate this origin. Yet even as the Western numerals developed along their own trajectory within the Christian European context, the Maghribi numerals survived for almost a full millennium. Ifrah (1998: 535) provides examples of arithmetical texts written using the system from as late as the eighteenth century, and suggests that the system may have survived into the nineteenth century before being completely replaced by the standard Arabic numerals.

Western Numerals From their origin as a foreign and suspicious novelty during the medieval period, the ten Western numerals, structured by the use of the positional principle, have become so familiar that it is easy for the nonspecialist to forget that there are other numerical notation systems. The ubiquity and universality of the Western numerals make understanding their origin and diffusion all the more important. Unfortunately, no monograph has dealt systematically with the topic since Hill (1915), whose work is rather outdated as a result of advances in paleography. The first example of Western numerals is generally held to be the Codex Vigilanus, written in 976 in the monastery of Albelda near the town of Logroño in northern Spain, in which the numerals are described (in Latin) as “Indian figures” (Hill 1915: 29; Burnett 2002b: 241). The nine units are listed, in descending order, but no zero-sign, probably because the signs were intended for use with a counting board. These signs are shown in Table 6.16. These figures are very similar to the Maghribi numerals shown in Table 6.11, and in fact there is no reason to consider them a separate system, except that they are used in a Latin and Christian text from northern Spain rather than in an Arabic one from Andalusia. Toledo was a major center for the transmission of Arabic knowledge to the Christian West in the tenth and eleventh centuries, and Lemay

Numerical Notation

220

Table 6.16. Western numerals (Codex Vigilanus, ad 976) 1

2

3

4

5

6

7

8

9

l

m

n

o

p

q

r

s

t

(1977: 444–445) believes that later scholars became aware of ciphered-positional numerals through reading Toledan texts. These numerals found their way into slightly more widespread usage through the writings of Gerbert of Aurillac (c. 945–1003), who was to become Pope Sylvester II in 1000. Gerbert traveled extensively and studied arithmetic in Islamic Spain in 967, at which time he almost certainly learned the Arabic numerals (Folkerts 2003: 1–2). Thereafter, he helped renew interest in computing boards in his Regulae de numerorum abaci rationibus (c. 980). Later authors such as his biographer Richer credited Gerbert with having introduced the use of the formae or notae, the nine numeral signs (excluding zero) on counting boards, replacing a large number of tokens placed in any column with a single token bearing one of these signs (Berggren 2002: 356–357; Folkerts 2003: 2).8 No zero-sign was needed because counting boards are positional by their nature, without the need for a placeholder, although over time, a symbol was added, first τ (for ‘terminus’) and later the familiar circle (Berggren 2002: 358). These marked tokens, called apices, were used as a teaching tool by medieval mathematicians, known as abacists, between the tenth and twelfth centuries (Evans 1977; Lemay 1977; Gibson and Newton 1995). Around thirty-five treatises on calculation with the “Gerbert” abacus survive from this period (Folkerts 2003: 3). Beaujouan (1947) has demonstrated that the apparent rotation of numeral-signs written in many tenth- through twelfth-century texts is explainable by the fact that the numerals on apices could be oriented in any direction when the tokens were placed on a board. Once the apices computational technique was no longer used, the numeral-signs were no longer rotated when written on paper. Yet this early period of Western numeral use was relatively restricted, and did not spread further than a limited group of mathematicians and astronomers. All of the eleventh-century manuscripts containing Western numerals, and the vast majority of the twelfth-century ones, are from the abacist tradition and are didactic in nature, describing the use of the counting board rather than using the numerals for performing calculations (Hill 1915: 29–31; Burnett 2006). 8

A remaining puzzle is why the Roman numerals for 1 through 9 were never employed on the counters; although Burnett (1997: 11) argues that this would have been impractical, I do not see any reason why even a long numeral-phrase like VIIII for 9 would have been too long to have been used in this fashion.

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The spread of Western numerals into the tradition of manuscript writing (in both mathematical and other texts) began in earnest in 1202, at which time the mathematician Leonardo of Pisa, better known as Fibonacci, promoted their use in his Liber Abaci (Book of the Abacus). Despite its name, the purpose of Fibonacci’s text was to promote not the use of the abacus, but rather the use of written numerals for computation, with nine unit-signs and a zero-sign (k), following Arabic practices advocated by scholars such as al-Uqlīdisī and al-Khwārizmī. Later thirteenth-century texts following in Fibonacci’s wake, such as Alexander de Villa Dei’s Carmen de algorismo and John de Sacrobosco’s Algorismus vulgaris, used the term algorismus, a corruption of the name al-Khwārizmī, to refer to this new art (Burnett 2006: 19). While he did not use the term himself, Fibonacci was thus the forerunner of the algorithmists who, in direct conflict with the abacists, promoted the use of written numerals for computation rather than the use of counting boards (cf. Evans 1977, Murray 1978). This technique is the precursor to modern computational techniques with pen and paper. Despite the unquestionable importance of Gerbert, Fibonacci, and other mathematicians in introducing ciphered-positional numerals to the West and promoting their use, their eventual adoption is not a vindication of a “great man” theory of history. Among the names given to the zero-sign in a late twelfth-century Latin manuscript (Cambridge, Trinity College R.15.16, Fol. Av) was “chimaera,” suggesting that it was assimilated only with difficulty into the conceptual system of medieval mathematical thought. Western numerals were not initially given the same conceptual status as letters of the alphabet, or even as Roman numerals. They were, instead, seen as characteres, signs to be made on physical artifacts, and only gradually assimilated into texts as written signs (Burnett 2006: 29). The diffusion of the Western numerals from Andalusia and North Africa to the West occurred slowly, numerous times and by several different routes, some of which were more fruitful than others (Gibson and Newton 1995: 316). In fact, Greek and Italian mathematical manuscripts contain the standard Arabic “Eastern” positional numerals, not ghubar-derived ones, through the twelfth century (Burnett 2002b). A late twelfth-century Latin manuscript from Bavaria contrasts the “Toledan figures” and the “Indian figures” (Burnett 2002b: 241). Only in the thirteenth century, when many of the important Toledan astronomers moved to northern Italy, were the standard Arabic forms fully abandoned. Contact between the Arab and Western cultural spheres followed several paths in the Middle Ages: through Spain, to be sure, but also through Norman Sicily, along main trade routes from African cities such as Tunis and Tripoli to Venice and Genoa, through the Crusader states such as Antioch, and through Byzantine Arabs (Burnett 2006). Far from being an instantaneous adoption, then, the Western numerals were used only by a small number of Western European scholars in the Middle Ages.

222

Numerical Notation

The ordinary populace of Western Europe used Roman numerals, if any, while Eastern Orthodox regions used alphabetic systems such as the Greek or Cyrillic alphabetic numerals. I have already discussed the various transitional and blended versions of Western and Roman numerals used from the twelfth through seventeenth centuries, and the various medieval prohibitions enacted against the use of Western numerals in Florence, Padua, and Frankfurt (Chapter 4). Most notable among these are the “Visigothic” Roman/Western blended systems of medieval Spain, where the forms of the Roman numerals affected the writing of similar-appearing Western numerals (Lemay 1982). Also as mentioned earlier, mixed Greek-Western and Hebrew-Western numerical structures were used in some late medieval mathematical and astronomical documents (Chapter 5). In astronomical texts, numbers up to 360 (i.e., degrees of the circle) continued to be written in alphabetic numerals (usually Greek, sometimes Hebrew or Arabic), while higher numbers were written in Western numerals (Burnett 2006: 20). Whether we regard these blends as hidebound efforts to retain fragments of an older notation or as progressive attempts to innovate, they illustrate the often haphazard manner in which the Western numerals came to be introduced into European scholarly circles. Table 6.17 demonstrates the slow transmission of Western numerals throughout Europe, including both their first occurrence in each region and the period in which they became more commonly known. In general, Latin and scholarly (particularly mathematical and astronomical) uses of the numerals preceded their vernacular and commercial use by several centuries (Murray 1978: 193–194). Most of the earliest examples of the numerals in any given region are found in mathematical treatises and texts designed specifically to explain the new numerals. Where Western numerals were used in nonscientific contexts, they often served cryptographic or secretive functions. One of the earliest such instances comes from the legal documents of a notary from Perugia dating from 1184 to 1206, in which the numerals indicated the numbers of lines of documents (Burnett 2006: 20). Similarly, the early thirteenth-century Genoese notary Lanfranco used Western numerals only in the margins of his private documents to make records of payments made to him by clients, while retaining Roman numerals otherwise (Krueger 1977).9 Western numerals were used, however, as assembly marks on timbers of the roof of Salisbury Cathedral in the 1220s; we do not know why this was done, and there is no obvious reason why they would have been preferred over Roman numerals (Tatton-Brown and Miles 2003). Only when the audience for 9

The quasi-alphabetic “Fez numerals” used by notaries in the Maghrib (Chapter 5) served similar functions, but in that case the nonpositional system was the obscure one and the positional system the more commonplace.

South Asian Systems

223

Table 6.17. Early Western numerals in Europe Location Spain

Italy

France

England

Germany/ Austria Greece Scandinavia

Iceland

a

First Attested Example 976: Codex Vigilanus (Hill 1915: 29) c. 1050–1075: Pandulf of Capua’s De Calculatione (Gibson and Newton 1995) mid to late 11th century: abacus treatises (Hill 1915: 29) c. 1130: Adelard of Bath’s translation of al-Khwārizmī 13th century: Wells Cathedral (first epigraphic use) (Morley 1947: 81) 1143: translation of al-Khwārizmī into Latin at Vienna (Menninger 1969: 411) 12th century: commentaries on Euclid’s Elements (Wilson 1981) c. 1275–1300: Valdemar’s yearbook (Kroman 1974: 120) c. 1310–1330: Haukr Erlendsson’s translation of Carmen de algorismo (Benedict 1914: 17; Bekken 2001)

Portugal

1415: Livro da Virtuosa Bemfeitoria (Barrados de Carvalho 1957: 124–125)

Russia

17th century: sundials (Ryan 1991)

Common Use 1490:a dating pages in texts c. 1325: banking records and account books in major cities (Struik 1968; Menninger 1969: 428) c. 1400: dating, accounting, etc. 1525–1550: archival records, accounting books (Jenkinson 1926) 1600–1630: probate inventories (Wardley and White 2003) 1525–1550 (Smith and Karpinski 1911: 133) c. 1400: Ottoman conquest of most Greek-speaking areas c. 1550 (books, manuscripts, records) c. 1550 (books, manuscripts, records) 1490–1510: travelogues, scientific documents (Barrados de Carvalho 1957: 125) 16th–17th century: general use (de Oliveira Marques 1996) early 18th century (reforms of Peter the Great)

Arabic documents from Spain used the ghubar numerals extensively from the tenth century ad onward; this date refers only to their common use in Christian Spain.

the numerals expanded from monks and astronomers did Western numerals begin to replace Roman numerals more generally, however. The common use of Western numerals in Europe was surely aided by the transmission of double-entry bookkeeping from Italy in the fourteenth and fifteenth centuries; this technique greatly benefits from ciphered-positional numerical notation (Jenkinson 1926: 267). Yet the Italian merchant Francesco Datini (c. 1335–1410), an early adopter of double-entry bookkeeping, switched his

224

Numerical Notation

accounts to Western numerals in 1366 but adopted double-entry accounts only in 1383, suggesting that the numerals may in fact have been the cause rather than the effect (Crosby 1997: 205–206). By 1600, however, Roman numerals were essentially absent in most accounting traditions. Continental Europeans started to date coins with Western numerals in the fifteenth century, the first being a Swiss coin from 1424; Austria followed in 1456, and France, Germany, and the Low Countries in the final quarter of the fifteenth century (Hill 1915: 94–105).10 The earliest Western numerals on English coins are those of an issue of Henry VIII (undated, but with the regnal name “Henric 8”) from 1526–44 (Wardley and White 2003: 15–16). The increasing complexity of the alloying of coinage, particularly the need for complex divisions involving fractions, between the thirteenth and seventeenth centuries made the adoption of Western numerals by goldsmiths and mint workers highly useful (Williams 1995). Once coins began to be minted and records kept using the new numerals, their spread to a large segment of the populace was inevitable. The rise in frequency of the Western numerals corresponds well with the birth of printing in the middle of the fifteenth century. The rise of literacy after the invention of the printing press, and the consequent expansion of the use of numeration to a broader range of people, correlated with a new willingness on the part of the middle class to use the new invention for a variety of purposes, including bookkeeping, inscriptions on coins and seals, foliation, and stichometry. Throop (2004) emphasizes that the integration of Western numerals into Western typography was both a technical and a conceptual issue; typographers had to determine not only how to distinguish 0 / o / O or 1 / l (a problem not yet fully resolved today) but also how to produce numerals that accorded with the aesthetic canons of the rest of the typeface, a task first accomplished by Claude Garamond (1480–1561). Doing so further helped in the standardization of numeral-forms, which had been extremely flexible prior to the fifteenth century (Hill 1915). Printed books were first paginated in Western numerals in 1470: an edition of Chrysostomus’s Homiliae in Rome (McPharlin 1942: 20–21) and Werner Rolewinck’s Sermo in festo praesentationis in Cologne (Archibald 1921: 423). Bibles began being printed using Western numerals in the mid sixteenth century (Williams 1997). In all of these cases, micro-scale processes and effects relating to users’ social status and proximity to major centers of learning affected the exact dates at which the transition from Roman to Western numerals occurred. Wardley and White (2003) have demonstrated that even within a single country (England) and considering 10

A copper coin of Norman Sicily dated to 533 ah (1138 ad) is the earliest positionally dated coin in Europe, but it is inscribed and dated using the Arabic script and numerals (Hill 1915: 16; Menninger 1969: 439).

South Asian Systems

225

only a single document type (probate inventories), regional differences in the dates of adoption of Western numerals were as much as a century – in Newcastle and some regions near the Thames, Roman numerals were still used commonly as late as 1700. Nevertheless, the general trend in all regions was always in favor of Western over Roman notation. By the eighteenth century, Roman numerals in Western Europe served primarily archaic and formal functions, ending two millennia of their effective domination. It is remarkable that a system that had not yet been established in its heartland in the fifteenth century could almost entirely replace not only the Roman numerals but also many other ancient systems in less than three centuries. The spread of the Western numerals throughout the world, and their eventual replacement of large numbers of indigenous numerical notation systems, occurred, however, only when European countries had become politically powerful, as the modern world-system began to form with the Western European nations at the core. The replacement of non-Western numerical notation systems began en masse in the sixteenth century.11 Within a few decades of Europeans reaching the New World, the Aztec and Maya systems had become obsolete and the Inka khipu greatly restricted in scope. At around the same time, the ciphered-additive systems of Eastern Europe and the Caucasus (Cyrillic, Glagolitic, Armenian, and Georgian) began to be replaced by Western numerals or Arabic positional numerals. The modern era of colonialism brought about the replacement of further systems starting in the nineteenth century. The Hebrew, Coptic, and Syriac alphabetic numerals all continue to be used for religious and formal purposes, but Western numerals are used in most other contexts. The indigenous numerical notation systems of South and East Asia have not been completely replaced, but they too have been supplanted for many purposes by the Western numerals. While there is no functional reason for the replacement of one ciphered-positional system by another, the dominance of the European nations, coupled with the desire to have a single, universally intelligible symbol system, have made the Western numerals an attractive option. Even in places like Japan and Thailand, which were never under direct political control by a European power, the Western numerals are usually preferred. By the late nineteenth century, the history of the Western numerals had been sufficiently obscured that considerable disagreement arose regarding their purported origins. While the correct notion that they originated in India and were 11

Curiously, the sixteenth century also marked the development of the attribution “Arabic” to the Western numerals, whereas earlier they had always been seen (correctly) as an Indian invention (Clark 1929: 217). It is possible that this occurred due to growing awareness of Arabic learning among early modern scholars.

226

Numerical Notation

transmitted through an Arabic intermediary was always prominent, Kaye (1919) and Carra de Vaux (1917), among others, espoused the theory that their origin was among Greek Neo-Pythagoreans, who transmitted the knowledge of positionality to the Persians and ultimately to India. This imperialistic theory has no redeeming virtues and no evidence in its favor; there is no evidence that the Neo-Pythagoreans knew of place value. This theory rests chiefly on the misguided notion that India has produced nothing of real scientific value. At the same time, the spread of Western numerals in the nineteenth and twentieth centuries has spawned a host of descendants among North American and African peoples, such as the Iñupiaq, Cherokee, Oberi Okaime, and Mende systems (Chapter 10). Many of these are structurally different from the Western numerals, and are not simply the ordinary system recast with new numeral-signs. Modern octal, binary, and hexadecimal numbers used for electronics make use of the Western numeral-signs and ciphered-positional structure, merely substituting different numerical bases. As well, while not structurally significant, graphic variations of many of the numeral-signs are quite common (e.g., ! vs. #, $ vs. %, & vs. *, ( vs. )). Despite the near-universality of the Western numerals, the history of numerical notation is thus far from complete.

Summary We end in the modern era with a host of local numerical notation systems and two (the Western and Arabic) that spread enormously on the heels of political conquests, but we should not forget the origins of these systems in the South Asian regional tradition originating with Brāhmī. The common feature of the South Asian systems is the set of nine Brāhmī unit-signs – which persist, though greatly altered, in the surviving numerical notation systems – and a decimal structure. Only the alphasyllabic systems use distinct signs, the letters of the Indian alphasyllabaries. It is common practice to end studies of numerical notation with the Western numerals (e.g., Guitel 1975, Ifrah 1998). Yet to do so portrays the spread of Western numerals throughout the world as the inevitable replacement of worse with better systems, in continuous progress from primitive beginnings to the perfection of the Western decimal positional system, an achievement that can never be surpassed. Most surviving modern systems are ciphered-positional, which indicates that they are useful, but this does not demonstrate that they are the inevitable conclusion of a teleological historical process. If technology truly spread only through the diffusion of what is functional and the replacement of what is not, one would expect that structurally identical systems should expand with equal rapidity and geographical reach. The South Asian phylogeny, with so many decimal cipheredpositional systems surviving and in use, provides a good testing ground for this

South Asian Systems

227

theory. Because only the Arabic and Western numerals (and, to a lesser extent, Nagari) have spread extensively, their diffusion must be due mainly to sociopolitical factors. Furthermore, the geographic distribution of the surviving nonpositional systems is no less than that of many positional ones. Why would the additive Tamil system be as widespread as, say, the Khmer system, if functionality is of supreme importance? Do we really expect Tibetan ciphered-positional numerals to survive and Chinese multiplicative-additive ones to decline? I do not mean to suggest that functionality has nothing to do with the spread of numerical notation systems, especially ones such as the Arabic and Western numerals that have been used extensively for accounting, arithmetic, and mathematics. Yet to proclaim the Western numerals’ spread as the triumph of functionality and reason over illogic and unwieldiness is to ignore the many ciphered-positional systems that have failed to spread – or failed to survive. While, owing to the political might of nations that use them, the Western numerals are very important, they are merely one branch of one phylogeny. In placing them in the middle of my study, I choose to emphasize that their present triumph is neither inevitable nor eternal.

chapter 7

Mesopotamian Systems

Numerical notation first developed in Mesopotamia around 3500 bc, contemporaneously with or slightly earlier than its development in Egypt. Scholars interested in the diffusion of Babylonian astronomy and mathematics to the Greeks have long studied Mesopotamian numeration (Neugebauer 1957, van der Waerden 1963). Yet to depict the Mesopotamian phylogeny of numerical notation systems as an archetypal case for the evolution of numerals, or to use it as the basis for a universal evolutionary pattern, is dangerous. While Mesopotamian mathematics is important for understanding later Greek developments (and, in turn, modern Western mathematics), Mesopotamian numeration is nearly a historical dead end. Although their history spans three millennia, the Mesopotamian numerals did not spread geographically far beyond their point of origin, and did not survive when placed under pressure from the numerical notation systems of later inhabitants of the region. The main signs of the Mesopotamian numerical notation systems are shown in Table 7.1. There are several ways to classify them, depending on which features we emphasize. Looking at the numeral-signs alone, the systems divide rather neatly into archaic systems, used prior to 2000 bc and written using curviform symbols made with a round stylus, and later cuneiform systems, written using wedgeshaped symbols. Both were written almost exclusively on tablets using a stylus to impress signs onto wet clay. A second important distinction is between systems that are primarily decimal and those that are primarily sexagesimal, or base-60. 228

229

Babylonian positional

Old Persian

Hittite

g

f f f

f

g g g

f

AssyroBabylonian

Mari

g

f

g

B

B B Z

A A S

A

10

1

Sumerian

Cuneiform Systems

Proto-Elamite decimal

Bisexagesimal 2

Bisexagesimal

Sexagesimal

System Archaic Systems

f

f

f

f

C C P

60

f i w

i

T

100

Table 7.1. Mesopotamian numerical notation systems

M Q

120

g

h

D

600

fm

gi

[

1000

N R

1200

f

j

E

3600

O

7200

x

]

10,000

g

k

F

36,000

230

Numerical Notation

Mesopotamia is the only region of the world where sexagesimal numerical notation is attested.1 Finally, comparing the interexponential structures of the systems, we can distinguish between additive systems, which include most of the systems, and positional systems, of which the only true example is the Babylonian positional system.

Proto-cuneiform Around 3200 bc or perhaps slightly earlier, the antecedent of the later Sumerian script arose at the city of Uruk in southern Mesopotamia, during what is now known as the Uruk IV period.2 This proto-script, which was probably read in Sumerian, lacked any means of expressing phonetic sounds. By the Uruk III period (c. 3000 bc), it had spread from Uruk (the primary Mesopotamian city at the time) to the north, to Jemdet Nasr, Khafaji, and Tell Uqair. The texts of this period of Mesopotamian history do not represent a true literate tradition but rather a protohistoric system of bookkeeping and administration. In total, about 5,600 clay tablets have been recovered that record this script, known as proto-cuneiform. Around sixty of the twelve hundred proto-cuneiform signs can be assigned numerical or metrological values (Nissen, Damerow, and Englund 1993: 25). Although Falkenstein (1936), who wrote the first comprehensive description of the Uruk tablets, thought the proto-cuneiform texts from Uruk were both decimal and sexagesimal, Friberg (1978–79, 1984) determined that there was no protocuneiform decimal system. In the 1980s, using computer-aided analysis of the entire corpus of texts, Nissen, Damerow, and Englund (1993) established that as many as fifteen distinct systems (of which five were particularly common) were used at Uruk, each used for enumerating a specific category of discrete objects or metrological quantity.3 By examining the maximum number of times each numeral-sign is repeated, they determined the relative values between signs in any given system (as if we were to infer that the Roman numeral V represents 5 by noting that I is repeated four times at most). This technique works because cumulative-additive systems order powers within each numeral-phrase from

1

2

3

However, Price and Pospisil (1966) claim that the Kapauku of Papua New Guinea derived their sexagesimal lexical numerals from the comparable Babylonian numerical notation. Over the past twenty years of research, the chronology of protohistoric Mesopotamia has been shifted backward; older sources tend to regard the Uruk IV period as representing the early third rather than the late fourth millennium bc. My discussion of the systems (including their functions) is derived almost entirely from the work of Nissen, Damerow, and Englund (1993: 25–29).

Mesopotamian Systems

231

highest to lowest and regularly replace lower power-signs with higher ones wherever possible. A difficulty is that a given numeral-sign may be found in several of the proto-cuneiform systems, but its value often varies from system to system. Thus, B is equal to 10 A in some systems, but to 6 A in others. Yet while we can identify the numerical ratio between the values of any two signs within a system, we often cannot identify the specific quantity represented by any one sign. For systems used for counting discrete objects, it is easy to identify the basic sign for 1, since fractions of humans do not normally occur in texts, but for systems that measure area or capacity, we can never ascertain with certainty which sign (if any) has the basic value of one unit. Following Nissen, Damerow, and Englund (1993), I present the values for these metrological systems as ratios, since we can only tell the value of a sign relative to the other signs of the system. Despite having different numeral-signs and different numerical values, all the proto-cuneiform numerical systems have much in common. All are cumulativeadditive, although some individual numeral-signs are formed multiplicatively – e.g., D (600) = C (60) × B\(10). Groups of identical signs were sometimes sorted into two or three rows for easy reading, but this is not a universal rule, and some tablets contain long strings of signs. Numerals were most often grouped with signs arranged from highest to lowest (although there are some rare exceptions, which may be scribal errors). A single numeral-phrase, together with one or more ideograms, was enclosed in a box in a section of the text.4 The proto-cuneiform texts are mainly accounting documents, often written on both sides – the obverse with a series of amounts of commodities, the reverse with a single total. The greater-than-expected prevalence of round or nearly round numerals in proto-cuneiform texts allows the identification of hypothetical problems that were used as training exercises and thus were not actual economic texts (Friberg 1998). Learning the various signs, the ratios among them, and how to construct texts would have required considerable scribal training, including “school mathematics” associated with the temple economy at Uruk (Robson 2007: 63–64). For instance, the late fourth-millennium tablet W20044,20 is an exercise in calculating the area of an irregular quadrilateral field (Robson 2008: 30). Throughout the history of the various systems’ use, only a very small portion of the populace would have had access to the training necessary to master the protocuneiform notations. 4

Note that the proto-cuneiform script was written vertically in columns reading from top to bottom, but I follow Assyriological convention (and that used by Nissen, Damerow, and Englund) in showing the signs rotated ninety degrees counterclockwise and thus read horizontally from left to right. This convention reflects a similar change in the direction of writing cuneiform signs around the middle of the third millennium bc.

Numerical Notation

232

Table 7.2. Sexagesimal numerals 36,000 a

Sexagesimal (S)

F

3600 =10 E

600 =6 D

60

1

1/2

=10 C =6 B

=10 A

( =6 &

=10 ^

Sexagesimal (S’) a

10

=2

L

The letters in parentheses in this table and the following ones are those assigned to each system by Nissen, Damerow, and Englund (1993) in their research.

Sexagesimal Systems The two sexagesimal systems shown in Table 7.2 alternate between factors of 6 and 10, and were the first and easiest to be deciphered because their structure is identical to that of the later Sumerian numerals. The main sexagesimal system (S) is employed in slightly less than half the Uruk texts (Damerow 1996: 292). It was used to enumerate most discrete objects: humans, animals, finished products, tools, and containers, which explains its frequency of use. The subsidiary S’ system was used to enumerate a much smaller category of discrete objects, such as dead animals and jars of some liquids.

Bisexagesimal Systems The two bisexagesimal systems shown in Table 7.3 are so named because an additional factor of 2 is interpolated among the factors of 6 and 10 used in the sexagesimal systems. While the regular bisexagesimal (B) system is identical to the regular sexagesimal (S) system up to 60, it has new signs for the values of 120 (60 × 2), 1200 (120 × 10), and 7200 (1200 × 6). It enumerated discrete numbers of grain products, cheese, and fresh fish, and is the second most common system found in the archaic texts. The function of the identically structured but much less common B* system is unclear, but it may have indicated discrete quantities of some kind of fish. Both systems appear to have been part of a rationing system, one for which a number-sign between 60 and 600 may have been useful.

Table 7.3. Bisexagesimal numerals 7200 Bisexagesimal (B) Bisexagesimal (B*)

O

1200 =6

120

60

10

1

N

=10 M

=2

C =6 B

R

=10 Q

=2

P =6 Z =10 S

=10 A =2

1/2

L

Mesopotamian Systems

233

Table 7.4. GAN2 numerals

E

=6

F

=10

B

W

=3

=6

A

=10?

L

GAN2 System The GAN2 system shown in Table 7.4 is used to represent area measures. While its signs are the same or similar to those of the common sexagesimal system (S), the GAN2 signs’ values differ from those of the main system. For instance, where F means 36,000 in sexagesimal numerals and is thus 3,600 times greater than B (10), in the GAN2 system it is only ten times greater. This similarity probably has something to do with the use of round-ended writing styli in all the proto-cuneiform numerical systems.

EN System This uncommon notation system, shown in Table 7.5, is known from only twenty-six texts, and may have represented weight measures. All but one of the tablets from Uruk on which the EN system was used were found at a single location, suggesting that whatever its function, it must have been very restricted in use.

ŠE Systems This relatively common group of numerical systems, shown in Table 7.6, denoted various capacity measures of grain. While its signs are similar to those of the sexagesimal systems, their order and the ratios between successive signs are quite different. For instance, while the ratio between A and B is 10 in the sexagesimal, bisexagesimal, and EN systems, it is only 6 in the ŠE systems. The regular Š system enumerated capacity measures of barley, the Š’ system for germinated barley for brewing beer, and the Š* system for barley groats.

U4 System This rather unusual numerical notation system becomes less so considering that its function is for recording time and calendrical units. By combining a single Table 7.5. EN numerals

B

=10

A

=2

Ä

=2

L

Numerical Notation

234 Table 7.6. ŠE numerals System Š

D =10 C =3

System Š’

ë =3

System Š*

E =10 B =6 A =5 J Æ =10 ô =6 ö =5 ò =5 û è =10 ï =6 î =5 ì

ideographic sign with numerical signs for 1 and 10, all the major divisions of the year could be expressed easily.

The Origin of Proto-cuneiform Numerals The most popular theory on the origins of Mesopotamian numeration is that it emerged from a system of clay tokens used for accounting in preliterate times. Throughout Mesopotamia and even further abroad, small clay objects of various shapes and sizes have been found in strata dating between 9000 and 2000 bc. Oppenheim (1959) assigned an administrative function to a hollow clay ball, or bulla, found at Nuzi inscribed with a brief cuneiform text enumerating forty-eight animals, and containing within it forty-eight small stone counters. Amiet (1966) showed that this technique was used much earlier than previously thought (since at least 3000 bc) and that the bullae were “double documents” through which transfers of goods such as livestock could be conducted while minimizing the risk of fraud or error. A literate official could see the quantity of goods from the inscription on the outside, but if there was any doubt, the bulla could be broken open and the clay or stone tokens inside counted to match them up with the actual quantity received. More recently, Denise Schmandt-Besserat (1984, 1987, 1992) has shown that the clay tokens are of even greater antiquity, and are ancestral to both the protocuneiform numerals and the proto-cuneiform script. She holds that the tokens represent a stage of “concrete counting,” fusing quantity (the number of tokens) and quality (different shapes representing different commodities), but do not represent abstract numbers (Schmandt-Besserat 1984: 55). A number of late fourthmillennium bc bullae, especially from Susa in modern Iran, are impressed with Table 7.7. U4 numerals \\+ * 10 months / 1 year

=10 =12

, 1 month

=3

10 days

=10

. 1 day

Mesopotamian Systems

235

signs resembling later archaic numerals and contain the correct total of tokens, suggesting that the systems are connected (Nissen, Damerow, and Englund 1993: 127–129). Furthermore, there are some similarities between the three-dimensional tokens and the proto-cuneiform ideograms, suggesting that the tokens developed into writing through the recognition that, if the total of a transaction is written on clay, one need not actually use the clay tokens but need only record their values. Schmandt-Besserat’s conclusions have been received with some skepticism (see especially Lieberman 1980, Zimansky 1993).5 Firstly, the scope in time and space of the token system is far greater than that of the proto-cuneiform numerals; it is implausible that such a widespread phenomenon represents a uniform system. Moreover, in Schmandt-Besserat’s study of tokens from the Uruk-Jemdet Nasr period (c. 3000 bc), around two-thirds of the tokens come from Susa in Iran, while only 10 percent come from the very thoroughly excavated site at Uruk (Lieberman 1980: 353). This suggests that the token system is unlikely to have given rise to numerals and writing at Uruk. Finally, some of the most common tokens are correlated with proto-cuneiform signs for rare objects such as nails and days of labor, whereas given the accounting function established for the tokens, we would expect livestock, people, and grain to be the most common tokens, as is the case in proto-cuneiform texts (Zimansky 1993: 316). This discrepancy points to a further problem. The archaic numeral systems always place a numeral-phrase in front of an ideographic sign; “16” + “sheep” = “16 sheep,” and so on. While the proto-cuneiform numerals partly fuse quantity and quality, because different systems represent different commodities, they do not do so completely, because one always needs a further sign to indicate exactly what is being counted. With the tokens, however, there is no separation of numerals and the objects being counted; to show sixteen sheep, one simply uses sixteen tokens for “sheep.” Thus, there is no correspondence between the archaic numeralsigns and the shapes of tokens. The use of tokens sealed within bullae appears to have been an accounting technology that pre-dated, but then later coexisted with, the proto-cuneiform numerals. While some early proto-cuneiform numerals are found on clay bullae, this is insufficient evidence that tokens led to numerals. Conversely, numerical signs resembling the proto-cuneiform ones have been found, not on bullae, but on ordinary clay tablets in late preliterate contexts at Uruk as well as at Jebel Aruda, Susa, and elsewhere (Nissen, Damerow, and Englund 1993: 127–130; see especially Figures 113, 114). These tablets have numerical signs only (no ideograms), and disobey the ordinary rule that once a certain 5

I cannot hope to address her claim that the tokens are ancestral to the proto-cuneiform script, and will restrict myself to the similarities and differences between the token system and the proto-cuneiform numerals.

236

Numerical Notation

number of lower-valued signs have been written, they are replaced with a single higher-valued sign. For instance, one tablet from Jebel Aruda contains twenty-two B signs, among others (Nissen, Damerow, and Englund 1993: 130). In any of the later systems, twenty-two signs would have to be replaced by a smaller number of higher-valued signs. Because these inscriptions are found in late preliterate contexts and are similar but not identical to the proto-cuneiform numerals, they are immediately ancestral to them and date from a period when the system was still being developed, weakening the hypothesis that they derived from tokens. While we do not know the language in which proto-cuneiform numerals were read, the Sumerian lexical numeral system is mainly sexagesimal, and furthermore, 10 is a sub-base in the lexical numerals just as it is in the proto-cuneiform numerals (Powell 1971, 1972a, 1972b). On this basis, Powell (1972b: 172) has correctly discerned that “the presence of a sexagesimal system of notation in the archaic texts from Uruk and Jemdet Nasr constitute [sic] the best – indeed irrefutable – evidence that Sumerian is the language of those texts.” Yet, as Høyrup (2006: 79–82) points out, proto-cuneiform notation was not meant to record any spoken language, but was an artificial recording system designed for limited purposes, and only very gradually became assimilated to Sumerian.6 The multiplicity of proto-cuneiform numeral systems and bases is not lexical in origin, but is likely based on Sumerian metrological systems in the Uruk period, for which we have minimal nontextual evidence. We do, however, have substantial textual evidence for the metrological systems of the Early Dynastic and later periods. The ratios between various signs in the proto-cuneiform numeral systems dealing with measures of capacity, area, and weight are similar to the ratios found in later Sumerian metrological systems. This supports the contention that the odd ratios of some of the older systems are due to unattested metrological systems that continued into better-documented periods. The increasing administrative demands associated with the rise of the Uruk city-state in the late fourth millennium bc created a new need for record keeping, metrology, and accounting, of which the numerals and the clay bullae are two distinct consequences.

Cognitive Consequences of Proto-cuneiform Numeration The analysis of the proto-cuneiform numerals has also led researchers to speculate on the possible cognitive correlates of the use of multiple numerical notation systems. Damerow (1996) has argued that the material record from the archaic period in 6

Høyrup attributes this purpose as being the administration of a multiethnic society of slaveholders and immigrant slaves, which we need not accept to recognize that the general principle is true.

Mesopotamian Systems

237

Mesopotamia directly reflects the numerical abilities of Mesopotamians, and represents a universal stage of concrete numeracy that precedes the modern abstract number concept. In this respect, his argument is similar to that of Hallpike (1979), who applies the insights of Piaget, Vygotsky, and others from developmental psychology to draw a parallel between individual cognitive development and the evolution of thought in societies. Damerow claims that the peoples of early Mesopotamia could not conceive of abstract numbers, but were only capable of concrete counting (Damerow 1996: 275–297). Because multiple proto-cuneiform numerical notation systems were used for representing different objects, and a single sign could have different relative values in different systems, he argues that Mesopotamian scribes could conceive of “8 sheep” or “8 jars of oil” but not simply “8.” Taken to its logical conclusion, this would imply that users of the proto-cuneiform numerals could see nothing in common between eight sheep and eight jars of oil. I cannot see how this can be the case; if so, it would be impossible to make any connection between eight sheep and eight marks on a clay tablet, and numeration would be impossible. Furthermore, in order for these context-dependent numerals to represent a stage of “archaic arithmetic” in the evolution of numeration, as Damerow (1996: 296) suggests, we would expect similar systems to be present in other civilizations. Yet nothing of the sort can be found in Shang, Predynastic Egyptian, or Zapotec inscriptions, the other early and independently invented systems. This is not to say that there are no cognitive consequences of the use of a dozen or more numerical notation systems, but whatever they are, they will not be universally applicable to every society. Finally, we have no idea how many of these systems would have been known to any individual official, and no evidence from the archaic period as to how numerals were manipulated and used arithmetically. We simply have values and totals, which do not tell us very much about how people were actually thinking about number.7 Even if individuals used many systems, this does not prove concreteness of thought. In contrast to the proto-cuneiform numerical notation, there was a single perfectly ordinary set of Sumerian lexical numerals (Powell 1971).8 Someone capable of abstract thought might well use multiple systems of numerical notation to 7

8

Liverani’s (1983) intriguing conclusion that a fragmentary Uruk IV-period clay tablet indented with holes may have served as a counting board has not been confirmed and must remain tentative unless further finds are made. Schmandt-Besserat (1984, 1992) has made much of the parallel between the many protocuneiform numerical notation systems and the use of “numeral classifiers” in Japanese, the Mayan languages, and others, where the set of numerals is modified depending on the class of object being counted. Numeral classifiers are not a feature of Sumerian. If taken to its logical conclusion, this would imply that the modern Japanese do not have a concept of abstract number.

238

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prevent confusion as to the type of thing being counted, or to correspond to metrological systems. There is no qualitative difference between the Uruk systems and the modern use of Roman numerals to distinguish the foreword of a book from its main text, or the use of hexadecimal numerals for computing purposes. Ironically, one of the principles behind the “new mathematics” movement in North America in the 1960s was the claim that teaching students to calculate using numerical systems of different bases would improve their understanding of abstract number concepts. I believe the Uruk scribes had an abstract number concept, but realized that abstract written numerals were not the most efficient solution to the problems they were facing. The theoretical importance attributed to the proto-cuneiform numerals as evidence of an evolutionary stage of cognition is entirely unwarranted.

Convergence and Decline While they are an interesting early example of numerical notation, the protocuneiform numerals did not diffuse extensively or last for an extended period. There are no significant resemblances between the proto-cuneiform numerals and the Egyptian hieroglyphic numerals (Chapter 2), which may precede the protocuneiform systems in any case. Their only descendants were the proto-Elamite systems used from about 3000 bc at the site of Susa and elsewhere in modern Iran, and the later Sumerian numerals. The start of the Early Dynastic period in Mesopotamian history marked a turning point in the history of its numerals. Beginning around 2900 bc, there was a marked decline in the frequency of almost all the proto-cuneiform numerical systems, while the sexagesimal system rapidly assumed the functions of the other systems. While the system for measuring area (GAN2) continued to be used as late as the Fara period (c. 2500 bc), it was in decline and considered archaic by that point (Nissen, Englund, and Damerow 1993: 137–138). While each metrological system had its own numerical notation system in the archaic period, eventually officials decided it was better to express all numbers, regardless of function, using a single notation. There are three plausible explanations for this convergence. The simplest is that the use of so many systems for so many different functions was cumbersome for administration, potentially confusing, and open to abuse. This may simply be a modern prejudice attributable to the Western use of only one set of numerals. While 200 years is a short time in the context of world history, it is a long time for a truly inefficient set of systems to persist. Secondly, while the archaic texts were used at only a very few locales (mainly at Uruk), the later numerals were used throughout Mesopotamia. If the Early Dynastic period marks the first era when Mesopotamian numerals were employed for long-distance communication, the use of a single system to facilitate communication among many individuals

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239

would be advantageous. Finally, changes in Sumerian metrological systems may have reduced the usefulness of the proto-cuneiform systems by eliminating the fit between metrology and numeration.

Proto-Elamite Around 3100 bc, a ideographic writing system developed in southern and western Iran, the region known as Elam in later Mesopotamian sources. This script, now known as “proto-Elamite,” is attested in over 1,500 texts, mainly from the major urban center of the region, Susa; most date from the Susa III period around 3000 bc. A few other proto-Elamite texts have been found at Tepe Yahya and elsewhere in modern Iran. It is a linear script, read from right to left and in lines proceeding from top to bottom. The language it was intended to represent cannot be identified, but proto-Elamite numerals can be read. While the proto-Elamite ideograms are different from those of early Mesopotamia, the proto-Elamite numerals are very similar to the proto-cuneiform systems. As with the proto-cuneiform numerals, confusion over the nature and number of proto-Elamite numeral systems has delayed their correct decipherment until recently. Brice (1962–63) provides a useful summary of several early twentiethcentury efforts to decipher the proto-Elamite numerals, all of which assume a single decimal and cumulative-additive numerical notation system. An adequate decipherment of the proto-Elamite numerals has been achieved recently through the mathematical analysis of the corpus of proto-Elamite texts by Robert Englund and Peter Damerow (Damerow and Englund 1989, Englund 1996). Damerow and Englund realized that, as with the proto-cuneiform numerals, not only were there multiple proto-Elamite numerical notation systems, but the relative values of individual numeral-signs vary from system to system. There are five major protoElamite systems: three for counting discrete objects, another for capacity measurements, and another for area measurements (Englund 1996: 162). The proto-Elamite numerical notation systems for counting discrete objects are shown in Table 7.8 (Englund 1996: 162; cf. Potts 1999: 78).9 The three systems are identical for 1 and 10, and the sexagesimal and bisexagesimal systems are further similar for 60. The sexagesimal system, like the proto-cuneiform sexagesimal numerals, is not a pure base-60 system; instead, each successive number alternates by factors of 10 and 6; that is to say, it has a sub-base of 10. In the bisexagesimal system, the value 120 comes after 60 (a factor of 2). 9

As with the proto-cuneiform numerals, I have represented the numerals as they would be read horizontally (following Assyriological convention; cf. Damerow and Englund 1989) rather than vertically (cf. Englund 1996).

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240

Table 7.8. Proto-Elamite numerals (discrete objects) 1

10 60 100 120 600 1000 1200 3600 10000 Function Inanimate E B C D objects

Sexagesimal

A

Bisexagesimal

A B

Decimal

A B

M

C

T

N

Grain products

[

]

Animate objects

The main signs of the systems for measuring capacity and area are shown in Table 7.9. Because they are not used for discrete objects, they are represented in terms of the ratios between values, not as discrete numerical values. These two systems are very similar (though not identical) to the ŠE and GAN2 proto-cuneiform systems, so, following Damerow and Englund, I have used those labels. The striking resemblances between the proto-cuneiform and proto-Elamite numerals make it certain that the latter were modeled on the former (Potts 1999: 76–77). In fact, while the respective scripts are entirely dissimilar, it is a matter of personal preference whether we regard the proto-cuneiform and protoElamite numerals as distinct sets of systems or as two regional variants of a single tradition. Because the first texts from Uruk date to the thirty-third century bc, while those found at Susa date to the late thirty-first century bc, the protoElamite ones cannot have been ancestral to those at Uruk, and must have diffused from west to east in the context of interregional trade. Given the importance of the Uruk city-state in the late fourth millennium bc, it is unsurprising that the numerals would spread to Susa, the other major polity at that time. The main difference between the two sets of numerical notation systems is the existence of a decimal system for counting discrete quantities of animals and humans in protoElamite. It may be that the language of the writers of the proto-Elamite texts had decimal lexical numerals, whereas we know that Sumerian numerals are primarily sexagesimal.

Table 7.9. Proto-Elamite metrological numerals Capacity (ŠE) Area (GAN2)

X =6 D =10 C =3 E =10

B

=6

A

=5

E =10?

B

=3

W

=6

J K A

=2

Ä

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241

The proto-Elamite numerals did not spread beyond Susa and a few other sites in modern Iran. Brice’s (1963) tentative identification of similarities between the proto-Elamite and Linear A (Minoan) numerals cannot be taken seriously as indicative of a historical connection, given the geographical and temporal distance between the two. The proto-Elamite numerals ceased to be used around 2900 bc, following the decline of Susa as a major urban polity in the early part of the third millennium bc and the subsequent rise of the various Mesopotamian city-states. The numerals in the Old Elamite cuneiform texts, which are roughly contemporaneous with the Old Akkadian texts in Mesopotamia, are derived from later Mesopotamian systems rather than from proto-Elamite (Potts 1999: 79). The proto-Elamite numerals are best seen as a brief florescence within a single citystate, rather than as part of a longer tradition.

Sumerian The only system among the multitude of proto-cuneiform systems to survive into the Early Dynastic period (2900 to 2350 bc) was the sexagesimal (or, more accurately, the decimal-sexagesimal) system. While it was originally used only for counting discrete objects, it began to be used for all numerical functions as the older metrological systems were abandoned. At the beginning of the Early Dynastic, significant changes were taking place in the script of the region. The older ideographic and curviform proto-cuneiform symbol system slowly transformed into a writing system that used wedge-shaped (cuneiform) signs and expressed phonetic as well as conceptual information. From this, we can tell that Sumerian was the language in which the script was read. Yet, despite these alterations to the script, the numeral-signs remained identical to the archaic sexagesimal ones. One important change occurred around the twenty-seventh century bc, when the numerals, like the entire script, underwent a ninety-degree rotation, so that they were written and read horizontally from left to right rather than vertically from top to bottom. The Sumerian numerals are shown in Table 7.10 (Nissen, Damerow, and Englund 1993: 28). These six numeral-signs were combined to make a cumulative-additive numerical notation system. Normally, groups of four or more signs were arranged in two rows to facilitate rapid reading. Because this system has signs for both 60 and 3600 (= 602), it has a sexagesimal component. In a purely sexagesimal system, one would need to repeat each sign up to fifty-nine times, which is impractical, but the Sumerian system also has sub-base signs for 10, 60 × 10 (600), and 3600 × 10 (36,000). The latter two signs are multiplicative combinations of the small circle for 10 with the sexagesimal signs for 60 and 3600. This decimal sub-base is similar, but not identical, to the use of the sub-base of 5 in the Roman numerals. While the figures of the

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242

Table 7.10. Sumerian archaic numerals 1

10

G B Horizontal A B 14,254 = EEE\DDD DD

Vertical

60

\A

3600

36,000

H I E F C D E F CCCC\BBB\AA CCC\ AA

(3 × 3600) + (5 × 600) + (7 × 60)

19 = BB

600

+ (3 × 10) + (4 × 1)

y

(20 LAL 1)

Roman sub-base (V, L, D) could occur only once in any numeral-phrase because 5, 50, and 500 are half of 10, 100, and 1000, respectively, the decimal signs in the archaic Mesopotamian numerals could be repeated up to five times. The sign for 60 is a large version of the sign for 1, just as the sign for 3600 is a large version of the sign for 10. Because the “big 1” is 60 times greater than the regular 1, but the “big 10” (3600) is 360 times greater than its counterpart, I cannot agree with Lieberman’s (1980: 343) suggestion that these signs represent the use of “size-value,” which then evolved into “place-value,” over time. This feature simply derives from the fact that two styli, one twice as large as the other, were used to impress numerical signs on clay tablets (Powell 1972a: 11–12). The Sumerian numerals provide the first evidence for the use of subtractive notation to express certain numbers, especially those that end in 8 or 9 in the Western numerals, using a Sumerian ideogram that corresponded with the phonetic value LAL. Thus, instead of writing 19 as one sign for 10 plus nine signs for 1, it could be written as 20 – 1, as seen in Table 7.10. A sign or signs placed inside the LAL sign indicated an amount to be subtracted from the signs preceding it. This technique was used at Fara (ancient Šuruppak), perhaps as early as 2650 bc (Jestin 1937: Pl. LXXXIV). There is no evidence of subtractive lexical numerals in Sumerian comparable to the Latin duodeviginti and undeviginti; the Sumerian words for 18 and 19 are etymologically ‘10 + 5 + 3’ and ‘10 + 5 + 4’, respectively (Powell 1971: 47). Rather, this innovation had its origin strictly in numerical notation and the desire to express numbers more concisely. As in the archaic period, Early Dynastic numerals are found overwhelmingly in economic or administrative texts, or in scribal exercises related to these functions. In the archaic period, there was no indication how calculations were being done (though calculations must have been made). At Fara, however, Sumerian “tables of squares,” geometrical and arithmetical exercises, and other arithmetical aids

Mesopotamian Systems

243

Table 7.11. Sumerian cuneiform numerals 1

10

60

600

3600

36,000

216,000

f

g

f

h

n

k o

nm

LAL:

u

have all been found (Powell 1976, Høyrup 1982). Nevertheless, the use of numerals for representation, especially in administrative contexts, greatly exceeds the frequency of their use for computation. There is nothing indicating the direct use of Sumerian numerals for computation (by lining up columns, etc.), as in Greek and Western arithmetic. Damerow (1996: 236–237) laments the fact that, despite the wealth of Early Dynastic economic records, we have no idea how multiplication was performed; he suggests that it must have been through a nonpermanent means, such as counting boards, finger reckoning, or mental calculation. By 2500 bc, the transition from the older Sumerian script to cuneiform signs had been completed, except for the numerals. Beginning in the Presargonic period (c. 2600–2350 bc), the older curviform numerals began to be replaced with a set cuneiform numeral-signs, while remaining virtually unchanged structurally (Powell 1972a: 13). This had the advantage of requiring only one stylus for all writing, whether lexical or numerical. While this trend appears to have been initiated by the Sumerians themselves, it was hastened considerably, starting around 2350 bc, by the rise of Akkadian hegemony over Mesopotamia. These new numeralsigns are shown in Table 7.11 (Powell 1971: 244). The signs for 1 and 60, which had previously been semicircular and horizontal, became vertical wedges. The earliest cuneiform numeral-sign for 60 was written as a “big 1,” just as it had been in the curviform numerals, but because the two signs were made with the same stylus, the size difference was always minimal, and soon the two signs became identical (Powell 1972a: 13). This feature does not mean that the system used the concept of place-value, although it may have played a role in the invention of the later sexagesimal positional system (Powell 1972a: 13–14). The old round sign for 10 was replaced by a Winkelhaken or corner wedge, made by impressing the stylus onto the clay tablet perpendicularly, while the large round sign for 3600 was represented visually by four (or occasionally five) wedges placed in a rough circle. In other respects – the writing of 600 and 36,000 as 60 × 10 and 3600 × 10, and the basic cumulative-additive structure – the cuneiform numerals were identical to the curviform ones. The phrase used for 216,000, not attested in the earlier numerals, is a combination of the sign for 3600 and the ideogram

244

Numerical Notation

GAL ‘big’, and is quite rare (Powell 1972a: 7). Powell also describes an even more complex phrase for 12,960,000 (216,000 × 60), šargal šunutaga, ‘big everything which hand cannot touch’. For such lexical phrases, we need to ask at what point a phrase ceases to become part of a numerical notation system. The subtractive ideogram LAL is used in this system, as in the archaic one, but it is depicted using two cuneiform wedges. The replacement of the curviform by cuneiform numerals was a gradual process. While the older system required additional styli to write numerals, and its numeral-signs generally took up more space, it stood out more clearly in a text of cuneiform characters, making totaling easier (Powell 1972a: 12). Additionally, a norm developed by which the two sets of numerals could be used side by side to indicate different functions. Possibly the older numerals indicated quantities directly counted, possibly using clay counters, while the cuneiform numerals enumerated quantities of objects not actually present to be counted (Lieberman 1980: 344–345). Alternately, Damerow (1996: 238) notes that some Early Dynastic economic texts from Girsu use the older numerals for amounts of grain and the cuneiform numerals for amounts of animals, and hypothesizes that this may have been done to avoid confusing the two different categories when taking sums. This use of multiple numerical notation systems is analogous to the modern use of Roman and Western numerals side by side. The round Sumerian numerals had been abandoned by the Ur III period (when Sumerian rulers regained control of Mesopotamia), and are not attested later than 2050 bc (Powell 1972a: 13). The Akkadian conquest, the most important political event of third-millennium bc Mesopotamia, had a minimal effect on numeration. The Akkadian kings and officials (c. 2350–2150 bc) were content to use the cuneiform and even the archaic numerals for most of the same purposes for which they had been used in the Early Dynastic period. More change in the numerals is visible in the Neo-Sumerian Ur III period (2150 to 2000 bc), during which the archaic numerals disappeared entirely. One slight modification that was tried in some Akkadian texts was to write multiples of 60 using units followed by the Akkadian lexical numeral for 60, šu-ši (:), using multiplicative notation (Labat 1952: 244–247). Thus, instead of writing 120 as ff, it would be written as ff:. This is a much more cumbersome representation, and probably was used in part to distinguish 120 (ff) from 2 (ff). For the higher decades – 70, 80, and 90 – the regular Sumerian forms were always used by the Akkadians (fg, fgg, and fggg, respectively). Regardless, many Akkadian inscriptions where šu-ši could have been used are written in the ordinary Sumerian fashion. The Sumerian cuneiform system is ancestral to all the later systems of Mesopotamia. The Semitic cuneiform decimal systems (Eblaite and Assyro-Babylonian) were directly derived from a Sumerian ancestor. The decimal structure of these

Mesopotamian Systems

245

systems reflected the lexical numerals of the Semitic languages of its users. While Thureau-Dangin (1939: 107) believed this tradition to have been developed in the Old Akkadian period (starting c. 2350 bc), it is now clear from the library at Ebla that it developed as early as 2500 bc (Pettinato 1981). The sexagesimal cumulativepositional system used in Babylonian mathematics and astronomy was also modeled on the Sumerian cuneiform system. It may have arisen in the Ur III period, and was used by the twentieth century bc at the very latest (cf. Powell 1976, Whiting 1984). The Sumerian cuneiform system continued to be used for most purposes until the Old Babylonian period (c. 2000–1595 bc). Around that time, the Assyro-Babylonian decimal system began to be used for most administrative, commercial, and literary functions, while the sexagesimal positional system was used for mathematics and astronomy – once again perpetuating the tradition of using multiple numerical notation systems for multiple purposes. Several Old Babylonian tablets provide translations from the old Sumerian additive numerals to the new positional system (Nissen, Damerow, and Englund 1993: 146–147), indicating either a need to learn the new positional system or, alternatively, that the older cuneiform system was already being forgotten. By the fifteenth century bc, it had disappeared from regular use. However, a peculiar vestige of the base-60 Sumerian system survived in certain late inscriptions, particularly those indicating the sizes of buildings (De Odorico 1995: 4). One such example is the Nameninschrift of the Assyrian king Sargon II (722–705 bc), which describes the dimensions of the fortress at Khorsabad as “16,283 cubits, the numeral of my name,” notated using Sumerian numerals rather than the Assyro-Babylonian system that would be expected (Fouts 1994: 207). In this case, the notation served literally to indicate a numeral corresponding to the royal name of Sargon by correlating the signs for 3600, 600, and 60 with the phonetic values šar, nêr, and šûš, which could not have been done using the Assyro-Babylonian decimal system. This inscription has been compared to later Greek, Hebrew, and Arabic gematria using alphabetic numerals (Chapter 5), but it seems to be an earlier and independent development.

Eblaite The inhabitants of the city-state of Ebla (in the western part of modern Syria) spoke a West Semitic language but were strongly influenced by Sumerian culture. A great library of thousands of Eblaite cuneiform texts dating certainly to the period prior to 2350 bc (the Akkadian conquest), and possibly as early as 2500 bc, provide us with ample evidence regarding the numerals used by the Eblaites (Pettinato 1981). This system is almost identical to that used by the Babylonians some centuries later, and reflects the shared Semitic language and culture of these

Numerical Notation

246 Table 7.12. Eblaite numerals 1

10

60

100

1000

10,000

mi-at li-im rí-ba A B C f g f AA\rí-ba AA li-im AAA mi-at CBBAA AA\\\\\\\ AAA

x

100,000 ma-i-at OR ma-i-hu

x

2

10,000

4

1000

6

100 60 10 10 1 1 = 24,682

two groups in contrast to those of the Sumerians. As indicated in Table 7.12, the Eblaite numerical notation system consisted of two sets of numeral-signs for numbers below 100, one curviform and the other cuneiform (corresponding with the Sumerian archaic and cuneiform systems), but only one set of expressions for the powers above 100 (Pettinato 1981: 183–184). The Eblaite system is cumulative-additive for values less than 100, and multiplicative-additive above that point. The signs for 1, 10, and 60 are ideographic signs identical to those used in the two Sumerian sets of numerals. The two sets of numeral-signs served quite separate functions: the curviform numerals were used for basic enumeration and counting discrete objects, while the cuneiform numerals were used only for capacity measures such as the mina and gubar, as well as for regnal years of kings (Pettinato 1981: 183–184). The sign for 60 was used to express the tens values in numbers between 60 and 99; its presence, a holdover from Sumerian, is the major irregularity in an otherwise perfectly decimal system. The “signs” for numbers above 100 are in fact the Eblaite lexical numerals and were combined multiplicatively with the unit-signs as necessary. Because it is decimal and multiplicative-additive above 100, this system required only one ideographic sign (the crescent or vertical wedge) for the higher powers; however, the repetition of intraexponential signs for the units, coupled with the use of complex two- and three-syllable words, meant that numerals were fairly long and cumbersome. To reduce this length, two features were often used. Firstly, just as in the Sumerian system, subtractive numerals were sometimes used for certain numbers to eliminate the need to write seven, eight, or nine unit-signs by placing the subtrahend after the syllable lal or lá. Secondly, the words mi-at for 100 and li-im for 1000 were often shortened to the single syllables mi and li, respectively. Thus, in one text, 7879 (expressing a number of gubar measures of barley) is written in cuneiform numerals as 7 li 8 mi 60 10 10 lá-1 (7 × 1000 + 8 × 100 + 60 + 10 + 10 – 1) (Pettinato 1981: 134). Such syllabic abbreviations are reminiscent of the Greek acrophonic numeral-signs (Chapter 4).

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247

Most Eblaite texts served economic or metrological functions. It is not clear whether the Eblaite numerical notation was ancestral to the later Assyro-Babylonian system or whether the latter developed out of the Sumerian cuneiform system in parallel to the Eblaite system. Because the two systems are very similar in structure (even including their common use of multiplicative structuring above 100 with abbreviated lexical numerals), the possibility that the earlier Eblaite notation was borrowed by the Babylonians is a good working hypothesis. The Eblaite system did not persist past about 2300 bc, after which point Ebla came under Akkadian, and later Amorite, control.

Assyro-Babylonian Common Because historians of mathematics are especially interested in the origins of our base-60 units of time and the division of the circle, enormous attention has been paid to the Babylonian positional numerals – the cumulative-positional, base-60 system used for astronomy and mathematics. The far more common decimal and additive numerals (which I call the Assyro-Babylonian common system), which were used for most economic, monumental, and literary purposes throughout Mesopotamia, are almost forgotten. This system came into common use in the Old Babylonian period (starting c. 2000 bc), a position it would maintain for over 1,500 years. The numeral-signs of this system are shown in Table 7.13 (De Odorico 1995: 4). The system is cumulative-additive below 100, multiplicative-additive above 100, and is always written from left to right. For the most part, it is purely decimal. The units were expressed cumulatively, except that 9 could be written using three overlaid vertical strokes ($) as an alternative to writing it with nine strokes (9) (De Odorico 1995: 4n). The tens values were usually expressed decimally using one through nine Winkelhaken corner wedges for 10. The vertical wedge for 60 is identical to that for 1, but unlike the Sumerian system, f was not normally used to represent 60 alone, which would have created ambiguity, but only in combination with signs for 10 and 1 to write numbers from 70 to 99. As in the Akkadian variety of the Sumerian cuneiform system, the lexical numeral šu-ši (:) was used to indicate 60 or multiples thereof; however, this phonetic form was not used to write 70, 80, or 90, and even for 60 its use declined over time. Therefore, there were as many as three different numeral-signs for 60, one decimal (ü) and two sexagesimal (: or f). Above 100, the multiplicative principle was used quite freely and could be combined in various ways to express very high numbers. The sign for 100 was the syllabic sign ME, an abbreviation of the Babylonian word for 100, me’at, while that for 1000 was a multiplicative combination of the signs for 10 and 100. For instance, a scribe from the period of Sargon II wrote 305,412 as shown in

Numerical Notation

248

Table 7.13. Assyro-Babylonian common numerals 1

10

60

f

g

f

305,412 =

or

:

100

1000

i

gi

3i5gi4ia2 ((3 × 100 + 5) × 1000) + (4 × 100) + 10 + 2

Table 7.13 (Ifrah 1998: 139). In theory, this system could be extended as far as one wished by juxtaposing signs for 100 and 1000 repeatedly, even though there was no sign for zero. Furthermore, unlike the Sumerian system, in which the signs for 1 and 60 were identical, this system presented no ambiguities to the reader. While this system was sometimes called “Akkadian” (Thureau-Dangin 1939), it was rarely used during the period of Akkadian control of Mesopotamia and began to predominate only during the Old Babylonian period. It originated in response to the increased power of Semitic peoples in Mesopotamia in the latter part of the third millennium bc: Akkadians, to be sure, but also Eblaites, Babylonians, and others. Its structure reflects the decimal lexical numerals of the Semitic languages rather than Sumerian lexical numerals, although the continued use of a special sign for 60 gives testament to its descent from the Sumerian numerals. All of the administrative, commercial, literary, and religious texts of the Babylonians and the Assyrians were written using this set of numerals. Perhaps the greatest significance of the Assyro-Babylonian common system is the large number of descendant systems it produced. Earliest among these is the system used at the city-state of Mari around 1800 bc, which blends features of this system and the Babylonian positional system. In the middle of the second millennium bc, both the Ugaritic and the Hittite cuneiform scripts began using numerals based on the Assyro-Babylonian ones, in the context of Mesopotamian trade with these polities. The Ugaritic texts written between the fifteenth and twelfth centuries bc at Ugarit on the Mediterranean coast use the cuneiform ideograms for 1 (f) and 10 (g); however, numerals in Ugaritic were normally written lexically (Gordon 1965: 42). This was likely borrowed from the Assyro-Babylonian system, but we have no idea whether higher numbers could be expressed through numerical notation. The Old Persian cuneiform numerical notation system, developed in the sixth century bc (by which time Mesopotamia was under Persian rule), also derived from the Assyro-Babylonian system rather than from any of the numerous other systems then used in the region. Finally, as I argued in Chapter 3, the earliest Levantine systems (Phoenician and Aramaic) developed around 800 bc as a blend of Egyptian hieroglyphic (or perhaps Hittite) and Assyro-Babylonian influences.

Mesopotamian Systems

249

Table 7.14. Graphic changes in numeral-phrases 4 Sumerian Babylonian

p 4

7

q 7

8

s 8

9

40

r t 9 d

The Assyro-Babylonian additive system flourished despite enormous political changes. It was the system used for administration and commerce by both the Babylonians and the Assyrians until the Persian conquest of Babylon in 539 bc. Afterward, it began to be supplanted by the Old Persian cuneiform system and, more importantly, by the Aramaic system that then became the principal Mesopotamian administrative and commercial system. Both these systems were indebted greatly to their Assyro-Babylonian ancestor. It is unclear when the Assyro-Babylonian system disappeared entirely, but it was used at least to a limited extent throughout the period of Achaemenid rule (539–332 bc). The latest cuneiform tablets date to the first century ad.

Babylonian Positional The Babylonian positional numeral system is assigned such great importance by many historians of mathematics that one could easily get the impression that it was the only noteworthy form of Mesopotamian numeration. Despite Neugebauer’s (1957: 17) warning that the positional numerals are a relatively minor part of the body of Babylonian numerals, these sexagesimal positional numerals, used for mathematics, have been assigned priority over much more widespread systems (Sumerian and Assyro-Babylonian). In fact, positional numerals were used in only a limited set of mathematical and astronomical contexts and over a much shorter period, serving primarily as a means of easing certain computations (Robson 2008: 75–6). The system uses only two basic numeral-signs, the vertical wedge f for 1 and the corner-wedge or Winkelhaken g for 10, to write any number between 1 and 59. Thus, small numeral-phrases were usually identical to those of the Sumerian cuneiform system. Nevertheless, certain graphic changes (shown in Table 7.14) were made to the numeral-phrases for 4, 7, 8, 9, and 40, so that, instead of grouping signs in at most two rows of up to five signs, three rows of no more than three signs were used. This shift eliminated any phrases that placed four or five signs side by side, and may have increased the system’s legibility (Powell 1972a: 16). Unlike the earlier cumulative-additive Mesopotamian systems, this system was cumulative-positional, combining the two basic signs in multiple positions to

250

Numerical Notation

express powers of 60. It was thus a base-60 system with a sub-base of 10. It had an additive structure within each power, because of the way that 10-signs and 1-signs combine together, but a positional structure among different powers. Just as in the Sumerian and Assyro-Babylonian systems, subtractive notation was frequently used to write numbers such as 9 (10 lal 1) and 19 (20 lal 1) (Thureau-Dangin 1939: 106). According to the rules of the system, 4,252,914 would be written as a9 d4 b a4 = (19 × 603) + (44 × 602) + (20 × 60) + 14. In addition to expressing integers, positional numerals could be used to express fractional powers of 60: 1/60, 1/3600, 1/216,000, and so on. During the Old Babylonian period, the positional numerals did not have any sign for zero to indicate an empty position within a numeral-phrase, nor was there any way to distinguish an integer from a fraction (i.e., there was no “sexagesimal point”). However, many texts list numbers in columns in which the positional values of all the numbers are lined up with one another, making misinterpretation less likely (Powell 1976: 421). When numbers are embedded in the middle of a text or occur alone, the lack of a zero leads to ambiguity; there is no way, except through contextual information, to determine which positional value expressed which power, and thus a single numeral-phrase could have an infinite number of readings. The simple phrase a2 3 could mean 723 (12 × 60 + 3), 43,380 (12 × 3600 + 3 × 60), 12.05 (12 + 3/60), and so on, depending on which positional values we assume are indicated. When the empty position was both preceded and followed by numerals, this difficulty was sometimes obviated by using a large empty space to indicate the empty position (Neugebauer 1957: 20). Thus, 1 b (80) could be distinguished from 1 b (3620). Yet this technique was not used universally, and in some texts what looks to be a large space does not bear any numerical significance. Unless numeral-signs were arranged in columns, there was no way during the Old Babylonian period to distinguish numbers where the empty position came at the end or beginning of the numeral-phrase. Nevertheless, by organizing numbers in columns, and through commonsense interpretations of texts, Babylonian mathematicians would not have experienced insurmountable difficulties in reading numbers despite these ambiguities. The Babylonian positional notation probably developed, in fact, in the twentyfirst century bc, during the Ur III (Neo-Sumerian) period. The late Sumerian system of weight units is purely sexagesimal and notated in a way that could be ancestral to positional notation (Powell 1972a: 14). Powell (1976: 420) also found positional numerals on several early texts, which led him to assert that the development of positional numerals occurred in the twenty-first century bc at the very latest. Robson (2007: 78–79) discusses twenty-first-century texts from the cities of Umma and Girsu that clearly depict sexagesimal place-value numerals. The development of the notation may have resulted from Ur III administrative reforms,

Mesopotamian Systems

251

which followed from managing much larger amounts of goods than had previously been the case (Powell 1976: 422; Høyrup 1985: 9). Nissen, Damerow, and Englund (1993: 142), however, remain agnostic regarding Ur III positional numerals, because most texts can be dated only paleographically, and the numerals do not show much variation throughout time. I am unconvinced by Whiting’s (1984) assertion that the positional numerals developed as early as the Old Akkadian period (i.e., the twenty-fourth or twenty-third century bc). Most of the early texts containing the positional numerals are mathematical texts of the Old Babylonian tradition, and thus date between 2000 and 1600 bc, with the majority from the latter part of that period (Powell 1976: 419). These range from simple multiplication tables and arithmetical exercises to complex problems that can legitimately be called algebra. Education in the positional notation and in “pure” mathematics was a significant component of scribal education, although of course only a small proportion of the total populace would have had access to such training. Figure 7.1 is a small square clay tablet on which a squaring exercise has been written in Babylonian positional numerals; the two numbers on the left are each 2, 30 (150), and their product (22,500) is depicted on the right as 6, 15 – the final position, whose value would be 0, must be understood from context (Nemet-Nejat 2002: 262–263). Many arithmetical exercises and texts for translating numerals into the new positional system date from the Old Babylonian period, indicating the existence of a vigorous process for teaching the system to scribes (Nissen, Damerow, and Englund 1993: 142–147). Yet, because nonmathematical texts did not contain positional numerals, scribes who did not write mathematical texts probably would not have been familiar with the positional numerals. The converse is not the case, however; mathematical texts containing positional numerals are often dated in Assyro-Babylonian numerals, showing that mathematicians also knew the common system (Neugebauer 1957: 17). The numerals were rarely attested to have been used to perform arithmetical calculations directly, as opposed to writing down results. However, it is possible that calculations were made on clay “scratch pads” that could be moistened and rewritten to record results, after erasing the preliminary work (Powell 1976: 420–421). It is equally probable that some sort of counting board or wax writing tablet was used, on which the intermediate steps of calculations were not preserved (Høyrup 2002). We do not know much about the precise steps by which arithmetical computations were performed. After the end of the Old Babylonian period around 1600 bc, attested texts containing sexagesimal positional numerals are extremely sparse for over a millennium. The degree to which there was an actual decline in the system’s use is unclear, however. The small number of post–Old Babylonian texts, in combination with the relatively small number of place-value mathematical texts regardless

252

Numerical Notation

Figure 7.1. An Old Babylonian clay tablet (YBC 7294); the numerals at left each signify 150 (2 × 60 + 3 × 10) and the product, 22,500, is at right. Courtesy Yale Babylonian Collection.

of period, may suffice to explain the decline, or it may have actually waned in use. Robson (2007: 154–156) has published a small number of mathematical texts from late second-millennium bc Babylonia that denote numbers using this notation. In Neo-Assyrian texts such as lnam ğišḫur ankia, an omen text dating to 712 bc, a technique called “downwards upwards, upwards downwards,” playfully altered sexagesimal numerals by swapping positional registers, so that, for instance, 140 (2 × 60 + 20) became 22 (20 + 2 × 1) (Robson 2008: 148). The positional numerals reemerged during the Seleucid period (the beginning of which is dated from the Alexandrine conquest of 332 bc) in new contexts and with several structural differences from the Old Babylonian system. First, subtractive expressions such as “20 lal 1” for 19 were no longer used in Seleucid texts (Neugebauer 1957: 5). Second, while in the Old Babylonian period the positional system was used exclusively in mathematical texts, by the Seleucid period it was used in astronomical texts as well (Neugebauer 1957: 14). The most important change was the introduction, in certain circumstances, of a sign serving some of the functions of zero, usually written as 0 or %, to fill in an empty position within a numeral-phrase. This sign could be used at the beginning of a numeral-phrase to indicate that the ones place was empty (i.e., to distinguish a fraction from an integer) or in a medial position to prevent misreading 3620

Mesopotamian Systems

253

as 80, as in the earlier example, but never phrase-finally (Neugebauer 1957: 20). Neugebauer (1941) emphasized that the primary role of the zero-sign was as much epigraphical as it was mathematical. He demonstrated that in a small number of texts, a “zero-sign” was inserted, apparently superfluously, in numeral-phrases preceded by an amount in tens and followed by an amount in units. This was done in order to preclude misreading b 7 (20 × 60 + 7, or 1207) as b7 (27); by writing the former as b07, the latter interpretation is prohibited. In such numeralphrases, the “zero-sign” does not indicate an empty position, but simply separates two consecutive positions (Neugebauer 1941: 213). In fact, the zero-sign was originally used to separate sentences, which may indicate that it originally merely indicated separation or space. It is unlikely that the Babylonians conceived of zero as an abstract number. No cuneiform positional text contains the bare numeralphrase 0; it always occurs in phrases along with other signs. Thus, 0 was not equivalent to 0 in the same way that b was equivalent to 20. The abstract concept of zero accompanied by a special sign for that concept developed independently among the Greeks and Indians, but probably never among the Babylonians. Despite its use of the positional principle, the system was restricted to an extremely small group of Babylonian mathematicians and astronomers in the Old Babylonian and Seleucid periods. It does not appear to have been known by merchants or most administrators, and certainly did not diffuse to other peoples of the Middle East, such as the Aramaeans, Phoenicians, and Persians. There is no reason to believe, as was formerly held by some, that the Babylonian positional numerals survived long enough to be ancestral to the Indian positional numerals (Février 1948: 585; Menninger 1969: 398–399). The only direct descendant of the Babylonian positional numerals is the sexagesimal Greek positional system used by classical mathematicians and astronomers to represent fractions. Around the second century bc, the Greeks combined the Babylonian system with their alphabetic numerals to produce a positional, base-60 numerical notation system (Chapter 5). Neugebauer (1975: 590) states that the use of a sexagesimal division of the circle into sixty parts by Eratosthenes (ca. 250 bc) is the earliest evidence of this borrowing, although Eratosthenes did not use the sexagesimal fractions. This Greek system was used only in mathematical and astronomical texts, and only for fractions. The quasi-positional cuneiform system used in a few texts in the city-state of Mari also appears to derive in part from the Babylonian positional system. While the Seleucid astronomical texts are important from the perspective of the history of science, they represent the work of a limited group of scholars whose knowledge was being surpassed by Greek mathematics and astronomy even in the fourth century bc. The Greek alphabetic numerals were those used for everyday purposes as well as for mathematics, and so, slowly, the Babylonian system fell into decay. The last example of positional cuneiform numerals dates from the first

Numerical Notation

254 Table 7.15. Mari numerals 1

10

100

1000

10,000

f

g

i

m

x

century ad (Powell 1972: 6a). That this system, the first positional system ever and one much lauded by modern scholars, should be replaced, after having nearly been abandoned once before by its own inventors, suggests that the advantages of positional systems do not correlate closely with their survival.

Mari The city of Mari, located on the Euphrates River at the border of modern Syria and Iraq, was an independent city-state between the twentieth and eighteenth centuries bc. During this time, it was engaged in extensive trade relations with Canaan and Babylonia. A large number of cuneiform tablets have been recovered from Mari, mainly dating to the eighteenth century bc, upon which a very unusual numerical notation system has been found. This system is shown in Table 7.15 (Durand 1987; Ifrah 1998: 142–146). Below 100, the system is purely cumulative-additive. For the units, it is identical to the Assyro-Babylonian system, but in the tens, there is no special sign for 60; rather, the higher decades are written as ü, é, â, and å, respectively. For the hundreds, the multiplicative ideogram i (ME) was often used (preceded by unit-signs for multiples). However, some texts omit the i sign entirely, turning the system into a quasi-positional one. Thus, 476 is written in one inscription as p\é\6 (Soubeyran 1984: 34). The sign for 1000 is a syllabic representation of “LI-IM,” while the sign for 10,000 is created by superposing the signs for 10 and 1000. For powers above 1000, the system is multiplicative-additive; there are no instances where higher power signs are omitted. The omission of the multiplicative sign for 100 is very interesting, but it does not represent a fully positional system. If it did, we would expect 476 to be written as p\7\6 (4 7 6), not as p\é\6 (4 70 6). The Mari system resembles the experiments with positionality in India in the seventh century ad, before the positional principle was fully adopted (Chapter 6). Most of the administrative and commercial texts from Mari use the AssyroBabylonian common system, while mathematical texts mostly use the Babylonian positional numerals. One text (M7857) uses both systems side by side (Robson 2008: 130). We thus know that the Mari scribes understood these systems

Mesopotamian Systems

255

Table 7.16. Hittite cuneiform numerals 1

10

60

f

g

f or : i

7169=

100

1000

10,000

;

\<

7;\fi:9 (7 ×

1000 +

1 × 100 + 60

+ 9)

perfectly well. What may have happened, at least in the case of mathematical texts, is that the aberrant system was used unofficially (perhaps for calculation) and then retranscribed for official purposes using the sexagesimal positional numerals (Soubeyran 1984: 34). It is possible that the decimal structure of the Mari numerals was borrowed from the Assyro-Babylonian ones, and that the idea of using positional notation for the hundreds was taken from the Babylonian positional numerals. This system should be considered as an aberrant and short-lived experiment with positionality, as the conquest of Mari by Hammurabi in 1755 bc ended its use.

Hittite Cuneiform In Chapter 2, I discussed the Hittite hieroglyphic system, which was probably borrowed from the Egyptian hieroglyphic system or the Linear B (Mycenaean) system. A separate Hittite script, written in cuneiform characters and related to the various Mesopotamian scripts, was used at the Hittite capital of Hattusha between the seventeenth and thirteenth centuries bc. The numerals used alongside this script, shown in Table 7.16, are similar to those of the Assyro-Babylonian common system (Rüster and Neu 1989: Table 7). The system is decimal and cumulative-additive below 100, while the multiplicative principle is used for higher powers. Numeral-phrases are written from left to right. To write 60 or multiples of 60, the Akkadian loanword šu-ši (:) was employed, just as it could be in other cuneiform systems. Yet, when writing numeral-phrases between 70 and 99, a simple vertical wedge represented 60 (Rüster and Neu 1989: 271). The sign for 100 is simply the ME syllable of Assyro-Babylonian numerals borrowed into Hittite, while the complex and apparently lexical expressions for 1000 and 10,000 appear to be indigenous. Thus, 7169 was written as shown in Table 7.16 (Rüster and Neu 1989: Table 7). There is no evidence for the use of subtractive notation in the Hittite cuneiform numerical notation system.

Numerical Notation

256

Table 7.17. Old Persian cuneiform numerals 1

10

100

f

g

w

604=

6wp

Because the royal archives at Hattusha are our main source for Hittite cuneiform inscriptions, we do not yet know a great deal about the range of functions for which Hittite numerals were used. We can be quite certain that the numerals were borrowed from the Assyro-Babylonian common system, which was widely used at the time the Hittite numerals are first attested, given the two systems’ close similarity in structure and numeral-signs. There does not appear to be any connection between the Hittite cuneiform and the Hittite hieroglyphic numerals, which are entirely different in structure. With the collapse of Hittite power in the thirteenth century bc, the cuneiform numerals ceased to be used.

Old Persian The Old Persian script was invented early in the domination of the Achaemenid Empire over Mesopotamia, probably near the beginning of the reign of Darius I (522 to 486 bc) (Testen 1996). It is an alphasyllabary; thus, while its letter-signs are cuneiform, it represents a distinct break from the older Assyrian and Babylonian scripts. The Old Persian numerals, shown in Table 7.17, closely resemble the Assyro-Babylonian common numerals (Testen 1996: 136). The system is decimal and cumulative-additive, with numeral-phrases written from left to right. The sign for 100 combines multiplicatively with preceding unitsigns in the one attested text containing a large numeral-phrase. No known Old Persian texts have numerals of 1000 or higher. Whereas the Assyro-Babylonian units could be grouped in sets of two or three, in up to three rows, Old Persian units and tens were arranged in at most two rows, with odd units represented at twice the size of paired ones. There is no trace of sexagesimal notation in Old Persian; 60 is expressed with six signs for 10 and 600 as 6 × 100. Although the Old Persian script has traditionally been regarded as strictly a display-oriented script used for prestige purposes related to the Achaemenid monarchy, the recent translation and interpretation of an Old Persian administrative record from the Persepolis Fortification Archive has cast some doubt on this notion (Stolper and Tavernier 2007). Figure 7.2 shows an Old Persian

Mesopotamian Systems

257

Figure 7.2. Old Persian tablet from the Persepolis Fortification Archive (Fort. 1208–101, obverse), which begins with the number 604, the only evidence for multiplicative notation for the Old Persian hundreds. Courtesy of Matthew Stolper.

cuneiform text (Fort. 1208–101, obverse) originally excavated in 1933 from Persepolis but published only in 2007, which begins with the numeral 604, probably denoting a quantity of a dry capacity measure (Stolper and Tavernier 2007: 12–13). This numeral is the highest found to date in any Old Persian inscription; it is one of only two to contain the sign for 100, and the only one to denote higher hundreds, thus attesting the hybrid multiplicative structuring. As an isolated document among many thousands of Elamite and Aramaic economic and other texts found at Persepolis, Fort. 1208–101 does not prove widespread Old Persian literacy or numeracy in the Achaemenid period, but it does provide tantalizing evidence that the script’s use was not as limited-purpose as was once believed. The invention of the Old Persian numerical notation system occurred in a context of cultural contact between Persians and Assyrians/Babylonians in the late sixth century. There are a number of Babylonian/Persian bilingual inscriptions, and the two systems existed side by side at that time (as evidenced by the thousands of cuneiform documents at Persepolis). The absence of any attested trace of sexagesimal notation is a unique feature that clearly differentiates this system from its Assyro-Babylonian ancestor, however. While Fort. 1208–101 shows that Old

258

Numerical Notation

Persian numerals (and indeed, the script) were used administratively in at least one instance, the Aramaic numerals (Chapter 3) were the system used much more commonly for international communication and commerce throughout Persia, and the Assyro-Babylonian numerals were used alongside cuneiform inscriptions in the Elamite language. The Old Persian system did not survive the Alexandrine conquest of Persia.

Summary The commonality among the Mesopotamian numerical notation systems is their use of cumulative notation with signs for 1 and 10. In other respects, there is considerable variation among these systems, whether due to linguistic (Sumerian vs. Semitic) or functional (administrative vs. mathematical) factors. The survival of the sexagesimal base over a period of nearly 3,500 years is testament to the Babylonians’ preservation of Sumerian traditions, but most of the numerals used after 2500 bc were primarily decimal. The mathematical functions of the various systems, while interesting to historians of mathematics, are minimal in comparison to their administrative and literary functions. The conquest of Mesopotamia by the Achaemenids and later the Seleucids sounded the death knell for native Mesopotamian traditions, after which Aramaic and then Greek numerals were used for most purposes. Even the positional numerals, the hallmark of Babylonian arithmetical achievement, quickly disappeared under conditions of cultural and political domination. The AssyroBabylonian common system was borrowed and modified in regions of the Middle East where Mesopotamian influence was strong, but the history of Mesopotamian numerals is largely linear rather than branching, with each system giving rise to its successor but not giving rise to many systems outside Mesopotamia. Multiple systems were often employed at the same time within Mesopotamia, the use of which was determined by their contexts in ways that remain unclear. With the sole exception of the Levantine systems, which derive from both Mesopotamian and Hieroglyphic ancestors, the Mesopotamian systems did not give rise to a large number of descendants either within Mesopotamia or without.

chapter 8

East Asian Systems

The East Asian numerical notation systems, like the region’s scripts, reflect the pervasive importance of Chinese civilization over the past three millennia. The “classical” Chinese system used from the Qin dynasty (221–206 bc) to the present day, and which spread throughout the region, is foremost in duration and significance. Still, the history of East Asian numeration is one neither of total Chinese hegemony nor of complete stasis, and these systems are more diverse in structure than any of the other phylogenies I have investigated. Attested historical connections and similarities in their numeral-signs allow us to identify their common ancestry. Table 8.1 indicates the most common numeral-signs of the East Asian systems.

Shang and Zhou Chinese mythical histories record that the Yellow Emperor directed the scholar Li Shou to create mathematics and the abacus (suan pan) in the twenty-seventh century bc (Li and Du 1987: 1); however, the first well-attested phonetic writing and numeration in East Asia dates to the latter part of the Shang Dynasty (ca. 1523–1028 bc). The most common Shang inscriptions are records of royal divinations written on bones and tortoise carapaces, called “oracle bone inscriptions” ( jiaguwen) by modern scholars, most of which were found at Anyang in Henan province. These brief texts, which date from 1300 to 1050 bc, often contain 259

260

1 2 3 4 5 6 7 8 9 0 /

! @ # $ % ^ & * ( ) Ñ ¢ £

Kitan

Jurchin

ℛ

ₘ

⥪

ሏ ℣

ሐ ⏼

ሑ

O

ₒ

ሒ

P

⏺

ሓ

Q ◐

f

f

⃬ ◐

ሔ

V

R

x

J

J

◒

f

f

ₖ

J

J

ₖ

₏

ሎ

T

Q

◒

~

Chinese positional

ል

S

P

䤍

(

10,000

ሌ

L

O

⃬ ◐

:

1000

Chinese commercial

K

N

⏺

J

100

J

M

ₒ

o

10

Late rodnumerals

L

⏼

n

9

K

℣

m

8

J

⥪

l

7

Rod-numerals

ₘ

j

6

ℛ

i

5

₏

h

4

Chinese classical

g

3

f

2

Shang / Zhou

1

Table 8.1. East Asian numerical notation systems

0

0

0

榅

0

East Asian Systems

261

Table 8.2. Shang numerals 1 1 10 100 1000 10,000 4539 =

2

f g q < : ( ~ ) * > o

3

4

5

6

h >

i ?

j $

k %

l

7

8

9

m

n &

o

numerical indications of tribute received, animals hunted, numbers of sacrificial victims, or counts of days, months, or other miscellaneous quantities relating to divinations (Takashima 1985: 45). The attested numeral-signs used on oracle-bone inscriptions are shown in Table 8.2 (Needham 1959: Tables 22, 23; Djamouri 1994: 39). The Shang numerical notation system combines the nine unit-signs with signs for the powers of 10; it is thus multiplicative-additive and decimal. The numeralsigns for 1 through 4 are cumulative combinations of horizontal strokes, while the signs for 10 through 40, only slightly less obviously, are ligatured combinations of vertical strokes. The numbers 20, 30, and 40 were never expressed using multiplicative expressions involving the signs for 2, 3, and 4. As in most multiplicativeadditive systems, there was no sign for zero; if a particular power was not needed, no sign indicated its absence in the numeral-phrase. These are perfectly regular combinations of the sign for 10 and various unit-strokes. While Needham (1959: 13–14) makes the case that the Shang numerals contain “place-value components” because of the regular highest-to-lowest ordering of the powers, the signs for 10 and its powers cannot be omitted, and so it is multiplicative, not positional. For the tens between 50 and 90, the unit-sign was placed below the sign for 10, which was normally a vertical stroke but could apparently be a cross when writing 60. For the hundreds, the unit-sign was placed above the power-sign, while for the thousands and ten thousands, the relevant unit-sign was superimposed upon the power-sign. There was also a symbol for ‘and’ which was placed between the hundreds and tens, or the tens and ones (Martzloff 1997: 182). The signs for 1 through 4 are simple ideograms, but otherwise most of the symbols have semantic or phonetic correspondences with non-numerical words.

262

Numerical Notation

The sign for 1000 is identical to the Shang character for ‘man’, probably due to a homophonic correspondence (Djamouri 1994: 15–16; Martzloff 1997: 180–181). By contrast, the use of identical graphs for ‘scorpion’ and ‘10,000’ may result from the association of an immensely large number with swarms of colonies of newborn scorpions (Martzloff 1997: 181). The highest Shang number expressed is 30,000 (Martzloff 1997: 182). Numeral-phrases were written in vertical columns read from top to bottom, with the highest power at the top. In almost all the oracle-bone inscriptions, numeral-phrases are not found alone, but are accompanied by a character for the object being quantified. On this basis, Djamouri (1994: 33) regards Shang numeral-phrases as determinatives of nounphrases, and argues that each sign was read as a single morpheme in the ancient Chinese language. Each numeral-sign corresponds to a single Chinese morpheme, an atypical correspondence between language and numerals that leads Djamouri to regard the Shang numerical notation system as a purely linguistic rather than a “graphic” phenomenon. This feature, which it shares with the Chinese classical system, raises the issue of whether we ought to consider such quasi-lexical formulations to be “real” numerical notation systems. Like the Shang script, the Shang numerical notation system was independently invented. Needham’s (1959: 149) tentative suggestion of stimulus diffusion from Babylonia rests on the dubious notion that the Shang numerals use place-value. Moreover, the correspondence of numeral-sign and numeral-word suggests that the Shang numerals have a linguistic origin. If the signs originated to represent Old Chinese morphemes, this further confirms their indigenous development (Djamouri 1994: 18–19). Some of the Neolithic marks on pottery and tortoise shells, such as those dating from 6600–6200 bc found at Jiahu in Henan province, resemble numeral-signs, which could extend this system’s history back several millennia further in China (Li et al. 2003). Yet there are no numeral-phrases – only single signs – among these marks, and in any case there is no way to be sure that their meaning remained constant. The signs probably have a mixed abstract and phonetic origin; more important than phonetic correspondences may be the fact that most of them are graphically quite simple as compared to the other Shang characters. The function of the Shang numerals is quite clear, however, in the context of royal divinatory inscriptions.1 There is nothing resembling a Shang ‘accounting text’ or ‘commercial inventory’ parallel to those found in Mesopotamia or Egypt. After the collapse of the Shang Dynasty, large parts of what is now China were controled by the Zhou Dynasty, first from its western capital at Hao (1027–770 bc) 1

Zhang and Liu (1981–82) go still further and argue that the oracle bones mark the beginning of the long-standing tradition of bagua milfoil divination in the pattern later exemplified in the Yijing.

East Asian Systems

263

Table 8.3. Zhou numerals 1

2

3

4

5

6

7

8

9

1

f

g

h

i

j

l

m

n

p

10

r

Ä

Å

æ

j r

l r

Unattested

n r

p r

100

û ù ;

1000 10,000

and then, after the failure of the Western Zhou state, by a more decentralized Eastern Zhou polity centered at Luoyang (770–256 bc). The Zhou kingdoms continued to employ the script and numerals of the Shang. In the early Zhou period, oracle-bone inscriptions continued to be written, but from the tenth to the third centuries bc, Zhou numerals were often stamped on bronze vessels and coins (Needham 1959: 5). The increasing complexity of Chinese society over this long period brought the numerals into use for a much wider range of functions than is documented to have previously been the case. While the Zhou numerals are structurally identical to the earlier Shang ones, except that the sign for 10 could also combine multiplicatively with the unit-signs for 2 through 4, the numeralsigns began to exhibit great graphic variability. Pihan (1860: 10) provides a comprehensive chart showing the various numeral-signs used between the sixth and second centuries bc, containing, for instance, no fewer than thirty-eight different signs for 10,000. Despite this extraordinary paleographic variability, the signs shown in Table 8.3 were the ones most commonly found on coins and bronzes until the third century bc (cf. Needham 1959: Tables 22, 23). The power-signs immediately ancestral to the classical Chinese ones were among the variants used in the late Zhou period. These signs, shown in Table 8.4, differ greatly from those in Table 8.3. By the late Zhou, multiplier-signs were no longer superposed onto or ligatured with the power-signs; instead, numerals began to be written more regularly with unit-signs preceding power-signs from top to bottom. As well, while the older signs for 20, 30, and 40 were retained, more typically the signs for 2, 3, and 4 were placed next to the sign for 10, just as with the rest of the tens. The system was still Table 8.4. Late Zhou power-signs 10

100

1000

10,000

a

{

|

}

264

Numerical Notation

multiplicative-additive, only less opaquely so than previously. Just as there is no sharp break in the forms of signs between the Shang and Zhou systems, neither is there a distinct break between the Eastern Zhou numerals and those of the Qin Dynasty; rather, the former gradually transformed into the latter. Yet, given the rather important changes in Chinese writing that took place after the unification of the country in 221 bc, I have chosen that point of demarcation to separate the earlier numerical notation system from the “classical” Chinese system.

Chinese Counting-Rod Numerals Before turning to the classical Chinese system, however, I will address a system that developed alongside the written numerals of the Warring States period (476– 221 bc). This system, known in Mandarin as suan zi, is both a numerical notation system and a computational technology, translated in English as “counting-rods.” Short rods known as chou or suan were used to compute on flat surfaces. While these rods were often made of bamboo, they could also be made of bone, wood, paper, horn, iron, ivory, or jade (Lam 1987: 369). Although it is poorly known in the West, counting-rod calculation was the primary computational technology used in East Asia before the sixteenth century, when the bead-abacus (Mandarin suan pan, Japanese soroban) began to supplant it. Yet rod-numerals were not simply computational aids, but could also be written using vertical and horizontal lines to represent the computing rods, as shown in Table 8.5. The system is quite simple to learn and use; vertical and horizontal lines are sufficient to write any number. For the units position, vertical strokes signified 1 and horizontal strokes 5; combinations of vertical and horizontal strokes indicated the value. Conversely, for the tens, the values of the individual strokes were reversed, so that horizontal strokes meant 1 and vertical strokes 5. Each successive position was modeled alternately on the ones and the tens; positions in which the sign for 1 is vertical (ones, hundreds) are called zong, while those in which it is horizontal are called heng (Needham 1959: 8–9). Thus, Chinese scholars learned the following rhyme (Li and Du 1987: 10): Units are vertical, tens are horizontal, Hundreds stand, thousands lie down; Thus thousands and tens look the same, Ten thousands and hundreds look alike.

In the earliest rod-numerals (fourth century bc to third century ad), the use of zong and heng numerals as appropriate to their position was not strict, so that horizontal strokes could be used for ones and vertical strokes for tens. However, the system had stabilized by the end of the Han Dynasty. No zero-sign was used at

East Asian Systems

265

Table 8.5. Early rod-numerals (Needham 1959: Table 23)

1s 10s 100s 1000s

1

2

3

4

5

6

7

8

9

J A J A

K B K B

L C L C

M D M D

N E N E

O F O F

P G P G

Q H Q H

R I R I

762 7008 905,920 6.49

I

G E

P

F

R

B

K Q O

D

R

this early date. Most authors presume that the numeral-signs were lined up strictly by position, leaving blank spaces as appropriate, obviating the need for a zero; however, Martzloff (1997: 187) notes that there is limited evidence for such spacing in written rod-numerals (as opposed to physical counting-rods). The rod-numerals constitute a cumulative-positional system with a base of 10 and a sub-base of 5. While it is possible to regard each sign – such as R for 9 – as a single sign, thus making this system ciphered-positional, the system’s true structure is best reflected by classifying it as intraexponentially cumulative, which allows us to recognize how the sign is constituted and to note its sub-base. While the numerals 6 through 9 are written using compounds of 5 and 1 through 4, the sign for 5 alone is always five strokes; if a horizontal stroke were used for 5 in a zong position, there might be more risk of confusion with the horizontal stroke for 1 in the next-highest (heng) position. Because the direction of the strokes alternates with each successive position, the rod-numerals are irregularly positional,2 since a sign takes its meaning from both its position and its horizontal or vertical orientation. To put the sign G in the tens position indicates 70, but to put it in the ones or hundreds position would have violated the system’s structure, except during the earliest phase of its history. The rod-numeral system was infinitely extendable by using these two alternating sets of numeral-signs in successively higher positions. Decimal fractions could be written by designating one of the places as the “units” position, with the places to the right of that one representing 0.1, 0.01, and so on (Volkov 1994: 81). In numeralphrases containing both whole and fractional positions, the ones position could be 2

Martzloff (1997: 205) coins the term “dispositional” to reflect this irregularity.

266

Numerical Notation

identified by the presence of a character beneath it to indicate what sort of thing was being counted (Libbrecht 1973: 73). Where numbers were arranged strictly by columns, however, it was not necessary to include this extra sign. In addition, as early as the Han Dynasty, negative numbers could be written, either by using different-colored rods (red for positive numbers, black for negative numbers) or by placing an extra rod diagonally across the last nonzero digit of the numeral (Lam 1986: 188). The earliest physical rods to be unearthed are several found at Fenghuangshan in Hubei province, which date to the reign of Wen Di (179–157 bc) (Mei 1983: 59). Textual and epigraphic evidence shows, however, that the rod-numerals were developed much earlier. Coins from the Warring States period frequently contain rod-numerals (Needham 1959: 5). Similarly, Warring States earthenware bearing rod-numeral signs has been found in Dengfeng County (Li and Du 1987: 8), so the system can hardly have been developed much later than 400 bc. Yet its acceptance was not automatic. The Daodejing (Tao Te Ching), written in the early third century bc, advises that “[g]ood mathematicians do not use counting-rods,” confirming that the system was in use at that time, while also showing that it was not yet universally accepted (Needham 1959: 70–71). Yet by the time of the writing of Sunzi suan jing (The Mathematical Classic of Master Sun) in the fourth or early fifth century ad, counting-rods were presumed to be the only foundation for arithmetical learning (Dauben 2007: 295). Counting-rods were not simply arithmetical tools, but served also as divinatory instruments, money, and even to hold food (Martzloff 1997: 210). While the rod-numerals originated as a means of computation, the late Zhou numerals also may have influenced their development. While the systems differ structurally, their signs are similar; the Zhou sign for 1 is a horizontal line and the sign for 10 a vertical line with a dot. Because the early rod-numerals did not have a regular orientation, a horizontal rod could indicate 1 and a vertical rod 10. Given that the inventor(s) of the rod-numerals were probably literate, they would have been familiar with the Zhou signs and may have borrowed them. The rod-numerals’ cumulative-positional structure and quinary sub-base allow a limited number of rods to express any number, though in practice, the use of physical rods would have limited the number of positions that could be managed easily. Although the rod-numerals are identical in structure to the Greco-Roman abacus (which predates the rod-numerals by at least two centuries), I attribute this similarity solely to the two technologies’ common function. Lam Lay-Yong (1986, 1987, 1988) hypothesizes that the rod-numerals were ancestral to the Hindu positional numerals, because the rod-numerals are positional and decimal, and because there was considerable cultural contact between China and India in the sixth century ad, when positionality developed in India. Because the rod-numerals were used in computation and commerce, she asserts

East Asian Systems

267

that it is inconceivable that the Indians would not have learned of this system from the Chinese, and, since it is so practical, they obviously would have borrowed it (Lam 1988: 104). Yet the Indian positional numeral-signs are those of the earlier Brāhmī numerals, not of the rod-numerals, and the rod-numerals have no zerosign (whereas the Indian system does). Moreover, the rod-numerals have a quinary sub-base that the Indian numerals lack, and the rod-numerals are intraexponentially cumulative, whereas the Indian positional numerals are ciphered. No Indian texts of the period mention rod-numerals or any other Chinese numeration. In the sixth or seventh century ad, the numerals and rod-computation were introduced into Japan at a time when Chinese cultural, religious, and political influence in Japan was enormous. The original rods were long, thin, round, made of bamboo, and called chikusaku; they were, however, quickly replaced by shorter square rods known as sangi (Smith and Mikami 1914: 23). There is no evidence of their use outside China, Japan, and Korea. The last coins to use rod-numerals are the five chu coins of the Liang Dynasty (502–557 ad), but these numerals are highly irregular (de Lacouperie 1883: 316–317). They continued to be written in Chinese texts and used directly for computation. In the twelfth and thirteenth centuries (the late Song Dynasty), the rod-numerals as written in texts – though not their computing-rod counterparts – transformed in three significant ways. Although this was a time of considerable political turmoil in China, due to invasions by groups such as the Jurchin and Mongols, it was also a time of considerable scientific achievement. Table 8.6 indicates the system as it was used at that time (Needham 1959: Table 22; Libbrecht 1973: 68). First, new signs for 4, 5, and 9 were introduced, while the original (cumulative) signs were retained. Because the only signs to change were those in which four or five cumulative strokes had previously been required, this was probably done to simplify the signs, though it meant that the written system differed from that used with physical rods. These changes made the system less cumulative than it previously had been, so that it approached a ciphered-positional structure. Second, written numeral-phrases sometimes were condensed into single glyphs, compressing the individual signs together so that they formed a monogram. Needham (1959: 9) attributes this development to the requirements of the new technology of printing books. Finally, a circle was introduced as a sign for zero. The first text known to use a zero-sign is the Shu shu jiu zhang (Mathematical Treatise in Nine Sections) of Qin Jiushao, published in 1247 (Libbrecht 1973: 69).3 Needham (1959: 10) suggests that the idea of a circle for zero may have been an endogenous development, based on the philosophical diagrams of twelfth-century Neo-Confucian scholars. I agree with 3

As I will discuss later, this text is also the first to use a circular sign for zero in conjunction with the classical numerals.

Numerical Notation

268 Table 8.6. Late rod-numerals

Zong (1, 100, ...) Heng (10, 1000, ...)

1

2

3

J

K

L

A

B

C

Old Style 5804

E Q

4

5

M S D S

N T E U

6

7

8

O

P

Q

F

G

H

9

R V I W

0

0 0

New Style

M

Æ

Martzloff (1997: 207), however, that this development was more likely related to the Indian zero, which had passed to China along with the transmission of Buddhism in the eighth century ad. We may never know, however, whether the exact route of transmission was through Southeast Asia, Tibet, or India proper. The rod-numerals were linked directly with arithmetical computation from the time of their invention. While they began as a system involving the physical manipulation of rods, Chinese mathematicians quickly adopted them for writing results of computations. The earliest strictly mathematical Chinese text, the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), which dates no later than the first century ad and summarizes the learning of earlier centuries, uses them extensively (Lam 1987: 367–368; Li and Du 1987: 33–37; Volkov 1994: 81). Thereafter the rod-numerals were a central part of the computation techniques used alongside most Chinese mathematical and astronomical texts until the sixteenth century.4 Most Chinese characters having to do with computation use the “bamboo” radical because of its association with bamboo computing rods (Needham 1959: 72). Tong (1999) asserts that overreliance on concrete, context-situated counting rods and rod-numerals acted as a “stumbling-block” preventing the development of propositional mathematics in the Song Dynasty. Yet the modifications to the system, including the addition of a zero-sign, suggest that the rod-numerals, as an infinitely extendable notation using the principle of place-value, could be used as objects of arithmetical thought independently of their materiality. Dauben (2007: 191) contends – rightly, in my view – that the perfectly regular and decimal character of counting-rod arithmetic greatly facilitated the extraction of roots and advanced work with linear equations. 4

We may never know the true extent of their use, since many printers considered the rodnumerals, with their vertical lines, to be insufficiently literary, and replaced them with the classical numerals (Needham 1959: 8).

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269

The introduction of the bead-abacus (suan pan) in the fourteenth or fifteenth century brought this novelty into direct competition with the rod-numerals. The earliest surviving suan pan dates from the sixteenth century, but the text Duixiang siyan zazi of 1337 indisputably depicts one (Martzloff 1997: 213–215). Textual sources indicate that the suan pan was perceived as being more efficient for computational purposes than the rod-numerals. The divorcing of rod-numerals from the physical manipulation of rods made their use in written form rather archaic. At first, the suan pan was an instrument for popular arithmetic, while computing rods were retained by mathematicians and the elite (Jami 1998: 4). Throughout the Ming Dynasty (1368–1644), they were used increasingly rarely in Chinese books, and they had become a historical curiosity by 1600 (Cheng 1925: 493).5 None of the many seventeenth- and eighteenth-century European scholars who mentions the abacus also notes the rod-numerals (Needham 1959: 80), and in fact they first came to the attention of the West in Biot’s (1839) antiquarian treatment. However, rod-numerals were used in Japan for some time after they had been abandoned in China, and were used actively through the nineteenth century in traditional Japanese mathematics (Menninger 1969: 368–369; Martzloff 1997: 210–211). A new Chinese computing technique developed in the seventeenth century in which computing rods were inscribed with classical numerals, probably under the influence of the system of numbered rods developed by the English mathematician John Napier (Needham 1959: 72). This technique (similar to a slide rule) need be given no attention here, since it is not a numerical notation system but simply a computing technology that uses the Chinese classical numerals. Two relatively recent numerical notation systems may be derived at least in part from the rod-numerals. The Chinese commercial numerals employed in Hong Kong and other regions since the sixteenth century use many of the rod-numeral signs, combined with the multiplicative-additive structure of the classical Chinese numerals (see the following discussion). It is less likely but still possible that some knowledge of the rod-numerals and/or the classical Chinese numerals among the inhabitants of the Ryukyu Islands south of Japan led to the development of the cumulative-additive sho-chu-ma numerals (Chapter 10). The manipulation of rod-numerals on boards appears to have been nearly as important to ancient and medieval Chinese scientific and commercial calculation as the bead-abacus would later be. Their origin and persistence must have had a great deal to do with their efficiency for computational functions. However, this supports rather than refutes my thesis that the history of numerical notation systems should be divorced from their use as mathematical tools. The rod-numerals 5

Wang Ling (1955: 91) reports that Chinese logarithmic tables were still written with rodnumerals in the early twentieth century, but I cannot substantiate this assertion.

270

Numerical Notation

and the classical Chinese numerals coexisted for nearly 2,000 years, and yet the former had no noticeable impact on the latter. If there truly existed a unilinear trend for positional systems to supplant additive ones, we would expect the rodnumerals to replace the multiplicative-additive classical numerals entirely, or at least to facilitate their transformation into a ciphered-positional system.

Chinese Classical The basic numerals associated with the Chinese script are perhaps the most stable symbol system currently in use; the numeral-signs of the Qin Dynasty (221–206 bc) are practically identical to those used in modern Chinese literature. While there are structural differences between that system and the way the numerals are normally used today, ancient numeral-phrases are still easy to read. The basic numeral-signs are shown in Table 8.7a, and a selection of numeral-phrases in Table 8.7b. In traditional writing, numerals, like the script, were arranged in columns from top to bottom, with the highest powers at the top. In modern writing, numerals are normally written in rows from left to right, although right-to-left writing is not unknown, in which case right-to-left numeration is employed. The basic system is multiplicative-additive; numbers are written by combining the signs for 1 through 9 with the appropriate signs for the powers of 10 to indicate their multiplication, and then taking the sum of these pairs of signs. There is no power-sign for the units; the unit-signs for 1 through 9 stand alone. When writing 11 through 19, the unit-sign attached to 10 is always omitted, although in numbers such as 214 the unit-sign for the tens is often included. Prior to the Tang Dynasty, it was optional to put the unit-sign 1 in front of 10, 100, 1000, and 10,000 in numeralphrases, but including the unit-sign 1 later became standard (Martzloff 1997: 185). However, in modern Chinese, the powers of 10 alone can be written without the unit-sign. When the multiplier of a power is zero, both the unit-sign and the power-sign are always omitted. There is no zero-sign in the classical system, although there is in modern Chinese numerals (to be described later). In addition to these standard signs, there are three nonstandard signs used for 20, 30, and 40, which have their origins in the Shang/Zhou cumulative signs for the lower decades (Needham 1959: 13). These signs were most often used in literary contexts, for paginating certain texts, and when denoting days of the month. In the fifth-century mathematical manuscripts found at Dunhuang in Central Asia, however, they were also used for mathematical purposes (Martzloff 1997: 185). They are still used occasionally, although the sign for 40 is very rare because it is not needed to enumerate days of the month. It was always acceptable (and now is preferred in most contexts) to use the standard multiplicative combinations of the unit-signs 2 through 4 and the power-sign for 10.

East Asian Systems

271

Table 8.7a. Classical Chinese numerals 1

2

3

4

5

6

7

8

9

₏ ℛor₳ ₘ

⥪

℣

⏼

ₒ

⏺

⃬

yi

er

san

si

wu

liu

qi

ba

jiu

10

100

1000

10,000

100,000,000

◐ 䤍

◒

嚻or ₖ

ⅎor ⎓or嚻嚻

shi

bai

qian

wan

yi

20

30

40

◓ ◔

◛

nian

xi

sa

or wan wan

Unlike Western numerals, which are grouped in chunks of three digits, Chinese numerals are grouped in sets of four, using the character wan (10,000, or, if you will, 1,0000) (Mickel 1981: 83). Any number from 10,000 to 100,000,000 could be written by placing a multiplicative numeral-phrase from 1 to 9999 before the sign for 10,000. The system did not stop there, however. By the first century ad, multiples of 100,000,000 could be written by placing a multiplicative numeralphrase in front of two signs for 10,000 or by using a sign for 100,000,000, either ⅎ or ⎓ (Martzloff 1997: 183). Another technique for expressing large powers of 10, which developed early in the history of Chinese numeration, involved a complex Table 8.7b. Chinese numeral-phrases 15

◐℣ 10

118

5

䤍◐⏺RU₏䤍₏◐⏺ 100

74,002

1

100

1

10

8

10,000 4

1000

2

b7d2c4ba9 100

4,703,600,854

8

ₒ嚻⥪◒ℛ 7

1,072,419

10

7 10,000 2 1000 4

100 10

9

4a7ƒ3b6ad8b5a4 4

10

7 100 mil. 3 100

6

10 10,000 8 100

5

10

4

Numerical Notation

272

Table 8.8. Chinese higher power-signs

Sign

Phonetic Value

Lower Series xia deng

Middle Series zhong deng

Upper Series shang deng

d ƒ á í £ ó Ñ ñ ¿ ¬ ½

wan

104

104

104

yi

105

108

108

zhao

106

1016

1016

jing

107

1024

1032

gai

108

1032

1064

zi

109

1040

10128

rang

1010

1048

10256

gou

1011

1056

10512

jian

1012

1064

101024

zheng

1013

1072

102048

zai

1014

1080

104096

system of power-signs that was assigned three different series of values, as shown in Table 8.8 (Needham 1959: 87; Martzloff 1997: 99).6 These power-signs combine multiplicatively with the nine basic unit signs, and thus extend the basic system. Needham (1959: 87) asserts that these signs first appeared in the Shu shu ji yi (Notes on Traditions of Arithmetic Methods) of Xu Yue, and dates this text to around 190 ad, but it may be a sixth-century forgery (Dauben 2007: 300). In any event, these techniques are well attested from the fifth century ad onward, and demonstrate a keen interest in extending the range of numeration far beyond that needed for any practical purpose. While this system may seem hopelessly complex and ambiguous, this confusion is identical to that resulting from the different values assigned to billion and trillion in American and European usage. In the lower series, each exponent is one greater than the one before it; in the middle series, each exponent (excepting wan) is eight greater than the one before it; and in the upper series, each exponent is double the one before it. The first sign in all three series is the standard sign for 10,000, and the second sign (yi) is one of the basic signs for 100 million (thus corresponding to the middle and upper series, but not to the lower one). Martzloff (1997: 97–99) holds that these systems may have been 6

The middle series (zhong deng) values in Martzloff and Needham differ somewhat; I use Martzloff’s values in Table 8.8.

East Asian Systems

273

borrowed from similar Sanskrit systems transmitted at the time of the introduction of Buddhism into China in the middle of the first millennium ad. None of these systems was ever in common use. The Chinese numerals began to take their modern form starting in the third century bc, developing directly out of the numerals used in the Warring States period. With the spread of a unified administrative apparatus under the Qin and Han Dynasties, they spread throughout areas under direct and indirect imperial control. The unification of China led to many efforts to standardize the forms of Chinese script and numeral-signs, although this was not accomplished to any significant extent until late in the Han Dynasty. Figure 8.1 depicts a Han Dynasty administrative calendar from 94 bc, found at Dunhuang (Gansu province), with numerals used to enumerate months and days (Chavannes 1913: Plate XV). Even as the signs of the system were being codified, however, Chinese writers began to use calligraphic variants and other modifications of the basic system for specific functions. These variants used different numeral-signs (ranging from mild paleographic variations to radically different signs), but their structure is identical to that of the basic system (decimal and multiplicative-additive). These variants were strictly functional, not regional. Perhaps the most important of these are the “accountant’s numerals” (da xie shu mu zi), which developed in the first century bc (Needham 1959: 5, Table 22). Structurally, they are identical to the classical numerals, but while the classical numeralsigns are quite simple, the accountant’s numerals were intentionally made very complex; thus they were considered more elegant and less susceptible to falsification. The signs are homophones of the phonetic values of the appropriate Chinese words, so they bear no graphic resemblance to the basic signs. Hopkins’s (1916) analysis of their origin as phonetic variations of the standard numerals is dated but quite thorough. Despite their name, they were used not only for accounting but also, for instance, on thirteenth-century coins (de Lacouperie 1883: 318–319) and even in a mathematical text, the Tongwen suanzi qianban of 1614 (Martzloff 1997: 184–185). Today, they are still used occasionally on checks, banknotes, coins, and contracts in order to prevent falsification. Another highly complex variant of the classical numerals are the shang fang da zhuan, a variant set of numeral-signs that developed in the Han Dynasty (Pihan 1860: 13; Perny 1873: 113). These numerals are highly stylized linear versions of the standard numeral-signs that were designed to be used on seals, and are still sometimes used for that purpose today. These signs are shown in Table 8.9. The diffusion of the Chinese classical numerals was associated with the spread of Chinese political influence throughout East Asia. In the late second century bc, the Chinese numerals were employed in tributary regions such as the Gansu corridor in Central Asia, the Vietnamese states, and the colony of Lelang (modern Pyongyang,

274

Numerical Notation

Figure 8.1. A Han Dynasty monthly calendrical document from the archive excavated from Dunhuang. The columns of the register at the bottom left begin with the numerals 23 and 22, with 20 indicated in each phrase using the ligatured sign for 20. Source: Chavannes 1913: Plate XV.

East Asian Systems

275

Table 8.9. Shang fang da zhuan numerals 1

2

3

4

5

6

7

8

9

Ç

ü

é

â

ä

à

å

ç

ê

10

100

1000

10000

ë

è

ï

ì

North Korea). The Chinese numerals were borrowed directly (without any transformation) by the Japanese as part of the kanji characters starting in the third century ad. The Korean hangul script developed in the fifteenth century has no corresponding numerical notation system, but Koreans often used the Chinese classical system. The numeral-signs associated with the chu’ nôm script of the state of Annam (in modern Vietnam) are simply the basic Chinese signs with additional phonetic notation; the basic Chinese system was also known and used in the region (Pihan 1860: 20–21). The numerals associated with the scripts of non-Chinese peoples of China, such as Tangut (Kychanov 1996) and Miao (Enwall 1994: 86), are also derived from the basic Chinese system, although sometimes with considerable paleographic modification. None of these systems is structurally distinct from the basic Chinese numerals. Starting in the tenth century, China began several centuries of intensive contact with its neighbors to the north and west; warfare with these nomadic groups and the conquest of China in turn by the Kitan and Jurchin led to the development of Chinese-inspired numerical notation systems among these two groups, which are structurally distinct and described later. The classical Chinese numerals were nonpositional and used no zero-sign for over a millennium after their development. The positional principle was known in China, however, through the cumulative-positional rod-numerals that had been used since 400 bc. Moreover, Chinese mathematicians became aware of Indian ciphered-positional numerals in the eighth century ad. Qutan Xida,7 an Indo-Chinese Buddhist astronomer working at the Tang capital at Changan, first reported the use of nine unit-signs with a dot for zero in his great astronomical compendium, Kaiyuan zhanjing, written between 718 and 729 ad (Needham 1959: 12; Guitel 1975: 630–631). This transmission reflects the enormous scientific contact that accompanied the introduction of Buddhism into China in the eighth century ad. Yet the knowledge of ciphered-positional numerals had no attested impact on traditional Chinese numeration for many centuries. 7

This name is the Sinicization of the author’s original name, Gautama Siddharta.

276

Numerical Notation

In the mid thirteenth century, a period of scientific vigor during the late Song Dynasty, the first zero-signs appeared alongside classical Chinese numerals in mathematical texts. The first such text was the Shu shu jiu zhang of 1247, the same document in which the zero-sign is first found with rod-numerals (Libbrecht 1973: 69). This modification allowed a circular zero-sign to be used whenever one of the decimal powers in the middle of a numeral-phrase was empty. In theory, this allowed Chinese mathematicians to use only the unit-signs from 1 through 9 in conjunction with the 0 to express any number – thus making the system cipheredpositional. Yet, during the Song Dynasty zero was used only to fill in empty medial positions, while retaining the power-signs, so that where 12,001 would be written in the classical style as ₏嚻ℛ◒₏, it is written as ₏嚻ℛ◒ᇲᇲ₏in the Shu shu jiu zhang, a less concise form that provides no other obvious advantage. Whether this resumption of the use of the zero-sign was a result of the continuation of its eighth-century use, or a reintroduction from India or the Middle East, is unknown. Starting in the late sixteenth and early seventeenth centuries, when Chinese mathematicians of the Ming Dynasty were in extensive communication with the West through the intermediary of Jesuit missionaries, this form of ciphered-positional Chinese notation was employed more regularly (Martzloff 1997: 185). Tables of logarithms appeared at this time, using the nine traditional unit-signs and a circle for zero in a way identical to the use of the Western signs 0 through 9 (Menninger 1969: 461). Before the sixteenth century, zero was employed only in mathematical and scientific texts. In the late Ming Dynasty, it began to be used more widely, but rather than using the circular sign for zero found in the Song texts, a character, ling (榅) ‘raindrop’, which had been used to designate remainders in division, was assigned the meaning “zero.” The first text in which it featured prominently is Cheng Dawei’s Suan fa tong zong (Systematic Treatise on Arithmetic) of 1592–93, which is also the first text to describe the Chinese commercial numerals or ma zi, and additionally contains the first complete description of the bead-abacus or suan pan (Needham 1959: 16, 75–78; Li and Du 1987: 185–187). In this and other early texts, ling was used in exactly the same way as the circle-sign had been used previously, with one ling sign for every missing power, so that 30,008 would be written as ₘ嚻榅榅榅⏺(3 10000 0 0 0 8). While the ling sign introduced an element of positionality into the system, it was not fully positional, since the power-signs were retained, and ling was used only in medial positions. Chinese writers soon realized that they could omit all but one ling when multiple consecutive powers are empty, so that one could write 30,008 simply as ₘ嚻榅⏺(3 10000 0 8). The classical Chinese system normally uses ling in this manner today. In modern China, any given number can be expressed in no less than six distinct ways, the choice of which depends greatly on context. Four of these forms are variants of the classical system. For literary and prestige purposes, the pure classical

East Asian Systems

277

Table 8.10. Modern Chinese expressions for 20,406 Classical

Classical with ling (zero)

2d4b6 2d[4b[6

Ciphered-positional

Western

20406

20,406

Chinese numerals (without any sign for zero) are used, thus representing continuity of the system from the Qin Dynasty to the present. In most ordinary prose writing, some sign for zero is usually introduced in the medial positions, while retaining the power-signs. The use of ling has even spread to spoken Chinese, so that the preferred way to say 203 is not simply er bai san but rather er bai ling san. Where conciseness or clarity is desired, and in most scientific contexts, the nine unit-signs along with a sign for zero are used positionally, as in the seventeenthcentury logarithm tables. In contexts where there is concern with forgery, the “accountant’s numerals” can be used. The final two options are to use the commercial or Hangzhou numerals, which I will describe later, or Western numerals. The Chinese classical numerals are ancestral to several numerical notation systems. The ciphered-positional variant Chinese numerals used in modern mathematics are, of course, one such descendant, as are, more indirectly, the Hangzhou numerals. The structurally distinct numerical notation systems used by the non-Chinese Kitan and Jurchin during the period in which they were in contact with (and later dominant over) the Chinese are also largely derivative of the Chinese classical numerals. Finally, two more obscure systems, the sho-chu-ma numerals used on wooden tallies on the Ryukyu Islands, and the Pahawh Hmong system developed recently for use among the Hmong of Laos, may also be derived in part from the Chinese system (Chapter 10). Western numerals, while known in China from the seventeenth century, were not widely used until the beginning of the twentieth century; the Shuxue wenda of 1901, an arithmetical primer for use in elementary schools, was one of the earliest such texts (Martzloff 1997: 35–36). In most scientific and technical contexts in China today, however, Western numerals are preferred. Because the ciphered-positional Chinese numerals with the circle for zero had been used for several centuries prior, this shift was strictly social and political, unrelated to structural considerations. Mao Zedong was amenable (at least initially) to the replacement of Chinese numerals by Western numerals, as indicated in a 1956 speech that was later suppressed (DeFrancis 1984: 262–263). Nevertheless, the replacement of Chinese with Western numerals has not been uninterrupted or uncontested. Some institutions reacted sharply to this trend, and anti-Western sentiment led to the replacement of Western numerals by the corresponding Chinese numerals in certain academic publications (DeFrancis 1984: 274–275). Western numerals are well known to all reasonably educated people in China. In Japan and South Korea, the dominance of Western numerals is considerably greater than it is in China. Nevertheless, the Chinese numerals continue to be known and taught in these countries, though

278

Numerical Notation

they are acquiring an archaic flavor. In China itself, however, the use of local numerals shows no sign of sharp decline, and there is every reason to believe Chinese numeration will persist into the foreseeable future.

Chinese Commercial The Chinese commercial numerals (often known as “Hangzhou numerals”)8 arose in the sixteenth century. The numeral-signs of the system are shown in Table 8.11 (Needham 1959: Table 22). Comparing these signs to those in Table 8.6, we see that all of the unit-signs, save that for 5, closely resemble the late forms of the rod-numerals used during the Ming Dynasty, although they have been borrowed haphazardly from the zong and heng forms. The unit-signs for 1, 2, and 3 use vertical rather than horizontal lines, showing that they are unrelated to the classical Chinese numerals. Hopkins (1916: 318) explains the aberrant form of 5 as a form of the character wu, which is a homophone of the Mandarin numeral word for five. The most common versions of the power-signs for 10 through 10,000 are obvious variants of the classical system’s power-signs. The circular sign for zero was in use in both the rod-numerals and the classical system. This evidence strongly suggests that the commercial numerals originated as a blend of the late rod-numerals and the Chinese classical numerals. The system is multiplicative-additive, with the zero used only to fill in empty medial positions, never at the end of numeral-phrases. Unlike the Chinese classical numerals, commercial numeral-phrases place the signs in two rows, with the unit multipliers of the various powers on the top row and the power-signs, zero-signs, and the signs for the ones position on the bottom row (Pihan 1860: 6). Numeralphrases were thus read in a zigzag fashion, starting at the top left, proceeding from top to bottom and then diagonally up and to the right. This basic system was made more complex by a large number of irregularities. When the number being expressed was a simple multiple of a power of 10 (e.g., 50, 800, 2000), the multiplier usually was placed to the left of the power-sign (as it would be in the classical system) rather than above it (Perny 1873: 101). When the number 10 occurred alone or in numbers such as 610 and 2010, the unit-sign 1 was always omitted, and the unit-sign could optionally be omitted when the sign for 10 was combined additively with unit-signs, as in numbers such as 18 and 212. Moreover, the special classical Chinese numeral-signs for 20 (,) and 30 (.) could be used in the commercial numerals where appropriate (Hopkins 1916: 319). When there were two consecutive zero-signs in a numeral-phrase, they could be placed one atop the other rather than side by side in the bottom row, as would 8

Other names for this system include “ma zi,” “Suzhou numerals,” and “hua ma.”

East Asian Systems

279

Table 8.11. Chinese commercial numerals 1

2

3

4

5

6

7

8

9

J

K

L

S

s

F

G

H

t

10

100

v

x

1000 or

w y

or

or

u

10,000 0

î z

0

be normal. Finally, the standard classical unit-signs for 1 through 3 (horizontal rather than vertical strokes) are sometimes used in the units position at the end of numeral-phrases, though they cannot be used as multipliers in conjunction with power-signs (Hopkins 1916: 319). The combination of all these irregularities and options means that almost any number may be expressed in several valid ways. Table 8.12 depicts a selection of numeral-phrases as written in this system. We do not know exactly when the commercial numerals were invented, but the earliest printed text that describes them is the Suan fa tong zong, published in 1593 (Needham 1959: 5). Because they were not used for prestige purposes – in literature or mathematics, for example – but were restricted to a limited set of commercial contexts (invoices, bills, signs for prices, and so on), sixteenth-century or earlier evidence of their use may not have survived. The rod-numerals, from which the commercial numerals are partly derived, were obsolescent by 1600, so it is unlikely that they would have been used as the basis for a new system as late Table 8.12. Chinese commercial numeral-phrases

40,709

26

162

917

3008 5000

S\\ G z0x0t K vF OR ,F JF JF xaK OR xag tJ t xvG OR xvG L0 y0H sy

4

7

10,000 2 10 1

0

6 OR 6

100 10 9 1 100 3

5

1000

0

20

6

9

2 9

10 0

1000 0

100

7

8

OR

100

10

7

280

Numerical Notation

as 1593. Yet early texts that mention them associate their invention and use with the great commercial city of Suzhou (in Jiangsu province). As this city came to prominence only in the sixteenth century, if the attribution of their invention to Suzhou is correct, a pre–sixteenth-century origin is unlikely. As is suggested by their name, the commercial numerals were (and are) used solely in commercial contexts. They are still used even today in some regions on bills, invoices, and signs in shops and markets (primarily to indicate prices), though their use is waning in favor of regular Chinese numerals or Western numerals. They are most common in regions where Cantonese is spoken, including Hong Kong.

Kitan The Kitan (or Khitan) were an Altaic-speaking people who ruled Manchuria and other parts of northern China between 907 and 1125 ad, a period now known as the Liao Dynasty (Kara 2005: 7). While there was no Kitan writing before their conquest of Manchuria, two scripts were developed shortly thereafter, the “large script” and the “small script,” both based largely on Chinese, and possibly also under the influence of the Central Asian Uygur script. Neither Kitan writing system is fully deciphered, because the Kitan language is only poorly known, but the meanings of the Kitan numeral-signs are understood. The numerals of the “large script” are identical to the classical Chinese numerals, but the “small script,” purportedly developed by the Kitan scholar Diela during the visit of an Uyghur delegation to the Kitan court in 924 or 925 ad, had a distinct numerical notation system. The attested signs of this system are shown in Table 8.13 (Kara 1996: 233). While the Kitan numeral-signs have a vaguely Siniform appearance, they are entirely dissimilar to the corresponding Chinese numerals, and must be of indigenous origin. Numeral-phrases are multiplicative-additive and are read vertically in rows from top to bottom and then right to left across the page, as in traditional Chinese writing. A slight ciphered element is introduced into the system by the existence of distinct characters for 20 and 30; this practice is probably derived from the analogous Chinese signs, ◓ and ◔, although the Kitan signs are not cumulative. It is not known how (if at all) numbers higher than 1000 were written. Numeral-signs could also serve as phonograms; for instance, the symbol for 5 (tau) was employed homophonically in the word t’ao-li ‘hare’ (Kara 2005: 13). Because Kitan writing is so poorly understood, it is difficult to know the total scope of contexts in which the numerals were used. Kitan texts are known from epitaphs on royal tombs, a text on a bronze mirror, some other stone monuments, and inscriptions on seals and vessels (Kara 2005: 9). Most texts were probably historical records of events, in which numerals are used primarily for dating. The Kitan script and numerals did not outlast the period of Kitan independence,

East Asian Systems

281

Table 8.13. Kitan numerals

1 10 100 473 =

1

2

3

4

5

6

7

8

9

1 0 /

2 ,

3 .

4

5

6

7

8

9

4 / 7 0 3

which ended in 1125 at the hands of the Jurchin. In 1191, the use of the Kitan script was forbidden by Chinese imperial order, after which time no further Kitan texts are attested (Kara 1996: 231).

Jurchin The Jurchin (also Jurchi or Jurchen) were the rulers of what is now known as the Jin Dynasty in the northern part of China (1115–1234), and one of the groups constituting the Manchu who conquered China in the seventeenth century. The Jurchin script, which consists of logograms and syllabograms in addition to a set of numeral-signs, is attested from inscriptions and texts from the twelfth through fifteenth centuries, but may have developed somewhat earlier (Kiyose 1977). The Jurchin numerals are shown in Table 8.14 (Grube 1896: 34–35; Kiyose 1977: 132–3). Jurchin numerals, like the script, were written in vertical columns read from top to bottom, with the highest-valued powers at the top. The system is primarily decimal; although the distinct numeral-signs for 11 through 19 suggest a vigesimal component, it is a product of the fact that the Jurchin lexical numerals have distinct words for 11 through 19 that are not connected to those for 1 through 9, but the irregularity extends no further than the teens (Kiyose 1977: 133). Because the Manchu language, in contrast to Jurchin, had no such words, Jurchin numeral-phrases could also be written using the sign for 10 additively with the signs for 1 through 9. For writing numbers from 20 to 99, unit-signs from 1 through 9 sometimes were combined with the power-sign for 10 as in the classical Chinese system, so the Jurchin numerals appear to be multiplicative-additive. Yet there were also ciphered, nonmultiplicative Jurchin numeral-signs for 20 through 90. For numbers above

Numerical Notation

282 Table 8.14. Jurchin numerals 1

2

3

4

5

6

7

!

@

#

HPX

ƲXZH

LODQ

$

%

^

&

GXZLQ

ģXQƲD

QLQJX

QDGDQ

11

12

13

14

15

16

17

Å

É

æ

Æ

ô

ö

ò

8

9

10

*

(

)

ƲDNXQ

X\XQ

ƲXZD

18

19

20

û

ù

ÿ

DPģR ƲLUKRQ JRUKRQ

GXUKRQ WRERKRQ QLOKXQ

GDUKRQ QL\XKXQ RQL\RKRQ RULQ

30

40

50

60

Ö

Ü

á VXVDL

JXģLQ WHKL

70

80

90

100

í

ó

ú

ñ

Ñ

QLQMKX

QDGDQMX MKDNXQMKX X\XQMX WDQJX

1000

10,000

¢

£

PLQJDQ

WXPDQ

100, the multiplicative principle was always employed. Thus, the Jurchin system is structurally closer to hybrid ciphered-additive/multiplicative-additive systems, such as the Ethiopic numerals (Chapter 5) and Sinhalese numerals (Chapter 6), than it is to Chinese. In the Sino-Jurchin texts from the Ming Dynasty published by Grube (1896), which date roughly to the period 1450–1525, only the unit-signs 1 through 9 and the power signs 10, 100, 1000, and 10,000 were used. A Jurchin “large script” was introduced in 1120 by Wanyan Xiyin, and was based on the Kitan script with significant Chinese influences; the script was officially introduced in 1145 by Emperor Xizong, with a number of “small script” characters added (Kara 1996: 235). The Jurchin numerals are found on many monuments of the Jin Dynasty and some manuscript fragments. The writings that survive are historical and literary in nature, and the numerals in them are mainly dates. Our best evidence for them comes from the Ming Dynasty (1368–1644), when Chinese translators produced a bilingual glossary and translated documents, in which the numeral-signs just described are found (Kara 1996: 235). The earlier signs differ paleographically but not structurally from the Ming ones. Although the Jurchin did not control large regions of China for very long, the Jurchin script survived for several centuries. It was used on a Ming inscription of 1413, suggesting that it was not simply a historical curiosity, but was being preserved because it was being used (at least by some people). It continued to be used until at least 1525, at which time Ming translators were still working with Jurchin documents. The Jurchin were one of the major constituent groups of the Manchu who conquered China in the seventeenth century (in fact, the ethnonyms “Jurchin” and “Manchu” may refer to a single group), but by this time they used

East Asian Systems

283

either the classical Chinese numerals or the ciphered-positional, Indian-derived Mongolian numerals.

Summary Chinese numerals are central to the history of the East Asian systems. Today, the classical Chinese numerals (along with positional variants) occupy a role parallel to the supremacy of the Roman numerals in Europe prior to 1500, despite the increasing use of Western numerals for science, technology, and commerce. This system’s continued strength (at least in China) suggests that it will continue to thrive, especially in nontechnical prose writing. We must also take into account the strong cultural preference for Chinese symbol systems when analyzing the present state of the Chinese numerals; functional considerations alone cannot account for it. The increasing rarity of the Chinese numerals in Japan and Korea represents not only the functional rejection of an “inefficient” system, but also resistance to a Chinese cultural feature in favor of the more international Western numerals. The Chinese numerical notation system as used today is enormously variable in structure, and employs a host of representational techniques. On the surface, this appears hopelessly nonfunctional, and we might question why such a system would survive. I think that its quasi-lexical nature – the fact that Chinese numerals act as both ideographic script-signs and graphic numeral-signs – renders this variability both comprehensible and rational. If Western numerals incorporated archaisms such as score, or accounted for the fact that 1400 can be one thousand four hundred but is more commonly fourteen hundred, similar eccentricities might arise. The Chinese classical numerals are well suited to being read because they account for the irregularities in spoken Mandarin. The basic multiplicative-additive structure of the system permits all sorts of structural manipulations, such as the occasional use of positionality or ciphered signs for the lower decades, without creating any ambiguity. The system’s flexibility and its correspondence with language are thus advantages rather than hindrances. The comparison of this phylogeny to the ones I have discussed previously is quite instructive. In Chapters 2 through 7, most systems employed a single common structural principle. By contrast, the East Asian systems display considerable structural variety: cumulative-positional (rod-numerals), ciphered-additive (Jurchin), ciphered-positional (Chinese positional variant), and multiplicativeadditive (Shang/Zhou, Chinese classical and commercial, Kitan). Yet there can hardly be any doubt that these systems comprise a cultural phylogeny. The historical connections among systems are well established, and the similarities in the numeral-signs are quite strong. If we were to rely on structural qualities alone, we would be at a loss to describe their cultural history.

chapter 9

Mesoamerican Systems

Mesoamerica was the homeland to a distinct family of numerical notations with two separate subtraditions. Mesoamerican written numerals were first developed in the lowlands (Yucatan, Belize, Honduras, Guatemala) by 400 bc at the latest, while a major set of variants developed around ad 1000 in the central Mexican highlands. These two interrelated traditions are associated most closely with the Maya and Aztec civilizations, respectively.1 Both these traditions flourished until the Spanish conquests of the sixteenth century. In past research, Mesoamerican numerals have provided clear New World examples of independent invention of features of numerical notation systems such as additive notation (Guitel 1958) and the zero (Kroeber 1948: 468–472). Along with calendrical signs, Mesoamerican numerals were the earliest aspect of the region’s representational systems to be deciphered, and thus are among the best understood, but misinterpretations of the data persist. The numeral-signs of the major Mesoamerican systems are shown in Table 9.1.

Bar-and-Dot The bar-and-dot numerals were the most commonly used system in lowland Mesoamerica, both on stone monuments (400 bc–910 ad) and the four surviving Maya bark-paper codices (1000–1500 ad). This system has long been an object of 1

I treat other, unrelated New World inventions, such as the Inka khipu and the Cherokee numerals, in Chapter 10.

284

Mesoamerican Systems

285

Table 9.1. Mesoamerican numerical notation systems System

1

Bar-and-dot (monumental)

V e V E V T U

Bar-and-dot (codices) Aztec Texcocan line-and-dot

5

20

400

8000

2 1 X v Y x y V

0

3\9 0\]

study (Bowditch 1910, Morley 1915). While used in all the lowland Mesoamerican polities, it is most commonly associated with the Maya. Bar-and-dot numerals are ubiquitous in most lowland Mesoamerican texts, reflecting both the strong interest in dating and calendrics and the practice of incorporating numerical values into the names of deities and individuals. The numbers from 1 to 19 are written by combining a dot sign for 1 and a bar sign for 5 additively. When the bars are vertical, as is most common on stone inscriptions, they are usually placed to the right of the dots, but they are placed below the dots when the bars are horizontal, as in the codices and a few monumental inscriptions, particularly early ones (Table 9.2). Short numeral-phrases such as these were normally combined with another glyph indicating the thing being enumerated, most often time periods. While the primary and original function of the signs was numerical, some bar-and-dot numerals could also be used syllabically in the Maya script (though not, as far as we know, in any of the other Mesoamerican scripts). For instance, four dots could mean 4, but could also indicate near or partial homonyms of Classic Maya kan ‘four’, such as ká’an ‘sky, height’ and the first part of ká’anhá’an ‘haughty’ (Macri and Looper 2003: 262). Mesoamerican hieroglyphic writing on stone was a very ornate art, and numerals could be altered or ornamented in various ways that can make reading a numerical value difficult. Ornamental crescents were often employed in order to “fill in” a numeral that would otherwise have an empty space, and these can easily be confused with dots (Thompson 1971: 130). Similarly, decorative lines were sometimes added to bars for aesthetic purposes, which can make it difficult to distinguish one from two bars. In addition, in Maya monumental inscriptions and also (with a different form) in Maya codices, a sign for 20 was also occasionally used, producing a base-20 cumulative-additive system with a sub-base of 5. In addition to the numerical value of 20, it is also a glyph meaning ‘moon’ or ‘lunar month’ (Lounsbury 1978: 764).2 2

Closs (1978: 691) notes that the central dot in the latter of these signs is found only on inscriptions where the glyph has the numerical value ‘20’, thus distinguishing it from the more generic ‘moon’, where the dot is missing.

Numerical Notation

286

Table 9.2. Bar-and-dot numerals

r

R

m

18: vertical vs. horizontal orientation

z

13 vs. 11 (with ornamental crescents)

2

1

(

20/‘moon’ (monumental)

20/‘moon’ (codices)

20/‘moon’ (?) (Isthmian)

3\ 9

0\]

0/‘completion’ (monumental)

0/‘completion’ (codices)

The sign for 20 can occur on its own or in conjunction with bar-and-dot numerals from 1 to 19, thus representing numbers as high as 39. However, it is never repeated in a numeral-phrase; that is, one would not write 60 with three 20-signs. The accompanying bar-and-dot numerals could be placed above, below, or to either side of a 20-glyph (Kelley 1976: 23). The 20-sign was mainly used to indicate intervals between dates between twenty and thirty-nine days, thus avoiding the use of combinations of uinals (periods of twenty days) and kins (one day) (Thompson 1971: 139). Very rarely, it was used in expressions for longer time intervals, such as the irregularly constructed date on Stela 5 at the Maya site of Pixoy, indicating a quantity of 20 tuns (periods of 360 days) (Closs 1978). In a few instances, to be discussed, the 20-glyph was used for noncalendrical counts as well. A sign for zero also accompanied the bar-and-dot numerals. There is considerable paleographic variation in the signs used, but a “shell” sign was commonly used in the codices, while different signs were used in monumental writing. The Mesoamerican zero-sign is not completely synonymous with its Western counterpart; normally it served as a placeholder within dates, with the rough meaning of “completion of a given cycle of time.” While this raises the issue of whether we should regard this sign as meaning ‘zero’ at all (Thompson 1971: 137), which I will discuss later in greater detail, I do not see any reason to deny the Maya their zero. The Maya zero-sign is definitely numerical in function; it is found in the same contexts as the regular bar-and-dot numerals, but not normally elsewhere, and so the meaning ‘zero’ is quite appropriate. While the Maya probably did not have a concept of zero as a whole number, as is present in Western mathematics, neither did Seleucid Babylonian astronomers (Chapter 7), for whom the zero-sign served as the marker of an empty medial position. While bar-and-dot numeration is most closely associated with the Classic period (ca. ad 150-900), it developed centuries earlier, in the latter part of the Middle

Mesoamerican Systems

287

Formative period (1000–400 bc). While Macri and Looper (2003: 4) insist that writing on perishable materials must have preceded the tradition of carved writing on monuments and portable objects, there is no iron law of script development that requires this to be true. Bar-and-dot numerals are among the first identifiable signs of Mesoamerican writing systems, and occur in all three of the major Formative script traditions: Isthmian (epi-Olmec), Zapotec, and Maya. All three of these scripts survived into the Classic period, but the Maya inscriptions are by far the most numerous and significant. Understanding the early history of bar-and-dot numeration, however, does not give clear priority to any of the three. The poorly understood Isthmian (or epi-Olmec) script, known from a handful of inscriptions starting in the Late Formative period (400 bc to 50 ad), is associated with the latter stages of the Olmec civilization along Mexico’s Gulf Coast. Recent claims would give Olmec writing a much longer and complex history, however. The Cascajal block, which appears to date to the early first millennium bc and to be associated with the Olmecs, does not contain bar-and-dot numerals (or any other apparent numerals), but as it is a unique artifact, its relevance to the history of Mesoamerican numeration and writing remains unclear (Rodriguez Martinez et al. 2006). There is apparently no connection between the Cascajal block and Isthmian or indeed any other Mesoamerican script. The San Andrés cylinder seal, found near the Olmec center of La Venta and dating to the seventh century bc, is asserted by its discoverers to contain the personal name “King Ajaw 3” using stylized dots (Pohl, Pope, and Nagy 2002). Isthmian writing itself is best known from Late Formative texts from the first century bc to the second century ad, a period when Olmec political fortunes were on the wane. In some of these texts, bar-and-dot numerals were arranged for the first time in vertical columns to indicate periods of time and specific dates in the famous “Long Count” system, the calendar expressing dates as a series of numerals indicating five time periods. We can thus assign absolute dates to these artifacts from the calendrical evidence alone. The earliest of these is Stela 2 (actually a fragment of a wall panel) from Chiapa de Corzo, dating to 36 bc; similarly, Stela C from Tres Zapotes has a Long Count date corresponding to 31 bc (Marcus 1976: 49–53).3 The longer second-century Isthmian inscriptions, such as the La Mojarra stela 1 (156 ad) and the Tuxtla statuette (162 ad), contain bar-anddot numeral-phrases including Long Count dates (Justeson and Kaufman 1993). The Long Count date 8.5.16.9.9 (corresponding to July 16, 156 ad) occurs on the La Mojarra stela, with standard bar-and-dot numerals arranged vertically in a single column. The stela also contains a sign identified by its decipherers as meaning 3

As with all such dates, there is a small possibility that they were inscribed at a date later than the one given textually.

288

Numerical Notation

‘20’ or ‘moon’, parallel to the later Maya practice (Justeson and Kaufman 1993). However, aside from the numerals, much about the decipherment of Isthmian is hotly contested, with Justeson and Kaufman (1993) arguing that their decipherment is well under way, but Houston (2004b: 297–298) arguing that because of the contextualized nature of the system, the script may be indecipherable. The Zapotec civilization of the Valley of Oaxaca in southern Mexico began to rise to ascendance in the late Middle Formative period, and developed a script tradition quite distinct from Isthmian. The earliest well-attested Zapotec numeral is found upon Monument 3 from San José Mogote (600–500 bc) in the Valley of Oaxaca, where the day-name “1 Earthquake” is written with a stylized dot (Marcus 1976: 44–45). If the Isthmian San Andrés cylinder seal is misdated or non-numerical, then this inscription is the earliest attested instance of bar-and-dot numeration. Stela 12 from Monte Albán provides the first example of a combined bar-and-dot phrase for 8, apparently indicating a day of the Zapotec month (Marcus 1976: 45–46). Both of these are Middle Formative and definitively Zapotec rather than Olmec, and their early date points to Oaxaca as a potential region of origination for the system. Colville (1985: 796), however, is agnostic as to whether the barand-dot system was invented by the Olmec or the Zapotec, since in his analysis, both used vigesimal lexical numerals with a quinary component, a structure common also to the bar-and-dot numerals.4 While it was once widely held that the Maya tradition was a latecomer in the history of Mesoamerican writing, there is some evidence that the Maya tradition may have emerged alongside the Isthmian and Zapotec scripts, perhaps as early as 400 bce. Monument 1 from El Portón is an extremely eroded stela with one partially readable column of glyphs that may be ancestral to later Maya glyphs, including numerals. Although the inscription has no date, archaeological evidence places it in the late Middle Formative (Harris and Stearns 1997: 122). This raises the possibility that the three Mesoamerican traditions were essentially contemporaneous, and may have emerged in a context of economic and diplomatic exchange. Yet there is a distinct scarcity of Maya or Maya-ancestral texts from the Middle and Late Formative. Starting in the second and third centuries ad, the Isthmian and Zapotec traditions began to wane, and Maya inscriptions predominate in the Mesoamerican lowlands thereafter. The first certain Maya inscription that uses bar-and-dot numerals is Stela 29 from Tikal, which dates to 292 ad (Lounsbury 1978: 809); however, Stela 5 from Abaj Takalik, which dates to 126 ad, may also be an early Maya inscription (Closs 1986: 327). 4

Colville accepts the idea that the Olmecs spoke a Mixe-Zoquean language, whose modern speakers have numerals of this structure; this is not a universally accepted conclusion, and in any case the numerals almost certainly changed in the intervening millennia!

Mesoamerican Systems

289

This sequence of inscriptions demonstrates conclusively that the bar-and-dot tradition was an independent Mesoamerican invention with a complex history that remains only partially understood. While Seidenberg (1986) and others have attempted to postulate diffusion from Babylonia as the source of Mesoamerican mathematics and numeration, this is highly improbable on the basis of the evidence just discussed. Aside from the use of place value (itself a contested issue, as I will show), there is no similarity between Babylonian and Maya systems; they use different bases, different directions of writing, different media, and serve extraordinarily different sets of functions. The early origin of bar-and-dot numeration alongside the Middle Formative Mesoamerican scripts, the quinary-vigesimal structure of the system, and the general increase in the frequency and complexity of numeral expressions over time all point to its indigenous development. There are many thousands of identified Maya inscriptions from the Classic period (150–900 ad), the vast majority of which contain at least some bar-and-dot numerals. There is no identifiable regional variation in the form or ornamentation of the numerals within the Maya sphere of influence. Very early in the Classic period, the bar-and-dot numerals spread through highland Mexico; the Mixtecs used bar-and-dot notation to write numbers up to 13 (Caso 1965: 955). It is not clear whether they borrowed them from the Zapotecs (who also lived in the Oaxaca region, and continued to use bar-and-dot numeration throughout the Classic period) or from the more influential Maya. Bar-and-dot numerals were used occasionally at the highland city-state of Teotihuacán; on about a dozen inscriptions, numbers smaller than 13 were written with bars and dots (Langley 1986: 139–142). There is some evidence that the bar-and-dot numerals survived in central Mexico until the Spanish conquest. In Mixtecan-Pueblan texts such as Codex FejervaryMayer and Codex Cospi, sets of bars and dots arranged vertically or horizontally could represent counted bundles of offerings (Love 1994: 61). Although it is an irregular formation by the normal rules of the system, a set of four bars from the Mixtec Codex Selden may represent a quantity of twenty bundles (Boone 2000: 43). However, the Aztecs never used bar-and-dot numerals, instead relying on their own additive base-20 numerals. These developments will be discussed in greater detail later in this chapter. After the collapse of classic Maya civilization in the tenth century ad, attested examples of bar-and-dot numerals become increasingly rare. The latest Maya monumental inscription dates to 909 ad (Closs 1986: 317). Many regions where bar-and-dot numerals had previously been used, such as Oaxaca and the Valley of Mexico, abandoned the old system in favor of the central Mexican dot-numerals (see the following discussion). Bar-and-dot numerals were retained during the Postclassic period (tenth to sixteenth centuries) in Guatemala and Yucatan, where they were used on bark-paper codices until the Spanish conquest (Urcid Serrano

Numerical Notation

290

Table 9.3. Maya period glyphs

!

@

#

$

%

kin 1 day

uinal 20 kins

tun 18 uinals

katun 20 tuns

baktun 20 katuns

2001: 3). The last text on which bar-and-dot numerals occur is one of the books of Chilam Balam, in which an annotated description of the system is dated 1793 (Thompson 1971: 130). Yet the system essentially had ceased to be used by 1600 and was replaced by Roman or Western numerals.

Maya Calendrics and Positional Bar-and-Dot Numeration In all three of the Formative period Mesoamerican script traditions, numerals were written with dots placed above horizontal bars and not directly linked to any other glyph (although we can tell that the quantities enumerated were periods of time). This technique would later be the standard practice in the Maya codices. However, in the Classic period, most Maya monumental numerals were written with vertical bars with dots to their left, and were linked to glyphs for various periods of time. A bar-and-dot numeral-phrase from 0 to 19 would be combined with one of five glyphs for time periods – kin (one day), uinal (one ‘month’ of twenty kins), tun (one ‘year’ of eighteen uinals), katun (twenty tuns), and baktun (twenty katuns) – by placing the numeral to the left of the period-glyph.5 Each successive period is twenty times the previous one, except for the tun of eighteen uinals, which comprises a sum of 360 days, corresponding roughly to the solar year.6 Some of the more commonly used period glyphs are shown in Table 9.3 (cf. Closs 1986: 304–305). To express a specific fixed Long Count date, five numeral-glyph combinations were required (one for each period, written from longest to shortest). These were normally written in pairs of columns, most commonly but not 5

6

The terms katun and baktun mean, literally, ‘20 tuns’ and ‘400 tuns’. The latter term is in fact a coinage of Mayanists; there is no evidence that this word was associated with the glyph in question in ancient times. There are several extremely rare glyphs for longer periods, again with coined names: pictun (8000 tuns), calabtun (160,000 tuns), and kinchiltun (3,200,000 tuns), which presume a purely vigesimal progression of dates (Closs 1986: 303). It appears, however, that the Yucatecan and Cakchiquel Maya may have had a purely vigesimal year of twenty months of twenty days, though their numerical notation does not reflect this fact (Satterthwaite 1947: 8–9).

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Table 9.4. Initial Series date (9.14.10.0.2) with period glyphs (9 baktuns) (10 tuns) (12 kins)

i% j# l!

n$ 9@

(14 katuns) (0 uinals)

exclusively read from left to right and top to bottom. These expressed the amount of time between the starting point of the Maya calendar (corresponding to the date August 10, 3113 bc, in the widely accepted Goodman-MartinezThompson correlation with the Gregorian calendar) and any other date. In addition, the amount of time between any two days could be expressed by a “Distance Number,” such as 12 tuns, 0 uinals, 4 kins. Modern scholars use a convention whereby time values are expressed by writing the five numerical coefficients separated by points; thus, the date shown in Table 9.4 would be written as 9.14.10.0.12. For both Long Count dates and Distance Numbers, if the coefficient of a time period was zero (e.g., “0 uinals” in Table 9.4), the Maya included both a zero coefficient and a period glyph for that value, even though it was not logically necessary to do so in order to interpret the phrase correctly. While it is not known exactly why the Maya did this, it was probably for aesthetic reasons. Only occasionally in Distance Numbers (though never in Long Count dates), a period with a coefficient of zero was suppressed (Thompson 1971: 139). In a few texts, period-glyphs were omitted entirely, and dates were written simply by placing the five coefficients in a single vertical column. As mentioned already, the technique was present in the Isthmian and Zapotec inscriptions by the first century bc, and continued to be used by the Preclassic Maya (Marcus 1976: 49–57). Although it was largely abandoned thereafter, Stela 1 at Pestac contains a date (9.11.12.9.0) written in this format, which corresponds to 665 ad (Closs 1986: 326–327). Most other Maya inscriptions include all the periodglyphs, although sometimes the glyph for the last position (kins) was omitted (Closs 1986: 308). Our best evidence for the omission of period-glyphs comes, however, from the Dresden Codex, a Postclassic text that was probably written in the early thirteenth century, though it may be a copy of a much earlier document (Marcus 1976: 35).7 It is the most astronomically sophisticated of the surviving Maya texts, and contains more vertical columns of numbers than any 7

One set of five numbers without period glyphs is found on the fifteenth-century Madrid Codex that may qualify, but otherwise no other codices have them.

Numerical Notation

292

Table 9.5. Initial Series date without period glyphs

I N J ] L

9 baktuns 14 katuns 10 tuns 0 uinals 12 kins

other. Table 9.5 shows the Long Count date 9.14.10.0.12 as it would be written in this manner. This system requires that all the relevant numerical coefficients be included, even for periods for which there is a zero coefficient, to ensure that the correct quantity of time is counted. The bottom value always represents kins, the second from the bottom uinals, and so on, preventing any misreadings. Because these units of time are arranged in a mainly vigesimal sequence – each higher value is equal to twenty of the next lower value, except the tun of eighteen uinals – Mayanists today agree that this system of writing dates is a base-20 cumulativepositional numerical notation system with a sub-base of 5 (Kelley 1976, Marcus 1976, Lounsbury 1978). If so, then when the Maya wrote number columns such as the one in Table 9.5, each position must have represented a particular component of a single number. Positional numerical notation systems do this by having each successive position represent the next higher power of a base. Thus, when I write the number 1942, I mean a single count of some quantity, of which there are 1942, consisting of one thousand, nine hundreds, four tens, and two ones. In the Maya case, where the lowest unit expressed is kins, it is quite natural to assume that it counts kins. It is easy to translate the five periods into counts of days and then to take the sum, as in Table 9.6. However, if the period-glyphs were meant to be inferred when reading these columns, then such numerals ought to be read as five separate values, each no greater than 19, just as they would be if the glyphs were included. How, then, can we tell whether the interpretation in Table 9.6 is one that the Maya themselves made, or whether they simply “read in” the missing period-glyphs? How do we decide whether the correct interpretation is “1,400,412 kins” or “9 baktuns, 14 katuns, 10 tuns, 0 uinals, 12 kins”? If bar-and-dot numerals were used for large quantities of things other than time, we would have clear instances where the higher positions represent powers

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Table 9.6. Positional Maya count of days

I N J ] L

9 × 144,000 days

1,296,000 days

14 × 7200 days

100,800 days

10 × 360 days

3600 days

0 × 20 days

0 days

12 × 1 day

12 days = 1,400,412 days

of a base, rather than large calendrical periods. Yet no Mesoamerican texts use “positional” bar-and-dot numerals to count noncalendrical amounts. One is struck, upon comparing Maya inscriptions to those of any other civilization, by the virtual absence of phrases indicating large quantities of captives taken in war, goods paid in tribute, wealth owned by individuals, or any other noncalendrical quantity. In the rare instances where the Maya wrote numbers of other quantities above 19, sometimes they used additive techniques, such as the moon-glyph for 20, which is used in counts of 20 and 21 captives, but this does not allow the writing of very large numbers (S. Houston, personal communication). In other cases, it is possible that multiplicative techniques were used. Houston (1997) suggests that on a mural from Bonampak, a bar numeral for 5 was combined with a glyph, pi, which may have meant ‘unit of 8000 cacao beans’, producing a quasinumerical expression for a count of 40,000 cacao beans (Houston 1997). The Yucatecan, Ch’olan, and Tzeltalan languages all use numeral classifiers – linguistic particles that obligatorily follow lexical numerals and indicate the thing being counted (Berlin 1968; Bricker 1992: 71–73). Macri (2000) contends on this basis that the period-glyphs ‘kin’, ‘uinal’, ‘tun’, ‘katun’, and ‘baktun’, as well as any other glyphs that follow numbers, should best be interpreted as numeral classifiers.8 If this interpretation is correct, then by analogy with the calendrical system, the Maya likely expressed large noncalendrical quantities by combining bar-and-dot numerals with a sign for a metrological unit that may have been a multiple of some smaller unit but was not expressed in terms of that unit (just as 1 tun = 360 kins but was not expressed in such terms). These different means of writing larger 8

For Macri, this also explains why the early epi-Olmec and Zapotec calendrical inscriptions – written in Mixe-Zoquean languages – do not use period glyphs but simply series of bar-and-dot numerals.

294

Numerical Notation

numbers, and the lack of noncalendrical “positional” vertical number columns, cast doubt on the entire existence of Maya positional numerals. In Chapter 8, I discussed the transformation of the Chinese multiplicativeadditive system into a ciphered-positional one by adding a zero-sign and deleting the power-signs for 10, 100, 1000, and so on, so that ₒ◒榅⥪◐⃬ (7 × 1000 + (0) + 4 × 10 + 9) becomes ₒ2◐⃬(7049). There are similarities between this transformation and the removal of the Maya period-glyphs, but in the Chinese case, the removed power-signs are numerical (representing the powers of 10), whereas in the Maya case the period-glyphs are calendrical. The assumption that the fourth position of the Maya numerals means “7200” is wrong. Even so, it could still have been read as “7200 days.” This is undemonstrated, however, and I do not consider it likely. When the period-glyphs are present, as they are in most of the inscriptions on stone, Mayanists do not consider the calendrical system to be a positional one, and do not treat dates as a sum of days. Why, then, should the removal of these period-glyphs be anything more than an abbreviatory convenience? One year is equal to 365 (or 366) days, but this does not mean that if I write the date “2005/06/14” I really mean a sum of days equal to 2005 years, 6 months, and 14 days, and certainly I do not calculate such a sum in my head. A neglected tradition in the study of Maya calendrics and numeration recognizes that Long Count dates (with or without period-glyphs) are capable of being read positionally or nonpositionally. Teeple (1931) and Thompson (1971) claimed that such dates should be considered as a count of tuns (years), in which the final two places (uinals and kins) represented two separate fractions of years. Satterthwaite (1947) held that they should be read as two separate counts, one of years (the first three positions), the other of days (the last two). Closs (1977), the most recent scholar to deal seriously with this issue, holds that there were in fact three counts: a tun count, comprising, first, a positional numeral indicating 1, 20, and 400 tuns; second, a nonpositional bar-and-dot numeral indicating uinals; and third, a nonpositional bar-and-dot numeral indicating kins. All agree that the highest three periods (baktuns, katuns, and tuns) were read and understood by the Maya as a single count of tuns. Moreover, they claim that the Long Count dates were understood in this way, whether or not the period-glyphs were present. These readings are made on the basis of several lines of evidence. Separating the higher values, which are purely vigesimal, from the lower ones, in which the 18 uinals = 1 tun irregularity occurs, renders the system more readable, given the purely vigesimal structure of the Maya lexical numerals. It also helps explain a number of texts where the glyphs for the tun and its multiples are distinguished (by color or ornamentation) from the other two (Closs 1977: 22–23). In the irregular “Tun-Ahau statement” from Xcalumkin, a Long Count date is expressed simply as “9 baktuns, 16 katuns, 2 tuns” without uinal or kin values, further suggesting that

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295

tuns (and multiples thereof ) occupy a special role in the Maya calendrical system (Closs 1983). Finally, the glyphs for ‘katun’ and ‘baktun’ often show an affiliation to the basic ‘tun’ sign. Yet there is no reason to think that the Maya wrote glyphs for the baktun and katun but then simply ignored them in reading. As an analogy, the English words ‘decade’, ‘century’, and ‘millennium’ etymologically refer to tens, hundreds, and thousands of years, but “9 millennia, 4 centuries, 3 decades, 6 years” is read and understood differently from “9436 years” even though both phrases refer to the same time value. I agree fully with Closs that the kin, uinal, and tun counts were read separately, but believe that he has not gone far enough, and regard the Maya Long Count dates as five separate nonpositional counts of five different time periods. The assumption that the numerals in the Dresden Codex must have been positional is linked with the belief that positional notation was highly useful for doing calendrical calculations. Since the Maya did do these calculations, and since these numbers look like positional notation, it is natural to infer that they were, even though Classic Maya dates were normally written as five different periods rather than as a single sum of days. When Mayanists interpret Mayan chronology, they must translate Maya dates into a single number of days in order to the correlate Maya and Western calendars (e.g., the Goodman-Martinez-Thompson correlation establishes the beginning of the Maya calendar as Julian day number 584,283). Yet, however the Maya may have read these columns of numbers, there is no evidence that they ever calculated with them. The Dresden Codex is a repository of calendrical data, including what appear to be multiplication tables, but there are no calculations on paper. There is specific ethnohistorical evidence concerning Maya computation, from Landa’s Relación de las cosas de Yucatan, which suggests that the sixteenthcentury Maya did not calculate directly using bar-and-dot numerals: Their count is by fives up to twenty, and by twenties up to one hundred and by hundreds up to four hundred, and by four hundreds up to eight thousand; and they used this method of counting very often in the cacao trading. They have other very long counts and they extend them in infinitum, counting the number 8000 twenty times, which makes 160,000; then again this 160,000 by twenty, and so on multiplying by 20, until they reach a number which cannot be counted. They make their counts on the ground or on something smooth. (Tozzer 1941: 98)

Computation was done on some sort of flat surface, suggesting that some sort of physical counting board was employed. Some Mayanists have turned their attention to what sort of physical counters the Maya might have used and whether the bars and dots used as Maya numerals had physical correlates in rods and beans, or some other such markers (Thompson 1941: 42–43; Tozzer 1941: 98;

296

Numerical Notation

Satterthwaite 1947: 30–31; Fulton 1979: 171). Sol Tax, working among the Maya of the Guatemala highlands at Panajachel in the 1930s, found that they computed using beans or stones in groups of five and twenty, supporting the idea that the ancient Maya may have done similarly (Thompson 1941: 42). Counting boards are often positional in structure, and some use special counters or markers for empty positions – signs that resemble a zero. On this basis, some suggest that numerals were written positionally in a purely vigesimal fashion for noncalendrical purposes – that is, with the third and fourth positions having the values of 400 and 8000 – in emulation of the mode of computation (Marcus 1976: 39; Lounsbury 1978: 764). Yet the host of speculations on the use of bar-and-dot numerals directly in calculation, without an intermediary computational device, is useless (Sanchez 1961, Bidwell 1967, Anderson 1971, Lambert et al. 1980, Mühlisch 1985). While, as Anderson (1971: 63) states, “it is not unreasonable to suggest that some attempt to use the numerals directly in computations might have occurred,” this pastime tells us much more about the ingenuity of modern scholars than it does about the actual practices of Maya mathematics. Just as the Romans and Greeks had a place-value abacus but no positional numerical notation system, the presence of a Maya abacus-like device does not presuppose that they had positional numerals. The columns of an abacus work just as well if they indicate distinct units of baktuns, katuns, tuns, uinals, and kins as they do if they represent the power-values 144,000, 7200, 360, 20, and 1. The manipulation of counters is identical, but the reading of the results is very different. Unfortunately, the great bulk of Maya codices is now lost to us forever due to the tragic destruction of manuscripts on Spanish orders in the early colonial period. It is far too easy to create hypotheses concerning lost positional inscriptions when huge quantities of evidence have literally gone up in smoke. Yet the surviving evidence does not support the hypothesis that the number columns in the Dresden Codex should be interpreted as sums of days, and thus as a cumulativepositional numerical notation system. The most parsimonious explanation is that the omission of period-glyphs was abbreviatory but did not entail a radical rereading of the numerical coefficients. In his analysis of Maya arithmetic, Fulton noted that “it is possible to have a strictly positional notation, not altogether different from our present one, without any zero whatsoever” (1979: 171). Positionality requires some way of avoiding ambiguity between 749 and 7049, but this may be simply an empty space. Inverting this insight, I believe that the Maya bar-and-dot system had a zero, but did not use the positional principle. This is not to say that the Maya zero or completion-sign was nonfunctional. While it was retained for aesthetic purposes in places where it was not strictly needed (when period-glyphs were present), the zero was needed whenever the period-glyphs were omitted and there was an “empty”

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period. But the purpose of a Maya zero in a number such as 1.0.4 does not appear to be to indicate that the first number should be multiplied by 360, but rather simply to indicate that the middle position is empty, and thus the 1 should be read as 1 tun rather than 1 uinal. While something like positionality is used to distinguish different units of time, there was no Maya positional numerical notation system. Once we abandon the notion that the presence of place-value is an eternal standard of utility in numeration, we can see that the bar-and-dot system was highly useful for recording dates even though it was not, strictly speaking, positional. The main bar-and-dot system is cumulative-additive, and when cumulative-additive numeral-phrases were combined to express large time periods, what results is a quasipositional calendrical notation, but not a true positional numerical notation.

Maya Head-Variant Numerals In place of the bar-and-dot numerals, the Maya occasionally used a set of complex glyphs for the numbers 0 through 19, many of which correspond to the heads of Maya deities.9 These head-variant glyphs are far more variable in form than are the very regular bar-and-dot numerals. Each head-variant replaced the corresponding bar-and-dot numeral-phrase in an expression for a Maya date. An example of each of the signs is shown in Table 9.7 (redrawn from Thompson 1971: Figure 24–25). Because the highest number expressed using head-variant numerals is 19, there is, strictly speaking, no base to this system. However, because they replace barand-dot numerals, head-variant glyphs are associated with the five calendrical coefficients (baktun, katun, tun, uinal, kin), and thus assume elements of a vigesimal structure. The head-variant numerals from 1 through 12 are written with elementary signs. The signs for 14 through 19 are additive combinations of a “bared jawbone” element that represents 10 and the upper head of the sign for the appropriate unit. There are two signs for 13; the more common one (13a) is an additive combination of the bare jawbone for 10 and the head-glyph for 3, while the other (13b) is a distinct glyph for some sort of monster, and possibly holds some lunar significance as well (Macri 1985: 74). Because individual signs are not repeated to signify their addition, the head-variant numerals have more in common with ciphered than they do with cumulative numerical notation systems, but since they never exceed 19, they cannot be said to be either additive or positional. Although the head-variants for 1 through 12 are elementary signs, the system does not have a base of 12, as stated by Kuttner (1986). The relevant subunit of the head-variant numerals is not 12 but 10, since the signs for 14 through 19 (and 9

For an analysis of the specific deities and other symbolism associated with each glyph, see Thompson (1971): 131–137; Macri (1985); Stross (1985).

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Numerical Notation

Table 9.7. Maya head-variant glyphs

sometimes also 13) are expressed additively using 10 (Macri 1985: 75). No other Mesoamerican numerical notation system uses a decimal sub-base. The origin of this feature probably lies with the lexical numerals of the Mayan language family, which uses decimal structuring to express the numerals from 13 to 19, but has rather opaque formations for 11 and 12 (in Classic Maya, buluc and lahca), just as the English ‘eleven’ and ‘twelve’ do not show any clear relation to ‘ten’ (Lounsbury 1978: 762). Additionally, Macri (1985: 48) suggests that it may have been important to have thirteen simple signs to correspond to the thirteen deities used to name days in the Maya sacred calendar. The head-variant numerals are relatively common on Maya inscriptions, though less common than the bar-and-dot numerals. They also appear occasionally in the Dresden Codex, though not in the other Postclassic codices (Thompson 1971: 131). Macri (1985: 55) hypothesizes that they may have had a Preclassic origin, but no pre-Maya inscription uses them. Macri (1985: 48), pointing to phonetic correspondences between the head-variant signs and the Eastern Maya lexical

Mesoamerican Systems

299

numerals, suggests an Eastern Maya origin for the system, but Stross (1985) points out that many of the same correspondences exist in the Mixe-Zoquean language family, to which the Olmec language may have belonged. Yet none of the Isthmian inscriptions contains head-variant numerals, and many centuries lie between the decline of the Olmec civilization and the appearance of head-variant glyphs. The head variants are extremely different graphically and structurally from the bar-and-dot numerals, and cannot have emerged directly from the bar-and-dot tradition. We had best think of them as a complex set of metaphors by which the numerical symbolism of deities was used as a code for numerical information, not as a numerical notation system in their own right. Given the destruction of so many Maya codices, as well as the imperfect state of Maya archaeology and hieroglyphic decipherment, it is difficult to say when the head-variant numerals ceased to be used. Since the Dresden Codex is the only surviving Postclassic codex to use them, and then only occasionally, it is possible that they declined in use during the Postclassic period.

Mexican Dot-Numerals During the Postclassic period (tenth to sixteenth centuries), many of the peoples of central Mexico began using a system of dots to represent small integers in their pictographic manuscript tradition. Since this means of representation lacks a base and relies only on one-to-one correspondence, strictly speaking it does not constitute a numerical notation system, but it deserves some mention here. In their early history, the Mixtec and Teotihuacáni used bar-and-dot numerals, borrowed from the Maya or the Zapotecs, but after the tenth century ad, the system fell into disuse (Caso 1965: 955; Langley 1986: 143). While bar-and-dot numerals were occasionally used in a few later Mixtec codices, apparently for archaic or sacred reasons, they were largely replaced by a system whereby dots alone were used for the numbers 1 through 19, representing day-numbers and other objects (Colville 1985: 839–841). The peoples of Oaxaca, the Valley of Mexico, and the Gulf Coast used this system until the time of the Spanish conquest. A numeral-phrase was composed of a series of dots in a single row. To facilitate reading and to save space, larger numbers were often grouped in segments of three to five units, sometimes connected by lines, and sometimes changing direction (e.g., horizontal to vertical) in the middle of a numeral-phrase. Numbers above 20 were never expressed in this system. Given that the central Mexican calendar is part of a Mesoamerican calendrical tradition, and given the common use of dots for units in both the Maya and dotonly systems, I think it is plausible that between the tenth and twelfth centuries ad, the use of bars for 5 was gradually abandoned, although the reason behind

300

Numerical Notation

this change is not clear. The influence of Toltec culture, which was becoming predominant in Mesoamerica at this time, has been cited as the cause of this shift (Caso 1965: 955). Yet this argument begs the question of why the Toltecs did not adopt bar-and-dot numerals. Dot-only numerals are not known from anywhere in Mesoamerica prior to the tenth century ad, so it is unlikely that there was such a tradition prior to that point. Thus, unless the use of dots for units developed independently in the two different parts of Mesoamerica, the dot-numerals must be descended from the bar-and-dot system. The dot-numerals were ancestral to the later Aztec numerals, a base-20 cumulative-additive system. Because the Aztecs, like the Maya and Mixtecs, used dots for units, but because, unlike the bar-and-dot numerals, the Aztec system has no quinary component, the dot-numerals are a likely intermediary between the lowland and highland Mesoamerican systems. Both the dot-numerals and the Aztec numerals use up to nineteen dots for units, the difference being that with the Aztec numerals, the dots were more regularly grouped in fives, and higher numbers were written using different signs for the powers of 20. It is generally believed that the Aztecs inherited their tradition of manuscript writing from the Mixtecs (Colville 1985: 839). Dot-numerals continued to be used in Aztec manuscripts even after the development of the cumulative-additive numerals in the fourteenth century. By the time of the Spanish conquest, the Aztec numerals had supplanted the dotnumerals in some areas outside their tributary area, and were used in many of the post-Conquest Mixtec codices (Terraciano 2001).

Aztec The name “Aztec” applies most precisely to the Nahuatl-speaking inhabitants of the region immediately surrounding the ancient city of Tenochtitlan (modern Mexico City), who controlled a substantial tributary system in central Mexico between the fourteenth and sixteenth centuries. More generally, the term often refers to the various Uto-Aztecan–speaking peoples of central Mexico who were under Nahua rule during this period. The Aztec tributary network, which embraced numerous small states, produced a large number of manuscripts, using a combination of ideographic and phonographic signs. The considerable debate concerning whether this Aztec manuscript tradition constituted true writing or simply served as a mnemonic aid is irrelevant to the study of Aztec numeration. The Aztecs most definitely possessed a vigesimal numerical notation system, whose signs are shown in Table 9.8. The sign for 1 is the dot that was commonly used for units throughout Mesoamerica. The signs for the vigesimal powers are depictions of objects: for 20, a flag (pantli); for 400, a feather (tzontli, literally ‘hairs’); and for 8000, a bag used to

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Table 9.8. Aztec numerals 1

20

400

8000

V

X v

Y x :

y

yyy

xxxxx vvvvv xxx vvvvv vvv

ttttt tttt

(3 × 8000) +

(8 × 400) +

(9 × 1)

27,469 =

(13 × 20) +

hold copal incense (xiquipilli) (Harvey 1982: 190). These signs were combined in a cumulative-additive fashion, written in horizontal rows with the highest powers on the left. Although the Aztec numerals, unlike the Maya bar-and-dot numerals, did not use a sign for 5, groups of more than five identical signs were arranged in sets of five for easier reading. Groups of five signs were sometimes joined to one another with a horizontal line underneath the set. The purely vigesimal structure of the Aztec numerical notation system and the shapes of its numeral-signs are quite different from those of the lowland Mesoamerican bar-and-dot system. Instead, the Mexican dot numerals are the most likely ancestor of the Aztec system. It is plausible that the Aztecs originally used dots alone, but then, as the administrative needs of their tributary system grew, invented new numeral-signs for 20 and its powers. As far as can be discerned, the inventors and early users of the Aztec system were not influenced directly by the lowland Mesoamerican systems of the Maya. The Mexican dot numerals do not constitute a numerical notation system according to my definition, because they lack a base, meaning that the Aztec system was invented relatively independently. The most important function of the Aztec numerals was to record the results of economic transactions, such as amounts of cacao beans, grain, clothing, and other goods received from different regions of their tributary system (Payne and Closs 1986: 226–230). Numerals were also used in Aztec annals and historical documents, such as the record of the massacre of 20,000 prisoners in the Codex Telleriano-Remensis (Boone 2000: 43). Sometimes, when recording amounts of goods, individual numeral-signs were attached to an equal number of pictographic signs for goods. Accordingly, one might record 1200 balls of incense not as the numeral 1200 followed by a picture of an incense ball, but rather using three balls of incense, each of which would be placed immediately underneath a sign for 400. The use of Aztec numerals to record large quantities of tribute and individuals stands in sharp contrast to the Maya bar-and-dot numerals, which were almost

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wholly calendrical in function. The Aztecs denoted their thirteen months using series of dots in rows, just as the Mixtecs did, but when they did so, they did not group dots regularly in groups of five, and thus this represents a continuation of the dot-numerals in Aztec manuscripts (Boone 2000: 43–44). Normally, the Aztecs did not record dates or other calendrical information using the larger numeral-signs. In a single text, the Vatican Codex, large periods of time seem to have been expressed using cumulative-additive combinations of different signs, the largest of which represents 5206 years with thirteen signs that probably represent 400 (the third sign in Table 9.8), above which six dots were written (Payne and Closs 1986: 234–235). After the Spanish conquest, the Aztec numerical notation system continued to be used in various colonial documents. In fact, its use spread well beyond the Valley of Mexico, as Nahuatl increasingly became a lingua franca used by indigenous highland Mesoamericans. For instance, Aztec numerals are common in the Mixtec Codex Sierra, a mid-sixteenth-century account book that uses Western, Roman, and Aztec numerals side by side (Terraciano 2001: 40–45).10 In a few postconquest manuscripts, fractions could be depicted by segments of 1/4, 1/2, and 3/4 of a dot, low multiples of five by filling in quarters of the pantli flag sign, and 100, 200, and 300 using segments of the tzontli sign for 400 (Vaillant 1950: 202). A few post-conquest Aztec codices use multiplicative rather than strictly additive numerical notation. Guitel (1958; 1975: 177) was the first to point out that one of the often-reprinted examples of Aztec numbers depicts a basket of cacao beans from which four signs for 400 emerge, above which a pantli or flag for 20 is placed. This numeral-phrase represents a total amount of 32,000 cacao beans multiplicatively, as 20 baskets of 1600 beans each, rather than additively, as 4 xiquipilli of 8000. In a circumstance where cacao beans come in baskets of 1600 beans, however, it is important to denote that there are 20 baskets of 1600 each, not simply “32,000 beans.” This does not certify that placing the numeral-phrases for 20 and 1600 together means “32,000.” However, Guitel was not aware of another text, a Texcocan document now known as the Codex Kingsborough, where multiplicative notation was used extensively (Paso y Troncoso 1912, Harvey 1982). I will treat this structurally distinct variant of the standard Aztec system later. As disease, warfare, and acculturation diminished the strength of Aztec traditions, the old numerals ceased to be used. I do not know of any documents from later than 1600 that use Aztec numerals. After this point, Roman and especially Western numerals were employed throughout highland Mexico. 10

Boone (2000: 254) indicates that Oaxacan texts do not contain signs for 400 or 8000; at least in the case of the tzontli sign for 400, she is incorrect, as this is found in the Codex Sierra (cf. Terraciano 2001: Figure 2.16). Yet the year-date, written as 1563 in Western numerals, is not transliterated in Aztec numerals but rather in Mixtec lexical numerals.

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Texcocan Line-and-Dot The city of Texcoco in the province of Tepetlaoztoc was one of the most powerful cities in the Valley of Mexico both before and after the Spanish conquest. While many sixteenth-century colonial documents continued to use the Aztec numerals just described, a handful of Texcocan documents contain a quite different system, which I will call the “Texcocan line-and-dot” system. These documents have been studied extensively by Herbert Harvey and Barbara Williams and are identified collectively as the “Tepetlaoztoc Group” (Harvey and Williams 1980, 1981, 1986; Harvey 1982; Williams and Harvey 1988, 1997). The numeral-signs of this system are shown in Table 9.9 (Harvey and Williams 1980: 500). This system is cumulative-additive, with a base of 20 and a sub-base of 5. The sign for 5 consists of five unit-strokes joined together by a curved line, so it is perhaps just a matter of personal preference whether we see it as a separate numeral-sign. A similar technique was occasionally used to group sets of five dots for 20 into a single unit of 100. Perhaps the most unusual feature of this system is that, whereas other Mesoamerican numerical notation systems used a dot for the units, here a vertical stroke denoted the units, while the dot took a value of 20. Numeral-phrases were written in a variety of directions, but were always arranged in a single line from highest to lowest sign (Harvey 1982: 191). This form of notation has been found in only three texts, all of which were written in the vicinity of Texcoco in the 1540s. Two of these, the Códice Vergara and the Códice de Santa María Asunción, were cadastral records written around 1545 to enumerate individuals and their land holdings. These two are in fact so similar that they may have been parts of the same manuscript at one point, or at least were drawn at the same time (Williams and Harvey 1997: 2). The third, the Oztoticpac Lands Map, was written around 1540, and is also a record of lands, though as a map rather than a census record (Cline 1966). The primary function of the numerals in all of these cases was to record land measures. Pictographic signs expressing fractional linear units sometimes accompanied the numerals, but their meanings are still unclear (Williams and Harvey 1997: 26). Several numeral-phrases on the Oztoticpac map were used to count sums of days, showing that this system was not restricted to one domain. The Códice de Santa María Asunción used a modified form of this system to express numbers positionally rather than additively. In studying this text, Harvey and Williams (1980) showed that line-and-dot numerals occurred in two different sections, but served very different functions. In one section, known as milcocoli, line-and-dot numerals were used in the regular manner, written along the edges of maps of plots of land owned by different individuals to indicate their lengths. In another section, known as tlahuelmatli, the plots of land

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Table 9.9. Texcocan line-and-dot numerals 1

5

20

T

U

V

100

W

from the milcocoli section were redrawn as rectangles (regardless of their original shape). This section also contained numerals indicating the areal measurement of each individual’s land holdings. Comparing the milcocoli values, which indicated the lengths of the sides of plots, and the tlahuelmatli values, which recorded their total area, Harvey and Williams showed that a form of positional notation was used to record land areas in the tlahuelmatli section using a set of three distinct registers within a rectangular depiction of a plot of land (Harvey and Williams 1986: 242). In the top right corner, dot-and-line numerals indicated values from 1 to 19 in a small protuberance. On the bottom line of the rectangle, units and groups of five indicated multiples of 20 units. No dots were ever used in either of these two registers. When dots were found, they occurred with or without units in the center of the rectangle. Strangely, this third register also counted multiples of 20 (i.e., lines equal 20 and dots equal 400). No plots of land show values both on the bottom line and in the center. When the twenties register and the units register were added together, a total area value was reached. Harvey and Williams found that in 71 percent of the land plots they examined, the tlahuelmatli value was within 10 percent of the projected area for that plot based on the milcocoli measures (1980: 501). While this may not seem remarkably accurate, the plots were often very erratic in shape, so that calculating area was not simply a matter of multiplying length by width. Where there is no value in the third (central) register, a corn glyph, or cintli, is drawn at the top of the rectangle (Harvey and Williams 1980: 501). This sign may have been to indicate that the third register is empty, and thus may have served one of the functions of a zero-sign. These numbers can be read as a base-20 cumulative-positional numerical notation system with a sub-base of 5. Unlike Western numerals, in which the positions are arranged in a straight horizontal line, the Texcocan system uses three registers, the last two of which have an identical positional multiplier. However, the cintli glyph is not used to indicate empty positions, but rather provides information as to where to find the twenties power (on the bottom line, rather than in the center of the rectangle), and thus is conceptually distinct from the Western zero (and, indeed, from other zeroes such as the Babylonian zero). While I think that the correlation established by Harvey and Williams demonstrates that the tlahuelmatli value represents an area value, I am not fully convinced that it is meant to be read as a single number; it may instead represent two values, one of which represents a

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Figure 9.1. Numerical phrase from the Codex Kingsborough enumerating the population of Tepetlaoztoc at 27,765 (3 × 8000 + 9 × 400 + 8 × 20 + 5). Source: Paso y Troncoso 1912: 218v.

larger area value that is twenty times another value. I do not know how this issue could be resolved at present. A unique Texcocan document from 1555, the Codex Kingsborough, also uses something like the line-and-dot numerals (Paso y Troncoso 1912). It was a record prepared as part of a legal plea made to the Spanish encomendero of the region, denoting the massive amount of tribute paid to Spanish officials by the inhabitants of the Tepetlaoztoc region in an effort to convince colonial officials that the populace was overworked; extensive description in Spanish confirms the numerical values (Harvey 1982: 193). Curiously, this text combines Aztec numerals and Texcocan line-and-dot notation. Lines and chunked groups of five lines indicate 1 and 5, respectively. To write larger numbers, dots organized in lines of five were placed beside the signs for 20, 400, and 8000. The dots were placed in a single row, with the signs for 20 and 400 above them and the 8000 sign below them. Thus, where the regular Aztec numerals use these three signs cumulatively, the Kingsborough numerals are written using just one of each sign, next to which units from 1 to 19 were expressed with dots. Figure 9.1 depicts the numeral-phrase 27,765, indicating the population of the district at the time, but replacing the standard Aztec sign for 8000 with a head above a sack (Paso y Troncoso 1912: 218v). Whereas the basic line-and-dot system is cumulative-additive, and the tlahuelmatli system is cumulative-positional, this system is multiplicative-additive. While the dots look like the ‘20’ dots of the line-and-dot system, they each stand for 1 in this system. The total value of the numeral-phrase is taken by multiplying the dots for units by the values of the power signs and taking the sum. To add to the complexity of this situation, in some cases the flag glyph for 20 could be omitted, retaining only the dots (Paso y Troncoso 1912: 261r, 238v, etc.). In these cases, we have the elements of a cumulative-positional system, since the value of the twenties power is determined by its position in the numeral-phrase through implied multiplication. Finally, in a couple of numeral-phrases, lines are placed to the left of dots, as where a number is written as II●●, which might be read from

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right to left as 42 (Paso y Troncoso 1912: 274v). The erratic nature of the system suggests that whoever wrote it was extremely inventive and was in the process of experimenting with different means of representation. The most important question regarding the line-and-dot numerals, their positional variant in the tlahuelmatli records, and their multiplicative variant in the Codex Kingsborough, is whether they existed before the Conquest, or if their development was stimulated by contact with the Spanish. Neither the Western or Roman numerals are cumulative-positional or multiplicative-additive, and neither uses a base of 20, so the Texcocan systems are structurally distinct from those of the Europeans. Thus, it would be premature to conclude that Spanish contact brought about the development of these systems. It would be a mistake to attach much importance to the use of a vertical stroke for 1 (parallel with both Western and Roman numerals), given the ubiquity of this notation worldwide. Harvey and Williams (1980: 503) argue that, while the tlahuelmatli numerals are positional and have something like a zero, the use of different registers around a rectangle is quite different from Western positionality, and the zero does not serve the same functions as the Western zero. On this basis, they regard these systems as a native invention. I agree that the Texcocan numerical notation systems are so different from Western and Roman numerals that the Spanish could not have introduced them. Nevertheless, these may be instances of stimulus diffusion, which the Texcocan scribes developed with an awareness of Western and/or Roman numerals but without adopting the form and structure of those systems. That the Texcocan systems occur in only a handful of documents in a single region in the generation immediately after the Conquest and cease to be used after only two decades suggests that this was not a system of great antiquity. I believe that the multiplicative (Kingsborough) and positional (tlahuelmatli) variants may well have been stimulated within the rapidly changing social and intellectual environment of the early colonial period, while the cumulative-additive lineand-dot numerals probably existed in the pre-Conquest period. After 1545, epidemic disease greatly diminished the indigenous population of the region, and it appears that the Texcocan numerals ceased to be used after the middle of the sixteenth century.

Other Systems Because our understanding of Mesoamerican numerals is imperfect, a number of Mesoamericanists have developed theories regarding other forms of written numeration. I think it quite likely that more numerical information has been recorded than we are currently able to read in the Maya, Zapotec, Teotihuacáni,

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and Aztec texts. Even if these hypotheses turn out to be incorrect, some elements of them may be salvaged in the reconstruction of as-yet unknown numerical notation systems. In the 1950s, Howard Leigh postulated that in addition to bar-and-dot numerals, some Zapotec inscriptions contained encoded astronomical data using a different set of glyphs (Urcid Serrano 2001: 49–50, 54). In addition to bars and dots, this supposed system had over twenty unique signs, including elements of base-10, base-13, and base-20 notation, culminating in a special sign for 1,186,380 (3 × 3 × 3 × 13 × 13 × 13 × 20). I am unconvinced that such a system actually existed in the form asserted, but the Zapotecs may have encoded numerical information in some of these glyphs, though not in the way Leigh imagined. An unusual cumulative-additive bar-and-dot numerical notation system may have existed at Teotihuacán, a system in which the bars did not have a fixed value but could mean 5, 10, or 30, depending on their configuration (Langley 1986: 141). The nature of the script of Teotihuacán is still controversial, though it is increasingly thought that there was a complex pictographic script of the type used later in highland Mexico (Taube 2000). However, because Teotihuacán never used phonetic writing, and because, unlike the Aztec situation, there is no body of colonial documents to explain the numerals, there is no way to confirm the values of any potential numeral-signs. Penrose (1984) asserts that in the almanac portions of the Dresden, Madrid, and Paris codices, the Maya used “cryptoquantum” numerations to represent an encoded quantity of days separately from the bar-and-dot or head-variant numerals. He argues that the Maya represented hidden counts of large numbers by assigning numerical values to special signs indicating the days of the “Sacred Round” 260day calendar, and then by manipulating them through multiplication. Mayanists do not appear to be aware of Penrose’s research, and his conclusions must be viewed as highly speculative and even pseudoarchaeological. The manipulations necessary to extract meaningful numerical information from these signs are probably no more than numerological play. An unusual form of numerical notation is employed on the Codex Mariano Jimenez, a sixteenth-century post-Conquest manuscript from Otlazpan (in the province of Atotonilco). It is cumulative-additive and uses dots for units, horizontal lines for twenties, and horizontally oriented tzontli (feather) glyphs for 400, with fractions of 400 depicted by showing partially denuded tzontli signs. Although treated by Harvey and Williams (1986: 251–253) as simply a variation on the Texcocan system described earlier, the differences between the two systems suggest that they are quite distinct. If more documents using this sort of notation are found, we would have yet another post-Conquest regional variant of the Aztec numerals.

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Summary The two features common to all the Mesoamerican numerical notation systems is that they have a vigesimal base and that they are all cumulative rather than ciphered. The Maya head-variant glyphs, a sort of ciphered symbolic code that expresses only units up to 19, constitute a partial exception to this rule. The presence of a quinary element is quite common, as is the use of dots for units, but neither of these features is found in all the systems. Like the East Asian phylogeny (Chapter 8), Mesoamerican numerical notation systems use a variety of basic principles, and our primary evidence for their commonality is historical rather than structural. When the bar-and-dot numerals were the only part of the Maya script to be deciphered, it must have seemed remarkable to be able to extract calendrical information from such otherwise inscrutable documents. Yet we have a less complete understanding of the cultural history of the Mesoamerican numerical notation systems than we do of most Old World families. As our reading of Maya and Aztec writings becomes more sophisticated, it is to be hoped that we will come to a clearer understanding of their numerical notation.

chapter 10

Miscellaneous Systems

Around twenty systems do not fit neatly into the phylogenetic classification presented in Chapters 2 through 9. A few, such as the Inka khipu numerals, the Indus (Harappan) numerals, and the enigmatic Bambara and Naxi numerals, apparently arose independently of any other system, but gave rise to no descendant systems. Others are cryptographic or limited-purpose systems used in the medieval or early modern manuscript traditions of Europe and the Middle East. The majority of this chapter, however, deals with systems that emerged in colonial settings under the influence of the Western or Arabic ciphered-positional numerals, in conjunction with the development of indigenous scripts. Most of these systems were developed in sub-Saharan Africa, but Asian (Pahawh Hmong, Varang Kshiti) and North American (Cherokee, Iñupiaq) indigenous groups have also developed their own numerical notation systems. Finally, a few systems are probably members of other phylogenies, but their exact affiliations remain inscrutable enough that no definite conclusions can be reached.

Inka The Inka civilization was an enormous state on the Pacific coast of South America that reached its pinnacle between 1438 and 1532. While writing is often (and mistakenly) seen as a sign of civilization, or at least as a necessity for large-scale bureaucracy, the pre-colonial Inka state operated in the apparent absence of any 309

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writing system capable of expressing phonetic values. Instead, the primary means of encoding information was a system of knotted cords of different colors, known as khipus,1 whose main purpose was to record numerical information to aid in the administration of the Inka state. About 500 to 600 Inka khipus survive, although accurate provenances cannot be established for most of them (Urton 1998: 410). As first established by Locke (1912), khipus encode information using a decimal positional numerical system of knots, and around two-thirds of all attested khipus encode information in this readily understood fashion. Around one-third, however, do not follow this structure; they remain completely undeciphered, and may well have encoded non-numerical information (Urton 1997, Quilter and Urton 2002). A khipu is a set of colored cotton or wool cords consisting of a main cord (ranging from 10 to 20 cm up to several meters in length) from which multiple cords are suspended. These numeral-bearing cords are subdivided into pendant cords, which hang directly down from the main cord when it is held horizontally and stretched taut; top cords, which hang from the main cord but are tied so as to lay on the opposite side of the pendant cords; and subsidiary cords, which hang from a pendant cord, top cord, or another subsidiary cord rather than the main cord (Ascher and Ascher 1980: 15–17). The designation that pendant cords hang “down” and top cords hang “up” is an artifice; while they naturally hang on opposite sides of the main cord, we do not know how they would have faced. In numerical khipus, pendant, top, and subsidiary cords may contain a numeral-phrase or, more rarely, two. The system used to encode information is cumulative-positional with a base of 10. In each position, the value of that power of 10 is encoded using one to nine knots or loops. There is no sign for zero; instead, a space was left on the cord in an empty position. The units position is the one farthest from the main cord (its loose end), while the highest power is found closest to the main cord. While a khipu theoretically could express any number (because the system is positional), in practice, five-digit numbers are the largest recorded, and these are quite rare (Ascher and Ascher 1972: 291). Despite this obvious numerical structure, khipus are often erroneously conflated with unstructured systems that use one knot for one object (cf. Ifrah 1998: 70). Khipus contain a numerical notation system (a positional one, in fact) and thus must be compared to written numerals rather than to simple tallies. Three different sorts of knots encoded numeral-phrases, as seen in Figure 10.1. To encode a value in the tens, hundreds, or higher powers, the khipu maker would tie an appropriate number of single knots in a line. For the ones power, however, two 1

The spellings ‘Inka’ and ‘khipu’ currently enjoy favor with Andeanists over the older ‘Inca’ and ‘quipu.’

Miscellaneous Systems

Single knot 10s, 100s, etc.

Long knot Units: 2-9

311

Figure-8 knot Units: 1

Figure 10.1. Khipu knots.

different types of knot were used. For all the units except 1, the cord was looped around itself an appropriate number of times for the number being expressed; the “long knot” shown in Figure 10.1 represents 4. Because a long knot cannot be made with fewer than two loops, a value of one in the units position required the use of a different knot, a figure-8. The use of different knots might appear to take away from the purely positional nature of the system. Yet, because there is no zero-sign, this technique greatly reduced the chance of misreading a cord. If a cord contained six single knots followed by two single knots, it could not be read as 62 but only as 620 (or possibly 6200). The use of long or figure-8 knots in the units position makes it much easier to tell which is the units position, and thus to identify the subsequent positions. Figure 10.2 depicts an unattested but plausible khipu. The main cord lies horizontally, with the pendant cords (P1 through P4) hanging down and the top cord (T1) facing up, and with subsidiary cords (S1 through S3) hanging off both pendant and top cords. On this cord, only a single value would have a figure-8 knot (the 1 in the units position on P4); the other units values (3 on P2, 6 on S1, 2 on P3, 6 on T1, and 6 on S3) would be made with long knots, and all the tens and hundreds figures with single knots. As is sometimes the case in attested khipus, the top cord value (776) is equal to the sum of the pendant cords (360 + 23 + 102 + 291), while the value on the top cord’s subsidiary (S3 = 26) is the sum of the subsidiaries of the pendant cords (20 + 6). Although we can read the numerical values on khipus, their origin and early history remain unclear. A set of twelve cotton strings twisted around sticks excavated at the late pre-ceramic pyramid complex of Caral, Peru (c. 2600–2000 bce) has been claimed by its excavator to be an early form of khipu; however, this claim is unsubstantiated, and full data on the artifact remain unpublished (Mann 2005). Bennett (1963: 616) notes that some Mochica vessels (Early Intermediate period, c. 200–600 ad) bear markings that are suggestive of khipus. The first wellsubstantiated evidence for khipu use comes from Middle Horizon sites (c. 600– 1000 ad) associated with the Wari civilization in coastal Peru (Conklin 1982).

Numerical Notation

312

T1 776

P1 360

P2 23

S3 26

S1 6

P3 102

Figure 10.2. Khipu structure.

P4 291

S2 20

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These khipus cannot be deciphered numerically because of their deteriorated condition (although they may have used nondecimal bases), and they use color in very different ways than the Inka khipus, but are nonetheless clearly of the same basic type. Most surviving khipus were collected haphazardly; prior to 2001, only two archaeological discoveries of khipus had adequate proveniences (Urton 2001: 131). The khipu system may have developed out of an earlier knot-based system using simple one-to-one correspondence, because knot tallies of this sort are widely distributed in the Circum-Pacific region (Birket-Smith 1966). There is no evidence of any connection between the khipu notation and any other numerical notation system, and thus it is definitive that the Andes was home to an independent development of the place-value principle. Khipus were a vital part of the Inka record-keeping system; they were employed in this capacity for censuses, tributary records, and similar administrative functions. Jacobsen (1964), noting the frequency with which the top cord equals the sum of the pendant cords, suggests reasonably (but unconfirmably) that such khipus may have been part of a double-entry accounting system. The decimal base of the khipu notation system corresponds to the decimal divisions of society by which the state was administered. Some khipus contained calendrical rather than administrative information (Ascher and Ascher 1989, Urton 2001). For instance, khipu UR6 from Laguna de los Cóndores contains a series of cords with values of 20 to 22 followed by cords with values of 8 or 9, and the sum total of these cords is 730 (365 × 2), strongly suggesting that it may have been a biennial calendar (Urton 2001: 138–143). Most surviving khipus with good provenience have been recovered from mortuary contexts. The Inka probably placed khipus in the graves of khipukamayuqs (khipu makers and users). It is unclear whether this implies that some of them should be read as “tomb texts,” because at present we are unable to extract non-numerical information from them (Urton 2001: 34). Much ink has been spilled recently about whether khipus constituted something more than a numerical notation system, approximating the functions of a writing system. Ethnohistorical data suggests that khipus recorded genealogical, historical, and literary information, which raises the question of what “code” was used to do so (Bennett 1963: 618). Gary Urton (1997, 1998, 2001) has argued forcefully that many khipus contain syntactic and semantic information far exceeding their numerical functions. He contends that purely numerical readings that translate khipu texts as Western numerals “inevitably mask, and eliminate from analysis, any values and meanings that may have been attached to these numbers by the Quechua-speaking bureaucrats of the Inka empire who recorded the information” (Urton 1997: 2). He argues against the idea that a khipu could have been interpreted only by its maker or those trained in an idiosyncratic private code (Urton 1998: 412). The khipus must have recorded some non-numerical information; a list of pure

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numbers is practically useless. In some way, at least the nature of what was being counted must have been recorded somehow. The most likely possibility is that this was done with color; the 1609 Comentarios of Garcilaso de la Vega (1539–1616) inform us that colored cords were used to record different commodities (Bennett 1963: 617). Yet many khipus use multiple colors of cord, and there exists no reliable means of reading the type of items counted. Recent scholarship has established that at least some khipus encoded toponymic information, associating particular records with the places to which they refer, which helps us to clarify how information was communicated within the Inka administrative hierarchy (Urton 2005, Urton and Brezine 2005). A cache of twenty-one khipus excavated from a single urn in the palace of Puruchuco (northeast of Lima, Peru) revealed many whose introductory cords begin with arrangements of three figure-8 knots (which normally represent the numerical value 1), suggesting that this served as an identifier with which any reader could associate the numerical data. Some of the khipus in this cache encoded identical or nearly identical information, suggesting that copies might be kept at the site of a khipu’s manufacture, with other copies distributed to the capital, Cuzco, or elsewhere. Urton has also identified three-term number sets that occur on some of the Puruchuco khipus that do not fit into the numerical structure of the remainder of the record, and suggests that these are labels, perhaps an ayllu (kinship group) with which the khipu was associated (Urton 2005: 162–163). The khipus encode at least as much information as the proto-cuneiform accounting signs of Mesopotamia (Chapter 7), which identify only items being counted and the quantity of each item, but which similarly served as a state-oriented bookkeeping system of “credits” and “debits” (Urton 2005: 164). Since the proto-cuneiform system is regarded as “proto-writing,” it is reasonable to attribute the same status to the Inka recording system (Salomon 2004). It is possible that the khipu system, over time, might have developed into a system for representing speech (though doing so would be more difficult for a knot-based notation than for a system based on inked or impressed signs). A single khipu cannot be at the same time both a record of numbers and of things being enumerated and a fully developed system for recording history and literature. Yet the roughly one-third of khipus that do not follow an ordinary decimal and positional structure may well have been non-numerical. It is equally possible that some numerical khipus recorded ideas or speech through some sort of code, but without a key, we cannot definitively conclude that this was the case. Urton’s (1997: 179) speculation that there might have been two pre-colonial khipu systems (one for recording quantity and another for recording narrative) is useful but at present unconfirmed. While post-Conquest chroniclers state explicitly that the khipus carried only numerical meanings, Urton postulates that the

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early colonial Spanish, in order to undermine traditional patterns of knowledge, rapidly transformed the khipu system from a full-fledged writing system into a purely numerical and non-narrative recording instrument (Urton 1998: 410–411). I admit that the Spanish may have wished to denigrate Inka knowledge, and also that there is an enormous issue of translation between indigenous concepts and what is claimed in early Spanish chronicles. Yet it would have been much simpler to replace the khipu system with European administrative techniques than to attempt such an alteration of its function. Moreover, analogies with the mathematical practices of modern Quechua speakers will not help us to interpret centuries-old khipus unless continuity between pre-colonial and modern ways of thinking can be demonstrated. We know that the pre-colonial khipu system was at minimum a “number + noun” information system, of which only the numerical component can usually be determined, but we do not know more than this with any certainty. The earliest Mesopotamian civilization did not require phonetic writing, nor did that of the Yoruba, to mention only two highly complex but nonliterate sets of polities. To infer a writing system out of nothing but an assumption that such a system would have been necessary is grossly anti-empirical. Regardless, khipus alone cannot have been used for performing arithmetical calculations. Khipus are even less amenable to physical manipulation than are written numerals (which can be lined up and crossed out). We do know, from sixteenthcentury documents, that the khipukamayuq were responsible not only for making and reading the khipus but also for calculating the results, and that they did so using a set of stone tokens (Urton 1998, Fossa 2000). While no archaeological evidence has confirmed the existence of such a system, there is limited documentary evidence for an “Inka abacus” in the Nueva corónica y buen gobierno, a document written between 1583 and 1613 by Don Felipe Guaman Poma de Ayala (ca. 1534– 1615), a descendant of an Inka princess who was an important chronicler of life in late sixteenth-century Peru and a critic of Spanish rule (Wassén 1931; Urton 1997: 201–208). In one corner of a page depicting a khipukamayuq at work, there is a grid of five rows by four columns, in each square of which is found a number of circles: five dots in the first column, three in the second, two in the third, and a single dot in the fourth. Moreover, some of the dots have been filled in, while others remain empty. Unfortunately, while the commentary that accompanies this picture notes that the Inka reckoners used computing boards, there is no description of how this system worked. Wassén’s (1931: 198–199) effort to infer this information assigns the rows values of the powers of 10 (starting with 1 at the bottom) and the values 1, 5, 15, and 30 to the columns (which were multiplied by the row-value), but he does so unconvincingly, solely on structural grounds. Nevertheless, it is unlikely that this board is a result of diffusion from Spain, since no comparable board was used in the sixteenth century anywhere in Europe (Wassén 1931: 204).

316

Numerical Notation

After the Spanish conquest in 1532, khipus continued to be used for the same administrative functions as they had been previously, and the Spanish, through Andeans who could read their values, used the data recorded on them (Loza 1998, Fossa 2000). Brokaw (2002) discusses the cognitive shift required of Quechua speakers in the sixteenth century with the transition from khipu notation to European literate conventions, as demonstrated through Guaman Poma’s Nueva corónica. This shift involved changes in how texts were organized, written, and read, and also forced pre-existing Quechua ideas about numbering and counting into conflict with Western textual conventions (e.g., regarding pagination). Also in the sixteenth century, the mestizo chronicler Blas Valera (1545–1597), who advocated for Quechua as a Christian liturgical language in addition to Latin, developed a system of forty syllabic knots to be used on so-called royal khipus, which reflected Valera’s theories about Quechua as a worthy language and his conviction that one could not rank societies based on the quality of their writing systems (Hyland 2003: 129–135). While Valera’s system has occasionally been regarded as a pre-colonial invention for which he took credit, thus making the khipus at least partly a phonetic writing system, it is substantially more likely that the royal khipus were a colonial invention that applied the notion of phoneticism to the existing pre-colonial system. The widespread use of khipus was curtailed in the 1580s, when they were declared to be idolatrous and the Spanish colonial administrators decreed that they should be destroyed. Yet in that same decade Mercedarian friars began using khipus to encode information about Christian life, using the principles outlined by Valera (Hyland 2003: 136–137). While it was once thought that the use of khipus had essentially ceased by the sixteenth century, it is now evident that their use for secular administration in the colonial period continued. Moreover, in local accounting contexts apart from state control, khipus have continued to be used by animal herders in parts of Peru and Bolivia for recording quantities of livestock up to the present day (Bennett 1963: 618–619; Ifrah 1998: 69–70; Urton 1998: 410; Salomon 2004). Nineteenth-century khipus found by the explorer Charles Wiener in Paramonga have systems of knots and bundles quite different from the pre-colonial khipu, but confirm that the practice continued in varying forms well after the colonial period (Hyland 2003). These were not simply tallying systems, however, but were cumulative-positional and decimal, and thus constitute a survival of the Inka numerical notation system. A significant part of khipu studies today, then, and of Inka ethnomathematics in general, relies on ethnographic work with the descendants of the Inka, for example, modern Quechua and Aymara (Quilter and Urton 2002, Salomon 2004). The modern “episteme of numbers” rests on different principles than Western arithmetic, in particular placing great emphasis on even numbers as “complete” and odd numbers as incomplete or even dangerous (Urton 1997; Brokaw 2002: 281–287). However, between the sixteenth and twenty-first centuries, significant

Miscellaneous Systems

317

changes almost certainly occurred in these ideas, much as the foundations of sixteenth-century European mathematics bear only a passing resemblance to modern practices. Further ethnographic, ethnohistorical, and archaeological data promises to help resolve some of the remaining mysteries concerning the khipu records, and the establishment of the Khipu Database Project will help facilitate computer analysis of khipus in museums and collections worldwide (Khipu Database Project 2004). A complete decipherment of the khipus as they were used in premodern contexts, however, may well be impossible.

Bambara One of the most peculiar African numerical notation systems was used by the Bambara of Mali in religious and divinatory contexts (Ganay 1950). Although details of the system’s history are sketchy, we have a fair idea of the numeralsigns and the structure of the system. The Bambara numeral-signs are shown in Table 10.1. The Bambara system is structurally irregular; while it is additive, it alternates between cumulative and ciphered notation, and while it is mainly decimal, it has vigesimal components. For instance, 1 to 19 are written primarily with vertical cumulative unit-strokes. The value of a set of vertical strokes is doubled if a horizontal line is crossed through it (effectively dividing the number into two registers, one above and one below the line). For odd numbers, an additional half-stroke can be placed at either end of the phrase, sometimes vertically and other times at an angle. Each of the tens from 20 to 170 has its own sign, which makes the system ciphered at this point. The signs for 180 and 190 are additive combinations of 100 + 80 and 100 + 90, respectively. To add a number of units from 1 to 9 to one of these ciphered signs, an appropriate number of strokes are attached to the sign for the multiple of 10 (or dots, when adding units to 60, 160, or 170). This means of representation is decimal – each decade has its own sign to which up to nine unit-signs were attached. Yet, because there are signs for 110, 120, and so on, it is unlike the Greek ciphered-additive alphabetic numerals (in which 100 is followed by 200, 300, and so on). Moreover, some of the decade-signs are similar enough to the ones preceding them (40 vs. 50, 100 vs. 110, 140 vs. 150, 160 vs. 170) to suggest an additional trace of a vigesimal base. For numbers higher than 200, the cumulative principle is again employed by repeating the sign for 100 (another decimal component) as many times as required in a vertical column, with any needed additional signs placed at the top of the column. Figure 10.3 shows some higher numeral-phrases (as reproduced from Ganay 1950: 300).2 2

Large numeral-phrases for 1935 and 4000 are also listed, but are highly irregular, and I cannot determine what principle has been used to determine their value.

Numerical Notation

318

Table 10.1. Bambara numeral-signs 1

2

3

4

5

a

aa

aaa

aaaa

aaaaa

6

7

8

9

10

aaaaaa

aaaaaaa

bbbb

bbbbc

bbbbb

11

12

13

14

15

bbbbbc

bbbbbb

bbbbbbc

bbbbbbb

ybbbbbbb

16

17

18

19

20

bbbbbbbb

bbbbbbbbz

bbbbbbbbb

ybbbbbbbb

d

30

40

50

60

70

e

f

g

h

i

80

90

100

110

120

j

k

l

m

n

130

140

150

160

170

q

r

s

o

p

180

190

t

u

The Bambara numerical notation system was used primarily in ritual contexts, especially those pertaining to divination using numbers (Ganay 1950: 298). Little is known of its origin, period of use, or decline. It shows no resemblance to any of the systems that would have been known by Bambara, who had considerable contact with the Muslim world. While the ciphered-additive Arabic abjad numerals commonly used for divination in the Maghreb are the most likely ancestor,

220

489

230

240

Figure 10.3. Bambara numeral-phrases.

Miscellaneous Systems

319

the Bambara system is quite different in most respects – its frequent use of the cumulative principle, the presence of a vigesimal component, and its numeralsigns. I have no idea whether this system continues to be used, though I suspect that it does not.

Berber The Berbers, or Imazighen, live in North Africa and speak a set of closely related Afro-Asiatic languages. For most of their history, the Berbers have been a marginal people living on the periphery of larger polities (Carthage, Rome, and various Muslim states), but they have nonetheless retained considerable cultural independence. The Berbers developed a consonantal script on the model of that used in Punic Carthage possibly as early as the sixth century bc, which was in continuous use until at least the third century ad; the Tifinigh script (still used by the modern Tuareg for love letters, domestic ornamentation, and games) is descended from it (O’Connor 1996). There is no numerical notation system associated with either the classical Berber script or its modern descendant. Nonetheless, a distinct numerical notation system was used by traders in the Berber city of Ghadames (on the border of Algeria and Libya) in the nineteenth century, and appears to be in use still (Rohlfs 1872, Vycichl 1952, Aghali-Zakara 1993). Vycichl (1952: 81–82) presents the system as described by two separate authors, Hanoteau and Si Mohammed Serif, while Rohlfs presents a third system; I reproduce all three in Table 10.2. The system is cumulative-additive and written from right to left, with the decimal powers repeated up to four times and the halved powers only once in any numeral-phrase. Sometimes, groups of signs could be placed in two rows to save space (Rohlfs 1872). In addition to these signs, Hanoteau claims that a horizontal line stood for the fraction 1/4, and that this sign could be grouped vertically to indicate 1/2 and 3/4 (Vycichl 1952: 81). The two sets of numeral-signs are identical, except for the signs for 500 and 1000. It is possible that both of these systems were actually used, either in different contexts or at different times. However, it is more likely that an error of interpretation created the discrepancy, because Hanoteau’s 1000-sign is essentially identical to Serif ’s 500-sign. The question of the Berber system’s ancestor (if any) is still open. It is possible that it was an entirely independent development. The similarities between certain numerical signs and letters of the Berber consonantary (r with 10, f with 500, and s with 1000) are interesting, but they do not correspond to the Berber lexical numerals in any obvious way. The Phoenician/Punic numerical notation system is quite different in its structure, lacking a sign for 5, and employing a special sign for 20 and a hybrid multiplicative-additive structure above 100. The use of | for 1 and > for 5 is superficially similar to the Roman system; Ghadames was an important

Numerical Notation

320 Table 10.2. Berber numerals 1

1 1 Rohlfs 1 1/4: E 1/2: J 3/4: O

Hanoteau

Si Mohammed Serif

5

10

50

100

í í í

ó ó ó

ú ú ú

ñ ñ š

500

1000

ª ª

Ñ º º

44 = ýý88 488 = 111íóóóúññññ

trading post (Cydamus) under imperial Roman control, and there are Roman numerals on some of the Latin inscriptions found there. However, the systems are written in different directions and have different signs for the higher values. Vycichl (1952: 83) suggests that the system derives from the South Arabian numerals. The Berber script may be somehow indebted to the South Arabian (O’Connor 1996: 112). If Hanoteau’s list of signs is correct, the Berber system, like the South Arabian, lacks a sign for 500; furthermore, both systems use O for 10. However, the South Arabian system ceased to be used in the first century bc and was never used in Africa, so to accept this theory requires that we believe in a two-thousandyear unattested history for this system. The system having the most promise as an ancestor is the Arabico-Hispanic variant Roman numerals (Chapter 4) used in a Spanish Inquisition document of 1576 (Labarta and Barceló 1988: 34). This system employed |, V, and O for 1, 5, and 10, was written from right to left, and was used in the same general region as the Berber system. Though three centuries is still a chronological gap that needs to be resolved, it is not nearly so great as the enormous leaps that need to be inferred to hypothesize alternate paths of diffusion. Ultimately, more data are needed for this system to be assigned unambiguously to any phylogeny. The Berber system was used in the nineteenth century for indicating the prices of trade goods. Rohlfs (1872) learned about this system as a traveler in the Ghadames region, but only ascertained the meanings of the signs through great effort and negotiation. He thus believed that the system was semi-cryptographic, restricting the flow of information concerning prices to a limited group of Berber traders in order to give them an advantage over Arab traders. The system is not especially difficult to decipher, however, and so I am unconvinced that this purpose was very important. Aghali-Zakara (1993: 151–153) reports that several numerical notation systems are still used in the region of Ghadames; one of these is the system just described; another simply repeats the sign for 10; and a third, inexplicably and surely incorrectly, is seen as having no signs for the powers of ten but only for the

Miscellaneous Systems

321

Table 10.3. Oberi Okaime numerals 1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

0

11

12

13

14

15

16

17

18

19

0

@ # $ % ^ 1938 = 4^* (4 × 400 + 16 × 20 + 18)

&

*

(

)

!

sub-bases, 5, 50, 500, and 5000. None of these systems is widely used, but they do appear to be in present use among at least some Tuareg.

Oberi Okaime In the late 1920s, a syncretic indigenous-Christian religious movement known as Oberi Okaime (or Obεri Vkaimε) arose among the Ibibio-Efik, speakers of a set of related dialects of the Niger-Congo language family in southeastern Nigeria. By 1931, the divinely inspired leaders of this movement had developed an alphabet (written from left to right) and a set of numeral symbols (Adams 1947; Hau 1961). The script represented an arcane revealed liturgical language of the sect, but was not used to write Ibibio. The Oberi Okaime numeral-signs are shown in Table 10.3 (Hau 1961: 295). The system is ciphered-positional and vigesimal; it is the only known cipheredpositional base-20 system with no sub-base, with the partial exception of the Maya head-variant glyphs. The vigesimal structure of the system is based on the similarly vigesimal Ibibio lexical numerals (Abasiattai 1989: 505–506). Numeral-phrases are written from left to right with the highest powers on the left. The inventors of the Oberi Okaime numerals were educated in Christian missionary schools in the 1920s, where they became literate in English and learned Western numerals. While none of the numeral-signs resemble the corresponding Western numerals except for 0, the script and its numerals were strongly influenced by Western traditions of writing (Dalby 1968: 160–161). Hau’s (1967) highly dubious suggestion that the Oberi Okaime script derives directly from Minoan Linear A, used thousands of kilometers away and over three millennia previously, cannot possibly apply to the numerals. The numerals were used in a relatively small number of liturgical texts and personal letters among the members of the Oberi Okaime sect. The system was still used by some individuals when Kathleen Hau corresponded with its leaders in 1961. In 1986, Sunday school classes were begun in order to revive the Oberi Okaime liturgical language along with the numerals and script, but this appears

Numerical Notation

322

Table 10.4. Bamum numerals (original) 1

2

3

4

5

6

7

8

9

¶

á

í

ó

ú

ñ

Ñ

¿

¬

1

10

100

1000

10000

«

»

½

¼ ¡ 76 = Ѽñ½ or Ѽ½ñ

to have been unsuccessful (Abasiattai 1989: 506). Western and sometimes Arabic positional numerals are used in the region today.

Bamum The Bamum live in part of southwestern Cameroon near the border with Nigeria. In the late nineteenth or early twentieth century,3 Sultan Ibrahim Njoya (ca. 1875–1933), a Bamum ruler, took it upon himself to develop a script for his people. Njoya, aided heavily by an assistant, Nji Mama, developed the script through several stages, starting with a large logosyllabary and gradually reducing the number of signs into an eventual syllabary of eighty characters (Tuchscherer 2005: 479). From its inception, Bamum writing made use of numerical notation. The earliest Bamum numerals are shown in Table 10.4 (Dugast and Jeffreys 1950: 6). This system is decimal and multiplicative-additive, with numeral-phrases written from left to right. Curiously, the power-sign for the units could either precede or follow the unit-sign (Dugast and Jeffreys 1950: 30). The unit-signs for 7, 8, 9, and 10 were not at this stage fully ideographic, but instead were constructed of two graphic parts, each of which represented a syllable in the two-syllable Bamum words corresponding to those numbers (Dugast and Jeffreys 1950: 98). At this point in the system’s history, we could well consider it to be a set of lexical numerals. This is the same problem we encountered with the Shang/Zhou and Chinese classical systems (Chapter 8), which, not coincidentally, also are multiplicativeadditive and associated with logosyllabic scripts in which some characters (including numeral-signs) are ideograms. Around 1921, Njoya supervised a transformation of the script into a form known as mf␧mf␧, which altered the numerals from multiplicative-additive to 3

Dugast and Jeffreys (1950: 4) place its invention in 1895 or 1896, although it may have been as late as the turn of the century.

Miscellaneous Systems

323

Table 10.5. Bamum numerals (mf␧mf␧) 1

2

3

4

5

6

7

8

9

0

ö

ò

û

ù

ÿ

Ö

Ü

¢

£

¥

ciphered-positional by removing the power-signs (Dugast and Jeffreys 1950: 30). The old sign for 10 took over the role of zero; numeral-phrases were written from left to right with digits for 0 through 9, like Western and Arabic positional numerals, the Bamum system’s primary rivals. The mf␧mf␧ numerals are shown in Table 10.5 (Dugast and Jeffreys 1950: 31). During its heyday in the first three decades of the twentieth century, the Bamum numerals were used quite widely, primarily due to Njoya’s political clout, and were employed on legal documents, census records, histories, and personal letters, both handwritten and printed. Njoya was deposed in 1931 and died two years later, after which time the Bamum script and numerals rapidly fell into obsolescence. Today, the script is preserved to a small extent as a source of ethnic pride among some Bamum, but there are very few surviving users (Tuchscherer 2005: 479). Nevertheless, because we are able to trace their rapid transformation from an additive to a positional structure, the Bamum numerical notation systems are more than just a historical curiosity and tell us a great deal about the way that numerical systems change.

Mende Around 1917, an Islamic scholar named Mohamed Turay developed a syllabic script known as Kikakui to represent graphically the Mende language spoken in southern Sierra Leone, almost certainly influenced by the Vai script of Liberia developed in the previous century (Tuchscherer 2005: 478). A few years later, Kisimi Kamara, Turay’s grand-nephew and student, expanded and revised the script in a second stage. While Western numerals were always used alongside the Vai script, the inventors of the Mende script developed a distinct set of numerical signs to accompany the syllabary, although it is unknown which of Turay or Kamara was responsible for the innovation. The Mende numeral-signs are shown in Table 10.6 (Tuchscherer 1996: 71–75). The system is decimal and multiplicative-additive, and numeral-phrases are constructed with the highest powers on the right. Because the system is multiplicative-additive, no sign for zero is needed or used. Unit-signs are placed above the corresponding power-signs, and so numeral-phrases are read from top to bottom and from right to left. There are two signs for 10. The first, 10 (+) in Table 10.6, combines additively with the units for 1 through 9 in order to write 11 through 19,

Numerical Notation

324

Table 10.6. Mende (Kikakui) numerals 1

2

3

4

5

6

7

8

9

b

c

d

e

F

g

h

i

10 (+)

10 (×)

100

1000

10,000

100,000 1,000,000

k

j

m n o P d k \\\\ba hjm \F io \\e\e\e\e\e\e e\j\m\n\o\p\ß

a

14 128 60,009

5,555,555

ß

while the other, 10 (×), is a multiplicative power-sign for 10 that combines with the unit-signs for 2 through 9, or with 10 alone by placing a dot rather than a sign for 1 above it (Tuchscherer 1996: 71–72). The higher power-signs use vertical strokes to indicate repeated multiplication by 10; the number of strokes represents the exponent of 10 corresponding to the number. This feature is quite distinct from the cumulative principle, which always refers to repeated addition of similar symbols, and is unique to the Mende system. In theory, the system could have been extended infinitely without using the positional principle, although there are practical limits to how many vertical-strokes could be read easily. Some scholars once thought that the numeral-signs for 1 through 10 derived acrophonically from the Kikakui signs for the first syllables of the numeral words for 1 through 10 (Tuchscherer 1996: 130–132). While the syllabic values and the numeral-signs correspond, Tuchscherer (1996: 140–142) has demonstrated that the Mende numeral-signs (at least those for 1 through 5) are also similar to certain signs (and variants) of the Arabic positional numeral-signs. From this, he argues that the Arabic numerals inspired the signs of the Kikakui syllabary for the first syllables of number words. While the similarities are not striking enough to prove the case conclusively, I am reasonably convinced that the Arabic positional numerals influenced the development of the Mende system. Yet the Mende numerals are multiplicative-additive, not ciphered-positional (like the Arabic positional system) or ciphered-additive (like the Arabic abjad-based system). The only other multiplicative-additive system used in West Africa is the earliest Bamum system, but it

Miscellaneous Systems

325

is a long way from Sierra Leone to Cameroon, and by the time the Mende system was developed in 1921, the Bamum had switched to ciphered-positional numerals. Moreover, the use of two different signs for 10 (one additive, one multiplicative) and the use of repeated strokes to indicate exponents are features that are not attested elsewhere. Thus, the structure of the Mende system should be regarded as largely indigenous. Curiously, the modern Mende lexical numerals are not decimal but vigesimal. While this might suggest that the base of the Mende numerical notation was borrowed from the Arabic numerals, in the nineteenth century the Mende had decimal lexical numerals (Tuchscherer 1996: 148–150). If this system survived (in even a vestigial form) into the first decades of the twentieth century, it, rather than a foreign numerical notation system, could have inspired the decimal base of the system. The Mende numerals were used for a wide variety of functions, and were taught in schools throughout the 1920s and 1930s. Some individuals used the system for accounting and record keeping, but it is not clear whether the numerals themselves were used directly for arithmetic (Tuchscherer 1996: 69). Dalby reports that the syllabary was used by some weavers and carpenters for recording measurements, which would presumably also require numerals (Dalby 1967: 21). Both the Kikakui syllabary and the numerals continue to be used for some purposes, including correspondence, record keeping, religious writings, and legal documents (Tuchscherer 2005: 478).

Sub-Saharan Decimal-Positional In addition to the African systems just described, which are structurally distinct from their ancestors, several of the indigenous scripts of sub-Saharan Africa have decimal and ciphered-positional numerical notation systems, and are thus structurally identical to their Western or Arabic ancestors. While these systems are of less interest from a structural point of view, they are noteworthy from a historical perspective. I list these systems in Table 10.7. The Bagam syllabary was invented early in the twentieth century in western Cameroon and used briefly by the Eghap (known in scholarly literature as the Bagam) of that region (Tuchscherer 1999). The only text to preserve Bagam writing and numerals is a recently discovered 1917 description of the system by a British colonial military officer, Captain L. W. G. Malcolm. The numerals probably were borrowed from the Bamum system rather than from the Western numerals, based on some graphic resemblances between the Bamum and Bagam sign sets. The Bagam numerals do not include a sign for zero, but do include a sign for 10. It is thus unclear whether it was a ciphered-positional system or how (if at all) it expressed larger numbers. In the early part of the century, the Bamum system was still multiplicative-additive, which

Numerical Notation

326

Table 10.7. Decimal systems of sub-Saharan Africa 1

2

3

4

5

6

7

8

9

Manding

q L Ç è A V

r M ü ï B W

s N é î C X

t O â ì D Y

u P ä Ä E Z

V Q à Å F ,

w R å É G .

x S ç æ H /

y T ê Æ I <

Osmaniya

+

,

-

.

/

:

;

<

=

>

Wolof

;

:

[

{

]

}

-

_ =

+

Bagam Bété Fula (Dita) Fula (Adama Ba) Kpelle

0

K ë ô

10

z U

J >

suggests that the Bagam system may also have had this structure. The Bagam script and numerals are now extinct, and recent ethnographic investigations in the region have revealed no knowledge of the numerals even among elderly Bagam (Konrad Tuchscherer, personal communication). The Bété numerals were invented in late 1957 or early 1958 by Frédéric BrulyBouabré, a native Bété from the western part of Ivory Coast, to accompany a syllabary of over 400 characters that he had invented a year earlier (Monod 1958). Bruly-Bouabré, who was fully literate in French, did not use Western models in developing his script-signs, as can be seen from the abstract nature of the numerals. However, the use of a dot for zero shows at least some influence from the Western numerals (or perhaps the Arabic numerals, although it is not clear whether BrulyBouabré knew Arabic at all). The unusual sign for 10 may have been used multiplicatively or additively in conjunction with the unit-signs. There is evidence of a quinary component to the Bété system in the fact that the signs for 6 through 10 are inverted forms of the signs for 1 through 5, with the exception of the extra dot atop the sign for 5 (Monod 1958: 437). Bruly-Bouabré’s efforts to have this system accepted among the Bété met with minimal success. I do not know whether it is still used at present. Two alphabets invented for the Fula of Mali have accompanying cipheredpositional decimal numerical notation systems. The first of these, known as Dita, was developed by Oumar Dembélé between 1958 and 1966; in keeping with his being a woodworker, his signs have a linear character (Dalby 1969: 168–173). Dembélé attended a Koranic school and spoke French, so the structure of the system was based on either Western or Arabic numerals. The second system, invented by Adama Ba, a Fula Muslim literate in French, before 1964, is identical

Miscellaneous Systems

327

in structure, but its signs are more curvilinear and perhaps show some influence from Western numerals (Dalby 1969: 173–174). Neither system was ever used except by its inventor. The Kpelle numerals were developed in the 1930s by Gbili, a paramount chief of the Kpelle in central Liberia, in conjunction with an indigenous syllabary (Stone 1990). Its numeral-signs include a sign for 10 but none for zero, so it is not clear how, if at all, higher numbers were written. Both Arabic and Western numerals were known in the region, and the Kpelle script was developed on the basis of the Vai script, which used Western numerals. Although the Kpelle signs vaguely resemble both Arabic and Western numerals, no definite historical ancestry can be assigned to them. The script was used traditionally for tax records as well as for official communication among chiefs, and was restricted to a small segment of the populace. Today, most Kpelle use Western numerals, although the indigenous system continues to be used for personal correspondence by a few individuals (Stone 1990: 141; Tuchscherer 2005: 478). A set of numerals was developed around 1950 by Souleymane Kantè, an educated trader who was literate in both French and Arabic, in conjunction with an alphabet known as N’ko (Dalby 1969: 162–165). It was designed for use among the many peoples whose dialects fall under the label “Manding,” most notably Mandinka, and was intended to provide a means of communication accessible without the need for formal schooling. The numerals are ciphered-positional and decimal, and perhaps are related graphically to the Western numerals; however, numeralphrases are written with the highest power on the right. Texts written in this script apparently included treatises on calculation, suggesting that the numerals may have been used for arithmetic (Dalby 1969: 163). N’ko continues to be used today, and probably has tens of thousands of users. Around 1920, an alphabetic non-Arabic script known as Osmaniya (also known as far soomaali and cismaanya) was developed by ‘Ismaan Yuusuf Kenadiid, brother of the sultan of Obbia, as an alternative to Arabic for writing the Somali language (Lewis 1958: 140–142). The Osmaniya decimal ciphered-positional numerals, like the script, were written from left to right. The fact that the script fully represented vowel sounds and was written from left to right shows influence from the Latin alphabet, so it is possible that the numerals were mainly of Western rather than Arabic origin, but the Osmaniya numeral-signs resemble neither Western nor Arabic positional numerals. While Osmaniya was declared an official script in Somalia starting in 1961, a Latin-derived orthography was adopted in 1972, after which Osmaniya was used far less regularly. Assane Faye developed a Wolof script around 1961 that has a set of cipheredpositional numerals (Dalby 1969: 165–168). Faye, who was literate in both French and Arabic, presumably drew more influence from the Western numerals in

328

Numerical Notation

creating this system, whose signs more closely resemble Western than Arabic numerals. Numeral-phrases were written from left to right. Curiously, Faye also assigned numerical values to nineteen of the letters of his script (1–9, 10–90, 100) in imitation of the ciphered-additive Arabic abjad system (Dalby 1969: 167–168). Neither the script nor the numerals survives today; most Wolof use either Arabic or Western numerals.

Miscellaneous West African While some African numerical notation systems (e.g., Mende, Bamum, Oberi Okaime) are structurally distinct from the Western and Arabic numerals, these systems probably would not have developed without contact with the West. Most of the prominent pre-colonial West African states, including the Yoruba and Benin civilizations, did not use numerical notation per se, although they were, like most other West African societies, quite numerate and capable of complex calculations using the cowrie currency ubiquitous to the region. At the same time, however, there is suggestive evidence that some pre-colonial West Africans occasionally used numerical notation. Unfortunately, we have only a handful of ethnographic details pertaining to the peoples of West Africa in the twentieth century concerning systems that may be considerably older. While we should not assume that these systems are of entirely indigenous origin, given extensive pre-colonial contact with Muslim traders from the north, neither should we discount the possibility. Historians of mathematics interested in African capabilities have not discussed these systems, doubtless because they were unaware of them (Zaslavsky 1973, Gerdes 1994). Because they are not attached to phonetic scripts, they have not been compared to other numerical notation systems and are often grouped inappropriately with unstructured tallying signs. I expect that a more thorough search of the ethnographic literature (especially from the early twentieth century) would reveal additional numerical notation systems. A. S. Judd (1917), reporting on the state of education in Nigeria, reported that the “Munshi” (the Tiv, speakers of a Niger-Congo language in central Nigeria) employed a numerical notation system. This system, which has “a thin line representing the units, a circle the tens, and a broad line made by the thumb representing a score,” was apparently used when drawing in sand or earth (Judd 1917: 5). Presuming that Judd’s description is accurate, this system was most likely cumulative-additive with a base of 20 and a sub-base of 10. The tradition of graphic symbolism practiced by the Dogon of Mali in rock paintings and sand drawings includes numerical signs that can be combined with one another. In one system, straight lines represent units and circles represent 5;

Miscellaneous Systems

329

a drawing of a man with four circles (each representing one of the limbs with five digits) joined with a cross means 22 (Griaule and Dieterlen 1951: 11–12; Flam 1976: 37). Another represents a period of sixty years by three rods of decreasing size, each with the value of 20 (Griaule and Dieterlen 1951: 28). There may not have been a regular system of correspondences between numbers and signs. In the context of reckoning and calculation, cowries representing 1, 5, 10, 20, 40, and 80 apparently were used (Calame-Griaule 1986: 232). The exact technique employed is unknown, however, and this may not have constituted a numerical notation system. While most systems of tally sticks use only one-to-one correspondence (thus lacking a base), Lagercrantz (1973: 572) reports that among the Ganda and Djaga of Uganda and Tanzania, tally sticks are also used in which units are marked by small notches, 10 by a larger notch and 100 by an even larger notch. It is not clear whether this system recorded cardinal numbers, or whether it is simply a series of marks equal to the number being counted, of which the tenth is large and the hundredth larger still. Another tallying system, possibly of more modern origin, was used on riverboats along the Ogowe River in Gabon in the twentieth century. When refueling steamships, a stroke on a piece of paper was written for every ten loads, and a cross for every hundred loads (Lagercrantz 1970: 52). Again, it is entirely possible that this system was not used to indicate cardinal numbers, but was simply an ordinal tally. The conceptual distinction between a system used only to mark items as they are counted, and one used to indicate whole sums after counting should not be underestimated; nevertheless, it is quite plausible that at least some of these African tallying systems did, eventually, transform into cumulativeadditive numerical notation systems. Regardless, even tallying systems that use specific abstract signs for powers of a base instead of one-to-one correspondence represent a considerable conceptual advance.

Cypriot Tallies Buxton (1920: 190) describes an otherwise undocumented numerical notation system used by nonliterate Greek speakers in Cyprus: The numbers are continually used as follows: a perpendicular stands for a unit, five is sometimes indicated by a cross and sometimes by a circle, ten either by a circle, by a theta, or by a cross inside a circle, twenty by a cross inside a circle, where that symbol has not already been utilized previously; if it has, there seems to be no alternative. Fifty is written by a loop on top of a perpendicular, and a hundred by two fifties. It will be seen that two of these symbols are not dissimilar to Arabic numerals, namely, the circle and the symbol for fifty. The Arabic symbol for five is, however, not circular, and it is possible that the two signs are connected, but the value of the looped line is in Arabic nine, not fifty. ... At Enkomi a man scores at cards in this way. He

330

Numerical Notation

chalks down units up to four, then he rubs them out and writes a circle, adds units to ten when he erases them, and draws a line through the circle, draws units up to fourteen then adds a circle; at twenty he erases the added nine and draws another line through the theta, which thus becomes a circle with a cross through it.

This evidence indicates that although numeral-phrases are constructed sequentially as tallies, rather than being written as a single sum, because intermediate values are erased and replaced with higher values, ultimately the result is a cumulativeadditive numeral-phrase with a base of 10, a sub-base of 5, and a special sign for 20. This sort of notation is qualitatively different from simple one-to-one correspondence, or tallying in which intermediate marks are not erased but simply continue on (e.g., XXVII vs. IIIIVIIIIXIIIIVIIIIXIIIIVII as two notations of 27). Other than this one brief description, however, we have no information on the origin, history, or use of the Cypriot system. While there are parallels between this system and the decimal cumulative-additive system used in ancient Cyprus (Chapter 2), there is no reason to think that this system is anything but a locally developed technique, one that is evidently idiosyncratic given the multiple numeral-signs used for 5, 10, and 20.

Indus The writing system of the Harappan civilization, centered in the Indus River valley, is one of the great remaining mysteries in the field of script decipherment. It was used from around 2500 bc to 1900 bc on several thousand very short inscriptions (averaging five signs per “text”), and was written primarily from left to right (Parpola 1996). Unfortunately, there is no reliable basis for deciphering the script, because the language it represents is unknown (though sometimes asserted to be a Dravidian language) and there are no bilingual inscriptions. The situation is even more grave than for scripts such as Linear A, where there are many easily readable numeral-phrases and associated ideograms (see Chapter 2). Many dubious interpretations of Indus numeration have been proposed (e.g., Subbarayappa 1996). We have barely enough evidence to confirm the existence of a numerical notation system in the ancient Indus Valley, much less determine its origin, history, or function. There have been several earnest attempts to decipher the Indus numerals, mostly relying on the very frequent occurrence of groupings of vertical strokes on the inscriptions. Table 10.8 shows these numerals as well as the frequency with which they are encountered in the texts (Fairservis 1992: 62).4 4

Fairservis (1992: 183) provides no count of single and double short strokes because these are also assigned grammatical functions (as genitive and locative case markers, respectively) in his decipherment.

Miscellaneous Systems

331

Table 10.8. Short and long Indus strokes and frequencies 1 Short strokes

Long strokes

2

3

4

5

6

7

8

9

10

111 111

1111 \111

1111 1111

11111 \1111

11111 11111

38

70

7

2

1

1

11

111

1111

111 \11

?

?

151

70

38

a

aa

aaa

aaaa

aaaaa

aaaaaa aaaaaaa

149

365

314

64

22

3

6

These signs probably represent low numbers in a cumulative fashion; the short strokes are grouped into sets of three, four, or five, just as the signs of most other cumulative systems. The longer ungrouped vertical strokes occur only in the early Indus inscriptions; during its mature phase, the shorter strokes were used exclusively (Parpola 1994: 82). Because these sets of strokes are paired interchangeably with non-numerical graphemes (e.g., the ‘fish’ sign + is attested in combination with three, four, six, and seven strokes), we are relatively confident that they were numeral-signs (Parpola 1994: 81). Yet Ross (1938) long ago pointed out that some groupings of vertical strokes pair noninterchangeably with other signs, which suggests that they may have had phonetic or grammatical values (Ross 1938; Fairservis 1992: 12). This is parallel to the frequent use of numeral-signs phonetically in Chinese writing, and resembles abbreviations such as “K-9” for canine in English. One enigmatic symbol consisting of three rows of four vertical strokes occurs frequently, but never in the same contexts as other putative numerals; Fairservis (1992: 71) argues that it should be read as ‘rain’, which may or may not be correct, but is far more likely than ‘12’. The Indus texts are so short and devoid of contextual information that we must be very careful not to read too much numerical information into them. This interpretive framework for the Indus numerals does little to establish whether this system had a base and used an interexponential principle to write larger numbers. Fairservis notes that there is a sharp drop-off in frequency after seven for both the long and short vertical strokes, and that in fact there are no attested instances of eight or more long strokes. From this, he concludes that the Indus numerals were probably octal or base-8 (Fairservis 1992: 61–62). Perplexingly, however, he then proceeds to assert that there are ‘pictographic’ signs for 8, 9, 10, and 11 that were simultaneously numerical and calendrical, indicating the eighth through eleventh months of the conjectural Harappan calendar, because these four signs, along with vertical strokes for 1 through 7, are found in association with a sign that he thinks represents ‘month’ (Fairservis 1992: 65). This theory has not been widely adopted by scholars of the Indus script (cf. Pettersson 1999: 103).

332

Numerical Notation

Figure 10.4. Inscription on artifact DK-7535 from Mohenjo-daro.

Our best evidence for a legitimate Indus numerical notation system comes from nine inscribed potsherds and copper and bronze tools found at Mohenjo-daro, Canhujo-daro, and Kalibangan, inscribed with sets of vertical strokes and crescents/ hooks (and sometimes other script-signs). These two signs are sometimes found in combination on the seal inscriptions, but never in large quantities and never clearly separated from the rest of the text. Pettersson (1999) adds that, in addition to vertical strokes and crescents, a distinction needs to be made between vertically and horizontally oriented strokes. One object, a chisel or axe blade (DK-7535)5 from Mohenjodaro, contains all three signs, as shown in Figure 10.4 (Parpola 1994: 108). While I am reasonably convinced that this inscription and similar ones on other Harappan tools are numerical in function, there is no agreement as to the specific structure and value of the signs. Fairservis (1992: 67–69) has constructed a convoluted argument whereby the vertical strokes (standing for units) can serve either an additive or multiplicative role in the numeral-phrase depending on whether they follow or precede the crescent sign(s). Pettersson (1999: 102–103) points out, however, that there is no case where a crescent sign is both preceded by and followed by vertical strokes. Fairservis (1992) and Pettersson (1999) argue that because none of these nine objects contains more than seven of any sign, the Indus numerals must be octal rather than decimal. Yet Parpola (1994: 82) argues that the crescents probably represent 10 rather than 8. Either of these interpretations of the system would mean that the Indus numerical notation system was cumulative-additive. At present, there is insufficient evidence to decide whether the crescent-sign had a value of ‘8’ or ‘10’. Nine numeral-phrases is a very limited corpus from which to conclude that, since no sign is repeated more than seven times, the numerical base

5

There is some confusion over the identification of this object, which is assigned different artifact numbers by Parpola (1994) and Pettersson (1999).

Miscellaneous Systems

333

must be 8. The limited linguistic reconstructions regarding Proto-Dravidian (even granting the controversial hypothesis that the Harappan language was a Dravidian tongue) are ambiguous and tenuous, but on balance best support a decimal interpretation, since Proto-Dravidian lexical numerals for ‘ten’ and ‘hundred’ have been reconstructed (Parpola 1994: 169). The linear measures of the Harappans appear to have been decimal (Sarton 1936a), and the system of weights is partly decimal and partly binary (Parpola 1994: 169; Pettersson 1999: 106). None of the Harappan weights bear any inscription, numerical or otherwise (Pettersson 1999: 91). Pettersson’s (1999) attempt to correlate the numerical signs on the metal tools with their weights showed only that no metrological interpretation of their meaning (either decimal or octal) was likely to be correct. The Indus numeral-signs are entirely unlike the Sumerian numerals (Chapter 7) used in Mesopotamia at the time of the invention of the Indus script. It is interesting that the Egyptian hieroglyphic numeral-signs for 1 and 10 are | and Ô, respectively, but, even if the Indus crescent-sign represented 10, this similarity could very easily have arisen by chance. While there are vague similarities in the metrological systems of Egypt and the Indus Valley, there was minimal cultural contact between the two regions (Petruso 1981). It is probably best to assume that the Indus numerals were independently invented. The fact that the Indus script is completely undeciphered, coupled with the limited number of surviving numeral-phrases, makes it nearly impossible to identify the function(s) for which they were used. There is no evidence of the numerals’ use for accounting or administration, which is abundant for other undeciphered scripts, such as Linear A and Proto-Elamite. The wide variety of materials on which numerals are found (clay seals, potsherds, metal tools) suggests that they were used widely among literate Harappans, but even this hypothesis requires caution. The Harappan civilization declined precipitously after 1900 bc, although it may have survived in certain regions for a century or two longer. There is no evidence that the Indus numerals had any influence on the Brāhmī numerals (Chapter 6), which arose almost 1,500 years later.

Naxi Naxi (also known as Nakhi and Moso) is a Tibeto-Burman language spoken by approximately 250,000 people in the northwestern part of Yunnan province in southwestern China. Naxi is written in three indigenous scripts: dongba (or tomba), a pictographic notation system, and two syllabaries (geba and malimasa), in addition to a more recent Latin-based orthography. The dongba script is highly idiosyncratic, consisting of 1,500–2,000 largely pictographic signs with some phonetic components, although there is not a regular correspondence of signs with

334

Numerical Notation

either words or phonemes. It is used primarily as a mnemonic aid or “promptbook” to assist priests in reciting memorized texts (Bockman 1989: 155). It was reputedly invented in the twelfth century ad (Coulmas 1996: 353). The earliest datable dongba texts, however, are from the middle of the eighteenth century (Bockman 1989: 153). The dongba script is actively used by some Naxi priests, and has even been part of modern literacy programs. Nevertheless, because dongba texts are pictographic and rely on oral and mental knowledge to draw meaning from them, their interpretation by Western scholars is incomplete and poor. In at least some dongba texts, numerical notation was used alongside the script. In the Nichols manuscript first made available to Western scholarship by F. H. Nichols in 1904, three repeated “South Asian” swastika-like signs precede six vertical strokes. These are interpreted by Rock (1937: 236) as representing 100 and 10, respectively, producing a sum of 360, indicating the 360 yu-ma deities of the Naxi. Groups of three or more signs are clustered in rows of three signs, where appropriate. In other dongba manuscripts, a simple cross rather than a swastika represents 100. This is a cumulative-additive numerical notation system with a base of 10, leaving open the question of how the number 1 was represented. Bockman (1989: 1952) suggests that the dongba signs X, +, and ᅹ were numerical, but does not assign them specific values. In other dongba manuscripts, however, vertical strokes or hooked vertical strokes mean 1, and X or + means 10; in one very clear instance, Hs. Or. Sim. 279 / R. 1912, Blatt 9r – 10r, eighteen consecutive panels depict gods, each enumerated using this system (Janert and Janert 1993: 2753). Like many cumulative-additive systems, the units 6 through 9 are represented in two or more rows of three to five strokes (3 + 3, 4 + 3, 4 + 4, 3 + 3 + 3); 5 is depicted with a single row of five strokes, but 15 is depicted with X (10) followed by 5 indicated in two lines of three and two strokes respectively. The variability in signs suggests that base-structured numerical notation was used only irregularly or idiosyncratically in the dongba texts, with local or even individual scribal tradition determining which signs represented which numbers. Against this position, however, a wide variety of dongba texts contain numeralsigns, and all appear to be cumulative-additive and decimal (i.e., signs are repeated, but no sign is repeated more than nine times). The origin of this numerical notation system, and its relation to any other system, remain obscure. It is possible that it was independently invented, as no other cumulative-additive systems were ever used alongside scripts of either East Asian or South Asian origin.

Varang Kshiti In the twentieth century, several scripts were developed for the various Munda languages of central and eastern India, of which Sorang Sompeng, Ol Cemet’,

Miscellaneous Systems

335

Table 10.9. Varang Kshiti numerals 1

2

3

4

5

6

7

8

9

d

e

f

g

h

i

j

k

l

10

20

30

40

50

60

70

80

90

o

p

q

r

s

t

u

m

n 61 = id

and Varang Kshiti are the primary ones to survive to the present day (Zide 1996). While these scripts have numerical notation systems, most are ciphered-positional and are derived from the Western numerals or the ciphered-positional systems of India (Chapter 6). I know of only one script, the Varang Kshiti script designed for the Ho of Bihar province, where a structurally distinct numerical notation system was developed for a Munda language. These numerals are shown in Table 10.9 (Pinnow 1972: 828). The system is ciphered-additive, as it has signs for 1 through 9 and 10 through 90. The signs for 10 through 30 do not resemble the signs for the corresponding units, but the higher decades do. Curiously, however, Pinnow (1972: 830) reports that only the unit-signs, combined in a ciphered-positional manner, were employed when writing numbers from 11 to 19, 21 to 29, 31 to 39, and so on. The separate signs for the decades from 10 to 90 may have obviated the need for a zero-sign. There may also have been signs for 100 and 1000, which presumably would combine multiplicatively with the unit-signs, but this cannot be confirmed (Pinnow 1972: 831). The Varang Kshiti script and numerical notation system were developed by a Ho shaman named Lako Bodra throughout the 1950s and 1960s. While various claims have been made concerning the antiquity of the script (such as that it was first developed in the thirteenth century and rediscovered by Lako Bodra in a vision), it is likely that it is a recent invention (Zide 1996: 616–617). The Varang Kshiti numeral-signs resemble those of various South Asian systems, but none of these resemblances proves a specific origin. Pinnow (1972) believes at least some of the script-signs to have been borrowed from ancient Brāhmī characters. Since both Varang Kshiti and Brāhmī numerical notation systems are ciphered-additive, I do not discount this possibility entirely, but there is no evidence that the Varang Kshiti system is of sufficient antiquity to have been influenced by Brāhmī. The Varang Kshiti script and numerals are still used in both primary and adult education, and efforts to make it the vehicle for strengthening Ho culture have had success. I strongly suspect that in most circumstances, Western, Devanagari, or Oriya numerals are used in place of the system just described.

336

Numerical Notation

Pahawh Hmong The Pahawh Hmong script was developed for speakers of the Hmong language of northern Laos. Its inventor, a Hmong peasant named Shong Lue Yang, though apparently illiterate when he developed this script, revised it constantly from 1959 until his assassination in 1971 and used it as a tool to promote Hmong cultural identity (Ratliff 1996). In addition to phonetic script-signs, Shong Lue Yang and his disciples developed a numerical notation system. The earliest (Source Version) of the Pahawh Hmong numerals are shown in Table 10.10 (Smalley 1990: 79). This system is primarily multiplicative-additive and decimal; unit-signs from 1 through 9 combine with power signs for 10, 100, and 1000. Numeral-phrases, like the script itself, are written from left to right. The only irregularity in the system is that 10 and 20 are not expressed through juxtaposition of the unit-signs 1 and 2 with the power-sign for 10, but with distinct signs, which also are combined additively with the unit-signs to write 11 through 19 and 21 through 29. The other sign for 10, shown as 10(×) in Table 10.10, is used to indicate multiples of 10 from 30 to 90 by placing it after the unit-signs 3 through 9. The use of two signs for 10 (one additive and one multiplicative) is analogous to the multiplicative-additive Mende system described earlier. There are no Pahawh Hmong numeral-signs for 10,000 or higher powers; these numbers were written multiplicatively by placing an entire numeral-phrase in front of the sign for 1000. The Source Version Pahawh Hmong numerals originated around 1959, but since Shong Lue Yang was apparently illiterate at the time, we cannot say whether some other numerical notation system had any influence on its invention. The standard Chinese numerals are multiplicative-additive, so they may have influenced the development of the Pahawh Hmong system. This is supported by the use of a special sign for 20 (ㆎ) in Chinese, though Pahawh Hmong, unlike the Chinese system, does not have distinct signs for 30 and 40. The Pahawh Hmong numeral-signs are entirely different from the Chinese ones, so we should not presume any influence from China. Within about ten years, Shong Lue Yang and his followers developed a new system based in part on the old numeral-signs. By this period, the script used was that known as the Second Stage Reduced Version, whose signs are shown in Table 10.11 (Smalley 1990: 80–81). This is a ciphered-positional, decimal system. Some of the numeral-signs from the Source Version are similar or identical to the newer ones (1, 3, 9), but many others are changed entirely. The addition of a sign for zero and the abandonment of the power-signs change the system’s structure radically. This transformation was likely a result of a growing awareness of Lao and/or Western numerals by Shong Lue Yang, although the numeral-signs (excepting the zero) are unlike those of any neighboring system. Despite the adoption of ciphered-positional numerals in the Second Stage Reduced Version,

Miscellaneous Systems

337

Table 10.10. Pahawh Hmong (Source Version) numerals 1

2

3

4

5

6

7

8

9

a

b

c

d

e

f

g

h

i

10(+)

20

10(×)

100

1000

j

k

l

m

n

36 =

clf

16 =

jf

150,000 =

ameln

multiplicative-additive notation was not abandoned but was in fact expanded by Shong Lue Yang. A new Pahawh Hmong multiplicative-additive system was used alongside the ciphered-positional system, which combined the unit-signs for 1 through 9 from the Second Stage Reduced Version with a new set of powersigns. Rather than creating a separate power-sign for each power of 10, Shong Lue Yang hit on the idea of using distinct signs only for the powers of 100 (100, 10,000, 1,000,000, etc.), using the power-sign for 10 multiplicatively with these signs to write the intermediate powers (1000, 100,000, etc.). This cut in half the number of new power-signs that needed to be invented. Because of this additional structural element, this form of Pahawh Hmong numeration, while still multiplicative-additive, is centesimal with a decimal sub-base, since powers of 100 (not just powers of 10) structure the system. As Smalley (1990: 81–82) points out, there is a considerable advantage in conciseness when writing large round numbers in this system as compared to using the ciphered-positional one. Both the ciphered-positional and the revised multiplicative-additive Pahawh Hmong systems continue to be used, although only the ciphered-positional system is used for arithmetical calculation. Because large numbers of Hmong have Table 10.11. Pahawh Hmong (Second Stage Reduced Version) numerals 1

2

3

4

5

6

7

8

9

0

p

q

r

s

t

u

v

w

x

o

10

100

1000

10,000

100,000 1,000,000 10,000,000 100,000,000

<

>

><

?

?<

+

+<

_

1,000,000,000 10,000,000,000 100,000,000,000

1,000,000,000,000

_<

}

{

{<

338

Numerical Notation

immigrated to the West (especially Australia), Hmong numerals are used not only in Laos and Hmong-speaking parts of Vietnam, but also in Western countries. At present, both the Lao and Western ciphered-positional numerals challenge the Hmong numerals, so it is not clear how long they will continue to be used. Regardless of the eventual success of any of these systems, it is noteworthy that three variants of the Pahawh Hmong system were developed with such rapidity, each with a different structure.

Ryukyu While the meagerly populated Ryukyu Islands seem an unlikely locus for numerical creativity, three different numerical notation systems have their origin in this tiny Pacific archipelago south of Japan. The first of these is a set of numeral-signs from 1 to 10, which are no more than slight paleographic variants of the Chinese numerals (Chapter 8) (Pihan 1860: 18–19).6 While we do not know how numbers higher than 10 were formed using this system, it was probably a multiplicativeadditive system. The second system was a form of knot-notation known as ketsujo, by which amounts of money were counted using series of knotted ropes that were strung perpendicular to a long cord, in a way that is analogous to the Peruvian khipu (Ifrah 1998: 543). This system was likely a cumulative-positional numerical notation system with a base of 10 and a sub-base of 5. Unfortunately, too little evidence is available to analyze the ketsujo system in detail. The third system was written on long wooden sticks (30 to 75 cm in length, and 2.5 to 4 cm in breadth), which were known in Okinawan as sho-chu-ma (Chamberlain 1898). It comprises several variants, each of which enumerated a particular commodity: money, bundles of firewood, bags of rice, and other goods. While these sticks have been described as “tallies,” the marks do not count objects in sequence (one mark for one object), but constitute a full-fledged numerical notation system used for recording amounts of goods. Tables 10.12 and 10.13 show two of the more common systems used in the late nineteenth century, the first for expressing quantities of money (in units of kwang and mung) and the second for counting bundles of firewood (Chamberlain 1898: 385, 388).7 The signs reflect those attested on the sho-chu-ma examined by Chamberlain, while the forms of most of the nonattested signs can easily be inferred on structural 6

7

While the Ryukyu Islands have been under Japanese control since the seventeenth century, the cultural influences in the archipelago have been at least as much Chinese as Japanese, given its location in the East China Sea. I have corrected a couple of errors in Chamberlain’s tables where numeral-signs were assigned incorrect values.

Miscellaneous Systems

339

Table 10.12. Ryukyu numerals (money) 1

2

3

10 mung 100 mung

A

1 kwang

L M C T E\G y : ;

10 kwang 100 kwang 1000 kwang 10,000 kwang

AA

4

5

Y AAA AAAA K N O B U V Z I\z ] |

6

7

8

9

Ç P [

ü Q î

é R ì

â S

- = 352 kwang, 250 mung grounds. Both systems are cumulative-additive and decimal, with a sub-base of 5. The multiples of each power from 1 to 4 are mainly cumulative (exceptions include the “100 kwang” money count and the hundreds value in the firewood count), and the multiples from 5 through 9 combine the appropriate sign for 5 with the required number of additional units. The numeral-signs are largely abstract. In some cases, the sign for five of a power is derived by halving the sign for one of the next higher power, (e.g., 5000 kwang vs. 10,000 kwang in Table 10.12, or 500 vs. 1000 in Table 10.13). Numeral-phrases were written in a roughly vertical fashion. Figure 10.5 depicts one of the sho-chu-ma studied by Chamberlain. We do not know when or how the sho-chu-ma were first used. Chamberlain states, “The custom may be traced to a hearsay knowledge of the Chinese written character among the Luchuan [Ryukyu] peasantry, who, not possessing sufficient learning to employ this character itself, and not being encouraged by their rulers to acquire the elements of an education deemed unsuitable to their lowly station, developed a make-shift of their own” (1898: 383). The sign for 10 bundles / 10 kwang, +, was probably borrowed from the identical Chinese sign for 10 (Chamberlain 1898: 384). Table 10.13. Ryukyu numerals (firewood bundles) 1

2

3

4

5

1

A

AA

AAA

AAAA

10

C E G

ê H

ë I

è J

B D F

100 1000

6

7

8

9

ï ä

= î à

ì å

Ä ç

340

Numerical Notation

Figure 10.5. Ryukyuan sho-chu-ma in which each register depicts units in the moneytallying system. Source: Chamberlain 1898: Pl. XXIII.

If Chamberlain is correct, the Ryukyu system was produced by stimulus diffusion from the Chinese classical numerals. Since the Japanese also used the Chinese numerals, the Okinawans may have learned the system from Japan rather than from China. Moreover, the “1 kwang” signs resemble the rod-numerals somewhat, and the rod-numerals are cumulative (though positional) and quinary-decimal, so the rod-numerals may have been ancestral to the Ryukyu tallies. Yet the rod-numerals had fallen out of use for most purposes by the seventeenth century, and the idea of using lines for units is nearly panhuman. More evidence is needed before such evidence could be considered conclusive. By the time these numerals were reported in the Western scholarly literature at the end of the nineteenth century, the Ryukyu numerals had already ceased to be used, having become a historical curiosity, or even an object of embarrassment, for the Ryukyuans (Chamberlain 1898: 383). Its sub-base of 5 and the use of halvings of the main power signs for the fives render the Ryukyu system structurally identical to the Etruscan and Republican Roman numerals (Chapter 4), which also have their origin in tally-style marks and make use of the principle of halving.8 We may never be able to learn more about the development of a remarkable and previously unacknowledged parallel invention.

Samoyed The Samoyedic peoples include speakers of several different but related languages of Siberia who traditionally relied on animal herding, hunting, and small-scale farming for subsistence. The ethnonym “Samoyed” is widely used 8

It is highly improbable that the Roman numerals are an ancestor of the Ryukyu system.

Miscellaneous Systems

341

Figure 10.6. Samoyed tallying stick showing different means of notation. The first two instances are primarily sequential/ordinal, while the third is a cumulative-additive notation for 333. Source: Jackson and Montefiore 1895: 403.

but underspecified. While Samoyedic peoples traditionally were nonliterate, there is evidence for an indigenous numerical notation system of notched sticks among at least some Samoyedic peoples. Figure 10.6 depicts a selection of tallying sticks found by the British explorer Frederick George Jackson (1860–1938) during his 1893 expedition by sledge (Jackson and Montefiore 1895: 403). These tallies reflect a diversity of numerical practices. The first two do not appear to be base-structured numerical notation, but rather to represent counting through one-to-one correspondence using notches across the width of a stick; the V-like signs may represent 5, 10, or some other value. The third stick, however, apparently indicates ones, tens, and hundreds with I, X, and ᅷ,if the annotations are correct. The fourth artifact may in fact be non-numerical (e.g., an ownership mark). The use of I and X for 1 and 10 suggests a possible link with Roman numerals, but the sign for 100 does not, and is instead identical to the “Tuscan tallies” used in nineteenth-century Chioggia (Chapter 4). This system of signs might thus be in some sense a pan-Eurasian cultural phenomenon (Ifrah 1998) or a borrowing from Russian users of Roman numerals; alternately, it may simply be that when using wood or bone as a tallying medium, signs of one, two, and three straight notches are an evident way to indicate successive decimal powers.

342

Numerical Notation

Easter Island The rongorongo script is the best known among a set of as-yet-enigmatic representational systems developed and used by the Polynesian inhabitants of Rapa Nui, or Easter Island. Decipherments of the Easter Island scripts are at best incomplete, and rival those given for the Indus and Minoan Linear A scripts in their use of conjecture.9 It is possible that rongorongo signs for various marine mammals symbolized numbers from 1 through 9, but if so, this tells us more about Rapanui numerology than about numerical notation (Barthel 1962, Schuhmacher 1974). Elsewhere, Barthel (1971: 1175) explicitly denies that there are rongorongo numeral-signs. The only plausible theory that has been raised concerning Easter Island numerical notation is that presented by Bianco (1990). Bianco’s proposed numeral-signs are shown in Table 10.14 (Bianco 1990: 41). The signs shown represent only a fraction of the variation that Bianco believes may have existed in the numerals; for instance, he lists twelve different possible signs for ‘1’. This putative system has a cumulative component in the use of multiple signs (often circles with vertical lines through them, as shown, but also sometimes diamonds, and also sometimes without lines through the signs). It is decimal in that it has a sign for 10, and at least minimally quinary in the use of the hand (with a circle and line) for 5. According to Bianco’s (1990: 46) interpretation, one combined sign represents (3 × 10) + 5, or 35. If so, this system would be multiplicative-additive and decimal, with an outlying special sign for 5. If this truly comprised a numerical notation system, we would expect to find frequent combinations of two or more numeral-signs, and of various numeralsigns with the same non-numerical signs (such as logograms for objects being counted). Yet such combinations are rare or nonexistent. Moreover, the claim that a grapheme that consists of a group of identical signs (in this case, circles) represents a cumulative numeral-sign is quite dubious, because there are no cumulative signs for 7 or 8 and because some such signs are not simple concatenations of identical elements, but, as in the alternate sign for 6, are constructed using various joining lines. The use of the hand as a sign for 5 makes sense only if it combines with the unit-signs for 1 through 4 to represent 6 through 9, which it never does. The relative frequencies of these signs in the tablets do not suggest that they are numerical. There are 150 examples of the twelve different signs for 1, 70 examples of 2, and 430 of 3, but none for 7 or 8, and only three for 9. Finally, Macri’s (1996) analysis of the rongorongo script assigns grammatical functions to many of these signs, which if correct makes it unlikely that they also served as numerals. 9

See Fischer (1997) for a remarkably complete summary of dozens of decipherment attempts from the 1860s to the present.

Miscellaneous Systems

343

Table 10.14. Putative Easter Island numerals 1

2

3

4

5

6

!

@

#

$

%

#\#

^ )

6 (?)

7

8

9

10

#\#\# &

35 = 3 × 10 + 5 (?)

In combination, these difficulties lead to the conclusion the rongorongo script had no numerical notation. Unlike other undeciphered scripts, such as Linear A, for which the numeral-signs are obvious and frequent, rongorongo texts lack any of the markers that would help identify such signs. Despite Bianco’s (1990: 39) statement that “[i]l est normal de trouver un système représentatif des nombres dans une écriture ancienne, les tablettes pascuanes ne pouvaient échapper à cette règle générale,” there exists no iron law that every script must have its own numerals. It is quite possible that the texts were not used for functions in which numerical notation was necessary or useful. Alternately, it may have been a ciphered system, in which case we would be unable to identify numeral-signs unless the script were deciphered more fully. A final possibility is that so few rongorongo texts survive (about two dozen) that any former system is now lost.

Cherokee One of the most famous instances of stimulus diffusion, in the form of the indigenous invention of a script by a nonliterate person on the basis of hearsay knowledge, was the creation of a syllabary for the Cherokee (Tsalagi) language around 1820 by Sequoyah. It is less commonly known that several years after inventing his syllabary, probably around 1830, Sequoyah also developed a decimal numerical notation system. This system is preserved only in two manuscripts now held at the Gilcrease Museum in Tulsa, Oklahoma, written by Sequoyah for John Howard Payne, the American dramatist and poet, who spent much time among the Cherokee and particularly with Sequoyah in the 1830s (Walker and Sarbaugh 1993: 77). The numerals of this system are shown in Table 10.15 (Holmes and Smith 1977: Appendixes II and III). The system is ciphered-additive for numbers from 1 to 99. There are distinct signs for 1 through 19 and each decade from 20 through 90. The signs for 1 through 20 are grouped graphically into sets of 5 (1–5, 6–10, 11–15, 16–20), but there is no structural relation among the signs in each subgrouping. This vigesimal element is very curious, since the Cherokee lexical numerals are purely decimal. Presumably, the signs for the tens between 20 and 90 combine additively with the unit-signs for 1 through 9, while the signs for 10 through 19 are

Numerical Notation

344

Table 10.15. Cherokee numerals 1

2

3

4

5

6

7

8

9

10

A

B

C

D

E

F

G

H

I

J

11

12

13

14

15

16

17

18

19

20

K

L

M

N

O

P

Q

R

S

T

30

40

50

60

70

80

90

100

×10

U

V

W

X

Y

Z

[

]

{

used only on their own. The documentary evidence neither confirms nor refutes this supposition, but if the signs for 10 through 19 were combined with the signs for the tens, the signs for 30, 50, 70, and 90 would have been redundant. For writing numbers above 100, the system is not ciphered-additive but multiplicative-additive.10 The sign indicated as “x 10” always combines multiplicatively with the sign for 100, and multiplies the value of the phrase by ten. Perhaps I am overinterpreting the first element of this sign, but it strikes me as being similar to the cursive English word ‘times’. While Sequoyah had hoped that his numerical notation system would be adopted, just as the syllabary had, when he laid it before the Cherokee tribal council, they voted against it and in favor of the Western numerals (Holmes and Smith 1977: 293). As a result, we know of the Cherokee system only from the two surviving Payne documents, only one of which transliterates the numeralsigns into Western numerals. One of these is dated 1839 (in Western numerals) by Payne, suggesting that Sequoyah may still at that time have been attempting to resuscitate his system’s fortunes (Holmes and Smith 1977: Appendix III). After Sequoyah’s death in 1843, the Cherokee numerals ceased to be used for the most part. However, some modern Cherokee are certainly aware of them; Figure 10.7 depicts a modern clock, purchased online, in which the Cherokee numerals for 1 through 12 indicate the hours.

Iñupiaq The newest numerical notation system, at the time of writing, was devised in 1995 by a group of Inupiat youth in Kaktovik, Alaska (located on Alaska’s Arctic coast about 100 km from the Alaska-Yukon border) as part of a middle school classroom 10

Strikingly, the Cherokee system is structurally identical to the Jurchin system (Chapter 8), including the use of distinct signs for 10 through 19, and is very similar to the Ethiopic (Chapter 5) and Sinhalese (Chapter 6) systems, both of which are ciphered-additive below 100 and multiplicative-additive above.

Miscellaneous Systems

345

Figure 10.7. Cherokee numeral clock. Author’s photo.

project, and has been adopted more widely among the Inupiat.11 The numeralsigns of this system are shown in Table 10.16. The system is cumulative-positional with a base of 20 and a sub-base of 5. The numeral-signs are written using slightly diagonal vertical strokes with a value of 1, above which slightly diagonal horizontal strokes are placed, each with a value of 5. When the numerals are handwritten, the vertical and horizontal strokes are of the same width, but sometimes in print the horizontal strokes are shown somewhat thicker than the vertical strokes. The zero-sign is reported to be symbolic of a human figure’s arms crossed over the chest, but is also similar to the Western zero-sign. Fortunately, we have enormous detail regarding the context of the system’s invention. The students, having completed work on binary notation, realized that the lexical numerals of the Iñupiaq language were base-20, and took it upon themselves to develop a vigesimal numerical notation system that would better

11

My information on this system is based entirely on very fruitful discussions with W. Clark Bartley, the non-Inupiat instructor of the mathematics class in which the system was developed.

Numerical Notation

346

Table 10.16. Iñupiaq numerals 1

Ç

ü

2

é

3

â

4

5

6

7

8

9

10

ä

à

å

ç

ê

11

12

13

14

15

16

17

18

19

0

è

ï

î

ì

Ä

Å

É

æ

Æ

ô

ë

correspond to their lexical numerals.12 At first, an attempt was made to develop ciphered signs for 10 through 19, but this was found to be taxing on the memory of users. The students instead developed a cumulative-positional system that requires only two different strokes (vertical for ones, horizontal for fives) and a zero. At the time, neither they nor their teacher were familiar with other cumulative-positional systems such as the Chinese rod-numerals (Chapter 8) or the quasi-positional Mesoamerican bar-and-dot numerals (Chapter 9). The students knew the Western numerals, and had had a brief introduction to Chisanbop finger computation, a quinary-decimal calculating technology that also inspired them. The Iñupiaq system is unusual in that its invention was specifically in the context of mathematical education; it was always meant to aid students in working with arithmetic. Although the choice of cumulative-positional notation with a sub-base was stimulated by the difficulty entailed in memorizing the twenty separate symbols that a ciphered system would have required, it had the added effect of facilitating arithmetic using physical counters. Techniques were quickly developed to manipulate numbers using popsicle sticks to represent the vertical and horizontal strokes of the written numerals, thus producing a computational device whose results could easily be written on paper thereafter. In some cases, the students found it more convenient to use this device in a purely base-5 fashion (i.e., with up to four horizontal sticks for 5 instead of only three, as in the numerical notation system). While the system is understood by a number of youth of northern Alaska, as well as by some educators, its eventual success is still very uncertain, as Western numerals are strongly preferred by many educators. Although Inupiat children trained in this system have had considerable success in their mathematics education, the very small number of users of this system limits its present value as a communication tool. It is too early to say whether the official adoption of this 12

While the Iñupiaq lexical numerals are vigesimal with a sub-base of 5, they deviate from the numerical notation system described here in the use of subtractive formations for 9 (10 − 1), 14 (15 − 1), and 19 (20 − 1), as well as in the use of a word for 6 that is not derived from that for 5.

Miscellaneous Systems

347

Figure 10.8. Zuni irrigation stick; the right side of the stick is a sequential-ordinal tally in which every fifth notch is diagonal and every tenth notch an X; the sum (24) is indicated in cumulative-additive fashion at left. Source: Cushing 1892: 298.

system by the Commission on Inupiat History, Language, and Culture will help its chances of survival.

Miscellaneous North American There is no evidence for numerical notation in the New World north of Mexico prior to the European conquest. However, just as in West Africa, a number of early reports regarding indigenous peoples of North America suggest that numerical notation systems were employed in certain circumstances, possibly deriving from earlier tallying systems. In all of these cases, it is possible that Roman numerals (or related tallying systems) used by European colonizers inspired North American native groups to develop these systems. By the 1890s, Zuñi farmers in the American Southwest used a cumulative-additive numerical notation system with a base of 10 and a sub-base of 5 (Cushing 1892). Figure 10.8 depicts what Cushing (1892: 300) calls an “irrigation tally stick.” On the right side of this object, reading from right to left, there are twenty-four marks, of which the fifth and fifteenth are marked with a slanted stroke, and the tenth and twentieth are marked with an X.13 On the left side, reading from left to right, there are two X marks, a vertical stroke, and a slanted line, which is amenable to the interpretation of 24 if a subtractive component to the system is assumed, as Cushing does (1892: 298). While the right side is a simple tally (it is grouped, but does not reduce multiple signs to a single one), the left side is a cumulativeadditive numerical notation system in which I represents 1, \ represents 5, and X represents 10. In addition, Cushing reports the use of a system of knot-numerals 13

To be precise, the two slanted notches differ slightly, as do the two X marks, but it is not clear from Cushing’s drawing exactly what the distinctions are.

348

Numerical Notation

(Cushing 1892: 300–302). Like the tally-stick system, it is cumulative-additive with a base of 10, a sub-base of 5, and uses subtractive notation for both 4 and 9. It uses a single knot for the units 1 through 3, a more complex knot – known as a “thumb-knot” – for 5, and an even more complex “double thumb-knot” for 10. These were combined in a cumulative-additive fashion, with 4 and 9 denoted by placing a single knot in front of a thumb-knot or double thumb-knot, respectively. While Cushing calls these knots “quippos” and finds them to be parallel to the cumulativepositional Inka numerals, the Zuñi system is additive, and has the additional features of a quinary sub-base and a subtractive component. I do not know how extensively the knot and tally numerals were used among the Zuñi, or whether they were used for other functions. The irrigation stick is strikingly similar to tally sticks used by Europeans, and both the tally and knotnumerals are essentially identical to Roman numerals (both are cumulativeadditive, have a base of 10 with a sub-base of 5, and use subtraction for 4). The early missionaries in the Southwest who worked among the Zuñi would have used Roman numerals. If this is an independent invention, it is a striking parallel; given the extent of the similarity, it is likely that this means of representation was borrowed from European sources. Among the Chickasaw living in Western Tennessee in the 1760s, a system of notation known as yakâ-ne talápha, or “scoring on the ground,” was used to undertake mercantile calculations, as reported by James Adair, who lived among them (Adkins 1956: 33). A single mark for a unit and a cross for 10 were marked on the ground and then added. Similarly, among the Passamaquoddy of Maine in the late nineteenth century, nonliterate shopkeepers used a ideographic system for keeping accounts in which I = 1, X = 10, and, in financial records, Q = $1 and X = 10¢ (Adkins 1956: 35–36). The alphabetic nature of these signs strongly suggests that this system was developed after contact with Euro-Americans, although I know of no similar Euro-American system that could be a potential ancestor.

Siyaq A very unusual set of numerical notation systems was employed by Arabic, Persian, Islamic Indian, and Ottoman administrators between the tenth and nineteenth centuries for representing numbers in financial transactions. While they are known by many names (dewani by the Arabs, siyaq by the Persians and Turks, and rokoum in India) and exhibit enormous paleographic variability, they all share a common origin and structure. Recognizing that it is slightly inappropriate to refer to all variants of the numerals as “siyaq,” I will nevertheless group them all here under this single term. The Persian siyaq numeral-signs are shown in Table 10.17 (Kazem-zadeh 1915: Plates I–III).

Miscellaneous Systems

349

Table 10.17. Siyaq numerals

1s 10s 100s 1000s 10000s

1

2

3

4

5

6

7

8

9

a j t C L

b k u D M

c l v E N

d m w F O

e n x G P

f o y H Q

g p z I R

h q A J S

i s B K T

The siyaq numerals have nine distinct signs for each power of 10, and thus the system is basically ciphered-additive and decimal. Numeral-phrases are written from right to left, although in numeral-phrases containing both units and tens, the unit-sign is found to the right of the tens-sign (i.e., before it rather than after it). Often, individual signs are ligatured together, making it difficult in some cases to distinguish the individual components of a numeral-phrase. There are similarities among the signs for different multiples of the same power. For instance, the signs for the tens all have a short diagonal stroke connected to a long hooked horizontal stroke followed by some additional component, while the signs for the hundreds all have a small curved stroke at the left followed by a separate additional component. Moreover, there are similarities among the signs for the same multiples of different powers. Thus, 9, 90, 9000, and 90,000 all have a common graphic element roughly resembling an obliquely slanted Western numeral 3. These similarities suggest that the system is multiplicative-additive and that each sign is composed of a unit-sign on the right and a power-sign on the left. Yet this classification would be overly simplistic, since there are many imperfections in the numeral-signs that defy a simple multiplicative explanation; for instance, the numeral 900 does not conform to the pattern just established. The solution to this taxonomic conundrum is that the siyaq numeral-signs were not originally abstract signs. Rather, the signs in Table 10.17 are extremely reduced cursive versions of the corresponding Arabic lexical numerals (Kazemzadeh 1915). Because the Arabic lexical numerals are multiplicative-additive (like those of English and most other languages), when they were cursively reduced into abstract and nonphonetic siyaq signs, they retained a visual vestige of their original multiplicative nature. Thus, the Arabic word for ‘thousand’, alf, which is written phonetically as , is reduced but still visible in the siyaq numeral 1000. This unusual origin also explains the odd structural features of the siyaq numerals, such as the placing of the units before the tens in numeral-phrases. Yet siyaq numeral-phrases could not be read phonetically; they are all too reduced to be understandable except to those trained in the system’s use. Especially in

350

Numerical Notation

non-Arabic-speaking areas, the association between numeral-words and numeralsigns was limited. Thus, the siyaq system is numerical notation, not a set of lexical numerals. Bagheri’s (1998) derivation of the siyaq numerals from the Pahlavi numerals used in early medieval Persia (Chapter 3), fails to account for the siyaq system’s irregular structure. The earliest document containing siyaq numerals (in fact, the dewani variant used by the Arabs) is a list of expenses and receipts presented to the Abbasid caliph Al-Moktadir Billâh by his minister, Ali ibn ‘Isa, dating to 306 a.h. (919 ad) (Kazem-zadeh 1915: 14; Bagheri 1998). The numeral-signs from this text are already quite impossible to read as lexical numerals, so these signs may have been used even earlier. The numbers expressed in this text and later ones normally represent monetary amounts, but in some cases expressed weights or discrete quantities of objects (Kazem-zadeh 1915: 31–32). The term siyaqat ‘style, arrangement, method’ was first applied to this system in the Kitab al-Fihrist (Catalogue Book) of Ibn alNadim, dating to 377 ah / 977–978 ad (Bagheri 1998). The siyaq numerals were likely chosen over the Arabic positional numerals or some other system in order to control who could read a particular document. They also prevented financial corruption by making forgery, falsification, and alteration more difficult (Bagheri 1998). The siyaq numerals were used for many centuries in all of the major successor states to the Abbasids. The system was especially popular in the Ottoman Empire from 1300 onward, and in Persia from the time of the Safavid Dynasty (1501–1736) onward. The Ottomans apparently stopped using siyaq numerals in the nineteenth century. However, it was still taught in Persian elementary schools until the 1930s (Bagheri 1998: 297).

Cistercian For most purposes, medieval European scribes used Roman numerals (in Western Europe) or Greek alphabetic numerals (in Eastern Europe), with the use of Western numerals becoming increasingly frequent from the eleventh century onward. Yet, beginning in the early thirteenth century, an unusual system began to be used in a limited number of manuscripts and marked on objects, primarily in contexts associated with the scribal tradition of the Cistercian monks. I therefore call this system “Cistercian numerals,” even though neither its earliest nor its latest users were Cistercians. While it has been ignored in synthetic works on numerical notation, thanks to the recent work of David King (1995, 2001), which supersedes all earlier research completely, we now have extensive information about it. The precursor of the full-fledged Cistercian numerals was a set of eighteen symbols introduced by John of Basingstoke (John of Basing), archdeacon of

Miscellaneous Systems

351

Table 10.18. Numerals of John of Basingstoke

1s 10s

1

2

3

4

5

6

7

8

9

A J

B K

C L

D M

E N

F O

G P

H Q

I R

:

75 =

Leicester, in the early thirteenth century (Greg 1924; King 2001: 51–57). These signs are shown in Table 10.18 (King 1995: 202). The symbols for the units can be grouped into three sets of three (1–3, 4–6, 7–9), based on the position of the short stroke to the left of the vertical stroke (at the top, middle, and bottom, respectively). Each of the tens signs is a horizontal mirror image of the corresponding unit-sign. This allowed the two powers to be combined into a single sign. There is no way to write numbers higher than 99. There are two valid ways to classify this system. It may be considered a ciphered-additive decimal system having nine distinct signs for the ones and nine more for the tens. Alternately, recognizing that the signs for the tens are mirror images of those for the ones, we may consider this system a very peculiar ciphered-positional system – one in which the positions are not arranged in a simple line, but in which the orientation of the numeral-sign around the vertical stroke determines its value. One theory holds that the Basingstoke numerals originated in Greece (King 2001: 57–65). Basingstoke’s biographer, Matthew Paris, reported in his Chronica maiora that Basingstoke spent much time in Greece and learned the system from Athenian scholars. Moreover, a fourth-century bc (!) tablet found on the Acropolis contains a form of cryptographic alphabetic shorthand whose signs are similar in shape to Basingstoke’s numerals. Yet the ancient Greeks never used this shorthand to express numbers, and there is no evidence of its survival in Byzantine scholarship. A more plausible theory is that a system of alphabetic shorthand known as the ars notaria, which developed and was used in England in the twelfth century, inspired Basingstoke’s invention (King 2001: 66–71). The ars notaria used all eighteen of Basingstoke’s numerals (plus a vertical stroke) to represent nineteen alphabetic signs. Moreover, while the ars notaria were not used to express numbers, when they are placed in Table 10.19. Alphabetic and numerical values of the ars notaria/Basingstoke’s system a

b

c

d

e

f

g

h

i

l

m

n

o

p

q

r

s

t

u

a

B E H K N Q C F I L O R A D G J M P

-

2

5

8

20 50 80 3

6

9

30 60 90 1

4

7

10 40 70

Numerical Notation

352

Table 10.20. Cistercian horizontal numeral-signs

1s 10s 100s 1000s

1

2

3

4

5

6

7

8

9

a j s 1

b k t 2

c l u 3

d m v 4

e n w 5

! # $ %

g p y 7

h q z 8

i r 0 9

alphabetic order and correlated with their numerical values in Basingstoke’s system, a clear pattern emerges, as seen in Table 10.19 (King 2001: 68–69). It is remotely possible that the graphic similarities and patterning of the ars notaria and Basingstoke’s system were developed independently. Nevertheless, since we know the former to have been invented in the twelfth century, the most parsimonious theory is that Basingstoke learned the ars notaria and then hit on the idea of assigning numerical values to these alphabetic signs. This raises the possibility that perhaps Basingstoke’s travels in Greece taught him the ciphered-additive Greek alphabetic numerals (Chapter 5), leading him to hit on the idea of using ars notaria letters as numerals. The use of alphabetic numerals was infrequent in Western Europe, but would have been common in early thirteenth-century Athens. While Basingstoke’s numerals appear in only two texts other than the Chronica maiora, one of these is a late thirteenth-century manuscript from a Cistercian monastery, Whalley Abbey in Cheshire. This is relevant because the next place we find a system like Basingstoke’s numerals is in late thirteenth-century Cistercian manuscripts from France and Belgium. While these signs were slightly different from his numerals, they were derived from the earlier English signs. The most common variant of this system is shown in Table 10.20 (King 2001: 102).14 Whereas Basingstoke’s numerals had signs only for the units and tens, this more developed system included signs for the hundreds and thousands as well, and used a horizontal base stroke rather than a vertical one. Nevertheless, the structure of the system is essentially the same, only with four positions instead of two, with the units in the top left, the tens in the bottom left, the hundreds in the top right, and the thousands in the bottom right. Thus, we may classify it as a ciphered-positional system based on orientation, or as a ciphered-additive system for which signs for the same multiple of different powers happened to resemble one another. Table 10.21 shows how several numbers would have been expressed using this system.

14

King (2001: 39) provides a chart illustrating the enormous variation in this system within what was, after all, a very limited manuscript tradition.

Miscellaneous Systems

353

Table 10.21. Horizontal Cistercian numeral-phrases

< . ; ,

157 2345 6666 9002

These numerals were used in a variety of Cistercian manuscripts from the thirteenth to the sixteenth centuries, primarily in the Low Countries and neighboring regions of northern France (King 2001: 95–130). They were used extensively in the pagination of Cistercian religious texts and the numbering of sermons, as well as for writing numbers (especially year-numbers) in the body of texts. They were usually much more compact than the corresponding Roman numerals, and, while they were in direct competition with the increasingly popular Western numerals, they spread widely within the Cistercian scribal tradition. These manuscripts were intended for a very limited audience, and, since they sometimes included charts in the margin of the text explaining their use, they cannot have been used cryptographically during this period. A variant on this system used vertical base-strokes; these signs were similar to those of the horizontal system, only rotated ninety degrees clockwise, so that the units occupied the top right position. A common version of these signs is shown in Table 10.22, indicating only the units (King 2001: 39). The vertical signs for the tens, hundreds, and thousands are simply those for the units, flipped and rotated as in the standard system. Notably, this system uses an additive framework within each power – the ciphered signs for 5, 7, 8, and 9 are additive combinations of the other signs. This adds a level of transparency to this variant that is not present in other ciphered systems, including the Western numerals and the standard Cistercian numerals described earlier. The vertical numerals first appear in a manuscript copied in Paris in the late thirteenth century, in which pages are foliated using this system (King 2001: 153–155). While they were not used Table 10.22. Vertical Cistercian numerals 1

2

3

4

5

6

7

8

9

S

T

U

V

W

X

Y

Z

[

(6 + 1)

(6 + 2)

(8 + 1)

(4 + 1) 5107 = >

354

Numerical Notation

as frequently as the horizontal numerals, they were employed in a wider set of contexts, including mercantile and scientific ones. They are inscribed on an astrolabe from Picardy, the only example we have for their use on an object rather than in a text (King 2001: 131–151). A fifteenth-century arithmetical text from Normandy describes a technique for writing numbers higher than 10,000 by placing a sort of bracket around a lower number using the multiplicative principle, so that 126,000 would be written as ? (King 2001: 159). They also occur outside of northern France on a late fifteenth-century astronomical table from Segovia, which belongs to a set of eclipse computations by the Jewish Spaniard, Abraham Zacuto (Chabás and Goldstein 1998). Unusually, a few manuscripts from Bruges describe their use as markings on wine barrels and wine gauges, as part of the mercantile practices of vintners starting in the late fourteenth century and used as late as 1720 (King 2001: 164–171, 239–242). Unfortunately, no marked wine barrels or related artifacts exist to complement the textual evidence. The advent of printing in the middle of the fifteenth century, and the decline in the Cistercians' fortunes that accompanied the Reformation, were disastrous for this system. The numerals ceased to be used regularly in the sixteenth century. After this point, interest in the numerals from an academic and mystical perspective increased. They appear as historical curiosities in many sixteenth-century texts, most notably De occulta philosophia (1531–33) by Agrippa of Nettesheim (King 2001: 190–202). They are also found in De numeris (1539) by Johannes Noviomagus and De subtilitate libri XXI (1550) by Girolamo Cardano, both of which cite Agrippa as an authority.15 In these texts, and frequently thereafter, the numerals were mistakenly thought to be “Chaldean,” an appellation often used in the Renaissance to refer to mystical learning supposedly diffused from the Near East, especially Babylonia. Even Cajori (1928: 68–69) cites the Chaldean theory of Agrippa, although skeptically. The use of the Cistercian numerals in well-known mystical and mathematical texts ensured that they were never completely forgotten, even though knowledge of their true origin was lost. They were described in various works on magic, the occult, and astrology, as well as in a variety of early works on numerical notation (King 2001: 210–238). Yet, apart from their use in wine gauging in Bruges mentioned earlier, they were used only rarely after 1550. A group of Parisian Freemasons used the numerals in some of their private correspondence with fellow members in the 1780s (King 2001: 243–246). The last nonscholarly mention of the numerals was by a number of German nationalistic authors in the early twentieth century, who saw the Cistercian numerals as a sort of proto-Aryan runic numeration (King 2001: 251–261). 15

Curiously, however, Noviomagus lists the horizontal rather than the vertical numerals, and some of Cardano’s vertical numeral-signs are more similar to Basingstoke’s thirteenthcentury numerals than they are to the later vertical signs.

Miscellaneous Systems

355

Table 10.23. Keutuklu numerals

1s 10s 100s 1000s

1

2

3

4

5

6

7

8

9

a j s 1

b k t 2

c l u 3

d m v 4

e n w 5

f o x 6

g p y 7

h q z 8

i r 0 9

Ottoman Cryptographic Throughout the period of Ottoman dominance in the Middle East, between roughly 1450 and 1900, the standard Arabic positional numerals (Chapter 6) were by far the most common system in use, while various ciphered-additive systems, most notably the Arabic abjad numerals (Chapter 5), were used in certain contexts. In addition, a number of quasi-cryptographic systems that bear little to no resemblance to the Arabic alphabetic or positional systems were used by Ottoman administrators (particularly military clerks). Four of these were reported in Western scholarly literature by M. J. A. Decourdemanche (1899). While their historical importance is not great, they were structurally ingenious, and three of them have unusual structural properties. The first system, known as keutuklu, was used by clerks to record data concerning the recruitment of Christian youth into the Ottoman army (Decourdemanche 1899: 261). The signs of this system are shown in Table 10.23 (Decourdemanche 1899: 260). This system has the appearance of a decimal ciphered-additive system with unique signs for each multiple of each power of 10. Yet the numeral-signs are not arbitrary, but are constructed by adding small circles to the set of nine basic unit-signs (one circle for the tens, two for the hundreds, and three for the thousands). An alternate way of looking at this system, then, would be to regard it as multiplicative-additive, with the unit-signs being the basic linear frames for 1 through 9 and the power-signs being one, two, or three circles. Additionally, there are graphic resemblances among the signs for 2 through 5 (which are based on the sign b and transpositions thereof ) and among the signs for 6 through 9 (based on f). However one chooses to look at it, there is a clear graphic resemblance between the signs in each column (e.g., 6, 60, 600, and 6000), which dispels one of the objections often leveled at ciphered-additive systems, namely the arduousness of memorizing many different signs. Because Decourdemanche does not describe how these signs combined with one another to produce numeral-phrases, we do not know whether numerals were written from left to right as in the Arabic

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356

Table 10.24. Ordouï cheïlu numerals

1s 10s 100s 1000s

1

2

3

4

5

6

7

8

9

A J S !

B K T @

C L U #

D N V $

E N W %

F O X ^

G P Y &

H Q Z *

I R ) (

positional system, from right to left as in the Arabic abjad and other alphabetic systems, or in some other manner. The second system, known as ordoui ‘army’, also constructs numeral-signs ingeniously. The most common variety of this system, known as ordouï cheïlu ‘army equipment’, was used by the Ottoman army for enumerating provisions, equipment, and other military supplies (Decourdemanche 1899: 262). The signs of the system are shown in Table 10.24 (Decourdemanche 1899: 263). Like the keutuklu system, the ordouï cheïlu is a decimal system that can be classified as either ciphered-additive or multiplicative-additive. Each sign consists of a vertical stroke with a number of diagonal strokes leading off it to the left and right. The left side represents the number of units, with no strokes for 1 up to eight strokes for 9, while the right side indicates the power, with one stroke for the units, two for the tens, three for the hundreds, and four for the thousands. While this system seems to have a cumulative component, the strokes on the left do not add up directly to the number of units, but rather to one fewer than the number of units represented (zero for 1, one for 2, ... eight for 9).16 Again like the keutuklu, it is possible to derive the value of a sign easily from a limited set of basic rules. In addition, this system might be rendered potentially infinite, even though it is nonpositional, using five strokes on the right side for the ten thousands, six strokes for hundred thousands, and so on. Decourdemanche does not report how the signs of this system were arranged into numeral-phrases. A variant of the ordoui system recorded the numerical strength of military units, and could additionally serve as a cryptographic script. The signs of this variant are shown in Table 10.25 (Decourdemanche 1899: 262). The signs of this system are far less regular than those of the ordouï cheïlu. Instead of indicating the power of the sign by the number of strokes on the right, the signs 16

While the total number of diagonals (left and right) equals the relevant number in the ones column, this pattern does not hold for the higher powers.

Miscellaneous Systems

357

Table 10.25. Ordoui numerals for personnel

1s 10s 100s 1000s

1

2

3

4

5

6

7

8

9

B V > _

C @ ?

D # [

E $ ]

K % +

L : ,

M ; -

T < .

U = /

are grouped erratically in sets of three or four (1–4: one right stroke; 5–7: two right strokes; 8–10: three right strokes ... 800–1000: eight right strokes). Moreover, there is no common feature among the multiples of different powers, so that 4, 40, and 400 have no inherent similarity. Finally, whereas the ordouï cheïlu could be used to write any number up to 10,000, the highest sign in this system was 1000. This variant originated from the structure of the Arabic abjad (Chapter 5). The twenty-eight-sign abjad was divided into eight mnemonic groups of three or four signs apiece, and the numerical values assigned to the abjad correlate perfectly with the divisions of this system. Moreover, the numeral-signs could be used not only in their numerical sense, but also to represent the appropriate Arabic letter. The third notable Ottoman cryptographic system, known as damgalu ‘inspection’, was used for marking numerals on military equipment, and also could be used as a cryptographic script (Decourdemanche 1899: 264–265). The signs of this system are shown in Table 10.26 (Decourdemanche 1899: 265). The damgalu numerals have twenty-eight signs, like the Arabic abjad numerals, corresponding to 1 through 9, 10 through 90, 100 through 900, and 1000, organized Table 10.26. Damgalu numerals 1

2

3

4

5

6

7

8

9

1s

Ç

ü

é

â

ä

à

å

ç

ê

10s

ë

è

ï

î

ì

Ä

Å

é æ ï æ

â æ

ä É

à É

å É

ç æ

ê æ

Ç æ ë æ

ü É è É

100s

1000s

358

Numerical Notation

into a decimal ciphered-additive system. Instead of simply using alphabetic signs, however, each sign has four registers, in each of which a line or a dot is placed. This allows for only sixteen (24) combinations, so two additional signs were used to represent the last twelve signs. The first, æ, was placed under any sign whose bottom register was occupied by a line, while the second, É, was placed under signs whose bottom register was a dot. The signs shown in Table 10.26 are the variety of damgalu used by the Ottoman navy, while a separate system was used in the army, which was identical except that it used the sixteen combinations in a different order. There is no correlation between the sequence of dots and lines and the numerical values in question, so the damgalu is particularly cryptographic. The additional signs for the last twelve numerical values are not structurally significant; their use was simply a necessity imposed by the lack of adequate signs available with the sixteen basic combinations. The damgalu numerals were each correlated with one of the twenty-eight signs of the Arabic abjad, and could be used to stand for phonetic as well as numerical values. We know remarkably little about these systems’ origin or history, save that they were employed in the nineteenth century, when Decourdemanche reported on their use. A system nearly identical to the keutuklu system is described in a sixteenthcentury Moroccan manuscript, where it is called, strikingly, qalam hindī ‘Indian figures’ (Sanchez Perez 1935). The inventors of the ordoui and damgalu systems were surely familiar with the Arabic abjad numerals, since they were organized according to the structure of the Arabic abjad and could stand either for a numeral or for the corresponding letter. The unusual structure of the keutuklu system may derive from the Arabic positional numerals. They are treated by Decourdemanche as already being obsolete at the turn of the twentieth century, and do not appear to have survived past the end of the Ottoman Empire.

Summary Because the systems described in this chapter are not part of a single phylogeny, they share little in common. Few have been studied in any histories of numeration. While systems such as the Inka khipu have been ignored or belittled because they are not associated with a script, a formal analysis of their properties shows them to be ordinary numerical notation systems. The failure to recognize other systems, such as the Cherokee, Pahawh Hmong, and various African systems, derives from the marginalization of these societies in the modern world-system. Such systems may also be thought to be unimportant because they are derived from Western numerals and because they typically have failed to be adopted on a widespread basis. This is unfortunate, because they are often structurally distinct from Western numerals, contradicting the assumption that African and other

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359

cultures influenced by Western imperialism are devoid of mathematical achievements, or that such achievements are at best purely derivative. Moreover, these newly invented systems help us understand the social contexts that lead to the development of new systems. In many of the cases described here, Western, Arabic, or other positional systems were available to be adopted. The fact that indigenous systems were invented indicates that the desire to resist imperialistic institutions or to produce local alternatives to foreign inventions often motivates the development of these and other systems. Even where systems are not in widespread use, they can remain important sources of cultural capital upon which script users can draw when necessary. This practice differs little from the use and retention of Roman numerals for prestige functions in Western societies, despite the near-ubiquity of Western numerals otherwise. Several ethnographically attested systems (such as Naxi, Oberi Okaime, Varang Kshiti, and Bambara) were invented and used specifically for ritual, liturgical, or divinatory purposes. This rationale parallels both the earliest Mesoamerican numerals (apparently used for calendrics and deity names) and the earliest East Asian numerals (used in divinatory contexts on oracle bone inscriptions). Several of the inventors of scripts and numerical systems were religious figures, and the association of script invention with religious visions or divine revelation has been noted elsewhere (Houston 2004: 235). Postgate, Wang, and Wilkinson (1995) argue that all early writing and numeration served bookkeeping/administrative functions, regardless of the attested evidence, claiming that the oracle-bones and Mesoamerican inscriptions are not reliable evidence of the initial functions of the scripts, and that the equivalent of proto-cuneiform tablets, long perished, once existed in those societies. Yet if colonial-period scripts and numerical systems can be invented for nonaccounting functions, why not ancient ones also? I have not discussed modern artificial numerical notation systems, such as the binary and hexadecimal numeration used in computing or the color-coded system of numeration used on resistors (International Electrotechnical Commission 2004). In these cases, there is a technological impetus toward the invention of particular forms of numeration.17 Yet these systems, in combination with the ethnographic evidence from the systems described here, conform to a pattern suggesting that the creation of numerical notation systems remains an ongoing project. More significantly, despite the predominance of Western and Arabic positional numerals worldwide, there exist technological, social, and cultural reasons for the continued invention of new numerical notation systems in the twenty-first century and beyond.

17

The same might be said of the use of I, V, and X shapes in tally-type systems!

chapter 11

Cognitive and Structural Analysis In Chapters 2 through 10, I described over 100 different numerical notation systems spanning over 5,000 years and every inhabited continent. While there are historically determined similarities among the systems of each phylogeny, the same structures and principles emerge independently multiple times. This situation creates a paradox only if we cling to the dichotomous assumption that historical explanations stand in stark contrast to universalizing ones. A set of interrelated cognitive factors help explain why systems are the way they are and why they change in the ways that they do. There are some domains of human experience for which the role of contingency is so great, or the functional constraints so minimal, that we cannot speak meaningfully of regularities or laws. Numerical notation is not one of them. In Chapter 1, I outlined the case that the study of cross-cultural regularities and universals is of equal importance to the study of unique or particular phenomena (Brown 1991). Here I will outline around thirty regularities that apply to numerical notation systems, while in the following chapter I will inject theoretical issues relating to social and historical context into this analysis. Numerical notation systems exhibit both synchronic regularities, which apply to numerical notation systems considered as static structures, and diachronic regularities, which apply to relations between systems over time. Synchronic or diachronic regularities can be either universals (for which there are no exceptions) or statistical regularities (which hold true only for a preponderance of cases). While

360

Cognitive and Structural Analysis

361

true universals are interesting, statistical regularities are also important, and may in fact be caused by cognitive factors similar to those that produce universals. Exceptions can help explain why the regularity exists in the first place, by clarifying the conditions under which it does not apply. One must also be aware that a universal may be only contingent or apparent – for instance, an exceptional case may not be well attested, or may simply not have been invented yet. Whenever there are exceptions, I have as an expositionary device stated the regularity in its universal (exceptionless) form and then discussed the exceptions in the text. Statistical regularities (general patterns that have exceptions) are quite different from implicational or conditional regularities, which take the form “If system A exhibits feature X, then it will also exhibit feature Y” and that apply to only a subsection of the universe of numerical notation systems. Frequently, systems to which an implicational regularity does not apply do not actually violate it; rather, the feature does not exist in outlying systems. They can be either universal (within that context), “If A, then B,” or statistically probable, “If A, then usually B.” The systems included in this study are those that are attested in the ethnographic or historical record. The clever skeptic can imagine systems that violate any of the regularities to be presented here. Such systems have already been invented by scholars (Dwornik 1980–81), cryptographers (Wrixon 1989: 103), and science fiction writers (Pohl 1966: 179–192). This does not demonstrate that these generalizations are not “true” regularities, or even that they are not universals, but merely proves that they are not logical necessities. Because existing numerical notation systems satisfy these constraints, even though it is not logically required that they do so, we must look instead to psychological and utilitarian constraints as the source of both universals and statistical regularities. That these constraints are apparently so great as to produce universals among 100 or more structurally distinct numerical notation systems confirms the constraining power of the mind, working in conjunction with the perceived environment. Greenberg’s (1978) examination of regularities in lexical numerals has been of particular use in formulating my list of regularities. Where appropriate, I have indicated the correlations between my regularities and those he found for lexical numerals, without confirming or denying the validity of the latter set. There are, however, many regularities for lexical numerals that do not apply to numerical notation systems, and vice versa. For every instance in which there is a parallel between lexical numerals and numerical notation, there is another in which there are significant differences between the two domains. Because of these differences, the regularities of numerical notation systems cannot possibly be derived from a biologically hard-wired “universal grammar.”

362

Numerical Notation

Synchronic Regularities Synchronic regularities describe features that are common to all systems, without reference to the time dimension. I have not included regularities that have many nontrivial exceptions and may well simply be coincidences. Because there are too few independently invented numerical notation systems to allow statistical analysis using only these independent cases, I instead judge the significance of statistical regularities by considering the nature of their exceptions. I begin with a brief list of axioms, which frame the phenomenon of numerical notation according to the basic guidelines set out in Chapter 1, before describing the general and implicational regularities I have been able to discover. I then list a small number of nonuniversals, statistical regularities whose exceptions are more interesting theoretically than are the systems that obey them.

Axioms A1. All numerical notation systems can represent natural numbers. A2. All numerical notation systems have a base. A3. All numerical notation systems use visual and primarily nonphonetic representation. A4. All numerical notation systems are structured both intraexponentially and interexponentially.

These features have been described fully in Chapter 1, and require no particular attention here, except insofar as they form the basis from which all other regularities are derived. Any representational system that does not conform to these four axioms is not a numerical notation system, by my definition.

General Regularities G1. Any system that can represent N+1 can also represent N, where N is a natural number.

This is a universal, which I call the Continuity Principle.1 It establishes the continuity of the sequence of natural numbers starting at 1, but does not imply that all numerical notation systems are infinite in scope. It also leaves open the question of the expression of zero, negative numbers, and fractions. A system might 1

Greenberg (1978: 254–255) offers a similar principle concerning lexical numerals, which he calls the “thesis of continuity.”

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conceivably be developed for the sole purpose of recording a set of nonsequential numbers with religious significance, or a group of users might use one system for representing odd numbers and an entirely different one for even numbers. Such unusual systems have never been implemented. One of the crucial functions of numerals is ordinal enumeration, for which only a continuous set of integers will suffice. G2. All systems use a base of 10 or a multiple of 10 for representing natural numbers.

This is a universal, which I call the Rule of Ten. It is possible that the Indus Valley civilization had an octal (base-8) numerical notation system (Chapter 10), but the base of this system is much more likely to have been decimal. Systems for representing fractions, which often use a different base than the systems for integers with which they are used, often have nondecimal bases, such as the base-2 Egyptian “Horus-eye” fractions (Chapter 2). The only widely used potential exceptions to the Rule of Ten are the binary, octal, and hexadecimal systems used in computing, but these show no sign of achieving wider currency as the general system of any society. The fact that some systems have sub-bases or extraneous structuring signs that are not multiples of 10 is irrelevant to the validity of this principle. The explanation of this feature requires that we consider several hypotheses (see the section “Fingers and Numbers”). G3. All systems form numeral-phrases through addition. G4. No system forms numeral-phrases through division.

These two regularities are universals. Addition will always be found among the arithmetical steps by which a system is used to derive the values of numeral-phrases, whether it is the only operation (as in cumulative-additive and ciphered-additive systems) or not (multiplicative-additive, cumulative-positional, and cipheredpositional systems). It is possible to imagine a purely multiplicative system – for instance, one that expresses all numbers as prime numbers or as the product of prime numbers – but this has never occurred. This does not imply, however, that every numeral-phrase in a system uses addition; the units and the powers of the base are expressed with single signs in many systems, and thus do not involve addition. Addition is frequently combined with multiplication, a very effective means of expressing large numbers. While addition and multiplication are common, subtraction is extremely rare (being found only in the Roman numerals and a few Mesopotamian systems), and division is absent entirely from the operations used to form numeral-phrases for integers. It is certainly possible to imagine 50 and 10 being expressed as “2 100” or “2 20,”

364

Numerical Notation

but this is not attested. Lexical numerals use division only in the form of multiplication by ½ or ¼, and this is very rare (Greenberg 1978: 261). Even this operation is never found in numerical notation. There is the physical division of Etruscan, Roman, and Ryukyu tallying-based signs (e.g., Roman V is the top half of X), but this is a nonarithmetical graphic technique governing the formation of signs, and can be interpreted as doubling or halving with equal validity. This regularity does not deny that fractions can be expressed in numerical notation. The absence of division (and the rarity of subtraction) may be a matter of representational convenience (since such numeral-phrases would need to use large divisors to express smaller numbers), or a consequence of the rarity of such operations in lexical numeral systems. G5. All numerical notation systems are ordered and read from the highest to the lowest power of the base.

This is a near-universal, which I call the Ordering Principle. Positional systems could be read from the lowest power to the highest, but this never occurs; in other words, interexponential ordering is respected throughout a numeral-phrase. While the powers of additive systems could be placed in any order (e.g., in classical Roman numerals, which do not normally use subtraction, 217 could be written as IICVCX), this occurs only when the writer has made an error. Subtractive forms such as the modern Roman numeral IX for 9 do not violate this principle, because they involve intraexponential structuring only. Many lexical numeral systems disobey this principle, including those of many of the major European languages (e.g., Italian sedici = 6 + 10 although diciasette = 10 + 7). However, numerical notation systems almost always reserve “low-high” forms for intraexponential subtraction or multiplication. Users of a system know, upon encountering a lower numeral-sign followed by a higher one, to be alert for the need to use an operation other than addition. The Ordering Principle also applies to sub-bases, which always precede signs for the next-lower full power. There are a couple of minor exceptions to the Ordering Principle; in certain alphabetic systems, including the Greek, Glagolitic, and Cyrillic alphabetic numerals, the numbers 11 through 19 are often written with the sign for 10 at the end of the numeral-phrase (e.g., Cyrillic 12 = bj (2 + 10), not jb (10 + 2)). These exceptions reflect the word order of the lexical numerals of the languages of these systems’ users. G6. No system uses signs for the arithmetical operations used to derive the value of a numeral-phrase.

This is a near-universal. Even though all systems form numeral-phrases through addition, and many of them also use multiplication, this is almost never directly represented with a sign. This is in contrast to lexical numerals, in which it is very

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common to express at least some operations with words, such as the German ‘sechshundert-fünf-und-vierzig’ ‘six hundred-five-and-forty’, Latin ‘duodeviginti’ ‘two-from-twenty’, and other phrases that are even more complex. In fact, lexical numerals almost never express subtraction without some indication of the operation (Greenberg 1978: 258–259). Such signs in numerical notation would render numeral-phrases less concise without providing any additional clarity because the system’s principle dictates the operations used. The (near-) universality of the Ordering Principle means that the operation to be used can be inferred easily from the context. When lexical numerals show arithmetical operations explicitly, it is often because unusual ordering is being employed, as in the two numerals just mentioned. There are, of course, systems that use graphemes that implicitly refer to a particular operation (e.g., the use of a hasta in Greek alphabetic numerals (Chapter 5) to indicate multiplication by 1000, or the ‘times 10’ sign in the Cherokee numerals (Chapter 10). These signs always primarily bear numerical values, and simply happen to be used only in the context of particular arithmetical operations. The only actual exception to this regularity is the use of the additive grapheme ‘and’ in the Shang Chinese numerals (Chapter 8). This may reflect the highly lexical nature of Chinese numerical notation and the linguistic origins of the system. G7. The only visual features used to determine the numerical value of figures in numerical notation systems are shape, quantity, and position.

This is universal or perhaps nearly universal. The relevant features for determining the value of a numeral-phrase are the shapes of the particular signs used, the quantity of those signs, and their position. The color of the signs, their relative size, and other extraneous graphic features do not affect the value of the phrase. It is possible to conceive of a system where different registers of sign sizes, rather than their position, would determine the value of signs within numeral-phrases, thereby eliminating the need for a zero-sign. We could, for instance, use Western signs from 1 through 9 to write 462 as 62 and 402 as 2 – no zero being required because the size of the 4 indicates its value. A similar system could apply different colors to different powers of the base. An exception to this pattern is that in the proto-cuneiform and archaic Sumerian systems (Chapter 7) the sign for 60 is a large version of the sign for 1. A second partial exception is the use of red- and black-colored rods in the Chinese rod-numerals (Chapter 8), but this was only done occasionally and only distinguished positive from negative numbers. Features whose different values are easily differentiated visually and that are easily represented in writing are highly desirable for numerical notation. Using size or color would be extremely difficult, requiring users to employ many differentcolored inks or to distinguish between different-sized registers of signs.

4

4

366

Numerical Notation

G8. There is never complete correspondence between the numeral-signs of a system and the lexical numerals of the language of the society where the system was invented. G9. There is always some correspondence between the numeral-signs of a system and the lexical numerals of the language of the society where the system was invented.

These two regularities complement each other, and at least the first of them has been pointed out by Menninger (1969: 53–55). There are enormous structural differences between lexical numerals and numerical notation. Most lexical numerals are multiplicative-additive in structure (six thousand four hundred seventy one), and while some numerical notation systems are multiplicative-additive, they are far less common than cumulative-additive, ciphered-additive, or cipheredpositional ones. Numerical notation is not simply a matter of reducing numerical morphemes into signs. Because lexical numerals are auditory in origin (they were spoken before they were written), while numerical notation is visual in origin, we ought to expect representational differences between them. In some East Asian systems such as Chinese (Chapter 8), numerical notation and lexical numerals closely parallel one another, because each character normally represents one morpheme. Even in Chinese, however, there are numerous variants and options for expressing most numerals, not all of which directly correspond to lexical representations. There is, however, a strong correlation between a language’s numeral words and the base of numerical notation systems developed by its speakers. It would be very surprising for a decimal numerical notation system to develop among speakers of a language with vigesimal lexical numerals, and when this occurs (e.g., in the Mende numerals [Chapter 10]) it is imperative to look for an explanation of the phenomenon – in this case, a radical change from decimal to vigesimal lexical numeration in the Mende language. Similarly, the fact that Greek and Cyrillic numerals express the teens using ‘1 + 10’ through ‘9 + 10’ in violation of the Ordering Principle (see the previous discussion) is best explained through comparison with the classical Greek and Slavonic numeral words.2 Imperialism and other forms of cultural hegemony frequently spread a numerical notation system far beyond the region of its original invention, however, and there is little to prevent such numerical notation systems from being adopted by speakers of languages with nondecimal lexical numerals. This suggests that, while the initial choice of a numerical notation system’s base may be constrained by its inventors’ lexical numerals, once this choice is made, there is much less flexibility for changes in base structure. 2

Note, however, that the modern Greek lexical numerals for 13 through 19 follow the inverse order: dekatreis ‘10 + 3’... dekaennia ‘10 + 9’.

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G10. No system uses an identical representation for two different numbers.

This is practically universal. It is possible to imagine a system where the numeral-phrases for 2 and 20 (or any other two numbers) are identical. Such ambiguity creates confusion and reduces the utility of a system, so few systems do so. The converse of this principle is not true: many systems use two or more representations for one number (e.g., Roman VIIII or IX for 9), but this never creates a numeral-phrase whose value is truly indeterminate. This regularity allows that there may be a single numeral-sign for two numbers, however. For instance, the Palmyrene system (Chapter 3) uses a single sign for 10 and 100, but the sign for 100 is always found in conjunction with one or more multiplicative signs, whereas the sign for 10 occurs as part of the cumulative-additive component of the system. Some of the proto-cuneiform signs (Chapter 7) have multiple values, but these occur in different subsystems representing different categories of counted objects, never together in a single system. A true exception is the Sumerian cuneiform system (Chapter 7), which uses a vertical wedge for both 1 and 60. The Sumerians were aware of the ambiguities this caused, however, and by the Ur III period (2150 to 2000 bc), a different sign for 60 was used whenever confusion could result. A similar issue arose in the related Old Babylonian positional cuneiform numerals, prior to the invention of zero in the Seleucid period. Except where numbers were lined up in columns, any numeralphrase could have an infinite number of interpretations, though the correct one was often evident from the context.

Implicational Regularities I1. If a system has a sub-base, the sub-base will be a divisor of the primary base.

This is a universal. While it is easy to imagine a system with a base of 10 and a sub-base of 3, and while such a system would be able to express every number uniquely (and, in fact, somewhat more concisely than with a sub-base of 5), this and similar nondivisor sub-bases are never attested in numerical notation systems. Greenberg (1978: 270) notes that at least two lexical numeral systems, Coahuilteco and Sora, have this feature, but it is extremely rare in lexical numerals as well. Numerical notation systems are sometimes structured by means of additional numbers that are not divisors of their primary bases (such as the use of a sign for 4 in the base-10 Nabataean and Kharoṣṭhī systems), but these additional numbers are not sub-bases, because they do not recur throughout the system (i.e., 40, 400, and 4000 do not have their own special signs).

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Numerical Notation

I2. No ciphered system has a sub-base.

This is nearly exceptionless. Of the twenty-three systems I have studied that have sub-bases, fifteen are cumulative-additive, five are cumulative-positional, and three are multiplicative-additive, but none are ciphered-positional or ciphered-additive. Because ciphered systems require only one sign per power of the base, introducing a sub-base does not reduce the number of signs required to write a number, as it does in cumulative systems. There are traces of a sub-base of 10 in the base-20 ciphered Maya head-glyphs (Chapter 9), since the signs for 14 through 19 (and sometimes 13) are expressed by combining the “bared jawbone” element for 10 with the rest of the sign for the appropriate unit; thus, one need not develop distinct signs for these numbers. However, because it is not used for 11 or 12, and rarely for 13, this is not a full exception. Apparently, it is not extremely advantageous to introduce a sub-base solely to obviate the need to develop new signs. We could avoid using the signs for 6 to 9 by introducing a sub-base of 5 into the Western numerals, using a horizontal line for 5, and thus replacing 6, 7, 8, and 9 with v w x y, but we have little difficulty remembering the ten digits we have. Potentially, doing so would create confusion due to the similarities between 1 and 6, 2 and 7, 3 and 8, and 4 and 9 that would result. I3. If a system is cumulative, it will group intraexponential signs in groups of between three and five signs.

This regularity, which I call the Rule of Four, is nearly universal but has a few minor exceptions. Humans are limited in their cognitive capacities, and work most efficiently when information is packaged in groups of three to five elements. Cumulative systems cope with this limit either by using sub-bases (e.g., Roman, Maya barand-dot), by using spacing to distinguish groups (e.g., Egyptian hieroglyphs, Aztec), or both (Babylonian sexagesimal). Probably the limit of five is even a bit too high for the human mind to grasp. Two full exceptions to the Rule of Four are the Inka and the Bambara numerals (Chapter 10), which always use groups of up to nine unitsigns. Partial exceptions include the Hittite hieroglyphs (Chapter 2) and the South Arabian numerals (Chapter 4), in which chunking in groups of three to five signs was an option, but in other cases groups of up to nine signs were used. I will discuss the Rule of Four further in the section “Subitizing and Chunking.” I4. If a system is multiplicative-additive for a particular power of its base, it will also be multiplicative-additive for all higher powers of the base.

This exceptionless regularity applies to hybrid systems – those that are cumulativeadditive or ciphered-additive for some powers, but multiplicative-additive for

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others. In such systems, it is always the higher rather than the lower powers that are multiplicative, and once the “switch” to the multiplicative principle has been made, it applies for all higher powers. No system is multiplicative-additive for lower powers while following some other principle for higher powers, or alternates multiplicative and non-multiplicative powers. Hybrid multiplication is primarily useful for extending a system further without developing increasingly large inventories of signs, but results in somewhat longer numeral-phrases. It is thus more useful for large powers than for small ones. Moreover, in many lexical numeral systems, multiples of lower powers are expressed with a single word, but multiples of higher powers separate the units and power components (e.g., Latin sex, sexaginta, sescenti vs. sex milia). The point at which this shift in lexical numerals occurs sometimes may affect whether or not a certain power is expressed multiplicatively in the corresponding numerical notation system, but it cannot be the only factor because many systems do not use multiplication at all. I5. Whenever the multiplicative principle is used in a system, the unit-sign or signs (multiplier) will precede the power-sign (multiplicand).

This is nearly universal. It is rarely permitted to express 300 as “100 3” in a multiplicative-additive system; regardless of the base of the system or other structural features, the units precede the power. Because of the Ordering Principle, expressions where the power-sign was placed first could be interpreted additively in many numerical notation systems, thereby creating ambiguity. Requiring the unit-sign(s) to precede the power-sign eliminates this risk. Most lexical numeral systems also place unit-signs first, and thus adherence to this pattern makes it easier to translate a graphic numeral-phrase into its lexical equivalent. In some alphabetic numeral systems, such as the Greek, Coptic, and Cyrillic systems (Chapter 5), a small diacritic mark to the left of (before) a sign indicates multiplication by 1000 (e.g., Greek /g = 3000). In these exceptional cases, there is no possibility of confusing multiplicative expressions with additive ones. I6. No multiplicative system uses 1 as a power-sign.

This regularity is virtually exceptionless. Multiplicative-additive systems combine unit-sign multipliers with power-sign multiplicands, but there is never a separate power-sign for the units. Rather, the numbers from 1 up to the base of the system are expressed through unit-signs alone. While a power-sign for 1 would be consistent with the principle of combining unit-signs with power-signs, it would be completely extraneous and provide no additional information. While most lexical numeral systems are multiplicative-additive, they similarly do not

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use power-signs for 1. The only exception to this rule is that the earliest Bamum numerals (Chapter 10) had a separate power-sign for 1, which was entirely distinct from the unit-sign for 1. This sign was used only for a brief time, however, before the Bamum system became ciphered-positional. In any event, the use of distinct graphemes for the unit-sign and the power-sign eliminated any ambiguity. In all other multiplicative-additive systems, there is no power-sign for 1. I7. All multiplicative expressions involve only bases or their powers as multiplicands.

This is a universal. No system uses multiplication involving sub-bases, multiples of powers of bases, or other additional structuring numbers. It would certainly be possible to have, say, a base-20 system with a sub-base of 5 in which 13 is written as (2 × 5) + 3, but pure addition (5 + 5 + 3) is always preferred in such circumstances. Similarly, a decimal system that regularly combines multipliers with 20 (as in the French lexical numeral quatre-vingt ‘four-twenties’) is never attested in numerical notation.3 Complying with this rule helps readers of multiplicative numeral-phrases to distinguish operations involving multiplication from those involving addition. I8. All composite multiplicands are strictly multiplicative.

This rule complements the previous one, and is also exceptionless. Various multiplicative-additive systems combine more than one power-sign with a unit-sign multiplier; for instance, the Chinese classical system often uses 嚻嚻 (10,000 × 10,000) to express 100 million, and the traditional Tamil system uses various combinations of 10, 100, and 1000 instead of developing new signs for 10,000 and higher powers. Doing so is the only way to make a multiplicativeadditive system infinitely extendable (see the following section, rule N1). This rule states that such composite multiplicands are always the product of their constituent signs; that is, they are always themselves multiplicative, never additive or subtractive. Without this rule, the Tamil numeral G L L could be misconstrued as 14,000 (7 × (1000 + 1000)) rather than 7,000,000 (7 × (1000 × 1000)). It therefore eliminates one source of potential ambiguity.

Nonuniversals While the search for cross-cultural universals is important, it can (and often does) go too far, postulating that a regularity is a universal when in reality it is not. The 3

In a few medieval French manuscripts, XX (20) is used as a multiplicand in emulation of this lexical formation, however.

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following regularities are nonuniversals whose interest lies not in their regularity but rather in the frequent assumption that they are universal, despite numerous significant exceptions. I once held many of these propositions to be universal, based on my intuition or preliminary reading, but under more careful scrutiny they have proved to be less regular than they first appeared. The existence of these exceptions does not make the generalizations irrelevant, but it does require an accounting of the processes that lead to exceptions. In all other respects, they are ordinary statistical regularities. N1. Some additive numerical notation systems are infinitely extendable without the need to invent new signs.

One of the primary benefits cited for positional notation is that any number, no matter how large, can be written without needing to develop new signs. However, several additive systems that use multiplication, such as the Ethiopic numerals (Chapter 5), the Armenian numerals of Shirakatsi (Chapter 5), the classical Tamil and Malayalam numerals (Chapter 6), the Babylonian common numerals (Chapter 7), the Chinese classical numerals (Chapter 8), and the Mende numerals (Chapter 10), are also infinitely extendable by using composite power-signs as multiplicands (see the previous section, I8). These techniques obviate the need to develop new signs for higher powers of the base, and thus produce an infinitely extendable system. It is sometimes far more concise to use these systems for writing large numbers than it is to use ciphered-positional systems; for instance, the Ethiopic expression for 100,000,000, , requires only two signs where nine Western numerals are needed. It is fair to say that all systems that are purely cumulative-additive or ciphered-additive are finite in scope. N2. Some positional systems are not infinitely extendable.

This is not a universal, even though it borders on being a logical necessity. The Cistercian system (Chapter 10), which is best understood as one based on orientational or rotational position, is not infinitely extendable; once the four positions are occupied (used for writing ones, tens, hundreds, and thousands), the system has reached its end. The same can be said for the Texcocan numerals (Chapter 9), which also use orientational position in the form of vertical and horizontal registers. Positional systems that are linear are all infinitely extendable, as one can keep adding new positions in front of the highest power. In practice, even in positional systems, artificial forms of notation, or expressions that combine lexical and nonlexical numerals, are used for the highest numbers. While my editorial style guide for this book tells me to write 900,000,000,000,000,000,000, I am much more

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likely to write 900 quintillion or, most likely of all, 9.0 × 1020, both for reasons of conciseness and because it is very difficult for humans to readily read such long strings of numbers (see the section, “Subitizing and Chunking”). N3. Some additive systems use a sign for zero.

It is often thought that systems that have a sign to indicate an empty place must, of necessity, be positional. However, in many inscriptions, the quasi-positional Maya zero simply indicates the absence of a numerical coefficient for time periods, even though both the sign and period glyph could have been omitted without ambiguity. Furthermore, the multiplicative-additive Chinese classical and commercial systems (Chapter 8) use signs for zero to indicate blank positions even though they are nonpositional and thus do not need to do so, strictly speaking. The zero in these cases adds redundancy to the system, which may clarify the meaning of a phrase. This function of zero is quite distinct from that in positional systems, where it is used to specify the position of nonzero digits, and thus to identify the power by which they should be multiplied. N4. Some systems are not written or read in a one-directional straight line.

While most numerical notation systems are purely linear, several systems are read in a more convoluted manner. The Chinese commercial system (Chapter 8) and the Texcocan numerals of the Kingsborough Codex (Chapter 9), both of which are multiplicative-additive, place unit-signs in a row beneath the corresponding power-signs, so that the phrase is written and read in a zigzag fashion. The same is true for the multiplicative component of the Greek alphabetic numerals (Chapter 5) above 10,000, except that the unit-signs are placed above the multiplicative sign for 10,000 (M). The most extreme nonlinear systems are the Cistercian numerals (Chapter 10) and the ordinary Texcocan numerals (Chapter 9), both of which use orientational rather than linear position, the former through four rotational orientations, the latter through horizontal and vertical registers. N5. Not all independently invented systems are cumulative-additive.

Based on limited evidence from the circum-Mediterranean region, or even based on nonempirical theoretical reasoning, some authors claim that cumulativeadditive systems are the most ancient or basic form of numerical notation (Hallpike 1979, Damerow 1996, Dehaene 1997). However, the evidence presented in this study refutes that assumption. The Shang numerals (Chapter 8) are multiplicative-additive,

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the Inka khipu numerals (Chapter 10) are cumulative-positional, and the Bambara numerals (Chapter 10) are ciphered-additive (with a cumulative component). If the Brāhmī numerals (Chapter 6) were developed independently of Egyptian or Greek influence, this would also be an independently invented ciphered-additive system. There is no evidence that any of these systems had cumulative-additive antecedents. It is probably not coincidental that several independently invented systems (Egyptian hieroglyphs, proto-cuneiform, Indus) and others that may have been independently invented (Etruscan) are cumulative-additive, because many nonliterate societies use unstructured tally marks, an obvious antecedent to cumulative-additive numerical notation. Yet this tendency does not reach the status of a universal evolutionary law or even a solid generalization.

Cognitive Explanations of Synchronic Regularities Synchronic regularities are features or patterns that hold true for all or most systems, or, in the case of implicational regularities, are true of systems that possess a certain feature. I have attempted to outline commonsensical or functional explanations for these regularities. They reflect common cognitive patterns among the inventors and users of systems, and thus have the potential to tell us a great deal about pan-human numerical cognition. Because some of these regularities have exceptions, we must exercise caution in attributing these regularities to neurological “hard-wiring.” Conversely, because these are not logical requirements – it is possible to conceive of systems that fulfill all the axioms A1 through A4 and yet violate one or more of the regularities just listed – we need to understand why the universe of attested systems is so much smaller than the universe of possible systems. Ideally, we would like to have more information about the decision-making processes and behavior that produced these regularities. However, since individual inventors were very rarely considerate enough to have left detailed records concerning their choice of a specific principle or base, and because very few systems were invented within living memory, indirect techniques are necessary. Moreover, since individuals may have been unaware of the advantages of different means of representation, it may be more productive to analyze representational efficiency without reference to individual decision making. The following cognitive factors are always relevant to some unknown degree in the processes relating to the development and use of numerical notation systems. It is simply impossible that so many specific regularities in the structure of numerical notation systems – ones that are not strictly determined by the logic of those systems – would emerge unless the various representational techniques were constrained by certain cognitive considerations.

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Phrase Ordering One of the central principles of the cognitive sciences (of which cognitive anthropology is an important branch) is that information is often most useful when it is structured. Since numerical notation is a representational system used to help record, remember, and use numerical information, several synchronic regularities relate directly to the ordering of signs within numeral-phrases. Most obvious among these is the Ordering Principle (G5), which orients all systems in a highestto-lowest direction of powers. Yet a number of other rules, including G6 (absence of signs for operations), G4 (absence of division as an operation), I4 (governing the switch from addition to multiplication in hybrid systems), I5 (unit-signs precede power-signs), and I6 (1 is not a power-sign), relate directly to the arrangement of numeral-signs within numeral-phrases. Something important is going on, from a cognitive perspective, to constrain the ordering of signs in numerical notation systems. The logical place to start in the analysis of the Ordering Principle is to look at similar attempts by linguists to explain ordering in lexical numerals (Salzmann 1950; Hurford 1975, 1987; Stampe 1976). Yet, while this research is important as a description of systems, it fails to explain the phenomena it describes, unless the assumption is made that descriptive rules map perfectly onto cognitive processes. While I have called the regularities that I have described “rules,” I do not mean that they are applied by individuals, either consciously or unconsciously. Rather, they are the outcomes of broader cognitive principles that must be examined in order to explain those regularities. In positional systems, signs must be put in their proper order to ensure that a numeral-phrase is interpreted correctly. A positional system that did not do this would be unworkable, since every numeral-phrase would have many equally valid readings. In many other systems, however, no logical requirement prohibits irregular ordering. In cumulative-additive and ciphered-additive systems, for example, the signs could be placed in any order without any ambiguity, since the values of the signs are simply added. In multiplicative-additive systems, as long as each unit-sign is associated with a specific power-sign, the resulting sign pairs could be placed in any order. Yet the Ordering Principle is nearly exceptionless, and such irregular phrases usually occur only where the writer has made an error. In cumulative systems, ordering ensures that identical signs are grouped together. If 327 could be written in Roman numerals as ICIXCVCX, the advantage of cumulation would be greatly reduced by the fact that identical signs are far apart from one another. Even requiring that the signs for each power be grouped together, one could still write CCCVIIXX, XXCCCVII, XXVIICCC, VIICCCXX, or VIIXXCCC instead of CCCXXVII, the only acceptable form.

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The Ordering Principle applies to all systems, and regulates intraexponential ordering in cumulative systems as well as interexponential ordering in both additive and positional systems. Greenberg (1978: 274) suggests that one cognitive principle favoring “larger + smaller” formations in lexical numerals is that, by beginning with the largest power, the first element closely approximates the final result, producing an expectation of the eventual size of the phrase in the listener/hearer. I think that explanations involving this factor of “successive approximation” also apply to numerical notation, and that the desire rapidly to approximate a value is in part responsible for the Ordering Principle. In particular, it explains why numeral-phrases that are in order, but read from the lowest to highest power, never occur. Note, however, that because numerical notation is a visual medium, numeral-phrases can be read in any order, regardless of the manner in which they are written, whereas spoken lexical numerals must be heard sequentially. Yet, while numerical notation systems are nearly always ordered from highest to lowest powers, lexical numeral systems are not. The few exceptions to the Ordering Principle in numerical notation (e.g., Greek, Cyrillic, and Glagolitic alphabetic numerals for 11 through 19) result directly from comparable irregular ordering in the corresponding lexical numerals. We then might expect to find violations of the Ordering Principle wherever the lexical numerals of a system’s users also do so. Since 17 in Latin is septendecim, we should expect the Roman numeral for 17 to be the unattested VIIX. Upon reading the number 16 aloud as sixteen, English speakers rapidly transform the “high-low” order numeral-phrase into a “low-high” lexical numeral. These differences between lexical and graphic numeration suggest that additional factors must be involved. In many circumstances, ordering constraints in numerical notation result from the omnipresent concern with avoiding ambiguity, coupled with the relative inflexibility of numeral-signs. Almost all numerical notation systems are designed to minimize the possibility that a reader will misinterpret signs; that is, each specific numeralphrase has only one numerical meaning (rule G10). Lexical numeral systems use a variety of techniques other than ordering to eliminate ambiguity. For instance, modern German (among many other European languages) uses stem alteration to distinguish 16 (sechzehn = 6 + 10) from 60 (sechzig = 6 × 10), and Classical Sanskrit uses pitch accent alone to distinguish 108 (aštáçatam = 8 + 100) from 800 (aštaçatám = 8 × 100), even though the numerical value and ordering of the two elements in each word are identical. Numerical notation systems lack this flexibility; one of their conveniences is that they use a relatively limited set of discrete signs. Therefore, the strict ordering of numeral-phrases is far more important. Ordering is also essential for unambiguously indicating which arithmetical operations are used to derive the values of numeral-phrases, because numerical notation systems do not explicitly express signs for the operations being used (rule G6).

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The rule that unit-signs precede rather than follow power-signs in multiplicative systems (I5) specifies that the two values are to be multiplied rather than added, allowing the unambiguous reading of numeral-phrases without the need for signs for operations. If it did not apply, and 300 could be written as “100 3,” then the numeral-phrase could be interpreted multiplicatively (as 100 × 3, or 300) or additively (as 100 + 3, or 103). Similarly, although subtractive operations are rare in numerical notation, an order that is the reverse of the ordinary additive phrase indicates that subtraction is to be used. No one of these factors fully explains the prevalence of the Ordering Principle, but in combination they are compelling. I suspect that the avoidance of ambiguity is foremost (being essential to ordering in positional systems and important to many additive ones), with the principle of successive approximation also being very important. This usually leads to considerable conformity in phrase-ordering with lexical numerals, and yet complete correspondence between numerical notation and lexical numerals is rare (cf. rule G8), because lexical numeral expressions are more flexible than their counterparts.

Subitizing and Chunking The Rule of Four (rule I4) is the nearly exceptionless regularity that cumulative systems group signs in three to five (4 ± 1) units, tending significantly toward the lower end of this range. This feature developed independently at least seven times (Egyptian hieroglyphic, Etruscan, proto-cuneiform, Chinese rod-numerals, Maya bar-and-dot, Indus Valley, Iñupiaq) and is very widespread (although not universal) in cumulative systems throughout history. The Rule of Four is explained most parsimoniously by reference to the process of subitizing, or the ability to enumerate rapidly small quantities of discrete objects without having to count them explicitly. Humans have been shown experimentally to be able to enumerate groups of between one and three dots rapidly and with almost no error, and groups of four dots with some error and slightly less quickly, but most individuals cannot count groups of five or more dots at a glance without significant error or considerable delay (Mandler and Shebo 1982). Subitizing probably results from the physiological constraints of the mechanism by which our visual system localizes objects in space (Dehaene 1997: 68). The implications of this principle for numerical notation are obvious: long groups of undifferentiated cumulative signs (e.g., |||||||, ||||||||, |||||||||) will take longer to read and result in more errors than if some technique is used to avoid them. A common way in which systems conform to the Rule of Four is by dividing long sets of signs into smaller groups. Cumulative systems that do not use 5 as a sub-base, including most of the systems of the Hieroglyphic, Levantine, and

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Mesopotamian families, as well as the Indus numerals, divide long sets of identical signs either by placing groups of three to five signs side by side with space between groups, or by using two or more rows of three to five signs each. The use of a quinary sub-base, found in most Italic systems, many Mesoamerican systems, the Chinese rod-numerals, and the Iñupiaq numerals, is another technique allowing systems to conform to the Rule of Four. In these systems, instead of using multiple groupings of three to five signs, the use of a sub-base means that there is never any need to use more than four signs of any one type (e.g., VIIII instead of III III III). Where the system’s primary base is 20 (Mesoamerican and Iñupiaq), a sub-base of 5 ensures both that the sign for 1 need only be repeated up to four times, and that the sign for 5 need only be repeated up to three times, again in conformity with the Rule of Four. Finally, in the base-60 systems of Mesopotamia, the use of a sub-base of 10 ensures that the sign for 10 never needs to be repeated more than five times. Even in systems that lack a sub-base, such as the Levantine systems, the Rule of Four applies. These decimal systems use additional structuring signs – always including 20, and sometimes also 4 and 5 (excepting Phoenician and Aramaic). By using signs for 4 (Kharoṣṭhī and Nabataean) and 5 (Hatran, Palmyrene, Syriac, and Nabataean), the sign for 1 need never be repeated more than three or four times. The additional sign for 20 need only be repeated four times at most in writing any number up to 100. Finally, because the Levantine systems are multiplicative above 100, the same principles allow any number up to 1000 (most of them go no higher) to be written without violating the Rule of Four. Further confirmation of the effects of subitizing is found in diachronic changes that reduce numeral-phrases that had four or five repeated signs to ones that only need three or four signs. The republican Roman numerals (Chapter 4) were purely additive, and required up to four cumulative signs for each power, but in the late republican period, the introduction of subtractive notation for 4 and 9 meant that a writer had the option of using phrasing that required only three signs of each type at most (Sandys 1919). In the Sumerian cuneiform numerals (Chapter 7), the numbers 7, 8, and 9 were written as q, s, and r, respectively. In its descendants, the Assyro-Babylonian common system and the Babylonian positional system, cumulative phrases grouped signs in sets of no more than three signs (7, 8, and 9). Finally, the early Chinese rod-numerals (Chapter 8) expressed 4, 5, and 9 as M, N, and R, which require four or five repetitions of single signs.4 This was necessary, because the system’s structure was partly a consequence of the use of physical rods as computational tools. However, in the Song Dynasty, when written rod-numerals were used extensively in mathematical texts, the older, purely 4

I have listed the zong forms only, but the heng forms are simply ninety-degree rotations of the former, and thus the same principle applies.

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cumulative forms were sometimes replaced with ciphered signs: S, T, and V, so that no phrase required more than three repeated signs. These three independent reductions strongly suggest that five signs is cognitively too many, and that even four signs may be difficult to perceive. In noncumulative systems, signs are not repeated intraexponentially and do not need to be counted, and thus subitizing is mostly irrelevant to them. A separate cognitive principle, called chunking, applies to ciphered and multiplicative systems. As first described by Miller (1956), humans have a limited ability to memorize and recall long sets of bits of information, with the maximum being the “magical number” 7 ± 2. In order to deal with long lists of information, it is much easier to recode input into a series of chunks, each of which contains a small number of bits. In practice, chunks of three or four bits effectively balance the limits of the memory, which restricts the maximum size of chunks, and the desire to minimize the number of chunks necessary. Chunking strongly affects the structure of noncumulative numerical notation systems. For instance, in Western numerals and many other ciphered-positional systems, it is typical to divide long numbers up into sets of three numbers (e.g., 123,456,789). Four-digit numbers are sometimes but not always grouped in this way (1000 vs. 1,000), but it is normal for all five-digit and longer numbers to be subdivided. Doing so not only groups large series of numbers into manageable chunks, it also accords precisely with the millesimal (base-1000) structure of American English lexical numerals (thousand, million, billion, trillion, etc.), and to a lesser degree with the mixed base-1000/base-1,000,000 lexical numerals of British English and many other European languages. Similarly, in the multiplicative-additive Chinese classical system, 10,000 and 100 million are specially structured to divide numeral-phrases into four pairs of unit-signs plus power-signs. Finally, one reason why most hybrid ciphered-additive/multiplicative-additive systems switch to multiplication at 1000 or 10,000 is that doing so groups signs into chunks of no more than three or four signs. For instance, in the Fez numerals (Chapter 5), which use a multiplier at 1000 by placing a horizontal line under a sign or signs, 658,379 is written (reading from right to left) as ipu-h\-n\-x , or (9 + 70 + 300) + (8 + 50 + 600) × 1000, thus dividing the numeral-phrase into two chunks of three bits on the basis of the subscript multiplier used. Chunking in numerical notation in all of these cases is at least partly correlated with the basestructure of the lexical numerals of associated languages, however. Whether there is a connection between subitizing and chunking cannot be resolved here. Subitizing may in fact be a specific example of how chunking affects humans' ability to perceive and encode patterns of discrete visual objects. Chunking has a much broader range of applications, as it is not restricted to visual information and applies to tasks other than simple enumeration. Subitizing, as the

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direct cause of the Rule of Four, has far more significant effects on numerical notation systems than does chunking, in general.

Fingers and Numbers The Rule of Ten (rule G2) is an exceptionless rule that all systems have 10 or a multiple of 10 as their primary base. Approximately 90 percent of all systems have 10 as their primary base, with 20 being the next most frequent at about 7 percent, while three systems (proto-cuneiform, Sumerian, Babylonian positional) have a primary base of 60 and two (Âryabhata’s numerals and the second-stage Pahawh Hmong numerals) a primary base of 100. Why should this be? In the pseudo-Aristotelian Problemata (Book XV.3, 910 b23–911 a4), the question is posed, “Why do all men, whether barbarians or Greeks, count up to ten, and not up to any other number? ... It cannot have been chance; for chance will not account for the same thing being done always: what is always and universally done is not due to chance but to some natural cause” (Heath 1921: 26–27). After discarding several fanciful suggestions, the author finally asks, “Or is it because men were born with ten fingers and so, because they possess the equivalent of pebbles to the number of their own fingers, come to use this number for counting everything else as well?” While the ultimate cause of decimal numeration may be that we have ten fingers, the proximate cause of decimal numerical notation is that the vast majority of the world’s languages have decimal lexical numerals. Where this is not the case, as in Mesoamerica (base-20) and early Mesopotamia (base-60), the numerical notation systems that develop are nondecimal, though they still comply with the Rule of Ten. Wherever numerical notation develops independently, the system that is developed has the same primary base as its inventors’ lexical numerals (rule G9). The existence of vigesimal and sexagesimal numerical notation systems refutes any simple causal relation between fingers and numerical notation. The evidence suggests an overwhelming influence of lexical numerals on the initial choice of base of a numerical notation system, which may occur millennia after the development of a numerical base in a language’s lexical numerals. While lexical numerals are constrained by the fingers, they are not determined by them, as seen in the host of nondecimal (and even non-base-structured) lexical numerals in the world’s languages.5 The development of a sub-base of 5 in at least three independent cases – the Etruscan numerals (Chapter 4), Chinese rod-numerals (Chapter 8), and the 5

This raises the interesting question, beyond the scope of this book, why apparently no speakers of languages whose lexical numerals violate the Rule of Ten have invented numerical notation systems.

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Mesoamerican bar-and-dot numerals (Chapter 9) – may also be related to the fingers. The existence of five handy cumulative-like digits on the end of each hand is too obvious a coincidence to overlook. Moreover, in the Iñupiaq system (Chapter 10), one of the stimuli to which its student inventors had been exposed was Chisanbop finger computation. Again, however, it is worthwhile to look to the lexical numerals of these regions for other possible explanations. The Iñupiaq lexical numerals have a quinary sub-base, which was part of the reasoning used by the students in designing their system. There was probably a quinary component, to the lexical numerals of the inventors of the Mesoamerican bar-and-dot numerals, who were probably Zapotec or Mixe-Zoquean speakers (Colville 1985: 796). In these cases, it is more parsimonious to presume that the lexical sub-base of 5 partly inspired the similar graphic sub-base. Nevertheless, the Etruscan lexical numerals probably had no quinary component, and the Chinese numerals certainly did not. In these cases, the fact that there are five fingers on each hand may have caused the adoption of quinary sub-bases. A further factor is that in a system with a sub-base of five, no one sign will need to be repeated more than four times, thus enabling the system to conform to the Rule of Four (rule I4), as discussed earlier. Because of rule I2, which states that sub-bases must divide evenly into bases, the only reasonable choice for a sub-base for a decimal system is 5. Yet this merely extends the causal chain further: ten fingers lead to decimal lexical numerals, which lead to decimal numerical notation, which then lead – in combination with the Rule of Four – to quinary sub-bases in numerical notation. Whether we are considering the origins of decimal primary bases or quinary sub-bases in numerical notation systems, the direct role of the fingers is not as great as might be thought. The particular effects of various factors, including – but not limited to – lexical numerals, the fingers, and chunking of visual information, are apparently complex, and we will probably never understand the causal relations precisely. A fuller examination of the bases of the lexical numeral systems of ancient civilizations, and the ways in which lexical numeral systems change their bases, is an important topic for future study.

Diachronic Regularities In contrast to synchronic regularities, which constrain the possible structural outcomes of numerical notation systems, diachronic regularities apply not to individual systems, but to temporal trends among systems. They exemplify change rather than stasis in numerical notation systems. To analyze diachronic regularities requires that we shift from the numerical notation system as the unit of analysis and toward the event of change (cf. Mace and Pagel 1994). Two processes of change, which I will simply call transformation and replacement, exhibit

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diachronic regularities. Transformation occurs when an older system gives rise to a structurally distinct descendant. It presumes a direct phylogenetic relationship between the ancestral and descendant systems, but does not tell us what happens to the ancestor after it gives rise to the descendant. We must establish the structures of both the ancestor and its descendant, correctly identify that the latter is derived from the former, and ideally understand the process by which the new system arose from the old. For my purposes, transformation does not include cases where the ancestral and descendant systems have the same basic structure. Replacement occurs when one system becomes extinct and is supplanted by another. It does not matter whether the system being replaced is directly related, indirectly related, or unrelated to its successor. For my purposes, a system that is essentially moribund but is understood by a very small group of specialists is considered to be extinct. Both transformation and replacement are severely constrained in their possible outcomes, and thus, while there are far fewer diachronic regularities than synchronic ones, their effects on the pattern of historical change in numerical notation over time are considerable. Diachronic regularities are nonrandom patterns of cultural change that can be meaningfully called evolutionary. To admit that the cultural evolution of numerical notation is real is not to concede that it is linear, however, nor does it require that such changes be regarded as adaptive.

Transformation of Systems Table 11.1 summarizes all the cases where a system uses a different intraexponential or interexponential principle than its ancestor. These comprise all cases of transformation of principle for which adequate evidence exists, considering only the five basic principles, but omitting other features (base, use of multiplication for higher powers, and other structuring signs). I have omitted cases, such as the Ryukyu sho-chu-ma tallies (Chapter 10), whose ancestor cannot be identified with confidence. In a few instances where a system is a blend of two ancestors, I simplify the relationship between ancestor and descendant, but this does not significantly affect the data because such cases usually have one clearly identifiable main ancestor, with the second system contributing far less. In all, twenty-two systems use a different principle than their ancestor.6 There is considerable but nevertheless constrained variability in the possible transformations of systems. Table 11.2 quantifies the frequencies of these structural transformations, first by graphing the changes according to both intraexponential 6

In the analysis that follows, I will treat the quasi-positional Maya numerals (with zero) as cumulative-positional even though, strictly speaking, this is not the case, as I have shown (Chapter 9).

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Ancestor(s)

Sumerian Maya bar-and-dot Roman Egyptian hieroglyphic Middle Persian Aztec Greek alphabetic Brāhmī Armenian alphabetic Brāhmī Brāhmī Western Western Indian positional Western/Arabic Western/Arabic Chinese classical Bamum Chinese classical Pahawh Hmong Âryabhata Malayalam

Chapter

7 9 4 2 3 9 5 6 5 6 6 10 10 10 10 10 8 10 8 10 6 6

Cumulative-additive Cumulative-additive Cumulative-additive Cumulative-additive Cumulative-additive Cumulative-additive Ciphered-additive Ciphered-additive Ciphered-additive Ciphered-additive Ciphered-additive Ciphered-positional Ciphered-positional Ciphered-positional Ciphered-positional Ciphered-positional Multiplicative-additive Multiplicative-additive Multiplicative-additive Multiplicative-additive Multiplicative-additive Multiplicative-additive

Principle

Table 11.1. Transformation of systems (by case)

Babylonian positional Maya “positional” Roman positional Egyptian hieratic Pahlavi Kingsborough Codex Greek positional Indian positional Armenian (Shirakatsi) Âryabhata Tamil/Malayalam Iñupiaq Cherokee Varang Kshiti Bamum Mende Jurchin Bamum (mfemfe) Chinese positional Hmong (second stage) Katapayadi Malayalam (modern)

Descendant Cumulative-positional Cumulative-positional Cumulative-positional Ciphered-additive Ciphered-additive Multiplicative-additive Ciphered-positional Ciphered-positional Multiplicative-additive Multiplicative-additive Multiplicative-additive Cumulative-positional Ciphered-additive Ciphered-additive Multiplicative-additive Multiplicative-additive Ciphered-additive Ciphered-positional Ciphered-positional Ciphered-positional Ciphered-positional Ciphered-positional

Principle

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Table 11.2. Transformation of systems Descendant’s Structure Ancestor’s Structure Cu-Ad Cu-Po Cu-Ad x 3 Cu-Po 0 x Ci-Ad 0 0 Ci-Po 0 1 Mu-Ad 0 0 Total 0 4 Intraexponential Changes Cu-Ci 2 Ci-Cu 1 Cu-Mu 1 Mu-Cu 0 Ci-Mu 5 Mu-Ci 6

Ci-Ad 2 0 x 2 1 5

Ci-Po Mu-Ad Total 0 1 6 0 0 0 2 3 5 x 2 5 5 x 6 7 6 22 Interexponential Changes Ad-Po 10 Po-Ad 4

and interexponential dimensions, and then by considering each dimension of change separately. Alternatively, these changes can be represented graphically as in Figure 11.1. Vertical arrows indicate intraexponential transformations; horizontal arrows indicate interexponential transformations; and diagonal lines involve both types of change,

Cumulative-Additive

3

Cumulative-Positional

2

1 2

1

Ciphered-Additive

2

1

2

3

Ciphered-Positional

5 Multiplicative-Additive

Figure 11.1. Transformation of systems (graphic representation)

384

Numerical Notation

with the numbers indicating the frequency of each change. Dotted lines indicate changes that are only attested in modern contexts (1800–present). Three important regularities can be extrapolated from these data: T1. Additive systems do not develop from positional ancestors.

While this is not a universal, it greatly constrains the evolutionary history of numeration. In ten cases, additive systems gave rise to positional ones, while in four cases the reverse occurred. This finding takes on greater importance when we examine the four exceptions to this rule: the ciphered-additive Cherokee and Varang Kshiti systems and the multiplicative-additive Bamum and Mende systems (all discussed in Chapter 10).7 These systems all originated in the colonial period, and their inventors had limited knowledge of the ciphered-positional antecedents of their systems (the Western, Arabic, and Indian systems). None of these systems has been notably successful: one (Cherokee) was rejected at the time of its invention, and another (Bamum) was transformed by its inventor into a ciphered-positional system within twenty years of its invention. When dealing with pre-modern numerical notation systems, this rule is truly universal; no attested additive system prior to the nineteenth century had a positional ancestor. T2. Cumulative systems do not develop from noncumulative ancestors.

This rule has one exception, the development of the Iñupiaq cumulativepositional numerals (Chapter 10) based on the Western system. This system originated very recently and in an educational context, and the numerals’ long-term survival is in doubt. No cumulative-additive system has emerged from any system other than another cumulative-additive one (again, the Ryukyu numerals may be an exception). Cumulative-additive systems gave rise to ciphered-additive ones twice (Egyptian hieroglyphic Æ hieratic; Middle Persian Æ Pahlavi) and once to a multiplicative-additive system (Aztec Æ Kingsborough Codex). T3. The only transformation that involves both intra- and interexponential change is the invention of multiplicative-additive systems from ciphered-positional ones, and vice versa.

This is an exceptionless rule. Of the nine unattested transformations in Table 11.2 (cells with a 0 value), six involve both an intraexponential and an interexponential 7

If the Ryukyu cumulative-additive numerals were developed on the basis of the Chinese rod-numerals (Chapter 8), this would constitute a fifth exception.

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change.8 These changes are presumably too radical alterations of principle to occur. Yet the rise of ciphered-positional systems based on multiplicative-additive antecedents occurs five times (albeit sometimes in conjunction with some externally introduced knowledge of positionality). While this transformation involves both intra- and interexponential change, it is nonetheless relatively simple, involving only the elimination of power-signs and the addition of a sign for zero. In two other cases (the Bamum and Mende numerals, already mentioned), the reverse change occurred, with ciphered-positional systems giving rise to multiplicativeadditive descendants. These three regularities tend over time to increase the frequency of noncumulative systems over cumulative ones, and of positional over additive systems. The reverse changes occur only in modern contexts, and the resulting systems have not been extremely successful. Among noncumulative systems, there is no trend favoring ciphered over multiplicative systems, or vice versa. Of the twenty possible transformations, only eleven are attested (only eight in pre-modern contexts), three of which (encompassing eight of the twenty-two examples of change in Table 11.2) result in ciphered-positional systems. The nine unattested transformations either result in cumulative-additive systems or involve both intra- and interexponential change. The trend toward ciphered-positional notation is partly explained by this transformational pattern. A second type of transformation does not involve changes to either the intraexponential or the interexponential principle, but only to the use of multiplication in higher powers of a system’s base. While only a limited number of systems use such a feature, it produces an important diachronic regularity. T4. When one system that uses the multiplicative principle gives rise to another, the power above which the descendant is multiplicative is not higher than that of the antecedent.

Figure 11.2 indicates all the systems that use the multiplicative principle for some powers and that have a multiplicative ancestor. Many other hybrid multiplicative systems (e.g., Cherokee, South Arabian, Roman multiplicative) have nonmultiplicative ancestors, but these are not relevant to this rule. The number in each box indicates the power(s) at which the multiplicative principle is first used (with ‘1’ indicating fully multiplicative-additive systems). Ancestral systems normally begin to use multiplication at a point equal to or higher than their descendants. Solid lines indicate cases that obey this rule, while dotted lines indicate exceptions. 8

The other three are all changes in principle resulting in cumulative-additive systems, which as I have already stated, is never known to have occurred.

386

Coptic 1000

Zimam 1000/10,000

Arabic abjad 1000

Greek Alphabetic 1000/10,000

Figure 11.2. Changes in hybrid multiplicative power.

Mari 1000

Jurchin 100

Hebrew 1000

Syriac 1000/10,000

Demotic 10,000

Aryabhata 1

AssyroBabylonian 100

Tamil/ Malayalam 1

Brahmi 100/1000

Chinese 1

Sinhalese 100

Meroitic 1000

Hieratic 100,000

Glagolitic Cyrillic Fez 1000

Ethiopic 100

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Both the Syriac and zimām cases are only partial exceptions, because they use multiplication at two distinct levels or stages; the lower of the two (1000) is shared in common with their ancestors, but they use an additional multiplicative technique for powers above 10,000. Notably, the two remaining exceptions, the Jurchin (Chapter 8) and Mari (Chapter 7) systems, are the only two cases outside the super-group encompassing the Hieroglyphic, Alphabetic, and South Asian phylogenies. This suggests that this rule may apply only within this larger group. I have been unable to find any regularities concerning changes in base structure, sub-base, or other features of systems. Changes in base are much less frequent than changes in principle. This is probably due to the overwhelming prevalence of decimal lexical numerals in languages worldwide; there is little reason to adopt a new base when developing a numerical notation system unless one’s lexical numerals differ in base from that of the ancestral numerical notation system.

Replacement of Systems The second diachronic process concerns the extinction of systems and their replacement by other systems, regardless of any phylogenetic relation between the two. I use the term “replacement” to refer only to systems that are supplanted more or less completely by other systems, even though doing so obscures cases where a system is replaced for many functions while continuing to be used regularly for others. The replacement of systems is far more frequent than the transformation of systems, because one system may replace many systems, but rarely does one system give rise to multiple systems that use a different principle (cf. Table 11.1). Moreover, a system may be replaced by one that has the same basic structure. Table 11.3 compares the structures of extinct systems with those of the systems that replace them (including cases where the two systems have the same structure). Two clear trends emerge from the examination of patterns of replacement: R1. Positional systems are not replaced by additive systems.

There are only two partial exceptions to this rule, both of which involve the replacement of cumulative-positional systems. The quasi-positional system used in the Mesopotamian city-state of Mari in the eighteenth century bc (Chapter 7), which was occasionally used in place of the Assyro-Babylonian system, was eventually replaced by that system after the Babylonian conquest of Mari. Positional numeral-phrases in the Mari system were used only rarely and only in the hundreds position, however; this system is actually best regarded as a shortlived experimental combination of the Babylonian positional (mathematical) and additive (scribal) systems. The second exception involves the replacement of the

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Table 11.3. Replacement of systems Extinct System Cu-Ad Cu-Po Ci-Ad Ci-Po Mu-Ad Total

Cu Ci Mu

Replaced by Cu-Ad 11 1 1 0 0 13

Cu-Po 0 0 0 0 0 0

Ci-Ad 15 1 3 0 1 20

Intraexponential Replacement Cu Ci Mu 12 29 1 1 34 1 0 7 2

Ci-Po 9 4 19 12 6 50

Mu-Ad 1 0 1 0 2 4

Total 36 6 24 12 9 87

Interexponential Replacement Ad Po Ad 35 34 Po 2 16

Babylonian cumulative-positional numerals used in mathematics and astronomy by the Greek alphabetic numerals following Alexander the Great’s conquest of Mesopotamia and the gradual domination of Greek over Mesopotamian learning in the exact sciences. Again, this is only a partial exception, because the Greeks borrowed and adopted Babylonian sexagesimal positional numerals in their own mathematics and astronomy, producing the sexagesimal ciphered-positional fractions (Chapter 5). All other positional systems were replaced by other positional systems (in fact, by ciphered-positional systems) or survive to the present day. R2. Noncumulative systems are not replaced by cumulative systems.

There is one exception to this rule. The Gothic numerals of Wulfila’s script (Chapter 5) were ciphered-additive and were based on the Greek alphabetic numerals. Yet, because they were used primarily in Western and Central Europe, they were replaced by the cumulative-additive Roman numerals. Gothic numerals were used in only a limited number of texts, and so, while other alphabetic systems such as the Greek alphabetic numerals survived and thrived in competition with Roman numerals, the Gothic numerals were overwhelmed. Despite the importance of Roman numerals as an instrument of Roman imperialism, they never totally displaced any of the ciphered-additive systems of Eastern Europe or the Middle East, although they replaced other cumulative-additive systems, such as the Etruscan numerals (Chapter 4) and various Levantine systems. Despite the historical importance of cumulative-positional systems, such as the Babylonian

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positional numerals and the Chinese rod-numerals, no cumulative-positional system has ever totally replaced any other system. The comparison of patterns of replacement with patterns of transformation is instructive, as the effects of the two processes overlap. Positional systems are rarely ancestral to additive systems (except in modern colonial contexts) and tend to replace additive systems over time (but not vice versa). The obvious effect is that positional systems become more frequent over time, while additive ones become less frequent. A similar effect is seen in the intraexponential dimension, where noncumulative systems are rarely ancestral to cumulative ones, and tend to replace cumulative systems over time, gradually decreasing the frequency of cumulative systems. Yet there are also considerable differences between patterns of invention and patterns of replacement. While the intraexponential transformation of cumulative systems into noncumulative ones is comparatively rare, the intraexponential replacement of cumulative systems by ciphered or multiplicative ones is very frequent. To understand why systems transform and are replaced in these patterned yet complex ways requires attention to the cognitive effects on individuals of changing the manner in which numbers are represented.

Cognitive Explanations of Diachronic Regularities In explaining synchronic regularities, it was necessary only to show that the presence of a feature was correlated with some cognitive factor that, if the feature were absent, would be inconvenient to a system’s users. Because these regularities were universal or near-universal, these explanations largely involved considerations of hypothetical exceptions. Their universality also means that social context is less relevant to explanations of their existence. In explaining diachronic regularities, we are considering patterned variability among systems, so we may also ask whether a descendant system is more or less convenient than its ancestor, or whether a successor is more or less convenient than the system it supersedes, with respect to a number of cognitive criteria. To explain diachronic patterns, I compare the observed trends with the cognitive advantages or disadvantages of particular features of systems. Where trends correspond to increased efficiency in some respect, theories can be developed connecting individual decision making to the observed patterns. Evaluations of efficiency can be derived from abstract principles of economy in some cases (e.g., a short numeral-phrase is better than a long one) or from principles derived from cognitive science. As with explanations of synchronic regularities, this is an indirect means of reconstructing cognitive processes, one made necessary by the limitations of the data. These explanations will be more complex than those of synchronic regularities simply because they must explain change rather than

390

Numerical Notation

stasis. Cognitive explanations for diachronic regularities explain not only why a feature came into existence, but also how one system compares to another in some respect. This also raises the possibility that advantages as well as disadvantages may be involved in the choice between any particular pair of systems.

Conciseness The conciseness of a numeral-phrase is simply the number of signs needed to write that particular number. It is thus a property of a numeral-phrase, not of a numerical notation system. All other things being equal, a system that requires many signs to write numbers is more cumbersome than one that requires few signs. Because many systems are infinitely extendable, it is impossible to state exactly the average number of signs needed to express numbers, and in any case, this would not necessarily be useful, since very large numbers are quite rare. Yet, because a system that regularly requires long numeral-phrases is going to be quite cumbersome to use, we wish to evaluate in general whether a system’s numeral-phrases are long or short. I consider the average length of numeral-phrases for all numbers from 1 to 999 to be a good rough measure of a system’s conciseness. Table 11.4 shows the conciseness of each principle (presuming a base-10 system with no sub-base for each case) for a variety of numbers. In general, ciphered systems are the most concise, requiring only one sign per power. For any natural number, no system is ever more concise than a purely ciphered-additive system. While ciphered-positional systems are usually more concise than nonciphered ones, for round numbers they are sometimes less concise because they require zero-signs in the empty positions (e.g., Roman numeral C = 100). Nevertheless, cumulative systems are almost always less concise than their ciphered and multiplicative counterparts, even for small and/or round numbers. Multiplicative-additive systems are slightly less concise than ciphered systems, because they often require two signs (a unit-sign and a power-sign) where the latter need only one. Yet, because they are additive, and thus do not require a zero-sign, they are more concise than ciphered-positional systems for round and nearly round numbers. Additive systems are only slightly more concise than positional systems that use the same intraexponential principle; the difference in their conciseness is equal to the number of empty positions present, for which positional systems require zero-signs. The comparative effect of this difference is far less than that between cumulative and noncumulative systems. The use of bases higher than 10 has variable effects on a system’s conciseness, depending on which principle it uses. Cumulative systems become far less concise through the use of higher bases; in a pure base-20 cumulative-additive system, such as the Aztec numerals, each sign may be repeated up to nineteen times, so

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Table 11.4. Conciseness of systems Number of Signs Required (base-10 system, no sub-base, no hybrid multiplication)

Number 6 27 70 100 400 649 870 2003 4268 9080 Average (1-999)

Cipheredadditive

Cipheredpositional

(Georgian) 1 2 1 1 1 3 2 2 4 2 2.70

(Western) 1 2 2 3 3 3 3 4 4 4 2.89

Multiplicative- Cumulative- Cumulativeadditive additive positional (Chinese (Egyptian Classical) Hieroglyphic) (Inka) 1 6 6 3 9 9 2 7 8 1 1 3 2 4 6 5 19 19 4 15 16 3 5 7 7 20 20 4 17 19 4.49 13.59 13.78

that 399 requires thirty-eight signs instead of only twenty in a base-10 system (average 22.82 signs/numeral-phrase from 1 to 999). Yet, for a noncumulative system, using a higher base makes numeral-phrases slightly more concise! In a cipheredpositional system like the Oberi Okaime numerals (Chapter 10), all numbers from 1 to 19 require only a single sign each, from 20 to 399 only two signs, and from 400 to 8000 only three signs (average 2.58 signs/numeral-phrase from 1 to 999). This advantage in conciseness, however, is offset by an increase in the sign-counts of these systems (see the following discussion). The use of sub-bases in cumulative systems improves their conciseness. By using a sub-base of 5 in a decimal cumulative-additive system, a number such as 870 requires only seven signs instead of fifteen (DCCCLXX vs. CCCCCCCCXXXXXXX), and for all numbers less than 1000 the average conciseness is reduced from 13.59 signs per numeral-phrase to 7.45 signs per numeral-phrase. While this reduces the disadvantage of cumulative systems as compared to noncumulative ones, it never eliminates it. A cumulative system with a sub-base has additional round numbers, which are often expressed more concisely; whereas only fourteen numbers less than 1000 are expressed as or more concisely in a cumulative-additive than in a ciphered-positional system, the introduction of a sub-base of 5 into the cumulativeadditive system raises that number to fifty-four. The absence of sub-bases in most

392

Numerical Notation

noncumulative systems (rule I2), and their relative frequency in cumulative ones, is due to the enormous advantage in conciseness that a sub-base provides to the latter but not to the former. The use of subtractive notation is rare in numerical notation systems, being found only in Roman numerals and some of the Mesopotamian cuneiform systems. It does carry a considerable advantage in conciseness, since 1999 in additive Roman numerals is MDCCCCLXXXXVIIII but MCMXCIX (or even MIM) when subtractive notation is used. However, because subtractive numeral-phrases do not group similar signs together, parsing and reading them may be more difficult. The relative paucity of subtractive numeral-phrases in Roman numerals in the classical period, coupled with the fact that subtractive notation is cross-culturally rare, suggests that its advantages were not perceived as being great. The use of a hybrid multiplicative component for higher powers of some additive systems has a slightly negative effect on the conciseness of systems that possess this feature. For example, a ciphered-additive system that has no multiplicative component, such as the Georgian alphabetic numerals (Chapter 5), expresses 4000 with one sign, while a similar system with a multiplicative component for higher powers, such as Sinhalese (Chapter 6), requires two signs. Similarly, where a purely cumulative-additive system, such as the Greek acrophonic numerals, requires four signs to write 4000 (XXXX), a hybrid multiplicative system, like the Phoenician system, requires five (μ\a\aaa). Using multiplicative expressions for higher powers obviates the need to develop distinct signs for each multiple of each higher power, but such expressions will contain both unit-signs and power-signs and thus be somewhat longer. The diachronic trend toward noncumulative systems over cumulative ones strongly accords with their far greater conciseness. The trend in favor of positional systems over additive ones, by contrast, does not have a basis in conciseness, since additive systems are slightly more concise than their positional counterparts.

Sign-count A system’s sign-count is the total number of signs its users must know in order to read and write numbers. A system with a smaller sign-count is generally easier to learn and use than one with a larger sign-count because of the decreased mnemonic effort involved. For systems such as the Western numerals, the sign-count is ten. Yet this seemingly simple definition produces many complexities when attempting to enumerate how many distinct signs a user of a system requires. For systems such as the Phoenician numerals (Chapter 3) and the republican Roman numerals (Chapter 4), there are multiple signs for many numbers, some of which represent regional or diachronic variability that would be irrelevant to individual

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users, while others may be multiple signs that every user needed to learn. Certain signs (normally for very large numbers) may have developed late in a system’s history or have been used by only a few writers. In other cases, in which a sign is composed of two largely undisguised other signs, a rather arbitrary decision must be made whether to count it as a separate sign. Should the Sumerian sign for 36,000, k, be counted as a sign separately from its constituent parts, j (3600) and g (10)? Finally, and perhaps most importantly, the issue of sign-count cannot be considered properly without also considering the numerical limit of a set of signs; the Indus numerals may have only two signs, but these can express only numbers from 1 to 99, whereas the Western numerals have ten signs but can express any number. Keeping these reservations in mind, it is nevertheless possible and useful to compare the sign-counts of different types of systems. Cumulative systems, which rely on the repetition of a small number of identical signs, have much lower sign-counts than ciphered ones, which use a wider variety of unrepeated signs. Positional systems, which do not require additional signs to be invented for successive powers, are more economical in sign-count than additive ones, although a sign for zero is highly useful. Cumulative-positional systems, which combine both of these advantages, have extremely small sign-counts (one to three distinct signs). The sign-counts of cumulative-additive systems are very low, but depend on their extendability; a decimal cumulative-additive system without a sub-base requires only one sign for each power of 10 that can be expressed (usually four to seven signs, with more if a sub-base is used). Multiplicative-additive systems have slightly larger sign-counts than ciphered-positional ones because, while cipheredpositional systems need only signs from 1 up to the system’s base, and 0, multiplicative-additive ones need signs for each power of the base. The sign-count for a ciphered-positional system is normally equal to its base, while that of a multiplicative-additive system equals its base plus one sign per power. Ciphered-additive systems, which require one sign for each multiple of each power of the base, have extremely large sign-counts, normally twenty or more, although using script-signs as numeral-signs can mitigate the inconvenience. From lowest to highest, the signcounts of systems of different principles are: cumulative-positional Æ cumulativeadditive Æ ciphered-positional Æ multiplicative-additive Æ ciphered-additive. Table 11.5 lists some systems whose sign-counts are unambiguous, thus allowing them to be compared. Base structure has variable effects on sign-count. For cumulative systems, a system’s primary base does not affect its sign-count, since a single sign per power will suffice, as long as it can be repeated as often as is necessary. For noncumulative systems, however, using a higher base than 10 is extremely detrimental, requiring many more signs to be developed. The only noncumulative systems with bases

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Table 11.5. Sign-counts (selected systems) Structure

Sign Inventory

Signcount

Babylonian positional 7

Cumulativepositional

f g

2

Egyptian hieroglyphic

2

Cumulativeadditive

q r s t u v w

7

Western

6

Cipheredpositional

1234567890

10

Tamil

6

Multiplicativeadditive

A\B\C\D\E\F\G\H\I J\K\L

12

5

Cipheredadditive

a b c d e f g h i j k l m n o p q r s t u v w x y z {

27

System

Gothic

Chapter

higher than 10 are the Maya head-glyphs (Chapter 9; base-20, but uses a subbase of 10 to decrease mnemonic effort), the Oberi Okaime numerals (Chapter 10; base-20, used only briefly and by few individuals), and Âryabhata’s numerals (Chapter 10; base-100, uses script-signs to decrease effort and has a sub-base of 10, and was used by only a single school of mathematical thought). The use of a sub-base also has variable effects on a system’s sign-count, depending on the system’s structure. For a cumulative system, introducing a sub-base increases its sign-count slightly. A cumulative-positional system requires only one extra sign (for the sub-base), while a cumulative-additive system requires one extra sign per power (e.g., the Roman numerals V, L, and D). In either case, this increase in sign-count is offset by an enormous saving in conciseness; it is safe to say that a base-60 cumulative-additive system, such as the Sumerian cuneiform numerals (Chapter 7), could not exist without a sub-base. Yet, in the one ciphered system that has a sub-base (see rule I3), the Maya head-variant glyphs (Chapter 9), introducing a sub-base actually decreases the sign-count; instead of signs for 0 through 19, it requires only fourteen signs (for 0 through 13) with 14 through 19 (and sometimes also 13) written with glyphs combining 10 with 4 through 9. Finally, the use of multiplication for higher powers (hybrid systems) greatly reduces the sign-count of ciphered systems, but has minimal benefit in cumulative systems. A single multiplicative power-sign can be combined with a set of existing ciphered unit-signs (1 through 9 in a decimal system) to avoid needing

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new signs for each multiple of each power. Most ciphered-additive systems of the Hieroglyphic (Chapter 2), Alphabetic (Chapter 5), and South Asian (Chapter 6) families, as well as systems such as the Jurchin (Chapter 8) and Cherokee (Chapter 10), use hybrid multiplication to express large numbers. By contrast, the use of hybrid multiplication in cumulative systems, such as the Levantine (Chapter 3) and many of the later Mesopotamian systems (Chapter 7), has a minimal effect on sign-count. Where each successive power has its own power-sign, and power-signs are combined only with signs from 1 up to the base (e.g., Aramaic F = 100, G = 1000, ± = 10,000, each of which is combined with up to nine cumulative strokes), there is no economy of sign-count; all that multiplication does is avoid repeating signs other than the unit strokes (e.g., Faaaa instead of FFFF for 400). Rarely, in systems such as the Assyro-Babylonian common system, whole cumulative-additive numeral-phrases, including both unit-signs and signs for higher powers, combine multiplicatively with large power-signs, so that 10,000 was written as 10 (g) times 1000 (gi), 100,000 as 100 (h) times 1000 (gi), and so on. Doing so eliminates the need for new signs for higher powers of 10. In such cases, there is a moderate savings in sign-count. Other than the reversal of the positions of ciphered-positional and multiplicativeadditive systems, there is an inverse correlation between a system’s conciseness and its sign-count. Thus, the observed diachronic trend from cumulative to noncumulative systems is unexpected if sign-count is an overwhelmingly important factor. However, the trend toward positional over additive systems may have such a basis, although ciphered-positional systems have larger sign-counts than most cumulative-additive systems and only slightly smaller ones than multiplicativeadditive systems.

Extendability A system’s extendability is measured by the largest number that can be written with it. Unlike conciseness and sign-count, both of which are relevant to the writing even of low numbers, infinite extendability, which is characteristic of most positional systems, becomes particularly important only when there is a strong societal inclination to express very large numbers (especially where numerals are used commonly for mathematics). However, any increase in extendability – even the addition of a new power-sign to an additive system – can be considered an increase in the capabilities of a system to represent numbers, regardless of the specific functions for which such developments are used. While some multiplicative-additive systems are infinitely extendable (rule N1), and some orientational positional systems are not (rule N2), the general rule that positional systems are infinite in scope while additive ones are not is largely

396

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correct. Even so, some additive systems are much more easily extended than others. Additive systems that use multiplication, whether throughout the system (fully multiplicative-additive systems) or only for larger powers (hybrids), are generally able to express larger numbers than non-multiplicative ones. This is because in many such systems, power-signs may be multiplied by entire numeral-phrases rather than single signs, and/or because multiple power-signs placed side by side can be used to indicate repeated multiplications. Most pure multiplicativeadditive systems can express numbers as large as 100,000, and many have limits as high as 10 million. In the abstract, there is no reason why ciphered systems should be more extendable than cumulative ones, but in actuality, they are slightly more extendable, usually having limits of 10,000 or higher whereas many cumulativeadditive systems are used only for numbers up to 1000 or 10,000 (such as the modern Roman numerals). This may be related to conciseness – beyond 1000 many cumulative-additive numeral-phrases become so long that the best alternative is to transform the system’s structure, or to adopt a new system entirely. The use of a particular base constrains but does not dictate the numerical limit of a system, because one must also take into account how many powers of the base the system can represent. All other things being equal in terms of sign-count and conciseness, a system with a base higher than 10 can represent larger numbers than a base-10 system (e.g., 203 = 8000, so the Aztec numerals can represent any quantity up to 160,000 using only four different symbols, whereas a similar base10 system could represent only numbers below 10,000). The use of sub-bases has no effect on extendability. The greater extendability of positional systems correlates with the trend over time toward positionality over addition. It should be noted, however, that practically any numerical notation system can be extended without great difficulty should the need arise, either by developing new numeral-signs or by introducing a structural change such as hybrid multiplication. Where such changes have not been made, there was no overwhelming need for them. A great preponderance of the numbers used in both pre-modern and modern contexts are below 1000, and nearly any numerical notation system can deal with such small quantities. Infinite extendability is really relevant only in mathematical contexts.

Effect of Cognitive Factors Table 11.6 summarizes the conciseness, sign-count, and extendability of systems using each of the five basic combinations of principle (presuming all other factors to be identical). Each principle is ranked on the three criteria I have discussed (1 being best, 5 being worst).

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Table 11.6. Ranking of systems by cognitive factors

Ciphered-additive

Ciphered-positional Multiplicative-additive Cumulative-additive Cumulative-positional

Conciseness

Sign-count

Extendability

1

5

2.70

18–30

4 Normally 10,000 – 1 million

2

3

1

2.89

10–11

Normally infinite

3

4

3

4.49

12–14

Normally 100,000 +

4

2

5

13.59

4–7

Normally 1000 – 100,000

5

1

1

13.78

1–3

Normally infinite

The conciseness and sign-count of a system are inversely correlated, except that ciphered-positional systems have a slightly smaller sign-count and are also slightly more concise than multiplicative-additive systems. This correlation is not a coincidence, because one of the most effective ways to increase conciseness is to reduce many signs (one-to-one correspondence) to one, which must involve inventing new signs. Yet there is no correlation between conciseness and extendability or between sign-count and extendability. Very concise systems may be highly extendable (ciphered-positional) or limited (ciphered-additive), just as systems with small sign-counts may be highly (cumulative-positional) or less (cumulative-additive) extendable. The reason for this is that conciseness, sign-count, and extendability are properties of dimensions of systems (intraexponential or interexponential), not of the systems themselves: ciphered systems are the most concise, cumulative systems have the smallest sign-counts, and positional systems are the most extendable. Because each system is structured both intra- and interexponentially, any system will be less than optimal in at least one of these dimensions. The Western numerals are less concise than the ciphered-additive Greek alphabetic numerals and has more signs than the cumulative-additive Roman numerals, both of which they replaced. It is a good compromise between maximum conciseness and minimum sign-count, but it is maximally efficient in neither respect. A major problem arises when we attempt to extend the analysis of cognitivestructural motivations of specific instances of invention or replacement to produce general rules. The diachronic trend toward positionality over addition, and toward

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ciphering and multiplication over cumulation, suggests that addition and cumulation should be seen as negative or inferior principles. To do so neglects an important consideration, which is that cumulative systems are more common than ciphered ones and additive systems more common than positional ones, and that for many millennia, cumulative-additive systems were the most common type. Moreover, if we explain the trend toward ciphering as a desire to maximize conciseness, we must deal with the fact that the trend toward positionality is in opposition to this desire, since positional systems are less concise than their additive counterparts. To explain long-term diachronic trends, we must acknowledge that the weighting of the cognitive advantages and disadvantages of different principles was not equal in all time periods or in all social contexts. Where there are diachronic trends – such as that favoring noncumulative systems over cumulative ones – they result from changing evaluations of the importance of various merits and defects of different principles. If we want to understand why those evaluations might have changed – for instance, why ciphering (and thus conciseness) came over time on a worldwide basis to be preferred over the small sign-counts of cumulative systems – we must understand the historical conditions under which such evaluations were made. The question of diachronic trends becomes even trickier when we examine the effects of features other than intraexponential and interexponential principle. Table 11.7 summarizes these effects. The presence of any of these features may mitigate any negative effects or reduce the advantages of the use of a principle. Moreover, their effects on conciseness and sign-count vary depending on a system’s intraexponential principle, adding an additional layer of complexity to the analysis of its merits and disadvantages. Although we can speak of the cognitive merits and disadvantages of a system’s base and/or sub-base, we would not expect diachronic trends to exist for this feature, because these are often a consequence of the lexical numerals of its users’ language(s) rather than the result of a conscious decision to alter a system. Similarly, hybrid multiplication is a flexible way of achieving greater extendability at little extra cost in sign-count or conciseness. Yet the only diachronic regularity concerning hybrid multiplication, rule T4, concerns the power above which multiplication is used, not its simple presence or absence. While the efficiency of systems is relevant to the diachronic patterns I have described, potential improvements in a system are not necessarily perceived automatically and regarded as relevant by its users. There are levels of difference too small to be relevant, and perhaps too small to be perceived. For instance, the minimal difference in conciseness between ciphered-additive and ciphered-positional systems, while recognizable, does not appear to exceed a minimum threshold level (above which, presumably, the additive would be preferred over the positional),

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Table 11.7. Overall effects of other features Conciseness

Sign-count

Extendability

Cumulative

Other

Cumulative

Other

Base > 10

Much less

Much more

No effect

Much higher

Sub-base

More

N/A

Higher

N/A

No effect

Slightly less

Usually none

Much lower

Higher

Hybrid multi- Slightly less plication

Higher

while more salient features such as the much smaller sign-count of cipheredpositional systems are probably quite relevant. Any change in a system that would result in ambiguous or poorly ordered numeral-phrases will not register as useful, even if such a change would bear some other benefit. Where significant social factors, such as political hegemony, are involved in the transformation and replacement of systems, otherwise important considerations of efficiency may be irrelevant to users. There is no single goal to be attained or variable to be maximized in numerical notation. Every principle has advantages and disadvantages, the choice of which is governed at least in part by considerations of those qualities. Explaining the diachronic trends observed from the data requires that we ask why certain qualities would be preferred over others. Because four of the five basic principles – the exception, ironically, being the “ideal” ciphered-positional system – have been developed independently multiple times, we may presume that these systems are perceived as being advantageous, and we can identify the circumstances that can lead to their adoption. Yet the changing functions for which systems are used will be extremely important in determining which features of systems will be valued most highly. Thus, any solely cognitive explanation of diachronic regularities will be incomplete.

Summary Because both synchronic and diachronic regularities relate to structural features of systems, cognitive factors must be involved in explaining them. Yet, because the unit of analysis for the two types of regularities is different, the types of explanations involved are quite distinct. A case can be made for the parsimonious explanation of many synchronic regularities using cognitive factors alone, since these regularities apply regardless of social context or the specific functions for which systems are used. Yet, even where this is so, we must be careful not to assume that

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synchronic regularities are consciously imposed rules for the construction of systems; rather, they are outcomes of cognitive processes that arise in specific social contexts, or else relate to lexical numeration. Furthermore, these patterns do not explain the variability among systems, which is still considerable despite the existence of many constraints. The existence of diachronic regularities is one way to begin to explain this variability. To ignore structural and cognitive features entirely would be ridiculous, given the trends toward particular sorts of systems (specifically, intraexponential ciphering and interexponential positionality). However, there is no perfect numerical notation system; all systems have advantages and disadvantages. To assume that every feature of a system is relevant to its retention or replacement, or that any difference in structure must have been perceived as important, is erroneous. The analysis of the structure of numerical notation systems is insufficient as a full explanation of these patterns (particularly evolutionary patterns of change), because social context plays a role in episodes in the history of numerical notation. In order to explain diachronic regularities fully, we must therefore turn to the question of how systems are used and how their functions change, which can only be answered by carefully comparing specific situations in the history of numerical notation.

chapter 12

Social and Historical Analysis

The primary function of numerical notation is to communicate numerical values. One cannot even lie effectively about how many enemies were killed in battle if the numerals being used are incomprehensible to the intended audience. Any attempt to explain the history of numerals without reference to the cognitive features underlying their structure is doomed to failure. Nevertheless, considerations of efficiency are not the sole or even the primary factor in the cultural evolution of numerical notation. While synchronic regularities may be explainable without reference to social context, diachronic regularities are not. Every cognitive advantage associated with a system is associated with disadvantages. The role of various social factors in explaining the history and development of numerical notation systems differs from case to case, depending on historical context, but they are always there. We cannot explain the replacement of Maya numerals by Western ones without consideration of the enormous social, political, and technological upheavals that were associated with the Spanish conquest of Mesoamerica. Numerical notation systems never exist as objects in isolation; their utility is not merely a function of their structure. By exploring the social contexts in which the transformation and replacement of numerical notation systems occur, it will be possible to evaluate the impact of social factors relative to purely cognitive and structural ones. I have identified seventeen factors that influenced the changes in numerical notation systems examined throughout this study, any of which may apply to a particular historical event. I list them in the following section, roughly in the order 401

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of their importance. They complement the transformations and replacements of systems that I discussed in Chapter 11, and also help to explain the cultural diffusion of systems without structural change and regardless of whether any existing systems were replaced. While some of these factors are more important and occur more frequently than others, it is not useful or possible to quantify their various effects on the history of numerical notation systematically, as I did with the differential effects of structural patterns. There are considerable complexities in the history of numerical notation that cannot simply be reduced to one or a few prime movers, and some of these factors have effects that are directly opposite to others. There is no contradiction implied in this; rather, it is to be expected, because multiple goals may be pursued by users, which may need to be reconciled in any given social situation.

Social Dimensions of Numerical Notation 1. A system may be transformed or replaced because its structural features are disadvantageous for new functions for which numerical notation is required.

Although numerical notation systems possess efficiency-related characteristics such as conciseness, sign-count, and extendability, the evaluation of these characteristics requires individual users to consider which of them are most relevant to the functions for which a system is used. The analysis of utility must therefore always be linked to the analysis of social context. No system is absolutely “efficient” in the way it might be absolutely positional or absolutely decimal. When a system changes or is replaced, changes in the needs of its users with respect to the writing of numbers are often the primary stimulus. For example, the development of the Babylonian positional numerals (Chapter 7) in the late Ur III period related to a renewed focus on mathematical and astronomical problems. Similarly, the development of a variant Armenian system by Shirakatsi (Chapter 5) was designed to facilitate the mathematical and astronomical work he was doing. The development of Texcocan variants of the Aztec numerical notation system (Chapter 9) was probably motivated by new demands relating to land mensuration and surveying in highland Mexico in the early colonial period. The changes involved need not be so drastic as to produce a system that employs entirely different principles. They may be as simple as the introduction of signs for higher powers in response to increasing administrative needs. As the late republican Roman polity grew in size and importance, new signs for powers of 10 were developed by simply adding additional lines to existing signs: Y for 1000, . for 10,000, ~ for 100,000. When even this did not suffice, Roman writers began using multiplicative notation with a horizontal bar (vinculum) for

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1000 and an enclosing box for 100,000. When expressions for 100,000 were no longer needed in the early Middle Ages (because of reduced social complexity in Western Europe), they disappeared. The need to represent larger numbers for administration and mathematics is responsible for the development of multiplicative notation above 100,000 in the Egyptian hieratic numerals (Chapter 2) and for the development of various sets of signs for very high powers of 10 in the Chinese classical numerals (Chapter 8). This principle is similar to that suggested by Divale (1999) for the development of higher lexical numerals under conditions of increased need for food storage and preservation. If a system is being used for a purpose for which it is unsuited, it may be replaced for that function, if a more suitable alternative is available. Thus, Roman numerals were not particularly well suited for double-entry bookkeeping when they were introduced in medieval Italy, because of their long numeral-phrases and the absence of place value, so the Western numerals, previously used mainly by mathematicians, were adopted instead. Similarly, the Arabic abjad numerals (Chapter 5) gradually were abandoned and replaced with the Arabic positional numerals (Chapter 6), as the exact sciences of the early medieval Islamic world became increasingly complex and the administrative needs of the Abbasid caliphate grew. A similar process is currently under way in East Asia, where Western numerals or modified Chinese positional numerals are normally used in scientific and technological contexts in place of the multiplicative-additive Chinese system. 2. A system may be adopted or rejected by individuals or groups because of the number of individuals or groups already using it.

Because numerical notation is a form of communication, the number of users of a system and the need to communicate with those individuals are relevant to its success. A system that is already used by a large number of individuals may be perceived to be useful by others, regardless of its structure or its usefulness for particular functions, because it lets one communicate with more people. The prevalence of Roman numerals throughout Western Europe can be explained in part by the Roman Empire’s domination of the region, but their geographic spread and continued use contributed to their continuing popularity throughout the Middle Ages. Thus, even though other systems known to Europeans had advantages in comparison to the Roman system, it staved off all its competitors until the sixteenth century. Similarly, the adoption of Chinese numerals throughout East Asia was partly a consequence of the advantages associated with adopting a well-known system. Conversely, systems are particularly vulnerable to extinction when they have few users, especially if there is already a popular system in use in a region. Thus, the failure of the Cherokee numerals to be adopted and

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the systems of West Africa to achieve widespread popularity is in part a consequence of the fact that they never achieved a critical mass of users. In all these cases, the role played by imperialism is also very important, since popular systems also tend to be those used by large and powerful states. Systems are not accepted or rejected solely according to the number of users they have; the choice to adopt a system may relate to the economic or social advantages of doing so within a system of hegemony, or the new system may be imposed externally. Once a system reaches a certain number of users, it becomes much more difficult to displace. The property of cultural systems that the current popularity of a cultural phenomenon affects the likelihood that other individuals will adopt it is known as a frequency dependent bias, and is a form of indirect bias, meaning that the characteristics of the trait itself are potentially irrelevant (Richerson and Boyd 2005: 120–123). Frequency dependence is similar in nature to the “QWERTY principle,” which explains the retention of the suboptimal QWERTY keyboard as a historical accident that it became very difficult to displace once it had achieved a critical mass of popularity (Shermer 1995: 74–75). This inertia is due in part to the difficulties involved in learning a new system and the fact that all the keyboards one is likely to encounter are of the QWERTY form. Similarly, popular computer operating systems may achieve near-ubiquitous (even monopolistic) popularity because users want to employ software packages that they are likely to encounter elsewhere. It is rational to continue using them even if they are inferior in some way, since their abandonment puts one at a significant disadvantage. Numerical notation systems are not difficult to learn, so the disadvantage of having to learn a new system is minimal, but because numerical notation is used for communication, this imposes a new constraint. Even if I should decide that some other system is advantageous, and am willing to make the switch, this will not be immediately useful, because I cannot be understood unless other people make the same decision and learn the new system. In other words, for communication systems like numerical notation, frequency dependent bias is not only an indirect bias, but also a direct measure of a system’s utility. I will return to this issue later in the section “Systemic Longevity and Phylogenetic Change.” 3. A numerical notation system may be imposed on a society under conditions of political, economic, or cultural domination.

The adoption of a numerical notation system is frequently stimulated by a society’s conquest or encapsulation in a tributary system. In several cases, a system was introduced into a region that previously had no numerical notation system after its conquest or subjugation by a more powerful polity. In other cases, political or economic domination led to the displacement of a society’s existing numerals by

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another system. Although new users are choosing to learn a particular system, it is not a free choice, but one guided by political, economic, and ideological circumstances. The Roman numerals did not come to be used throughout Western Europe because every society needed such a system, but because the numerals were an administrative tool of the Roman Empire. Similarly, the spread of Egyptian numerals among the early Hebrews was facilitated by the economic domination of Egypt over the Levant around 1000 bc. The most notable example is the prevalence of Western numerals throughout the world, accompanying Western European colonialism and imperial domination in regions that previously had no need for numerical notation. In these cases, there was no or only minimal competition with other systems. While in some cases (as in West Africa), indigenous systems were developed on the model of that of the hegemonic power, these were rarely successful. In the case of the pre-Columbian numerical notation systems of the New World, there is no need to compare the relative merits of the indigenous systems and the Western numerals; for political reasons, there was very little possibility that the Maya, Inka, or Aztec systems would survive for long or replace the systems of their conquerors. The effect of imperialism can be seen most clearly when the systems of the dominant and subordinate powers are structurally identical, thereby eliminating differential efficiency for specific functions as an explanation. Thus, the replacement of the Etruscan numerals by Roman numerals during the late republican period can be explained only by Rome’s rising political and economic fortunes and the decline of those of the Etruscan polities. Similarly, the replacement of the Egyptian demotic system by Greek and later Coptic alphabetic numerals was a consequence of Ptolemaic rule, followed later by Christian missionization. It is no coincidence that the Western numerals have billions of users while the (similarly ciphered-positional) Tibetan numerals have only a few million, but the structure of the two systems is completely irrelevant. This factor in combination with the previous two is an extremely powerful explanatory tool. The transformation and replacement of numerical notation systems often depends on social needs relating to administration, bookkeeping, and the exact sciences. These functions also allow large and complex societies to dominate less complex ones. Thus, systems that are well suited for a set of functions related to the exercise of power will tend to be those of large states with many users of numerical notation, and will thus tend to replace the systems of smaller, less politically complex societies. This potentially explains why cumulative and additive systems tend to be replaced over time – their inferiority for some administrative and scientific purposes leads them to be replaced when they come into contact with the numerical notation systems of more powerful societies

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with greater administrative needs. While it is probably going too far to claim that numerical notation is directly an instrument of hegemony, it is an adjunct system that supports hegemonic institutions, and a useful tool for many tasks relating to the exercise of power. 4. A numerical notation system may be invented when a region is integrated into larger socioeconomic networks or by elites in emulation of another society.

In some sociopolitical contexts, the ancestral system is not used directly by the adopting society, but instead a new system is invented for local administrative use as the adopting society becomes more complex. In such cases, the context surrounding the system’s invention is probably administration rather than longdistance trade, since the latter circumstance might make it advantageous simply to adopt one’s partner’s system wholesale. For instance, several ancient eastern Mediterranean systems developed when those societies (Minoan/Mycenaean, Hittite, Phoenician/Aramaic) increased in social complexity upon entering into regional socioeconomic networks that included Egypt and Mesopotamia. The nature of the long-distance trade that resulted was not such that the adoption of a foreign numerical notation system was particularly advantageous, but the need to control production locally and to extract surpluses made it imperative that some such system should exist. In other cases, a system develops when local elites desire to emulate other, usually more powerful states. This was one of the factors behind the development of the Armenian and Georgian numerical notation systems (Chapter 5), under the influence of Greek-speaking missionaries during the fourth and fifth centuries ad. The Brāhmī numerals (Chapter 6) may have developed in the early Mauryan Empire on the model of the Egyptian demotic system (Chapter 2) for a similar reason. 5. A system may be transformed or replaced if it is incompatible with the computational techniques used in a society.

Throughout this study, I have downplayed the role of computational efficiency in measuring the usefulness of numerical notation systems, because they were rarely used directly for computation in pre-modern contexts. Yet they are often used indirectly to record the results of computations performed using some other technology. Where the structure of a society’s numerical notation systems and computational technologies are consonant (for instance, in base structure or in principle), the survival of one system may be correlated with the survival of the other. The continued use of Roman numerals in medieval Europe and of rod-numerals in China is due in part to the utility of the counting board and rod

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computation, respectively, for arithmetical calculations. The connection between computational technologies and numerical notation was so strong in these cases that the replacement of the former (by pen-and-paper calculation and the suan pan or bead-abacus, respectively) actively contributed to the replacement of the latter (in favor of Western numerals and Chinese positional numerals). Similarly, one of the factors behind the replacement of the multiple proto-cuneiform systems of the Uruk IV period in Mesopotamia by a single system, Sumerian (Chapter 7), may have been the abandonment of older metrological systems. In the Early Dynastic period, once those systems of weights and measures were no longer used, the corresponding numerical notation systems declined. Another case that may be a result of computational techniques is the development of the Etruscan numerals (Chapter 4) or the Ryukyu sho-chu-ma (Chapter 10) out of tallying marks. Both of these systems have sub-bases of 5 even though the corresponding lexical numerals of their inventors have no such component, but in accordance with the Rule of Four, tallying systems work best when divided into groups of no more than four units. The development of hexadecimal and binary numerals transforms ordinary Western numerals (and, in the case of hexadecimal, the letters A–E) into systems with bases better suited to electronic computation. Yet the congruity of computational and representational techniques is not universally important; for instance, the use of the soroban (bead-abacus) has not declined significantly in Japan despite the widespread use of Western numerals there. Despite the consonance between the cumulative-additive Italic systems with a sub-base of 5 and the use of the abacus, systems such as the Greek acrophonic numerals were replaced with the ciphered and nonquinary alphabetic numerals, even though the use of the abacus continued. Finally, the use of hexadecimal and binary numbers in electronics, while convenient, is not likely to lead to the replacement of the Western numerals. 6. A system may be used for limited purposes in which it is useful to distinguish one series of numbers from another.

Many societies retain older systems for limited purposes so that the two systems, when used together, help distinguish two types of objects, each of which is enumerated using a different system. Doing this may reduce ambiguity or indicate the function of a numeral-phrase by the system that it uses. For instance, in the modern West, Roman numerals are retained for prefaces to books, volume numbers for multibook series, certain lists (especially those with subcategories), and sometimes even in dates (6.vii.2002 instead of 6/7/02). In modern Greece, the same principle governs the occasional use of the alphabetic numerals for numbered lists, even though Western numerals are used in most contexts.

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Numerical Notation

Coincidentally, in ancient Greece, the acrophonic numerals were retained for stichometry as late as the third century ad, even though they had been superseded by the alphabetic numerals centuries earlier. A more ancient example is the employment of multiple systems in Mesopotamia (Chapter 7). From their inception, various proto-cuneiform systems were used to express different types of quantity. While Nissen, Damerow, and Englund (1993) have interpreted this as evidence of the absence of abstract numeration at that time, it is just as likely to have been a simple functional division based on the employment of several different metrological systems. Similarly, in the second half of the third millennium bc, linear-style Sumerian numerals and the newer cuneiform signs were used in the same texts to indicate different types of object, possibly to avoid confusing the different categories. 7. At the time of the diffusion of a numerical notation system into a region, the principle of the ancestral system may be adopted, but using an indigenous set of numeralsigns.

The principle of frequency dependence or “strength in numbers” (#2) suggests that the need to be understood by a wide range of users reinforces the spread of already-popular systems. Yet in many cases, even when the structure of a system is adopted precisely, the numeral-signs are indigenously invented, even though this change renders them unreadable to users of other systems. Adopters of numerical notation systems may wish to express a different cultural identity than that held by those who transmitted the system, possibly in the process obscuring the new system’s origin. The clearest examples of this are systems such as those of West Africa (Chapter 10), where ciphered-positional systems were developed on the model of Western or Arabic numerals, but with new, indigenous numeral-signs. Similarly, in a case such as the possible development of Linear A numerals from the Egyptian hieroglyphs (Chapter 2), it would not have made much sense for the Minoans to adopt the Egyptian signs, which were also phonetic signs of the hieroglyphic script. While the Linear A system is structurally identical to its ancestor, it uses simple abstract signs. Each of the many different alphabetic systems (Chapter 5) uses a distinct inventory of signs using its own letters as numeral-signs. It would not have made any sense to retain foreign numeral-signs, since the very point of alphabetic numerals is the need to learn only one set of symbols. This factor must be differentiated from the paleographic divergence of systems that once were unified, such as the changes seen in the Brāhmī-derived systems of India. In such cases, the divergence of systems occurred well after the time of a system’s invention, in response to the migration of peoples or the separation of regions that were once politically unified (see factor #14).

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8. A descendant system may be structurally distinct from its ancestor because of differences in the lexical numerals associated with them.

Systemic transformations sometimes result from efforts to adapt a system to the structure of the lexical numerals associated with the adopting society, particularly to the base of the new system. In certain modern instances, a system’s inventors explicitly stated their intention to fit a numerical notation system to their language’s lexical numerals, as in the Iñupiaq and Oberi Okaime (Chapter 10) systems, which are both vigesimal even though they were derived from the Western numerals. In pre-modern cases, usually we can infer only that such a decision was made by comparing a group’s numerical notation system and lexical numerals. The shift from sexagesimal to decimal numerical notation in Mesopotamia corresponds well with the shift in political dominance from Sumerian to Semitic speakers (although sexagesimal elements were retained in AssyroBabylonian numeration for millennia thereafter). In some cases, the additional signs of a system rather than its major features are affected. The use of special signs for 11 through 19 in the Jurchin numerals (Chapter 8) corresponds to the fact that in the Jurchin language, the corresponding lexical numerals are not directly related to the word for ‘ten’. 9. When an established system is challenged by a newly introduced one, the older system may be defended for cultural or political reasons.

It is virtually inevitable that when a new numerical notation system is introduced into a society, it will have both proponents and detractors, leading to a fluid situation in which, depending on social position, personality, or other factors, individuals and groups adopt the novelty at differential rates (Rogers 1940, Hägerstrand 1967). I have already discussed situations where the new system is imposed through conquest or cultural hegemony (#3). In some cases, active local resistance can prevent or delay the new system from achieving a foothold. The effect of cultural inertia cannot be predicted based on the relative merits of the competing systems. For instance, while Western numerals quickly took hold in Japan and Korea, in China, the cultural associations of the classical numerals (together with their correspondence to the lexical numerals) effectively prevented Western numerals from replacing the older system. Where strong religious connotations are attached to the use of a particular system – as with the Hebrew alphabetic numerals (Chapter 5), for instance – it may be almost impossible to displace them, even when the system’s users are encapsulated in larger polities. One reason why the Malayalam, Tamil, and Sinhalese systems (Chapter 6) remained nonpositional for a long time was that the new invention was perceived to be associated with the culturally distinct northern

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Indian states. Yet, in other cases (e.g., the Mesoamerican systems), a generation or two suffices to eliminate a system, and any resistance is overcome relatively quickly. Often, resistance takes the form of invention of an entirely new system – witness the creation of the Varang Kshiti and Pahawh Hmong numerals in the twentieth century, or the invention of quasi-positional Roman numerals in reaction to the Western numerals. Such systems have rarely been widespread or long-lived. Even where the arguments defending one system against another purport to be concerned with efficiency, the role of tradition in resisting new and/or foreign inventions can be important. Such sentiments may have led to the prohibition of Western numerals in Florence in 1299 and to similar derogatory statements about their ease of forgery in Western Europe between the thirteenth and sixteenth centuries (Struik 1968; Menninger 1969: 426–427). Although much of the discourse disparaging the Western numerals in late medieval Europe focused on their potential to be used for illicit purposes, other factors were also at work. These new numerals were a foreign invention and could be seen as undesirable by xenophobic administrators. They were also associated with merchants and moneylenders, so class interests were certainly relevant (Swetz 1987). 10. A system may be borrowed or invented for use in a limited sector of society to control the flow of information.

In rare instances, a system was developed primarily to conceal information in a code understood by only a limited number or else to protect information against forgery. The siyaq numerals and Turkish cryptographic numerals (Chapter 10) appear to have their origins in the desire of certain categories of individuals to control the flow of information. The Cistercian numerals (Chapter 10) also may have occasionally been used cryptographically, particularly in the latter part of their history. The Fez numerals (Chapter 5), originally used quite widely, were eventually used only in contracts in order to conceal values and thus prevent forgery and modification. A similar function is served by the da xie shu mu zi accounting numerals used in China (Chapter 8); the complexity of the numeral-signs makes altering these numerals for fraudulent purposes nearly impossible. Here, in effect, we have the opposite of a frequency dependent bias; a system’s obscurity is what makes it desirable for users. 11. A system may be retained for prestige or literary purposes even after it has been supplanted by another system.

The retention of Roman numerals in the West is the best-known example of such a situation. They are often used today, in contexts such as clock faces,

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monumental inscriptions, copyright dates of films, and ordinal numbering (e.g., of monarchs, world wars, and Super Bowls), to assign prestige value to something by denoting it in Roman instead of Western numerals. They carry with them a connotation of age and classical education, and their retention into the foreseeable future thus seems likely. Similarly, the retention of Greek and other alphabetic numerals, especially in liturgical contexts, reinforces the venerable status of texts that use them. Particularly elegant forms of the Chinese numerals, such as the shang fa da zhuan used on seals, are treasured for their age and beauty. Finally, the retention of Sumerian numerals (Chapter 7) in certain Assyrian royal inscriptions as late as the eighth century bc served the purpose of associating the kings mentioned in those inscriptions with the traditions of ancient Mesopotamia. 12. A system may be invented on the model of two or more existing systems.

I slightly oversimplified a few processes of transformation in Chapter 11 by treating all ancestor-descendant relationships as ones with a single ancestor and a single descendant. While this is usually accurate, occasionally a system blends important features of two ancestral systems. The Phoenician-Aramaic systems (Chapter 3) may well have originated from the interaction of the hieroglyphic systems of the eastern Mediterranean (probably the Egyptian hieroglyphs) and the AssyroBabylonian cuneiform system (Chapter 7) in the context of interregional trade through the intermediary of the Levantine peoples. Other cases do not involve a change in basic structural principle. The cursive zimām numerals used in medieval Egypt (Chapter 5) combined structural and paleographic elements from the ciphered-additive Arabic abjad numerals, the Coptic numerals, and possibly the Greek alphabetic numerals. The Syriac alphabetic numerals similarly combine features from the Greek and Hebrew alphabetic numerals. In other cases, the basic structure of an existing system is altered slightly because of knowledge of another system. This was the case with the development of positional variant Roman numerals after the introduction of Western numerals into medieval Europe (Chapter 4) and the transformation of Âryabhata’s numerals into the ciphered-positional katapayâdi system on the model of the Indian cipheredpositional numerals (Chapter 6). In these cases, the signs of an older system were combined in a new way based on the intraexponential and interexponential structure of another one. In some such instances, these alterations may reflect an attempt to resist the newer system by altering the structure of an existing (and culturally valued) system. Finally, in blended systems such as the Chinese commercial numerals (Chapter 8), which combine the classical system and the rod-numerals, functional considerations, such as the need to do rapid calculations, may have been the most important stimulus for combining the two systems.

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13. A system may be transformed or replaced because of changes relating to the media on which or the instruments with which it is written.

The transformation of the Egyptian hieroglyphic numerals into the hieratic system (Chapter 2) and of Middle Persian into Pahlavi (Chapter 3) are the only occasions when cumulative-additive systems gave rise to ciphered-additive ones. These developments did not occur due to new social contexts, but rather due to the transfer of old systems onto new media (e.g., ink on papyrus, in the Egyptian case). The development of a cursive script tradition and a media on which distinct cumulative signs could be reduced gradually to ligatured ciphered ones was a significant development. I believe that the hieratic ciphered-additive system would never have developed solely within the context of Egyptian monumental writing. I think it probable that the cursive reduction of many signs to one sign is more likely than the reverse (the division of previously ciphered signs into cumulative ones); this correlates with the trend away from cumulative structuring. Similarly, changes in writing style in Mesopotamia were responsible for the rotation of signs from top-to-bottom to left-to-right in direction, and a later change in stylus shape produced the shift from the Sumerian archaic to cuneiform numerals (Chapter 7). The paleographic differences between the ordinary Arabic positional system and the North African Maghribi “ghubar” numerals (Chapter 6) may (disputably) relate to the former system’s use on stone inscriptions and in texts, whereas the latter was used in “dust-board” calculation. A quite different instance where this factor applied was in the gradual replacement of Roman numerals by Western ones in early modern Europe. Western numerals can potentially be used with greater ease than Roman numerals in printed books and on dated coins because of their greater conciseness. The adoption of Western numerals was somewhat influenced by the increasing use that the burgeoning middle classes of Western Europe were making of books and coins in the fifteenth and sixteenth centuries. 14. A system used in multiple politically independent or geographically diverse regions may diverge over time into several systems.

A system may diverge over time into multiple systems, usually when a previously unified region becomes politically fragmented or because of geographical separation caused by migration. This paleographic drift may or may not produce structural changes, but over time usually results in systems that are related to one another phylogenetically but are not mutually intelligible. The best-known example of such a divergence is that by which the Brāhmī numerals developed into the various Indian systems (Chapter 6), a process that was ongoing throughout the

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history of Indian writing but was hastened by the fragmentation of the Gupta empire in the sixth century ad. The numerals’ spread into the Arab world and eventually to Western Europe continued the process of paleographic divergence, so that today it is difficult to see any resemblance among the numeral-signs of Europe, the Middle East, and South Asia. This process also led to the divergence of the Egyptian demotic and “abnormal” hieratic numerals (Chapter 2) that were used in Lower and Upper Egypt, respectively, in the politically fragmented Late Period. Similarly, the fragmentation of Achaemenid Persia after the Alexandrine conquest, and the relatively loose Seleucid rule thereafter, led to the divergence of the older Aramaic system (Chapter 3) into its many structurally distinct variants used in the Levantine city-states (Palmyrene, Nabataean, Hatran) and other Middle Eastern polities. 15. A system may diverge structurally from its ancestor due to factors related to the society’s writing system.

Occasionally, the structure of a society’s script was inconsistent with the way in which an ancestral numerical notation system formed numeral-signs, thus requiring or enabling changes in a locally developed descendant system. For instance, the Greek alphabetic numerals have twenty-four signs plus three episemons, but the Hebrew consonantary has only twenty-two signs (Chapter 5). When the Hebrew numerical notation system was invented, a new technique had to be invented to represent the numbers 500 through 900, which was to additively combine the twenty-second sign (for 400) with other signs for 100 through 400 as necessary. In contrast, in the Greek-derived Armenian and Georgian systems, whose corresponding alphabets had more than thirty-six signs each, unique signs could be developed for 1000 through 9000 instead of using hybrid multiplication (Gamkrelidze 1994). Similarly, the failure of the various Indian alphasyllabic numerical notation systems (Chapter 6) to achieve widespread acceptance outside South Asia is also due in part to this factor. These systems could not have spread to regions that lacked alphasyllabaries because their structure requires corresponding alphasyllabic scripts. 16. An existing system may be retained after its replacement for limited purposes for which it is more useful than the system replacing it.

This uncommon circumstance is the inverse of #1, and occurs when a system is replaced for most purposes, but retained for a limited set of functions because the newer system is inadequate in some way. In both the South Asian and Alphabetic families, riddles exist that use the assignment of numerical values to phonetic signs to embed dates or other numerical values in words or phrases. When the Hebrew

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alphabetic numerals, Arabic abjad, and other systems were replaced by positional numerals for most purposes, the older systems were retained for number-magic because the newer systems did not assign numerical values to letters. For the same reason, the varnasankhya systems of India (Âryabhata’s system, katapayâdi, aksharapallî) were used by astrologers and in literature for centuries after they had been superseded by ciphered-positional numerals. 17. A systemic transformation may result from factors relating to the symbolic, religious, or metaphysical conventions of the society in which it is used.

This rare circumstance has nevertheless in two cases resulted in important structural changes. The invention of the ciphered Maya head-variant glyphs as alternatives to the cumulative bar-and-dot numerals was motivated by the symbolic association of gods with numerical values, probably related to phonetic correspondences between their names and Maya lexical numerals (Macri 1985). The complexity of the head-variant glyphs and the consequent difficulty in inscribing them on monuments refutes any simple functional explanation for their development. The development of positionality in India and the shift from cipheredadditive notation to ciphered-positional numerals with a zero (Chapter 6) similarly related to metaphysical and literary rather than practical concerns. Positionality has clear literary antecedents in Hindu philosophy of the late Gupta period, including the development of the concept of śûnya ‘emptiness, void’ and the subsequent naming of the zero-sign śûnya-bindu. While there were also functional correlates to this development, this philosophical prefiguring of positionality and zero is nonetheless highly intriguing. In summary, each of these factors is relevant in multiple systems examined in this study, and no system’s evolution can be analyzed without taking account of the effects of various social circumstances. None of these factors refutes the findings of Chapter 11, where I demonstrated powerful multilinear diachronic trends favoring ciphered and positional systems over time. It thus becomes imperative to explain how these social and cognitive factors interacted to produce the attested historical patterns. The increasing need for numeration for administration and the exact sciences in large and complex states, combined with the greater potential that such functions allowed for dominating other societies, is extremely important. Once this process had begun, the number of users of such systems increased, which made it more likely that these systems would be perceived as useful by members of other societies. While resistance to introduced systems might be partially successful and might result in the retention of older systems for limited purposes, the diachronic trend favored systems whose users were associated with larger-scale and socially complex societies.

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Systemic Longevity and Phylogenetic Change Events of transformation and replacement are extremely important from a theoretical perspective, since they help us understand why numerical notation systems are invented, altered, and replaced. Episodes of transformation of numerical notation systems are extremely rare, however, and the replacement of systems is only slightly less so. Numerical notation has existed for 5,500 years, but there have only been around 22 attested instances of a system giving rise to one that uses a different basic principle (an average of one event every 250 years), and only about 80 instances of a system going extinct (approximately one every 70 years). Thus, in contrast to many other sociocultural phenomena, numerical notation systems are remarkably durable. Table 12.1 lists all twenty-eight numerical notation systems that have been used for periods of 1,000 years or more, including systems still in use (with bc dates indicated using negative numbers). The systems on this list comprise nearly one-third of all those examined in this study. Longevity is not exclusive to either ancient or modern systems, as is shown by the presence of both very old systems, such as the Egyptian hieroglyphs, and relatively recent ones, such as the Arabic positional numerals. The duration of some of these systems may be slightly exaggerated, since many systems survive for several centuries after they fall out of common use. Yet the effect of such vestigial survivals is small, and under any calculation the Egyptian hieroglyphic and hieratic numerals have the longest period of use.1 There is no correlation between a system’s principle and its longevity; all five combinations of principle are found multiple times among long-lived systems. The fact that thirteen of the twentyeight systems are ciphered-additive is an artifact of the larger number of such systems overall. Few ciphered-positional systems have yet reached their millennial anniversary only because they are mostly of relatively recent invention. The systems of Egypt and Mesopotamia are among the longest-lived because the cultural traditions of the Egyptian and Mesopotamian civilizations were stable, there was little impetus to develop new systems, and cultural contact with regions that might offer alternative systems was limited. Other systems (e.g., the Roman and Chinese classical systems) persisted owing to their use in enormous empires and their subsequent use as shared numerical notation systems over large regions. Still others, such as the long-lived alphabetic systems, were developed and persisted in the context of specific liturgical and literary traditions. In all of these cases, change in numerical notation systems is the exception rather than the rule. Systems can persist for millennia even in the face of competition from others that 1

If we consider the Shang numerals and the Chinese classical numerals to be a single system, however, their total life span is 3,300 years so far.

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Table 12.1. Long-lived systems System

Principle

First

Last

Duration

Egyptian hieroglyphic

Cu-Ad

−3250

400

3650

Egyptian hieratic

Ci-Ad

−2600

200

2800

Greek alphabetic

Ci-Ad

−575

2000

2575

Roman (classical)

Cu-Ad

−400

2000

2400

Chinese (traditional)

Mu-Ad

−250

2000

2250

Assyro-Babylonian common

Cu-Ad

−2300

−200

2100

Hebrew alphabetic

Ci-Ad

−100

2000

2100

Babylonian positional

Cu-Po

−2000

0

2000

Maya (bar-and-dot)

Cu-Ad

−400

1600

2000

Chinese rod-numerals

Cu-Po

−300

1600

1900

Ethiopic

Ci-Ad

350

2000

1650

Coptic

Ci-Ad

350

2000

1650

Sinhalese

Ci-Ad

500

2000

1500

Tamil

Mu-Ad

500

2000

1500

Syriac alphabetic

Ci-Ad

500

2000

1500

Indian

Ci-Po

575

2000

1425

Sumerian

Cu-Ad

−2900

−1500

1400

Malayalam

Mu-Ad

500

1850

1350

Maya “positional”

Cu-Po

−50

1250

1300

Armenian

Ci-Ad

400

1650

1250

Egyptian demotic

Ci-Ad

−750

450

1200

Arabic positional

Ci-Po

800

2000

1200

Georgian

Ci-Ad

450

1600

1150

Maghribi

Ci-Po

875

2000

1125

Brāhmī

Ci-Ad

−300

800

1100

Cyrillic

Ci-Ad

900

2000

1100

−1300

−250

1050

900

1925

1025

Shang/Zhou

Mu-Ad

Siyaq

Ci-Ad

may seem more efficient for some functions. Reconciling the many factors that can lead to systemic change with the reality that such changes are comparatively rare is a challenge. At least three separate factors combine to ensure the relative stability of numerical notation systems. First, there is no strong selection against systems, even when they are insufficient for the purposes for which they are being used. Numerical notation systems

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often seem to operate on the principle of the “survival of the mediocre” (Hallpike 1986: 81–145) – a system will tend to persist unless it is obviously maladaptive for the functions for which it is being used. Until the rise of early modern mathematics, double-entry bookkeeping, and widespread printed books, Roman numerals were reasonably well suited to any of the purposes for which they were needed in either classical Rome or medieval Europe. Even where they were perceived to be inefficient, options other than replacement were available. For mathematics, the Greek alphabetic numerals could be used. For mensuration, metrology, and arithmetic, multiplication tables and similar arithmetical charts could be introduced, or other computational techniques such as finger arithmetic and the abacus could be used. If the Roman numeral-phrases were insufficiently concise, subtractive phrases could be used. The abandonment of an existing and time-tested system is a drastic step. An alternative system not only must exist, but must also be perceived as sufficiently useful to justify abandoning an existing one. It is simply not the case that the history of numerical notation can be explained in strongly selectionist terms. Second, even though it is not difficult for one individual to learn and use a new numerical notation system, the complete replacement of an older system throughout a large social network is extremely difficult because numerical notation is used for communication, and one of the primary factors governing a system’s usefulness is how many users it has (social factor #2). Even if a newly introduced numerical notation system has some advantage, it must overcome the disadvantage that it initially has few users and that it is not very effective for communication until some critical mass is reached. This is unlikely to occur unless there is a significant shift in social conditions – conquest, for instance, or the integration of a society into a large interregional trade network. In such situations, it may be advantageous for the new system to be adopted by certain groups of specialists (traders or astronomers, for instance), by which means it may gradually acquire the critical mass necessary to displace the older system. This circumstance describes quite closely the replacement of the Egyptian systems (Chapter 2) by the Greek alphabetic and Roman numerals. Even though alphabetic numerals were known and used throughout the region by the fourth century bc, they did not displace the Egyptian systems for many centuries. Greeks and Romans in Egypt employed their own systems, while Egyptian scribes used their indigenous ones, until the number of users of the introduced systems so greatly outnumbered those of the Egyptian ones that there was no other option. Finally, because numerical notation systems are written rather than verbal, their stability is directly comparable to the stability of scripts, which also can persist without major change for millennia despite radical social and linguistic changes. The Roman alphabet and Chinese logosyllabary have changed little over

418

Numerical Notation

the past two millennia, even though the spoken languages associated with them have changed radically. Numerical notation systems, like scripts, are stable because retaining an existing representational system ensures that older texts and inscriptions can continue to be read.2 As long as there is a continuous literary tradition in a region, abandoning an established writing system means that older texts may become confusing or unreadable. Because numerical notation systems are translinguistic written systems, an additional factor accounting for their stability is that they may spread very widely, and even if they cease to be used in one region or among one group of users, may be retained elsewhere. Nevertheless, although numerical notation systems are generally stable, minor changes are not uncommon. Paleographic alterations in the shapes of numeralsigns happen regularly in numerical notation systems, as they do in scripts, particularly cursive ones. Even in modern Western numerals, there are variant forms for many numeral-signs (0 vs. , 2 vs. *, 4 vs. 4, 7 vs. 7). These changes, while seemingly inconsequential, sometimes can have great effects (as witnessed by the cursive reduction of Egyptian hieratic numerals from their hieroglyphic ancestors). Minor structural changes, discounting transformation and replacement, also occur with some frequency. These include a) changes in nonbase numeral-signs that contribute to a system’s structure; b) the introduction of subtractive notation; c) the invention of new signs for higher powers of a system’s base; d) changes in the point above which hybrid multiplication is used in a system; e) changes in the direction of writing of a system; and f ) changes in the way in which cumulative systems chunk groups of signs. These minor diachronic changes represent the vast majority of changes that occur in numerical notation systems. Yet it is exactly these minor changes in the structure of systems that distinguish similarly structured systems within each phylogeny discussed in Chapters 2 through 9, and allow us to identify particular ancestor-descendant relationships within them. Most systems use the same base and the same structural principles as their descendants. Families of systems represent yet further stability in numerical notation, as they are composed of long chains of ancestor-descendant relationships. Every phylogeny in this study has a total life span greater than 1,500 years. Individual systems can go extinct after only a short time, and several independently invented systems (e.g., Inka, Bambara, Indus) gave rise to no descendants and thus represent abortive phylogenies. Moreover, some phylogenies are far more unified than others. While the Italic systems share many structural features, others (such as the East Asian and Mesoamerican 2

This is one reason why various attempts at Chinese script reform have met with only limited success, even though the existing script is widely regarded both in China and elsewhere as being very difficult to learn.

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systems) can be identified as being descended from a common ancestor only through historical context. Even so, the fact that such traditions can be identified at all highlights the remarkable stability of numerical notation systems.

Civilization and Systemic Invention There is no reason to postulate a qualitative gulf between cases of independent invention and “secondary” developments that have an ancestor. The functional and social needs that govern the adoption of externally invented numerical notation systems or the invention of new ones using an external model are similar to those governing the invention of systems in the absence of such a model. As Julian Steward (1955: 182) maintained, every borrowing must be construed as an independent recurrence of cause and effect. Moreover, because other representational systems (e.g., unstructured or minimally structured tally systems, lexical numerals, metrological systems) precede the independent development of numerical notation, when we speak of “independent invention,” we are not simply talking about an invention that springs into the mind of its creator out of nothing. Nevertheless, the process by which independently invented systems arise is somewhat different from that by which systems are modeled on a specific ancestor. Seven numerical notation systems were almost certainly invented independently of any specific influence from other systems: the Egyptian hieroglyphic (Chapter 2), Mesopotamian proto-cuneiform (Chapter 7), Shang Chinese (Chapter 8), Maya bar-and-dot (Chapter 9), and the Indus, Inka, and Bambara (all Chapter 10) systems. In three additional cases – the Etruscan (Chapter 4), Brāhmī (Chapter 6), and Naxi (Chapter 10) numerals – the hypothesis of independent invention could not be rejected entirely. In yet two more cases – the Chinese rodnumerals (Chapter 8) and the siyaq numerals (Chapter 10) – it was clear that their inventors knew other numerical notation systems, but these other systems played no obvious role in their development. Finally, the Aztec numerals (Chapter 9) are historically related to the Maya bar-and-dot numerals only through the intermediary of the unstructured highland Mexican system of using dots alone to represent numbers. The development of independently invented numerical notation systems coincides very closely with the rise of civilizations in Egypt, Mesopotamia, East Asia, the Indus Valley, Mesoamerica, the Andes, and elsewhere. Early civilizations are qualitatively distinct from the less complex societies that precede them, being characterized by great socioeconomic inequality, surplus extraction, a reduction in the social importance of kinship, and a complex administrative apparatus (Trigger 2003: 40–52). Moreover, if the Etruscan and Brāhmī cases are truly independent creations, these also developed in the context of the emergence of civilizations in

420

Numerical Notation

Italy and India, respectively. Yet in the pre-colonial West African Yoruba civilization, there was no numerical notation system.3 The Bambara system was used further north, but although we know almost nothing about its history, there is no evidence of its use among the Yoruba. There is likewise no evidence of the employment of numerical notation systems in many other African or early New World civilizations. There is thus a strong but imperfect correlation between the origin of numerical notation and the emergence of complex civilizations. This suggests that the initial development of numerical notation may frequently be a response to new social needs that arise at a certain level of social complexity. This could also help to account for the development of numerical notation systems in colonial situations. A difficulty with this proposition is that the functions for which numerical notation was used in these societies are variable. Among the Mesopotamians, Inka, and Aztecs, numerical notation was first used for administration and record keeping. In Shang China, however, the first attested numerical notation is found on oracle-bones and was used for divinatory purposes. In Egypt, the tomb-tags at Abydos and artifacts such as the Narmer mace-head seem mostly to have been used as displays of royal authority. From Ganay’s (1950) ethnographic work, our best guess is that the Bambara system was also used for divination. In lowland Mesoamerica, the earliest numerical notation indicated month and day names and periods of time (as did most later Maya numerical notation). It is possible that the Shang, lowland Mesoamerican, and Bambara systems were originally used for administrative functions (Postgate, Wang, and Wilkinson 1995). However, evidence is lacking for this proposition, and indeed Postgate, Wang, and Wilkinson emphasize that they believe that such evidence has all perished. It is impossible to identify a specific function that is universally correlated with the development of numerical notation. Another way to approach this question is to treat the origins of numerical notation as the consequence of a general need for visual representational techniques, without regard to the specific functions for which these systems were used. In four cases – the Egyptian, proto-cuneiform, Shang, and Maya – numerical notation developed just prior to, or nearly simultaneously with, the indigenous development of phonetic scripts, and this may also be true of the Harappan system. In Egypt, the numerical tags found at Abydos also provide the earliest attested instances of proto-hieroglyphs. The earliest Mesoamerican inscription with a sound provenience and clear dating (San Jose Mogoté, Monument 3) contains only the day-name 3

The status of several of the pre-colonial West African states as “early civilizations” is increasingly accepted by archaeologists (Connah 1987; Trigger 2003).

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421

“1 Earthquake.” Most of the earliest Shang oracle-bone inscriptions record numerical values (e.g., indicating sacrifices to be made). Finally, the Mesopotamian proto-cuneiform tablets are simply numerical systems combined with pictorial signs for commodities. Thus, despite my reservations about Schmandt-Besserat’s (1992) arguments concerning the origins of writing (see Chapter 7), I agree with her that writing emerges as an outgrowth of, or alongside, independently invented numerical notation systems. Yet this generalization is not a universal law. Among the Bambara and Inka, no known phonetic script was associated with the numerical notation systems that developed, and the Aztec pictographic system was not capable of representing speech directly. While every instance of independent script development followed or accompanied the development of a corresponding numerical notation system, the converse is not true. At present, we can say with some certainty that the independent development of numerical notation is strongly correlated with both the rise of civilizations and the independent development of scripts. Yet we do not know exactly why numerical notation should coincide with these developments, since it served different functions in different civilizations, and since not all civilizations developed either scripts or numerical notation systems. The pursuit of answers to this question thus requires the accumulation of new data by scholars of individual civilizations.

The Macrohistory of Numerals The phylogenetic study of structural transformations of systems and the nonphylogenetic analysis of the replacement of systems are powerful tools for examining diachronic patterns in numerical notation. At best, however, these tools can study relations between pairs of systems (as opposed to large regional and worldwide networks of cultural contact) and focus on single events or episodes of change. Adding in the numerous social explanations for diachronic regularities that I have just discussed does not help much. We still want to know whether broad changes in the types of societies in the world and the nature of the interactions among them affected how numerical notation systems were invented, transformed, and replaced over the past 5,500 years. While I reject simple unilinear macrohistories of numerical notation involving the gradual replacement of cumulative-additive and other “crude” systems with cipheredpositional ones, especially the Western numerals, it is nonetheless true that ciphered and positional systems tend to replace other types. Yet the large-scale history of numeration is not so simple. There are macrohistorical patterns to be explained, but they are not the ones to be expected if the unilinear theory of the evolution of numerals were correct.

422

Numerical Notation

Figure 12.1 graphs the invention and extinction of all systems used over a period of 100 years or more,4 which encompasses 78 of the approximately 100 systems examined in this study, and from these figures I derive the total number of systems in use in every century from 3000 bc to 2000 ad. This approach of necessity neglects local chronologies of regions that are not in contact with one another and does not take into account the number of users of each system. There are more users of numerical notation today (even when considered as a percentage of the world’s population) than at any other point in history, because of high literacy rates, but those individuals are using far fewer systems than in the past. Nevertheless, the number of systems in use at any given point in time is a relatively good measure of worldwide variability among systems, and the patterns from one period to another are certainly not random fluctuations. When these figures are aggregated, it becomes clear that from 3000 bce to around 1500 ad, there was a nearly linear increase in the number of systems in use from century to century; conversely, from 1500 ad to the present, there was a precipitous decline in the worldwide diversity of numerical notation systems. Despite the linear increase in the number of systems prior to 1500 ad, it is useful to divide this period into four subperiods in order to understand the interrelationships among cultural contact, imperialism, and the invention and diffusion of different sorts of numerical notation systems. 3000–800 bc: This period of slow growth and relative stability saw the invention of the earliest systems of the Old World civilizations, first in Egypt and Mesopotamia, but also in the Indus Valley and China, and including systems used by secondary or peripheral civilizations (Minoan, Hittite, Eblaite, etc.). Most, but not all, were cumulative-additive. Numerical notation was infrequently used in interregional trade, and trade networks were poorly integrated; hence, the opportunities for cross-cultural contact were not as great as they would later become. The contacts that did occur (between Egypt and Mesopotamia, for instance) were not conducive to the transmission of ideas about numerical notation. Numerical notation systems were usually developed at the same time as local indigenous scripts, and few societies adopted the numerals of another society wholesale. The replacement of systems during this period was largely due to the gradual transformation of older systems (proto-cuneiform Æ Sumerian Æ Assyro-Babylonian; Linear A Æ Linear B), and thus had no net effect on the number of systems in 4

Systems of less than 100 years’ duration are too short-lived to be analyzed using macrohistorical techniques and are therefore ignored. If I had included them, the only major effect on Figure 12.1 would have been that the decline after 1500 would have leveled off in the twentieth century due to the invention in colonial contexts of many systems that were quickly abandoned or replaced.

423 Figure 12.1. Number of actively used systems.

424

Numerical Notation

use. The most significant and rapid change in the use of systems during this period occurred in the twelfth-century bc, when three systems (Hittite, Ugaritic, Linear B) ceased to be used during a period of sociopolitical upheaval in the eastern Mediterranean. 800 bc–bc/ad: This period of rapid increase in the number of systems might be called the “axial age” of numerical notation systems, although it ends slightly later than Jaspers’s (1953) traditional definition of that period (800–200 bc) as it related to the development of world religions. While I reject the teleological or mystical notions associated with the “axial age” concept, I believe that the processes involved in the rapid formation of new systems during this period were akin to those leading to the somewhat similar but distinct world religions across Eurasia. The formation of complex networks of interregional trade, coupled with the expansion of literate traditions into several previously nonliterate or mostly nonliterate regions (Italy, Greece, India, and the Levant) inspired the rapid development of new scripts and corresponding numerical notation systems – notably, most of the systems of the Levantine and Italic families. This period also saw the development of the first New World numerical notation systems in Mesoamerica. Cumulative-additive systems were more common than other types, but all combinations of principle except ciphered-positional were used. In general, political fragmentation was more typical of this period than large empires. Since each small polity or group of polities tended to develop its own script, and because of the continuation of the pattern where each new script had its own distinct numerical notation, there was a substantial increase in the rate of invention of such systems. Near the end of this period, many of the older cumulative-additive systems of the circum-Mediterranean and Middle East were replaced by the ciphered-additive Greek alphabetic numerals, a direct consequence of the spread of Greek learning in the Hellenistic period. bc/ad–800 ad: This period was marked by a slightly slower increase in the number of systems used, with the rate of episodes of invention only slightly exceeding the rate of extinctions. Many new systems, a large plurality of them cipheredadditive, were invented in the Alphabetic and South Asian families, derived from the Greek alphabetic and Brāhmī systems. The first ciphered-positional system appeared in India around ad 600. In this period, ciphered systems first came to outnumber cumulative systems. Many of the cumulative systems used for millennia in Egypt and Mesopotamia were replaced by these new ciphered systems. Another important effect was the expansion of the Roman Empire, leading to the replacement of many of the cumulative systems of Europe and the Levant by Roman numerals. Many of the systems invented in this period survive to the present day.

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800–1500 ad: This period was one of rapid expansion in the number of systems used, a consequence not of extraordinarily high rates of invention but rather of extremely low rates of replacement. New systems continued to be invented in this period, especially in the Alphabetic and East Asian families, but also including the two ciphered-positional systems – Western and Arabic – that are most widely used today. Nevertheless, the continued use of older systems throughout this seven-century period was primarily responsible for the increase in the number of systems from twenty to thirty-two. This finding contradicts any simplistic notions concerning the decline of knowledge systems in the early Middle Ages. In this period, ciphered and multiplicative systems came to be used much more frequently than cumulative systems in all regions except Mesoamerica, although certain Old World cumulative systems, such as the Roman numerals and Chinese rod-numerals, continued to be used quite widely. 1500 ad–present: This was the only period in history when there was a prolonged decline in the number of systems in use. This decline was particularly steep between 1550 and 1650, when eleven systems went extinct and several others were reduced to vestigial use (for instance, in archaic or strictly liturgical contexts). Particularly hard hit were the systems of the New World, which all went extinct, but many alphabetic systems were also replaced, though less dramatically, by the Western or Arabic ciphered-positional systems. Virtually no new systems were invented that survived for as long as 100 years. In earlier times, it was normal for each script to have its own numerical notation system. Over the past five centuries, although local scripts have been retained and many new scripts have been invented, Western or Arabic ciphered-positional numerals have supplanted older systems and been adopted by the users of newly invented scripts, so that there is no longer anything close to a one-to-one ratio of scripts to numerical notation systems. Today, for the first time, there are more positional systems in use than additive systems – though just barely, since many ciphered-additive systems are still used in limited contexts. Of course, users of positional systems vastly outnumber users of additive systems, and other than the Chinese system, there are very few regular users of additive systems left. The simplest explanation for this drastic decline – that it is a consequence of European imperial expansion – has some truth to it, especially in explaining the extinction of the New World numerical notation systems as a result of Spanish conquest. Similarly, the failure of the various colonial-era African systems to achieve widespread acceptance is largely a product of the overwhelming social and economic dominance of users of the Western and Arabic numerals. Yet simply invoking imperialism as a prime mover is overly simplistic; there has, after all, been imperialism for as long as there have been empires. Even granting the unusually

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powerful nature of the Western imperial project, the rise of the imperial powers of Western Europe was a development primarily of the eighteenth and nineteenth centuries, whereas most of the decline in the number of systems took place in the Old World between 1500 and 1650. Many systems that went extinct or became obsolescent, such as the Glagolitic, Armenian, and Georgian numerals, did so because of the expansion of the Ottoman Empire into southeastern Europe and the Caucasus, not because of European expansion. The decline of the Roman numerals and their variants (calendar numerals and Arabico-Hispanic numerals), as well as of the Glagolitic and Cyrillic alphabetic systems, can hardly be explained by European conquests, since these systems were used by high-status, well-educated Europeans. The period of great decline in the number and variety of numerical notation systems in use worldwide between 1500 and 1650 corresponds to the “long sixteenth century” demonstrated by Wallerstein (1974) to mark the formation of the capitalist world-system. The rise of capitalism and the concomitant development of superior transportation and communication technologies in Western Europe, best explain the sharp falloff in the number and variability of systems. If we view numerical notation as a communication system and an administrative tool, it is evident that a dramatic expansion in the need for communication and administration on a worldwide basis would alter dramatically the fates of the systems in use before that time. One major reason why the reduction in the number of numerical notation systems worldwide corresponded to the rise of the capitalist world-system is that no earlier interregional network had nearly the same scope or strength. While there were certainly multiple “world-systems” (in the sense of relatively closed hierarchical networks of interregional socioeconomic interaction) before 1500, their ability to overwhelm older knowledge systems was not nearly as great (Abu-Lughod 1989). The capitalist world-system was and is an agent of an entirely different order of magnitude. By 1650, Western numerals were being introduced to new users in China, India, North and South America, and Africa, through both economic transactions and the implementation of European secular and religious educational institutions in missions and ports of trade. The role of the Jesuits in the spread of Western numerals remains an understudied but very interesting topic. At a very basic level, since a system’s perceived usefulness is related to the number and status of its users, the development of the capitalist world-system increased the number of people exposed to numerical notation and made it overwhelmingly likely that the system associated with core states would be widely adopted. The situation is, nonetheless, rather more complex than a simple accounting of the number and status of the users of various systems. We must also examine the rise and rapid spread of new functional contexts and media in which Western

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numerals predominated. In the Middle Ages, in both Europe and the Islamic world, literacy and higher education were relatively restricted, and in Europe, moreover, literacy was strongly associated with the Latin language, which was thoroughly tied to the persistence of Roman numerals. The invention of the printing press in the middle of the fifteenth century encouraged a significant increase in literacy among the middle classes of Western Europe, and in turn was associated with the shift away from Latin toward vernacular languages. Printers were unencumbered by the tradition of Roman numeral usage of the earlier scribal tradition and frequently employed Western numerals for pagination and for representing numbers in text. Bibles and other religious texts first began to use Western numerals alongside or in place of Roman numerals in the sixteenth century (Williams 1997). Similarly, the use of dated coinage expanded dramatically starting around 1500, a function for which Roman numerals were not really suited due to the length of their numeral-phrases. The spread of coinage as a medium of international trade within the world-system would have exposed most individuals to Western numerals. Moreover, Western numerals were better suited than either the Roman or alphabetic systems for double-entry bookkeeping, which was invented in the thirteenth century but did not become overwhelmingly popular outside Italy for a couple of centuries. Accounting systems are formidable tools for the administration of large polities, so, as mercantile activities became more central to the burgeoning early capitalist societies of Western Europe, Western numerals achieved a position of greater prominence. This set of social and technological changes accompanying the rise of the capitalist world-system in Western Europe made the widespread adoption of Western numerals a near-inevitablity. The rise of the capitalist world-system was the most significant event in the history of numerical notation. By comparison, the shift to ciphered-positional numerals and the invention of zero in medieval India are relatively insignificant. Once Western European states became core states in the world-system, it was highly desirable for the numerals associated with the administration of these states to be applied and adopted elsewhere, voluntarily or otherwise. As more and more societies adopted Western numerals, a process of positive feedback began, because a system with many users is more useful for communication than one with few users. Moreover, Western numerals (and other ciphered-positional systems such as the Arabic system) were very useful for a set of new and emergent functions (such as bookkeeping and mathematics) that aided core states in maintaining their hegemonic position – and, in turn, needed to be adopted if peripheral societies hoped to resist domination. What can be said, then, about the prospects for the currently surviving numerical notation systems? This study is not an exercise in futurology, but I think that

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some provisional conclusions can be drawn from events of the past. At present, no system is remotely likely to replace the Western numerals. The Arabic and various South Asian systems continue to enjoy some degree of health, as do the multiplicative-additive Chinese numerals, and if the fortunes of the Western countries were to change dramatically, and other countries became core states in the world-system, such a shift would be possible. Because most of these systems are ciphered-positional, the effort required to learn them is minimal. As for the various alphabetic systems that have survived (e.g., Hebrew, Greek, Arabic, Cyrillic, Coptic, Syriac), because they continue to be used in extremely conservative religious texts, their complete replacement is unlikely, but their expansion to new contexts is equally improbable. Similarly, while Roman numerals are used in only a few contexts, the prestige associated with them (and the practical function served by having an alternative to Western numerals) is likely to ensure their continued use in those contexts in the foreseeable future. As for the invention of new systems, it is altogether premature to proclaim the end of numeration history. In the twentieth century, no fewer than six systems were invented that are not of the predominant decimal ciphered-positional structure (Bamum, Mende, Oberi Okaime, Pahawh Hmong, Varang Kshiti, Iñupiaq). Even if these systems prove to be short-lived, or transform rapidly into decimal ciphered-positional systems, new systems will continue to be invented. One factor militating against such developments, however, is that several recent inventions were undertaken by individuals whose knowledge of Western numerals was limited (i.e., situations that might be described as stimulus diffusion). Because most people today (even if nonliterate) can use Western numerals, we might expect such innovations to become less frequent. Another source of innovation in numeration might be systems designed for use in electronics or mathematics, such as binary, octal, and hexadecimal numbers, scientific (exponential) notation, or even the system of colored bars used to designate the electrical resistivity of resistors. Because they are most useful in limited contexts, however, and often do not correspond to lexical numeral systems, it is unlikely that they would ever displace Western numerals. Such new developments might, however, increase the variability among numerical notation systems worldwide, without reducing the value of having a single worldwide representational system for numbers. Finally, the prospect exists that at some point in the future, a “post-positional” system might be developed, one that does not conform to any of the five combinations of principles that I have outlined in my typology or that violates the regularities I have described in a new and nontrivial way. One cannot predict what such a development might look like or what its cognitive advantages and disadvantages might be. Unless this new system has a very significant advantage over ciphered-positional numeration for a set of specific functions, however, it will not

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be widely adopted. The Western numerals are so prevalent as a representational system that they would have to be practically useless for such specific functions before any alternative system would replace them, just as the Roman numerals did not become obsolescent until they became inadequate for new functions. Even then, the demise of the Roman numerals was hastened by the introduction of an entirely new class of users of numerical notation (the newly literate middle classes) who were not necessarily familiar with the Roman system. There is no such class of individuals today, since one can find Western numerals virtually anywhere. We might expect, however, that new systems – whether additive, positional, or something else – might play a role auxiliary to ciphered-positional numerals if they were perceived to be useful in particular contexts. In this way, new systems may continue to be invented and propagated, even if no system is likely to displace the Western numerals in the foreseeable future.

chapter 13

Conclusion

Out of the darkness, Funes’ voice went on talking to me. He told me that in 1886 he had invented an original system of numbering and that in a very few days he had gone beyond the twenty-four-thousand mark. He had not written it down, since anything he thought of once would never be lost to him. His first stimulus was, I think, his discomfort at the fact that the famous thirty-three gauchos of Uruguayan history should require two signs and two words, in place of a single word and a single sign. He then applied this absurd principle to the other numbers. In place of seven thousand thirteen, he would say (for example) Máximo Pérez; in place of seven thousand fourteen, The Railroad; other numbers were Luis Melián Lafinur, Olimar, sulphur, the reins, the whale, the gas, the caldron, Napoleon, Agustín de Vedia. In place of five hundred, he would say nine. Each word had a particular sign, a kind of mark; the last in the series were very complicated. ... I tried to explain to him that this rhapsody of incoherent terms was precisely the opposite of a system of numbers. I told him that saying 365 meant saying three hundreds, six tens, five ones, an analysis which is not found in the ‘numbers’ The Negro Timoteo or meat blanket. Funes did not understand me or refused to understand me. Jorge Luis Borges, “Funes, the Memorious” (1964)

In Borges’s story, the character Funes, blessed with a limitless memory, constructs an alternative system for representing numbers in which order and structure are irrelevant. In so doing, however, he creates a system whose symbols are so arbitrary as

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to render it useless to those of us whose memories are less prodigious than his own. What Funes can ignore – and what Borges sought to convey – is that, given human cognitive limitations, structure is necessary to communicate and retain information. Number is easily amenable to such structuring, and in fact, beyond a very basic level, requires it. Whether we write 7013 or hggcaaa or Máximo Pérez is not simply a stylistic choice, but a decision that has important social and cognitive consequences. Structure reduces chaos to ordered simplicity and constrains a domain of activity within well-defined and understandable rules. Numerical notation is useful because it imposes structure on the series of abstract natural numbers in a way that allows humans to manipulate them more effectively. Over the past 5,500 years, more than 100 different systems and hundreds of paleographic variants thereupon have been developed for representing numbers in a visual and primarily nonphonetic manner. Very few systems are completely identical in structure to any other system. There is thus considerable variability among the systems used worldwide. Even so, they are all structured by only three intraexponential and two interexponential principles, and are further constrained in the way they use bases, hybrid multiplication, phrase ordering, and arithmetical operations. As additional systems come to light, I expect that they will fit into the typology I have constructed, because no attested system is so aberrant that it cannot be described within it. A multidimensional typology better reflects the various features of numerical notation systems than do one-dimensional schemes that regard the transition from additive to positional systems as the only basis for classification. It lets us ask new and important questions about the synchronic and diachronic patterns visible among systems. Even though numerical notation systems have been independently invented multiple times and have existed in a wide variety of societies across many millennia, they are easily learned and understood, and often can be interpreted even in the absence of other contextual clues. We can read Etruscan numerals even though we do not fully understand the Etruscan language. We can read Minoan numerals without being able to decipher other aspects of the Linear A script. We can read numerical values from Inka khipus even though our knowledge of how they were used and read is mostly lost. If there were no patterning in numerical notation – if there were no cognitive rules constraining how numbers could be written – we would not be able to perform such acts of translation. These patterns thus refute radically relativistic notions concerning the way in which concepts are determined by culture. Recognizing the significant differences in how societies think about numbers (especially number symbolism), the core of comparable features common to all numerical notation systems demonstrates that these differences are surmountable.

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The gap between systems that are imaginable and those that are actually attested is substantial, and can be explained only in terms of constraints imposed by human cognitive abilities. Therefore, the examination of synchronic regularities in numerical notation systems allows a partial reconstruction of the mental processes of members of past societies. Given the limitations of the data from most numerical notation systems, such reconstructions are necessarily incomplete, but nonetheless important. At the same time, some of the universalistic claims that have been made regarding numerical notation are untrue, and some rules do have important exceptions. These exceptions are theoretically important, as they allow the testing of hypotheses about the underlying causes of generalizations. A smaller set of diachronic regularities governs patterns of invention and replacement of systems over time, describing events of change rather than single systems. It is possible to determine these rules inductively because complete historical sequences of systems can be demonstrated, thus making it possible to trace phylogenetic and diffusionary relationships among systems. The universal diachronic methodology I have adopted thus allows the empirical demonstration of historical relations, as opposed to other inferential techniques that require many assumptions. Yet the patterns discerned are multilinear rather than unilinear. Because both synchronic and diachronic regularities are strongly correlated with the structural principles of numerical notation systems, purely particularistic explanations for them will not suffice. It is not coincidental that cumulative and additive systems tend to be replaced over time by ciphered and positional ones. Systems can be evaluated in terms of various criteria such as conciseness, extendability, and sign-count, each of which has advantages and disadvantages. While there is no one single goal that humans universally seek to achieve when using numerical notation, a constellation of related goals can be identified, and various features of systems can be evaluated as to how well they reflect them. The existence of diachronic regularities and the commonalities among independent events of systemic transformation and replacement refute, or at least redefine, the commonly held anthropological dichotomy between independent invention (analogy) and diffusion (homology). Nevertheless, social factors play a major role in determining how systems are invented, transmitted, and accepted. A decision to maximize conciseness rather than sign-count in a system, for instance, is not made on that basis alone, but in relation to one or more functions for which the system is to be used. We do not know a priori what specific functions will be most important, and thus we cannot evaluate how a system’s users will assess its utility, except through the empirical demonstration of its specific contexts of use. Even so, considerations other than purely structural or cognitive ones are often very important. The evaluation of a system also requires that we take into account the medium on which it is used,

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the linguistic affiliation of its users, the desire to emulate a powerful neighbor, and similar factors. By understanding the functions for which systems were used, and the reasons why their users may have perceived them to be useful, we achieve a much more thorough understanding of how people think with numbers than we could from studying the systems alone. Numerical notation systems are first and foremost representational systems used to communicate. They often arise alongside, or slightly earlier than, writing systems, and exist because their users feel a need for a visual and durable communication method. Because they are used in interregional trade, the administration of colonies, missionary writings, and other contexts of intercultural communication, they can spread more rapidly than phenomena that are not primarily communicative. Yet the path of transmission of numerical notation systems differs significantly from that of both writing systems and lexical numerals. Numerical notation systems are translinguistic, and as such can spread in a way that is entirely divergent from patterns of script diffusion, since all scripts must to some extent represent certain phonemes and not others. While numerical notation systems correlate structurally with their inventors’ lexical numerals, speakers of other languages can easily learn them. The fact that number has two very different visual representational systems (written lexical numerals and numerical notation) is very interesting and warrants more extensive study. While numerical notation is vitally important as a representational system, it has been largely irrelevant as a computational system, except in the recent past. A wide variety of computational techniques, including mental calculation, finger counting, tallies, and abaci, allow users to do arithmetic in various societies. Numerical notation may record the results of computations, but is rarely used directly for calculating values. Hence, any analysis that treats computational efficiency as the prime mover behind the evolution of numerical notation is fatally flawed. The modern use of numerical notation in pen-and-paper calculation is largely an emergent function associated with the rise of mathematics and capitalism. With the advent of digital technologies over the past half-century, the direct manipulation of numeral-signs for doing arithmetic will not persist much longer. Your computer does not greatly care if you are entering spreadsheet figures in Roman numerals. In general, most systems are very similar in structure to their ancestors, and systems tend to be quite long-lived. Changes in systems are the exception rather than the rule. When changes do occur, it is usually because of dramatic changes in the functions for which, or the social contexts in which, a system is used. Systems can be linked together into families with minimal difficulty, using evidence from their structural features and sign-forms, together with non-numerical evidence of cultural contact. As long as a system is minimally adequate for a set of functions,

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it will rarely be modified significantly except for changes in the forms of numeralsigns. In ancient societies, in particular, there was little competition from other systems and little reason for a system’s users to alter their behavior. Yet, since 1500, the number of systems used worldwide has decreased dramatically, and ciphered-positional systems have replaced nonciphered and nonpositional ones throughout most of the world. This is not simply a coincidence, but is the outcome of broad social changes related to the rise of capitalism in Western Europe and throughout the modern world-system: mercantilism, printing, double-entry bookkeeping, dated coinage, and vernacular religious literature. The replacement of Roman numerals in core societies followed, as did the subsequent adoption of Western numerals in peripheral societies into which Western institutions spread. Because the functions for which Western numerals were most useful were also functions that helped Western societies to dominate others, they have spread, relatively unopposed. Once their diffusion had begun in earnest, the adoption of Western numerals in peripheral areas was a rational strategy for those who wanted to be able to communicate numerically with large numbers of powerful individuals. Yet there is no direct “cultural selection” in favor of the structure of ciphered-positional numerals; otherwise, we might expect the identically structured Western and Tibetan numerals to have spread with equal effectiveness. The conclusion that place-value is the ultimate goal of numerical notation systems, or represents a “perfect” development, is a disastrously poor explanation of the present near-universality of ciphered-positional systems like Western numerals. The history of numerical notation is a multi-millennial, complex pattern of nonlinear yet directional cultural change. Explaining this history without recourse to reductionism requires synchronic and diachronic cross-cultural comparison that accounts for the cognitive constraints of the human mind as well as the pervasive effects of social factors on individual choices. As previously lost systems come to light – and as new systems are invented – the findings of this study will no doubt be revised and refined. Yet, because human minds work with numbers in similar ways – notwithstanding the memorious Funes – and because similar social factors affect decisions, there is no reason to think that the future of numeration will be greatly different from its past. We do not stand at the end point of a linear historical sequence, but in the midst of a branching and complex yet patterned and explainable world of written numbers.

Glossary

Acrophonic principle [Greek akro ‘tip’ + phonia ‘voice, sound’]: The use of phonetic signs that correspond to the first sounds of words; for instance, Greek Δ = ΔEKA (deka) = 10. Additive: An interexponential structuring principle where the total value of a numerical phrase is equal to the sum of the signs for each power expressed. Cf. positional. Alphabet: A writing system whose signs represent both consonant and vowel phonemes. Alphasyllabary: A writing system whose signs represent consonant + vowel (CV) combinations, with the basic sign indicating the consonant and diacritics indicating the vowel. Also known as abugidas. Base: A natural number whose powers are specially denoted within a numerical system. Blended system: A numerical notation system that incorporates features of two ancestral systems. Contrast with hybrid system. Boustrophedon [Greek boustrophos ‘ox-turning’]: Having a direction of writing alternating from left to right and right to left on successive lines. Cardinal number: A natural number used to denote the quantity of some set of objects, but not its order. Also known as counting numbers. Cf. ordinal number. Centesimal: Having a numerical base or sub-base of 100. Chronogram: A word, phrase, or verse in which some or all graphemes have corresponding numerical values whose sum produces a date corresponding to the event described by the text. Chunking: The structuring of long strings of information in groups, usually of three to five elements, to aid in comprehension. Cf. subitizing. Ciphered: An intraexponential structuring principle where the numerical value of a power is expressed in a single sign. Cf. cumulative and multiplicative.

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Consonantary: A writing system whose symbols denote consonantal phonemes but not normally vowels. The Arabic and Hebrew scripts are consonantaries. Also known as abjads. Counting board: Any of various flat surfaces or artifacts on which beads, pebbles, or counters could be placed and moved in order to perform arithmetical calculations. An abacus is a form of counting board. Cumulative: An intraexponential structuring principle where the numerical value of a power is expressed by taking the sum of multiple identical signs. Cf. ciphered and multiplicative. Cuneiform [Latin cuneus ‘wedge’]: Of writing or numerical notation in Mesopotamia, impressed on clay using a wedge-shaped stylus. Cursive: Of writing or numerical notation, having a ligatured or curved quality typical of writing with ink on parchment, papyrus, or pottery. Decimal: Having a numerical base or sub-base of 10. Diacritic: An accent or other ancillary mark added to a grapheme to modify its phonetic or numerical value. Epigraphy: The study of inscriptions engraved into stone or other durable materials. Cf. paleography. Grapheme: Any discrete graphic sign in a script or numerical notation system. Hieroglyph: Informally used to describe signs in writing systems that are largely nonphonetic (logograms and ideograms), especially Egyptian, Hittite, and Maya. Hybrid system: A numerical notation system that is cumulative-additive or ciphered-additive for lower powers, but multiplicative-additive for higher powers. Ideogram: A grapheme that indicates an abstract idea rather than a sound or word in a language. Interexponential: The principle governing how the values of different powers in a numerical notation system are arranged and combined to derive the total value of numeralphrases. Intraexponential: The principle governing how the signs within each power of the base of a numerical notation system are arranged and combined. Lexical numeral: A spoken or written representation of number in a specific language. Logogram: A grapheme that represents a specific word in a language but is essentially nonphonetic. Mathematics: The science that deals with the logic of quantity, shape, and arrangement. Metrology: The science or practice of weighing or measuring. Multiplicative: An intraexponential structuring principle where the numerical value of a power is expressed using two signs, a unit-sign and a power-sign, whose values are multiplied. Cf. cumulative and ciphered. Number: An abstract concept used to designate quantity. Numeral-phrase: A group of one or more numeral-signs used to represent a specific number. Numeral-sign: A grapheme that represents a numerical value. Numerical notation: Graphic, relatively permanent, and primarily nonphonetic representations of numbers. One-to-one correspondence: The association of a collection of discrete set of identical marks with a set of objects of the same quantity (e.g., IIIIIII = 7). Cf. tallying.

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437

Ordinal number: A natural number used to denote the order or place of an object within a sequence of objects (e.g., first, second, third …). Cf. cardinal number. Ostracon (pl. ostraca): A potsherd (pottery fragment) containing text, usually written with ink. Paleography: The study of pre-modern handwriting, usually on perishable materials, including the study of changes in the forms of graphemes. Cf. epigraphy. Phylogeny: The study or description of relationships between ancestors and descendants. Pictogram: A grapheme, usually an ideogram, that is a visual depiction of the object being represented. Positional: An interexponential structuring principle where the value of a power’s intraexponential signs is affected by their position or place within the numeral-phrase. Also known as place-value. Power: The result of a natural number (e.g., a base) being multiplied by itself some number of times. Quinary: Having a numerical base or sub-base of 5. Sexagesimal: Having a numerical base or sub-base of 60. Stela (pl. stelae): A stone slab or pillar on which an inscription is placed, often as a funerary monument or boundary marker. Stichometry [Greek stichos ‘row, line’ + metria ‘measuring’]: The enumeration of lines of prose or poetry in texts; the measurement of the length of texts by lines of fixed or average length. Stimulus diffusion: The transmission and adoption of an innovation where the basic principle of the innovation diffuses, but where there is some obstacle to its transmission or acceptance. Sub-base: A natural number which, when multiplied by the base of a numerical system or its powers, is specially denoted within the system. Subitizing: The cognitive capacity (in humans and other species) to immediately perceive quantities of no more than three or four objects without counting. Syllabary: A writing system whose signs denote whole syllables rather than single phonemes. Tallying: The practice of making a series of relatively permanent marks to indicate an ongoing total of some objects. Cf. one-to-one correspondence. Translinguistic: Of graphic symbols, not linked inextricably to a particular language; able to be read in multiple languages. Uncial [Latin unciales ‘inch-long’]: Writing in Greek, Roman, or sometimes other writing systems in late antiquity, characterized by large, rounded majuscule letters. Unit fraction: A fraction of the form 1/N, whose numerator is 1. Vigesimal: Having a numerical base or sub-base of 20.

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Index Numerical notation systems are indexed alphabetically by name of system (e.g., “Aramaic numerals”). Number words are indexed under “lexical numerals, Language-name” (e.g., “lexical numerals, English”). Writing systems are indexed under “writing system, Script-name” (e.g., “writing system, Greek”). abacists 123, 147, 220 abacus 97, 104, 115–6, 123–4, 144, 147, 182, 218, 220, 259, 264, 266, 269, 315 Gerbert’s abacus 220 ghubar 218 Greco-Roman 266 Inka 315 pebble-board 97, 104, 144 and Roman numerals 115–16, 124 schety 182 soroban 264 suan pan 259, 264, 269 Abaj Tabalik 288 Abbasid caliphate 219, 403 Abraham ibn Ezra 159, 184, 216, 237 abstract numbers 18, 237 abugida See alphasyllabary Abusir papyri 43, 47 Achaeans 64 Achaemenid Empire 73, 76, 86, 83, 92, 142, 249, 256–8, 413 acrophonic principle 102–3, 107–8, 128, 435 Adab al-kuttāb 215 Adair, James 348 additive principle 12, 371–2, 384, 387, 389, 392, 395–6, 405, 425, 432, 435 definition 12, 435 extendability 371–2, 395–6 Adelard of Bath 120, 223

Aegean numerical notation systems See Linear A / Linear B numerals Afro-Asiatic languages 319 Agabo 152 Agora (Athens) 100, 142 Agrippa of Nettesheim 354 Ahar stone inscriptions 197 Akinidad stela 52, 53 Akkadians 71, 243–4, 248 Akko 76 aksharapallî numerals 205, 210–13 Aksumites 153 Albelda (Logrono), Spain 219 al-Biruni 170 Alexander the Great 143, 388 Alexandria 153, 192 Alexandrine conquest 252, 258, 413 Alexandrine period 104, 143, 192 Algorismus vulgaris 221 algorithmists 123, 147, 221 al-Jāhiz 215 al-Khwārizmī, Muhammad ibn Mūsā 128, 166, 214, 217 al-Mansur (Abbasid caliph) 214 al-Moktadir Billâh (Abbasid caliph) 350 alphabet 75, 92, 157, 187, 435 See also writing system alphabetic systems 35, 67, 132, 133–87, 222, 387, 395, 413, 424–5, 428

471

472

Index

alphasyllabary 133, 152, 189, 256, 435 alphasyllabic numeration See Indian alphasyllabic numerals Alphonsine tables 169 al-Sijzi 214, 218 al-Sūlī 215 Altaic languages 280 Altintepe 64 al-Uqlīdisī 215 al-Ya’qūbī 214 Amenhotep III 41 Amharic language 152 analogy vs. homology 23–4, 432 Anatolia 63–4, 72 Ancient Letters (Manichaean) 87–8 Andalusia 219 Antioch 216, 221 Anyang (Shang capital) 259 Anza stela 152 Apollinius 144 Aqaba 78 Araba 80 Arabia 68, 78 Arabic abjad numerals 91, 118, 127, 133, 146, 149–51, 162–7, 169, 171–2, 213, 215, 217–18, 318, 328, 357–8, 403, 411, 414 decline and replacement 165–6 diffusion and transmission 149–51 fractions 165–6 functions 165–7 origins 162–5 Arabic positional numerals 91, 118, 147–8, 166, 185–6, 213, 215, 227, 309, 324–8, 384, 403, 408, 412, 416, 425, 428 influence on African systems 325–8 Arabico-Hispanic numerals 94, 127–9, 426 Arad 50, 52, 73 Aramaeans 70–1, 73, 78, 158 Aramaic language 71, 73, 83 Aramaic numerals 50, 65, 67–8, 70–4, 83, 86, 91–2, 96, 156, 248–9, 258, 377, 406, 411, 413 diffusion and transmission 72, 74, 413 functions 73–4 origins 71–3, 406, 411 Archimedes 144 Ardubarius See Âryabhata Argos acrophonic numerals 94, 102, 132 Arifmetika 182 Aristarchus 138 arithmetic 2, 30–1, 47–9, 53, 55, 74, 92, 97, 104, 115, 117, 138, 144, 150–1, 159, 165–6, 170, 214–15, 220–2, 251, 268, 295, 315, 324, 337, 346, 431 with Arabic abjad numerals 165 with Arabic positional numerals 166, 214–15 with Aramaic numerals 74

with astronomical fractions 170 with Egyptian demotic numerals 55 with Egyptian hieratic numerals 47–9 with Etruscan numerals 97 with Greek acrophonic numerals 104 with Greek alphabetic numerals 138, 144 with Hebrew alphabetic numerals 159 with Levantine systems 92 with Maghribi numerals 165 with Meroitic numerals 53 with Roman numerals 115, 117 with Western numerals 220–2 with zimām numerals 150 See also abacus; computational technologies; finger-reckoning; tallying Arjabhad See Âryabhata Armenia 145 Armenian numerals 173–5, 185–6, 225, 406, 413, 416, 426 decline and replacement 175, 225, 426 ars notaria 351 Arte del Cambio 123 Âryabhata 177, 205–9 Âryabhata’s numerals 205–9, 379, 381–2, 394, 411 Âryabhatiya 205 Asia Minor 64–6, 68 Aśoka 83, 188–9, 192 Assyrians 65, 70–1, 76, 248, 257 Assyro-Babylonian common numerals 44, 63, 65, 71–2, 91, 229, 245, 247–51, 254–6, 258, 371, 377, 387, 395, 409, 411, 416, 422 astrolabe 150, 183 Astronomia 183 astronomical fractions 167–70, 185 Athens 99, 102, 104, 142–3, 105–6 Augsburg 125 Augustan period 111–12 Aurelian (Roman emperor) 78 Austria 223–4 Axial Age 424 Aymara 316 Aztec, Aztecs 284, 289, 300 Aztec numerals 225, 285, 300–2, 305, 368, 381–2, 384, 390, 402, 405, 419 Babylonia, Babylonians 194, 248, 252, 254, 257, 289 Babylonian positional numerals 11, 167, 229, 248–55, 367–8, 377, 379, 381–2, 387–9, 402, 416 origins 250–1, 402 use in mathematics 251–2, 388 Bactria 192 Bagam numerals 325–6 Baghdad 166, 214, 217, 219 Balinese numerals 200 Bambara numerals 309, 317–19, 368, 373, 418–20

Index Bamum numerals 322–4, 325, 370, 381–2, 384–5 mf␧mf␧ variant 323, 381–2 Bangka 195 Bangladesh 198, 212 bar-and-dot numerals (Mesoamerican) 225, 284–97, 301, 368, 376, 380–22, 401, 416, 419 decline and replacement 290, 401 origins 287–8, 419 quasi-positional 290–7, 381–2 Rule of Four 368, 376 use in mathematics 295–6 base (numerical) 4, 435 Basingstoke numerals 350–2 Bede 146 Belize 284 ben Gerson, Levi 169 Bengali numerals 198 Benin civilization 328 Berber numerals 319–21 Berlin Papyrus 49 Bété numerals 326 bhutasaṃkya 195 Bianco, Jean 342–3 bisexagesimal numeration 229 blended systems 222, 411, 435 Bodra, Lako 335 Bogazkoy 64 bolorgir 173 Bonampak mural 293 Borges, Jorge Luis 430 Borough, Christopher 182 boustrophedon 67, 99, 107, 134, 435 Brāhmī numerals 56, 67, 84, 115, 146, 189–97, 204, 208, 212, 226, 267, 335, 373, 381–2, 406, 408, 412, 416, 419, 424 diffusion and transmission 192–3, 204, 412 origins 191, 406 transition to ciphered-positional 193–7 Bronze Age 58 Bruges 354 Buddhism 88, 197, 202–3, 268, 273, 275 Bulgaria 180–1 bullae 234–5 Bungus, Petrus 125 Bürgermeisterbuch (Frankfurt) 124 Burma 212 Burmese numerals 200 Byzantine Empire 118, 147, 165, 172, 181, 185 Byzantium 86 Cairo 150 caldéron 129 calendars 36, 46, 126–7, 129–31, 159–60, 195, 233–4, 270, 273, 280, 285–99, 302, 307–8, 313, 331–2, 353 Chinese 270, 273 Egyptian 36, 46

473

European 126–7, 129–31, 353 Hebrew 159–60 Indian 195 Indus (Harappan) 331–2 Inka 313 Kitan 280 Mesoamerican 285–99, 302, 307–8 Mesopotamian 233–4 calendar numerals 94, 129–31 Cambodia 212 Cameroon 322, 325 Canaan 254 Canhujo-daro 332 Cantonese language 280 capacity system submultiples See Horus-eye fractions capitalism 126, 225, 426–7, 433–4 Cardano, Girolamo 354 cardinal numbers 4, 435–6 Caria 55, 134, 140 Carmen de algorismo 221 Carolingian Renaissance 118 Carthage 76, 319 Cascajal block 287 Cecil, William (Lord Burghley) 124 centesimal numeration 435 Central Asia 68 Central Asian systems 199 Chamberlain, Basil Hall 338–40 Changan (China) 275 Char Bakr 167 Cherokee numerals 27, 226, 309, 343–5, 358, 365, 381–2, 384–5, 395, 403–4 Chickasaw numerals 348 chikusaku 267 Chilam Balam 290 Childe, Gordon 14 China 197, 202, 262, 273, 275, 281–2, 333, 403, 409, 422, 426 Chinese classical numerals 177, 191, 202, 227, 260, 269–78, 281, 283, 336, 340, 370–2, 378, 380–2, 403, 409, 411, 415–6, 428 diffusion and transmission 273, 277, 281, 336, 340, 403 modern persistence 409, 415–16, 428 positional variants 260, 270, 283, 381–2, 403 variant forms 273–8 Chinese commercial numerals 260, 269, 277–80, 283, 372, 411 Chinese counting-rod numerals 264–70, 269, 283, 365, 376–7, 379, 389, 411, 416, 419, 425 diffusion and transmission 267, 269 origins 266, 419 Chinese language 262

474

Index

Chioggia 98 Chisanbop finger computation 380 Ch’olan languages 293 Christianity 81, 321, 355–8, 405 Chronica maiora 351 Chronicon Paschale 206 chronograms 119, 159, 166–7, 196, 210, 224, 413, 435 Chrysostomus, Johannes 224 chunking 15, 376–9, 435 cifre chioggiotte 98 ciphered principle 11, 48, 133, 142, 168, 368, 378, 389, 393, 396, 421, 425, 432, 435–6 ciphered-additive systems 12–13, 75, 146, 165, 185–7, 374, 382–3, 384, 388, 390–1, 393–5, 397, 424 ciphered-positional systems 12–13, 193–5, 378, 382–4, 388, 390–1, 393–4, 397, 424, 425, 428 Cistercian numerals 350–4, 371–2, 410 Codex Cospi 289 Codex Fejervary-Mayer 289 Codex Kingsborough See Kingsborough Codex numerals Codex Mariano Jimenez 307 Codex Puteanus 119 Codex Reginensis 119 Codex Selden 289 Codex Sierra 302 Codex Telleriano-Remensis 301 Codex Urbinas Latinus 146 Codex Vigilanus 219, 223 Codex Vindobonenisis 795 155 Códice de Santa María Asunción 303 Códice Vergara 303 cognitive psychology 14 coins See money Cologne 224 colonialism 225, 309, 316, 347, 384, 389, 402, 405, 420, 425–6 Columna rostrata 112 Commentary on the Sentences of Peter Lombard 122 Commission on Inupiat History, Language and Culture 347 comparative ethology 15 comparativism 6–8 composite multiplicands 370–1 computational technologies 18, 124–6, 215–16, 218, 220, 234–5, 242, 264, 266–7, 268–9, 338, 346–8, 348, 379–80, 315, 406–7, 412, 417, 433 apices 220 body counting 18 bullae 234–5 chikusaku 267 Chisanbop 346, 380 dust-board 215–16, 218, 412

electronic 407 finger reckoning 215, 379–80, 417, 433 ketsujo 338 kitāb al-takht 215 LoDagaa cowrie shells 18 sangi 267 tables of squares 242 takht 218 See also abacus, Chinese counting rods, tallying conciseness 395, 397–9, 402, 432 concrete counting 18, 234, 237 consonantaries 70, 81, 89, 319, 435 conspicuous computation 112 Constantinople 147, 182 constraints 6, 404, 431 Continuity Principle 362 Coptic language 148, 151 Coptic numerals 44, 54, 56, 145, 148–9, 185–6, 225, 369, 405, 411, 416 Copts 149–50, 153 Cordoba 219 counting board See abacus cowrie shells 18 Cretan hieroglyphic numerals 34, 43, 58–61 Crete 56, 98 Croatia 178 cryptography 39–40, 216, 410 cryptoquantum numeration, Maya 307 cultural evolution 401, 419–20 cultural identity 187, 408–9 cumulative principle 11, 22, 142, 368, 374, 376–8, 384, 388–9, 391, 393, 394, 405, 425, 432, 435–6 definition 11, 435–6 cumulative-additive systems 12–13, 144–6, 372–4, 382–4, 388–97, 422, 424 cumulative-positional systems 12–13, 382–3, 387–8, 390–7, 394 cursive 44, 47, 90–1, 111, 118, 145, 149, 172–3, 349, 412, 418, 436 definition 436 effect on system development 91, 412 Cushing, Frank 347–8 Cypriote syllabic numerals 34, 65–6, 72, 96 Cypriote tallies 329–30 Cyprus 58, 63, 65–6, 329–31 Cyrenaic numerals 100–1 Cyrene 100–1 Cyrillic numerals 146, 180–2, 185–6, 225, 364, 366, 369, 375, 416, 426 decline and replacement 182, 225, 426 divergence from Ordering Principle 364, 375 da xie shu mu zi numerals 277, 410 Dadda III 194

Index Damascus 76, 78–9 Damerow, Peter 236–40, 243, 408 damgalu numerals 357–8 Daodejing (Tao Te Ching) 266 Dark Age (Greece) 63 Datini, Francesco 223 Dawei, Cheng 276 De calculatione 223 De inventione litterarum 184 De numeris 354 De occulta philosophia 354 de Sacrobosco, John 221 De subtilitate libri XXI 354 De temporum ratione 146 de Villa Dei, Alexander 221 Dead Sea Scrolls 156 decimal numeration 436 Decourdemanche, M. J. A. 355–8 Dehaene, Stanislas 17, 29 Deir el Medina 47 Dengfeng County (China) 266 denotation 2, 30–1 determinism 6 Devanagari numerals 198 Devendravarman 197 dewani See Siyaq numerals Dhyāna text 88 diachronic analysis 7, 32, 360–1, 380–401, 418, 432 diacritic 436 Diela 280 Distance Numbers (Maya) 291 divination 210 Djaga people 329 Djamouri, Redouane 262 Dogon people 328 Dravidian languages 201–2, 204, 330 Dresden Almagest 183 Dresden Codex 291, 295, 297–9, 307 Duixiang siyan zazi 269 Dunhuang (Gansu province) 270, 273 Durandus of Saint-Pourcain 122 dust-boards 166, 215, 412 Early Dynastic period (Mesopotamia) 236, 238, 242, 244, 407 East Asian systems 259–83, 359, 366, 418, 425 East Germanic languages 154 Easter Island 342 Easter Island numerals 342–3 Eastern Roman Empire 145–6, 185 See also Byzantine Empire Ebla 245, 248 Eblaite numerals 244–7, 422 Edessa 80–1, 160 Eghap people 325

475

Egypt 49, 55, 64, 68, 71–3, 79, 149–51, 163, 333, 405–6, 417, 419, 422, 424 Egyptian demotic numerals 34, 40, 49, 54–6, 73, 140–1, 192, 405–6, 413, 416 decline and replacement 56, 405 diffusion and transmission 55–6, 140–1 functions 54, 55 origins 49, 54 use in mathematics 40, 55 Egyptian hieratic numerals 34, 40–9, 66, 73, 79, 91, 156, 192, 381–2, 384, 403, 412–13, 415–16, 418 diffusion and transmission 49, 413 fractions 46 functions 41, 43–4, 48 multiplicative 42, 46–7, 403 origins 41, 44, 46–8 relationship with hieroglyphs 39–42, 46–8 use in mathematics 40, 46–7, 49 Egyptian hieroglyphic numerals 34–44, 66, 71–4, 91, 96, 109, 238, 248, 333, 368, 373, 376, 381–2, 384, 408, 412, 415–19 comparison to hieratic 39–42 cryptographic ciphered numerals 39, 40 decline and replacement 44, 47, 417 diffusion and transmission 39, 40, 43–4, 71–4 fractions 42–3 functions 40–2 multiplicative 41–2 origins 35–8, 419 Egyptian language 35, 37, 148 Egyptian Mathematical Leather Roll 49 Elagabalus (Heliogabalus) 115 Elamite language 258 Elements 147 Elephantine 71, 73 Elizabeth I (English monarch) 124 El Portón 288 emporion 141 England 124, 130, 133, 223–5, 351 Englund, Robert 239–40, 408 Epidaurus numerals 94, 132 epigraphy 436–7 episemons 134 Eratosthenes 253 erkat’agir 173 Ethiopia 133, 148, 152–4, 169 Ethiopic numerals 54, 145, 148, 152–4, 186, 282, 371, 416 Etruria, Etruscans 93, 96–7 Etruscan “abacus-gem” cameo 97 Etruscan numerals 62, 67, 76, 94–8, 100, 103–4, 113–14, 132, 364, 373, 376, 379, 380, 388, 405, 419–20 decline and replacement 97–8, 388, 405 origins 95–8, 103, 419–20 Euclid 147

476 Euphrates River 254 Evans, Arthur 58 Exchequer 126 extendability 31, 371, 387, 395–9, 402, 432 Fairservis, Walter 331–2 Faliscan language 96 Fara (Ṧuruppak) 242 Fara period 238 Farsi language 86, 91 Fasti Danici 131 Faye, Assane 327 Fenghuangshan, Hubei province 266 Fez numerals 151, 165, 171–3, 185–6, 378, 410 Fibonacci (Leonardo of Pisa) 221 figure indice, “Indian figures” 216, 219, 221 figure toletane, “Toledan figures” 216, 221 fingers, finger-reckoning 116, 215, 346, 379–80, 417, 433 Flanders 184 Florence 123, 222, 410 fractions 362–4 Arabic abjad numerals 165–6 Aramaic numerals 71 astronomical (sexagesimal) 167–70, 253, 388 Aztec numerals 302 Babylonian numerals 388 Berber numerals 319 Chinese counting-rod numerals 265 Cretan hieroglyphic numerals 59 Egyptian hieratic numerals 46 Egyptian hieroglyphic numerals 42 Fez numerals 171 Greek acrophonic numerals 104, 139 Greek alphabetic numerals 139, 141 Horus-eye fractions 42–3, 363 Linear A numerals 57 Phoenician numerals 75 proto-cuneiform numerals 231 Roman numerals 115–16, 125 Sogdian numerals 87–8 Syriac alphabetic numerals 160 Texcocan line-and-dot numerals 303 unit-fractions 42, 139, 437 Western numerals 224 France 121, 223–4, 353–4 Frankfurt 222 frequency dependent bias 404, 408 Fula (Adama Ba) numerals 326–7 Fula (Dita) numerals 326 Funes, the Memorious 430 Galton’s problem 5 Ganda 329 Gandhara 83 Gangâ dynasty 189, 196 Ganges River 198

Index Gansu corridor 273 Gaon, Saadia 158 Garamond, Claude 224 Garni 174–5 Gbili (Kpelle chief ) 327 Geertz, Clifford 9 Ge’ez language 103, 108, 152 gematria 159 Genoa 221 Georgian numerals 177–8, 185–6, 225, 392, 406, 413, 416, 426 Gerbert of Aurillac (Pope Sylvester II) 123, 220 Germany 124, 127, 130, 133, 223–4, 354 Ghadames 319–20 ghubar numerals See Maghribi numerals Gilcrease Museum (Oklahoma) 343 Girsu 244, 250 Glagolitic numerals 146, 178–80, 185–6, 225, 364, 375, 426 decline and replacement 180, 426 divergence from Ordering Principle 364, 375 Goldenweiser, Alexander 6 Goody, Jack 18 Gothic language 154 Gothic numerals 145, 154–6, 185–6, 388 Grahacāranibandhana 209 Grantha numerals 199 graphemes 3, 19, 195, 218, 365, 436 Greco-Roman period (Egypt) 44 Greece, Greeks 76, 93, 98–9, 102–4, 108, 142–5, 223, 351–2, 407–8 Greek acrophonic numerals 3, 62, 94, 96, 99–106, 128, 132, 134, 139, 141–2, 144–5, 246, 392, 407–8 archaic variants 94, 100–2 arithmetic with 144–5, 407 decline and replacement 104–5 diffusion and transmission 103 epichoric variants 100–2 fractions 104, 139, 141 functions 100, 103, 105, 144, 408 origins 103 use with abacus 104, 144 Greek alphabetic numerals 54–5, 62, 67, 76–7, 79, 82, 86, 115, 117–18, 126–7, 134, 138–47, 151–7, 164, 171–2, 178–9, 181, 185–6, 192, 208, 253, 258, 317, 352, 364–6, 369, 372, 375, 381–2, 388, 397, 405, 411, 413, 416–17, 424 abacus 144 arithmetic with 144–5 decline and replacement 142, 147 diffusion and transmission 142, 145–6, 151, 155–7, 172, 178–9, 181, 424 episemons 134, 142 fractions 139 functions 144–5 hasta 138, 365

Index modern persistence 147, 411 Ordering Principle in 364, 369, 375, 413 origins 55, 134, 140–1 positional variants 253, 381–2 Greek language 65 Greenberg, Joseph 7, 361, 375 Griffith, F. Ll 52, 53 Guaman Poma de Ayala, Don Felipe 315–16 Guatemala 284, 289, 296 Guitel, Geneviève 10–11, 29 Gujarati numerals 198 Gupta Empire 84, 86, 198, 205, 413–14 Hadramauti language 107 Halicarnassus 140, 142 Hammurabi 255 Han Dynasty 264, 273 Hangzhou numerals See Chinese commercial numerals Hao (China) 262 Harappan civilization 189, 330, 333 Harappan numerals See Indus numerals Haridatta 209 Harris Papyrus 44, 45, 47 Harris, Marvin 24, 28 Harvard University 126 Harvey, Herbert 303, 307 hasta 138, 148, 365 Hastivarman 196 Hatra 80 Hatran numerals 68, 69, 80–1, 86, 377, 413 Hattusha 255–6 Hau, Kathleen 321–2 Hebrew alphabetic numerals 133, 145, 149, 156–9, 185–6, 225, 409, 411, 413, 416 origins 156–7 positional variants 158–9 Hebrew hieratic numerals 50–2 Helcep Sarracenicum 120 Hellenistic period 132, 144 heng and zong registers 264, 278 Henry VIII (English monarch) 224 heqat 46 Hermann of Carinthia 183 Herodianic numerals See Greek acrophonic numerals Herodotos 144 Hibeh papyrus 142–3 hieroglyph 34, 436 Hieroglyphic systems 34–67, 132, 376, 387, 395 ḥisāb abjad 162 ḥisāb al-djummal 166 ḥisāb al-ghubar 216–18 ḥisāb al-hindī 166, 213–15, 218 ḥisāb al-qalam al-Fāsī 171 ḥisāb al-ūqūd 165 Histoire comparée des numérations écrites 10

477

Hittite cuneiform numerals 229, 255–6 Hittite hieroglyphic numerals 34, 43, 63–4, 71, 96, 255–6, 368, 406, 422, 424 Hittite language 63 Hittites 63–4 Hmong See Pahawh Hmong numerals Ho (Bihar province) 335 Homiliae 224 homology 23–4 Honduras 284 Hong Kong 269, 280 Horus-eye fractions 42–3, 363 ḥurūf al-zimām 149 hybrid systems 12–13, 70, 113, 132, 181, 257, 368, 378, 392, 394, 396, 431, 435–6 Hypsicles 168 Iberian peninsula 128 Ibibio-Efik people 321 Ibn al-Banna 173 Ibn al-Nadim 350 Ibn ‘Isa, Ali 350 Ibn Khaldun 151, 173, 218 Iceland 121, 223 iconicity 18 ideograms 63, 88, 235, 436 Ifrah, Georges 29–30 Imazighen See Berber numerals imperialism 112, 359, 388, 404–5, 425–6 See also colonialism implicational regularities 367–70, 373 Inam gišhur ankia 252 India 84, 166, 193–4, 197–8, 202, 212, 254, 420, 424, 426 Indian alphasyllabic numerals 205–13, 413 Indian numerals 166, 196–200, 253, 266–7, 381–2, 384, 408, 411–2, 414, 416 Indo-European languages 63 Indonesia 216 Indus numerals 191, 309, 330–3, 373, 376–7, 393, 418–20 Indus Valley 166, 191, 198, 330, 333, 363, 419, 422 Inka Empire 312 Inka numerals 8, 225, 309–17, 358, 368, 373, 405, 418–19 abacus 315 prohibition of use 316 status as numerical notation 312–15 See also khipu interexponential structure 11–12, 364, 368, 375, 381–4, 388, 397, 411, 431, 436 definition 11–12, 436 Iñupiaq language 345 Iñupiaq numerals 226, 309, 344–7, 376–7, 380–2, 384, 409

478

Index

Ionia 55, 140–1 Ionian numerals See Greek alphabetic numerals Iran 68, 74, 86, 161 Iranian languages 87 Iraq 86, 161, 254 Iron Age 52 Islam 88, 92, 146, 166, 215, 218–19, 403 See also Muslims Israel 52, 159 Italic systems 35, 66–7, 93–132, 377, 407, 418 Italy 93, 95–6, 102–3, 105, 109, 118, 129, 147, 216, 221, 223–4, 403, 419, 420 Jackson, George 341 Jacobites 161 Jains 212 Janneus, Alexander 157 Japan 225, 267, 269, 338, 407, 409 Japanese numerals 340 Java 212 Javanese numerals 200 Jebel Aruda 235 Jemdet Nasr 230 Jesuits 276, 426 Jews, Judaism 50–2, 149–50, 156–9 See also Hebrew alphabetic numerals jiaguwen 259 Jiahu, Henan province 262 Jin Dynasty 281–2 Jiushao, Qin 267 Jiuzhang suanshu (Nine Chapters on the Mathematical Art) 268 John of Basingstoke (John of Basing) 350–2 Johnston, Alan 100–3 Judaean weights 51 Judah 52, 73, 157 Jurchin people 267, 275, 281 Jurchin numerals 260, 281–3, 277, 381–2, 387, 395, 409 Kadesh-barnea 50–2 Kahun Papyrus 44–5, 47, 49 Kaiyuan period 197 Kaiyuan zhanjing 275 Kaktovik, Alaska 344 Kalacuri era 194 Kalibangan 332 Kamara, Kisimi 323 Kannada numerals 199 Kantè, Souleymane 327 Karatepe inscription 75 Kartir 86 katapayâdi numerals 205, 209–11, 381–2, 411 Kenadiid, Ismaan Yuusuf 327 Kensington Rune Stone 131 Kerala, India 202

ketsujo 338 keutuklu numerals 355 Khafaji 230 Kharoṣṭhī numerals 68–9, 83–4, 377 Khasekhem 38 khipu 21, 310–12, 315, 338 See also Inka numerals khipukamayuq 312, 315 Khirbet el-Qôm ostracon 157 Khmer numerals 200, 227 Khorsabad 245 Kingsborough Codex numerals 302, 305–6, 372, 381–2, 384 Kitāb al-Fihrist 350 Kitāb al-fusūl fi al-ḥisāb al-hindī 215 Kitāb al-mu’allimin 215 Kitāb al-takht 215 Kitan language 280 Kitan numerals 260, 275, 277, 280–1, 283 Kluge, Theodor 7 Kobel, Jacob 124 Korea 409 Kotakapur 195 Kpelle numerals 326–7 Krakow 182 Kroeber, Alfred 27–8 Kuhn, Thomas 29 Kululu lead strips 64 Kunitzsch, Paul 218 Kusumapura (Bihar) 205 La Mojarra stela 287 La Venta 287 Lachish 50 Lake Van 174 Lam Lay-Yong 266 Lamasba tablet 115 Landa, Diego de 295 Landnámabók 121 Lanfranco 222 Lao numerals 200, 338 Late Period (Egypt) 54, 413 Latin alphabetic numerals 146, 183–6 Latin language 105, 216, 369, 427 Layard, A. H. 71 Lebanon 161 Leigh, Howard 307 Lelang (Pyongyang), North Korea 273 Leonardo of Pisa (Fibonacci) 123, 221 Letoon 105 Levant 52, 68, 70–1, 73–6, 78, 157, 405, 411 Levantine systems 35, 44, 52, 65–92, 106, 132, 248, 258, 376–7, 388 diffusion and transmission 68, 70, 91–2 origins 35, 44, 52, 68, 258 Lex de Gallia Cisalpina 112

Index lexical numerals 3–4, 7–8, 10, 17, 20, 22, 361, 364–7, 369, 374–5, 378–80, 409, 414, 436 Arabic 349 Aramaic 70–1, 74, 86 Armenian 174 auditory aspect 20, 366, 375 Bamum 322 Cherokee 343 Chinese 262, 273, 277 Coahuilteco 367 definition 3, 436 Eblaite 246 Egyptian 35–7 English 22, 378 Etruscan 94, 96 Georgian 177 German 10, 375 Gothic 155 Greek 100, 138–9, 366, 375 Hebrew 50 Iñupiaq 345 Jurchin 409 Latin 109, 111, 242 Maya 288, 298–9, 414 Mende 325 Middle Persian 86 Nabataean 79 Nepali 211 proto-Dravidian 333 quasi-lexical numerical notation 262, 283 relation to Ordering Principle 365, 369, 374 Sanskrit 196, 207, 273 Slavic languages 180 Slavonic 366 Sogdian 87 Sora 367 Sumerian 236–7, 242, 244, 248 Turkish 88 Ugaritic 248 Li Shou 259 Liao Dynasty 280 Liber Abaci 221 Liber Mamonis 183, 216 Linear A numerals 34, 43, 56–8, 321, 406, 408, 422 Linear B numerals 34, 43, 58, 61–3, 95–6, 255, 406, 422 linen lists (Egypt) 37, 39 ling sign 276 literacy 224, 226, 424, 427 Livro de Virtuosa Bemfeitoria 223 LoDagaa people 18 logogram 35, 56, 61, 281, 436 logosyllabary 322 Long Count (Maya) 287, 290, 292, 294 lost-letter theory 101, 114 Lower Egypt 49, 55, 73 Luoyang 263

479

Luwian language 63–4, 105 Luwian numerals See Hittite hieroglyphic numerals Lycian numerals 65, 76, 103, 105–7 Macedonia 178 macrohistory 25, 422–9 Madrid Codex 307 Maghreb 172, 217–9, 318 Maghribi numerals 151, 198, 216–18, 412, 416 magic number 7±2 14, 378 magic squares 175 Magna Graecia 98 Magnitskii 182 Mahrnâmag 89–90 Malabar Coast 212 Malayalam language 212 Malayalam numerals 201–4, 371, 381–2, 409, 416 Malcolm, Captain L. W. G. 325 Maldives 204 Mali 317 Mama, Nji 322 Manchu people 280–2 Mandarin language 283 Manding language 327 Manding numerals 326–7 Mani 89 Manichaean numerals 68–9, 89–90, 92 Mankani copper plate 194–5 Mao Zedong 277 Marathi numerals 198 Mari 248, 253–4, 387 Mari numerals 229, 254–5, 387 Mari Schad Ormizd 89 Maronite Christians 161 mathematics 4, 29–30, 402, 417, 436 See also arithmetic, computational technologies Mauryan Empire 191–2, 406 Maximus Planudes 147 Maya civilization 194, 284–5, 299–300 Maya head-variant numerals 297–9, 308, 368, 394, 414 Maya numerals See bar-and-dot numerals Mende language 323, 366 Mende numerals 226, 323–5, 366, 371, 381–2, 384–5 Mercedarian friars 316 Meroe 52, 54 Meroitic numerals 52–4, 66 Mesoamerica 284, 401, 419 Mesoamerican calendar 290–7, 299, 307 Mesoamerican systems 31, 284–308, 359, 377, 379, 410, 418 Mesopotamia 68, 72, 73, 86, 163, 169, 228, 237, 241, 243, 247–8, 256, 258, 313, 333, 388, 406–9, 412, 419, 422, 424, 436 Mesopotamian systems 228–58

480

Index

Mesrop Mashtots 173–4 Methen 40 Methodius 146, 178–81 Metonic cycle 131 metrology 239–40, 436 Mexican dot-numerals 299–300 Mexico 288, 402 Mexico City 129 mf␧mf␧ numerals See Bamum numerals Miao people 275 Microcosmographia 122 Middle Formative Period (Mesoamerica) 287–9 Middle Kingdom (Egypt) 42, 49 Middle Persian language 86, 89 Middle Persian numerals 68–9, 80, 86–7, 381–2, 384, 412 cursive reduction 87, 412 Middle Persian period 90 Midrash 159–60 Milesian numerals See Greek alphabetic numerals Miletus 63, 134, 140 Miller, George 14, 378 mina 71 Minaea, Minaeans 107–8 Ming Dynasty 269, 276, 278, 282 Minoan numerals See Linear A numerals Minoans 57, 408 Mixe-Zoquean languages 299, 380 Mixtecs 289, 299–300, 302 Mochica 311 Mohenjo-daro 332 Mommsen, Theodor 101, 114 money 30–1, 412, 427, 434 alphabetic numerals 142–3 Arabic abjad numerals 165 Arabico-Hispanic numerals 129 Brahmi numerals 189 Chinese numerals 266–7, 273, 279 Cyrenaic numerals 101 Etruscan numerals 95–6 Greek acrophonic numerals 100–1, 104 Greek alphabetic numerals 147, 158, 164 Hebrew alphabetic numerals 157–8 Nabataean numerals 79 Passamaquody numerals 348 Phoenician numerals 76 Roman numerals 114, 116, 118 Ryukyu numerals 338–9 Shang and Zhou numerals 263 Western numerals 224 Mongols 267 Monte Albán 288 Moravians 178 Morocco 133, 165, 171–3, 216 Moscow Papyrus 49

Moscow School of Mathematics and Navigation 182 Moso See Naxi numerals Mozarabs 171–2 multiplication 18 multiplicative principle 11, 73, 84, 435–6 definition 11, 436 multiplicative-additive systems 12–3, 368, 371–2, 374, 378, 382–97, 425 cognitive factors 390–7 composite multiplicands 371 decline and replacement 388–9 phylogenetic analysis 385–7 structure 368, 374 transformation of 384–7 Munda languages 334–5 Munshi numerals 328 Muqaddimah 151, 218 Muslims 118, 149–50, 172, 318 See also Islam Muziris 192 Mycenae, Myceneans 63–4, 95 Mycenean numerals See Linear B numerals Mysticae numerorum significationis 125 Nabataean numerals 68, 69, 78–80, 92, 107, 158, 213, 377, 413 Nagari numerals 213, 218, 227 Nahuatl language 300, 302 Nana Ghat 190 Napier, John 269 Narmer mace-head 38, 112, 420 Nāsik Cave 190 Natural History 110 natural numbers 10 Naukratis 55, 141 Naxi language 333 Naxi numerals 309, 333–4, 419 Negev 73 Nemea acrophonic numerals 94, 102, 132 Neo-Babylonian empire 76 Neo-Hittite kingdoms 63, 65–6, 71–2, 105 Nepal 198, 212 Nepali numerals 198 Nestorian Christians 87, 161 Neugebauer, Otto 253 New Kingdom (Egypt) 41 New Mathematics 238 Newcastle (England) 225 Nichols, F. H. 334 Nickerson, Raymond 19 Nigeria 322, 328 Nimrud 71 Ninni, A. P. 98 Njoya, Ibrahim 322–3 noncumulative systems 378–9, 384–5, 388–9, 391 nonuniversal regularities 370–2 Norman, Donald 32

Index Normandy 354 North Africa 217 North America 347, 426 North Indian numerals 198 Noviomagus, Johannes 354 Novum Testamentum in Linguam Amharicam 154 Nueva corónica y buen gobierno 315–16 number 4, 20, 436 numeral classifiers 293 numeral phrases 4, 8, 11, 22, 66, 111, 133, 363–7, 370–2, 374–6, 390–2, 396, 436 conciseness 371–2, 390–1 definition 4, 436 interexponential ordering 364 multiplicative 363–4, 370 structure 11, 22, 363, 367, 375–6, 390, 364 subtractive 363, 392 numeral words See lexical numerals numeral-signs 2–3, 11, 19, 39, 393, 396–7, 418, 436 definition 2–3, 436 numerical notation 2–3, 5, 7, 9–33, 38, 236–8, 245, 309, 316, 359–434, 436 biological prerequisites 16–17 and capitalism 426–7, 433–4 cognitive analysis 14–16, 236–8, 360–401, 432 and colonialism/imperialism 309, 316, 384, 389, 402, 404–6, 409, 420, 425–7 computation with 2, 29–33, 402–3, 406–7, 433 decline and replacement 387, 405, 417, 432 definition 3, 436 diachronic analysis 32, 360, 380–9, 399–400, 421, 431 diffusion and transmission 23–8, 402, 405, 408, 414, 432–3 functions 2, 29–33, 38, 245, 402, 404, 407, 411, 413–14, 433 longevity 415–19 medium of writing 412, 432 origins 18, 23–8, 359, 419–21, 432–3 perceived efficiency 29–33 phylogenetic analysis 7, 26–8, 408, 411–12, 433 relationship to lexical numerals 19–22, 361, 366, 379, 409, 433 sociopolitical factors 17, 24, 359, 366, 398, 401–30, 432 structure 3, 360–401, 431 synchronic analysis 32, 360, 362–7, 399–400, 431 transformation of 384, 405, 414, 432 typology 9–14 universals 5, 360–1 and writing systems 19–23, 372, 413, 417–8, 420–1, 433 numerology 151, 159, 210, 414

481

Oaxaca 288 Oberi Okaime numerals 226, 321–2, 391, 394, 409 Ocreatus 120 Ogowe River, Gabon 329 Old Akkadian period 245, 251 Old Babylonian period 245, 248, 250–3 Old Chinese language 262 Old Church Slavonic language 178, 180 Old Kingdom (Egypt) 37, 39, 42, 46 Old Persian numerals 86, 229, 248–9, 256–8 Old Syriac language 81 Old Syriac numerals 68, 69, 78, 81–3, 92, 146, 377 Olmec 287–8, 299 Olynthus numerals 94, 101 one-to-one correspondence 3, 15, 23, 115, 299, 328–9, 397, 436–7 open vs. closed notation 20 oracle bone inscriptions 259 Ordering Principle 364–6, 369, 374–5 ordinal numbers 2, 4, 435–6 ordouï cheïlu numerals 356 ordoui numerals 356–7 Orissa 196 Oriya numerals 198 Oscan language 96 Osmaniya numerals 326–7 ostraca 43, 49, 73, 437 Otlazpan 307 Ottoman cryptographic numerals 355–8, 410 Ottoman Empire 151, 216, 350, 358 Oztoticpac Lands Map 303 Padua 123, 222 pagination 124, 127, 147–8, 150, 161, 167, 178, 182, 210, 270, 427 Pahawh Hmong numerals 277, 309, 336–8, 358, 379, 381–2, 410 Pahlavi numerals 68, 69, 90–2, 350, 381–2, 384, 412 Pakistan 74, 83, 198 Palembang 195 paleography 436–7 Palermo Stone 39 Palestine 164 Palmyra 78, 158 Palmyrene numerals 68–9, 76–8, 92, 366–7, 377, 413 Palsgrave, John 123 Panajachel 296 Pandulf of Capua 223 Panini 205 papyrus 43 Paris 354 Paris Codex 307 Paris, Matthew 351

482

Index

Parnavaz (Georgian monarch) 177 Parthian Empire 80, 86, 192 Passamaquoddy numerals 348 Payne, John Howard 343–4 peasant numerals See calendar numerals period-glyphs (Maya) 291–7 Persepolis Fortification Archive 256 Persia 86, 166, 249, 257, 350 Peter the Great 182 Petra 78–9 Philae 44, 56 Phoenician numerals 43–4, 65–6, 68–9, 71, 74–6, 81, 91, 96, 102–3, 140, 248, 319, 377, 392, 406, 411 diffusion and transmission 65–6, 76, 103 multiplicative 74–5, 96, 103 origins 43–4, 74–6, 406, 411 phoneticism 20, 133 phonograms 35, 280 phrase ordering 374–6, 431 Phrygians 64 phylogeny 25, 437 Piaget, Jean 237 Picardy 354 pictography 58, 114, 437 Pisa 216 Pithom 39 place value See positional principle Pliny the Elder 110 Pompeii 112 Ponce de Leon papers 129 Portugal 124, 223 positional principle 10, 11–12, 122, 124, 167, 193–7, 253–5, 266–7, 365, 384–5, 387–9, 421, 432, 434–7 increase in frequency 122, 124, 226, 385, 389, 425 modern additive descendants 384 nonlinear orientation 306, 351–2, 371–2 quasi-positionality 39, 77, 108, 290–7, 372, 387 Postclassic period (Maya) 289 post-positional systems 428 power (mathematical) 4, 437 Predynastic era, Egyptian 42 Presargonic period 243 principle of limited possibilities 6 printing press 124, 224, 354, 427 Problemata 379 proto-cuneiform numerals 230–8, 240, 365, 367, 373, 376, 379, 419, 422 bisexagesimal systems 232 cognitive correlates 236–8 computer-aided decipherment 230–1 concrete counting 234 double documents 234 EN system 233

GAN2 system 233, 240 origins 234–6 Rule of Four 376 Rule of Ten 379 ŠE systems 233–4, 240 U4 system 233–4 proto-Elamite numerals 57, 229, 238–41 Ptolemaic era 39, 41, 48, 54–6, 104, 141–2, 145, 157, 192, 405 Ptolemy (mathematician) 168 Ptolemy II Soter (Egyptian pharaoh) 142 Ptolemy Philadelphos (Egyptian pharaoh) 39 Punjabi numerals 198 Puruchuco 314 Pylos 62–3 Qa’ba inscription 86 qalam hindī 358 Qatabanian language 107 Qin Dynasty 264, 270, 273, 277 qoppa 134 Quechua language 8, 312, 315–16 quinary numeration 380, 437 Qutan Xida 197, 275 QWERTY principle 404 Rapa Nui See Easter Island Rechenbiechlin 124 Reconquista 127 Regulae de numerorum abaci rationibus 220 Reisner Papyrus 49 Relación de las cosas de Yucatan 295 relativism 5 replacement 380–1, 387–9, 401, 402, 405, 407, 413, 421 Restivo, Sal 9 Rhind Mathematical Papyrus 49 rod-numerals See Chinese counting-rod numerals rokoum See Siyaq numerals Rolewinck, Werner 224 Roman Empire 54, 56, 93, 98, 118, 132, 143, 154, 388, 402–3, 405, 424 Roman numerals 10, 31, 56, 67, 78, 80, 82, 93–6, 101–2, 104, 109–32, 146, 184, 222–5, 241–2, 283, 290, 302, 306, 320, 341, 347, 364, 368, 374, 377, 381–2, 385, 388, 392, 394, 397, 402–3, 405–7, 410–12, 415–17, 424, 426–7, 429 and abaci 115–16, 124 Arabico-Hispanic numerals 127–9 arithmetic with 31, 115–18 calendar numerals 129–31 cursive variants 111, 118 decline and replacement 116, 120, 122–7, 222–5, 412, 426–7, 429 diffusion and transmission 115, 132, 403, 405

Index fractions 115–16, 124 longevity 415–6 lost-letter theory 101, 114 medieval 118–19, 120–21 modern persistence 126–7, 410–1 multiplicative 94, 111–13, 116, 121, 132, 385 Ordering Principle in 364, 374 origins 113–15 positional variants 120, 122, 381–2 prestige functions 126–7, 410–1, 428 Rule of Four 368 sub-base 110–11, 241–2 subtractive principle in 109, 111, 377, 392 variant forms 109, 116, 128–9, 320 Roman Republic 94, 96–7 Rome 86, 93, 96–7, 224, 319, 405, 417 Rule of Four 368, 376, 380, 407 Rule of Ten 363, 379–80 rūmī 171 runic numerals See calendar numerals Rus 182 Russia 133, 179, 182, 223 Russian language 180 Ryukyu numerals 269, 277, 338–40, 364, 384, 407 Saba 107 Sabaean language 107–8 Sacred Round 307 sade 134 Safavid Dynasty 350 Saint Cyril 146, 178–81 Saint Frumentius 153 Saka calendar 195 Salamis tablet 104 Salisbury Cathedral 222 Samaria ostraca 50, 79 Samos 140 Samoyed numerals 340–1 san (sampi) 134 San Andrés cylinder seal 287–8 San José Mogote 288 sangi 267 Sankheda copper plate 194–5 Sanskrit language 195, 375 Saqqara 71, 73, 103 Šargal Šunutaga 244 Sargon II (Assyrian king) 245, 248 Sasanian Empire 86, 160 Saussure, Ferdinand de 8 Scandinavia 129–31, 223 schety 182 Schitanie udobnoe 182 Schmandt-Besserat, Denise 234–5, 421 scripts See writing systems Sebokht, Severus 213–14 Second Intermediate Period 49

483

Secret of Secrets 182 Sefer ha-Mispar 159, 184 Segovia 354 Seleucid Empire 81, 86, 92, 104, 145, 157, 252–3, 258, 367, 413 Semitic languages 73, 245, 248, 409 Sequoyah 27, 343–4 Serbia 178, 181 Sermo in festo praesentationis 224 sexagesimal systems 167–8, 185, 229–30, 232, 379, 437 Shalmaneser V 71 Shang and Zhou numerals 31, 259, 260–2, 264, 270, 283, 365, 372, 415–16, 419 quasi-lexical 262, 365 Shang Dynasty 259, 262 shang fang da zhuan numerals 276, 411 Shapur I (Middle Persian monarch) 80 Shirakatsi, Anania 174–7, 402 Shirakatsi’s numerals 174–7, 185–6, 317, 381–2, 402 sho-chu-ma numerals See Ryukyu numerals Shu shu ji yi 272 Shu shu jiu zhang (Mathematical Treatise in Nine Sections) 267 Shuxue wenda 277 Siberia 340 Sicily 62, 95, 102–3, 221, 224 Siddhantam grant 197 Sidon 74–6 Sierra Leone 323, 325 sign-count 31, 392, 394–5, 397–9, 402, 432 Sinhalese language 204 Sinhalese numerals 204, 282, 392, 409, 416 siyaq numerals 348–50, 410, 416, 419 size-value 365 Sogdian language 87 Sogdian numerals 68, 69, 87–8 Somali language 327 Song Dynasty 267–8, 276, 377 South Arabian numerals 94, 103, 107–9, 132, 154, 320, 368, 385 divergence from Rule of Four 368 quasi-positional 108 South Asian systems 67, 92, 132–3, 188–227, 335, 387, 395, 413, 424, 428 South Semitic languages 78, 107 Southeast Asian numerals 199–200 Spain 76, 116, 118, 127–9, 147, 151, 171–3, 216–23 Spanish New World conquests 284, 289, 299– 300, 316, 401, 425 Sphujidhava 195 Sri Lanka 193, 200, 204 Sriwijaya 195 statistical regularities 361, 370 Stein, Aurel 84 stela 437

484 Stephen of Pisa 183 Steward, Julian 419 stichometry 105, 148, 178, 437 stimulus diffusion 27, 437 Suan fa tong zong 276 sub-base 4, 437 Subhandu 196 subitizing 14–15, 376–9, 435, 437 sub-Saharan decimal positional numerals 325–8 subtractive principle 106, 111–12, 119, 128, 242, 244, 250, 391–2, 418 successive approximation 375 Sumerian language 241, 409 Sumerian numerals 229, 232, 238, 240–50, 333, 365, 367, 377, 379, 381–2, 394, 407–8, 411–12, 416, 422 cuneiform vs. curviform 243, 408 use in mathematics 245, 407 śûnya-bindu 196, 208, 414 Sunzi suan jing 266 survival of the mediocre 417 Susa (Iran) 234–5, 238–41 Susa III period 239 Suzhou (China) 280 Switzerland 127, 224 syllabary 64, 437 syllabograms 56, 64, 281 synchronic regularities 7, 32, 360–80, 399–401, 432 axioms 362 cognitive analysis 373–80 Syntaxis 168 Syria 70, 76, 78, 82, 163–4, 254 Syriac alphabetic numerals 82, 87, 146, 160–1, 164, 185–6, 225, 387, 411, 416 arithmetic with 161 blended system 411 expression of large numbers 160 fractions 160 longevity 416 multiplicative 160, 387 origins 160–1 tables of squares 242 Tajikistan 87 tallying 15, 21, 58, 62, 95, 97–8, 102–3, 130, 132, 310–11, 316, 329–32, 347–8, 419, 436–7 and Roman numerals 115 connection to cumulative-additive systems 373 definition 437 Rule of Four 407 Samoyed numerals 341 sho-chu-ma (Ryukyu) 277, 381, 407 while inebriated 15 Zuni 347

Index Talmud 159 tamgas 179 Tamil numerals 200–3, 227, 370, 381–2, 409, 416 composite multiplicands 371 connection to Malayalam 203, 381–2 origins 200–1 Tang Dynasty 270 Tangut people 275 Tannery, Paul 145 Tarikh 214 Tax, Sol 296 taxograms 61 Tell Qasile 71 Tell Uqair 230 Telugu numerals 199 Tenochtitlan 300 Teotihuacan 289, 299, 307 Teotihuacani numerals 307 Tepe Yahya 239 Tepetlaoztoc 303, 305 Texcocan line-and-dot numerals 285, 303–6, 371–2, 402 fractions 303 non-infinite 371 nonlinear 372 origins 306, 402 Thai numerals 200 Thailand 212, 225 Thames River 225 Theon of Alexandria 168 Thespiae 100 Tibet 212 Tibetan numerals 227, 405 Tikal 288 Tocharian language 193, 199 Tod, Marcus Niebuhr 99–101 Toledo (Spain) 171, 219 Toltecs 300 Tomb U-j (Abydos) 37, 420 Tongwen suanzi qianban 273 transformation 380–7, 401–2, 405, 407, 409, 411, 421 constraints 383–4 definition 380–1 transformational grammar 117 translinguistic notation 22, 418, 433, 437 Trigger, Bruce 6 Tripoli 221 Tsenhor papyrus 55 T’uabant’iwn 176 Tuareg people 319 Tunis 221 Tunisia 218 Turay, Mohamed 323 Turkey 161 Turkish language 88 Tuscany 93

Index Tuxtla statuette 287 Tyre 74–6 Tzeltalan languages 293 Ugaritic numerals 248, 424 Umayyad caliphate 87 Umbrian language 96 Umma 250 uncial 148, 437 unilinear evolution 2, 29, 421 unit-fractions 42, 437 universal grammar 361 universalism 5, 361 universals 6, 360 Upper Paleolithic 16, 23 Ur III period 244, 250–1, 367, 402 Urartian language 64 Urartians 64 Urton, Gary 8, 312–13 Uruk 230, 235, 240 Uruk period 37, 230, 235–6, 407 Uto-Aztecan languages 300 Uygurs 88 Uzbekistan 87 Vâkâtaka grants 190 Valera, Blas 316 Valley of Mexico 302 Varang Kshiti numerals 309, 334–335, 381–2, 384, 410 varnasankhya systems 205, 212 See also Indian alphasyllabic numerals Vasavadatta 196 Vatican Codex 302 Venice 221 vernacular languages 427 Vietnam 273 vigesimal numeration 379, 437 Vināyakapalā 197 vinculum 402 Vygotsky, Lev 237 Walid I (Umayyad caliph) 164 Wallerstein, Immanuel 426 Wari civilization 311 Warring States Period 264, 266, 273 Wells Cathedral 223 Wen Di 266 West African systems 328, 404, 408 Western numerals 2, 10, 22, 29–32, 67, 118, 123–4, 146–8, 159, 182, 185, 188, 193, 198, 213, 216–27, 271, 277, 283, 290, 302, 306, 309, 321–3, 325–8, 335, 338, 344, 346, 365, 381–2, 384, 397, 401, 403, 408–12, 418, 426–9 arithmetical use 29–30, 222 in Bibles 224, 427

485

and Christianity 219–20 diffusion and transmission 22, 219, 221–3, 224–7, 321, 409, 426–7 near-universality of 2, 219, 226, 428–9 origins 146–48, 219, 222 positive characterizations of 29, 31, 346 prohibition of use 123–4, 222 Whalley Abbey (Cheshire) 352 Wiener, Charles 316 Williams, Barbara 303, 307 Winkelhaken 243, 247 Wolof numerals 326–8 Worm, Ole 131 writing, scripts 19–22, 413, 417–18 writing system Arabic 89, 133, 149, 198, 435–6 Aramaic 71, 73, 76, 80, 84 Armenian 173 asomtavruli 177 Bamum 322–3 Berber 319 Bhattiprolu 200 Book Pahlavi 87, 90, 91 Brāhmī 84, 188, 200 Canaanite 74 Cham 195, 200 Chinese 270, 417 chu’ nom (Vietnam) 275 cismaanya 327 Coptic 44, 148–9 Cretan hieroglyphic 57–9 cuneiform 73, 167, 228, 248–9, 436 Cypriote syllabary 65 Cypro-Minoan 65 dongba (Naxi) 333–4 Egyptian demotic 35, 54, 148 Egyptian hieratic 35, 49, 55 Egyptian hieroglyphic 35, 44 Elamite 73 Ethiopic 133, 152 Etruscan 94–5, 109 far soomaali 327 geba (Naxi) 333 Georgian (asomtavruli) 177 Georgian (mxedruli) 177 Glagolitic 178 Gothic 154, 388 Grantha 199, 202, 204 Greek 44, 63, 96, 99, 101, 105, 134, 148, 437 Greek (Chalcidic) 114 Greek (Euboean) 94 Greek (Ionic) 140–1 Greek epichoric scripts 99, 101, 102, 105 Gupta 199 hangul (Korean) 275 Hasmonean 157 Hatran 80

486 writing system (cont.) Hebrew 50, 156, 435–36 Hittite cuneiform 63, 248, 255 Indus (Harappan) 330–1, 333 Isthmian 287, 291 Jurchin 281–2 kanji (Japanese) 275 Kannada 199 Kawi 200 Kharoṣṭhī 83, 84, 188, 193 Kikakui (Mende) 323–4 Kitan 280–1 Kpelle 327 Latin 22, 95, 109, 115, 183, 327, 417, 437 Linear A 56–7 Linear B 64 Luwian 63 Lycian 105 Malayalam 199–200, 202 malimasa (Naxi) 333 Manichaean 87, 89 Maya 285, 287 Meroitic cursive 49, 52 Meroitic hieroglyphic 35, 52 Mesoamerican 255, 289 Middle Persian 86 Minaeo-Sabaean 152 mxedruli 177 Nabataean 79 Nestorian 160 N’ko 327 North Semitic 108 Oberi Okaime 321 Ol Cemet’ 334–5 Old Khmer 195 Old Malay 195 Old Persian 73, 256 Old Syriac 80 Pahawh Hmong 336 Phoenician 70, 74–5, 99, 156 proto-cuneiform 230 rongorongo 342 runic 129–31 Sanskrit 195 Semitic 108, 140 Serto 160 Sinhalese 199, 200, 204 Sogdian 87–8 Sorang Sompeng 334–5 South Arabian 107–8 Sumerian 230, 241, 243 Syriac 81–3, 87, 89 Tamil 199–200 Telugu 199 Teotihuacan 307 Tifinigh 319 Tocharian 199 Ugaritic 248 Uygur 280

Index Vai 323, 327 Varang Kshiti 334–5 Visigothic 118 Wolof 327 Zapotec 287, 291 Wulfila 154, 388 Xanthus 105 Xcalumkin 294 Xiyin, Wanyan 282 Xizong (Chinese emperor) 282 yakâ-ne talápha 348 Yang, Shong Lue 336 Yavanajātaka 195 Yellow Emperor 260 Yezdigird III (Persian monarch) 87 Yoruba civilization 328, 420 Yucatan 284, 289 Yucatecan language 293 Yue, Xu 272 Yunnan province (China) 333 Zacuto, Abraham 354 zā’irajah technique of divination 151 Zapotec civilization 288, 299 Zapotec language 380 Zapotec numerals 288 Zenobia 78 zero 22, 362, 365, 371–2, 390–2, 414 alphabetic numerals 147, 159, 169, 186–7 Arabic numerals 169, 213–14 Âryabhata’s numerals 208–9 astronomical fractions 169 Babylonian numerals 250–3 Bété numerals 326 Chinese classical numerals 275–7 Chinese commercial numerals 278 Chinese counting-rod numerals 267–8 Indian positional numerals 194–7, 427 Inka numerals 310 Iñupiaq numerals 345–6 katapayâdi numerals 209 Linear A numerals 57 Malayalam numerals 203 Maya numerals 286, 292, 294, 296–7 Oberi Okaime numerals 321 Pahawh Hmong numerals 336–7 Roman numerals 120 Tamil numerals 202 Texcocan numerals 304, 306 Western numerals 123–4, 221, 224 Zhang, Jiajie 32 Zhou Dynasty 262–3 Zhou numerals See Shang and Zhou numerals zimām numerals 149–52, 171–2, 186, 218, 387, 411 Zoroastrianism 89, 91 Zuñi numerals 347–8