Numerical Notation: A Comparative History

This page intentionally left blank

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


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

Levantine Systems

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.

Levantine Systems

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.

Levantine Systems

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

Levantine Systems

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

Levantine Systems

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

Italic Systems

105

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

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).

East Asian Systems

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).

Mesoamerican Systems

291

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

Mesoamerican Systems

293

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

Mesoamerican Systems

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”

Mesoamerican Systems

297

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).

298

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

Mesoamerican Systems

301

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

302

Numerical Notation

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.

Mesoamerican Systems

303

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

Numerical Notation

304

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

Mesoamerican Systems

305

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

306

Numerical Notation

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,

Mesoamerican Systems

307

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.

308

Numerical Notation

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

310

Numerical Notation

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

Miscellaneous Systems

313

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

314

Numerical Notation

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

Miscellaneous Systems

315

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


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




>
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

400

Numerical Notation

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

402

Numerical Notation

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

Social and Historical Analysis

403

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

404

Numerical Notation

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

Social and Historical Analysis

405

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

406

Numerical Notation

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

Social and Historical Analysis

407

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.

408

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).

Social and Historical Analysis

409

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

410

Numerical Notation

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,

Social and Historical Analysis

411

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.

412

Numerical Notation

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

Social and Historical Analysis

413

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

414

Numerical Notation

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.

Social and Historical Analysis

415

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.

416

Numerical Notation

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

Social and Historical Analysis

417

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.

Social and Historical Analysis

419

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).

Social and Historical Analysis

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.

Social and Historical Analysis

425

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

426

Numerical Notation

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

Social and Historical Analysis

427

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

428

Numerical Notation

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

Social and Historical Analysis

429

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

430

Conclusion

431

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.

432

Numerical Notation

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,

Conclusion

433

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,

434

Numerical Notation

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.

435

436

Glossary

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.

Glossary

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.

Bibliography

Abasiattai, Monday B. 1989. The Oberi Okaime Christian mission: towards a history of an Ibibio independent church. Africa: Journal of the International African Institute 59(4): 496–516. Abgarian, G. W. 1962. The Matenadaran. Erevan: Armenian State Publishing House. Abramian, R., ed. 1965. Pekhleviisko-persidsko-armiano-russko-angliiskii slovar. Erevan: Mitk. Absolon, Karl. 1957. Dokumente und Beweise der Fähigkeiten des fossilen Menschen zu zählen im mährischen Paläolithikum. Artibus Asiae 20(2/3): 123–150. Abu-Lughod, Janet. 1989. Before European Hegemony: The World System A.D. 1250–1350. New York: Oxford University Press. Acharya, Subrata Kumar. 1993. The transition from the numerical to the decimal system in the inscriptions of Orissa. Journal of the Epigraphical Society of India 19: 52–62. Adams, R. F. G. 1947. Ob␧ri Okaim␧: a new African language and script. Africa 17: 24–34. Adkins, Julia E. 1956. Historical and analytical study of the tally, the knotted cord, the fingers, and the abacus. Ph.D. dissertation, Ohio State University. Aghali-Zakara, Mohammed. 1993. Les lettres et les chiffres: écrire en berbère. In A la croisée des études libyco-berbères: mélanges offerts à Paulette Galand-Pernet et Lionel Galand, ed. Jeannine Drouin and Arlette Roth, pp. 141–155. Paris: Librairie Orientaliste Paul Geuthner. Aharoni, Yohanan. 1966. The use of hieratic numerals in Hebrew ostraca and the shekel weights. Bulletin of the American School of Oriental Research 184: 13–19. Allen, James P. 2000. Middle Egyptian: An Introduction to the Language and Culture of Hieroglyphs. New York: Cambridge University Press. Allrik, H. L. 1954. The lists of Zerubabel (Nehemiah 7 and Ezra 2) and the Hebrew numeral notation. Bulletin of the American Schools of Oriental Research 136: 21–27. Amiet, Pierre. 1966. Il y a 5000 ans les Elamites inventaient l’écriture. Archéologia 12: 16–23.

439

440

Bibliography

Anderson, W. French. 1956. Arithmetical computations in Roman numerals. Classical Philology 51(3): 145–150. Anderson, W. French. 1958. Arithmetical procedure in Minoan Linear A and in MinoanGreek Linear B. American Journal of Archaeology 62: 363–369. Anderson, W. French. 1971. Arithmetic in Maya numerals. American Antiquity 36(1): 54–63. Archibald, R. C. 1921. 1321, 1471, 1521, 1571, 1621, 1671, 1721, 1771, 1821. American Mathematical Monthly 28(11–12): 423–427. Arnett, William S. 1982. The Predynastic Origin of Egyptian Hieroglyphs. Washington, DC: University Press of America. Arnold, William R. 1905. Additional Palmyrene inscriptions in the Metropolitan Museum of Art, New York. Journal of the American Oriental Society 26: 105–112. Ascher, Marcia. 1991. Ethnomathematics. Pacific Grove, CA: Brooks/Cole. Ascher, Marcia, and Robert Ascher. 1972. Numbers and relations from ancient Andean quipus. Archive for History of Exact Sciences 8: 288–320. Ascher, Marcia, and Robert Ascher. 1980. Code of the Quipu: A Study in Media, Mathematics and Culture. Ann Arbor: University of Michigan Press. Ascher, Marcia, and Robert Ascher. 1989. Are there numbers in the sky? In Time and Calendars in the Inca Empire, ed. Mariusz S. Ziolkowski and Robert M. Sadowski, pp. 35–48. BAR International Series 479. Avigad, N. 1975. A bulla of Jonathan the High Priest. Israel Exploration Journal 25: 8–12. Babajanov, Bakhtyar, and Maria Szuppe. 2002. Corpus inscriptionum iranicarum, part IV, vol. XXXI (Uzbekistan), Les inscriptions persanes de Chār Bakr, nécropole familiale des Khwāja Jūybārī près de Boukhara. London: School of Oriental and African Studies. Bag, A. K. 1990. Indian arithmetic in Central Asia. In Interaction between Indian and Central Asian Science and Technology in Medieval Times, vol. 1., ed. W. H. Abdi et al., pp. 288–296. New Delhi: Indian National Science Academy. Bagheri, Mohammad. 1998. Siyaqat accounting: its origin, history, and principles. Acta Orientalia Academia Scientiarum Hungaricae 51(3): 297–301. Baines, John. 2004. The earliest Egyptian writing: development, context, purpose. In The First Writing, ed. Stephen D. Houston, pp. 150–189. Cambridge: Cambridge University Press. Barradas de Carvalho, Joaquim. 1957. Sur l’introduction et la diffusion des chiffres arabes au Portugal. In Bulletin des études portugaises 20: 110–151. Barthel, Thomas S. 1962. Zählweise und Zahlenglaube der Osterinsulaner. Abhandlungen und Berichte des staatlichen Museum für Volkerkunde (Dresden) 21: 1–22. Barthel, Thomas S. 1971. Pre-contact writing in Oceania. In Current Trends in Linguistics, vol. 8, part 2, ed. Thomas Sebeok, pp. 1165–1186. The Hague: Mouton. Basham, A. L. 1980. Early imperial India. In The Encyclopedia of Ancient Civilizations, ed. Arthur Cotterell, pp. 184–191. London: Penguin. Bayley, Clive. 1882. On the genealogy of modern numerals. Journal of the Royal Asiatic Society 14: 335–376. Beaujouan, Guy. 1947. Étude paléographique sur la “rotation” des chiffres et l’emploi des apices du Xe au XIIe siècle. Revue d’histoire des sciences 1: 301–313. Beeston, A. F. L. 1984. Sabaic Grammar. Manchester: Journal of Semitic Studies. Bekken, Otto B. 2001. Algorismus of “Hauksbok”: An Old Norse text of 1310 on HinduArabic numeration and calculation. http://home.hia.no/~ottob/ALGORISMUS.pdf. Accessed on May 8, 2008.

Bibliography

441

Bendall, Cecil. 1896. On a system of letter-numerals used in South India. Journal of the Royal Asiatic Society 28: 789–792. Bender, Marvin L., Sydney W. Head, and Roger Cowley. 1976. The Ethiopian writing system. In Language in Ethiopia, ed. M. L. Bender, J. D. Bowen, R. L. Cooper, and C. A. Ferguson, pp. 120–129. London: Oxford University Press. Benedict, Suzan Rose. 1914. A Comparative Study of the Early Treatises Introducing into Europe the Hindu Art of Reckoning. (Ph.D. dissertation, University of Michigan.) Concord, NH: The Rumford Press. Bennett, Emmett L. 1950. Fractional quantities in Minoan bookkeeping. American Journal of Archaeology 54: 204–222. Bennett, Emmett L. 1996. Aegean scripts. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 125–133. New York: Oxford University Press. Bennett, Wendell C. 1963. Mnemonic and recording devices. In Handbook of South American Indians, vol. 5., ed. Julian Steward, pp. 611–619. New York: Cooper Square. Berdnikov, Alexander, and Olga Lapko. 1999. Old Slavonic and Church Slavonic in TEX and Unicode. In EuroTEX ,99 Proceedings, ed. Günter Partosch and Gerhard Wilhelms, pp. 1–32. http://www.uni-giessen.de/partosch/eurotex99/berdnikov2.pdf. Berggren, J. Lennart. 2002. Medieval arithmetic: Arabic texts and European motivations. In Word, Image, Number: Communication in the Middle Ages, ed. John J. Contreni and Santa Casciani, pp. 351–365. Florence: Edizioni del Galluzzo. Berlin, Brent. 1968. Tzeltal Numeral Classifiers. The Hague: Mouton. Bianco, Jean. 1990. Une numération de Polynésie Occidentale dans les tablettes de l’Ile de Pâques? In Circumpacifica: Festschrift für Thomas S. Barthel, vol. 2., ed. Bruno Illius and Matthias Laubscher, pp. 39–53. Frankfurt: Peter Lang. Bidwell, James K. 1967. Mayan arithmetic. Mathematics Teacher 74: 762–768. Biella, Joan Copeland. 1982. Dictionary of Old South Arabic, Sabaean Dialect. (Harvard Semitic Series, no. 25.) Chico, CA: Scholars Press. Billington, James H. 1968. The Icon and the Axe: An Interpretive History of Russian Culture. New York: Alfred A. Knopf. Biot, Edouard. 1839. Notes sur la connaissance que les Chinois ont eue de la valeur de position des chiffres. Journal Asiatique (Dec. 1839): 497–502. Birket-Smith, Kaj. 1966. The circumpacific distribution of knot records. Folk 8: 15–24. Bischoff, Bernard. 1990. Latin Palaeography. Cambridge: Cambridge University Press. Boas, Franz. 1940. Race, Language and Culture. New York: The Free Press. Bockman, Harald. 1989. The typology of the Naxi Tomba script. In Ethnicity and Ethnic Groups in China, ed. Chien Chiao and Nicholas Tapp, pp. 149–156. (New Asia Academic Bulletin, vol. 8.) Hong Kong: New Asia College. Bonfante, Larissa. 1990. Etruscan. Berkeley: University of California Press. Bonfante, Larissa. 1996. The Scripts of Italy. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 297–311. New York: Oxford University Press. Boone, Elizabeth Hill. 2000. Stories in Red and Black: Pictorial Histories of the Aztecs and Mixtecs. Austin: University of Texas Press. Borges, Jorge Luis. 1964. Labyrinths, ed. Donald A. Yates and James E. Irby. New York: New Directions. Bowditch, Charles P. 1910. The Numeration, Calendar Systems and Astronomical Knowledge of the Mayas. Cambridge: Cambridge University Press.

442

Bibliography

Boyer, A. M., Rapson, E. J., and Senart, E. 1920. Kharosthi Inscriptions Discovered by Sir Aurel Stein in Chinese Turkestan. Oxford: Clarendon. Boyer, Carl B. 1944. Fundamental steps in the development of numeration. Isis 35: 153–165. Boyer, Carl B. 1959. Note on Egyptian numeration. Mathematics Teacher 52: 127–129. Boysen, S. T., and G. G. Berntson. 1989. Numerical competence in a chimpanzee Pan troglodytes. Journal of Comparative Psychology 103: 23–31. Boysen, S. T., and G. G. Berntson. 1996. Quantity-based interference and symbolic representation in chimpanzees (Pan troglodytes). Journal of Experimental Psychology 22: 76–86. Branigan, Keith. 1969. The earliest Minoan scripts – the pre-palatial background. Kadmos 8: 1–22. Brashear, William. 1985. The myrias-symbol in CPR VII 8. Zeitschrift für Papyrologie und Epigraphik 60: 239–242. Braune, Wilhelm, and Ernst Ebbinghaus. 1966. Gotische Grammatik. Tübingen: Max Niemeyer Verlag. Brice, William C. 1962–63. The writing system of the proto-Elamite account tablets of Susa. Bulletin of the John Rylands Library 45: 15–39. Brice, William C. 1963. A comparison of the account tablets of Susa in the proto-Elamite script with those of Hagia Triada in Linear A. Kadmos 2: 27–38. Bricker, Victoria R. 1992. Noun and verb morphology in the Maya script. In Supplement to the Handbook of Middle American Indians, Vol. 5, Epigraphy, ed. Victoria R. Bricker, pp. 70–81. Austin: University of Texas Press. Brokaw, Galen. 2002. Khipu numeracy and alphabetic literacy in the Andes: Felipe Guaman Poma de Ayala’s Nueva corónica y buen gobierno. Colonial Latin American Review 11(2): 275–303. Brown, Donald R. 1991. Human Universals. Philadelphia: Temple University Press. Brugsch, Karl Heinrich. 1968 [1883]. Thesaurus Inscriptionum Aegyptiacarum, vol. 3. Graz: Akademische Druck-u. Verlagsanstalt. Brunschwig, Jacques, and Geoffrey E. R. Lloyd, eds. 2000. Greek Thought: A Guide to Classical Knowledge. Cambridge, MA: Harvard University Press. Bryce, T. R. 1976. Burial fees in the Lycian sepulchral inscriptions. Anatolian Studies 26: 175–190. Bühler, Georg. 1896. Indische Palaeographie. Strassburg: Trubner. Bühler, Georg. 1963 [1895]. On the Origin of the Indian Brahma Alphabet. Varanasi: Chowkhamba Sanskrit Series Office. Buonamici, Giulio. 1932. Epigrafia Etrusca. Firenze: Rinascimento del Libro. Burge, Len. 1994. Numerals on Cornish church sundials. Cornish Archaeology 33: 206–222. Burgess, James, and Bühler, Georg. 1883. Report on the Buddhist Cave Temples and Their Inscriptions. (Archaeological Survey of Western India, vol. 4.) Bombay: Government Central Press. Burnell, A. C. 1968 [1874]. Elements of South Indian Palaeography. Varanasi: Indological Book House. Burnett, Charles. 1996. Algorismi vel helcep decentior est diligentia: the arithmetic of Adelard of Bath and his circle. In Mathematische Probleme im Mittelalter, ed. Menso Folkerts, pp. 221–331. Wiesbaden: Harrassowitz. Burnett, Charles. 1997. The Introduction of Arabic Learning into England. London: The British Library.

Bibliography

443

Burnett, Charles. 2000a. Latin alphanumerical notation, and annotation in Italian, in the twelfth century: MS London, British Library, Harley 5402. In Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, ed. Menso Folkerts and Richard Lorch, pp. 76–90. Wiesbaden: Harrassowitz. Burnett, Charles. 2000b. Why we read Arabic numerals backwards. In Ancient and Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr, ed. Patrick Suppes, Julius M. Moravcsik, and Henry Mendell, pp. 197–202. Stanford, CA: Center for the Study of Language and Information. Burnett, Charles. 2000c. Antioch as a link between Arabic and Latin culture in the twelfth and thirteenth centuries. In Occident et Proche-Orient: Contacts scientifiques au temps des Croisades, ed. I. Draelants, A. Tihon, and B. van den Abeele, pp. 1–78. Louvain: Brepols. Burnett, Charles. 2002a. Indian numerals in the Mediterranean basin in the twelfth century, with special reference to the ‘eastern forms’. In From China to Paris: 2000 Years Transmission of Mathematical Ideas, ed. Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen, pp. 237–288. Stuttgart: Franz Steiner Verlag. Burnett, Charles. 2002b. The abacus at Echternacht in ca. 1000 AD. Sciamus 3: 91–108. Burnett, Charles. 2002c. Learning Indian arithmetic in the early thirteenth century. Boletín de la Asociación Matemática Venezolana 9(1): 15–26. Burnett, Charles. 2006. The semantics of Indian numerals in Arabic, Greek and Latin. Journal of Indian Philosophy 34: 15–30. Burton, Michael L., Carmella C. Moore, John W. M. Whiting, and A. Kimball Romney. 1996. Regions based on social structure. Current Anthropology 37: 87–123. Busch, Joergen. 2004. The mysterious Roman numeral VI in the Old Icelandic Saga “Landnámabók.” Journal of Navigation 57: 479–89. Butterworth, Brian. 1999. What Counts. New York: The Free Press. Buxton, L. H. Dudley. 1920. The anthropology of Cyprus. Journal of the Royal Anthropological Institute of Great Britain and Ireland 50: 183–235. Cagnat, René. 1964. Cours d’épigraphie latine, fourth edition. Rome: L’Erma di Bretschneider. Cajori, Florian. 1919. The controversy on the origin of our numerals. Scientific Monthly 9: 458–464. Cajori, Florian. 1922. Spanish and Portuguese symbols for “thousand.” American Mathematical Monthly 29(5): 201–202. Cajori, Florian. 1928. A History of Mathematical Notations. 2 vols. LaSalle, IL: Open Court. Calame-Griaule, Genevieve. 1986. Words and the Dogon World. Philadelphia: Institute for the Study of Human Issues. Cantineau, Jean. 1930. Le nabatéen. Paris: Ernest Leroux. Cantineau, Jean. 1935. Grammaire du palmyrénien épigraphique. Le Caire: Imprimerie de l’Institut Français. Cappelli, Adriano. 1901. Lexicon Abbreviaturarum. Leipzig: J. J. Weber. Carra de Vaux, Bernard. 1917. Sur l’origine des chiffres. Scientia 21: 273–282. Caso, Alfonso. 1965. Mixtec writing and calendar. In Handbook of Middle American Indians, vol. 3, part 2, ed. Robert Wauchope and Gordon R. Willey, pp. 948–961. Austin: University of Texas Press. Chabás, José, and Bernard R. Goldstein. 1998. Some astronomical tables of Abraham Zacut preserved in Segovia. Physis 35(1): 1–11.

444

Bibliography

Chace, Arnold Buffum, Ludlow Bull, and Henry Parker Manning, eds. 1929. The Rhind Mathematical Papyrus. 2 vols. Oberlin, OH: Mathematical Association of America. Chamberlain, Basil Hall. 1898. A quinary system of notation employed in Luchu on the wooden tallies termed Sho-chu-ma. Journal of the Anthropological Institute of Great Britain and Ireland 27: 383–395. Chavannes, Edouard. 1913. Les documents chinois découverts par Aurel Stein dans les sables du turkestan oriental. Oxford: Oxford University Press. Cheng, David Chin-Te. 1925. The use of computing rods in China. American Mathematical Monthly 32: 492–499. Childe, V. Gordon. 1956. Society and Knowledge. New York: Harper. Chrisomalis, Stephen. 2003. The Egyptian origin of the Greek alphabetic numerals. Antiquity 77(297): 485–496. Chrisomalis, Stephen. 2004. A cognitive typology for numerical notation. Cambridge Archaeological Journal 14(1): 37–52. Clagett, Marshall. 1989. Ancient Egyptian Science, Volume 1: Knowledge and Order. Philadelphia: American Philosophical Society. Clark, Walter Eugene. 1929. Hindu-Arabic numerals. In Indian Studies in Honor of Charles Rockwell Lanman, pp. 217–236. Cambridge, MA: Harvard University Press. Cline, Eric H. 1994. Sailing the Wine-Dark Sea: International Trade and the Late Bronze Age Aegean. (BAR International Series 591.) Oxford: Tempus Reparatum. Cline, Howard F. 1966. The Oztoticpac Lands Map of Texcoco, 1540. Quarterly Journal of the Library of Congress 23(2): 77–115. Closs, Michael P. 1977. The nature of the Maya chronological count. American Antiquity 42: 18–27. Closs, Michael P. 1978. The Initial Series on Stela 5 at Pixoy. American Antiquity 43: 690–94. Closs, Michael P. 1983. A truncated Initial Series from Xcalumkin. American Antiquity 48(1): 115–122. Closs, Michael P. 1986. The mathematical notation of the ancient Maya. In Native American Mathematics, ed. Michael P. Closs, pp. 291–370. Austin: University of Texas Press. Closs, Michael P., ed. 1986. Native American Mathematics. Austin: University of Texas Press. Coedès, Georges. 1931. A propos de l’origine des chiffres arabes. Bulletin of the School of Oriental Studies 6: 323–328. Cohen, Rudolph. 1981. Excavations at Kadesh-barnea, 1976–78. The Biblical Archaeologist 44: 93–107. Colin, Georges S. 1933. De l’origine grecque des “chiffres de Fès” et de nos “chiffres arabes.” Journal Asiatique 222: 193–215. Colin, Georges S. 1960. Abdjad. In Encyclopedia of Islam, vol. 1, pp. 97–98. Leiden: Brill. Colin, Georges S. 1971. Hisab al-djummal. In Encyclopedia of Islam, vol. 3, p. 468. Leiden: Brill. Colville, Jeffrey. 1985. The structure of Mesoamerican numeral systems with a comparison to non-Mesoamerican systems. Ph.D. dissertation, Tulane University. Conant, Levi L. 1896. The Number Concept. New York: Macmillan. Coningham, R. A. E., F. R. Allchin, C. M. Batt, and D. Lucy. 1996. Passage to India? Anuradhapura and the early use of the Brāhmī script. Cambridge Archaeological Journal 6(1): 73–97. Conklin, William J. 1982. The information system of Middle Horizon quipus. Annals of the New York Academy of Sciences 385: 261–281.

Bibliography

445

Connah, G. 1987. African Civilizations. Cambridge: Cambridge University Press. Cooke, G. A. 1903. A Text-Book of North Semitic Inscriptions. Oxford: Clarendon. Copans, Laurie. 1999. Israeli court rules Latin numerals can be inscribed on tombstones. Associated Press Newswire, July 7, 1999. Coulmas, Florian. 1996. The Blackwell Encyclopedia of Writing Systems. Oxford: Blackwell. Cracraft, James. 2003. The Revolution of Peter the Great. Cambridge, MA: Harvard University Press. Crosby, Alfred W. 1997. The Measure of Reality: Quantification and Western Society, 1250– 1600. Cambridge: Cambridge University Press. Crump, Thomas. 1990. The Anthropology of Numbers. Cambridge: Cambridge University Press. Cubberley, Paul. 1996. The Slavic alphabets. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 346–355. New York: Oxford University Press. Cushing, Frank H. 1892. Manual concepts: a study of the influence of hand usage on culture-growth. American Anthropologist 5: 289–317. d’Errico, Francesco. 1998. Paleolithic origins of artificial memory systems: an evolutionary perspective. In Cognition and Material Culture: The Archaeology of Symbolic Storage, ed. Colin Renfrew and Chris Scarre, pp. 19–50. Cambridge, UK: McDonald Institute for Archaeological Research. d’Errico, Francesco, Christopher Henshilwood, Graeme Lawson, Marian Vanhaeren, Anne-Marie Tillier, Marie Soressi, Frédérique Bresson, Bruno Maureille, April Nowell, Joseba Lakarra, Lucinda Backwell, and Michèle Julien. 2003. Archaeological evidence for the emergence of language, symbolism, and music – an alternative multidisciplinary perspective. Journal of World Prehistory 17(1): 1–70. Dalby, David. 1967. A survey of the indigenous scripts of Liberia and Sierra Leone: Vai, Mende, Loma, Kpelle and Bassa. African Language Studies 8: 1–51. Dalby, David. 1968. The indigenous scripts of West Africa and Surinam: their inspiration and design. African Language Studies 9: 156–197. Dalby, David. 1969. Further indigenous scripts of West Africa: Manding, Wolof and Fula alphabets and Yoruba “holy” writing. African Language Studies 10: 161–181. Damerow, Peter. 1996. Abstraction and Representation: Essays on the Cultural Evolution of Thinking. Dordrecht: Kluwer. Damerow, Peter, and Robert Englund. 1989. The Proto-Elamite Texts from Tepe Yahya. (American School of Prehistoric Research Bulletin 39.) Cambridge, MA: Harvard University Press. Daniels, Peter T. 1996. Aramaic scripts for Aramaic languages. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 499–514. New York: Oxford University Press. Daniels, Peter T., and William Bright, eds. 1996. The World’s Writing Systems. New York: Oxford University Press. Dantzig, Tobias. 1954. Number, the Language of Science. New York: Macmillan. Das, Sukumar Ranjan. 1927a. The origin and development of numerals. Indian Historical Quarterly 3: 97–120. Das, Sukumar Ranjan. 1927b. The origin and development of numerals II. Indian Historical Quarterly 3: 356–375. Das Gupta, C. C. 1958. The Development of the Kharosthi Script. Calcutta: Firma K. L. Mukhopadhyay. Datta, Bibhutibhusan, and Avadhesh Narayan Singh. 1962 [1935]. History of Hindu Mathematics. Bombay: Asia Publishing House.

446

Bibliography

Dauben, Joseph W. 2007. Chinese mathematics. In The Mathematics of Egypt, Mesopotamia, China, India, and Islam, ed. Victor Katz, pp. 187–384. Princeton, NJ: Princeton University Press. De Lacouperie, Terrien. 1883. The old numerals, the counting-rods and the swan-pan in China. Numismatic Chronicle, third series, 3: 297–340. De Odorico, Marco. 1995. The Use of Numbers and Quantifications in the Assyrian Royal Inscriptions. Helsinki: Neo-Assyrian Text Corpus Project. De Oliveira Marques, A. H. 1996. L’introduction des chiffres arabes dans les documents médiévaux portugais. In Graphische Symbole in mittelalterlichen Urkunden, ed. Peter Rück, pp. 503–508. Sigmaringen: Jan Thorbecke Verlag. Decourdemanche, M. J. A. 1899. Note sur quatres systèmes turcs de notation numérique secrète. Journal Asiatique, ninth series, 14: 258–271. DeFrancis, John. 1984. The Chinese Language: Fact and Fantasy. Honolulu: University of Hawaii Press. Dehaene, Stanislas. 1997. The Number Sense. New York: Oxford University Press. Derolez, René. 1954. Runica manuscripta. Brugge: De Tempel. Destombes, Marcel. 1987. Les chiffres coufiques des instruments astronomiques arabes. In Marcel Destombes (1905–1983), ed. Günter Schilder, Peter van der Krogt, and Steven de Clercq, pp. 127–140. Utrecht: HES Publishers. Detlefsen, M., D. Erlandson, H. J. Clark, and C. Young. 1975. Computation with Roman numerals. Archive for History of Exact Sciences 15(2): 141–148. Dilke, Oswald Ashton Wentworth. 1987. Mathematics and Measurement. Berkeley: University of California Press. Diller, Anthony. 1995. Sriwijaya and the first zeros. Journal of the Malaysian Branch of the Royal Asiatic Society 68(1): 53–66. Diller, Anthony. 1996. New zeros and old Khmer. Mon-Khmer Studies 25: 125–132. Diringer, David. 1949. The Alphabet. London: Hutchinson. Divale, William. 1999. Climatic instability, food storage, and the development of numerical counting: a cross-cultural study. Cross-Cultural Research 33(4): 341–368. Djamouri, Redouane. 1994. L’emploi des signes numériques dans les inscriptions Shang. In Sous les nombres, le monde, ed. Alexei Volkov, pp. 13–42. (Extrême-Orient, ExtrêmeOccident 16). Paris: Université de Paris. Dobbs-Allsopp, F. W., J. J. M. Roberts, C. L. Seow, and R. E. Whitaker. 2005. Hebrew Inscriptions: Texts from the Biblical Period of the Monarchy with Concordance. New Haven, CT: Yale University Press. Dow, Sterling. 1952. Greek numerals. American Journal of Archaeology 56(1): 21–23. Dow, Sterling. 1958. Mycenaean arithmetic and numeration. Classical Philology 53: 32–34. Dow, Sterling, and John Chadwick. 1971. The Linear Scripts and the Tablets as Historical Documents. Cambridge: Cambridge University Press. Drewes, A. J. 1962. Inscriptions de l’Ethiopie antique. Leiden: Brill. Dreyer, Günter. 1998. Umm el Qaab I. Mainz: Verlag Philipp von Zabern. Drijvers, Han J. W., and John F. Healey. 1999. The Old Syriac Inscriptions of Edessa and Osrhoene. Leiden: Brill. Driver, Harold E. 1966. Geographical-historical versus psycho-functional explanations of kin avoidances. Current Anthropology 7: 131–182. Dugast, I., and M. D. W. Jeffreys. 1950. L’écriture des Bamum: sa naissance, son évolution, sa valeur phonétique, son utilisation. Paris: Memoires de l’Institut Français d’Afrique Noire.

Bibliography

447

Durand, Jean-Marie. 1987. Questions de chiffres. Mari, annales de recherches interdisciplinaires 5: 605–610. Duval, Rubens. 1881. Traité de grammaire syriaque. Paris: Vieweg. Dwornik, Henryk. 1980–81. A 2n-number system in the arithmetic of prehistoric societies. Organon 16–17: 199–222. Ebbinghaus, Ernst A. 1978. The Gotica of Codex Vindobonensis 795. In Germanic Studies in Honor of Otto Springer, ed. Stephen J. Kaplowitt, pp. 93–102. Pittsburgh: K & S Enterprises. Ebbinghaus, Ernst. 1996. The Gothic alphabet. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 290–293. New York: Oxford University Press. Eisenstein, Elizabeth L. 1979. The Printing Press as an Agent of Change. New York: Cambridge University Press. Englund, Robert K. 1996. The proto-Elamite script. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 160–164. New York: Oxford University Press. Enwall, Joakim. 1994. A Myth Become Reality: History and Development of the Miao Written Language. 2 vols. Stockholm: Institute of Oriental Languages. Evans, Arthur J. 1906. Minoan weights and mediums of currency from Crete, Mycenae and Cyprus. In Corolla Numismatica: Numismatic Essays in Honour of Barclay V. Head, ed. George Francis Hill. London: Oxford University Press. Evans, Arthur J. 1909. Scripta Minoa. Oxford: Clarendon. Evans, Arthur J. 1935. The Palace of Minos, vol. IV. London: Oxford University Press. Evans, Gillian R. 1977. From abacus to algorism: theory and practice in medieval arithmetic. The British Journal for the History of Science 10(35): 114–131. Everett, Daniel L. 2005. Cultural constraints on grammar and cognition in Pirahã. Current Anthropology 46(4): 621–646. Everson, Michael. 1998. On Greek letter koppa. ISO/IEC JTC 1/SC 2/WG 2 working group document N1938. http://std.dkuug.dk/jtc1/sc2/WG2/docs/n1938.pdf. Fabretti, Ariodante. 1867. Corpus inscriptionum italicarum. Turin: ex officia regia. Faddegon, J.-M. 1932. Note au sujet de l’aboujad. Journal asiatique, ser. 12, 220: 139–148. Fairbanks, Sydney, and F. P. Magoun, Jr. 1940. On writing and printing Gothic. Speculum 15(3): 313–330. Fairman, H. W. 1963. Two Ptolemaic numerals. Journal of Egyptian Archaeology 49: 179– 180. Fairservis, Walter A. 1992. The Harappan Civilization and Its Writing. Leiden: Brill. Fales, F. M. 1995. Assyro-aramaica: the Assyrian lion-weights. In Immigration and Emigration within the Ancient Near East: Festschrift E. Lipinski, ed. K. van Lerberghe and A. Schoors, pp. 33–55. Leuven: Peeters. Falk, Harry. 1993. Schrift im alten Indien: Ein Forschungsbericht mit Anmerkungen. Tübingen: Narr. Falkenstein, Adam, ed. 1936. Archaische Texte aus Uruk. Berlin: Mann. Fedosov, Dimitry. 1995. A Scottish mathematician in Russia: Henry Farquharson (c. 1675– 1739). In The Universities of Aberdeen and Europe: The First Three Centuries, ed. Paul Dukes, pp. 102–118. Aberdeen: University of Aberdeen Press. Fernald, L. D. 1984. The Hans Legacy: A Story of Science. Hillsdale, NJ: Erlbaum. Février, James. 1948. Histoire de l’écriture. Paris: Payot. Feyel, Michel. 1937. Études d’épigraphie béotienne. Bulletin de correspondance hellenique 61: 217–235.

448

Bibliography

Fischer, Stephen. 1997. Rongorongo: The Easter Island Script. Oxford: Clarendon. Flam, Jack D. 1976. Graphic symbolism in the Dogon granary: grains, time, and a notion of history. Journal of African Studies 3: 35–50. Fleet, J. F. 1911a. Âryabhata’s system of expressing numbers. Journal of the Royal Asiatic Society 1911(1): 109–126. Fleet, J. F. 1911b. The Katapayadi system of expressing numbers. Journal of the Royal Asiatic Society 1911(2): 788–794. Foat, F. W. G. 1902. Sematography of the Greek papyri. The Journal of Hellenic Studies 22: 135–173. Foat, F. W. G. 1905. Tsade and sampi. Journal of Hellenic Studies 25: 338–365. Folkerts, Menso. 2001. Early texts on Hindu-Arabic calculation. Science in Context 14(1–2): 13–38. Folkerts, Menso. 2003. Essays on Early Medieval Mathematics: The Latin Tradition. Aldershot, UK: Ashgate. Fossa, Lydia. 2000. Two khipu, one narrative: answering Urton’s questions. Ethnohistory 47(2): 453–468. Fossey, Charles, ed. 1948. Notice sur les caracteres étrangers, anciens et modernes. Paris: Imprimerie Nationale. Fouts, David M. 1994. Another look at large numbers in Assyrian royal inscriptions. Journal of Near Eastern Studies 53(3): 205–211. Fowler, D. H. 1999a. The Mathematics of Plato’s Academy, second ed. Oxford: Clarendon. Fowler, D. H. 1999b. Inventive interpretations. Revue d’histoire des mathématiques 5: 149–153. Fowler, D. H., and E. G. Turner. 1983. Hibeh Papyrus i 27: an early example of Greek arithmetical notation. Historia Mathematica 10: 344–359. Frei, Peter. 1976. Die Trilingue von Letoon, die lykischen Zahlzeichen und das lykische Geldsystem. Revue suisse de numismatique 55: 5–17. Frei, Peter. 1977. Die Trilingue von Letoon, die lykischen Zahlzeichen und das lykische Geldsystem. Revue suisse de numismatique 56: 65–78. Friberg, Jöran. 1978–79. The Third Millennium Roots of Babylonian Mathematics. Göteborg: University of Göteborg. Friberg, Jöran. 1984. Numbers and measures in the earliest written records. Scientific American 250(2): 110–118. Friberg, Jöran. 1998. Round and almost-round numbers in proto-literate metro-mathematical field texts. Archiv für Orientforschung 44–45: 1–58. Frye, Richard N. 1973. Sasanian numbers and silver weights. Journal of the Royal Asiatic Society 1: 2–11. Fulton, Charles C. 1979. Elements of Maya arithmetic. The Epigraphic Society Occasional Publications 6(133): 167–180. Fuson, Karen C. 1988. Children’s Counting and Concepts of Number. New York: Springer-Verlag. Gallistel, C. R. 1990. The Organization of Learning. Cambridge, MA: MIT Press. Gamkrelidze, Thomas V. 1994. Alphabetic Writing and the Old Georgian Script. Delmar, NY: Caravan Books. Ganay, Solange de. 1950. Graphies Bambara des nombres. Journal de la Société des Africanistes 20: 295–305. Gandz, Solomon. 1931. The origin of the Ghubar numerals, or the Arabian abacus and the articuli. Isis 16: 393–424.

Bibliography

449

Gandz, Solomon. 1933. Hebrew numerals. Proceedings of the American Academy of Jewish Research 4: 53–112. Gandz, Solomon. 1970. Saadia Gaon as a mathematician. In Studies in Hebrew Astronomy and Mathematics, by Solomon Gandz, pp. 477–506. New York: Ktav Publishing House. Ganguli, Saradakanta. 1927. The elder Âryabhata and the modern arithmetical notation. American Mathematical Monthly 34(8): 409–415. Gardiner, Sir Alan. 1927. Egyptian Grammar. Oxford: Clarendon. Gardiner, Sunray Cythna. 1984. Old Church Slavonic: An Elementary Grammar. Cambridge: Cambridge University Press. Gardner, Ernest A. 1888. The inscriptions. In Naukratis, Part I, by W. M. Flinders Petrie, pp. 54–63. London: Trübner & Co. Gasparov, Boris. 2001. Old Church Slavonic. Munich: Lincom Europa. Gasperini, Lidio. 1986. Il sistema numerale cirenaico e una nuova epigraphe dall’agorà di Cirene. Annali della Facolta di Lettere e Filosofia, Universita di Macerata 19: 357–366. Geertz, Clifford. 1965. The impact of the concept of culture on the concept of man. In New Views of the Nature of Man, ed. John R. Platt, pp. 93–118. Chicago: University of Chicago Press. Geertz, Clifford. 1984. Anti anti-relativism. American Anthropologist 86: 263–278. Gelb, Ignace. 1963. A Study of Writing, second edition. Chicago: University of Chicago Press. Gelman, R., and C. R. Gallistel. 1978. The Child’s Understanding of Number. Cambridge, MA: Harvard University Press. Geraty, Lawrence T. 1975. The Khirbet el-Kôm bilingual ostracon. Bulletin of the American School of Oriental Research 220: 55–61. Gerdes, Paulus. 1994. On mathematics in the history of sub-Saharan Africa. Historia Mathematica 21: 345–376. Gibson, Craig A., and Francis Newton. 1995. Pandulf of Capua’s De calculatione: An illustrated abacus treatise and some evidence for the Hindu-Arabic numerals in eleventhcentury south Italy. Medieval Studies 57: 293–335. Gibson, John C. L. 1971. Textbook of Syrian Semitic Inscriptions. Oxford: Clarendon. Gillings, Richard J. 1972. Mathematics in the Time of the Pharaohs. Cambridge, MA: MIT Press. Gillings, Richard J. 1978. The mathematics of ancient Egypt. In Dictionary of Scientific Biography, vol. 15, supplement 1, ed. Charles Coulston Gillespie, pp. 681–705. New York: Scribner. Glass, Andrew. 2000. A preliminary study of Kharoṣṭhī manuscript paleography. Unpublished M.A. thesis, University of Washington. Glautier, M. W. E. 1972. A study in the development of accounting in Roman times. Revue internationale des droits de l’antiquité 19: 311–343. Gmür, Max. 1917. Schweizerische Bauernmarken und Holzurkunden. Bern: Stämpfli. Goedicke, Hans. 1988. Old Hieratic Paleography. Baltimore, MD: Halgo Inc. Goerwitz, Richard L. 1996. The Jewish scripts. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 487–497. New York: Oxford University Press. Goitein, Shelomo Dov. 1967–88. A Mediterranean Society: The Jewish Communities of the Arab World as Portrayed in the Documents of the Cairo Geniza. 6 volumes. Berkeley: University of California Press. Gokhale, Shebhana Laxman. 1966. Indian Numerals. Poona: Deccan College.

450

Bibliography

Goldenweiser, Alexander. 1913. The principle of limited possibilities in the development of culture. Journal of American Folk-lore 16: 259–290. Goldstein, Jonathan A. 1966. The Syriac bill of sale from Dura-Europos. Journal of Near Eastern Studies 25(1): 1–16. Goody, Jack. 1977. The Domestication of the Savage Mind. New York: Cambridge University Press. Gordon, Arthur E. 1983. Illustrated Introduction to Latin Epigraphy. Berkeley: University of California Press. Gordon, Cyrus. 1965. Ugaritic Textbook. Rome: Pontifical Biblical Institute. Gordon, Joyce S., and Arthur E. Gordon. 1957. Contributions to the Palaeography of Latin Inscriptions. Berkeley: University of California Press. Gordon, Peter. 2004. Numerical cognition without words: evidence from Amazonia. Science 306: 496–499. Gow, James. 1883. The Greek numerical alphabet. Journal of Philology 12: 278–284. Grace, Virginia R. 1940. A Cypriote tomb and Minoan evidence for its date. American Journal of Archaeology 44(1): 10–52. Graham, J. Walter. 1969. X = 10. Phoenix 23: 347–358. Greenberg, Joseph H. 1978. Generalizations about numeral systems. In Universals of Human Language, ed. J. H. Greenberg, vol. 3, pp. 249–297. Stanford, CA: Stanford University Press. Greg, W. W. 1924. John of Basing’s “Greek numerals.” Library (Transactions of the Bibliographic Society), fourth ser., 4: 53–58. Grenet, Frantz, Nicholas Sims-Williams, and Etienne de la Vaissière. 1998 [2001]. The Sogdian ancient letter V. Bulletin of the Asia Institute 12: 91–104. Grenfell, Bernard P., and Arthur S. Hunt, eds. 1906. The Hibeh Papyri, part 1. London: Egypt Exploration Fund. Griaule, Marcel, and Germaine Dieterlen. 1951. Signes graphiques soudanais. Paris: Hermann et Cie. Griffith, F. Ll. 1898. Hieratic Papyri from Kahun and Gurob. London: Bernard Quaritch. Griffith, F. Ll. 1909. Catalogue of the Demotic Papyri in the John Rylands Library, Manchester, vol. 3. Manchester: University of Manchester. Griffith, F. Ll. 1916. Meroitic studies. Journal of Egyptian Archaeology 3: 22–30. Griffith, F. Ll. 1925. Meroitic studies V. Journal of Egyptian Archaeology 11: 218–224. Griffith, F. Ll. 1937. Catalogue of the Demotic Graffiti of the Dodecaschoenus. Oxford: Oxford University Press. Grohmann, Adolf. 1952. From the World of Arabic Papyri. Cairo: Al-Maaref Press. Grube, Wilhelm. 1896. Die Sprache und Schrift der Jučen. Leipzig: Harrassowitz. Guedj, Denis. 1996. Numbers: The Universal Language. Translated by Lory Frankel. New York: Harry N. Abrams. Guergour, Youcef. 1997. Les différents systèmes de numérotation au Maghreb à l’époque ottomane: l’exemple des chiffres rumi. In Sciences, Technology and Industry in the Ottoman World. Proceedings of the XXth International Congress of History of Science, vol. 6, ed. E. Ihsanoglu, A. Djebbar, and F. Günergun, pp. 67–74. Burlington, VT: Ashgate/Variorum. Guitel, Geneviève. 1958. Comparaison entre les numérations aztèque et égyptienne. Annales ESC 13: 687–705. Guitel, Geneviève. 1975. Histoire comparée des numérations écrites. Paris: Flammarion. Gunn, Battiscombe. 1916. Notices of recent publications. Journal of Egyptian Archaeology 3: 279–286.

Bibliography

451

Gupta, R. C. 1983. Spread and triumph of Indian numerals. Indian Journal of History of Science 18(1): 23–38. Haarmann, Harold. 1996. Early Civilization and Literacy in Europe. Berlin: Mouton de Gruyter. Hackl, Rudolf. 1909. Merkantile Inscriften auf attischen Vasen. In Münchener Archäologische Studien dem Andenken Adolf Furtwänglers Gewidmet, ed. Adolf Furtwängler, pp. 1–106. Munich: C. H. Beck. Hagen, S. N. 1950. The Kensington runic inscription. Speculum 25(3): 321–356. Hägerstrand, Torsten. 1967. Innovation Diffusion as a Spatial Process. Chicago: University of Chicago Press. Haile, Getatchew. 1996. Ethiopic writing. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 569–576. New York: Oxford University Press. Halévy, Joseph. 1873. Études sabéennes. Journal Asiatique, seventh ser, 1: 434–521. Halévy, Joseph. 1875. Études sabéennes. Paris: Imprimerie Nationale. Halliwell, James Orchard. 1839. Rara Mathematica. London: J. W. Parker. Hallpike, C. R. 1979. The Foundations of Primitive Thought. Oxford: Clarendon. Hallpike, C. R. 1986. The Principles of Social Evolution. Oxford: Clarendon. Hamilton, Alistair. 2006. The Copts and the West, 1439–1822. Oxford: Oxford University Press. Hans, Nicholas. 1959–60. Henry Farquharson: pioneer of Russian education, 1698–1739. Aberdeen University Review 38: 26–29. Harris, John F., and Stephen K. Stearns. 1997. Understanding Maya Inscriptions: A Hieroglyph Handbook, second ed. Philadelphia: University of Pennsylvania Museum. Harris, Marvin. 1968. The Rise of Anthropological Theory. New York: Thomas Crowell. Harris, R. A. 1905. Numerals for simplifying addition. American Mathematical Monthly 12: 64–67. Harris, Roy. 1995. Signs of Writing. London: Routledge. Harris, Zelig. 1936. A Grammar of the Phoenician Language. New Haven, CT: American Oriental Society. Harvey, Herbert R. 1982. Reading the numbers: variation in Nahua numerical glyphs. In The Indians of Mexico in Pre-Columbian and Modern Times, ed. M. E. R. G. N. Jansen and Th. J. J. Leyenaar, pp. 190–205. Leiden: Rutgers. Harvey, Herbert R., and Barbara J. Williams. 1980. Aztec arithmetic: positional notation and area calculation. Science 210: 499–505. Harvey, Herbert R., and Barbara J. Williams. 1981. L’arithmétique aztèque. La Recherche 126: 1068–1081. Harvey, Herbert R., and Barbara J. Williams. 1986. Decipherment and some implications of Aztec numerical glyphs. In Native American Mathematics, ed. Michael P. Closs, pp. 237–260. Austin: University of Texas Press. Hau, Kathleen. 1961. Oberi Vkaime script, texts and counting system. Bulletin de l’Institut Français d’Afrique Noire 23(1–2): 291–308. Hau, Kathleen. 1967. The ancient writing of Southern Nigeria. Bulletin de l’Institut Français d’Afrique Noire 29(1–2): 150–191. Hawkins, J. D. 1986. Writing in Anatolia: imported and indigenous systems. World Archaeology 17(3): 363–375. Hawkins, J. D. 2000. Corpus of Hieroglyphic Luwian Inscriptions. 2 vols. Berlin: de Gruyter. Heath, Thomas L. 1921. A History of Greek Mathematics. 2 vols. Cambridge: Cambridge University Press.

452

Bibliography

Hering, D. W. 1939. Numerals on clock and watch dials. Scientific Monthly 49(4): 311–323. Hewsen, Robert H. 1968. Science in seventh-century Armenia: Ananias of Shirak. Isis 59(1): 32–45. Hill, George Francis. 1915. The Development of Arabic Numerals in Europe. Oxford: Clarendon Press. Hoffman, A. G. 1858. The Principles of Syriac Grammar. Translated by B. Harris Cowper. London: Williams and Norgate. Holisky, Dee Ann. 1996. The Georgian alphabet. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 364–369. New York: Oxford University Press. Holmes, Ruth Bradley, and Betty Sharp Smith. 1977. Beginning Cherokee, second ed. Norman: University of Oklahoma Press. Hommel, Fritz. 1893. Süd-arabische Chrestomathie. Munich: Franz. Hopkins, L. C. 1916. The Chinese numerals and their notational systems. Journal of the Royal Asiatic Society 68: 314–333, 737–771. Hopper, Vincent. 1938. Medieval Number Symbolism. New York: Columbia University Press. Houston, Stephen D. 1997. A king worth a hill of beans. Archaeology 50(3): 40. Houston, Stephen D. 2004a. The archaeology of communication technologies. Annual Review of Anthropology 33: 223–250. Houston, Stephen D. 2004b. Writing in ancient Mesoamerica. In The First Writing, ed. Stephen D. Houston, pp. 274–312. Cambridge: Cambridge University Press. Høyrup, Jens. 1982. Investigations of an early Sumerian division problem, c. 2500 b.c. Historia Mathematica 9: 19–36. Høyrup, Jens. 1985. Varieties of mathematical discourse in pre-modern socio-cultural contexts: Mesopotamia, Greece, and the Latin Middle Ages. Science and Society 49(1): 4–41. Høyrup, Jens. 2002. A note on Old Babylonian computational technologies. Historia Mathematica 29: 193–198. Høyrup, Jens. 2006. Artificial language in ancient Mesopotamia – a dubious and a less dubious case. Journal of Indian Philosophy 34: 57–88. Hurford, James R. 1975. The Linguistic Theory of Numerals. Cambridge: Cambridge University Press. Hurford, James R. 1987. Language and Number. Oxford: Basil Blackwell. Hyland, Sabine. 2003. The Jesuit and the Incas. Ann Arbor: University of Michigan Press. Ibn Khaldun. 1958. The Muqadimmah: An Introduction to History. 3 vols. Translated by Franz Rosenthal. New York: Pantheon Books. Ifrah, Georges. 1985 [1981]. From One to Zero: A Universal History of Numbers. Translated by L. Bair. New York: Viking Penguin. Ifrah, Georges. 1998. The Universal History of Numbers. Translated by David Bellos, E. F. Harding, Sophie Wood, and Ian Monk. New York: John Wiley and Sons. Imhausen, Annette. 2003. Egyptian mathematical texts and their contexts. Science in Context 16(3): 367–389. Imhausen, Annette. 2006. Ancient Egyptian mathematics: new perspectives on old sources. The Mathematical Intelligencer 28(1): 19–27. Imhausen, Annette. 2007. Egyptian mathematics. In The Mathematics of Egypt, Mesopotamia, China, India, and Islam, ed. Victor Katz, pp. 7–56. Princeton, NJ: Princeton University Press. Indraji, Bhagavánlál. 1876. On ancient Nagari numeration: from an inscription at Naneghat. Journal of the Royal Asiatic Society, Bombay Branch 12: 404–406.

Bibliography

453

Indraji, Bhagavánlál. 1877. On the ancient Nâgarí numerals. The Indian Antiquary 6: 42–48. International Electrotechnical Commission. 2004. IEC 60062: Marking codes for resistors and capacitors. Geneva: International Electrotechnical Commission. Irani, Rida A. K. 1955. Arabic numeral forms. Centaurus 4(1): 1–12. Jackson, F. G., and Arthur Montefiore. 1895. Notes on the Samoyads of the great tundra. Journal of the Anthropological Institute of Great Britain and Ireland 24: 388–410. Jacobsen, Lyle E. 1964. The ancient Inca empire of Peru and the double entry accounting concept. Journal of Accounting Research 2(2): 221–228. Jami, Catherine. 1998. Abacus (Eastern). In Instruments of Science: An Historical Encyclopedia, ed. Robert Bud, pp. 3–4. London: The Science Museum. Janert, Klaus Ludwig, and Ilse Janert. 1993. Nachitextedition, vol. 6(1). Bonn: VGH Wissenschaftsverlag GmbH. Jaspers, Karl. 1953. The Origin and Goal of History. London: Routledge and Kegan Paul. Jeffery, Lilian H. 1990. The Local Scripts of Archaic Greece. Oxford: Clarendon. Jenkinson, Hilary. 1926. The use of Arabic and Roman numerals in English archives. The Antiquaries Journal 6: 263–275. Jestin, Raymond. 1937. Tablettes sumériennes de Shuruppak conservées au musée du Stamboul. Paris: E. de Boccard. Jha, Parmeshwar. 1988. Âryabhata I and his Contributions to Mathematics. Patna: Bihar Research Society. Johnston, Alan W. 1973. Two-and-a-half Corinthian dipinti. Annual of the British School at Athens 68: 181–189. Johnston, Alan W. 1975. A South Italian numeral system. La Parola del Passato 30: 360–366. Johnston, Alan W. 1979. Trademarks on Greek Vases. Warminster, UK: Aris and Phillips. Johnston, Alan W. 1982. Two numerical notes. Zeitschrift für Papyrologie und Epigraphik 49: 205–209. Johnston, Alan W. 2006. Trademarks on Greek Vases: Addenda. Oxford: Aris and Phillips. Jones, Alexander. 1999. Astronomical Papyri from Oxyrhynchus. (Memoirs of the American Philosophical Society vol. 233.) Philadelphia: American Philosophical Society. Jorgensen, Joseph G. 1979. Cross-cultural comparisons. Annual Review of Anthropology 8: 309–331. Judd, A. S. 1917. Native education in the northern provinces of Nigeria. Journal of the African Society 17: 1–15. Justeson, John S., and Terrence Kaufman. 1993. A decipherment of epi-Olmec hieroglyphic writing. Science 259 (5102): 1703–1711. Kak, Subhash C. 1990. The sign for zero. Mankind Quarterly 30: 199–204. Kak, Subhash C. 1994. Evolution of early writing in India. Indian Journal of History of Science 29(3): 375–388. Kara, György. 1996. Kitan and Jurchin. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 230–238. New York: Oxford University Press. Kara, György. 2005. Books of the Mongolian Nomads. Bloomington: Indiana University Bloomington Research Institute for Inner Asian Studies. Kaufman, Ivan Tracy. 1967. New evidence for hieratic numerals on Hebrew weights. Bulletin of the American Schools of Oriental Research 188: 39–41. Kavett, Hyman, and Phyllis F. Kavett. 1975. The eye of Horus is upon you. Mathematics Teacher 68: 390–394. Kaye, G. R. 1907. Notes on Indian mathematics – arithmetical notation. Journal of the Asiatic Society of Bengal, n.s. 3: 475–508.

454

Bibliography

Kaye, G. R. 1919. Indian mathematics. Isis 2: 326–356. Kazem-zadeh, Husayn Iranshahr. 1915. Les chiffres siyâk et la comptabilité persane. Paris: Ernest Leroux. Keil, Bruno. 1894. Eine Halikarnassische Inschrift. Hermes 29: 249–280. Kelley, David H. 1976. Deciphering the Maya Script. Austin: University of Texas Press. Kennedy, James G. 1981. Arithmetic with Roman numerals. American Mathematical Monthly 88(1): 29–32. Keyser, Paul. 1988. The origin of the Latin numerals 1 to 1000. American Journal of Archaeology 92: 529–546. Kharsekin, A. I. 1967. On the interpretation of Etruscan numerals. Soviet Anthropology and Archaeology 5(3–4): 39–50. Khipu Database Project. http://khipukamayuq.fas.harvard.edu/index.html. Last accessed August 6, 2008. King, David A. 1995. A forgotten Cistercian system of numerical notation. Citeaux Commentarii Cistercienses 46(3–4): 183–217. King, David A. 1999. World-Maps for Finding the Direction and Distance to Mecca. Leiden: Brill. King, David A. 2001. Ciphers of the Monks: A Forgotten Number-Notation of the Middle Ages. Stuttgart: Franz Steiner Verlag. Kiyose, Gisaburo N. 1977. A Study of the Jurchen Language and Script: Reconstruction and Decipherment. Kyoto: Horitsubunka-sha. Klein, Jeffrey J. 1974. Urartian hieroglyphic inscriptions from Altintepe. Anatolian Studies 24: 77–94. Kletter, Raz. 1998. Economic Keystones: The Weight System of the Kingdom of Judah. Sheffield: Sheffield Academic Press. Kluge, Theodor. 1937–42. Die Zahlenbegriffe. 5 vols. Berlin: Steglitz. Knorr, Wilbur R. 1982. Techniques of fractions in ancient Egypt and Greece. Historia Mathematica 9(2): 133–171. Koehler, O. 1951. The ability of birds to count. Bulletin of Animal Behaviour 9: 41–45. Kozlov, Vladimir A., and Vladimir M. Khrustalev, eds. 1997. The Last Diary of Tsaritsa Alexandra. New Haven, CT: Yale University Press. Kraus, Jürgen. 2004. Die Demographie des alten Ägypten. D. Phil. dissertation, GeorgAugust University of Göttingen. Krenkel, Werner. 1969. Das Rechnen mit römischen Ziffern. Altertum 15: 252–256. Kroeber, Alfred L. 1948. Anthropology, second ed. New York: Harcourt, Brace and World. Kroman, Erik. 1974. Taltegn. In Kulturhistorisk Leksikon for Nordisk Middelalder. Bd. XVIII. København: Rosenkilden og Bagger. Krueger, Hilmar C. 1977. The Arabic numerals of a Genoese notary, 1202–1226. Bollettino linguistico per la storia e la cultura regionale 27(3–4): 51–64. Kuhn, Thomas. 1962. The Structure of Scientific Revolutions. Cambridge, MA: MIT Press. Kunitzsch, Paul. 2003. The transmission of Hindu-Arabic numerals reconsidered. In The Enterprise of Science in Islam: New Perspectives, ed. Jan P. Hogendijk and Abdelhamid J. Sabra, pp. 3–21. Cambridge, MA: MIT Press. Kuttner, Robert E. 1986. Convenience does not explain the use of the Mayan duodecimal numeral system. Perceptual and Motor Skills 63: 930. Kychanov, E. I. 1996. Tangut. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 228–230. New York: Oxford University Press.

Bibliography

455

Labarta, Ana, and Carmen Barceló. 1988. Numeros y cifras en los documentos arábigohispanos. Córdoba: Universidad de Córdoba. Labat, René. 1952. Manuel d’épigraphie akkadienne. Paris: Imprimerie Nationale. Lagercrantz, Sture. 1970. Tallying by means of lines, stones, and sticks. Paideuma 16: 52–62. Lagercrantz, Sture. 1973. Counting by means of tally sticks or cuts on the body in Africa. Anthropos 68: 569–588. Lam Lay-Yong. 1986. The conceptual origins of our numeral system and the symbolic form of algebra. Archive for History of Exact Sciences 36: 183–195. Lam Lay-Yong. 1987. Linkages: exploring the similarities between the Chinese rod numeral system and our numeral system. Archive for History of Exact Sciences 37: 365–392. Lam Lay-Yong. 1988. A Chinese genesis: rewriting the history of our numeral system. Archive for History of Exact Sciences 38: 101–108. Lambert, Joseph B., Barbara Ownbey-McLaughlin, and Charles D. McLaughlin. 1980. Maya arithmetic. American Scientist 68(3): 249–255. Lancy, David F. 1983. Cross-Cultural Studies in Cognition and Mathematics. New York: Academic. Lang, Mabel. 1956. Numerical notation on Greek vases. Hesperia 25: 1–24. Lang, Mabel. 1957. Herodotos and the abacus. Hesperia 26: 271–287. Langley, James C. 1986. Symbolic Notation of Teotihuacan. Oxford: BAR. Larfeld, W. 1902–07. Handbuch der griechischen Epigraphik. Leipzig: Reisland. Laroche, Emmanuel. 1960. Les hiéroglyphes hittites. Paris: Editions du CNRS. Laroche, Emmanuel. 1971. Les hiéroglyphes d’Altintepe. Anatolia (Anadolu) 15: 55–61. Lassère, Jean-Marie. 2005. Manuel d’épigraphie romaine, vol. 1. Paris: Picard. Lattin, Harriet Pratt. 1933. The origin of our present system of notation according to the theories of Nicholas Bubnov. Isis 19: 181–194. Lawall, Mark L. 2000. Graffiti, wine selling, and the reuse of amphoras in the Athenian Agora, ca. 430 to 400 BC. Hesperia 69(1): 3–90. Lejeune, Michel. 1981. Procédures soustractives dans les numérations étrusque et latine. Bulletin de la Societé de Linguistique de Paris 76: 241–248. Lemaire, André. 1977. Inscriptions hébraiques, vol. 1. Paris: Les Éditions du Cerf. Lemaire, André, and Pascal Vernus. 1980. Les ostraca paléo-hébreux de Qadesh-Barnéa. Orientalia 49: 341–345. Lemay, Richard. 1977. The Hispanic origin of our present numeral forms. Viator 8: 435–462. Lemay, Richard. 1982. Arabic numerals. In Dictionary of the Middle Ages, ed. Joseph R. Strayer, vol. 1, pp. 382–398. New York: Charles Scribner’s Sons. Lemay, Richard. 2000. Nouveautés fugaces dans des textes mathématiques du XIIe siècle: un essai d’abjad latin avorte. In Sic itur ad astra: Studien zur Geschichte der Mathematik und Naturwissenschaften, ed. Menso Folkerts and Richard Lorch, pp. 376–389. Wiesbaden: Harrassowitz. Levi Della Vida, G. 1934. Numerali greci in documenti arabo-spagnoli. Rivista degli Studi Orientali 14: 281–283. Levine, Baruch A. 2004. Tracing the Biblical account register: terminology and the signification of quantity. In Commerce and Monetary Systems in the Ancient World: Means of Transmission and Cultural Interaction, ed. Robert Rollinger and Christoph Ulf, pp. 420–443. Stuttgart: Franz Steiner. Lewis, I. M. 1958. The Gadabuursi Somali script. Bulletin of the School of Oriental and African Studies 21(1/3): 134–156.

456

Bibliography

Li Yan and Du Shiran. 1987. Chinese Mathematics: A Concise History. Translated by John N. Crossley and Anthony W-C. Lun. Oxford: Clarendon. Li, Xueqin, Garman Harbottle, Juzhong Zhang, and Changsui Wang. 2003. The earliest writing? Sign use in the seventh millennium BC at Jiahu, Henan Province, China. Antiquity 77(295): 31–44. Libbrecht, Ulrich. 1973. Chinese Mathematics in the Thirteenth Century: The Shu-shu Chiuchang of Ch’in, Chiu-shao. Cambridge, MA: MIT Press. Lidzbarski, Mark. 1898. Handbuch der Nordsemitischen Epigraphik. Weimar: Emil Felber. Lieberman, Stephen J. 1980. Of clay pebbles, hollow clay balls, and writing: a Sumerian view. American Journal of Archaeology 84: 339–58. Littmann, Enno, and David Meredith. 1953. Nabataean inscriptions from Egypt. Bulletin of the School of Oriental and African Studies 15(1): 1–28. Liverani, Mario. 1983. Fragments of possible counting and recording materials. In Origini: Preistoria e Protostoria delle civiltà antiche, part 2, ed. Marcella Frangipane and Alba Palmieri, pp. 511–521. Rome: Università degli studi la sapienza. Livshitz, V. A. 1970. A Sogdian alphabet from Panjikant. In W. B. Henning Memorial Volume, ed. Mary Boyce and Ilya Gershevitch, pp. 256–263. London: Lund Humphries. Locke, L. Leland. 1912. The ancient quipu, a Peruvian knot record. American Anthropologist 14: 325–332. Loprieno, Antonio. 1986. Zahlwort. In Lexikon der Ägyptologie, ed. W. Helck and W. Westendorf, vol. 6, pp. 1306–1319. Wiesbaden: Harrassowitz. Loprieno, Antonio. 1995. Ancient Egyptian: A Linguistic Introduction. Cambridge: Cambridge University Press. Lounsbury, Floyd G. 1978. Maya numeration, computation, and calendrical astronomy. In Dictionary of Scientific Biography, ed. Charles Coulston Gillespie, vol. 15, supplement 1, pp. 759–818. New York: Scribner. Love, Bruce. 1994. The Paris Codex: Handbook for a Maya Priest. Introduction by George E. Stuart. Austin: University of Texas Press. Lowe, S. K. 1943. Paleographic Guide for Spanish Manuscripts, Fifteenth–Seventeenth Centuries: Roman Numerals. (Middle American Research Institute pub. 12: 1–12.) New Orleans: Tulane University Press. Lowie, Robert H. 1920. Primitive Society. New York: Bonie and Liveright. Loza, Carmen Beatriz. 1998. Juger les chiffres. Histoire et Mesure 13(1–2): 13–37. Lunt, Horace G. 2001. Old Church Slavonic Grammar, seventh rev. ed. Berlin: Mouton de Gruyter. Mace, Ruth, and Mark Pagel. 1994. The comparative method in anthropology. Current Anthropology 35(5): 549–564. Mackenzie, D. N. 1971. A Concise Pahlavi Dictionary. London: Oxford University Press. Mackenzie, D. N., ed. 1976. The Buddhist Sogdian Texts of the British Library. Acta Iranica, third ser., vol. III. Leiden: Brill. Macri, Martha J. 1985. The numerical head variants and the Mayan numbers. Anthropological Linguistics 27: 46–85. Macri, Martha J. 1996. Rongorongo of Easter Island. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 183–188. Cambridge: Cambridge University Press. Macri, Martha J. 2000. Numeral classifiers and counted nouns in the classic Maya inscriptions. Written Language and Literacy 3(1): 13–36.

Bibliography

457

Macri, Martha J., and Matthew G. Looper. 2003. The New Catalog of Maya Hieroglyphs, Vol. 1: The Classic Period Inscriptions. Norman: University of Oklahoma Press. Maher, David W., and John F. Makowski. 2001. Literary evidence for Roman arithmetic with fractions. Classical Philology 96(4): 376–399. Maisels, Charles Keith. 1987. Models of social evolution: trajectories from the Neolithic to the state. Man (n.s.) 22: 331–359. Mallon, Jean. 1948. Pour une nouvelle critique des chiffres dans les inscriptions latines gravées sur pierre. Emerita 16: 14–45. Mandler, G., and B. J. Shebo. 1982. Subitizing: an analysis of its component processes. Journal of Experimental Psychology: General 11: 1–22. Mangalam, S. J. 1990. Kharosthi Script. Delhi: Eastern Book Linkers. Mann, Charles C. 2005. Unraveling khipu’s secrets. Science 309 (August 12, 2005): 1008–1009. Marcus, Joyce. 1976. The origins of Mesoamerican writing. Annual Review of Anthropology 5: 35–67. Marcus, Joyce. 1992. Mesoamerican Writing Systems. Princeton, NJ: Princeton University Press. Marshack, Alexander. 1972. The Roots of Civilization. New York: McGraw-Hill. Marstrand, Vilhelm. 1949. The runic inscription on No Man’s Land. The New England Quarterly 22(1): 85–92. Martzloff, Jean-Claude. 1997. A History of Chinese Mathematics. Berlin: Springer-Verlag. Masson, Oliver. 1983. Les inscriptions chypriotes syllabiques. Paris: Editions E. de Boccard. Matsuzawa, T. 1985. Use of numbers by a chimpanzee. Nature 315: 57–59. Mazaheri, A. 1974. Formes ‘sounnites’ et formes ‘chi’ites’ des chiffres arabes au les avatars de chiffres indiens en islam. Proceedings of the 13th International Congress of the History of Science (1971, Moscow): 60–63. McCarter, P. Kyle. 1975. The Antiquity of the Greek Alphabet and the Early Phoenician Scripts. Missoula, MT: Scholars Press. McLeish, John. 1991. The Story of Numbers. New York: Fawcett Columbine. McPharlin, Paul. 1942. Roman Numerals, Typographic Bases and Pointing Hands: Some Notes on Their Origin, History, and Contemporary Use. New York: The Typophiles. Megally, Fuad. 1991. Numerical system, Coptic. In The Coptic Encyclopedia, ed. Azis S. Atiya, pp. 1820–1822. New York: Macmillan. Mei Rongzhao. 1983. The decimal place-value numeration and the rod and bead arithmetics. In Ancient China’s Technology and Science, pp. 57–65. Compiled by the Institute of the History of Natural Sciences, Chinese Academy of Sciences. Beijing: Foreign Languages Press. Melchert, H. Craig. 1996. Anatolian hieroglyphs. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 120–124. New York: Oxford University Press. Menninger, Karl. 1969. Number Words and Number Symbols. Translation by Paul Broneer. Cambridge, MA: MIT Press. Messiha, Heshmat. 1994. Les chiffres coptes. Le Monde Copte 24: 25–28. Meuret, Christophe. 1996. Le règlement de Lamasba: des tables de conversion appliquées à l’irrigation. Antiquités africaines 32: 87–112. Mickel, Stanley L. 1981. 10,000 and more: teaching large numbers in Chinese. Chinese Language Teachers Association Journal 16: 83–93. Millard, A. 1995. Strangers from Egypt and Greece – the signs for numbers in early Hebrew. In Immigration and Emigration within the Ancient Near East, ed. K. van Lerberghe and A. Schoors, pp. 189–194. Leuven: Peeters. Miller, G. A. 1933. Our common numerals. Science 78 (2020): 236–237.

458

Bibliography

Miller, G. A. 1956. The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychological Review 63: 81–97. Millet, Nicholas B. 1990. The Narmer Macehead and related objects. Journal of the American Research Center in Egypt 27: 53–59. Millet, Nicholas B. 1996. The Meroitic script. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 84–86. New York: Oxford University Press. Mingana, A. 1936. Catalogue of the Mingana Collection of Manuscripts, Vol. 2. Cambridge: W. Heffer and Sons. Möller, Georg. 1936 [1909–12]. Hieratische paläographie. 4 volumes. Osnabruck: Otto Zeller. Mommsen, Theodor. 1965 [1909]. Gesammelte Schriften, vol. 7. Berlin: Weidmann. Monod, T. 1958. Un nouvel alphabet ouest-africain: le bété (Côte d’Ivoire). Bulletin de l’Institut Français d’Afrique Noire, ser. B, 20(3–4): 432–553. Monteil, Vincent. 1951. La cryptographie chez les Maures. Bulletin de l’Institut Français d’Afrique Noire 13: 1257–1264. Morley, H. T. 1947. Notes on Arabic numerals in mediaeval England. Berkshire Archaeological Journal 50: 81–86. Morley, Sylvanus. 1915. An Introduction to the Study of the Maya Hieroglyphs. Washington, DC: Smithsonian Institution. Mühlisch, Alfred. 1985. Maya-Arithmetik. Ethnographisch-archäologische Zeitschrift 26(2): 223–239. Mukherjee, B. N. 1993. The early use of decimal notation in Indian epigraphs. Journal of the Epigraphical Society of India 19: 80–83. Mukherjee, R. N. 1977. Background to the discovery of the symbol for zero. Indian Journal of History of Science 12: 224–231. Müller, F. W. K. 1912. Ein Doppelblatt aus einem manichäischen Hymnenbuch (Mahrnâmag). Abhandlungen der Königlich preussischen Akademie der Wissenschaften, philosophisch-historische Classe, Jahrgang 1912, Abh. V: 1–40. Murray, Alexander. 1978. Reason and Society in the Middle Ages. Oxford: Clarendon. Naroll, Raoul. 1968. Some thoughts on comparative method in cultural anthropology. In Methodology in Social Research, ed. Hubert M. Blalock, Jr., and Ann B. Blalock, pp. 236–277. New York: McGraw-Hill. Nau, F. 1910. Notes d’astronomie indienne. Journal Asiatique, tenth series, 16: 209–225. Naveh, Joseph. 1968. Dated coins of Alexander Janneus. Israel Exploration Journal 18: 20–25. Needham, Joseph, with the assistance of Wang Ling. 1959. Science and Civilization in China. Volume 3: Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press. Nemet-Nejat, Karen R. 2002. Square tablets in the Yale Babylonian Collection. In Under One Sky: Astronomy and Mathematics in the Ancient Near East, ed. John M. Steele and Annette Imhausen, pp. 253–281. Münster: Ugarit-Verlag. Nenci, Giuseppe. 1995. Il sistema numerale acrofonico pseudoascendente nella Sicilia Greca. Annali della scuola normale superiore di Pisa, classe di lettere e filosofia, ser. III, 25(1–2): 1–5. Neugebauer, Otto. 1941. On a special use of the sign “zero” in cuneiform astronomical texts. Journal of the American Oriental Society 61: 213–215. Neugebauer, Otto. 1952. Tamil astronomy: a study in the history of astronomy in India. Osiris 10: 252–276. Neugebauer, Otto. 1957. The Exact Sciences in Antiquity, second ed. Providence, RI: Brown University Press.

Bibliography

459

Neugebauer, Otto. 1960. Studies in Byzantine astronomical terminology. Transactions of the American Philosophical Association 50(2): 3–45. Neugebauer, Otto. 1975. A History of Ancient Mathematical Astronomy. 3 vols. New York: Springer-Verlag. Neugebauer, Otto. 1979. Egyptian Astronomy and Computus. Vienna: Verlag der Osterreichischen Akademie der Wissenschaften. Nickel, Helmut. 1973. Tamgas and runes, magic numbers and magic symbols. Metropolitan Museum Journal 8: 165–173. Nickerson, Raymond S. 1988. Counting, computing, and the representation of numbers. Human Factors 30(2): 181–199. Nielsen, Richard. 1986. The Arabic numbering system on the Kensington Rune Stone. The Epigraphic Society Occasional Publications 15: 47–61. Ninni, A. P. 1888–89. Sui segni prealfabetici usati anche ora nella numerazione scritta dai pescatori Clodiensi. Atti del Reale Istituto Veneto di Scienze, Lettere e Arti 7 (ser. VI): 679–686. Nissen, Hans J., Peter Damerow, and Robert K. Englund. 1993. Archaic Bookkeeping. Translation of Frühe Schrift und Techniken der Wirtschaftsverwaltung im alten Vorderen Orient by Paul Larsen. Chicago: University of Chicago Press. Nöldeke, Theodor. 1904. Compendious Syriac Grammar. Translated by James A. Crichton. London: Williams and Norgate. Novum Testamentum in Linguam Amharicam. 1852. C. H. Blumhardt, ed. London: Watts. O’Connor, M. 1996. The Berber scripts. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 112–116. New York: Oxford University Press. Okenfuss, Max J. 1973. Technical training in Russia under Peter the Great. History of Education Quarterly 13(4): 325–345. Oliverio, Gaspare. 1933. Documenti antichi dell’Africa Italiana. Bergamo: Istituto italiano d’arti grafiche. Olivier, Jean-Pierre. 1986. Cretan writing in the second millennium bc. World Archaeology 17(3): 377–389. Olivier, Jean-Pierre, Louis Godart, and Jean-Claude Poursat. 1996. Corpus Hieroglyphicarum Inscriptionum Cretae. Athens: Ecole française d’Athènes. Oppenheim, A. Leo. 1959. On an operational device in Mesopotamian bureaucracy. Journal of Near Eastern Studies 18: 121–128. Palsgrave, John. 1530. Lesclarcissement de la langue francoyse. London: Richard Pynson. Palter, Robert. 1996. Black Athena, Afrocentrism, and the history of science. In Black Athena Revisited, ed. Mary K. Lefkowitz and Guy Maclean Rogers, pp. 209–266. Chapel Hill: University of North Carolina, Press. Pankhurst, Richard K. P., ed. 1985. Letters from Ethiopian Rulers (Early and Mid-Nineteenth Century). Translated by David L. Appleyard and A. K. Irvine. Oxford: Oxford University Press. Parker, Richard A. 1972. Demotic Mathematical Papyri. (Brown Egyptological Studies VII.) Providence, RI: Brown University Press. Parpola, Asko. 1994. Deciphering the Indus Script. Cambridge: Cambridge University Press. Parpola, Asko. 1996. The Indus script. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 165–171. New York: Oxford University Press. Paso y Troncoso, Francisco del. 1912. Codice Kingsborough: Memorial de los Indios de Tepetlaotzoc al monarca español contra los encomenderos del pueblo. Madrid: Fototipia de Hauser y Menet.

460

Bibliography

Payne, Stanley E., and Michael P. Closs. 1986. A survey of Aztec numbers and their uses. In Native American Mathematics, ed. Michael P. Closs, pp. 213–236. Austin: University of Texas Press. Peet, T. Eric. 1923. The Rhind Mathematical Papyrus. London: British Museum. Peet, T. Eric. 1931. Mathematics in ancient Egypt. Bulletin of the John Rylands Library 15: 409–431. Pennington, A. E. 1967. A sixteenth-century English Slavist. The Modern Language Review 62(4): 680–686. Penrose, Thomas F. 1984. Mayan Cryptoquantum Numerations. Franklin Park, NJ: Liberty Bell Associates. Perny, Paul. 1873. Grammaire de la langue chinoise orale et écrite, vol. 1. Paris: Maisonneuve. Peruzzi, Emilio. 1980. Myceneans in Early Latium. Rome: Edizioni dell’Ateneo & Bizzarri. Pestman, P. W., ed. 1994. Les papyrus démotiques de Tsenhor (P. Tsenhor). Vol. II: Paléographie et planches. Leuven: Peeters. Petrie, W. M. F. 1926. Ancient Weights and Measures. Warminster, UK: Aris and Phillips. Petruso, Karl M. 1978. Marks on some Minoan balance weights and their interpretation. Kadmos 17: 26–42. Petruso, Karl M. 1981. Early weights and weighing in Egypt and the Indus Valley. Bulletin of the Museum of Fine Arts (Boston) 79: 44–51. Pettersson, J. S. 1999. Indus numerals on metal tools. Indian Journal of History of Science 34(2): 89–108. Pettinato, Giovanni. 1981. The Archives of Ebla: An Empire Inscribed in Clay. Garden City, NY: Doubleday. Piaget, Jean. 1952. The Child’s Conception of Number. London: Routledge and Kegan Paul. Pihan, Antoine Paulin. 1860. Exposé des signes de numération usités chez les peuples orientaux anciens et modernes. Paris: Imprimerie Impériale. Pinnow, Heinz-Jürgen. 1972. Schrift und Sprache in den Werken Lako Bodras im Gebiet der Ho von Singbhum (Bihar). Anthropos 67: 822–857. Plofker, Kim. 2007. Mathematics in India. In The Mathematics of Egypt, Mesopotamia, China, India, and Islam, ed. Victor Katz, pp. 385–514. Princeton, NJ: Princeton University Press. Pohl, Frederik. 1966. Digits and Dastards. New York: Ballantine. Pohl, Mary E. D., Kevin O. Pope, and Christopher von Nagy. 2002. Olmec origins of Mesoamerican writing. Science 298: 1984–1987. Posener-Kriéger, P. 1977. Les mesures des étoffes à l’Ancien Empire. Revue d’Égyptologie 29: 86–96. Postgate, Nicholas, Tao Wang, and Toby Wilkinson. 1995. Evidence for early writing: utilitarian or ceremonial? Antiquity 69(264): 459–480. Pot, Hessel. 1999. Roman numerals. The Mathematical Intelligencer 21(3): 80. Potts, Daniel T. 1999. The Archaeology of Elam: Formation and Transformation of an Ancient Iranian State. Cambridge: Cambridge University Press. Powell, Marvin A. 1971. Sumerian numeration and metrology. Ph.D. dissertation, University of Minnesota. Powell, Marvin A. 1972a. The origin of the sexagesimal system: the interaction of language and writing. Visible Language 6: 5–18. Powell, Marvin A. 1972b. Sumerian area measures and the alleged decimal substratum. Zeitschrift für Assyriologie 62: 165–221.

Bibliography

461

Powell, Marvin A. 1976. The antecedents of Old Babylonian place notation and the early history of Babylonian mathematics. Historia Mathematica 3: 417–439. Preston, Jean F. 1994. Playing with numbers: some mixed counting methods found in French medieval manuscripts at Princeton. In Medieval Codicology, Iconography, Literature and Translation: Studies for Keith Val Sinclair, ed. Peter R. Monks and D. D. R. Owen, pp. 74–78. Leiden: Brill. Price, Derek J. de Solla, and Leopold Pospisil. 1966. A survival of Babylonian arithmetic in New Guinea. Indian Journal of History of Science 1(1): 30–33. Prinsep, James. 1838. Examination of inscriptions from Girnar in Gujerat, and Dhauli in Cuttack. Journal of the Asiatic Society of Bengal 7: 352–353. Priskin, Gyula. 2002. The Eye of Horus and the synodic month. Discussions in Egyptology 53: 75–81. Priskin, Gyula. 2003. Cryptic numerals on cubit rods. Göttinger Miszellen 192: 61–66. Quibell, James Edward. 1900. Hierakonpolis, Part I. London: Bernard Quaritch. Quilter, Jeffrey, and Gary Urton, eds. 2002. Narrative Threads: Accounting and Recounting in Andean Khipu. Austin: University of Texas Press. Ratliff, Martha. 1996. The Pahawh Hmong script. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 619–624. New York: Oxford University Press. Rebstock, Ulrich. 1990. Arabic mathematical manuscripts in Mauretania. Bulletin of the School of Oriental and African Studies 53(3): 429–441. Reif, Stefan C. 2002. A centennial assessment of Genizah studies. In The Cambridge Geniza Collections: Their Contents and Significance, ed. Stefan C. Reif, pp. 1–35. Cambridge: Cambridge University Press. Reineke, Walter F. 1992. Zur Entstehung der ägyptischen Bruchrechnung. Altorientalische Forschungen 19: 201–211. Renou, Louis, and Jean Filliozat. 1953. L’Inde Classique: Maneul des Études Indiennes, vol. 2. Hanoi: École Française d’Extrême-Orient. Restivo, Sal. 1992. Mathematics in Society and History. Dordrecht: Kluwer. Richerson, Peter, and Robert Boyd. 2005. Not by Genes Alone: How Culture Transformed Human Evolution. Chicago: University of Chicago Press. Ritner, Robert K. 1996. Egyptian writing. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 73–84. New York: Oxford University Press. Ritter, Hellmut. 1936. Griechisch-koptische Ziffern in arabischen Manuskripten. Rivista degli studi orientali 16: 212–214. Ritter, Jim. 2002. Closing the Eye of Horus: the rise and fall of ‘Horus-eye fractions’. In Under One Sky: Astronomy and Mathematics in the Ancient Near East, ed. John M. Steele and Annette Imhausen, pp. 297–323. Münster: Ugarit-Verlag. Rix, Helmut. 1969. Buchstabe, Zahlwort und Ziffer im alten Mittelitalien. In Studi linguistici in onore di Vittore Pisani, vol. 2, pp. 845–856. Brescia: Paideia Editrice. Robson, Eleanor. 2007. Mesopotamian mathematics. In The Mathematics of Egypt, Mesopotamia, China, India, and Islam, ed. Victor Katz, pp. 58–186. Princeton, NJ: Princeton University Press. Robson, Eleanor. 2008. Mathematics in Ancient Iraq: A Social History. Princeton, NJ: Princeton University Press. Rock, Joseph F. 1937. The Nichols Mo-so manuscript of the American Geographical Society. The Geographical Review 27: 229–239.

462

Bibliography

Rödiger, E. 1862. Die syrischen Zahlzeichen. Zeitschrift der deutschen morganländischen Gesellschaft 16: 577–578. Rodríguez Martínez, Ma. del Carmen, Ponciano Ortíz Ceballos, Michael D. Coe, Richard A. Diehl, Stephen D. Houston, Karl A. Taube, and Alfredo Delgado Calderón. 2006. Oldest writing in the New World. Science 313(5793): 1610–1614. Rogers, Everett. 1940. Diffusion of Innovations. New York: The Free Press. Rohlfs, Gerhard. 1872. Die Zahlzeichen der Rhadamser. Ausland 45: 695–696. Rollston, Christopher A. 2006. Scribal education in ancient Israel: the Old Hebrew epigraphic evidence. Bulletin of the American Schools of Oriental Research 344: 47–74. Ronen, Yigal. 1996. The enigma of the shekel weights of the Judean kingdom. The Biblical Archaeologist 59(2): 122–125. Ross, Allan S. C. 1938. The “Numeral-signs” of the Mohenjo-daro Script. (Memoirs of the Archaeological Survey of India, vol. 57.) Delhi: Archaeological Survey of India. Rossi, Corinna. 2004. Architecture and Mathematics in Ancient Egypt. Cambridge: Cambridge University Press. Rouillon, André. 2006. Au Gravettien, dans la grotte Cosquer (Marseille, Bouches-duRhône), l’homme a-t-il compté sur les doigts? L’anthropologie 110: 500–509. Rouse, Irving. 1986. Migrations in Prehistory. New Haven, CT: Yale University Press. Rowe, John H. 1966. Diffusionism and archaeology. American Antiquity 31(3): 334–337. Rüster, Christel, and Erich Neu. 1989. Hethitisches Zeichenlexikon. Wiesbaden: Harrassowitz. Ryan, W. F. 1991. Scientific instruments in Russia from the Middle Ages to Peter the Great. Annals of Science 48: 367–384. Sahlins, Marshall. 1976. Culture and Practical Reason. Chicago: University of Chicago Press. Saidan, Ahmad S. 1966. The earliest extant Arabic arithmetic: Kitab al-Fusul fi al Hisab alHindi of Abu al-Hasan, Ahmad ibn Ibrahim al-Uqlidisi. Isis 57(4): 475–490. Saidan, Ahmad S. 1974. The arithmetic of Abu’l-Wafa’. Isis 65(3): 367–375. Saidan, Ahmad S. 1996. Numeration and arithmetic. In Encyclopedia of the History of Arabic Science, vol. 2, ed. Roshdi Rashed, pp. 331–348. London: Routledge. Salama, Pierre. 1999. Anomalies paléographiques des chiffres sur les inscriptions africaines de l’antiquité tardive. Antiquité Tardive 7: 231–254. Salomon, Frank. 2004. The Cord Keepers: Khipus and Cultural Life in a Peruvian Village. Durham, NC: Duke University Press. Salomon, Richard. 1995. On the origin of the early Indian scripts. Journal of the American Oriental Society 115(2): 271–279. Salomon, Richard. 1996. Brāhmī and Kharoshthi. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 373–383. New York: Oxford University Press. Salomon, Richard. 1998. Indian Epigraphy. New York: Oxford University Press. Salzmann, Zdenek. 1950. A method for analyzing numerical systems. Word 6: 78–83. Sanchez, George I. 1961. Arithmetic in Maya. Austin, TX: privately printed. Sanchez Perez, Jose A. 1935. Sobre las cifras rumies. Al-andalus 3: 97–125. Sander, Lore. 1968. Paläographisches zu den Sanskrithandschriften der Berliner Turfansammlung. Wiesbaden: Franz Steiner. Sandys, Sir John Edward. 1919. Latin Epigraphy. Cambridge: Cambridge University Press. Sanjian, Avedis K. 1996. The Armenian alphabet. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 356–363. New York: Oxford University Press. Sarma, Sreeramula Rajeswara. 1999. Katapayadi notation on a Sanskrit astrolabe. Indian Journal of History of Science 34(4): 273–287.

Bibliography

463

Sarton, George. 1936a. A Hindu decimal ruler of the third millennium. Isis 25: 323–326. Sarton, George. 1936b. Minoan mathematics. Isis 24: 375–381. Satterthwaite, Linton, Jr. 1947. Concepts and Structures of Maya Calendrical Arithmetics. (Joint Publication No. 3, Museum of the University of Pennsylvania and the Philadelphia Anthropological Society.) Philadelphia: University of Pennsylvania. Saussure, Ferdinand de. 1959. Course in General Linguistics. New York: Philosophical Library. Saxe, Geoffrey B. 1981. Body parts as numerals: a developmental analysis of numeration among the Oksapmin in Papua New Guinea. Child Development 52(1): 306–316. Schanzlin, G. L. 1934. The abjad notation. The Moslem World 24: 257–261. Schapiro, Meyer. 1942. A note on an inscription of the cathedral of Santiago de Compostela. Speculum 17(2): 261–2. Schärlig, Alain. 2001. Les deux types d’abaques des anciens grecs et leurs jetons quinaires. Archives des Sciences 54(2): 69–75. Schenker, Alexander M. 1996. The Dawn of Slavic. New Haven, CT: Yale University Press. Schmandt-Besserat, Denise. 1978. The earliest precursor of writing. Scientific American 238(6): 38–47. Schmandt-Besserat, Denise. 1984. Before numerals. Visible Language 18(1): 48–60. Schmandt-Besserat, Denise. 1987. Oneness, twoness, threeness. The Sciences 27(4): 44–49. Schmandt-Besserat, Denise. 1992. Before Writing. Austin: University of Texas Press. Schmookler, Jacob. 1966. Invention and Economic Growth. Cambridge, MA: Harvard University Press. Schroder, Paul. 1869. Die Phönizische Sprache. Halle: Buchhandlung des Waisenhauses. Schub, Pincus. 1932. A mathematical text by Mordecai Comtino. Isis 17: 54–70. Schuhmacher, W. W. 1974. Die zoomorphen Zahlzeichen der Osterinselschrift: eine mengentheoretische Notiz. Anthropos 69: 271–272. Segal, J. B. 1954. Some Syriac inscriptions of the 2nd–3rd century ad. Bulletin of the School of Oriental and African Studies 16(1): 13–36. Segal, J. B. 1983. Aramaic Texts from North Saqqara. London: Egypt Exploration Society. Seidenberg, Abraham. 1986. The zero in the Mayan numerical notation. In Native American Mathematics, ed. Michael P. Closs, pp. 371–386. Austin: University of Texas Press. Sen, S. N. 1971. Mathematics. In A Concise History of Science in India, ed. D. M. Bose, S. N. Sen, and B. V. Subbarayappa, pp. 136–212. New Delhi: Indian National Science Academy. Senart, E. 1905–6. The inscriptions in the caves at Nasik. Epigraphia Indica 8: 59–95. Sesiano, Jacques. 1989. Koptisches Zahlensystem und (griechisch-)koptische Multiplikationstafeln nach einem arabischen Bericht. Centaurus 32: 53–65. Sesiano, Jacques. 2002. A reconstruction of Greek multiplication tables for integers. In From China to Paris: 2000 Years Transmission of Mathematical Ideas, ed. Yvonne DoldSamplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen, pp. 45–56. Stuttgart: Franz Steiner Verlag. Sethe, Kurt. 1916. Von Zahlen und Zahlworten bei den alten Ägyptern, und was für andere Völker und Sprachen daraus zu lernen ist. Strassburg: Karl J. Trübner. Shafer, Robert. 1950. Lycian numerals. Archiv Orientalni 18(4): 251–261. Shaw, Allen A. 1938–39. An overlooked numeral system of antiquity. National Mathematics Magazine 13: 368–372. Shea, William H. 1978. The inscribed Late Bronze jar handle from Tell Ḥalif. Bulletin of the American Schools of Oriental Research 232: 78–80.

464

Bibliography

Shermer, Michael. 1995. Exorcising Laplace’s demon: chaos and antichaos, history and metahistory. History and Theory 34(1): 59–83. Shipley, Frederick W. 1902. Numeral corruptions in a ninth century manuscript of Livy. Transactions of the American Philological Association 33: 45–54. Simonov, Rem Aleksandrovic. 1993. Die Archäologie bestätigt 900jähriges Alter des altrussischen Abakus (des “Zählens mit Kernen”). Jahrbücher für Geschichte Osteuropas 41: 111–113. Singh, A. K. 1991. Development of Nagari Script. Delhi: Parimal Publications. Sircar, D. C. 1964. Select Inscriptions Bearing on Indian History and Civilization, vol. 1. Delhi: V. K. Publishing House. Skjaervø, P. Oktor. 1996. Aramaic scripts for Iranian languages. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 515–535. New York: Oxford University Press. Smalley, William A., Chia Koua Vang, and Gnia Yee Yang. 1990. Mother of Writing: The Origin and Development of a Hmong Messianic Script. Chicago: University of Chicago Press. Smith, David Eugene. 1908. Rara Arithmetica. Boston: Ginn and Co. Smith, David Eugene. 1926. The Roman numerals. Scientia 40: 1–8, 69–78. Smith, David Eugene, and Jekuthiel Ginsburg. 1937. Numbers and Numerals. Washington DC: National Council of Teachers of Mathematics. Smith, David Eugene, and L. C. Karpinski. 1911. The Hindu-Arabic Numerals. Boston: Ginn. Smith, David Eugene, and Yoshio Mikami. 1914. A History of Japanese Mathematics. Chicago: Open Court. Smith, David Eugene, and Salih Mourad. 1927. The dust numerals among the ancient Arabs. American Mathematical Monthly 34(5): 258–260. Smith, Grafton Elliot. 1923. The Ancient Egyptians and the Origin of Civilization. London: Harper. Smyly, J. Gilbart. 1905. The employment of the alphabet in Greek logistic. In Mélanges Nicole, pp. 515–530. Geneva: W. Kündig and Son. Soubeyran, Denis. 1984. Textes mathématiques de Mari. Revue d’Assyriologie 78: 19–48. Souissi, M. 1971. Hisab al-ghubar. In Encyclopedia of Islam, vol. 3, pp. 468–469. Leiden: Brill. Stampe, David. 1976. Cardinal number systems. Papers from the 12th Regional Meeting, Chicago Linguistic Society, pp. 594–609. Ste. Croix, G. E. M. de. 1956. Greek and Roman accounting. In Studies in the History of Accounting, ed. A. C. Littleton and B. S. Yamey, pp. 14–73. London: Sweet and Maxwell. Steele, Robert, ed. 1922. The Earliest Arithmetics in English. London: Early English Text Society. Stenhouse, William. 2005. Reading Inscriptions and Writing Ancient History: Historical Scholarship in the Late Renaissance. London: Institute of Classical Studies. Steward, Julian. 1955. Cultural causality and law: a trial formulation of the development of early civilizations. In Theory of Culture Change, ed. Julian Steward, pp. 178–209. UrbanaChampaign: University of Illinois Press. Stieglitz, Robert R. 1978. Minoan mathematics or music? Bulletin of the American Society of Papyrologists 15: 127–132. Stolper, Matthew W., and Jan Tavernier. 2007. From the Persepolis Fortification Archive Project, 1: An Old Persian Administrative Tablet from the Persepolis Fortification. ARTA 2007.001. http://www.achemenet.com/document/2007.001-Stolper-Tavernier.pdf. Stone, Lawrence. 1949. Elizabethan overseas trade. The Economic History Review, n.s. 2(1): 30–58.

Bibliography

465

Stone, Ruth M. 1990. Ingenious invention: the indigenous Kpelle script in the late twentieth century. Liberian Studies Journal 15(2): 135–144. Stross, Brian. 1985. Maya head variant numerals: the Olmec case. Anthropological Linguistics 27: 1–45. Struik, Dirk J. 1964. The Kensington stone mystery. Mathematics Teacher 57: 166–168. Struik, Dirk J. 1968. The prohibition of the use of Arabic numerals in Florence. Archives Internationales d’Histoire des Sciences 21: 291–294. Struik, Dirk J. 1982. Minoan and Mycenaean numerals. Historia Mathematica 9: 54–58. Subbarayappa, B. V. 1996. Indus Script: Its Nature and Structure. Madras: New Era Publications. Sundermann, Werner. 1997. The Manichaean texts in languages and scripts of Central Asia. In Languages and Scripts of Central Asia, ed. Shirin Akiner and Nicholas Sims-Williams, pp. 39–45. London: School of Oriental and African Studies. Sundermann, Werner, and Peter Zieme. 1981. Soghdisch-Türkische Wortlisten. In Scholia: Beiträge zur Turkologie und Zentralasienkunde, ed. Klaus Röhrborn and Horst Wilfrid Brands, pp. 184–193. Wiesbaden: Otto Harrassowitz. Swetz, Frank. 1987. Capitalism and Arithmetic: The New Math of the 15th Century. LaSalle, IL: Open Court. Swiggers, Pierre. 1996. Transmission of the Phoenician script to the West. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 261–270. New York: Oxford University Press. Swinton, John. 1753–54. An Explication of All the Inscriptions in the Palmyrene Language and Character Hitherto Publish’d. In Five Letters from the Reverend Mr. John Swinton, M. A. of Christ-Church, Oxford, and F. R. S. to the Reverend Thomas Birch, D. D. Secret. R. S. Philosophical Transactions (1683–1775) 48: 690–756. Syamalamma, K. 1992. The Origin and Evolution of Telugu-Kannada Numerals. Tirupati: Master Publications. Taisbak, C. M. 1965. Roman numerals and the abacus. Classica et Mediaevialia 26: 147–160. Takashima, K. 1985. On the quantitative complement in oracle-bone inscriptions. Journal of Chinese Linguistics 13(1): 44–68. Tatton-Brown, Tim, and Dan Miles. 2003. Salisbury Cathedral. Current Archaeology 16(188): 364–369. Taube, Karl. 2000. The Writing System of Ancient Teotihuacan. (Ancient America 1.) Barnardsville, NC, and Washington, DC: Center for Ancient American Studies. Teeple, John E. 1931. Maya Astronomy. (Contributions to American Anthropology and History 1(2). Carnegie Institute of Washington, Publication 403.) Washington, DC: Carnegie Institute. Temple, R. C. 1891. Notes on the Burmese system of arithmetic. Indian Antiquary 20: 53–69. Terraciano, Kevin. 2001. The Mixtecs of Colonial Oaxaca: Ñudzahui History, Sixteenth through Eighteenth Centuries. Stanford, CA: Stanford University Press. Testen, David D. 1996. Old Persian cuneiform. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 134–137. New York: Oxford University Press. Thomas, Ivor. 1962. Selections Illustrating the History of Greek Mathematics, vol. 1. Cambridge, MA: Harvard University Press. Thomas, J. David. 1977. Avoidance of theta in dating by regnal years. Zeitschrift für Papyrologie und Epigraphik 24: 241–243. Thompson, J. Eric S. 1941. Maya Arithmetic. Washington, DC: Carnegie Contributions to American Anthropology and History pub. 528, vol. 403.

466

Bibliography

Thompson, J. Eric S. 1971. Maya Hieroglyphic Writing: An Introduction. Norman: University of Oklahoma Press. Thomson, Robert W. 1989. An Introduction to Classical Armenian. Delmar, NY: Caravan Books. Threatte, Leslie. 1980. The Grammar of Attic Inscriptions, Vol. 1 (Phonology). Berlin: de Gruyter. Throop, Liz. 2004. Thinking on paper: Hindu-Arabic numerals in European typography. Visible Language 38(3): 290–303. Thureau-Dangin, François. 1939. Sketch of a history of the sexagesimal system. Translated by Solomon Gandz. Osiris 7: 95–141. Till, Walter C. 1961. Koptische Grammatik. Leipzig: Veb Verlag. Tod, Marcus Niebuhr. 1911–12. The Greek numeral notation. Annual of the British School at Athens 18: 98–132. Tod, Marcus Niebuhr. 1913. The Greek numeral systems. Journal of Hellenic Studies 33: 27–34. Tod, Marcus Niebuhr. 1926–27. Further notes on the Greek acrophonic numerals. Annual of the British School at Athens 28: 141–157. Tod, Marcus Niebuhr. 1936–37. The Greek acrophonic numerals. Annual of the British School at Athens 37: 236–257. Tod, Marcus Niebuhr. 1950. The alphabetic numeral system in Attica. Annual of the British School at Athens 45: 126–139. Tod, Marcus Niebuhr. 1954. Letter-labels in Greek inscriptions. Annual of the British School at Athens 49: 1–8. Tod, Marcus Niebuhr. 1979. Ancient Greek Numerical Systems. Chicago: Ares Publishers. Tolstoy, Paul. 1972. Diffusion: as explanation and event. In Early Chinese Art and Its Possible Influence in the Pacific Basin, ed. Noel Barnard, vol. 3, pp. 823–841. New York: Intercultural Arts Press. Tong, Anthony K. K. 1999. The language in Chinese mathematical texts before AD 1300: a case for logical abstraction. Interchange 30(4): 433–458. Tozzer, A. M. 1941. Landa’s Relacion de las Cosas de Yucatan. (Papers of the Peabody Museum of American Archaeology and Ethnology, Harvard University, vol. 18.) Cambridge, MA: Harvard University Press. Travaini, Lucia. 1998. Un sistema di conto poco consosciuto: la ‘mano da quattro’. Revue numismatique 153: 327–334. Trigger, Bruce G. 1991. Constraint and freedom – a new synthesis for archeological explanation. American Anthropologist 93: 551–569. Trigger, Bruce G. 2003. Understanding Early Civilizations: A Comparative Study. Cambridge: Cambridge University Press. Troupeau, G. 1974. À propos des chiffres utilisés pour le foliotage des manuscrits arabes. Arabica 21(1): 84. Tuchscherer, Konrad T. 1996. The Kikakui (Mende) syllabary and number writing system. Ph.D. dissertation, London School of Oriental and African Studies. Tuchscherer, Konrad T. 1999. The lost script of the Bagam. African Affairs 98: 55–77. Tuchscherer, Konrad T. 2005. History of writing in Africa. In Africana: The Encyclopedia of the African and African American Experience, second ed., ed. Kwame Anthony Appiah and Henry Louis Gates, Jr., vol. 5, pp. 476–480. New York: Oxford University Press. Turner, Eric G. 1975. Four obols a day men at Saqqara. In Le monde grec: Hommages à Claire Préaux, ed. J. Bingen, G. Cambier, and G. Nachtergael, pp. 573–577. Bruxelles: Éditions de l’Université de Bruxelles.

Bibliography

467

Tylor, E. B. 1958 [1871]. Primitive Culture. 2 vols. New York: Harper and Brothers. Urcid Serrano, Javier. 2001. Zapotec Hieroglyphic Writing. (Studies in Pre-Columbian Art and Archaeology, no. 34.) Washington, DC: Dumbarton Oaks Research Library. Urton, Gary. 1997. The Social Life of Numbers. Austin: University of Texas Press. Urton, Gary. 1998. From knots to narratives: reconstructing the art of historical record keeping in the Andes from Spanish transcriptions of Inka khipus. Ethnohistory 45(3): 409–437. Urton, Gary. 2001. A calendrical and demographic tomb text from northern Peru. Latin American Antiquity 12(2): 127–147. Urton, Gary. 2005. Khipu archives: duplicate accounts and identity labels in the Inka knotted string records. Latin American Antiquity 16(2): 147–167. Urton, Gary, and Carrie Brezine. 2005. Khipu accounting in ancient Peru. Science 309 (Aug. 12, 2005): 1065–1067. Vaillant, André. 1948. Manuel du vieux slave. Paris: Institut d’Études Slaves. Vaillant, G. C. 1950. The Aztecs of Mexico. Harmondsworth: Pelican Books. Van den Branden, Albertus. 1969. Grammaire phénicienne. Beyrouth: Librairie du Liban. Van den Brom, Lourens. 1969. Woher stammt das 60-system? Janus 56: 210–214. Van der Waerden, B. L. 1963. Science Awakening. Translated by Arnold Dresden. New York: John Wiley. Van der Waerden, B. L. 1983. Geometry and Algebra in Ancient Civilizations. Berlin: SpringerVerlag. Ventris, Michael, and John Chadwick. 1973. Documents in Mycenean Greek, second ed. Cambridge: Cambridge University Press. Verdan, Samuel. 2007. Systèmes numéraux en Grèce ancienne: description et mise en perspective historique. CultureMATH (http://www.dma.ens.fr/culturemath/). Last accessed July 24, 2007. Verma, Thakur Prasad. 1971. The Palaeography of Brahmi Script in North India. Varanasi: Siddharth Prakashan. Volkov, Alexei. 1994. Large numbers and counting rods. In Sous les nombres, le monde, ed. Alexei Volkov, pp. 71–91. (Extrême-Orient, Extrême-Occident 16.) Paris: Université de Paris. Vycichl, Werner. 1952. Das berberische Ziffernsystem von Ghadames und sein Ursprung. Rivista degli Studi Orientali 27: 81–83. Walker, Willard, and James Sarbaugh. 1993. The early history of the Cherokee syllabary. Ethnohistory 40(1): 70–94. Wallerstein, Immanuel. 1974. The Modern World-System, Vol. 1: Capitalist Agriculture and the Origins of European World-Economy in the Sixteenth Century. New York: Academic Press. Wallis, Faith, ed. 1999. The Reckoning of Time, by the Venerable Bede. Liverpool: Liverpool University Press. Wang Ling. 1955. The Chiu chang suan shu (Jiu zhang suan shu) and the history of Chinese mathematics during the Han Dynasty. Ph.D. thesis, University of Cambridge. Wardley, Peter, and Pauline White. 2003. The Arithmeticke Project: a collaborative research study of the diffusion of Hindu-Arabic numerals. Family and Community History 6(1): 5–18. Was, Daniel A. 1971. Numerical fractions in Minoan Linear script A. Kadmos 10: 35–51. Wassén, Henry. 1931. The ancient Peruvian abacus. In Comparative Ethnographical Studies, ed. Erland Nordenskiöld, vol. 9, pp. 189–205. New York: AMS Press.

468

Bibliography

Wells, David. 1986. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin. White, Leslie. 1949. The Science of Culture. New York: Farrar, Strauss. White, Leslie. 1959. The Evolution of Culture. New York: McGraw-Hill. Whiting, Robert M. 1984. More evidence for sexagesimal calculations in the third millennium B.C. Zeitschrift für Assyriologie 74: 59–66. Wiese, Heike. 2003. Numbers, Language, and the Human Mind. Cambridge: Cambridge University Press. Wiese, Heike. 2007. The co-evolution of number concepts and counting words. Lingua 117: 758–772. Williams, Barbara J., and H. R. Harvey. 1997. The Códice de Santa María Asunción. Salt Lake City: University of Utah Press. Williams, Barbara J., and H. R. Harvey. 1988. Content, provenience, and significance of the Codex Vergara and the Codice de Santa Maria Asuncion. American Antiquity 53(2): 337–351. Williams, Burma P., and Richard S. Williams. 1995. Finger numbers in the Greco-Roman world and the early Middle Ages. Isis 86: 587–608. Williams, J. 1995. Mathematics and the alloying of coinage 1202–1700. Annals of Science 52: 213–263. Williams, Jack. 1997. Numerals and numbering in early printed English Bibles and associated literature. Journal of the Printing Historical Society 26: 5–13. Williams, John R. 1934. The Microcosmographia of Trier MS. 1041. Isis 22(1): 106–135. Wilson, Nigel. 1981. Miscellanea paleographica. Greek, Roman and Byzantine Studies 22: 395–404. Wingate, J. S. 1930. The scroll of Cyprian: an Armenian family amulet. Folklore 41(2): 169–187. Woepcke, F. 1863. Mémoire sur la propagation des chiffres indiens. Journal Asiatique, sixth ser, 1: 27–79, 234–290, 442–459. Woodruff, Charles E. 1994 [1909]. The evolution of modern numerals from ancient tally marks. In From Five Fingers to Infinity, ed. Frank Swetz, pp. 53–60. Reprinted from American Mathematical Monthly 16: 125–133. Chicago: Open Court. Worm, Ole. 1643. Fasti Danici. Copenhagen: Hafnia. Wright, W. 1870. Catalogue of Syriac Manuscripts in the British Museum, part II. London: Longman’s & Co. Wrixon, Fred B. 1989. Codes, Ciphers and Secret Languages. New York: Bonanza Books. Wroth, Warwick. 1966. Imperial Byzantine Coins. Chicago: Argonaut. Wynn, Karen. 1992. Addition and subtraction by human infants. Nature 358: 749–751. Yadin, Yigael. 1961. Ancient Judaean weights and the date of the Samaria ostraca. Scripta Hierosolymita 8: 9–25. Yadin, Yigael, Jonas C. Greenfield, Ada Yardeni, and Baruch A. Levine, eds. 2002. The Documents from the Bar Kokhba Period in the Cave of Letters. Jerusalem: Israel Exploration Society. Yano, Michio. 2006. Oral and written transmission of the exact sciences in Sanskrit. Journal of Indian Philosophy 34: 143–160. Zabilka, Ivan Lee. 1968. Ancient Near Eastern number systems and their relationship to the Hebrew number system during the biblical Period. M.Th. thesis, Asbury Theological Seminary.

Bibliography

469

Zangemeister, Karl. 1887. Entstehung der römischen Zahlzeichen. Sitzungberichte der königlich preussischen Akademie der Wissenschaften zu Berlin 1887(2): 1011–1028. Zaslavsky, Claudia. 1973. Africa Counts: Number and Pattern in African Culture. Boston: Prindle, Webber & Schmidt. Zaslavsky, Claudia. 2003. The influence of ancient Egypt on Greek and other numeration systems. Mathematics Teaching in the Middle School 9(3): 174–176. Zhang Yachu and Liu Yu. 1981–82. Some observations about milfoil divination based on Shang and Zhou bagua numerical symbols. Early China 7: 46–55. Zhang, Jiajie, and Donald A. Norman. 1995. A representational analysis of numeration systems. Cognition 57: 271–295. Zide, Norman. 1996. Scripts for Munda languages. In The World’s Writing Systems, ed. Peter T. Daniels and William Bright, pp. 612–618. New York: Oxford University Press. Zimansky, Paul. 1993. Review of Denise Schmandt-Besserat, Before Writing. Journal of Field Archaeology 20: 513–517. Zimmer, Szczepan, Ludwik Krzyzanowski, Irene Nagurski, and Krystyna M. Olszer. 1983. The Beginning of Cyrillic Printing in Crakow. New York: Social Science Monographs; distributed by Columbia University Press.

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