Numerical Notation: a Comparative History

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Th is 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 defi es progressive, unilinear evolutionary models of change, Stephen Chrisomalis identifi es fi ve 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. Th e transformation and replacement of numerical notation systems relate to specifi c 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 Uni- versity 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 Jour- nal, 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 vi Contents

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 fi gures 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. Th is book had its genesis during my time at McGill University. Th e late Bruce Trigger was the shepherd and guiding hand behind this book, beginning in its formative stages and continuing almost to the fi nal draft. Th e 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 refi nements 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 psychologi- cally oriented anthropology and linguistics. I have benefi ted 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 Th e Shape of Script was the key to moving my work into its fi nal 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 infl uences. 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 chiff res (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 Shap- ardanis prepared the index and read proofs. Many thanks to the fourteen anony- mous 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 fi rsthand 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 Th roughout 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

Th e 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 eff ects, many unanswered questions remain concerning how humans use numerals. Why do societies enumerate? How does the representation of numbers today diff er from their representation in the past? Why does the visual representation of number fi gure 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 Th e 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 diff er 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”; ordina- tion – “1. Wash dishes, 2. Sweep fl oor, 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. Th ese 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). Th at 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 devia- tions from the abstract entity comprised by these signs. Th is view is erroneous, and rests on the confusion of a mental concept (signifi ed) with its symbolic representation (signifi er). Our numerical notation system has an extensive history, as do the more than one hundred systems that have existed over the past fi ve 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. Th is 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. Th e 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-fi rst- century critiques of unfettered scientifi c 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 signifi cant contributions to the literature. Th ese 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 sub- Saharan Africa, North America, and Central Asia. Similarly, social scientists such as the anthropologist Th omas 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. Th is 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, identifi es evolutionary regularities, and yet remains cog- nizant 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 effi ciency for performing specifi c 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. Th ough 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-specifi c 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 fi rst letters of the appropriate number-words in a language. However, since such systems are still comprehensible without having to understand a specifi c 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. Th ey rely on one- to-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 diff erent 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 specifi c 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 defi nition, 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 infi nite number of powers of B (e.g., 10, 100, 1000, 10,000, ... ad infi nitum), most numerical notation systems are not infi nitely extendable. It is suffi cient 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. Th e Roman numeral system has a primary base of 10 with a sub-base of 5. Un- like 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) defi nition will suffi ce. 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. Th e 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 diff erent series of words (e.g., two versus second ) for the two concepts. Th is distinction also has implications for our understanding of the origin of numerals and numerical concepts in humans (Crump 1990: 6–10), but has little infl uence on numerical notation. In defi ning mathematics as the science that deals with the logic of quantity, shape, and arrangement, I am consciously employing a simple defi nition 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 Th e 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 dif- fused from any other case. Th is issue, known as Galton’s problem, is the thorniest methodological issue in statistical cross-cultural research (Naroll 1968: 258–262). Th e 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. Th is 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. Th irdly, I wish to explain structural variation among historically related systems, which frequently diff er considerably from their relations. Th is 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. Th e 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). Th e 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 diff er. While some aspects of human existence are truly universal, and others are almost infi nitely 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 reju- venated 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, infor- mational, 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 infl uence of various factors to be more important than positive (enabling) eff ects. 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 fi nd 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 signifi cant regularities in lexical numeral systems presents a list of fi fty-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 Th eodor 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 fi nite5 and has no gaps (Greenberg 1978: 253–255). Similarly, no natural language expresses “two” as “ten minus eight” or “twenty” as “one-fi fth 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 specifi c and unique responses to local needs. We do not live in such a world. Th ere 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). Th is does not preclude the existence of binary and hexadecimal numerical notation for specialized computing purposes. Similarly, while there are only fi ve 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 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). 5 Th is is not true of numerical notation systems, some of which (like our own) are truly infi nite. 6 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 dif- ferent 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 fl ourish 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 diff erent 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. Th e 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 suffi ciently similar to warrant their theoretical analysis as variations on a single theme. I regard numerical notation as translatable cross-culturally without signifi cant loss of information or change of meaning. Th e number 1138 is practically identical in referent to MCXXXVIII or t\s\rrr\qqq\qqq\qq or any other representation. Th ese systems have very diff erent structures, but, in Saussurean terms, the various signifi ers refer to the same signifi ed (Saussure 1959). Although the linguistic and symbolic signifi ers for numbers may diff er 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 criti- cized 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 diff erent 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 diffi culty. 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 diff erences 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). Th e arrival of the year 2000 was not simply another cause for cel- ebration (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 signifi cance 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. Th ese interesting diff erences do not aff ect 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 Th e systematic classifi cation 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. Th e goal of typology is not simply to develop a scheme into which every case fi ts, 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 classifi catory scheme is inherently theory-laden, and answers only some of the questions that might be asked of a set of data. Th e 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 diff erent 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 diff erent 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: MMMMD- CXXXVII – 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. Th e 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. Th e 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 MMMMD- CXXXVII 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 aff ect our understanding of such systems. Th e only major systematic attempt to date to classify numerical notation systems is Geneviève Guitel’s Histoire comparée des numérations écrites (1975).7 Guitel classifi es approximately twenty-fi ve 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 West- ern 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 classifi cation, 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 signifi cant 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 diff erences among numerical notation systems.8 Th ere is another way to approach the subject, however. Consider a diff erent 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. Th e resolution to this diffi culty 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. Th us, 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. Th e 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 diff erent signs being used to represent diff erent multiples of the power. eTh 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. Mul- tiplicative 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

Th e product of the two signs determines the value of that power. Th e 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 numeral- phrase. 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. Th e 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 fi ve 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, multiplicative- additive 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 fi ve 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. Th ese 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 Positional Th e sum of the values of each Th e value of each power power is taken to obtain the to- must be multiplied by a tal value of the numeral-phrase. value dependent on its position before taking the sum of the numeral-phrase. Cumulative Classical Roman Babylonian cuneiform Many signs per power of the base, 1434 = MCCCCXXXIIII 1434 = b3 e4 which are added 1000 + (100 + 100 + 100 + 100) + (10 + 10 + 1 + 1 + 1) × 60 + to obtain the total (10 + 10 + 10) + (1 + 1 + 1 + 1) (10 + 10 + 10 + 10 + 10 + 1 + value of that power. 1 + 1 + 1) × 1 Ciphered Greek alphabetic Khmer Only one sign per power of the base, 1434 = /auld 1434 = which alone repre- ((1 × )1000 + 400 + 30 + 4) (1 × 1000 + 4 × 100 + 3 × sents the total value 10 + 4 × 1) of that power. Multiplicative Chinese (traditional) LOGICALLY Two components EXCLUDED per power, unit- 1434 = ₏◒⥪䤍ₘ◐⥪ sign(s) and a power- (1 × 1000 + 4 × 100 + 3 × 10 + 4) sign, multiplied together, give that power’s total value.

use two principles are not exceptions to my typology. Th ey 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 fi ve 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 clari- fi cation 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 Th is 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 (cumulative- cumulative)-positional system. However, no system uses a diff erent intraexponential principle for its sub-base than for its base, so this elaboration is mostly unnecessary. Again, none of this aff ects the interexponential structure of these systems. Because it refl ects 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 eff ects 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 refl ect 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 eff ects, 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 fi ve 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 fi gures or objects are perceived directly, while larger quantities must be encoded Introduction 15 by counting, a more time-consuming process. Th is experiment involved showing groups of dots to subjects for one-fi fth of a second, after which they would indicate how many were present; up to fi ve 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 quantifi cation by the brain. eTh perception of larger units as gestalts thus maximizes the brain’s effi ciency within the limits of its biological evolution. A third element to be considered, related to the fi rst two, is the principle of one-to-one correspondence defi ned earlier. Th is 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 fi ngers to represent eight coconuts, put aside twenty-seven pebbles to count their fl ock 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 fi fth pint, I might place a horizontal stroke through the four existing strokes to indicate a group of fi ve; my twelve pints would thereby be rendered as two groups of fi ve 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, specifi cally 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 Th ere is much skepticism about the ability of animals to count, after it

10 Th ese 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 cal- culators 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 pri- mate 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 fi ve units. It is not yet known whether animal quantifi cation is a homology inherited from an ancestral species, a specifi c 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 diff erent species have confi rmed that something more than a Clever Hans phenomenon is being observed. Th e 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 suff ering from certain types of brain damage or other serious mental defi ciencies) 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 eff ects 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 fi ve from six bananas is related to the human visual capacity to distinguish fi ve 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. Th e 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. Th e 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. Th e 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 effi ciently, 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 signifi cant historical documentation. eTh contention that the history of technology can be understood as a sequence of ever-superior inventions “the better to fi t the human mind and improve the usability of numbers” is untested at best, ethnocentric at worst (Dehaene 1997: 117). Th ere are three sociocultural features that are likely prerequisites for the development of numerical notation. Th ese 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. Th is 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. Th ere 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. Th e 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 specifi c purpose – is probably necessary to its development. Related to this is the need to communicate number outside a local community. While verbal numbers suffi ce 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 specifi c 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 eff ects on its users. Th ese 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). Th e LoDagaa separate large groups of cowries into smaller groupings of fi ve 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 effi cient 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. Th e 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. Th e 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 specifi c contextual information about how the numerals were used. However, it may be possible to determine whether diff erent types of numerical notation have diff erent cognitive eff ects 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 infi nite extendability (Hallpike 1986: 121–122; Damerow 1996). Th e 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 diff erent groups used numerical notation, keeping in mind that it is only one part of a cluster of techniques that includes mental calculation, lexical numerals, fi nger 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. Th is 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)

Th ere 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. Th ere 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 Th e scholarly analysis of numerical notation has often been pursued by scholars interested in writing systems. Th erefore, 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. Th e 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. Th ere 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 signifi cant than the similarities between the two lexical representations, then the connection between numerical notation and writing is strong. However, four diff erences 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 chrono- logically, and could not have occurred in a nonlinguistic species, but they are not inherently linked to any language structurally or semantically. Th e 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. Th e 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 classifi es pictographic representational systems, numerical notation, and even pictorial art under the rubric “writing.” Gelb’s classic defi nition 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 defi nition 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. Be- cause 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 diff erences. Th is does not imply that they exist completely outside of culture. As Houston (2004a: 226) and others have noted, quantifi cation systems exist within cultural contexts, which is why, even though we can identify Inka khipu (Chapter 10) as encoding specifi c 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. Th is 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. How- ever, 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. Th roughout 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. Th irdly, numerical notation systems and scripts exhibit very diff erent patterns of geographical distribution and historical change. In part, this may be because scripts are largely phonographic, so their diff usion can be constrained by patterns of 22 Numerical Notation language use. Numerical notation, by contrast, is largely nonphonetic and translinguistic, and may diff use more readily than scripts. Th e Western numerals diff used initially from India and passed through the Arab world before reaching Europe, while the Roman alphabet is of Greek and Phoenician ancestry. Th is historical dif- ferentiation is not uncommon; the path of diff usion of numerical notation is often radically diff erent from that of the diff usion 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 coin- cided with the development of a full-fl edged 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 diff erent. Th e simple fact of being denoted visually is not as important as the diff erent principles used in the two symbol systems. Lexical numerals (whether written or verbal) share a common structure that is very diff erent 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 multiplicative- additive. To take a familiar example, let us compare Western numerical notation with English lexical numerals. Th e 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 base- 10 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 infi - nitely extendable – one can add zeroes to the right of a number ad infi nitum – English lexical numerals are only potentially infi nite, since one needs to develop new words to express higher and higher values. Th e highest number in many English diction- aries is decillion (1033 in American English, 1060 in British English).

11 Th e 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 defi nitional issue in Chapter 8. Introduction 23

Th e 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). Th is may in turn have been related to the early use of the fi ngers 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 strong- est 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 diff erent and far-more-ancient origin,” and this may be correct. However, numerical notation (as opposed to non-base- structured 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 specifi c relationship between numerals and scripts.

Diffusion and Invention Numerical notation systems develop out of purposeful human eff orts to perform tasks related to the visual representation of number. In a handful of instances, they developed independently of infl uence from other numerical notation systems, while in the vast majority of cases systems were borrowed wholesale or with modifi cation 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. Th is 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 eff ect – 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. Th ere 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 eff ort 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 diff usion is a sterile “nonprinciple” that is “not only superfl uous, but the very incarnation of antiscience” (1968: 378). Harris is right that simply classifying an innovation as representing either diff usion or independent invention is insuffi cient, 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 diff usion leads him to assume that cultural adaptation is a unitary process and that analogical explanations are the only ones worthy of scientifi c consideration. Yet if the social consequences of, and motivations for, adopting diff used numerical notation systems and adopting independent invention are diff erent – as I believe them to be – then we must instead compare the two diff erent circumstances while keeping an open mind as to potential diff erences. “Diff usion” 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. Th is 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 long- distance 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 aff ect 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 specifi cally 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 diff usionary explanations only when abundant evidence of colonies, trading posts, or traded objects independently confi rms contact between two regions, while Tolstoy (1972) deemed it suffi cient to show that a particular combination of features is probabil- istically unlikely to have occurred independently. Th is question is unresolvable in the abstract, because the ease of demonstrating cultural transmission depends on the nature of the specifi c 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. Th us, I must build a circumstantial case for the origins of most systems. In order to demonstrate cultural affi liations between numerical notation systems, I use both internal (structural and graphic) resemblances between systems and external (contextual and historical) considerations. Th e main criteria I use are as follows: 1. Use of the two systems at the same point in time. Th is 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 modifi ed by a later society (for instance, on the basis of old inscriptions), but this is hardly a suffi cient 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 suffi cient 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. Th is 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 fi t because they were used much later and have diff erent numeral-signs. Th e use of structural features as evidence of contact suff ers 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. Th e 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 diff erent 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. Th is 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. Th ere is no general paleographic principle for identifying relations among graphically similar signs; hence, such eff orts 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. Th us, 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 suffi cient to postulate the diff usion of a numerical notation system is always a tricky matter and involves an evaluation of various lines of evidence. For instance, one of the diffi culties 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. Th ere must always be some structural or graphic resemblance between postulated ancestor and descendant systems. Th is problem is made more complex by stimulus diff usion, a complex blend of inventive and diff usionary 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. Th e most widely cited example of stimulus diff usion 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 diff usion (e.g., the abortive 28 Numerical Notation

Cherokee numerals, never used in the syllabary), no principles exist to help identify stimulus diff usion. It is tempting to postulate stimulus diff usion even when the basic fact of incomplete transmission cannot be established. However, I use stimulus diff usion as an explanation only when it can be established that the form of cultural contact that occurred between two regions fi ts 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 confi rmatory evidence that the two systems are related historically. Th is factor, while useful, is never suffi cient 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 confi rmed not only by the striking similarities in the systems but also by the systems’ use in Bibles and other liturgical texts. Th e 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 specifi c 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. Th is 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 diff erent 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. Th ese are, however, analytical descriptions, which allow the explanation of evolutionary patterns of change in numerical notation systems. Th ese explanations are analogical, because they describe independent recurrences of cause and eff ect. However, they are also explaining homological processes resulting from cultural contact and the transmission of knowledge among many societies. Th is 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. Diff usion 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 Th omas Kuhn (1962) have eff ectively demol- ished the myth of linear progressivism in science. While in some fi elds, 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 scientifi c 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 nu- mération 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 al- gorithms 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 effi cacious in that it allows everyone to do arithmetic. . . . In short, the invention of our current number-system is the fi nal 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 effi cient ever developed, and are not only the “best” in existence but also “perfect” – the best that could ever be conceived. Th eir 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. Th e 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 aff ects the development of new systems, their acceptance after being transmitted, their modifi cation over time, and their eventual abandonment. Th is pattern of long-term sociocultural change can meaningfully be called evolutionary. Th e primary diffi culty with the assumption of the evolutionary progress of numerical notation is not the notion of evolution itself. Th e problem is that the effi - ciency 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 infl uential 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)

Th is 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. Th e effi ciency of any technology can be evaluated only in terms of the purposes for which it was developed and/or used. Th ere is thus no eternal abstract standard of effi ciency for any technology. It smacks of teleology to argue that Western numerical notation is wonderful because it enabled modern mathematics to develop. Th e 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. Th e 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 fi ngers, or with an abacus. Even in industrial- ized societies, computation remains a secondary function of numerical notation. I am looking at a not-so-crisp Canadian fi ve-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 fi ve-dollar bills received at some event). None of these numeral-phrases was actually ever used to compute.13 Num- bers denote far more often than they reckon, even in our highly numerical society. Th is was doubly true in pre-industrial contexts. In defi ning 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. Th e 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 fi rst used in the context of Chinese divination. We should not expect to fi nd a single specifi c 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 effi ciency 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 effi cient than one that requires many signs. One could then argue that the Roman numerals are not as effi cient 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 defi nable 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 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 fi ve-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. Th e numeral on the bill denotes its value, but is not used in calculation. 14 See Nickerson (1988: 189–197) for a diff erent, 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, effi ciency 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. Th ey analyze how specifi c 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 diff erent 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. Th e situation is analogous to denigrating screwdrivers for being ineffi cient hammers. Th e 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). Th ey 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. Th e 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 defi nite answers to these effi ciency-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 ineffi cient system, so long as it is not entirely unworkable, may be retained, despite its “obvious” inferiority. Th ere 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 specifi c individuals adopt it, as would be the case with a more effi cient 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. Th us, 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 fi rst, leading forward to systems developed more recently. Th e fi rst fi ve 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. Th e 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 proto- cuneiform 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 fi t 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 fi ndings 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 Hit- tite 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. Th e hieroglyphic systems are sum- marized 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, superfi cially similar ones helps explain the diff usion of numerical notation throughout the ancient Mediterranean region.

1 Th e 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 r s t u\‹ v w Cretan hieroglyphic ù\= • 0 ÿ Minoan Linear A Å É\• æ Æ Mycenean Linear B ÅÉæÆô Cypriote syllabic ÅÉ Hittite hieroglyphic qÉ()

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

Egyptian Hieroglyphic Th e 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 diff usion 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). Th e script is mixed in principle, with both phonograms (consisting of one, two, or three consonants) and logograms indicating words nonphonetically (Ritner 1996: 74). Th e 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 diffi cult 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 numeral- signs rather than words, however, with separate signs corresponding to each power. Th ese signs are shown in Table 2.2 (cf. Gardiner 1927: 191; Allen 2000: 97). 36 Numerical Notation

Table 2.2. Egyptian hieroglyphic numerals

1 10 100 1000 10,000 100,000 1,000,000 L-R qrs t u v\w R-L qr„ … † įİ Lex. w̒ mḏ št ḫ ḏb̒ ḥfn ḥḥ

68,257 = qqqq rrr „„\\\…………\\\\††† qqq\ \\rr\\\\\\\ \\\\\\\…………\\\\†††

Th e system is purely decimal and cumulative-additive, with each sign repeated up to nine times as necessary, and ordered from highest to lowest rank. Th e direction in which a numeral is read is always the same as the direction of writing, but varies depending on the inscription in question. Th e 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, fi ve or more identical signs were usually grouped in sets of three or four rather than placed on a single line. Th us, 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 fi ve above a row of four or as three rows of three.2 Th e 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. Th e 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). Th e higher power signs also have specifi c representational qualities and can also represent phonetic values in Egyptian apart from their use as numerical symbols. Th e 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 fi nger; and that for 100,000 ( fn) is a tadpole. Th ese 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 fi ve separated signs (as in the numeral-phrase mentioned earlier), multiple “lotus plants” were depicted as emerging from a single bush (e.g., 3000 = ‰). Th e 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 nonspecifi c lexical sense predominated over the specifi c numerical value. In most other respects, Predynastic hieroglyphic numerals would have been completely intelligible to Late Period scribes. Th e 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). Th is 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 fi rst two of these diff erences simultaneously by noting that on some Old Kingdom linen-lists, horizontal strokes stand for 10. Th e 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). Th e very early date of the tags suggests that the system was developed independently of Mesopotamian infl uence, although the U-j tags are essentially contemporaneous with the Uruk IV tablets. Th e margin of error and discrepancies in the diff erent radiocarbon dates from Tomb U-j are large enough that no defi nite 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 Th e 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 unifi cation 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). Th e mace-head indicates an exaggerated tally of 400,000 oxen, 1,422,000 goats, and 120,000 humans (Quibell 1900: 8–9, Pl. XXVI). Th is 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-identifi er and suggests that the numbers are purely artifi cial, 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). Th ese large numerical values fi gure prominently in early hieroglyphic inscriptions, further supporting the idea that early Egyptian monumental writing was primarily oriented toward display purposes relating to the ideological justifi cation 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 fab- ric 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. Th e 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. Th ese 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. Th ey 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). Th e most common of these signs is that for 5, a fi ve-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. Th e origin of this sign is almost certainly pictorial, from the five 40 Numerical Notation

Table 2.3. Ptolemaic-era cryptographic hieroglyphic numerals

5 Q

7 U

9 Z

60 A

80 B

points of the star. Other common signs were a human head for 7, from the Egyp- tian understanding of the head as having seven orifi ces, 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). Th ese 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. Th e hieroglyphic numerals were rarely if ever used for mathematics and calculation. Th e 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. Th e hieroglyphic system is cumulative-additive, while the hieratic system is ciphered- additive, but since this diff erence in structure was underemphasized by Egyp- tologists 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 defi ciency (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 diff erent in structure and used in entirely dif- ferent 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 Th e 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. Th e 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 fi gure as 100,270, the fi gure 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

Th e question of when this borrowing took place remains open; the fi rst hieratic documents to use this structure date to the Middle Kingdom (2040 to 1652 bc), while the fi rst 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. Th ere 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. Th e 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. Th e fi rst 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. Th is 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). Th e last two of these are not unit-fractions, and are thus exceptions to the general rule. Th is 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. Th e 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). Th is 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. Th e sum of these signs is only 63/64; symbolically, the remaining 1/64 would be supplied magi- cally by the god Th oth 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 2/3 3/4 5 x 23 “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 mis- leadingly become standard among historians of mathematics. Th e 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). Th is system’s binary structure was probably most useful for dividing and multiplying by two, a standard operation needed when manipulating volumes of goods. Th e 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). Th e 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. Th ere 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 diff usion 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). Th is development marks the formation of the Levantine systems (Chapter 3), refl ecting 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. Th e latest dated hieroglyphic inscription is on the temple of Isis at Philae, and dates from August 24, ad 394 (Griffi th 1937: I, 126–127). By the fifth century, knowledge of how to read and write hieroglyphs had disappeared. Th e hieroglyphic numerals, as well as their immediate descendants, were replaced by the Coptic alphabetic numerals (Chapter 5).

Egyptian Hieratic Th e 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. Th e 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. Th e hieratic numeral-signs, like the script itself, changed considerably over the system’s extensive history. Th e 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 fi rst and earliest from the Kahun papyrus (Twelfth Dynasty / 2000–1800 bc), the second from Pap. Louvre 3226 (fi fteenth century bc), and the third from the Harris papyri (twelfth century bc) (Möller 1936, vol 1: 59–63, vol. 2: 55–59). Th ese 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)

123 456789 1s abcdefghi 10s jkl mnopqr 100s stuvwxyz0 1000s 123456 89 10,000s : :: ::: = 100,000s >

4367 = gou4

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

123456789 1s ABCDEFGHI 10s JKLMNOPQR 100s STUVWXYZ[ 1000s \] 657 = GNX

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

123456789 1s abcdefghi 10s jkl mn opqr 100s stuvwxyz{ 1000s |}1234567 10,000s 890:;<=>? 100,000s ,

56,207 = gt4; 46 Numerical Notation comparative purposes. Th e 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 Ptole- maic era. Many of the numeral-signs are very similar to others from the same period; for instance, it is very diffi cult 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 refl ect 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). Th e 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. Th e 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 diff erent 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. Th ese 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. Th is 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 fi rst 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

Kahun papyrus P. Louvre P. Harris Hieroglyphic Old Kingdom (Dyn. 12) 3226 (Dyn. 18) (Dyn. 20) qqq 6 qqq FfFf qqq 9 qqq IiIi qqq 300 „„„ Du Uu ostracon from the New Kingdom workers’ village of Deir el Medina, a scribal exercise in which 600,000, 700,000, 800,000, and fi ve, six, and seven million are expressed multiplicatively using the hieratic signs for 100,000 and one million, respectively. Th e 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. Th e 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 (Griffi th 1898: 16). e Th 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. Th e 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 fi rst appear in the First Dynasty (Peet 1923: 11). Th eir use became widespread from the Old Kingdom onward, with the two systems (hieroglyphic and hieratic) being used for parallel purposes. Th e 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 diff erent 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. Th e hieratic numerals were fi rst used in Egypt for arithmetic and mathematics in the late Middle Kingdom and the early Second Intermediate Period (Twelfth and Th irteenth 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 Mathemat- ical 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 fi eld 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 diff usion of the hieratic numerals is quite diffi cult. 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). Th e 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 Griffi th (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’. Th e 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 diff er only paleographically from ordinary Egyptian hieratic numerals (Kletter 1998: 142). Th e earliest Hebrew inscriptions containing hieratic numerals are the Samaria ostraca from the late ninth or early eighth century bc. Th e 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 Fig- ure 2.2, with the numerals transcribed in Table 2.9. Th e 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). Th e 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. Th e 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). Th e 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 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. 6 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. Th e 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.

Th e Hebrew hieratic numerals were also used on Judaean inscribed limestone weights, a metrological system that refl ected 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

123456789 10s

100s

1000s 52 Numerical Notation ciphered-additive numerals like those on the ostraca (Aharoni 1966). Th e 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. Th is 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). Th e 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. Th is period marks the fi rst 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. Th e Hebrew hieratic numerals are not directly ancestral to the cumulative-additive Levantine systems that emerged in the eighth century bc, among Phoenicians and Aramaeans. Th ese systems were used contemporane- ously with the hieratic numerals and, for instance, occur in the same contexts on ostraca from Arad. Th ere 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 Th e kingdom of Meroë, which fl ourished from roughly 300 bc to 350 ad, made use of two distinct scripts. Th e fi rst, Meroitic hieroglyphs, were based on Egyptian hieroglyphs and were used on some stone monuments. Th e attested Meroitic hieroglyphic inscriptions contain no numerals. Th e 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. Griffi th, 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. Griffi th (1916: 22) fi rst 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. Th is system is ciphered-additive and decimal, and written from right to left (like the Meroitic cursive script). Th e number 2348 as shown in Table 2.10 appears on the stela of Akinidad, which dates to the late fi rst century bc (Griffi th 1916: 22). Hieroglyphic Systems 53

Table 2.10. Meroitic numerals

123456789 1s abcdefghi 10s jklmn p 100s stuvwx z 1000s ABC 2348 = hmuB

As with the hieratic numerals, the signs for the units, low hundreds, and possibly 1000 through 3000 are somewhat cumulative. Th ere is only one case (again, from the Akinidad stela) where a number greater than 10,000 is expressed; interest- ingly, 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 (Griffi th 1916, 1925). Griffi th (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), Griffi th (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. Th e Meroitic numerals were used for administrative purposes such as tax records and mensuration, as well as in funerary and monumental contexts indicating year- dates and quantities of individuals. Griffi th 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). Th ere is no evidence that the Meroitic numerals were ever used for arithmetic or mathematics. Even on ostraca upon which multiple numerals have been written, Griffi th 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). Th e 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. Th e 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 Th e demotic script developed in the late eighth century bc (Twenty-fi fth Dy- nasty) 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. Griffi th (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. Th e demotic numerals are a base-10, ciphered-additive system, written from right to left. Th ey 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. Th e 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

123456789 1s ABCDEFGHI 10s JKL MN Op QR 100s STUVWXY Z! 1000s @#$% & *(

6268 = HOT& systems were used side by side in Egypt for nearly a millennium. Th is long coexis- tence 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 diff erent parts of Egypt. Th e 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, confi rming 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 (Griffi th 1909). An extensive set of demotic numerals is found in P. Tsenhor, the private archive of a sixth-century woman (Pestman 1994). Th e 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. Th e demotic numerals are almost certainly ancestral to the Greek alphabetic numerals (Chapter 5). Th ese numerals, which are structurally identical to the demotic numerals, fi rst 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). Sec- ondly, 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 suff ered a fatal decline. Th e last text with demotic numerals is a graffi to on the temple of Isis at Philae (the same temple that contains the last evidence for hieroglyphs), which dates to December 2, ad 452 (Griffi th 1937: I, 102–103). The last known demotic text of any sort (dated in Greek) was written nine days later.

Linear A (Minoan) Th e 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 eff ort 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). Th e 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 fi ve signs each rather than placing them in an uninterrupted row. Th e variant dot symbol for 10 is found only in Hieroglyphic Systems 57

Table 2.12. Linear A numerals

1 10 100 1000 ÅÉ\• æ Æ 7659 = ÆÆÆÆ æææ ÉÉÉ qqqqq ÆÆÆ æææ ÉÉ 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 (Ben- nett 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 unit- fractions, no real resemblances exist between the two fractional systems. Th e 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 simplifi ed. 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). Th e 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. Th e 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. Th e 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 graffi to 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 signifi cant 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. Th e 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. Th e Linear B Mycenean script used on Crete and the Greek mainland defi nitely derived from Linear A. Its numerals (to be discussed later) are nearly identical to those of Linear A. Th e 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 fi fteenth century bc. Presumably, during this period, the Greek-speaking Myceneans adapted Linear A for their own language, resulting in Linear B. Th e two scripts coexisted in Crete from about 1550 to 1450 bc, after which time Linear B replaced Linear A completely.

Cretan Hieroglyphic Th e Cretan Hieroglyphic or Pictographic script was fi rst identifi ed 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). Th e 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 fi ve 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 diff erent fractional system with signs for 1/2, 1/4, and 1/8. Since the Cretan hieroglyphs are largely undeciphered, it is diffi cult 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. Th ere 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 diff erences 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. Th e Cretan hieroglyphic numerals are probably a local variation of the Linear A numerals or, less plausibly, a direct borrowing from the Egyptian hieroglyphic numerals. Th e 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. Th e 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) Th e 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 (classifi ers), including a numerical notation system. Th e Linear B numerals are shown in Table 2.14. Th e 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. Th e 10,000 sign is probably a multiplicative combination of the signs for 10 and 1000. Th e structure of the system is cumulative-additive and decimal, with the highest powers on the left, written in descending order and with fi ve or more identical signs divided into two rows. Unlike the Linear A numerals, Linear B lacks a separate system for expressing fractions; instead, specifi c 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 specifi c ratios of two types of units, which further shows the indebtedness of Linear B to its Minoan forerunner. Th e Linear B system defi nitely originated through direct contact with the Minoan civilization and the Linear A numerals. Th e earliest Linear B inscriptions date from the sixteenth century bc, so the two scripts coexisted on Crete for about a century. Th eir numerical notation systems are so similar that some authors do not distinguish between the two (Ventris and Chadwick 1973: 53; Struik 1982). Th e distinction between the two is not nearly as great as between the two scripts, which record diff erent languages. Th roughout 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 fi nancial 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 62 Numerical Notation

Table 2.14. Linear B numerals

1 10 100 1000 10,000 ÅÉæÆô

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 suff ers 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 fi ve 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 diff erent sizes. Rather than proving that the Myceneans used numerical notation for arithmetic, however, these examples indicate that tallying by units and in groups of fi ve, 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. Th ere 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 fi rst 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. Th e 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. Th e contemporaneity of the two systems makes this scenario plausible, if not proven. Th e 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 Py- los, 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 Th e Hittites lived in central Asia Minor from about the end of the third millennium bc. Th e Hittite and closely related Luwian languages are the rstfi Indo-Euro- pean 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). Th is 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). Th is script is known as Hiero- glyphic 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). Th e 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 fi ve unit-signs. Laroche (1960: 395) indicates that 9 was variously written using three rows of three strokes, a row of fi ve above a row of four, or simply with nine strokes in sequence on a single line. Th e 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 fi rst Hittite hieroglyphic inscriptions are found, but 64 Numerical Notation

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. Th e Linear A, Linear B, and Hit- tite 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. Th ere was a signifi cant degree of intercultural contact between the Aegean and Asia Minor during this period. Th e Myceneans had settlements in western Anatolia and traded throughout the eastern Mediterranean, and were possibly the “Ah- hijawa” (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. Th e 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). Th e Hittite numerical notation is used far more frequently than lexical numerals, which is also true of the Egyptian hieroglyphs and Aegean scripts. Th ere 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.” Th e numerals that accompany the Cypriote syllabary, which was invented around 800 bc, are also potentially derived from the Hittite hieroglyphic numerals. Th e 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. Th e 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. Th is rudimentary system was decimal and cumulative-additive and, like the syllabary itself, was written from right to left. Th e 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. Th is system parallels the Aegean Linear systems from which the Cypriote numerals are probably derived. Th is 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 66 Numerical Notation

Table 2.16. Cypriote numerals

110 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 diffi cult 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 fi nal 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 Cypri- ote numerals are unconnected except by their temporal and geographic proximity on the island of Cyprus. Th e 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. Th e connections between the Egyptian hieroglyphic, hieratic, and demotic systems are well established, but more data are needed to establish the specifi c links between the Egyptian and Aegean systems. Nevertheless, on the basis of a shared set of features that distinguish it from other, superfi cially 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. Th ird, large numbers of cumulative signs in a numeral-phrase are grouped in sets of three to fi ve. 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

Th e fi rst millennium bc was an era of considerable interregional commerce, warfare, and colonization in the Levant. Th is region was peripheral to both Egypt and Mesopotamia and thus exposed to multiple cultural infl uences. Th e various Levantine numerical notation systems that developed in the fi rst millennium bc share several common features that refl ect 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. Th e 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, Sog- dian, Manichaean, and Pahlavi systems. Th e 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 Levan- tine 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 specifi c societies. Issues of diff usion and

68 Table 3.1. Levantine numerical notation systems

1 2 4 5 10 20 100 500 1000 10,000 Aramaic a H A C F\ ² G ± Phoenician aAB@DäÇâ¶Eμ Palmyrene aHAJA Nabataean aRNKJ I

69 Kharoṣṭhī agàJ Lå– ç— Hatran a>AêJü Old Syriac aë >A@ è ï Middle Persian 1245 7 Sogdian G HI J 7 Manichaean A BA CD E F Pahlavi abd f g s u 70 Numerical Notation

Table 3.2. Aramaic numerals

1 10 20 100 1000 aACFG

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 Th e Aramaeans, who originally inhabited a large portion of modern-day Syria, are fi rst 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. Th e 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. Th e 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. Th e 10-sign appears to have originally been a simple horizontal stroke, with a tail added cursively. Th e 20-sign is almost certainly a ligatured combination of two 10-signs, as shown by the occasional use of a variant form B. Th ere 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. Th e same principle was followed for the thousands. Th ere were apparently two signs for 1000; the fi rst, G, is actually no more than the fi nal 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. Th e fi rst 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. Th e earliest uncon- testable examples are from the Assyrian bronze lion-weights found at Nimrud by Layard in the nineteenth century. Th ese 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 fi fteen minas in three diff erent ways on its three lines of text: in Aramaic lexical numerals, as fi fteen ungrouped single strokes, and according to the structure detailed above (aa\aaa\A) (Fales 1995: 35). Th is threefold repetition using diff erent 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). Th e conquest of the Aramae- ans in 732 bc by the Assyrian empire establishes a clear historical context in which this transmission could have taken place. Th e 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. Th ere 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 diff erent from the cuneiform Assyro-Babylonian 72 Numerical Notation

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

424 = \aaaa\C\F\aaaa Aramaic 4 20 100 4 Å

424 = Hieroglyphic qqqqrr„„„„ 4 20 400 Å

Assyro-Babylonian 424 = 4i\b4 Æ 4 100 20 4

system. Th e 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. Th ese diff erences 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. Th e 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 cumulative- additive 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. Th e Aramaic numerals were likely developed under a dual cultural infl uence from Egypt and Mesopotamia. Th e 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 fi rst 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 fi rst 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 fi nd the failure of the Aramaeans to adopt the hieratic numerals to be rather curious. Th e existence of a distinct sign for 20 in Aramaic, and the recombination of features of two quite diff erent 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). Th is may explain why the graphic “etymology” of the Aramaic numeral-sign for 20 is two ligatured 10-signs. Th is 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 afi eld throughout the second half of the fi rst millennium bc. Segal (1983) gives ample evidence for the use of the system among Aramaic texts from Saqqara in Lower Egypt throughout the fi fth and fourth centuries bc, and Aramaic papyri found at the fi fth-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. Th e 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 offi cial royal proc- lamations 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. Th roughout 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. Th e end of the Achaemenid Empire did not spell the end of Aramaic infl u- ence over the Middle East; however, it did result in the fragmentation of what previously had been a unifi ed 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. Th e 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. Th e 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 Th e Phoenicians, who inhabited various cities (Tyre and Sidon foremostly) along the Levantine coast in the fi rst millennium bc, were perhaps the greatest mercantile people of the ancient Mediterranean. Th e 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 Phoeni- cian 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. Th e Phoenician numerical notation system is similar in structure to the Aramaic, with distinct signs for 1, 10, 20, 100, and 1000. Th ese 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 10 20 100 1000 aAB@DäÇ â¶Eμ 697 = 0\aaa\aaa\ADDDD\â\aaa\aaa one exception (CIS.87,ph) in which left-to-right ordering is used, probably in error. Th e 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 diff ering individual scribal styles, while others may refl ect regional or diachronic variation. I list only the more common forms for the sake of brevity. Th e 1000-sign is extremely rare; Lidzbarski (1898: 201) reports only a single instance from Tyre. Th ere 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 numeral- signs 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). Th e myth of Phoenician alphabetic numeration has been repeated for more than a century, but there is no foundation for this assertion. Th e fi rst alphabetic numerals were developed by the Greeks in the late sixth century bc (Chapter 5). Zabilka (1968: 118) claims that the fi rst 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 diff erent from the practice of modern writers who label points of discussion A, B, C, and so on. Th e fi rst example of numerical notation in a Phoenician inscription is the Kara- tepe 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 fi rst centuries bc, Phoenicia was politically dominated in turn by the Assyrian, Neo-Babylonian, Achaemenid, and Alexandrine Greek empires. Th e 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 signifi cant regional variation, until Roman and Greek conquests in the second century bc eff ectively 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). Th e 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. Th e lack of a symbol for 5 in Phoenician numerals suggests that most of these later systems were Aramaic-based. Phoeni- cian 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 aff airs despite Greek and Roman infl uence in the area. Palmyrene inscriptions are found dating from the fi rst 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

151020 aHAJ 178 = 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 fi rst analyzed over a quarter- millennium ago by Swinton (1753–54). Th e Palmyrene system had distinct signs for 1, 5, 10, and 20, as shown in Table 3.5. Th ese 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. Th e 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. Th e 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 fi nal 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. Th us, political factors, rather than criteria of function and effi ciency, 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 Th e Nabataeans were a South Semitic people of Arabian ancestry who inhabited the area between Syria and Arabia in the southeastern Levant in the late fi rst millennium bc and into the Christian era. Th ough not Aramaeans, they came under considerable Aramaean infl uence and adapted the Aramaic script for their South Levantine Systems 79

Table 3.6. Nabataean numerals

1 4 5 10 20 100 AR N KJ I 178 = aaa\\NA\\J\J\J\\I\a

Semitic language, including a variant of its numerical notation. Th is 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. Th e 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. Th e 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 (Litt- mann and Meredith 1953: 16). Th e 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. Th e 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

Th e 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. Th roughout its history, it was used in inscriptions on edifi ces, 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. Th e 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). Th e Greek alphabetic numerals (Chapter 5) and Roman numerals (Chapter 4) were also well known and frequently used in otherwise Nabataean inscriptions. Th ough 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. Th e 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, cumulative- additive for numbers less than 100, multiplicative-additive above 100, and written from right to left. Th e 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. Th e 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 20 100 a>AêJü

697 = 11>AJJJJü1>

Th e Hatran numerals were probably ancestral in some way to the Middle Persian numerals, at least giving rise to some of the Middle Persian numeral- signs. 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 Th e 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. Th e 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. Th e sign for 2, as defi ned 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. Th e sign for 5 is identical to that of the Hatran system, and the sign for 20 to one variant form used in Phoenician. Th e sign for 5 was not consistently used in numeral

1 Th e earliest dated Syriac inscription is from ad 6, although Drijvers and Healey (1999: 17) argue that it may have originated earlier. 82 Numerical Notation

Table 3.8. Old Syriac numerals

1251020100500 Aë >A@ ‚ è’ï

697 = ë>A@@@@èï 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 diff erent ways, as indicated in Table 3.9 (Drijvers and Healey 1999). Th e earliest dated Old Syriac inscription (As55, dating to ad 6) expresses the year-date (317) using seven unit- strokes (Drijvers and Healey 1999: 141). Th is 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 modifi ed form of the 10- sign, resembling the Palmyrene numerical notation in this respect. Th e 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 power- sign for 100, rather than with a sign for 500. Th e 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 specifi c relationship to the other Levantine numerical systems remains unclear. Th ree 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 Transliteration (1 + 1 + 1 + 1) × 100 + 20 + 20 + 20 + As29 a>A\‚‚‚\’\aaaa 10 + 5 + 1 (1 + 1 + 1 + 1) × 100 + 20 + 20 + 20 + As36 aaaaaaA\‚‚‚\’\aaaa 10 + 1 + 1 + 1 + 1 + 1 + 1 (1 + 1 + 1 + 1) × 100 + 20 + 20 + 20 + As37 ëaëaA\‚‚‚\’\aaaa 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 fi fth 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. Th is gradual obsolescence corresponds to the development of the indigenous Syriac Chris- tian 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. Th e 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

Th e 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. Th is 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 cumulative- additive 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 84 Numerical Notation

Table 3.10. Kharoṣṭhī numerals

1 4 10 20 100 1000 agàJLå–ç—

697 = aaa\R\à\JJJJ\å\aaR usually ligatured together in groups of two and three. Th e sign for 1000 is found only in the late (perhaps fi fth 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). Th e 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). Th e 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 defi nitely part of the Levan- tine 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. Th e 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. Th ese 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-fi ve 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. Th e 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). Th roughout its history, it was in competition with its rival, Brāhmī numerals (Chapter 6), the system used on the Mauryan inscriptions of the Indian heartland. Th e 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. Th e 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 Th e Persians of the Achaemenid, Seleucid, and Parthian Empires used the Old Per- sian numerals (Chapter 7), the Greek alphabetic numerals (Chapter 5), or the common Aramaic numerals already described. Th e Sassanian dynasty began in ad 228 when Ardashir I destroyed the Parthian Empire, which had ruled much of the ter- ritory 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. Th e Middle Persian language (the ancestor of modern Farsi), the language of Sassanian administration and commerce, was written in an Aramaic-derived consonantary, refl ecting the legacy of Achaemenid, Seleucid, and Parthian rule in the region. Th e 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 cumulative- additive and decimal for numbers below 100, and written from right to left with the highest powers at the right. Th e 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. Th e 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. Th us, 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 affi liation of the Middle Persian system to the other Levantine systems, other than to note that it is most defi nitely descended from the Aramaic system in some way. Th e 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. Th e Middle Persian script is most closely affi liated 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 12345 67 6798 = 1111\1111\24444\5\111\1111\7\111\111 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. Th e 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. Th e Book Pahlavi numerals are suffi ciently diff erent from their Middle Persian ancestor to warrant separate treatment.

Sogdian Th e 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 Mid- dle 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). Th is religious and scriptal pluralism greatly complicates the history of Cen- tral Asian Iranian scripts and numerals. Th e Sogdian script is fi rst attested from the “Ancient Letters” dating to ad 312–313 found by Stein in Chinese Turkestan, but may have originated in the third century. Th e 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). Th ere has been no systematic comparative treatment of Sogdian numerals to date, and minimal paleographic work. Th e Sogdian numeral-signs are shown in Table 3.12 (cf. Sundermann and Zieme 1981).

Table 3.12. Sogdian numerals

1 10 20 100 1000 GHIJ7 697= G\GGG\GGGHIIII\JGGG\GGG 88 Numerical Notation

Th e 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. Th e 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. Th ere 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). Th e 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. Th e 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 ‘fi ve’. Numeral-signs could be combined with lexical numerals, as in the phrases 100 ‘št ‘108’ and δwy 100 20 ‘220’ in the Padmacintāmaņi-dhāraņī- sūtra dating to around ad 700 (Mackenzie 1976: 12–17). Multiplicative power- ideograms 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 Th e development of a specifi cally 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. Th e 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 fi rst. Th e 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 fl ourishes 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). Th e 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 fl ourished 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 dignifi ed and prestigious but also arcane quality (Sundermann 1997). Th e 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. Th e Zoroastrian Persians, however, continued to use 90 Numerical Notation

Table 3.13. Manichaean numerals

1251020100 ABACDEF 697 = BACDEEEEFAC 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). Th e 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. Th e 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 Man- ichaean Mahrnâmag of ad 761–62, with the numeral 546 spanning the fi rst 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

123 4 5 6 7 8 9 1s a b c d bc cc cd dd ccc 10s fjlm n o p q r 100s s 1000s u 4697 = cdrsccud 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. Th e 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. Th e Pahlavi system is thus, for all intents, ciphered-additive rather than cumulative-additive below 100. Th is 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. Th is 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. Th ese two instances, in fact, are the only two known cases where a cumulative system directly gave rise to a ciphered one. Th e 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 Th e 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 fi rst 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 signifi cantly, Achaemenid and Seleu- cid 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. Th ey persisted the longest in Central Asia, where Islam was somewhat slower to take hold. Th e 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 10- signs); 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. Th e 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. Th e 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. Th ere 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 signifi cant role in the script traditions of these societies than would otherwise have been the case. Moreover, in comparison to the incredibly wide diff usion of Aramaic-derived scripts throughout Europe, the Middle East, and South Asia, Levantine numerical notations spread only sporadically, and their imprint was impermanent. Th e 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

Th e Roman numerals are undoubtedly one of the better-known numerical notation systems, and have received a tremendous amount of scholarly attention. Neverthe- less, 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. Th e name “Italic” refers only to this geographical origin, and thus does not refl ect any shared linguistic or cultural affi liation. Italic systems fl ourished 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. Th e most common variants of the Italic numeral-signs are shown in Table 4.1.

Etruscan Th e 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). Th e 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 signifi cantly infl uenced Roman culture

93 94 Numerical Notation

Table 4.1. Italic numerical notation systems

System 1 5 10 50 100 500 1000 5000 10K 50K 100K Etruscan 1QR÷;\¬\ì¡Ä Greek – archaic 1ÂÈÁÀ Greek acrophonic 1bcdefghij Greek –Argos/Nemea •ód\Üg Greek – Epidaurus •2Üg Greek – Olynthus 1gá¥ Lycian 1<óÐ1Ò Roman 1PRS\UWY\,.½~ Roman multiplicative 1PRSUWabdeg Arabico-Hispanic ØÙ Ú ÛÝ Calendar numerals AE J South Arabian 1!@#$ % throughout the Republic and even later. Th e 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). Th e 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 /\ ¡ Ä ?\ 1378 = aaaQRR÷;;;ì Italic Systems 95

Th is 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. Th e numeral-signs for 1, 5, 10, and 50 are quite regular throughout the system’s history. Th e 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 fi rst in Latin inscriptions and found its way into Etruscan only when the Etruscan system was already declining. Th e signs for 1000 and 10,000 are encountered only rarely (Keyser 1988, Bonfante 1990). Th e 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). Th is theory is given some support in that the earliest Roman sign for 500 is X (later to become W). Further- more, two Etruscan inscriptions contain an unidentifi ed 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. Th eir invention may have been entirely independent of other base-10, cumulative-additive systems used around the Mediterranean in the early fi rst 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. Th e 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). Th is 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 infl uenced 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. Th e 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 infl uence 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. Th e 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 numeral- signs of the mature systems are quite diff erent, they are structurally identical. eTh 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. Th e 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 Etrus- can 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 fi fth 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 graffi ti inscribed on potsherds contain Etruscan numerals. ese Th graffi ti 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. Th e Etruscan “abacus-gem” (CII 2578 ter) showing a fi gure 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 fi fth 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). Th is 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. Th ere 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 fi ngers, or on a counting board. Large numbers and long numeral-phrases are very rarely encountered; even the sign for 100 is relatively uncommon. Th e demise of the Etruscan numerical notation system was a direct consequence of the rising fortunes of the Roman republic. Th e 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 98 Numerical Notation

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. Th is 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 diffi culty in making the change to the new system. Th e last certainly dated examples of Etruscan numerical notation are from the second century bc. However, A. P. Ninni (1888–89) fi rst presented the theory that the Etruscan numerals survived into the nineteenth century. While studying the tally marks used by fi shers 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). Th is potential vestige is cumulative-additive, with a mixed base of 5 and 10, like both the Roman and Etruscan systems. Its numeral- signs 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 fi shers? At present, there is simply not enough surviving data to speculate on the possibility of such long- term 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 fi rst 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 1b c de f g h i j ΠΕΝΤΕ ΔΕΚΑ ΗΕΚΑΤΟΝ ΧΙΛΙΟ∑ ΜΡΙΟ ∑

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 ciphered- additive 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 Th e acrophonic system as used in classical Athens is shown in Table 4.4 (Tod 1911–12: 100–101). Th e 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 numeral- phrases in descending order of numeral-sign value. Th e acrophonic system is so named because the signs for many numbers are taken from the fi rst 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). Th e 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 defi nition, and does not substantially aff ect how we classify the entire system. Similar acrophonic signs were used in large portions of the Hellenic world, the only diff erence 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. 100 Numerical Notation

Table 4.5. Non-acrophonic archaic Greek numerals

151050100 aÂÈÁÀ throughout its geographic and temporal range. Th is degree of variation among local systems is far greater than the variety of lexical numerals used in the Greek dialects. However, the diff erences in sign-forms were probably not great enough to aff ect their comprehensibility (Tod 1936–37: 246). For expressing monetary values, the acrophonic numerals were often modifi ed to refl ect 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) (Th reatte 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 fi t easily into the standard system. For instance, a system found in inscriptions from Th espiae (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 graffi ti from the Athenian Agora from the last quarter of the fi fth 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 fi fth centuries bc. Th e 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). Th is 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 fi ve 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 20 4 2 1 1/5 1/10 1/50 ¿ƒZ íb2R¿ƒZ í2 c

(in modern Libya). Th ese 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). Th ey present a dual series of fi gures in which each numeral- sign has both a higher and lower value; the specifi c 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. Th e relative values of diff erent 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. Th e 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). Th is 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). Th ere, a system was used that is nonacrophonic and lacks a sub-base of 5 (Tod 1936–37: 248–249; Graham 1969). Th e 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 Chalcid- ian colony at Cumae (in southern Italy). Th is theory, while attractive, has several fl aws, many of which derive from Mommsen’s (1965 [1909]) fl awed “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 modifi ed their signs into alphabetic ones for mnemonic purposes.

2 Th e 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. 3 No signifi cance 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. 102 Numerical Notation

Table 4.7. Epichoric Greek numerals

System 1 5 10 50 100 500 1000 Standard Acrophonic 1bcdefg Olynthus 1 None g None á None ¥ Epidaurus • None 2 None Ü None g Argos and Nemea • None ó\\« d\\b Ü None g

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). Th e 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 diffi cult. 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 fi fth 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 diff erent 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 fi fth and fourth centuries bc, suggesting an initial period of experimentation followed by consolidation and agreement on a single form of the numerals. Th e 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 diffi cult 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. Th ere may also have been some infl uence 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). Th e 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. Th e Lycians of southern Asia Minor used a nonacrophonic numerical notation system in the late fi fth and fourth centuries bc that is probably an epichoric variant of the acrophonic system, although their language was not Greek (see the following discussion). Th e 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 fi fth century bc, derive from the acrophonic system. Th e South Arabian numerals are cumulative- additive, base-10 with a sub-base of 5, and use acrophonic numeral-signs. How- ever, 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, graffi ti or other marks on pottery often indicate quantities for commercial purposes. Th e 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. Th e Greeks never expressed dates in acrophonic numerals. eTh 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. Th e custom of dating using regnal years did not arise until the Alexan- drine period. Documents in connected prose (decrees, for instance) rarely contain acrophonic numerals, except to indicate the price of executing the inscription (Th reatte 1980: 112). Th ere 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 fi fth century bc (Menninger 1969: 299–303). Th e 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. Th e 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 aff airs, 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 fl ourish. Th ere are only a handful of known fi rst-century bc Italic Systems 105 examples from Athens (Th reatte 1980: 113). By this time, Greece was rmlyfi 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 Latin- speaking 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). Th is 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 fi rst millennium bc, centered around the city of Xanthus. Th e 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 infl uence, and was intimately involved in interregional commerce and confl ict. Th e 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 fi fth and fourth centuries bc. Th e Lycian numerical notation system is still very poorly understood. Th e 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. Th e signs of this system are shown in Table 4.8 (cf. Shafer 1950, Bryce 1976). Th ere is also a sign, 2, that probably represents ½, although Shafer (1950: 260) argues that it may represent an additional one-half of any numeral-sign that precedes it; ó2 would be 15 and <2 would be 7½ according to this theory. Th e numeral-signs for 50 and 100 are found only on a few inscriptions. Th e value 50 is assigned to Ð primarily by default; its value is certainly between 10 and 100 (it is found after 1Ò but before ó). I follow Frei (1976: 15) in assigning it the value of 50. We can be fairly certain about the value of the sign for 100, because it is found in the Lycian portion of a trilingual Greek-Lycian-Aramaic inscription found at Letoon and dating from 358 bc (Frei 1976: 13–15). Shafer (1950: 258–259) suggests, 106 Numerical Notation

Table 4.8. Lycian numerals

151050100 1<óÐ1Ò 127 = 1Òóó<11 based on one inscription, that the Lycians may have used the subtractive principle to express the number 4 as a<. However, in other inscriptions, 4 is expressed as aaaa. Th e Lycian numerals are very likely a previously unidentifi ed variant of the Greek cumulative-additive systems. Lycian numerals arise in the early fi fth century bc at the time of the peak use of the Greek acrophonic numerals in Athens and throughout the Hellenic world. Both systems were cumulative-additive, had a base of 10 with a sub-base of 5, and were used in the fi fth century bc in the Aegean region. While the Lycian numerals are not acrophonic, this is true of many of the epichoric numerical notation systems of the classical Greek world. Alter- natively, the Lycian system may have been based on an Aramaic model, with the sign for 100 (1Ò) being in fact multiplicative (1 × 100) rather than constituting a single numeral-sign (Frei 1976, 1977). No numbers higher than 120 are expressed in any Lycian inscriptions, so we do not know whether, for instance, the Lycian numeral-phrase for 200 was additive (1Ò\ 1Ò) or multiplicative (11Ò). Because none of the Semitic systems had separate signs for 50, but all of them had signs for 20, we would need to modify the value of the Lycian Ð to 20, which is consistent with the numeral-phrases known from inscriptions. Th ere is little similarity, however, between the numeral-signs of Lycian and either Phoenician or Aramaic, and the Lycian system uses a sign for 5, which is very rare in Aramaic. As well, Lycian, like the acrophonic numerals, is written from left to right, whereas the Levantine systems are all written from right to left. Th e Lycian numerical notation system apparently did not diff use outside Lycia. Although Shafer (1950) argues that the similarities between the Roman and Lycian numerals are suffi cient to indicate the derivation of the former from the latter, this likeness is no greater than that between Lycian and the Greek acrophonic system. Furthermore, while there is some similarity between the Lycian numeral-signs and other systems of the Italic phylogeny (especially South Arabian), these similarities do not correspond to any plausible circumstances of cultural contact. In Asia Minor, scripts such as Phrygian and Lydian, both of which are closely related to Lycian and were used in the fi fth and fourth centuries bc, used numerical notation based on the Phoenician-Aramaic model rather than on the Greek or Lycian. Lycian numerals are found primarily in a single context – sepulchral epitaphs indicating monetary amounts, normally including a numeral-phrase preceded by the word ada, now considered to be a monetary unit (Bryce 1976: 175). Th e Italic Systems 107 monetary values may have stipulated a penalty to be paid should the tomb in question be violated (Shafer 1950), or they may indicate fees paid in advance by the family for a tomb site (Bryce 1976). Th e only nonfunereal context in which Lycian numerals are used is the trilingual inscription found at Letoon, a public legal regulation (Frei 1976). As Lycian numerals are not found on coins or on fi nancial inscriptions, they are quite distinct from the numerals of the rest of Asia Minor and the Aegean. As the Lycians became increasingly caught up in imperial confl icts between the Persians and the Greeks, their script was used increasingly infrequently. By 300 bc, the Lycian script had assimilated to the Greek, and its numerical notation ceased to be used, replaced by the Greek alphabetic numerals.

South Arabian Th e Old South Arabian scripts are of a very ancient origin, fi rst appearing around the turn of the fi rst millennium bc in the southern part of the Arabian peninsula (modern Yemen). Th ey are consonantal and are characterized by large, well-formed letters and by their extremely varied direction of writing (left-to-right, right-to-left, or boustrophedon, depending on the inscription). Th ey were used to write South Semitic languages such as Minaean, Sabaean, Qatabanian, and Hadramauti. Dur- ing their early history, these scripts did not possess any numerical notation system, but following the rise of the kingdoms of Minaea and Saba in the fi fth century bc, numerical notation began to be used in monumental inscriptions. Th e numeral- signs used are shown in Table 4.9, including both left-to-right and right-to-left sign forms, where appropriate (Halévy 1873: 511–512; Hommel 1893: 8). Th e system is cumulative-additive, with a base of 10 and a sub-base of 5, and is written in whichever script direction is used in the inscription as a whole. Th e sign for 1 is, as in all Italic systems, purely iconic. Th e signs for 5, 10, 100, and 1000, however, are acrophonic; each is simply the fi rst letter of the appropriate South Arabian lexical numeral (Beeston 1984: 8). Th e sign for 50 is non-acrophonic, but is simply a halved version of the sign for 100. In one inscription (Biella 1982: 531), the sign X is apparently used with the numerical value 4, possibly in imitation of the Nabataean system (Chapter 3). Th ere are no signs for 500 or 5000 known from any South Arabian inscriptions; these numbers were written with fi ve signs for 100 and 1000, respectively (Biella 1982: 1, 265). Normally, numeral-phrases were placed between large hatched bars to avoid confusing numeral-phrases with words, given the use of the acrophonic principle (Halévy 1875: 78). Large sets of unit-signs were not divided into smaller groups, which creates a legibility issue because the sub-base of 5 is not used throughout the system; one inscription from Sirwah denotes 12,000 using twelve signs for 1000 (Ifrah 1998: 187). 108 Numerical Notation

Table 4.9. South Arabian numerals

1 5 10 50 100 1000 Left to right 1!@#$% Right to left 1Ê@ËÌÍ 3697 = Î\%%%$$$$$$#@@@@!11\Î (L-R) Î\11Ê@@@@ËÌÌÌÌÌÌÍÍÍ\Î (R-L)

In some inscriptions, the South Arabian numerals used an unusual technique of implied multiplication that resembles positional notation. Most often, this was done when expressing values greater than 10,000, by placing signs to the left of a sign for 1000 (when reading from left to right), which were implicitly taken to represent multiples of 1000. For instance, one inscription (R3943/2) has Î\@@@%\Î for 31,000, in which the three @ signs have the value of 10,000 instead of 10, while % retains its ordinary value of 1000 (Biella 1982: 349; Ifrah 1998: 187). Apparently this technique was also sometimes used for multiples of 100; Halévy (1875: 79) notes an inscription that has Î\111\Î instead of Î\$$$\Î for 300. We know that the multiplied value is correct because of contextual information and because South Arabian inscriptions commonly list the appropriate lexical numeral beside the numeral-phrase. Ifrah sees in this use of implied multiplication “what might be called the germ of our place-value notation” (Ifrah 1985: 232). However, without contextual information, such numeral-phrases would simply be confusing and ambiguous, as there is no sign for zero. Th ese formations are very rarely attested throughout the system’s history. Th e South Arabian system is derived from the Greek acrophonic system (Ifrah 1998: 186; Fevrier 1948: 579). Both systems are decimal and cumulative-additive, and both have a sub-base of 5. Additionally, both systems use the acrophonic principle, a feature that is otherwise uncommon during this period. Th e numeral-signs are not similar to those of any other system, but this is unsurprising, since the system is acrophonic. In the fi fth century bc, when the South Arabian numerals were fi rst used, the Minaeans and Sabaeans were actively engaged in trade with the Greeks at a time when the acrophonic numerals were the only ones the Greeks were using for monetary and metrological purposes. While one would expect the South Arabian scripts to have numerical notation systems similar to those used in North Semitic scripts at the time (Aramaic or Phoenician), this is not borne out by comparing the systems. Th ere is no sign for 20 in the South Arabian system, while the Aramaic and Phoenician systems did not normally use signs for 5 and 50. Th ere is no evidence that the South Arabian numerals ever diff used outside the Arabian peninsula. Th e Ge’ez script used for the Ethiopic languages, which is Italic Systems 109 derived from a South Arabian model, used numerals based on the Greek alphabetic system (Chapter 5). Although some South Arabian cursive inscriptions on wood have been found, these contain no numerals. Th e system just described is documented only in monumental contexts. Th e functions of the numerals included recording details of sacrifi ces or off erings to gods, quantities of booty obtained, numbers of military troops, and information on construction projects such as monuments and irrigation systems. Th e South Arabians did not use an enumerated dating system, nor do South Arabian coins contain numerical notation of any kind. By the second century bc, instances of the South Arabian numerals were normally preceded by the appropriate lexical numeral written out in full. While this aids modern scholars in their interpretation, doing so also removed any incentive to continue to use the system. By the fi rst century bc, although the South Arabian scripts continued to be used, the numerical notation system had become extinct, and it was not replaced until the seventh century ad, when the Islamic conquest brought alphabetic and later positional numerals to southern Arabia.

Roman Despite the importance and continued use of Roman numerals, the early history of the system was very poorly understood until Keyser’s (1988) study. Th ere are several diff erent classical variants of the Roman numerals, while the sign forms and structure of the Roman numerals used today are medieval in origin. Th e Roman alphabet was developed on an Etruscan model around 600 bc at a time when much of Italy was under Etruscan political domination; it was written from left to right, as it is today. Like its Etruscan precursor, the Roman numerical notation system has a base of 10, with a sub-base of 5, and is essentially cumulative- additive; unlike it, the Roman numerals are written from left to right and are sometimes used subtractively. Th e great variety of numeral-signs used throughout two millennia of its history contrasts strongly with the highly static quality of the equally long-lived Babylonian cuneiform and Egyptian hieroglyphic numerals. Table 4.10 presents the numeral-signs used during the republican period (Ifrah 1985: 132; Keyser 1988). Th ese numerals were cumulative-additive in structure in most inscriptions. Only the signs for 1 and 10 remain unchanged throughout the entire history of the system. Th e “inverted V” sign for 5, Q, is found only in early contexts, and is evidence of the system’s indebtedness to its Etruscan ancestor, but P is used exclusively thereafter. Th e signs for 50 in Table 4.10 are roughly in chronological order; the “inverted arrow” forms are earliest, with the “inverted T” forms prevalent until about ad 25, and L most common thereafter (Gordon and Gordon 1957: 181). 110 Numerical Notation

Table 4.10. Roman numerals (republican period)

1 5 10 50 100 500 1000 5000 10,000 50,000 100,000 1Q Rõ\V X ¤ , - ½~\ P\ § U W Y\\ ® .\ ° T _ ¯ \¸ \© S ¦ | 19,494 = ¯®YYYYUUUUTRRRRaaaa 19,494 = ¯Y¯UXRUaP

Th ere is no sign for 100 in any early Roman inscription although there surely must have been one to complete the series. Th e V form for 100 is extremely rare; Ifrah (1998: 188) lists only a single inscription where it is found. Keyser indicates that the fi rst Roman C = 100 whose date is secure is from 186 bc, but he postulates a third-century bc origin for the symbol as a reduction of the Etruscan ;, even though there is no example of any sign for 100 at this early date (Keyser 1988: 542). Th e number 500 is expressed using X in all early contexts, with assimilation to the alphabetic D occurring around the transition to empire. Th e familiar M = 1000 used from the Middle Ages to the present occurs only in one classical Latin inscription as part of a numeral-phrase, although it is also found in various places where M is simply an abbreviation for mille or milia (Gordon and Gordon 1957: 181–182; Gordon 1983: 45). Th e signs for 5000, 10,000, 50,000, and 100,000 are rarely encountered, though they are all attested as early as the third century bc. Adding arcs on either side of the most common sign for 1000, Y, indicates successive powers of 10, while the right half of the appropriate base-10 sign represents the quinary component. In a very limited set of texts, ´ is used to represent 500,000 (Mommsen 1965 [1909]: 788–791; Gordon 1983: 45). Th e sign is probably derived from alphabetic Q and is thus an abbreviation of quingenta milia. Its use was limited to the later Republic, and it was certainly not familiar a century later to Pliny the Elder, who, in his Natural History, wrote “Non erat apud antiquos numerus ultra centum milia” or “Among the ancients there was no numeral larger than 100,000” (Natu- ral History 33.47.133). Alongside the Roman numerals, the Romans had a duodecimal fractional system based on the as of twelve unciae (Menninger 1969: 158–162; Cagnat 1964: 33). Italic Systems 111

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

4 In the tradition of modern Roman numeral hour-numbers on clocks, 4 is normally denoted additively (IIII), while 9 is denoted subtractively (IX), possibly because IIII aesthetically balances the left and right sides of the clock face (Hering 1939: 319). 112 Numerical Notation rostrata of the consul Gaius Duilius, originally erected in Rome in 260 bc, which celebrated a naval victory over Carthage in which over two million aes worth of loot was plundered. Th e column is inscribed with at least twenty-two signs for 100,000, and possibly as many as thirty-two, as the inscription is fragmentary (Menninger 1969: 43–44). Although it was recut, probably in the Augustan period when other means of expressing this number were available, the original structure of the expression was retained (Stenhouse 2005: 59–60). One is struck, in looking at this inscription, at the sheer enormity of the numeral-phrase, and thus by the impressive amount of booty obtained, and in fact this may have had something to do with the reason it was written at all. Gordon and Gordon (1957: 180–181) suggest that one of the factors working against the widespread acceptance of subtractive notation was the desire of public offi cials to impress and indulge. is Th “conspicuous consumption of numerals” is cross-culturally frequent, for example, in the “Narmer macehead” (Chapter 2) and other royal commemorative inscriptions. Starting in the late republican period, a horizontal bar (vinculum or virgula) placed above a numeral-phrase or some portion thereof indicated that the number under the bar should be multiplied by 1000 (Smith 1926: 76–78; Gordon and Gordon 1957; Cagnat 1964: 31–32; Gordon 1983). Th is principle was fi rst used in the Lex de Gallia Cisalpina written between 49 and 42 bc (Gordon 1983: 47). For most numbers, doing so did not improve conciseness; both gedddda\SROOO and °½....YSROOO for 191,063 require twelve symbols. Th e main advantage is that one need no longer remember so many numeral-signs or invent new ones; the signs for 1, 5, 10, 50, and 100 are suffi cient to express any number up to 500,000, whereas eleven diff erent signs would be needed under the purely additive system. One slight complexity is that starting around the same time, barred numerals were often used to distinguish ordinal from cardinal numbers, as in abbreviations such as aaVIR for duumvir (Gordon and Gordon 1957: 166–176). In one irregular inscription from Pompeii from the fi rst century ad, the numeral-phrase LXXXX¦ apparently indicated 90,000 using a regular sign for 1000 multiplicatively, a practice not encountered again until the Middle Ages (Smith 1926: 7). Starting around the early Imperial period, three vertical bars enclosing a numeral-phrase on the top and sides signifi ed multiplication by 100,000. us,Th instead of the thirty-two signs for 100,000 found on the Columna rostrata, one would need only to write mdddaan. Th is technique was fi rst used in the late rst fi century bc, according to Gordon (1983: 47), but was employed rather sparsely until the second century ad.5 In this way, any number less than 500 million could

5 According to Suetonius (Galba, 5), the emperor Tiberius willfully read the will of Livia in such a way as to read a three-barred numeral min as i in order to reduce by a factor of 100 the amount of inheritance owed to the future emperor, Galba (Cagnat 1964: 32). Italic Systems 113

Table 4.11. Roman numerals (multiplicative)

1 5 10 50 100 500 1000 Regular signs aPRSU W Y 1000 5000 10,000 50,000 100,000 500,000 1,000,000 Multiplicative (1000) abdeg j k 100,000 500,000 1,000,000 5,000,000 10,000,000 50,000,000 100,000,000 Multiplicative (100,000) man mbn mdn men mgn mjn mkn

35,863,120 = mgggebaaan edaaa URR

be expressed with just the lowest numeral-signs plus two types of bar to express multiplication. Th is revised system is still a decimal system with a sub-base of 5; however, instead of being purely cumulative-additive, it is a hybrid system using cumulative-additive structuring for numbers up to 1000 and multiplicative- additive thereafter. Th e entire system (up to 100,000,000) as used in the Imperial period is shown in Table 4.11. Gordon (1983: 47) claims that the largest number expressed using this hybrid cumulative and multiplicative system is 35,863,120, though an inscription at Ostia from ad 36 apparently indicates 100 million as mkn (Menninger 1969: 245). Most of the higher signs are attested only rarely. Curiously, Guitel (1975: 215) regards Roman multiplicative notation as an evolutionary dead end, because, she argues, they no longer needed to develop a more effi cient positional system. eTh teleology of this argument is immediately apparent, as it regards this development only with respect to its failure to lead to a “superior” system. Th e Romans themselves likely perceived it as a means of improving conciseness, while reducing the number of signs one needed to memorize. Although the use of the 1000 “bar” continued among some post-Roman scribes, the use of the 100,000 “box” did not outlast the empire. While the origin of the Roman numerals is a common topic of inquiry, unfortunately, as Cajori (1928: 31) noted, “the imagination of historians has been unusually active in this fi eld.” Fortunately, Keyser’s (1988) panoptic essay on the origin of the Roman numerals, which examines a variety of theories, ranging from the sixth-century theories of the grammarian Priscian to the twentieth-century theories of modern classicists, has fi rmly settled the issue. Th e popular belief that the Roman numerals originated as alphabetic signs is false. While the modern Roman numeral-signs are also letters, the signs I, V, and X mean 1, 5, and 10, rather than U, Q, and D, for unus, quinque, and decem. C is the fi rst letter of centum, but this is coincidental, since C is a reduction of the older Etruscan sign ; or VaU (Keyser 114 Numerical Notation

Table 4.12. Etruscan and Roman numerals

1 5 10 50 100 500 1000 Etruscan 1 Q R ÷\: U\\; \¬ ì\\\/ Roman 1 Q\P\ R õ\§\T U \X Y\\\¤

1988: 542). Th e signs for 50 and 500 were not associated with the letters L and D until the late Republic, and M was not used for 1000 until the Middle Ages. It is fortuitous that the older sign for 1000 (Y) could be easily transformed into an M. Similarly, theories employing the pictographic principle (for instance, I = 1 from a single fi nger; V = 5 from an outstretched hand, and X = 10 from two hands together) were proff ered by many early modern antiquarians, and later by classicists such as Sandys (1919), but, while they are imaginative, there is no evidence to support them. Until recently, the most widely accepted theory was that of Th eodor Mommsen (1965 [1909]), who argued that the signs for 50 (õ), 100 (U), and 1000 (Y) were taken from letters of the Chalcidic Greek alphabet that were not needed to transliterate the Latin language: chi, theta, and phi, respectively, which somewhat resemble the numeral-signs. Unfortunately, this attractive theory has several fl aws: the sign for 100 does not really resemble the Chalcidic theta; these “lost letters” were sometimes used in Etruscan and Roman inscriptions; and special pleading is required to derive the origin of X = 500. While it was once plausible, it is no longer a parsimonious theory. In fact, the Roman numerals up to 1000 developed through direct diff usion from the Etruscans. Th e astonishing similarity between the Etruscan and archaic Roman numeral-signs, as shown in Table 4.12, ought to be enough to prove a relationship between the two systems. Th e Etruscan numerals have temporal priority over the Roman numerals, which do not appear until well into the fi fth century bc, and are not frequently encountered until the third century bc. In fact, the similarities between the two systems are so great that one could treat them as a single numerical notation system; they are identically structured, and many of their numeral-signs are similar or identical. I treat them separately because the two systems are written in opposite directions and because the Roman system used signs for much larger powers at an early date. Th e Roman numerals were used in a broader range of contexts than any other cumulative-additive system. In its earliest forms, they were used on coins, on pottery, and on inscriptions on stone. Dates, monetary values, and measures were all frequently expressed in the Roman numerals. Th e Roman numerals could be employed to express both cardinal and ordinal values. Th eir use in administration and literature was widespread from the republican period onward. In texts, Roman numerals were used to enumerate page and line numbers. Accounts, inventories, Italic Systems 115 and legal documents also occasionally provide us evidence of their use in commercial and institutional contexts. While Roman numerals certainly were used in the contexts of arithmetic and calculation, there is minimal evidence that they were ever used for calculation. Glautier (1972) discusses the Roman account records, which he characterizes as primitive from an accounting standpoint because of the lack of positionality. A similar point is raised by Meuret (1996) in his discussion of the Lamasba tablet, an irrigation regulation from North Africa during the reign of Elagabalus (218–222 ad), which contains a multiplication table to enable quick calculation of water supplies, thus overcoming the computational defi ciencies of the system. Maher and Makowski (2001) demonstrate persuasively, however, that the assumption that Roman mathematics was poor because of Roman numerals cannot be correct, given the complexity of the arithmetical calculations attested in Latin texts. Nevertheless, their stronger claim – that Roman mathematicians actually used written numerals to do arithmetic, in particular for calculations involving fractions – remains unproven. Roman numerals were used for computational functions – if not directly for computation, then certainly to mark the rows on the abacus. While few Roman abaci survive, Taisbak claims that the Romans did all their calculations with them, and even that “the notation of Roman numerals originates from the abacus reckoning” (Taisbak 1965: 158). Th is fi nding is contradicted by the derivation of the Roman numerals from the Etruscan system and ultimately from an older tallying system. Because cumulative systems use one-to-one correspondence intraexponentially, just as one counter equals one multiple of a power on the abacus, the Roman techniques of numeration (Roman numerals) and computation (the abacus) complement one another. Th is correlation is confi rmed by the quinary (base-5) component of the abacus (there are rows not only for the powers of 10 but also for their halvings). No row on the abacus ever would have contained more than four counters, which would have facilitated reading and working with them. However, because the original Etruscan system probably emerged from a system of tallying, it is more likely that the structure of the abacus emerged out of the structure of the numerals than vice versa. Despite the enormous infl uence of Roman civilization on Europe, North Africa, and the Middle East, and despite the extraordinary chronological duration of the Roman numerals (almost 2,500 years), they produced relatively few descendants. While other systems in use over similar periods, such as the Brāhmī numerals and Greek alphabetic numerals, changed their form greatly as they spread across time and space, the Roman numerals of antiquity spread largely unmodifi ed throughout Western Europe and other areas where the Roman alphabet was used. While the Indian and Greek systems spread throughout many diff erent scripts, changing 116 Numerical Notation the forms of signs as they were transmitted, Roman numerals were infrequently adopted by users of other scripts. Of the few descendants of the Roman numerals, I have already discussed the hybrid multiplicative-additive system used occasionally from the fi rst century bc onward. In the medieval period, Roman numerals were essentially the same as classical ones, though with slight diff erences in form and structure. In Arab- infl uenced Spain, certain variant Roman numeral systems were used starting in the tenth century ad. Around the same time in northern Europe, certain types of medieval calendars contained unusual Roman numerals. Finally, as the Roman numerals came increasingly under assault from the rival Western system, certain positional variants of the numerals were occasionally used, combining features of both systems.

Computation with Roman Numerals Th e computational effi ciency of the Roman numerals is a common subject in the history of mathematics. Th e consensus of these arguments, with which I am in general agreement, is that the Roman numerals are poorly suited to performing arithmetical calculations. Th is accords with the historical fi nding that people used abaci or fi nger computation to actually manipulate numbers, with the Roman numerals being used only to express the result. Yet the rejection of the effi ciency of the Roman numerals for computation has been used to draw two unwarranted implications. Firstly, it is unlikely that the use of the Roman or other cumulative-additive numerals has any simple or unilinear correlation with developmental stages of psychology, as Murray (1978), Hallpike (1979), and Dehaene (1997) have suggested, either as the cause or eff ect of a less abstract way of thinking about number. Although many ancient numerical notation systems are cumulative-additive, this is not evidence that the Roman number concept is less abstract than the modern one. Other evidence, such as the use of abaci and fi nger reckoning, refutes any simple correlation between the cultural evolution of numerical notation and cognition. Th ere may be some correlation, but it must be demonstrated, not assumed from the ineffi ciency of the Roman numerals for a task for which they were never intended. Secondly, some scholars claim that the Roman numerals prevented the Romans from developing other useful institutions or techniques. Glautier (1972) asserts that the Romans did not develop an effi cient accounting system due to the lack of a suitable numerical notation system, while Williams (1995) bemoans the limitations of Roman numerals for doing the complex calculations with fractions needed for alloying coinage. Yet the Romans evidently had suffi cient accounting techniques Italic Systems 117 to administer their empire, and while double-entry bookkeeping is made easier by a ciphered-positional numerical notation system, it must have functional equivalents, or else no society lacking such numerals could administer a large political entity. Th e inferiority of Roman numerals is used by Guitel (1975) to explain why the Romans were poor mathematicians as compared to their Greek subjects. How- ever, both Greek and Roman mathematicians relied considerably on Babylonian knowledge, and educated Romans knew the Greek alphabetic numerals. Maher and Makowski (2001) show that the Romans were, in fact, quite good at arithmetical computation. If, in fact, the Romans were poorer mathematicians than the Greeks, evidence other than their numerals must be sought. In contrast, a small body of research within the subdiscipline of the history of mathematics holds that the Roman numerals are not less effi cient for computation than the Western numerals. At least four modern scholars claim to show how the Roman numerals could have been used in written calculations without the aid of an abacus or similar technology (Anderson 1956, Krenkel 1969, Detlefsen et al. 1975, Kennedy 1981). Th ese analyses, apparently derived independently of one another, diff er in the exact technique used in performing calculations, but all conclude that even if the Romans never used their numerals in such a fashion, the Roman numerals are in fact amenable to computational functions. While this is superfi cially true, I regard this argument as highly spurious. Th e proposed techniques are often more complicated than they are presented to be, and certainly more complicated than the standard arithmetical techniques using Western numerals. For instance, the technique proposed by Detlefsen and colleagues (1975) involves a complex “transformational grammar” that is far removed from the knowledge systems of Roman or medieval scholars. Additionally, none of these studies establishes that the Roman numerals are equally or more effi cient for computation than Western or other numerals. A system’s minimal utility for a function is hardly proof that it was or should have been used for it. Th ese studies presume that it is natural for numerals to be manipulated for computation, even in the absence of historical evidence. Anderson (1956: 145) suggests that “any reader, once he discovers how simple the operations are, will be inclined to imagine that some Roman engineers and surveyors, in building their great projects, did occasionally do their computations very much in the way described below, even though they left no records of their work.” Detlefsen and colleagues (1975: 147) go so far as to blame the Romans for not recognizing the computational potential of their numerals. While the possibility that the Roman numerals were used in this way cannot be ruled out, the argument ex silentio is highly implausible. We have persuasive evidence that ancient and medieval scholars used the abacus, fi nger computation, and other techniques (Lang 1957, Taisbak 1965, Murray 1978). Th e best way to counteract the denigration of the Roman numerals is not to show that they are 118 Numerical Notation mildly (or even greatly) useful for a function for which they were never known to have been used. Th e issue is not simply a mathematical game to see whether a system can serve some arithmetical function. By understanding the functions for which the Roman numerals were not used, we may better understand the circumstances of their eventual replacement.

Medieval – Additive As early as the second century ad, but most prominently in the fi fth and sixth centuries, Roman numerals inscribed on stone inscriptions frequently took on a cursive quality, and some signs became ligatured together (Gordon 1983: 46). A special sign for 6, ³, which is nothing more than a cursively written and ligatured vi or ui, occurs on many inscriptions from late antiquity (Lassère 2005: 57–59). On sixth-century Byzantine imperial coins, ³ was as common as VI (Wroth 1966: cx). Th e use of this sign probably ceased in the eighth century (Bischoff 1990: 176). Similarly, in late antique and early medieval inscriptions, 40, 60, and 500 could be written cursively, even in inscriptions on stone. Th e Visigothic script used between the seventh and thirteenth centuries was particularly characterized by these and other ligatured Roman numerals, particularly for 40 (XL) (Schapiro 1942, Bischoff 1990). eseTh sporadic shifts toward ligatured notation would later play some role in the paleographic modifi cation of Arabic signs into what would become the Western numerals (Lemay 1982: 393; see also Chapter 6, of this volume). Yet these ligatures did not represent an inexorable diachronic trend toward ciphering in Roman numerals, and the majority of early medieval Roman numerals remained cumulative-additive. After the fall of the western Roman Empire, literate knowledge was distributed rather sparsely – for instance, among Western monks, Byzantine bureaucrats, and Middle Eastern scholars. Th e great early medieval Mediterranean polities – the Byzantine Empire and the early Muslim caliphates – mainly used ciphered- additive numerical notation systems such as the Greek and Arabic alphabetic numerals (Chapter 5). In Western Europe during the Middle Ages, the Roman numerals were the only ones in common use. Knowledge of other systems was restricted to peripheral regions such as Spain and southern Italy, and to a tiny well-educated elite. Even at the height of the Carolingian Renaissance (around 800 ad), arithmetic was the province of a learned few, and was acquired late in the scholar’s education (Murray 1978). Still, while the need for large-scale bureaucracy and the corresponding need to express large numbers had declined since the height of the Roman Empire, Roman numerals were still frequently encountered, and even expanded in the range of functions they served. Th e medieval Roman numeral-signs diff er from the classical ones in several aspects. Rather than being written solely in majuscule characters, Roman numerals Italic Systems 119 were frequently written in the lowercase script on perishable materials. When written in minuscule form, placing a stroke through the last numeral-sign of a numeral-phrase indicated that one-half was to be subtracted from the represented value. As a measure against fraud, the last i in cursive numeral-phrases was often extended into a j, preventing anyone from adding additional signs to the end of the phrase (Menninger 1969: 285). Th is practice originated around ad 900 in South Italian manuscripts, although it may have had an antecedent in the upward rather than downward extension of phrase-fi nal cursive i into I in a few classical texts (Smith 1926: 74). Alphabetic forms of the numerals for 500 (D) and 1000 (M) replaced the earlier X and _, marking the end point of a long process of alphabetization. Th is would have had the advantage of some mnemonic convenience; even if most of the numeral letters did not correspond to the numbers they represented, at least literate learners of the system would not need to learn an entirely new set of signs. Alphabetization of the numeral-signs also enabled one to use them for numerical riddles, particularly chronograms, in which the total value of the Roman numerals in a line of verse expressed the date of an event described in that verse (Menninger 1969: 281). Th e multiplicative vinculum bar for 1000 used in classical antiquity continued to be used under a new name, the titulus, but the three-sided box symbol for multiplying by 100,000 was no longer used (Menninger 1969: 281). Large numbers were very rarely needed, and even the need for the titulus was limited. Subtractive forms were used more frequently in the Middle Ages, though purely additive forms (e.g., IIII) were still common. Finally, starting in the tenth century, numeral-phrases for ordinal numbers were often adverbialized using superscript endings such as -o and -mo – for instance, Mmo CCmo Lmo IVto, incorporating aspects of the medieval Latin lexical numerals into numeral-phrases (Smith 1926: 75). Shipley (1902) suggests that the changing use of Roman numerals between classical antiquity and the ninth century ad led to transcription errors in medieval manuscripts. Comparing the fi fth-century ad Codex Puteanus, containing sections of the works of Livy, and the ninth-century ad Codex Reginensis, a copy of the former, the analysis of copying errors reveals much about the Roman numerals used at the time of copying. Where the classical Roman form for 1000 was _, the medieval scribe was more accustomed to using Z, and thus _ was often transcribed as R. Where the classical manuscripts contained X for 500, medieval scribes used W, and thus often omitted the X symbols entirely on the theory that the horizontal stroke indicated that the scribe had crossed out an error. Finally, because the subtractive form XL for 40 was used in medieval times as opposed to RRRR, instances of RRRR were abbreviated to RRR to correspond to correct medieval numeral-phrases. Shipley’s analysis confi rms the increased use of both subtractive structuring and alphabetic Roman numeral-signs. 120 Numerical Notation

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

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

6 Smith (1926: 72) argues that this practice was attested in a few classical inscriptions as well, such as XII.L.D for 1,250,500 (!), but I know of no inscriptions where this was actually done. Italic Systems 121

Table 4.13. Medieval multiplicative Roman numerals

Year Numeral-Phrase Roman with Western Source Multiplication 1220–25 II•DCCC•XIIII aaWUUUROP 2814 Menninger 1969: 285 1231 IX.XX.XVI URUPO 196 Smith 1926: 6 1258 Mijc lviii ZUUSPOOO 1258 Steele 1922: xvii 1340 IILXXIII ZZSROOO 2073 Crosby 1997: 208 1388 IIIIxx et huit SRRRPOOO 88 Guitel 1975: 225 M XX 1392 aaaUSRRROOO 3183 Cajori 1928: 33 III C IIII III 1437–38 VIIXXXIX USOR 159 Preston 1994 1505 I•Vc•V ZWP 1505 Menninger 1969: 285 IIC UU 200 1514 Cajori 1928: 34 IIIIC.LX UWSR 460 1514 XVCXIV ZWROP 1514 Smith 1926: 7 1550 CCCM ggg 300000 Menninger 1969: 285 1554 viM viiC xiii baWUUROOO 6713 Menninger 1969: 283 c m c 1771 i xxiij iiij lvj gddaaaUWSPO 123456 Cajori 1928: 33 made between the Roman and Arabic systems. Nevertheless, this system was not taken up more widely in medieval scribal or mathematical texts. Th e speculation of Busch (2004) that a diffi culty with seafaring travel times in the fteenthfi century Icelandic Landnámabók can be resolved by interpreting VI (6) as “V I” (e.g., 51) is highly dubious. While the use of barred numerals to indicate multiplication by 1000 was of early origin and continued throughout the Middle Ages, new multiplicative forms began to be used starting in the twelfth and thirteenth centuries. Some examples of such numeral-phrases are listed in Table 4.13, along with the transcriptions of the appropriate number both in the classical Roman system (including the use of subtractive and multiplicative forms, where appropriate) and in the Western system. Th ese numeral-phrases primarily express multiplication by 100 or 1000 by juxtaposing C or M either immediately beside the appropriate multiplier, above it, or in superscript, and sometimes interposed with a dot. Often, the number 20 occupies a special role as a multiplier; such phrases are almost all from France and result from assimilation to the partly vigesimal structure of the French lexical numerals and/or to the monetary system (Preston 1994). Th ese numeral-phrases have a multiplicative component, but they are not positional – the value of the 122 Numerical Notation

Table 4.14. Partially positional Roman numerals

M•CCCC•8II 1482 CC2 202 ICC00 1200 1•5•IIII 1504 15X5 1515 MDZ4 1624 MCCCC4XVII 1447 IV0II 1502 numeral-signs does not change due to their position, but rather due only to their juxtaposition with another sign. It might be thought that multiplicative forms were adopted in order to write numerals more concisely or with a smaller set of numeral-signs – as was the case with the initial use of multiplicative forms in classical Rome. However, as seen by comparing the numeral-phrases in Table 4.7 to their equivalents in standard Roman numerals, there is no such benefi t. In addition to these multiplicative forms, we fi nd many cases where the signs of the Roman system were intermingled with the positional principle of the newer Western system, as well as its actual numeral-signs. Th e earliest known example is from the late twelfth-century Microcosmographia, in which the only known copy (from the thirteenth or fourteenth century) is dated mclxxviii (1178) but in another place as mc87 (an error for 1178) (Williams 1934: 107). Preston (1994) describes a number of such mixed uses in Durandus of Saint-Pourçain’s Commen- tary on the Sentences of Peter Lombard of 1336, such as xxx3 for 33 and xl7 for 47. In one example from an English almanac from 1386, Western numerals both precede and follow Roman numerals in a numeral phrase for 52,220, written as 52mcc20 (Halliwell 1839: 116). Menninger (1969: 287–288) provides examples of such admixtures starting in the late fi fteenth century, as indicated in Table 4.14. In these partly ciphered-positional numeral-phrases, conciseness is greatly increased over their Roman counterparts. It is unknown whether this was being done consciously as a compromise between the two systems, as a misunderstand- ing of the Western system, or as numerical playfulness. In at least one case, the blending is obviously erroneous; in an astronomical table, the scribe wrote MCC6 for 1269, then crossed out the entire phrase and wrote 1269 in Western numerals (Steele 1922: xvii). Such combinations are not necessarily advantageous; the idiosyncratic nature of these numeral formations almost certainly decreased their comprehensibility. None were used frequently or consistently enough to create a true variant system. Th ese hybrid formations no longer appear after about 1650. Th e roughly contemporaneous expansion of Roman multiplicative notation and the introduction of hybrid forms employing ciphering and positionality in Italic Systems 123 late medieval European scribal traditions meant that any number could be written in several ways. Th e wide range of textual genres containing such variants, and their geographical and chronological breadth, confi rm that this variety was not isolated, but neither was it standardized. In any event, a parallel process was ongoing from the twelfth century onward that would eventually lead to the nearly complete replacement of Roman numerals by Western ones throughout Europe and eventually throughout the world.

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

7 Th is statement must be considered in light of the fact that a) it is a sample sentence rather than part of an evaluation of the two methods and b) Palsgrave’s book was paginated throughout using Roman numerals. 124 Numerical Notation inventories “not by ciphers but by plain letters” (Berggren 2002: 361). As late as 1494, the Frankfurt Bürgermeisterbuch (a mayoral book of edicts) ordered reckoners not to compute using Western numerals (Menninger 1969: 426–427). One can certainly imagine the consternation of merchants and bookkeepers upon learning that one can simply add zeroes to the end of a numeral ad infi nitum to multiply its value by ten each time, but the role of xenophobia and traditionalism in all of these cases should not be underestimated. Th e Western numerals may have been rejected in part because of practical considerations, but they were, after all, a foreign and newfangled invention that involved the unusual principle of positionality and that (because of their novelty) could be used to conceal information or deceive others. Th e rarity of paper in the earlier Middle Ages may also have contributed to the continued use of the Roman numerals (Smith and Ginsburg 1937: 29). Until pen- and-paper calculation became feasible, the “Roman numeral–friendly” abacus was the computational technology of choice, whereas the switch to pen and paper made the conciseness of Western numerals more attractive. Th is attractiveness increased with the introduction of printing presses, where long strings of movable- type Roman numerals would have been unwieldy in comparison to Western numerals. Yet despite Eisenstein’s contention that, “[t]he use of Arabic numbers for pagination suggests how the most inconspicuous innovation could have weighty consequences – in this case, more accurate indexing, anotation, and cross- referencing resulted” (1979: 105–6), most incunabula contained no Western numerals. Even where the utility of Western numerals for computation was recognized, the Roman numerals were not always abandoned entirely, as in Italian metrological documents studied by Travaini (1998), in which computations were undertaken using Western numerals, but totals were then written in Roman numerals. William Cecil (Lord Burghley), England’s Lord High Treasurer from 1572 to 1598 and a chief advisor of Elizabeth I, would frequently transcribe economic documents from Western back into Roman numerals, and was decidedly uncomfort- able with the new system (Stone 1949: 31). Despite such resistance, the Western numerals had taken hold by around 1300 in the Italian city-states. Elsewhere in Western Europe, particularly in Germany and England, the Roman numerals predominated until the late fi fteenth century or even later. Jenkinson (1926) fi nds limited evidence for the use of Western numerals in English archives before 1500, and notes that Roman numerals were forbidden from use in state accounts only in the nineteenth century. Barradas de Carvalho’s (1957) study of fi fteenth- and sixteenth-century Portuguese texts showed that the Roman numerals were replaced there only around 1500. Several important sixteenth-century arithmetical texts continued to use Roman numerals in preference to Western ones. Jacob Köbel’s (1470–1533) Rechenbiechlin, published in Italic Systems 125

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

Augsburg in 1514, an important commercial arithmetic book intended for mercantile instruction, uses Roman numerals throughout and strongly advocates the use of the counting board, although Köbel frequently used multiplicative phrasing for hundreds and thousands, and his text is the source of the unorthodox “260/400” Roman numeral fraction in Table 4.13 (Smith 1908: 100–107). However, by the seventeenth century, the battle was essentially fi nished, and the Roman numerals ceased to be used in most contexts. Figure 4.2, from the Mysticae numerorum signifi cationis of Petrus Bungus (d. 1601), fi rst published in 1583–84, depicts a wide variety of archaic and curious Roman numeral phrases, but is essentially a text on number mysticism and symbolism rather than a manual for practical use (Smith 1908: 380–383). Why, after over a millennium of essentially unchallenged use in Western Europe, should the Roman numerals have ceased to be used for most purposes? Th e traditional answer given – that Roman numerals were less effi cient for computation – is correct but incomplete. Th e Roman numerals were used alongside the Greek 126 Numerical Notation alphabetic numerals for well over a thousand years, despite the great increase in conciseness that could have been achieved by abandoning the Roman system. In medieval Western Europe, debates over the use of Roman numerals lasted into the seventeenth century, with the relative usefulness of the Roman numerals for calculation being greatly increased by the accompanying use of counting boards. However, two developments were fundamental in rendering the Roman system obsolete in the West. Firstly, the development of the printing press and the consequent rise in literacy after 1450 correlates very well with the rapid adoption of Western numerals throughout Europe (particularly in northern regions). Th e newly literate middle classes of Western Europe were learning to read and calculate, unconstrained by centuries of traditional use of the Roman numerals. At the same time, the rise of mercantile capitalism in the Renaissance changed how numbers were viewed and used (cf. Swetz 1987). Th e functional needs of Western society to represent number changed dramatically between 1300 and 1700, and the computational functions that are better served by Western than by Roman numerals increased in importance. Nevertheless, Crosby’s (1997: 41) insistence that Roman numerals were “adequate for the weekly market and for local tax collection, but not for anything grander” is misplaced if intended as an explanation for their replacement. In fact, as Crosby himself notes, Roman numerals persisted for centuries in the record books of institutions as grand as the British Exchequer (Crosby 1997: 115–116). Double-entry bookkeeping was facilitated to some degree by the transition to Western numerals, but it was by no means a requirement. Th e introduction of a more fully mercantile and urban economy in Western Europe did, however, expose more people, particularly members of the newly wealthy and increasingly literate middle class, to a new form of numeration. Rather than examining this change in notation simply by comparing the two notations, it is more productive to examine the question from a social perspective. If Roman numerals were truly so inferior, how did they persist for two millennia without being replaced? Why did it take six centuries even after the introduction of Western numerals for this shift to take place? Th e most likely scenario is that the new users of writing and written numeration, unconstrained by past practices, provided a critical mass of users for the Western numerals that facilitated certain mercantile functions. Even today, the Roman numerals have not disappeared. Th e use of the Roman system on clock faces, in the enumeration of kings and popes, on many dated inscriptions, and to mark the copyright dates on fi lms is rather archaic, but the cultural importance of its symbolic connotations of antiquity, tradition, and prestige likely guarantee its future survival. In media where archaism is desired, Roman numerals are often preferred; this is the case among the Cornish church sundials from 1670–1850 studied by Burge (1994), for instance. An inscription on a gate- way at Harvard uses archaic Roman numeral-signs, noting the college’s 264 years of Italic Systems 127 existence in 1900 as “ANN. DOM. _WUUUU. COLL. HARV. UUTROOOO” (McPhar- lin 1942: 18). Pot (1999) notes the existence of several twentieth-century postage stamps in which additive Roman numerals rather than the ordinary subtractive ones are employed – for instance, XXXXI instead of XLI on a 1990 German issue. Pre- sumably these variants are not errors per se, but are mobilized for aesthetic reasons, just as on Roman numeral clock faces, 4 is normally written IIII instead of IV, even though 9 is IX (Hering 1939). Th e fi fteenth-century accountant Luca Pacioli advocated the use of Roman numerals for the year-dates in accounting books for aesthetic reasons (although Western everywhere else), recommending “[u]se the ancient letters in making this entry, if only for the sake of beauty” (Crosby 1997: 222). At the other end of the class and prestige spectrum, Roman numerals were often retained in peasant or rural traditions and trades long after they had ceased to be used in formal texts and administration. Gmür (1917) describes a wide variety of wooden tally sticks of Swiss provenience, dating from the sixteenth to the nineteenth centuries, containing Roman numeral-phrases. Th e same principle applies to the much earlier “calendar-numerals” used on North- ern European calendrical documents (see the following discussion). With the inexorable spread of industrialization and capitalist commercial practices throughout Europe, the use of wooden tallies containing Roman numerals has essentially disappeared. When used to paginate the introductions of books, or to enumerate items in sublists or subsections, Roman numerals continue to serve the very practical function of providing an alternate form of enumeration wherever two diff erent series of things must be listed. I would not expect that any industrial society could function solely with a cumulative-additive system such as the Roman numerals. Con- versely, however, I would not expect them to be replaced entirely. Th e failure of over fi ve hundred years’ worth of predictions of the Roman numerals’ imminent demise, and complaints regarding their utility, suggests that functional considerations, while important, do not tell the entire tale with respect to the history of numerical notation systems.

Arabico-Hispanic Variants Arabic and Western European knowledge systems interacted intimately on the Iberian peninsula between the tenth and sixteenth centuries – the period of the Reconquista and somewhat beyond. At this time, the Roman numerals were well known in Spain and Portugal, as in the rest of Europe. Arab mathematicians and astronomers had, since around 800 ad, used a ciphered-positional system much like the one used today (Chapter 6), along with older ciphered- additive abjad numerals related to those of the Greeks (Chapter 5). Medieval 128 Numerical Notation

Spain thus presents us with a remarkable case where three coexistent systems each had a diff erent structuring principle (cf. Labarta and Barceló 1988). But rather than a simple case of the two Arabic systems replacing the Roman numerals, medieval Spanish astronomers, bookkeepers, and scribes experimented and mixed principles, incorporating ciphering and positionality into the basic structure of the Roman numerals in innovative ways. It was, as described by Berg- gren (2002: 358), a “promiscuous blend of systems,” a mélange that persisted for several hundred years. In the tenth and eleventh centuries ad, Spanish astronomers wrote extensive astronomical texts using primarily the Roman numerals. Even the use of the subtractive principle was very infrequent in Spain at this time. Despite the use of Arabic positional numerals in the region for some time previous and familiarity with works such as the Arithmetic of Al-Khwarizimi, Spanish astronomers did not adopt the system directly (Lemay 1977: 458). Instead, the Roman numerals were modifi ed to increase their conciseness. Instead of using the cumulative principle to express the numbers 3 through 9 (III, IIII, V, etc.), they used new acrophonic symbols based on the fi rst letters of the Latin numeral words: 3 = t = tres; 4 = q = quatuor; 5 = Q = quinque; 6 = s = sex; 7 = S = septem; 8 = o = octo; 9 = N = novem. Of these, only the symbols for 4, 8, and 9 were common, probably because they are the longest numeral-phrases below 10 in Roman numerals (IIII, VIII, VIIII). In addition, a special sign for 40 was employed: Ó, a cursive ligatured version of XL, and the only exception to the abandonment of subtractive forms (Lemay 1977: 459). Bischoff (1990: 176) reports that this sign for 40 is also found in some Visigothic manuscripts, suggesting considerable antiquity for the sign on the Ibe- rian peninsula. Th ese changes altered the classical Roman numerals into a partly ciphered-additive system. However, these modifi cations were not accompanied by the adoption of positional notation, and the system still had a mixed base of 5 and 10 and was purely additive. Th ese developments increased the conciseness of numeral-phrases considerably. Th e number 99, which would have been LXXXXVIIII, could now be expressed as LÓN. Th is system was used in mathematical and astronomical texts until about the mid twelfth century, at which time the Western numerals took hold in astronomy, as in the rest of Western Europe. Yet while the Arabic positional numerals had fi rmly established themselves and were later transformed into the Western numeral-signs familiar to us, the Roman numerals did not cease to be used on the Iberian peninsula, though their use became increasingly limited. Labarta and Barceló discuss two curious off shoot Roman numeral systems found in early modern Spanish documents, as shown in Table 4.15 (Labarta and Barceló 1988: 32–34). Both systems are cumulative-additive, decimal systems with a sub-base of 5, unlike those of the earlier astronomical texts. Unlike most Roman numerals, Italic Systems 129

Table 4.15. Arabico-Hispanic numerals

151050100 ØÙÚÛÝ 1Vó numeral-phrases are written from right to left (highest values at the right), which is curious because even in the Arabic script, which has a right-to-left direction, the numerical notation system has a left-to-right direction. Th e fi rst system is found in a few late sixteenth- and early seventeenth-century documents to indicate monetary quantities, and its numeral-signs are somewhat similar to letters of the Ara- bic script. Th e second system is found only in a single Inquisition document from 1576 (Labarta and Barceló 1988: 34).8 From the sixteenth century onward, Western numerals became increasingly common throughout the Iberian peninsula, but certain anomalies remained in the Roman numerals of the early modern period. Subtractive numerals remained rare in comparison to the rest of Europe. A symbol known as the caldéron was sometimes used; it meant 1000 but was used multiplicatively, preceded by unit-signs, not additively (Cajori 1922, Lowe 1943). Th e caldéron was most often shaped like a U, sometimes with one or two diagonal bars across it (™). Th us, a manuscript from the Ponce de Leon papers from 1501 has the year dated as J™DJ (1 1000 500 1), reminiscent of the use of M for that purpose in other parts of medieval Western Europe (Lowe 1943: 9). Cajori (1922) hypothesizes that the caldéron originated in Italy in the fi fteenth century, possibly as a modifi cation of the older Y for 1000 that was not fully replaced by M until the modern period. Curiously, even after the Roman numerals had been abandoned entirely, it continued to be used multiplicatively with Western numerals, as in a contract written in Mexico City in 1649 in which a sum of 7291 pesos is expressed both as 7U291 and VIIUCCXCI (Cajori 1922: 201).

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

8 However, see Chapter 10 for a discussion of a potential descendant of this system in the form of Berber numerals used in North Africa. 130 Numerical Notation

Table 4.16. Calendar numerals

1234567891011121314151617181920 abcdefghijklmnopqrst ABCDEFGHIJKLMNOPQRS8 1234555 5 5 666 6 6 777 7 7 8 1 2 3 4 1 2 3 4 1 2 3 4 them.9 Likewise, the term “peasant numerals” tells us who may (or may not) have been using them, but lacks the precise functional association of “calendar numerals.” Some examples of these signs are shown in Table 4.16 (cf. Kroman 1974: 121; Ifrah 1985: 146–147). Th ese systems are all cumulative-additive and have a base of 10 with a sub-base of 5. Units are marked by strokes or dots; fi ves are marked by angled or curved lines or loops to create “U” or “V” shapes, and tens are marked by transecting the vertical line perpendicularly, creating a cross or X. Although these numeral- signs are often joined together into single fi gures resembling digits by using a vertical stroke, the system is not ciphered. Th ere are no signs for 50, 100, or higher values, and I am not aware of calendar numerals being used for numbers higher than 31 (the number of days in the longest months). Other than their unusual numeral-signs, the calendar numerals are identical to ordinary Roman numerals. While they were used in rural traditions that suggest to some that they were part of the large cultural substratum of tallying and notching in medieval and early modern Europe (Menninger 1969: 249–251; Ifrah 1985: 146–147), the calendar numerals are not tallies in this sense. A tally is used ordinally – one places marks as necessary on some material (wood, paper, stone, etc.) in sequence in order to keep a running total, rather than, as with numerical notation systems, marking an already totaled value. Th e calendar numerals are simply a variant of Roman numerals in which numeral-phrases are written vertically and attached to a line. By the late fi fteenth century, the Western numerals were fairly well known in Germany and were becoming much more common in England and Scandi- navia. Calendar numerals were used in a very delimited set of contexts, namely

9 Th e medieval antiquarian tradition of manuscripts discussing runic writing contains minimal evidence of any specifi cally “Runic” numerical system. The clophruna (Old Norse klapprúnir) system of notating numbers by dots (a = 1 = .; b = 2 = ..; c = 3 = … ) discussed by Derolez (1954: 134) is of obscure origin and is not clearly related to any other numerical notation. Italic Systems 131 on documents of wood, stone, or horn designed to assist the largely nonliterate populace in determining the dates of festivals, especially Easter. Th roughout the medieval period, the Metonic cycle of nineteen years, after which the moon’s phases recur on the same day of the year, was used as a rough-and-ready guide to calculate the date of Easter. Th e function of the calendar numerals was solely to denote the various years of this cycle on stylized perpetual calendars, known as “runic calendars” in German- and Norse-speaking areas and as “clog almanacs” in England. Because of this specialized function, there was almost never any reason to express numbers higher than 19. While these texts were used for computation, the numerals were not used directly for arithmetic. One merely needed to line up the appropriate days and years to get the correct value. Calendar numerals were, however, reasonably compact and easily understandable by anyone who knew the Roman numerals. Th e calendar numerals lie at the heart of one of the major pseudoscien- tifi c controversies of New World archaeology – the so-called Kensington Rune Stone of Minnesota, which purportedly contains a Viking inscription left in 1362 by Norse explorers who had traveled westward from Vinland. At issue are the calendar or runic numerals used in the inscription, particularly the numbers 14, 22, and 1362, expressed as ad, bb, and acfb, respectively (Hagen 1950; Nielsen 1986: 51). Th ese numeral-phrases are ciphered-positional, like our Western numerals but unlike the cumulative-additive calendar numerals (in which 14 would be written as n). It is improbable that a Norse explorer in “Vinland” would have been familiar with the Western numerals in 1362, which were known only by the very educated in northern Europe at the time (Struik 1964: 167). Even if this knowledge is presumed, we need to explain why the savant substituted the cumulative-additive runic numeral-signs for the appropriate Western fi gures. While some fourteenth-century runic inscriptions used Western numerals (Nielsen 1986), calendar numerals were never used in such a positional fashion. Th ere is no evidence for the use of calendar numerals except in calendrical texts, which the Kensington stone is not. Ohman, the Swedish-American “discoverer” of the stone, was likely familiar with the calendar numerals (easily found in any book on runology) but not with the proper structure of the system. Similar inscriptions with runic numerals, such as that on No Man’s Land, an island south of Martha’s Vineyard, Massachusetts, are undeniable forgeries (Marstrand 1949). Th e calendar numerals were a brief and local phenomenon in northern Europe. As the Western numerals came increasingly to be used for calendrical purposes, their use waned. By 1643, when Ole Worm wrote his Fasti Danici describing the runic numerals used in calendar tables, it was primarily as a curiosity and to aid the transition to newer methods of calculation (Worm 1643). 132 Numerical Notation

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

Alphabetic Systems

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

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

133 134 Numerical Notation cases important historical questions remain unresolved. While they are mostly ciphered-additive, they are not structurally identical. We can learn much more from these structural diff erences than from the paleographic curiosities of the signs of various systems. I hope in this chapter to illuminate areas of study where our knowledge is less than perfect in order to draw attention to the need for further specialized research. Table 5.1 shows the most common variants of the major alphabetic systems.

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

2 Th ere was no general name for the three signs in classical Greek; the term episemon originally referred only to the sign for 6, but following its etymology, some classicists use the term in a more generic sense (J. Kalvesmaki, personal communication; Foat 1905). Table 5.1. Alphabetic numerical notation systems

Greek Coptic Ethiopic Gothic Hebrew Syriac Arabic Fez Armenian Georgian Glagolitic Cyrillic Latin 1 a .a. a a a a a a a 2 b .b. b b b b b b b 3 g .c. c c c c c c c 4 d .d. d d d d d d d 5 e .e. e e e e e e e 6 v .f. f f f f f f f 7 135 z\ .g. g g g g g g g g 8 h .h. h h h h h h h h 9 q .i. i i i i i i i i 10 i .j. j j j j j j j k 20 k .k. k k k k k k k l 30 l .l. l l l l l l l m 40 m .m. m m m m m m m n 50 n .n. n n n n n n n o 60 x .o. o o o o o o o p 70 o .p. p p p p p p p q (continued ) Table 5.1. Alphabetic numerical notation systems (continued)

Greek Coptic Ethiopic Gothic Hebrew Syriac Arabic Fez Armenian Georgian Glagolitic Cyrillic Latin 80 p .q. q q q q q q q r 90 , .r. r r r r r r r s 100 r .s. s s s s s s s t 200 s .t. t t t t t t t u 300 t .u. u u u u u u u x 400 u .v. v v v v v v v y 500 f .w. sv w w w w w w z 136 600 c .x. tv x x x x x x θ 700 y .y. uv y y y y y y φ 800 w .z. vv z z z z z z 900 . .{. svv : , ! 1 / { 1000 /a /a ; @ 2 1 2000 /b /b # 3 2 3000 /g /c $ 4 4000 /d /d % 5 5000 /e /e ^ 6 6000 /v /f & 7 Greek Coptic Ethiopic Gothic Hebrew Syriac Arabic Fez Armenian Georgian Glagolitic Cyrillic Latin 7000 /z /g * 8 8000 /h /h ( 9 9000 /q /i ) 0 10,000 a -a - 20,000 b -b 30,000 g -c 40,000 d -d 137 50,000 e -e 60,000 V -f 70,000 z -g 80,000 h -h 90,000 q -i 100,000 i -j 1,000,000 p -s 138 Numerical Notation

Table 5.2. Greek alphabetic numerals (archaic)

123456789 1s ab c de f g h i 10s jklmnopqr 100s stuvwxyz, 562 = wob

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

3 Heath also discusses techniques such as that of Heron’s Geometrica, where two dots placed over a sign indicate multiplication by 10,000; that of Apollonius, using “tetrads,” turning the system into a mixed base-10/10,000 ciphered-additive system; and that of Nicholas Rhabdas, a fourteenth-century scholar who used Heron’s technique, except that additional pairs of dots above a number indicated successive powers of 10,000. None of these systems was ever widely used. 4 For the ancient Greeks, ‘ten thousand times ten thousand’ – 100 million, the fi rst uncountable number in the alphabetic system – was of symbolic signifi cance, as in Revela- tion 5:11: “and the number of them was ten thousand times ten thousand.” Alphabetic Systems 139

Table 5.3. Greek alphabetic numerals (classical)

123456789 1s Aa Bb Gg Dd Ee Vv Zz Hh Qq 10s Ii Kk Ll Mm Nn Xx Oo Pp <, 100s Rr Ss Tt Uu Cc Ff Yy Ww >. 1000s /A /B /G /D /E /V /Z /H /Q 10,000s ABGDEVZHQ 562 = CXB hendeka, dodeka, treis kai deka, tettares kai deka, and so on. For instance, Th reatte (1980: 114) provides a number of examples from Attica where GI, ZI, HI, and QI appear for 13, 17, 18, and 19. Starting in the Roman period, the signs became more rigidly fi xed in highest-to-lowest order. Th ere was a mild taboo against the use of theta (Q) for 9 in some texts because it is the fi rst letter of θανατοσ ‘death’; circumlocutions were used instead, such as writing the number lexically, or as an additive combination of alphabetic numerals such as EΔ (5 + 4) or AH (1 + 8) (Smith 1926: 69; Th omas 1977). Because alphabetic numerals could easily be confused with written words, classical Greek alphabetic numerals were sometimes distinguished from the rest of the text with special signs, most commonly a horizontal stroke above the numeral- phrase, but occasionally with dots placed to either side of it. One of the problems in identifying earlier Greek alphabetic numerals is the lack of such marks, meaning that any single letter could be an alphabetic numeral or a non-numerical label. Even in later periods, numerals frequently appear without any indicator mark whatsoever (Th reatte 1980: 115). In most Greek monumental inscriptions, the only fractions used are acrophonic signs for fractions of diff erent monetary units (Tod 1950: 134). In mathematical and literary texts, an entirely diff erent system was used in which small accents or strokes placed above and to the right of a numeral indicated unit-fractions (Th omas 1962: 43). Special signs existed for 1/2 (<’ and U’) and 2/3 (ω’), in addition to standard unit-fractions (Th omas 1962: 45). From the second century ad onward, the requirement of using only unit-fractions was lifted, and fractions were expressed with both numerators and denominators using alphabetic numerals. Finally, a special system for fractions was used in astronomy, combining the alphabetic numerals with sexagesimal structures borrowed from the Babylonians. Th ere is no evidence for an early (eighth century bc or earlier) origin of the numerals, although prominent classicists such as Larfeld (1902–07) endorsed this 140 Numerical Notation theory, arguing for descent from the Phoenician alphabetic system, because the Greek alphabet was borrowed from a Phoenician ancestor and because many Se- mitic scripts have alphabetic numerals (Brunschwig and Lloyd 2000: 388). Yet no Semitic consonantal numerical notation systems existed before the second century bc, and they were based on Greek rather than the other way around (Gow 1883). None of the very earliest Greek alphabetic inscriptions contains numerals. Con- versely, however, the once-popular theory of a very late origin (late fourth or even third century bc) cannot be sustained in light of evidence from earlier periods (Gow 1883). Th e fi rst epigraphic evidence for alphabetic numerals comes from a vase, dating to around 575 bc, found at Corinth, which contains the inscription “SYM g,” which Johnston reads as “mixed batch of 7” (Johnston 1973: 186). Th ere is good evidence from Attica and Corinth for the system’s use on mercantile vases in the late sixth and early fi fth centuries bc (Hackl 1909). Yet it is unlikely that the numerals actually developed in either of these localities. Rather, the numerals probably developed in western Asia Minor, in the regions of Ionia and Caria, especially in the cities of Miletus5 and Halicarnassus, where several early instances of the numerals have been found (Heath 1921: 32–33). Another early example is found in the “tun- nel of Eupalinos” on the Ionian island of Samos, constructed around 550 bce, in which distances are noted using letters that are probably alphabetic numerals (as opposed to non-numerical or ordinal labels) (Verdan 2007: 12–13). All the early examples of the alphabetic numerals, even those found outside Asia Minor, are written using the Ionic script, which was used in Ionia, Caria, and various Ionic colonies throughout the Mediterranean. Th is refl ects the predominance of Ionia in regional and international commerce during the sixth and early fi fth centuries bc. While vau and qoppa were used phonetically in the early Ionic script, san was not, and so had lost its place between pi and qoppa in the numerical order (Jeff ery 1990: 327). Later, it was reincorporated into the script for the purpose of providing a twenty-seventh sign for the numerals to function, but was placed at the end of the system, with the numerical value 900. Th e principle of the Greek alphabetic numerals was borrowed directly from the Egyptian demotic numerals in the early sixth century bc, only using alphabetic signs as numeral-signs (Chrisomalis 2003; see also Zaslavsky 2003). Boyer (1944: 159) regarded the similarity between the two systems as indicative of a historical connection, but his paper was not primarily oriented toward such an argument. Because the alphabetic numerals were the fi rst to use phonetic signs as numeral- signs, there will be no paleographic similarity between the Greek numeral-signs

5 Miletus, from whence the adjective “Milesian,” was the most important Ionian city in Caria, the region of Asia Minor immediately to the south of Ionia proper. Alphabetic Systems 141 and any other system. Yet the alphabetic numerals are structurally similar to the demotic numerals. Th ey are both ciphered-additive, base-10 systems. While it has yet to be established whether the alphabetic numerals used multiplicative notation at an early date, both systems are multiplicative-additive above 10,000. Unlike the demotic numerals, the alphabetic system is also multiplicative for the thousands. Th e Greeks could simply have continued the series above 1000 using 10 through 90 and 100 through 900 (/I = 10,000; /K = 20,000; etc.), so the use of two-stage multiplication may be a clue to the alphabetic numerals’ history. It would be reasonable for the Greeks to adopt the multiplicative principle at the same level as in the demotic numerals, namely 10,000. However, it would not have been feasible to fi nd nine extra signs for the values 1000 through 9000. Consequently, the inventor(s) of the alphabetic numerals may have had the idea of using multiplication for the thousands values as well as the ten thousands. Th e only remaining problem is to explain why the Greeks, recognizing this irregularity, did not then abandon the higher multiplicative series. Furthermore, Greek arithmetical techniques for dealing with fractions strongly resemble the Egyptian unit-fraction (1/x) tradition of computation (Knorr 1982, Fowler 1999a). Both systems used unit-fractions formed by placing a small mark above numeral-signs to indicate the appropriate unit-fraction. As well, both used non-unit-fractions for specifi c values such as 1/2 and 2/3. Historians of mathematics are unanimous that the Greeks borrowed the unit-fraction technique from the Egyptians, and I see no reason to doubt that the Greek use of special signs for 1/2 and 2/3 is also a result of Egyptian infl uence. Turning to the historical evidence, the demotic numerals were the predominant ones in use in Egypt (especially Lower Egypt) in the early sixth century bc, when Greeks were just starting to encounter Egyptians in large numbers for the purposes of international trade. Most notable among the Greek traders in Egypt were Ionian colonists from Miletus, who set up an important emporion (port of trade) at Naukratis in the western Nile delta in the seventh century bc. Naukratis quickly became the central locus for cultural contact between Greece and Egypt, a position that it held until the Ptolemaic era. Inscriptions in the Ionic Greek script dating as early as 650 bc have been found at Naukratis (Heath 1921: 33). It should be noted, however, that no known inscriptions from Naukratis contain alphabetic numerals, and there are later (fourth-century bc) inscriptions with acrophonic numerals (Gardner 1888). Since the earliest examples of the alphabetic numerals are from containers for commercial goods, the context of the system’s development was probably in mercantile activity. Th e only alternative to the hypothesis of Egyptian origin is that the Ionians independently developed a ciphered-additive, decimal numerical notation system within a few decades of coming into contact with Egyptians in large numbers, 142 Numerical Notation founding a colony at Naukratis, and being exposed to the demotic numerals used throughout Lower Egypt. Th ese connections are too signifi cant to be coincidental. Th is should not be taken as a denial of the Greeks’ inventiveness, however, because the alphabetic numerals have several distinctive properties. Firstly, while some of the demotic numeral-signs use the cumulative principle, the alphabetic numerals use purely ciphered signs. Secondly, as mentioned earlier, the alphabetic numerals use the multiplicative principle for 1000 through 9000, obviating the need for nine more signs for those values, as one could simply write a hasta before a unit- sign. Finally, correlating numeral-signs with the ordered set of alphabetic signs meant that anyone who knew the order of the alphabet could determine the signs’ values as long as the episemons were taken into account. Th e often-mentioned “weakness” of the alphabetic numerals, that too many signs needed to be learned, is thus illusory. In learning to read and write, Western pupils must learn twenty- six alphabetic signs (in their proper order) plus ten digits in order, making thirty- six total signs in two separate series, while the ancient Greeks needed to learn only twenty-seven alphabetic signs and two auxiliary signs (/ and :), and needed only twenty-nine total signs in one series. After a period of Ionian cultural dominance between 575 and 475 bc, when alphabetic numerals were commonly found, alphabetic numerals are found only rarely in a period starting in 475 bc and lasting around 150 years (Johnston 1979: 27). During this period, the height of Greek civilization, Athens came to the fore- front as an Aegean power, while Ionia’s power waned after the Milesian-led Ionian revolt of 499 to 494 bc against Achaemenid Persia. Th e system did not disappear entirely; at Halicarnassus, there is evidence of its continued use (Keil 1894; Heath 1921: 31–33). A curious inscription from Athens (IG I2 760) from the middle of the fi fth century bc contains a long series of alphabetic numerals, written with Ionic letters (Tod 1950: 137). Th ere is, as well, good reason to believe that some of the graffi ti on amphorae from the Athenian Agora had weight and volume marks notated in alphabetic numerals (Lawall 2000). In general, however, acrophonic numerals were used in most Greek texts during this period. Th e resurgence of the alphabetic numerals in Greece around 325 bc corresponds precisely with the rise of the Ptolemies in Egypt. In this renewed period, some of the earliest instances of the numerals come from Egypt. Figure 5.1 depicts a portion of a Greco-Egyptian astronomical text including a calendar for the Saite nome (Hibeh Papyrus i 27) dating to around 300 bc, one of these very early instances (Grenfell and Hunt 1906, Fowler and Turner 1983). Ordinary (unmarked) Greek letters stand for whole numbers, while 1/x “unit-fractions” are represented by signs modifi ed with long oblique strokes. Th is is the earliest attested example of alphabetic numerals used in a scientifi c context. Coins dating to 266 bc indicating the regnal year of Ptolemy II Soter are, to my knowledge, the fi rst coins bearing Alphabetic Systems 143

Figure 5.1. Hibeh papyrus i 27, with Greek alphabetic numerals indistinguishable from letters, but with unit-fractions clearly distinguished through the addition of long upward slanting strokes. Source: Grenfell and Hunt 1906: Plate VIII. any ciphered-additive numerals (Tod 1950: 138). It is interesting that while these examples come from Egypt, we have no record of the alphabetic numerals’ use in Egypt during the interlude of the fi fth and fourth centuries bc. Th e evidence, at present, is simply too scanty to conclude what stimulus caused the rejuvenation of the alphabetic numerals. From the third century bc onward, the alphabetic numerals were preferred over the acrophonic numerals throughout most of the Greek-speaking world, with only Athens retaining the acrophonic system until around 50 bc (Th reatte 1980: 117). While the acrophonic numerals used in diff erent city-states varied quite widely, the alphabetic numerals had no regional variants. As such, they could be used as an eff ective instrument of cross-cultural communication and trade among diverse regions (Dow 1952: 23). Whereas Greece before Alexander was highly fragmented, rendering the development of a universal Greek numerical notation system unlikely, Alexandrine and especially Roman Greece provided a suitable environment for the development of a single pan-Hellenic notation. Th at the system was a very concise way to represent numbers, and that it relied on alphabetic symbols that had become invariant throughout Greece by this period in history, cannot have hurt this process. 144 Numerical Notation

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

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

Before the Hellenistic period, we have no evidence that the alphabetic numerals were used for arithmetic or mathematics. Early writers such as Herodotus instead used the acrophonic numerals to record the results of computations performed with a pebble-board or abacus (Lang 1957). Th is situation changed once the alphabetic numerals began to be used more widely. Most surviving Greek mathematical manuscripts use the alphabetic numerals, which were used by all Greek mathematicians beginning with Archimedes and Apollinius (third century bc). Users of alphabetic numerals had a wide variety of computational techniques available. Schärlig (2001) suggests that while the earliest abaci are most conducive to working with acrophonic numerals, certain types of abaci were well suited to working with alphabetic numerals also. Furthermore, the Greeks developed a complex system of computing on the fi ngers up to 9999, reminiscent of the alphabetic system’s structural shift above 10,000 (Williams and Williams 1995). To facilitate multiplication and division, which were diffi cult to undertake using the abacus or fi nger computing, the Greeks used multiplication tables written with alphabetic numerals (Dilke 1987: 16; Sesiano 2002). Finally, Greek arithmetic was sometimes done in a manner analogous to modern Western practice, by writing down numbers and manipulating numeral-signs directly; the primary diff erence between the Alphabetic Systems 145 two techniques is that in the Greek case, because the numerals are nonpositional, there was no need to line up numbers in columns (Smyly 1905). One would think, as Boyer (1944: 160) comments, that the adoption of the alphabetic numerals by such prominent Greek mathematicians would curb the criticism of modern scholars that the system was dysfunctional for mathematics. Yet many modern scholars denigrate the alphabetic numerals in comparison not only to ciphered-positional systems but also to cumulative-additive systems such as the acrophonic numerals (cf. Boyer 1944: 160–166). In particular, the need to learn many signs and the lack of resemblance between numeral-signs for common multiples of diff erent powers (e.g., 5, 50, and 500) are perceived as serious weak- nesses. In the only instance where a modern scholar actually attempted to learn and use the numerals, however, the system fared very well. Th e classicist Paul Tan- nery found that calculation with alphabetic numerals took little more eff ort than with Western numerals, with which Tannery, despite all his eff orts, surely had far greater experience and familiarity (Boyer 1944: 160–161). In any event, only a small fraction of the Greek texts containing alphabetic numerals serve mathematical functions. Like virtually all numerical notation systems, numerals are primarily for representation and only secondarily for computation. Th roughout their history, the Greek alphabetic numerals were used primarily in Greek-speaking areas or in regions under the control of Greek speakers. During its early history, the system was used in the eastern Mediterranean (particularly the Aegean), Ptolemaic Egypt, and Seleucid Persia. Cursive varieties of the alphabetic numerals on Greco-Egyptian papyri show a great deal of paleographic variability (Foat 1902, 1905). A large number of variant multiplicative signs for 1000 and 10,000 were used in these papyri, most notably Ô for 10,000, which was the normal form starting in the second century ad (Brashear 1985). While alphabetic numerals were not used for administration during the height of Roman power, from the fourth century ad onward, they were used as the primary numerals of administration, law, literature, and mathematics in the Eastern Roman Empire. Whenever and wherever the Greek alphabet was used in the Middle Ages, the alphabetic numerals followed. Additionally, the Greek alphabetic numerals were the most common system used in Arabic papyri for several centuries after the Islamic conquest for recording the results of fi nancial transactions (Grohmann 1952: 89). Most of the descendants of the Greek alphabetic numerals substituted the letters of their own scripts for the Greek signs. In the late second century bc, the Israelites developed the Hebrew alphabetic numerals along a Greek model. In the mid fourth century ad, the Goths developed alphabetic numerals along with their Greek-infl uenced script, while in regions of Africa under Greek infl uence, the Coptic script of Egypt and the Ge’ez script used in Ethiopia both developed alphabetic numerical notation systems based on a Greek model. In Armenia and 146 Numerical Notation

Georgia, right on the border of the Eastern Roman Empire, scripts and alphabetic numeral systems developed in the fi fth century ad at around the time they were Christianized. In the sixth century ad, the Syriac script abandoned an earlier cumulative-additive numerical notation system (Chapter 3) in favor of one based on the Greek. Around the same time, the Arabic abjad numerals used following the Islamic conquest of the Middle East were derived at least in part from the Greek alphabetic system. Th e Glagolitic and Cyrillic numerals, developed in Slavic regions in the late ninth century ad, under the auspices of the missionary-related script development of Cyril and Methodius, closely resemble their Greek ancestor. Finally, knowledge of the Greek alphabetic numerals in Western Europe led to the development of a set of Latin alphabetic numerals in the twelfth century ad. It is remotely possible that the Greek numerals are ancestral to the Brāhmī numerals (Chapter 6), which were used from the late fourth century bc onward in India, and which eventually gave rise to Western numerals. Th e Brāhmī numerical notation system is ciphered-additive and decimal, and used a variety of the multiplicative principle. Th e chronology of its invention, corresponding almost exactly with the Alexandrine conquests and journeys in India, is also suggestive. However, the Brāhmī system is more similar in structure and numeral-signs to the Egyptian demotic numerals than to the Greek alphabetic numerals. Th e large number of descendants of the Greek alphabetic numerals is due in part to their longevity. Th is cannot be a full explanation, however; other systems, such as the Egyptian hieroglyphs, were used over a much longer period, yet gener- ated few direct descendants. Other long-lived systems, such as the Roman numerals, spread very widely over large parts of the world due to Roman imperial power, but they were often accepted by colonized or subordinate societies unchanged, and did not replace indigenous systems entirely. Because the Greek system was alphabetic, cultures borrowing the principle of alphabetic numeration tended to modify the signs to fi t their own scripts (whether alphabets or consonantaries) rather than adopting the Greek alphabetic numerals directly, and also made minor structural changes to the system. In the early Middle Ages, when the Eastern Roman Empire’s fortunes were prosperous, the numerals were widely used throughout Greece, the Balkans, Egypt, the Levant, and Asia Minor, and were incorporated into the learning of all European mathematicians. For instance, they were known to the English scholar Bede, who described them in his De temporum ratione (Th e Reckoning of Time) in the early eighth century ad (Wallis 1999). Many Western European manuscripts include numerical glossaries, for instance, a tenth- to eleventh-century codex (Vatican Library, Codex Urbinas Latinus 290) that lists (among other information concerning non-Latin alphabets) the Greek lexical numerals, letter names, and alphabetic numerals along with their translations into Roman numerals Alphabetic Systems 147

(Derolez 1954: 106–109). Such tables would have aided medieval scribes in understanding texts, but the Greek alphabetic numerals were not actively used except in rare instances. For instance, in one ninth-century Latin manuscript, the Greek numerals 1 (A) to 21 (KA) stand for the letters A through X in a cryptogram to indicate that it was written by Irish writers at the Welsh court of Merfyn Frych (824–844) (Derolez 1954: 97–99). Such cryptographic uses confi rm rather than refute the obscurity of the system, however. In the Western European debate between the “abacists” and “algorithmicists,” the Greek alphabetic numerals, used by all the great mathematical minds of antiquity, did not rate a mention. Th e eventual fate of the Greek numerals was directly tied to the fortunes of the Byzantine Empire. Byzantine mathematicians and astronomers used alphabetic numerals exclusively until the twelfth century, and numerals such as regnal years were often stamped on Byzantine coins. By 1300, however, the geographical extent of the numerals’ use had become very limited, and mathematicians were using Western numerals under the infl uence of Arab learning in Spain and Italy. In the Byzantine Empire, mathematicians used Arabic positional numerals in marginal notes on Euclid’s Elements in the twelfth century (Wilson 1981). Th e fi rst major Byzantine mathematician to recommend the switch to the Arabic numerals was Maximus Planudes (c. 1260–1310) (Schub 1932). In some Byzantine astronomical texts, Arabic numerals were used to write year numbers while Greek alphabetic numerals were used for all other functions; even then, the Greek equivalent of the Arabic numeral-phrase was often written in the margin of the page (Neugebauer 1960: 5). In a few late Byzantine mathematical texts, combinations of Greek alphabetic numerals and Arabic positional numerals with a zero occurred, such as υνο (400 + 50 + 0) instead of υν for 450 (Neugebauer 1960: 5). Th e ultimate extension of this principle was the use by some writers of only the fi rst nine Greek letters (A through θ) to indicate the units, then adding a dot, a circle, or a special sign, Ч, to indicate a zero position (Schub 1932: 59; Menninger 1969: 273–274). In 1453, with the fall of Constantinople, the Greek numerals ceased to be used administratively. Nevertheless, they continued to be used thereafter for restricted purposes, such as paginating the introductions to scholarly texts and enumerating ordinal lists, just as the Roman numerals were used in Western Europe. Start- ing in the seventeenth century, the sign for 90 (qoppa), little used in alphabetic contexts, underwent a palaeographic alteration, so that a new, roughly Z-shaped sign served only as a numeral, while the older Q-shaped sign could serve both alphabetic and numerical functions, but was regarded as a diff erent sign (Everson 1998). Alphabetic numerals are still used today in many Greek Bibles to enumerate chapter and verse numbers, in certain legal contexts, and in pagination. Given the prestige associated with these functions, it is unlikely that the alphabetic numerals will cease to be used entirely. 148 Numerical Notation

Coptic Th e Coptic language is the last descendant of the Egyptian languages. It was written using the Greek alphabet until the fourth century ad, when a distinctive Coptic alphabet originated based on a Greek model, but using six additional characters taken from the demotic script to express uniquely Egyptian phonemes. Unlike the earlier Egyptian scripts, it is written from left to right and has signs for vowels. Th e adoption of the Coptic script was accompanied by the introduction of a numerical notation system based on the model of the Greek alphabetic numerals (Megally 1991, Messiha 1994). Th e Coptic numeral-signs are shown in Table 5.4. Th e numerals, like the script itself, were written from left to right, and were ciphered-additive and decimal. Th e numeral-signs closely resemble the Greek uncial signs used between the fourth and ninth centuries ad. In addition, as in the Greek alphabetic numerals, a horizontal stroke above the numeral-phrase indicates that it is a numeral rather than a word, and a slanted subscript stroke under a unit- sign (the Greek hasta) indicates multiplication by 1000 (Megally 1991: 1821). Th ere is no known sign or multiplier for 10,000 or higher values for these numerals. Th e classical age of Coptic lasted from the fourth to the tenth centuries ad, during which time the script and numerals were used extensively, surviving the seventh-century ad Muslim conquest of Egypt. Th ere may have been a geographical division in the frequency of their use, with northern Egyptian scribes using them frequently, while southern writers tended to write out numbers using lexical numerals (Till 1961: 80). While the Coptic numerals were generally used in formal manuscripts, the ordinary cursive Greek alphabetic numerals were used for calculation and administration, possibly because the Coptic numerals, being uncials without tails, were less practical for rapid writing (Megally 1991: 1821). It is unclear whether the Ethiopic numerals (used to write the Ge’ez language from the fourth century ad onward) were based directly on the Greek alphabetic numerals or derived through a Coptic intermediary. While the Ethiopic system is generally said to derive directly from Greek, the Coptic uncial letter-signs are similar enough to the Greek to render such a determination premature, and given the geographic proximity of Egypt and Ethiopia, this possibility cannot be dismissed outright. While the primary function of Coptic numerals has always been religious, in the context of the Coptic Church, their administrative and arithmetical functions should not be discounted. Despite the control of the population of Egypt by a suc- cession of foreign powers, and despite the decline of Coptic as a spoken language from the twelfth century onward, the use of Coptic numerals continued as late as the fourteenth century. Th ey are still used today in Coptic Christian liturgical texts for pagination and stichometry, although for most ordinary purposes, either West- ern or Arabic positional numerals are preferred by those familiar with Coptic. Alphabetic Systems 149

Table 5.4. Coptic numerals

123456789 1s 10s 100s 1000s

6085 =

ZIMĀM Numerals In Egypt in the tenth century ad, when the Coptic script was being replaced by Arabic for most administrative purposes, a diff erent Coptic cursive numerical notation system developed, known as “numerals of the Epakt” (named after the computation used to resolve the discrepancy between the solar and lunar calendars) (Messiha 1994: 26) or, more generally, as ḥurūf al-zimām ‘account book; register’ numerals. Often simply labeled “Coptic numerals” in modern scholarship, it seems prudent to distinguish them from their uncial antecedents, particularly since the zimām numerals are often found alongside Arabic or Hebrew texts, not ones in the Coptic language or the Coptic alphabet. Th eir long-standing use in a wide variety of contexts renders their importance far greater than the restricted- function Coptic uncials. While doubtless many users of the zimām numerals were Coptic Christians from Egypt, this was certainly not an ironclad rule, and Egyp- tian Jews and Muslims writing in Arabic script frequently used Coptic numerals instead of the Arabic abjad or Hebrew alphabetic numerals. Th e zimām system is shown in Table 5.5. Th is system is structurally identical to the classical Coptic system, but the numeral-signs are cursive minuscule letters rather than uncial ones. Many of these signs bear little or no resemblance to the classical signs. Some of them are probably taken from the signs of the Arabic abjad numerals, which I will describe later, while others are of indigenous development. Th is system also uses two stages of multiplication (at 1000 and 10,000), like the Greek alphabetic system. A horizontal stroke and two dots placed below a sign indicated multiplication by 10,000 (Sesiano 1989: 64). A fi fteenth-century multiplication table includes instructions on writing “numerals of the Epakt,” and indicates that this sign for 10,000 could be used in conjunction with any of the twenty-seven letters, thus allowing any number less than ten million to be expressed (Sesiano 1989: 54–55). It may be that the two-stage multiplicative principle at 1000 and 10,000 existed even in 150 Numerical Notation

Table 5.5. Zimām numerals

123456789 1s 10s

100s

1000s

7104 = the earlier uncial numerals, and that we simply have no paleographic evidence to confi rm this. Th e paleographic relation between the fi rst and second Coptic systems remains unclear; although some of the signs are related, more are not, and in fact many are modifi cations of Greek minuscule letters rather than Coptic uncials. eseTh numerals were often used in early bilingual Arabic-Coptic documents, suggesting that the Arabs were making concessions to local administrators. Th is situation is quite extraordinary, because many Egyptian Arabs by this time were employing the ciphered-positional Arabic numerals used today. Th is suggests that the advantages of ciphered-positional systems over ciphered-additive ones may not have been perceived to be important. In addition to Arabs, many users of the zimām system were Egyptian Jews who used the system alongside their own alphabetic numerals (Goitein 1967–88: II, 178). Th eir primary functions were commercial, including contracts, bills of sale, payments, and accounts. Th ere is no evidence for or against the position that numerals were manipulated for arithmetic; this surely would not have been done on permanent media, even perishable ones like papyrus. Although zimām numerals originated in commercial contexts and are rare in attested astronomical and mathematical texts, Lemay (1982: 384) has found one early text dating to ad 938 (MS Garullah Efendi 1508, Süleymanie Library, Istanbul) in which the numbers in the text are written lexically, the Arabic abjad numerals used in astronomical tables, and the zimām numerals used in pagination. King (1999: 76–78) discusses an astrolabe made in Cairo in ad 1282–83 with an Arabic inscription but zimām numerals. At this period, particularly in Egypt, the Coptic calendar was used for various purposes by Muslims, potentially explaining this artifact’s unusual blend of representational systems. Zimām numerals were also used very widely for pagination and foliation of Arabic texts written by adherents of all three Abrahamic faiths, throughout Egypt and more broadly throughout the Maghreb (Troupeau 1974). Mingana (1936: II, 183–184) reports that an Arabic Alphabetic Systems 151 letter of bequest from 1586 relating to a Syriac Christian manuscript (Mingana Syriac 617) is dated in Coptic numerals (presumably zimām rather than uncial), suggesting that even at this late date, their use was widespread and not restricted to Coptic Christians and Egyptian Muslims. Although Coptic began to become obsolescent as a language in the twelfth century, and Coptic Christians made the transition to the Arabic language and script, the zimām numerals continued to be used widely in economic documents otherwise written in Arabic, and for divinatory purposes. Th eir obscurity lent itself to cryptographic uses; a letter sent in 1139 from the Egyptian Jewish trader Halfōn ben Nethanel to the poet Isaac ibn Ezra uses zimām numerals as a code for Hebrew characters that in turn denote Arabic words (Goitein 1971: II, 303). Th eir opacity and rarity could also be useful in avoiding fraud, for instance, on a thirteenth-century check in which the amount was written in Arabic words and zimām numerals, presumably because their obscurity made it less likely that they would be altered fraudulently (Reif 2002: 26).6 In the fourteenth-century Muqadimmah of Ibn Khaldūn, he mentions the use of “zimâm ciphers, the same as those used for numerals by government offi cials and accountants in the contemporary Maghrib” alongside the Arabic abjad numerals and the ghubar positional numerals (Chapter 6) in the zā’irajah technique of numerical divination (1958: I, 239). Th e zimām numerals played a considerable role in the development of the so- called Fez numerals used in Spain and parts of North Africa from the twelfth century onward (see the following discussion). Both the Greek alphabetic numerals and the Arabic abjad numerals were also involved in the development of the Fez numerals, however, and the paleographic lines of descent among these systems are complex. Th e problem is made more complex by the fact that the later users of zimām numerals were otherwise writing Arabic texts, not Coptic ones. Moreover, the Fez numerals are fi rst attested in Spain, not in Africa. However, the similarities of some of the sign-forms for particular numerals suggest some relationship. Th e zimām numerals were used for several centuries, and survived the conquest of Egypt by the Ottoman Empire in 1517. Egyptian Coptic children in the sixteenth century learned Arabic before Coptic in kuttāb schools, and their practical education emphasized their assimilation to Ottoman practices, but they continued to learn and use zimām numerals for bookkeeping and accounting, thereby acquiring a reputation for competent and secure fi nancial practices associated with

6 Th e use of highly ligatured Roman numerals in late medieval Europe, and of special complex numeral-signs in Chinese banking, are additional examples of the practice of using unusual numerical systems to avoid fraud, and are somewhat paralleled by the modern requirement of writing out numerals lexically in addition to using Western numerals on checks. 152 Numerical Notation what was by this time a highly cryptic numerical code (Hamilton 2006: 38). Th ey ceased to be used only in the seventeenth century, after which time Arabic positional numerals were used for these functions (Ritter 1936; Messiha 1994: 26).

Ethiopic Th e Ethiopic script developed in the late third century ad, primarily on the model of the Minaeo-Sabaean consonantary used in South Arabia, but also infl uenced by the Greek and Coptic alphabets used to the north. It was used (and continues to be used) for writing various languages of Ethiopia, especially Ge’ez, the liturgical language of the Ethiopian church, and modern Amharic. While the earliest Ethiopic script has no signs for vowels, by the fourth century ad, the script was an alphasyllabary, in which each individual sign represents a consonant + vowel. Th e direction of writing is always left to right. While the Ethiopic script is based on South Arabian writing, its numerical notation is strictly Greek and Coptic in origin. Th e earliest Ethiopic numerical notation, such as that found on third- and fourth-century inscriptions from the kingdom of Aksum (on which the signs for 70 and 90 are not found) is shown in Table 5.6 (Drewes 1962; Ifrah 1998: 247). Th e earliest inscription with Ethiopic numerals is on the late third-century Anza stela erected by Bazat, negus (local king) of Agabo, in which the highest number expressed is 20,620 (loaves of bread) (Drewes 1962: 65–67). Th e modern Ethiopic script uses a system derived from these early inscriptions, structurally unchanged but slightly modifi ed, as shown in Table 5.7 (Fossey 1948: 99; Haile 1996: 574). Th e Ethiopic numeral-signs are not letters of the Ethiopic script, but rather are derived from the Greek or Coptic letters. Even though the signs have no non- numerical meaning in Ethiopic, the signs normally have marks both above and below them to indicate that their value is numerical. Th is practice was universal only from the fi fteenth century onward, and it is not found at all in the Aksum inscriptions (Ifrah 1998: 246–247). In addition to these signs, modern Amharic texts use a sign for 1000 ( ), which lacks marks above and below it (Bender et al. 1976: Table I). It may be that the minimal demand for writing very large numbers led to the abandonment of the higher multiplicative formations and the introduction of an indigenous 1000-sign. Th e system has a hybrid structure: it is ciphered-additive for the units and tens but multiplicative-additive for numbers over 100. For 10,000, a multiplicative sign consisting of two ligatured 100-signs was used, as shown in Table 5.7. Th us, because there are special signs for 100 and 10,000, the Ethiopic system has a mixed base of 10 and 100, with the base-10 formations governed by eighteen ciphered characters for 1 through 9 and 10 through 90 and the hundreds and ten thousands governed by multiplicative power-signs that combine with the ciphered signs. Alphabetic Systems 153

Table 5.6. Ethiopic numerals (Aksumite period)

123456789 1s 10s 100s 1000s etc. 10,000s etc.

20,620 =

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

Table 5.7. Ethiopic numerals (modern)

123456789 1s 10s 100s 1000s 10,000s

647,035 = (60 + 4) × 10,000 + 70 × 100 + 30 + 5 100,000,000 = (10,000 × 10,000) 154 Numerical Notation

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

Gothic Th e Gothic alphabet was developed around 350 ad by Wulfi la, a bishop who translated the Bible into his native language. Gothic was an East Germanic language Alphabetic Systems 155

Table 5.8. Gothic numerals

123456789 1s abcdefghi 10s jklmnopqr 100s stuvwxyz{ 665 = .xoe. spoken by the Germanic tribes who migrated throughout Europe in the latter years of the Western Roman Empire (Ebbinghaus 1996: 290). Th e script was alphabetic and written from left to right. Along with the script, an alphabetic numerical notation system was employed, as indicated in Table 5.8 (Braune and Ebbinghaus 1966: 10). Like the Greek alphabetic numerals, the Gothic numerals were usually distinguished from the rest of the text either through dots to either side of the numeral- phrase or by placing a horizontal stroke above the phrase (Braune and Ebbinghaus 1966: 10). Th e Gothic numerals never expressed quantities higher than 1000, and thus there is no evidence of the use of the multiplicative principle. Larger numbers were always written out in full (Menninger 1969: 260). Th e system is therefore ciphered-additive and decimal throughout. Th e numeral-signs are related to the Greek uncial letters. Of the episemons, f (6) was the sixth letter of the Gothic alphabet, and had the phonetic value [kw]. Th e other two episemons, qoppa r( ) and san ({), had no phonetic value and were simply used to fi ll out the full complement of twenty-seven signs using Greek models (Fairbanks and Magoun 1940: 319). Th e possibility has been considered, but now largely rejected, that the Gothic script owes its ancestry at least in part to either the Latin alphabet or the Germanic runes (Ebbinghaus 1996: 290–291). However, since neither of these scripts uses alphabetic numerals, the Gothic numerals must be of Greek origin. Th e Gothic alphabet is attested in only a limited set of documents, mostly translations of parts of the New Testament, but also in a small number of secular texts. Most numerals in Gothic texts indicate chapter and verse numbers in Bibles. Additionally, they were used within the text to indicate numerical values, while in Greek Bibles such numbers were always written out in full (Menninger 1969: 260). Th ere is no evidence that the Goths ever did arithmetic or mathematics using these numerals. Th e sign for 900 is only attested in the Codex Vindobonensis 795, an antiquarian text of the ninth century, by which time the script had already fallen out of use (Fairbanks and Magoun 1940, Ebbinghaus 1978). 156 Numerical Notation

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

Hebrew Alphabetic Th e earliest Hebrew scripts began to diverge from the earlier Phoenician consonantary in the ninth or tenth century bc. Th en as now, Hebrew consisted of twenty-two consonantal signs, written from right to left, and placed in a customary order. Early Hebrew inscriptions used a variant of the Egyptian hieratic numerals (Chapter 2). Somewhat later, particularly in the fi fth and fourth centuries bc, some Hebrew speakers used the Aramaic numerals (Chapter 3) for administrative and commercial purposes. Only at a much later date, probably in the second century bc, did a uniquely Hebrew set of numerals develop. Th is system is indicated in Table 5.9. Th e fi rst twenty-two signs, indicating 1 through 400, are the letters of the Has- monean Hebrew script circa 125 bc, at the time of the writing of the Dead Sea Scrolls and also when the numerals were fi rst used (Goerwitz 1996: 488). eTh system is decimal and ciphered-additive, and written from right to left. Values between 500 and 900 were represented using the sign for 400 in conjunction with one or more signs for the lower hundreds (i.e., 500 = 400 + 100, 600 = 400 + 200 ... 900 = 400 + 400 + 100). Th is structural irregularity exists because there are too few letters in the Hebrew consonantary to fi ll out the twenty-seven signs needed to extend the system to 900. Th e number 400 occupies a special structural role, but is not a base of the system, as its powers (16,000, 6,400,000, etc.) do not receive any special treatment. While the very earliest Hebrew inscriptions contain no signs for numbers above 1000, the need to do so quickly arose, as the numerals began to be used for dating on grave inscriptions using the Hebrew calendar. For multiples of 1000, a mark – either a small curved stroke to the left of a numeral-sign or two dots placed above it – could be used to indicate multiplication by 1000; thus, would signify 9000 and 90,000. Th is feature is similar to, but distinct from, the Greek alphabetic system, which adds a stroke above or below a numeral to indicate multiplication by 1000, but begins again at 10,000 by placing the multiplicand above the sign Z. Th us, while the Greek system could express any numeral up to ten million, as opposed to one million for the Hebrew numerals, the Hebrew system is arguably easier to use because it has only one value at which the multiplicative principle is employed. Alphabetic Systems 157

Table 5.9. Hebrew alphabetic numerals (Hasmonean)

123456789 1s 10s 100s 369 =

Th e Hebrew alphabetic numerals were derived from the Greek alphabetic numerals between 125 and 100 bc for use on coins inscribed with the Hasmonean script (Gandz 1933: 76; Millard 1995: 192). Th eories of a much earlier (ninth- to seventh-century bc) independent origin are no longer credible (Smith and Karpinski 1911: 33; Gandz 1933: 75–76; Schanzlin 1934; Zabilka 1968: 176–178). A bilingual ostracon from the Palestinian village of Khirbet el-Qôm and dating to 277 bc uses both Aramaic and Greek alphabetic numerals to denote a fi nancial transaction (Geraty 1975). Although it is not a Hebrew text, the village was in Judah and hundreds of contemporary Hebrew inscriptions have been found there. If the Hebrew alphabetic numerals had been in use at that time, they likely would have been used instead of the Aramaic numerals in a situation where the Greek inscription a few lines below used a similar system. Th e fi rst safely dated instance in which the use of Hebrew alphabetic numerals is certain is on coins from the reign of the Hasmonean king Alexander Janneus (103 to 76 bc), some of which were stamped with Greek script and alphabetic numerals, others with the Hebrew script and alphabetic numerals, in both cases using alphabetic signs (Naveh 1968). Avigad (1975) suggests that a clay seal reading “Jonathan high priest Jerusalem M)” might refer to the fortieth (m = 40) year of the Hasmonean kings, and thus dates it to 103 bc, early in the reign of Alexander Janneus, whose Hebrew name was Jonathan. Th at these early Hebrew alphabetic numerals would both be found in the context of the same man is strong circumstantial evidence. Despite the special formation of Hebrew numerals from 500 to 900, the similarities between the Greek and Hebrew numerical notation systems are striking. Th e two systems share not only a similar structure (decimal and ciphered-additive) but also the alphabetic principle for forming the numeral-signs. Th e notion that the Hebrew numerals were independently developed can no longer seriously be sustained, despite the agnostic attitude of some scholars, including Ifrah (1998: 239). Th at Hasmonean coins were struck in both languages and using both systems provides specifi c contextual evidence that the Hasmonean kings adopted the technique 158 Numerical Notation from the Greeks. Th e cultural infl uence of the Ptolemaic and Seleucid kingdoms in the Levant at this time was enormous; Greek alphabetic numerals were used on coins from the Phoenician cities of Sidon, Tyre, Byblos, and Akon from the mid third century bc onward (Millard 1995: 193). At the same time, the Hebrew use of 400 as a “stepping-stone” for representing the higher hundreds is an important innovation, as it did not require that Hebrew speakers learn and adopt additional nonphonetic signs. Despite the development of the Hebrew numerals, most Semitic peoples of the ancient Levant (Nabataeans, Aramaeans, Palmyrans) never used alphabetic numerals, but continued to use their own hybrid cumulative-additive/multiplicative- additive systems (Chapter 3), which coexisted with the Hebrew alphabetic system for several centuries, and were replaced only over a long period. Until the seventh century ad, inscriptions on Jewish graves throughout the Mediterranean region were often written, not with the Hebrew numerals, but rather in the Greek alphabetic numerals (Ifrah 1998: 238–239). Gradually, the Hebrew numerals replaced the Greek and Levantine systems, until they were fi rmly established as a distinctive system peculiar to the Jewish populations of Europe, North Africa, and the Levant. Also, in the sixth century ad, the Hebrew numerals were partly or wholly used by the creators of the Syriac alphabetic numerals (to be discussed), which are also ciphered-additive and decimal and have the same break at 400 as the Hebrew system. Th e modern Hebrew alphabetic numerical notation system has the same structure as the ancient system, but uses modern script-signs, as shown in Table 5.10. Around the beginning of the tenth century, fi ve additional Hebrew characters, the signs used for kof, mem, nun, pe, and tsade in word-fi nal position, were sometimes used to complete the sign set for 500 through 900. Th ese signs occur in some Masoretic commentaries on the Old Testament, but were never the regular forms used (Gandz 1933: 96–102). Th e older formations using additive combinations of hundreds-signs are the usual means of representation. Th e newer forms probably were not widely accepted because the word-fi nality of these signs is inconsistent with the principle of the numerical notation system that the highest values should come fi rst in a numeral-phrase (Gandz 1933: 98). To put a word-fi nal letter for 500 through 900 at the head of a numeral-phrase would be inconsistent with its original purpose; to put it at the end of the numeral-phrase would be inconsistent with the rule of decreasing ordering of the powers. Also in the tenth century, Hebrew scholars became aware of positional numerals, and occasionally experimented using combinations of the alphabetic numerals and the positional principle. In a Masoretic poem by Saadia Gaon (ad 882–942), numbers are written quasi-positionally in two columns; the rightmost represents the thousands, while the leftmost column represents the ones, so that 42,377 was Alphabetic Systems 159

Table 5.10. Hebrew alphabetic numerals (modern)

123456789 1s 10s 100s OR written as , i.e., (7 + 70 + 300) + ((2 + 40) × 1000) (Gandz 1970: 487–488). Some medieval Hebrew writers simply used the fi rst nine alphabetic symbols in place of the ordinary Western or Arabic signs, supplementing them with a circle for zero, which made the system ciphered-positional (Gandz 1933: 110). Th is practice is fi rst attested in the mid twelfth-century Sefer ha-Mispar of Abraham ibn Ezra (Burnett 2000c: 14). Th e ordinary Hebrew numerals were always used in the regular text of such works, with the positional variants used only for mathematical expressions (Schub 1932). Th is technique was not commonly used, however, and later Hebrew mathematicians and astronomers simply used Western or Arabic numerals. One of the more important functions for which the Hebrew numerals have been used historically is gematria, the art of number-magic (Ifrah 1998: 250–256). Because every letter of the Hebrew script has a numerical value, every Hebrew word has a numerical value equal to the sum of its letters’ values. Among medieval and early modern scholars, this practice was commonly employed for interpreting passages from the Talmud and the Midrash and for fi nding symbolic associations among words that share the same numerical value. For instance, two of the terms associated with the Messiah, shema ‘seed’ and menakhem ‘consoler’, have the same numerical value (8 + 40 + 90 = 40 + 8 +50 + 40 = 138). A related practice is the construction of chronograms, in which a specifi ed set of words has a numerical value equal to the date of an event (e.g., a person’s death) to which the words refer. Because these practices (also used with the Arabic abjad, described below) can only be done with a system for correlating phonetic signs with numerical values, they probably contributed to the retention of the alphabetic numerical notation systems long after ciphered-positional systems had been adopted. While the Western numerals are used in modern Israel for most purposes, the alphabetic system is regularly used for dates using the traditional Jewish calendar, especially in religious texts and on graves. In 1999, the Israeli Supreme Court ruled that gravestones in orthodox Jewish cemeteries could henceforth record numbers using Western numerals rather than the Hebrew alphabetic system, whose use had previously been mandatory in that context (Copans 1999). It is unclear whether this ruling will have any long-term eff ect on the use of the Hebrew numerals for 160 Numerical Notation dating. However, the fact that such a ruling needed to be made at all shows the continued health and vibrancy of Hebrew alphabetic numerals, albeit in a limited set of religious contexts. Th e Hebrew alphabetic numerical notation system is now over two millennia old, and is one of the oldest systems in continuous and regular use.

Syriac Alphabetic In Chapter 3, I described the numerals used alongside the Old Syriac script between about 50 and 500 ad. Around the fi fth century ad, this script diverged into two forms: an eastern variety, Nestorian, used by the Christians of Persia, and a western variety, the Serto script used by the Jacobite Christians of Syria (Dan- iels 1996). Th is split was precipitated by the expulsion of the Nestorian Chris- tians from the Byzantine city of Edessa and their subsequent migration into the Sassanian Empire (Duval 1881: vii). Soon thereafter, both the Nestorian and Serto scripts began to use an alphabetic numeral system akin to those used elsewhere in the Middle East. Th e basic signs of this system (sign-forms are those of the Serto script) are shown in Table 5.11. Th e system is decimal and ciphered-additive, and written from right to left. As in many alphabetic numeral systems, numerals were sometimes distinguished from letters by placing a horizontal stroke above a numeral-phrase, but often no special mark was present. Like the Hebrew alphabetic numerals, values from 500 to 900 were usually written using the signs for the lower hundreds with the sign for 400 in various additive combinations. Alternately, the upper hundreds were occasionally expressed multiplicatively, by placing a small dot above the signs for 50 through 90 to indicate multiplication by 10 (Duval 1881: 15; Noldeke 1904: 316–317). For values above 1000, a slanted stroke placed beneath a unit-sign indicates multiplication by 1000, while a horizontal stroke placed beneath a sign indicates multiplication by 10,000 (Hoff man 1858: 6–7). In this way, any number below ten million could be expressed. Even higher values could be expressed by placing two small strokes beneath a sign to indicate multiplication by ten million, and placing one small stroke above and one small stroke beneath a sign indicated multiplication by ten billion; however, these techniques were used extremely rarely and not followed uniformly (Duval 1881: 15–16). As in several other alphabetic systems, placing an oblique stroke above a numeral-sign indicated the appropriate (1/x) unit fraction (Duval 1881: 16). Th e Syriac alphabetic numerical notation system was probably invented in the sixth century ad, and gradually replaced the Old Syriac system over the next two hundred years (Duval 1881: 15). Independent invention of this system can be ruled out, given its strong similarity to others used in the region. Two likely possibilities Alphabetic Systems 161

Table 5.11. Syriac alphabetic numerals

123456789 1s abcdefghi 10s jklmnopqr 100s s t u v sv tv uv vv svv OR w\ x y \z , 1000s /a /b /c /d /e /f /g /h /i 10,000s -a -b -c -d -e -f -g -h -i 369 = iou are that it was modeled on the Greek alphabetic numerals prevalent in the Byzan- tine Empire or else on the Hebrew alphabetic numerals. Th e Hebrew and Syriac numerals are the only two systems in which 400 occupies a special structuring role, in that the higher hundreds are expressed using additive combinations of 400 and the lower hundreds. Th e ordering of the Syriac numerals follows the letter-order shared by the Syriac and Hebrew scripts, not that of Greek. Finally, the Syriac system was written from right to left like the Hebrew system, but unlike the Greek. On the other hand, the Syrians were closely affi liated with Eastern (Greek) Christianity, and many Syrians lived under Byzantine rule. Th e hypothesis of Greek ancestry is supported by the shared feature of the two systems that the multiplicative principle was used at two diff erent powers of the base, 1000 and 10,000, whereas the Hebrew numerals did so only at 1000. Th e inventor(s) of the Syriac numerals probably would have been familiar with both the Greek and Hebrew numerals, so it is possible that features of both systems were blended in the Syriac system. Th e Syriac scripts were never used as the offi cial script of any polity, and thus Syriac numerals are rarely found on stone monuments or coins. However, their use in religious texts is extremely prevalent from the time of their invention to the present day, a situation aff orded by the relative separation of the Jacobites and Nestorians from both Roman and Greek Christianity. Most Syriac liturgical texts are dated and paginated using the numerals, making it easy to examine paleographic changes. Th ere is no evidence that the Syriac numerals were ever used in mathematical texts, nor, in contrast to the Hebrew numerals, were they used for letter-magic or numerology. Although the heyday of the Syriac scripts came and went before ad 1000, both the Nestorian and Serto scripts survive to this day, the former in Iraq, Turkey, and Iran among a small number of Nestorian Christians, the latter in Lebanon 162 Numerical Notation among the Maronite Christians of that country. Both scripts retain their distinctive numerical notation systems, albeit restricted in the contexts of their use to the same liturgical functions for which they have been employed for nearly 1,500 years. Arabic or Western numerals serve most other purposes. Nevertheless, there is no reason to think that the Syriac numerals are about to disappear, particularly given the special status accorded to Maronite Christianity in Lebanon’s 1990 con- stitution.

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

7 Th e fi rst three letters in the modern Arabic script are ‘alif, ba, and ta. Alphabetic Systems 163

Table 5.12. Arabic abjad numerals

123456789 1s abcdefghi 10s jklmnopqr 100s stuvwxyz: 1000 ; 7642 = bmx;g (7 × 1000 + 600 + 40 + 2) and Mesopotamia. Under Byzantine rule, this region used the Greek alphabetic numerals for administrative and commercial functions; furthermore, both the Syriac and Hebrew alphabetic numerals were used in their respective scripts. Th e abjad numeral-signs have exactly the same order as the corresponding Syriac and Hebrew characters up to 400 (above which point the other two systems are structurally irregular). Th e Arabic script adopted a diff erent letter-order starting in the eighth century ad, but the older order was retained for the numerals. Th e Greek system follows an order similar to that of the Syriac and Hebrew numerals up to 80, but diverges thereafter by putting sampi (equivalent to the Hebrew tsade) at the end of the system, rather than in the middle. Th us, either the Hebrew or Syriac systems inspired the development of the Arabic abjad numerals (Guitel 1975: 276–278; Ifrah 1998: 243). Table 5.14 illustrates how the Arabic order is directly parallel to the Hebrew and Syriac, while the Greek numerals diverge from them starting at 90.

Table 5.13. Arabic abjad numerals (Eastern vs. Maghreb)

Sign Letter-name Eastern Value Maghreb Value o sin 60 300 r sad 90 60 u shin 300 1000 z dad 800 90

: ẓa 900 800 ; ghayn 1000 900 164 Numerical Notation

Table 5.14. Arabic, Greek, Hebrew, and Syriac numeral-signs and letters

Arabic Greek Hebrew Syriac 60 o sin X xi samekh o semkat 70 p ayin O omicron ayin p ‘e 80 q fa P pi pe q pe 90 r sad < qoppa tsade r sode 100 s qaf R rho quf s quf 200 t ra S sigma resh t rish 300 u shin T tau shin u shin 400 v ta U upsilon tav v taw 500 w tha F phi w 600 x kha C chi x 700 y dhal Y psi y 800 z dad W omega z 900 : ẓa >\ sampi ,

However, the Arabic abjad was rarely used in the same texts as either the He- brew or the Syriac numerals. In contrast, Greek alphabetic numerals are found in Arabic documents from the seventh to the ninth centuries ad. In ad 706, Caliph Walid I dictated that, although his Greek fi nancial administrators in Damascus should no longer use the Greek alphabet, they could continue to use the Greek alphabetical numerals (Menninger 1969: 410). In an eighth-century ad Arabic tax record, numbers are expressed in both Greek alphabetic numerals and abjad numerals (Cajori 1928: 29). Because many of the regions conquered by the Arabs – even those such as Syria and Palestine in which the Syriac or Hebrew numerals were found – were under Greek rule, Greek numerals were the normal system used for administration, on coins, and in inscriptions. It is unlikely that the Greek system played no role in the development of the abjad numerals. Th e phonetic values of the fi nal six Arabic characters do not correspond to the Greek, however. Furthermore, the Greek system has only twenty-seven rather than twenty-eight signs (lacking a sign for 1000). While the Arabs were aware that the Greek system had signs for the higher hundreds, and may thus have attached Alphabetic Systems 165 numerical values to the remaining signs in their own script, the use of a special sign for 1000 is unique to the Arabic system among all four systems under consideration. More likely, the Arabic system was based on the Semitic letter-order but employed the structural advantages of a system, such as the Greek, with a full complement of numeral-signs. Th is feature would have been particularly important, since the administrative needs of the new Islamic caliphate were growing rapidly. By the late eighth century ad, the Arabic abjad numerals had spread throughout the Middle East and into the Maghreb, including southern Spain. Th ey were used on administrative documents, in literary and scientifi c texts, on dated coins, and on monuments, though not for the most part on ephemeral media such as ostraca. In areas in which an existing administrative apparatus was retained from the Byzantine Empire (such as Egypt), the Greek and Coptic alphabetic numerals were used much more frequently on administrative and fi nancial documents than were the abjad numerals (Grohmann 1952: 89). While the systems are structurally similar to the abjad numerals, it would have taken some eff ort for Arabic writers to learn an entirely new set of twenty-seven signs, so its failure to be used more widely is somewhat surprising. Issues of identity and ethnicity may have played a signifi cant role in determining the scope of their use. For instance, Ifrah describes a ninth-century ad Christian manuscript written in Arabic but in which the verses are numbered in Greek (Ifrah 1998: 243). In this case, the writer was a Christian who associated himself with Greek Christianity through the alphabetic numerals even though he wrote in the Arabic language and script. Abjad numerals were rarely if ever used directly for arithmetical calculation. Instead, a system of fi nger computing borrowed from the Byzantine Greeks, and known as h. isab al-’uqūd (fi nger-joint arithmetic), was employed (Saidan 1974: 367–368), as well as computation on dust-covered boards. Despite their use over a wide area, the abjad numerals did not give rise to a large number of descendants. In part, this must be due to the remarkable stability of the Arabic script itself. Th e Coptic zimām numerals used in Egypt from the tenth century ad onward are an interesting blend of Greek and Arabic infl uences, although relatively few of the numeral-signs are evidently based on those of the abjad, and it is unclear to what extent the Arabic abjad aff ected the structure of the system. A similar situation probably arose in Morocco, where “Fez numerals,” incorporating elements of the Arabic abjad as well as Greco-Coptic alphabetic numerals, were used until very recently (Colin 1933). Finally, like Greek and He- brew scholars, Arabic astronomers used a very unusual system for writing fractions, which combines a ciphered-additive system with a base-60 (sexagesimal) positional notation. Shortly after they were invented, the abjad numerals began to be supplanted by the well-known ciphered-positional Arabic system (Chapter 6) borrowed from 166 Numerical Notation the Hindus, and known in Arabic as al-h. isāb al-hindī (Indian arithmetic). Th e Islamic conquest of enormous territories to the east brought the Arabic and Indian spheres of infl uence into close contact by the mid seventh century. As Islam spread eastward throughout the eighth century ad as far as the Indus River, the Indian style of numeration began to diff use westward and supplant the Arabic abjad, which itself was still a novelty in western regions such as North Africa. Th is replacement was hastened by the arrival in 773 ad of Hindu astronomers and astronomical knowledge at the court of Caliph al-Mansur in Baghdad, which formed the basis for the early ninth-century ad writings of the famed Arab mathematician al-Khwārizmī, who popularized the Arabic positional system (Menninger 1969: 410–411). By the late ninth century, the positional system was being used in administrative and fi nancial documents, and by the late tenth century, on inscriptions (Grohmann 1952: 89). Th e latest Arabic papyrus in which abjad numerals express a date is from 517 ah, or 1123/4 ad (Destombes 1987: 131). Despite the introduction of positional notation, abjad numerals survived in a variety of contexts. Astronomers continued to use ordinary abjad numerals much later than other writers, probably because they had also adapted them for use in the quasi-positional sexagesimal fractions to be described later. Until the thirteenth century all astronomical and astrological texts used abjad numerals alone, and for several centuries thereafter, abjad and positional numerals were often used side by side in any given text (Lemay 1982: 385–386). Abjad numerals were commonly used on most Arabic astrolabes (both for marking gradations on the instrument and for recording dates of construction) until the sixteenth century (Destombes 1987). Mauritanian mathematicians did not adopt Arabic positional numerals until the end of the eighteenth century (Rebstock 1990). Th at positional numerals would be adopted more rapidly in nontechnical manuscripts than in scientifi c ones is seemingly paradoxical, but only because it is customarily assumed that positional numerals would be accepted owing to their effi ciency for arithmetic. In fact, however, Arabic arithmetic was largely conducted using dust-boards, fl at surfaces covered with a layer of fi ne dust on which calculations could be made using any numerical system, but of course no record of this would survive; in contrast, permanent records were often made in more traditional and authoritative numerical systems like the abjad numerals (Lemay 1982: 383–384). We do not know to what extent pre-modern Arabic astronomers may have performed computations using Arabic positional numerals. Th e abjad numerals survive to the present for very limited cryptographic, literary, and magical functions, such as chronograms (described earlier for Hebrew numerals), known in Arabic as h. isāb al-djummal (Colin 1971: 468). Chronograms using the abjad numerals were common throughout the Middle Ages, particularly in Persia and Islamized parts of India (Ahmad 1973). Babajanov and Szuppe Alphabetic Systems 167

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

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

8 Curiously, one of these systems is known as el-Yunani (Ionian!), suggesting that its users were aware of the Greek origin of such notations. 168 Numerical Notation

Table 5.15. Astronomical numerals (Greek)

123456789 1s abg d e v z hq 10s iklmn /afie k ie 1515 + 20 (× 1/60) + 15 (× 1/3600) = 1515.3375 mathematical astronomer to use sexagesimal notation was Hypsicles in the second century bc (Fowler 1999). Th e fi rst major text in which this new system appeared was the Syntaxis of Ptolemy, written in the second century ad (Heath 1921: 44–45). In place of the cumulative Babylonian signs in each position, fourteen of the Greek alphabetic numerals were used (the units 1 through 9 and the decades 10 through 50) to write any number from 1 through 59, as shown in Table 5.15. Un- like the Babylonian system, however, the Greek sexagesimal system was never used for expressing integers. Numbers greater than 60 were always written with the ordinary alphabetic numerals. Th e sexagesimal numerals correspond to Greek astronomy, in which, as today, the circle is divided into 360 degrees, each degree into 60 minutes, each minute into 60 seconds, and so on. Th us, in Th eon of Alexandria’s (fourth-century ad) commentary on Ptolemy’s Syntaxis, the numeral-phrase /afie k ie expresses 1515 (/afie) degrees, 20 (k) minutes, and 15 (ie) seconds (Th omas 1962: 50–51). Th e degrees value uses the decimal alphabetic numerals,9 while the latter two positions are sexagesimal. Sexagesimal fractions could be used to express any fractional value; successive positions represent 1/60, 1/602, 1/603, and so on, to express as small a value as desired. Within each sexagesimal position, decimal ciphered-additive numeral-phrases represent the value from 1 to 59. However, the primary base of the system – the one involved in the positional aspect of the system – is 60; thus, 10 (the base of the ordinary alphabetic numerals) becomes the sub-base of the sexagesimal fractions. Th e system’s base and sub-base thus follow the Babylonian system, but the intraexponential principle used in forming the signs of each position is ciphered rather than cumulative. Th is system is thus both ciphered and positional, but it is not identical to ordinary ciphered-positional systems such as the Western numerals. Rather, because it has a sub-base, it is intraexponentially ciphered- additive rather than simply ciphered. It is thus a (ciphered-additive)-positional system.

9 If this value were expressed sexagesimally, we would expect it to be written as ke ie, or (25 × 60) + 15. Alphabetic Systems 169

Th e astronomical numerals used a special sign as a placeholder to indicate an empty position in order to avoid ambiguity. In some late (fourth- to fi rst-century bc) Babylonian texts, a placeholder sign that served some of the functions of zero was used (see Chapter 7). Greek astronomers from the fi rst century ad onward took up this practice, originally using a sign (, which in later manuscripts was written as ) (Irani 1955; Jones 1999: 61–62). Th e latter sign is sometimes held to represent omicron, the fi rst letter of the Greek word ouden, ‘nothing’, with a stroke added above to distinguish it from the appropriate letter (Ifrah 1998: 549). Yet it is unlikely that the Greeks would have chosen a sign that already had a numerical value (o\= omicron = 70) in the alphabetic system (Neugebauer 1957: 14; Jones 1999: 61). Th e zero-sign was probably a paleographic outgrowth of the earlier form, which was, like the Babylonian placeholder sign, purely ideographic. As far as can be discerned, however, it is completely unrelated to the later Hindu zero, which emerged in the sixth century ad in India. Th e principle of sexagesimal fractions could, in theory, be combined with any alphabetic numeral system, replacing the Greek signs for 1 through 59 with those of any other system. Th e Arabs, who inherited the bulk of Greek astronomical knowledge when they took control of Mesopotamia in the mid seventh century ad, used the signs of the Arabic abjad rather than the Greek alphabetic numeral-signs. Unlike the abjad numerals, the Arabic sexagesimal fractions are written from left to right, following the Greek practice, and the Arabic symbol for zero (), or later () is derived from the Greek symbol (Irani 1955). Similarly, medieval Hebrew astronomers adopted sexagesimal fractions, as in the fourteenth-century astronomical writings of Levi Ben Gerson (Ifrah 1998: 158). From the Arabic translations of Greek astronomical texts, sexagesimal positional numeration reached the Byzantine Greeks and, by ad 1000, Western Europe (Neugebauer 1960, Berggren 2002). Medieval European astronomers used sexagesimal notation for fractions thereafter, and in the Alphonsine tables edited in Paris in 1327, even whole numbers were written sexagesimally, the only case I know of where this was done (Berggren 2002: 364). Medieval Ethiopian astronomers used sexagesimal fractions in their own computations and in translations of Greek and Arabic documents (Neugebauer 1979). Th e use of the Greek variant waned after the end of the Byzantine Empire in the mid fi fteenth century, after which Western numerals were used for most purposes. Th e Arabic version of the system survived even longer; Irani (1955: 3) lists many texts from the sixteenth and seventeenth centuries and one from as late as 1788, although I suspect that this latter text is delib- erately archaic. Astronomical numerals often occur in the same texts as pure decimal systems (either ciphered-additive ones such as alphabetic numerals, or ciphered- positional ones such as Arabic or Western positional numerals). In eff ect, sexagesimal notation is not so much a distinct numerical notation system as it is a technique 170 Numerical Notation that can be used with any numerical notation system as a means of combining numeral-phrases for 1 through 59 into a positional notation system for fractions. Th e sexagesimal fractions were used only by astronomers and mathematicians working with astronomical problems and writing in manuscripts. Even so, when writing non-astronomical material, or when paginating and dating astronomical texts, they used other numerical notation systems. Yet, on astronomical instruments and in astronomical manuscripts, sexagesimal notation alongside alphabetic numerals was employed continuously for well over a millennium. Th at sexagesimal notation was used solely for astronomy suggests that the demands of the discipline led to its retention. Th e division of the circle into 360 degrees (with subdivisions of 60 minutes per degree and 60 seconds per minute) is very useful, since 60 has a large number of divisors.10 Faddegon (1932) showed that this feature enables quick and easy multiplication and division using sexagesimal fractions. Th e utility of sexagesimal fractions must therefore be evaluated in relation to computations involving this specifi c metrological system. Th e tenth-century Persian mathematician al-Biruni reported, however, that because it was inconvenient to multiply using a 60 × 60 multiplication table, Islamic astronomers would convert sexagesimal fractions into decimal numbers, multiply them, and then convert them back into sexagesimal notation (Berggren 2002: 362). Although sexagesimal fractions are no longer used, modern astronomers still use the sexagesimal division of the circle, and anyone who can read a digital clock uses a kind of sexagesimal numeration. While we no longer mix additive and positional principles in notating time and angles, astronomers continue to restrict themselves to values under 60 for the division of the sky into segments, just as everyone is able to realize that thirty minutes pass between 1:50 and 2:20. Th ese vestiges come to us, via Greek and Arabic sexagesimal fractions, from the Babylonian custom of numbering by 60. In this way, a peculiar custom of numeration, useful for astronomy but not much else, infl uenced how humans perceive and structure time throughout most of the world today. However, these do not represent a sexagesimal numerical notation system, but simply a sexagesimal division of various metrological units (for angles and for time) that are then represented with decimal numerals. Th e notation “11:05” does not mean 665 minutes (11 × 60 + 5), but simply 11 hours and 5 minutes. Th at we continue to measure time and angles in this way is a fascinating issue in the history of astronomy and of timekeeping.

10 In number theory, 60 is a “highly composite” number, defi ned as a natural number that has more divisors than all the numbers below it (Wells 1986: 127–128). Th e number 60 has twelve divisors: {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}. Alphabetic Systems 171

Table 5.16. Fez numerals

123456789 1s a A b B c d D e f g G h H i I 10s j J k l L m n N o p q Q r 100s s t T u v V w W x X y z , < 766 = foy

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

11 See also Sanchez Perez (1935) and Guergour (1997) for diff erent paleographic variants of this system. 172 Numerical Notation

Table 5.17. Arabic abjad, Greek, zimām, and Fez numerals

Arabic Greek Zimām Fez Numerals Arabic Greek Zimām Fez Numerals 1 a A a A 60 r x o 2 b b\ b B 70 p o p 3 c G c 80 q p q Q 4 d D d D 90 z < r 5 e E e 100 s r s 6 f V f 200 t s t T 7 g z\ g G 300 o t u 8 h E h H 400 v u v V 9 i Q i I 500 w f w W 10 j I j J 600 x c x X 20 k k k 700 y y y 30 l l l L 800 : w z 40 m m m 900 ; >\ , < 50 n n n N

All four systems are written cursively and have an enormous amount of variation. It is possible that the Mozarabs’ numerals are paleographic variants of the Greek alphabetic numerals, and thus came to the Arabs of Spain via direct diffusion from the Byzantines (Levi Della Vida 1934: 283). Th e attribution of these signs as rūmī (Byzantine) would tend to confi rm this origin. However, many of the numeral-signs bear no resemblance to the Greek alphabet. Colin suggests, rather, that the Fez numerals (and their Spanish antecedent) were borrowed, not directly from the Greek alphabetic numerals used in the Byzantine Empire, but by way of the Egypto-Coptic zimām numerals (Colin 1933: 213). Th e paleographic similarities between several of the Coptic and Fez numeral-signs (e.g., 8, 80, and 500) suggest that some connection must exist between the two. Yet while the Fez numerals are multiplicative at only one level (1000), both of these candidates for its origin are multiplicative at both 1000 and 10,000. An alternate ancestor is the Arabic abjad system used for number-magic and astronomy in the Maghreb at that time. Some of the paleographic resemblances between the Fez numerals and the zimām numerals (e.g., the signs for 7, 30, and 90) can be explained by both systems’ connection to the appropriate letters of the Arabic abjad. Furthermore, unlike the other two Alphabetic Systems 173 systems, the Arabic abjad is multiplicative only in combination with 1000. Most likely, the Fez numerals are an unusual blend of the Greek, zimām, and Arabic alphabetic systems adopted among a very unusual group of users, highly educated Arabized Christians living in Muslim-dominated southern Spain. Th e Fez numerals did not last long in Spain; I know of no texts from the fourteenth century or later in which they are used. Th ey were described by the Moroccan mathematician Ibn al-Banna in the early fourteenth century, and later by the great historian Ibn Khaldun in the late fourteenth century, both of whom lived and worked at Fez (Colin 1933: 206; Guergour 1997: 68). Th eir use in Morocco began in earnest only in the sixteenth century, however, after the expulsion of the Moors from Spain in 1492. Th ey were used frequently throughout the sixteenth and seventeenth centuries in accounting and other commercial contexts, after which time they began to be replaced by the Arabic positional numerals. While they were known to scholars, their use was always mercantile and legal, never mathematical or astronomical (Guergour 1997: 74). Th e Fez numerals were still used in the early twentieth century for indicating monetary values in wills and in related legal documents (Colin 1933, Sanchez Perez 1935). Because the meaning of the numerals was known to only a few learned notaries by this time, the system had become a cryptographic notation to prevent fraudulent modifi cations and forgeries, and generally to restrict access to information (Colin 1933: 195). Political changes in post-colonial Morocco have ended this system’s use.

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

12 Th e modern Armenian script has thirty-eight letters, the last two of which (o and fé) were introduced in the medieval period and have no numerical value. 174 Numerical Notation

Table 5.18. Armenian erkat’agir numerals

123456789 1s AB CDE FG H I 10s JKLMNOPQR 100s STUVWX YZ1 1000s 23 45 6 7 8 90 346 = UMF

Because the ancient Armenian alphabet had thirty-six letters, it had enough signs to express the complete series from 1000 to 9000 as well as all the units, tens, and hundreds. Th e system could thus denote any number less than 10,000. However, unlike many ciphered-additive alphabetic systems, the Armenian system does not use multiplication to express higher values, which were written in full using lexical numerals. Very little epigraphic or paleographic evidence survives from the earliest centuries of the system’s use. Th e Armenian numerals were probably developed on the model of the Greek alphabetic numerals, just as the Armenian script itself was derived from the Greek. Many other scripts have been suggested as possible ancestors of the Armenian script, based on resemblances in the shapes of certain characters (Gamkrelidze 1994: 37), while there are few resemblances to the Greek alphabet. However, of these likely ancestors, only the Greek alphabet used appropriate alphabetic numerals. Th us, regardless of the origins of the script-signs, the principle of alphabetic numeration was certainly borrowed from Greece. It is unclear whether the Armenian alphabetic numerals were developed by Mesrop Mashtots himself (or his assistants) in the early fi fth century ad, or whether they were produced later in the century. Figure 5.2 is a monumental grave inscription from the temple of Garni east of Yerevan, which commemorates a ninth-century Armenian Catholicos, also named Mashtots; the fi rst three signs are numerals (300 + 40 + 6), indicating his death-year to be 346 according to the Armenian calendar, or 897 ad. Although a connection is sometimes asserted to exist between the Armenian and Georgian alphabetic numerals, the evidence for this is too tenuous to suggest any defi nite link. Th e primary similarities between the two are that they were used in the same region and had distinct signs for 1000 through 9000. Th e only system that is derived from the Armenian alphabetic numerals is the variant Armenian system developed in the seventh century ad by Anania Shirakatsi. Th e Armenian numerals did not spread beyond the limited area around Lake Van where Arme- nian was spoken, nor do they appear to have inspired the creation of any foreign Alphabetic Systems 175

Figure 5.2. Grave inscription of the Armenian Catholicos Mashtots (897 ad); the fi rst three visible signs of the inscription are 300 + 40 + 6, the year of his death. Courtesy Gabriel Kepeklian. systems. After the development of the minuscule Armenian script, these signs were also used numerically in the same manner. Ciphered-positional numerals – the Arabic system used by the neighboring Seljuk Turks – were fi rst used in Armenia in the twelfth century (Shaw 1938–39: 368). Yet Armenian writers retained the alphabetic numerals for most ordinary purposes long afterward. Only in the mid seventeenth century, when Armenia had been fi rmly under Ottoman control for some time, did ciphered-positional numerals (Arabic, then later Western) replace the alphabetic system. Wingate (1930) has published an unusual, undated, and unsolved “magic square” arithmetical puzzle using both Arabic positional and Armenian alphabetic numerals, part of a scroll contained within a Armenian “family amulet” designed to be worn upon the person. Th e Armenian alphabetic system is still sometimes used for numbering chapters of the New Testament, although page and verse numbers are most often written using Western numerals. Otherwise, the numerals used in modern Armenia are the standard Western numerals.

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

13 Also known as Ananiah Shiragooni, or Ananias of Shirak. 176 Numerical Notation

Table 5.19. Armenian numerals: Shirakatsi’s notation

123456789 1s AB CDE FG H I 10 j 100 s 1000 2 of Persian, Arabic, Greek, and other scientifi c knowledge. In addition to these ac- complishments, Shirakatsi developed a very interesting numerical notation system in his “Book of Arithmetic” (T’uabant’iwn), which is a collection of arithmetical tables designed for the instruction of pupils. Th e basic form of this system uses twelve signs, as shown in Table 5.19 (Shaw 1938–39: 270). Th e individual signs are identical to those used for the appropriate numbers in the traditional Armenian system. However, Shirakatsi showed how these signs could be combined to express numbers through multiplication as well as addition. In this system, a unit-sign followed by one of the three power-signs (for 10, 100, or 1000) indicates that the values of the two should be multiplied; these pairs of signs were combined into numeral-phrases through addition. Instead of writing 9642 as 0XMB (9000 + 600 + 40 + 2), as in the traditional Armenian alphabetic numerals, Shirakatsi would write the same number as I2FSDjB (9 × 1000 + 6 × 100 + 4 × 10 + 2). Th us, where the traditional Armenian system is ciphered-additive, Shirakatsi’s system is multiplicative-additive. Any numeral-phrase can be written more compactly with the traditional alphabetic numerals than with Shirakatsi’s variant – so why would Shirakatsi advocate its use? Firstly, it requires knowing fewer symbols (twelve versus thirty-six) in order to express any number less than 10,000. More importantly, numbers greater than 10,000 could be expressed using multiplicative combinations of two or three signs. To do so, however, one needs the entire repertoire of Armenian numerals from 1 through 9000, as described earlier. For numbers from 10,000 through 90,000, Shirakatsi juxtaposed the signs for 10 through 90 with the sign for 1000. Similarly, the numeral-phrases for 100,000 through 900,000 combined the signs for 100 through 900 with the sign for 1000. Alternatively, the hundred thousands could be expressed using unit-signs followed by a 100-sign and then a 1000-sign. Th us, one could write 460,000 as VO2 – (400 + 60) × 1000 – or DSO2 – ((4 × 100) + 60) × 1000. Th is system is no longer a purely decimal system, but has a mixed base of 10 and 1000. For values below 1000, it is purely multiplicative- additive, but above 1000, the multiplicand that is juxtaposed with the sign for 1000 (2) is not a single sign, but rather a ciphered-additive numeral-phrase. In the “Book of Arithmetic,” numbers up to the ten millions are expressed relatively compactly (Abgarian 1962: 46; Hewsen 1968: 42). Alphabetic Systems 177

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

Georgian Like the Armenians, the Georgians developed a script and numerical notation system modeled after the Greek alphabet shortly after they converted to Chris- tianity. While the creation of this fi rst Georgian alphabet is often attributed in folklore to King Parnavaz in the third century bc, there is no direct evidence of Georgian writing until the fi fth century ad, at which time the asomtavruli or majuscule script began to be used (Holisky 1996). More familiar to modern scholars, however, are the mxedruli characters developed in the eleventh century ad, which continue to be used to write the Georgian language today. Th e numerals associated with this script are shown in Table 5.20 (Holisky 1996: 366). Th e system is decimal and ciphered-additive and, like the Georgian script, is written from left to right. Like the Armenian script, the Georgian script had enough letters to serve for all numerical values up to 9000. Some later inscriptions even include a special sign for 10,000 (-). Th ere is no evidence that the Georgian alphabetic numerals were ever used to express larger numbers than this, either through multiplication or through additional signs. Presumably, such numbers were always written out in full using lexical numerals. Th ere is an undeniable structural similarity between the Georgian and Arme- nian systems, which both, unlike the Greek alphabetic numerals, have enough 178 Numerical Notation

Table 5.20. Georgian numerals

123456789 1s abcdefghi 10s jklmnopqr 100s st uvwx yz 1 1000s 234567890 4808 = 5zh additional letters to represent the values from 1000 through 9000. However, while the Georgian and Armenian scripts both use thirty-six signs for 1 through 9000, the letter-order of the two scripts is vastly diff erent. Th e Georgian letter-order was modeled very closely on the Greek, with additional signs added as necessary at the end of the series, while the uniquely Armenian signs in that script were in- terspersed randomly within the original Greek letter-order. It is unlikely that the Georgian numerals would be modeled on the Armenian numerals but retain the Greek letter-order for their values (Gamkrelidze 1994: 77). Th ere may have been some mutual infl uence between the two numerical notation systems, given certain similarities in the sign-forms, but the direction of this infl uence remains unclear (Gamkrelidze 1994: 81–82). For now, the hypothesis of direct diff usion from the Greek alphabetic numerals, while using distinctly Georgian signs and using distinct signs for the thousands, is most strongly supported. Th e Georgian numerals were used in literary and religious texts throughout the medieval period, particularly for pagination, dating, and stichometry, as well as in monumental inscriptions. Th eir regular use ended in the sixteenth century, when Georgia came under Ottoman control, after which Arabic positional numerals were used for administrative and commercial purposes, although the alphabetic numerals may have been retained for religious functions. However, Paolini and Irbach’s 1629 Georgian-Italian dictionary, the fi rst book printed in Georgian, does not contain any mention of the alphabetic numerals alongside its list of Georgian letters. Since the eighteenth century, when Georgia fell under the Russian sphere of infl uence, the Western numerals have been those normally used for all purposes in written Georgian.

Glagolitic Th e Glagolitic script was probably developed between 860 and 870 by the brothers Cyril and Methodius, who, while on a mission to the Moravian Slavs of what is today modern Serbia, Croatia, and Macedonia, created an alphabet for liturgical Alphabetic Systems 179

Table 5.21. Glagolitic numerals

123456789 1s ab cd efghi 10s jk lmnopqr 100s stuvWxyz/ 708 = yh writings in the language now known as Old Church Slavonic (Schenker 1996: 166–167). Th ere may have been a pre-Christian script in the region, which might explain why many of the Glagolitic letters have no correlation with the Greek alphabet, but no pre-Christian numerical notation existed (Cubberley 1996). Th e numeral-signs of the Glagolitic numerical notation system are shown in Table 5.21 (Vaillant 1948, Gardiner 1984). As with the Greek and many other systems, Glagolitic numerals were frequently distinguished from words in texts by placing dots to either side of a numeral- phrase or by placing a mark of some sort above it (Vaillant 1948: 24; Schenker 1996: 182). In addition to these twenty-seven signs, additional signs for 1000, 1, and 2000, 2, were purportedly used in some texts. Th e system is ciphered-additive and decimal, and is always written from left to right. However, for the numbers 11 through 19, the ordinary sign order is reversed (e.g., bj instead of jb for 12), which refl ects the Slavic lexical numerals for the teens (Schenker 1996: 182). Yet none of the surviving Glagolitic manuscripts apparently use numerals higher than 1000 (Gardiner 1984: 15; Lunt 2001: 28). Schenker (1996: 182) contends that the Glagolitic thousands were expressed by placing a small diagonal or curved stroke (like the Greek hasta) to the left of a numeral-sign to indicate that its value should be multiplied by 1000; if so, Glagolitic is a hybrid multiplicative-additive system above 1000. Gamkrelidze (1994: 39–40) and others contend that, because the earliest Glagolitic script had thirty-six characters, the last nine letters of the alphabet (most of which were later dropped from the script) originally had the values 1000 through 9000. Th e issue remains unresolved and contentious. Th e Greek alphabetic numerals were the sole and direct ancestor of the Glagolitic numerals. Th e similarities in structure between the Greek and Glagolitic systems are considerable, and Cyril, Methodius, and their followers were Greeks. Among other possible ancestors, the Gothic numerals were long defunct by the ninth century ad, and the Cyrillic numerals were not invented until later in the century. Nickel’s (1973) suggestion that the Glagolitic numerals may have originated from the tamgas (clan identifi ers) used by Turks and Iranians in southern Russia as early as the fi rst century is problematic. While a variety of scripts, such as the Latin, 180 Numerical Notation

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

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

14 While some scholars call this system “Slavonic,” I use the term “Cyrillic” to prevent confusion with the Glagolitic system. Alphabetic Systems 181

Table 5.22. Cyrillic numerals

123456789 1s Abcde fg h i 10s Jkl mnopqr 100s Stu/wxyz0 708 = yh placing dots on either side of the signs (Lunt 2001: 28). Placing a small stroke to the left of a number indicated that its value should be multiplied by 1000 (Vaillant 1948: 24; Schenker 1996: 182). Th e Cyrillic numerical notation system is thus a hybrid: purely ciphered-additive below 1000 and multiplicative-additive for higher powers. In some cases, higher Cyrillic numerals were expressed by using the signs preceded by an unusual sign, [, to indicate multiplication by 1000 (Gardiner 1984: 15). Apparently in some cases the multiplier-stroke could be repeated two or three times to indicate multiples of 1,000,000 and 1,000,000,000, respectively (Berdnikov and Lapko 1999: 16). In the earliest phase of the script’s history, there were a number of ideograms for powers of 10 from 10,000 up to 1,000,000,000, but these rare notations are poorly studied, and the range of their use is unknown (Berdnikov and Lapko 1999: 17). While there are only twenty-seven signs listed in Table 5.22, there are more than twenty-seven signs in all varieties of the Cyrillic script; modern Russian Cyrillic uses thirty-two letters, and earlier Cyrillic scripts used a number of older signs that have now fallen into disuse. Th e signs that are assigned numerical values in Cyrillic are those that are directly derived from Greek, including the otherwise rarely used signs for xi (o), psi (y), and theta (i). Yet numerical values were never assigned to the commonly used but non-Greek characters (Gardiner 1984: 14–15). Th us, the Cyrillic numerical values do not correspond to the customary order of letters, remaining faithful to the Greek order instead. Th e Cyrillic script and numerals originated around 890 ad, at which time Slavs and Greeks who had been infl uenced by Cyril and Methodius were extremely active in the Christianization of the Slavs in the region of modern Bulgaria and Serbia. Th at this missionary work was undertaken under the auspices of the Byz- antine Empire confi rms what is clear from the paleographic evidence – that the sole external infl uence on the Cyrillic script and numerals was the Greek uncial alphabet used at the time. Th e non-Greek signs for additional consonantal Slavic phonemes were never assigned numerical values, further confi rming the Greek origin of the Cyrillic numerical notation system. 182 Numerical Notation

Th e Cyrillic numerical notation system spread to Kievan Rus in the tenth century. Th e earliest printed books in Cyrillic script, those printed in Kraków from 1491 onward, were paginated using Cyrillic rather than Roman or Western numerals (Zimmer et al. 1983). Th e Balkans fell under Ottoman infl uence in the fifteenth century, after the fall of Constantinople, and the alphabetic numerals largely ceased to be used there by around 1500. In Russia, the Cyrillic numerals were used much longer. Th e late sixteenth-century English Slavist, Christopher Borough, authored a Russian copy of the pseudo-Aristotelian Secret of Secrets (Bodleian MS Laud misc. 45 (SC 500)), which was paginated in Western numerals but used Cyrillic numerals and Arabic positional numerals elsewhere in the text (Pennington 1967: 681–682). Western numerals were known in Russia in the seventeenth century; a seventeenth-century sundial from Mangazeia (in Siberia, near the Arctic coast) is numbered using Western numerals, showing that they were known even in the Russian hinterland (Ryan 1991: 375). Yet the popular Schitanie udobnoe ‘ready reckoner’ arithmetic text of 1682, intended for merchants, contained only Cyrillic numerals (Okenfuss 1973: 329). For most purposes, the schety or Russian bead- abacus, which had been used in Moscow since the eleventh century, was perfectly adequate for the computational needs of pre-Petrine Russia (Ryan 1991: 373–374; Simonov 1993). Not until the reforms of Peter the Great around 1700, and the introduction of technical training in mathematics by scholars from Western Europe, were the Western positional numerals introduced into Russia on a widespread basis (Hans 1959–60, Fedosov 1995). Magnitskii’s Arifmetika of 1703 used both systems side by side (Ryan 1991: 373). In the same year, logarithm tables published for the use of students at the Moscow School of Mathematics and Navigation were printed in Western numerals only. In 1710, however, Peter deferred to the clergy in decreeing that church texts were to be printed using the traditional Cyrillic numerals (Crac- raft 2003: 103). Th roughout the eighteenth century, aristocrats and literate offi cials would have needed to be familiar with Roman, Western, and Cyrillic numerals in order to read the full range of textual genres used in Russia (Billington 1968: 209). As late as 1918, Tsaritsa Alexandra (Alix of Hesse) was learning Cyrillic numerals and using them to paginate her fi nal diary, demonstrating that their use was still relevant, if increasingly formal, in the late tsarist period (Kozlov and Khrustalev 1997: 2–3). Many texts were paginated using the older system even as Western numerals increasingly were employed for a wider variety of functions. Unlike the transition in Western Europe, where the Western numerals took centuries to be adopted fully, the change from the Cyrillic to the Western numerals took place relatively smoothly and rapidly, however. Today, the Cyrillic numerals are occasionally used in modern Church Slavonic texts (especially for numbering chapters and verses in Bibles), but never in ordinary Cyrillic writing (Gasparov 2001: 17–18). Alphabetic Systems 183

Latin From the fi fth through the twelfth centuries, knowledge of the Greek exact sciences in Western Europe was relatively limited. Starting in the eleventh century, as Arabic translations of Greek astronomical works began to be retranslated into Latin, Western European scholars became increasingly aware of the Greek numerals. In a handful of texts, an attempt was made to convert the Greek alphabetic numerals or Arabic abjad numerals for use with the Latin alphabet, as shown in Table 5.23 (cf. Lemay 2000, Burnett 2000c). Th e system is ciphered-additive and written from left to right, and simply employs twenty-two letters of the Latin alphabet as it existed at the time to represent 1 through 9, 10 through 90, and 100 through 400 (e.g., xlh = 328). Additionally, in the Dresden Almagest of 1121, z, θ, and ϕ represent 500 through 700 (Burnett 2000c: 61). Th ere is, however, no known way to write 800 and 900, and there is no known multiplicative technique to express numbers higher than 1000. In one of the texts containing such an alphabetic system (MS Cambrai Bibliothèque Mu- nicipale 930, a copy of Hermann of Carinthia’s Astronomia), even the sign for 400 is not used (Lemay 2000: 378–379). Yet, because the texts containing the Latin alphabetical numerals are astronomical treatises, in which numbers higher than the 360 degrees of the circle are rarely needed, this was not a serious weakness of the system, which was perfectly adequate for such values. While minuscule letters were used most of the time, in one text, majuscule letters for K, M, and N, and occasionally for G, R, Q, and L were used, in order to prevent confusion with similar-appearing Western numerals (Burnett 2000a: 82). Burnett (2000c: 61–62) briefl y discusses a few tenth- to twelfth-century astrolabes and manuscripts with Latin letters serving as numerals, but these are direct transliterations of Arabic or Hebrew numerals that follow the Semitic alphabetic order rather than the Latin one. Th e earliest documents containing the Latin alphabetic numerals in Table 5.23 date to 1121 and 1127 and were copied in Antioch in the Crusader States, probably in association with the work of Stephen of Pisa, an early twelfth-century translator of Arabic scientifi c texts. Th ese contain no key, suggesting that the system would already have been understood by its intended audience (Burnett 2000a: 76). Th e copy of Hermann of Carinthia’s Astronomia similarly contains no key, and simply uses Latin alphabetic numerals to annotate astronomical diagrams (Lemay 2000). A later manuscript (MS London, British Library, Harley 5402), dated 1160 and written in a mixture of Latin and Italian, does, however, have a key for translating Roman numerals into the Latin alphabetic system (Burnett 2000a: 76). All of these manuscripts use Roman numerals in the text and for nonastronomical purposes such as column labels, and some of them use positional numerals as well. Th e Liber Mamonis uses Roman numerals 184 Numerical Notation

Table 5.23. Latin alphabetic numerals

123456789 1s a b c d e f g h i 10s k l m n o p q r s 100s t u x y z θϕ for single digits and some low compounds, Latin alphabetic numerals for many two- and three-digit compound numbers, and represents larger numbers using Arabic positional numerals (not Western numerals) (Burnett 2000c: 64–65). Th e Latin alphabetic numerals represent a brief and abortive attempt to adapt the Greek and Arabic systems to the Latin alphabet for the sole purpose of translating astronomical documents effi ciently. Th ey did not, however, lead to a consistent or long-lived tradition of Latin alphabetic numeration. By the time the Latin alphabetic system was developed, ciphered-positional numeration was already being widely adopted by Western European mathematicians and scholars, rendering the alphabetic system obsolete. I know of no thirteenth-century or later documents that use the system described here. However, two late thirteenth- century manuscripts probably written in Flanders use the letters A through I in place of 1 through 9, in conjunction with 0, in a ciphered-positional alphabetic system written from right to left (e.g., B0G = 702), parallel to but probably independent from ibn Ezra’s use of Hebrew alphabetic numerals in the same way, discussed earlier (Burnett 2000b: 200; 2000c: 62–63). In some medieval Latin texts from Western Europe, a set of unusual letter- symbols were associated with numerical values in a sporadic and nonsystematic way, of which one example is shown in Table 5.24 (Cappelli 1901: 413–421). In some respects, this notation resembles the ciphered-additive Latin notation. Th ese letter-symbols occur in very diff erent contexts, however – in the same texts as Roman numerals rather than in translations of Greek or Arabic astronomical texts. Th e only letters that were not assigned values in this system were I/J, U/V, X, L, C, D, and M, for obvious reasons – these already had numerical values in the Roman numeral system. Like the corresponding Roman numerals, placing a bar above any of these abbreviations indicated multiplication by 1000; e.g., Ō = 11,000. Unlike the Latin alphabetic numerals or the Roman numerals, however, these abbreviations could not be combined to form numeral-phrases – that is, one

Table 5.24. Medieval Latin numeral abbreviations

ABEFGHKNOPQRSTYZ 500 300 250 40 400 200 151 90 11 400 500 80 70 160 150 2000 Alphabetic Systems 185 could not write HN for 290 and be understood. Some numbers are represented twice (G and P both equal 400); many multiples of powers of 10 (30, 60, 600) are not represented; and strange numbers such as 151 are assigned letters. In other manuscripts (e.g., the tenth-century De inventione litterarum in Strasbourg, Bib- liothèque Nationale et Universitaire MS. 326), the numerical associations of the letters are completely diff erent than the ones in Table 5.24 (Derolez 1954: 332–335). In fact, this form of notation is not related to either the Greek or Latin alphabetic numerals, but instead is a complex but unstructured quasi-numerical abbreviatory system.

Summary Alphabetic numerical notation systems originated with the Greeks in the sixth century bc, who combined the structure of the Egyptian demotic system with the idea of using phonetic signs as numeral-signs. Th e political and ecclesiasti- cal authority of Greek speakers, coupled with the brevity and adaptability of ciphered-additive numeration, led to the development of other alphabetic systems using numeral-signs specifi c to each script. Th is phylogeny expanded tremendously between the fourth and seventh centuries ad (the time of greatest Eastern Roman/ Byzantine power), with eight new systems arising during this period. Yet most systems of this phylogeny had died out, or had at least been greatly reduced in the contexts of their use, by the sixteenth century ad, during which time the Arabic and Western positional numerals replaced them. Many alphabetic systems are still used today, but only in limited contexts such as liturgical texts, numbered lists, and divinatory magic. Th ere is no one feature common to all the alphabetic systems. Most are ciphered- additive and decimal, with or without the use of multiplicative structuring for the higher powers. However, the Armenian notation of Shirakatsi is multiplicative- additive and sometimes uses a base of 1000, while the astronomical fractions are positional and involve a sexagesimal base. Th e number of signs used, the degree to which multiplication is used, and the correspondence of numeral-signs with script-signs are all variable. One of the great advantages of alphabetic systems is that, if the signs are ordered using a local script, one need not learn both a set of script-signs and a set of numeral-signs; one merely superimposes the decimal structure of the numerals onto the script, thereby lessening the mnemonic burden on both new learners and experienced users. Th e Greek alphabetic, Coptic, Gothic, Hebrew, Syriac, Arme- nian, Georgian, Glagolitic, and Latin systems all take advantage of this feature. However, the values assigned to Arabic and Cyrillic letters do not correspond to the customary letter-order, thus reducing this benefi t. Th e Fez numerals and the 186 Numerical Notation zimām numerals are blended alphabetic systems, combining the numeral-signs of two or more existing alphabetic systems to create a new system that does not correspond with the sign-order of any script. Finally, in the Ethiopic system, the users of one script adopted the ordered numeral-signs of another (in this case, the Greek alphabet) rather than adopting both the script and numerals. Despite the advantage of combining phonetic and numerical representation systems, alphabetic numerical notation systems require many more signs than the cumulative-additive systems of Chapters 2 through 4. Even the Ethiopic system, which is multiplicative above 100, uses nineteen separate signs, more than any cumulative-additive system, and the Armenian and Georgian systems require thirty-six signs. Decimal ciphered-additive systems require nine signs for each power: twenty-seven signs to express all numbers up to 1000. In the case of the Hebrew and Syriac systems, whose scripts had only twenty-two signs, numerals above 400 were expressed through cumulative combinations of hundred-signs. While this solves the problem of having only twenty-two phonetic signs, it makes numeral-phrases longer and more complex. As it is inconvenient to develop nine new signs for each higher power of 10, many alphabetic systems are ciphered-additive for lower powers but use multiplicative-additive structuring above some specifi c point, a feature originally borrowed by the Greeks from the Egyptian demotic numerals. Th e Gothic, Arme- nian, Georgian, and Latin systems do not use multiplication at all, and express only numbers below 1000 (Gothic) or 10,000 (Armenian and Georgian). Th e Ethiopic system is multiplicative above 100, a feature that can exist only because the signs of the system are not Ethiopic script-signs. A large plurality of systems – Cyrillic, Hebrew, Fez numerals, Coptic, Arabic abjad, and possibly Glagolitic – use multiplication above 1000, a natural way to proceed in systems with twenty- seven ordinary signs. Th ree other systems – the Greek, the Syriac, and zimām numerals – are multiplicative at both 1000 and 10,000; that is, after 8000 (8 × 1000) and 9000 (9 × 1000), one uses a new sign for 10,000 (1 × 10,000) rather than (10 × 1000) as in systems that are only multiplicative at 1000. Interest- ingly, only the obscure Armenian system developed by Shirakatsi takes the step of making the entire system multiplicative-additive. Because multiplicative-additive numeral-phrases are usually longer than ciphered-additive ones, doing so may not have been appealing. While there were occasional eff orts to “positionalize” Greek, Hebrew, and Latin alphabetic numerals by simply using the fi rst nine letters along with a zero, these attempts were sporadic and short-lived. Th e longevity of the alphabetic systems is remarkable. Seven alphabetic numerical notation systems were regularly used for a millennium or longer (Greek, Cop- tic, Ethiopic, Arabic, Hebrew, Syriac, Armenian, and Georgian). In cases such as the Greek system, this can be explained by the political importance of the Alphabetic Systems 187 system’s users, but in others, such as Hebrew and Armenian, the systems’ users have largely been politically and culturally marginalized. Th at these systems could survive in such circumstances and where, in many cases, ciphered-positional systems were available, requires explanation. Structural features may partly explain it: alphabetic systems’ “alphabeticity” means that one need not learn a set of numerals in addition to a script, and ciphered-additive systems are more concise than ciphered-positional systems – for any Western numeral-phrase containing zeroes, the corresponding ciphered-additive numeral-phrase will be shorter. A more satisfying explanation, however, is that alphabetic numerical notation systems, like scripts, can be important markers of cultural identity. In many cases (e.g., Coptic, Gothic, Armenian, Georgian, Glagolitic, and Cyrillic), a group of people developed a unique set of alphabetic numerals and developed their own script at the same time. Th e point of alphabetic numerals is not to be comprehensible translinguistically, but rather for each system to serve for one script alone. Under these circumstances, an alphabetic numeral system becomes an integral part of a script, and thus marks ethnic identity. Even when these systems ceased to be used regularly, many of them continued to be used in restricted functions, particularly in the domain of religion (e.g., Hebrew, Syriac, Coptic, Greek, Cyrillic). Of course, it must not be forgotten that ultimately, in the face of massive globali- zation over the past fi ve centuries, Western and Arabic positional numerals have become the earliest and standard systems learned and used by almost everyone. Th e future of alphabetic numerals seems likely to be one of increasing vestigiality and obsolescence. chapter 6

South Asian Systems

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

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

188 South Asian Systems 189

Table 6.1. Brāhmī numerals

123456789 1s ABCDe F G H I 10s JKLMNOP QR 100s STUVw 1000s 12345

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

1 Most Western epigraphers and archaeologists accept that, aside from the Harappan script, there was no pre-Mauryan writing in India (Salomon 1995, 1998). 190 Numerical Notation

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

2 Th is opinion contradicts that of Guitel (1975: 605), who sees the Aśoka numerals as being non-multiplicative on the basis that the sign for 200 does not suffi ciently resemble a ligatured multiplicative 100 × 2. South Asian Systems 191 inscriptions are found all over India, the Brāhmī system must have been developed much earlier than the paleographic evidence would indicate, perhaps between 1000 and 600 bc. Th is is a spurious use of the discredited “age-area” method (determining the age of features by their geographical distribution). In the early Mauryan Empire, an enormous region was quickly encapsulated within a single polity, so it is unsurprising that Mauryan administrative inscriptions are widely distributed. While it was certainly plausible for nineteenth-century Indologists to hope to fi nd earlier paleographic evidence for the numerals, such hopes now seem remote. I agree with Salomon (1996, 1998) and many other Indologists that a mid- third-century origin for the Brāhmī numerals and script is probable. Th e question of the ultimate origin of the Brāhmī numerals – specifi cally, whether or not they constitute a case of independent invention, and if not, on which ancestor(s) they were modeled – is unresolved, and is made more complex by the politicization of the matter. Previous scholars have emphasized the paleographic comparison of individual signs. I believe that the consideration of the system’s structural features and historical context of origin – supplemented by paleography, where appropriate – will be a more fruitful approach. One set of theories regarding the origin of the Brāhmī numerals derives them from existing representational systems used in South Asia. Borrowing from the letters of the Brāhmī script to create an alphabetic numeral-system, while once a popular theory, is not really sustainable (Prinsep 1838, Woepcke 1863, Indraji 1876, Datta and Singh 1962 [1935], Gokhale 1966, Verma 1971). While a few Brāhmī numeral-signs resemble phonetic signs, if one accepts certain paleographic transformations, these correspond neither to the standard Brāhmī letter-order nor acrophonically to any language’s lexical numerals. Renou and Filliozat (1953) note that in texts containing both purported “letter-numerals” and the corresponding signs used phonetically, the two varieties are quite diff erent. Th e derivation of the Brāhmī numerals from the Kharoṣṭhī letters is even more improbable, and has not been seriously proposed for over a century (Bayley 1882). Ifrah (1998) proposes but discards the theory that the Brāhmī numerals derive from the Kharoṣṭhī numerals, which can easily be dismissed by noting the temporal priority of the former. Finally, a more recent set of theories derives the Brāhmī numerals from those of the Indus Valley civilization (Sen 1971, Kak 1994), but there are no examples of any writing from India between the latest Harappan inscriptions (around 1700 bc) and the fi rst Brāhmī inscriptions (around 250 bc), and only limited and confl icting evidence for the nature of the Harappan numerical notation system (Parpola 1994; cf. Chapter 10). If not derived from any South Asian system, the Brāhmī numerals could have developed independently. Woodruff (1994 [1909]: 53–60) speculated that both the Chinese and Brāhmī numerals derived from a hypothetical ancient set of cumulative tally signs for 1 through 9, which would then have spread to both China and 192 Numerical Notation

India. Kaye (1919) argued that the Brāhmī numerals developed independently during Aśoka’s time, with their structural features representing three diff erent stages of development, but inexplicably then argues against Indian creativity. Ifrah (1998: 390), arguing that there are “universal constants caused by the fundamental rules of history and paleography,” postulates nonattested cumulative signs for 1 through 9 which later became abbreviated and ligatured into the Brāhmī system. Salomon’s position is more sensibly agnostic; recognizing the problems involved with many other theories, he simply notes that numerical signs are sometimes “cursive reductions of collocations of counting strokes,” citing the hieratic and demotic systems as examples (Salomon 1998: 60). Finally, a number of theories argue for a foreign origin of the Brāhmī numerals. Falk (1993: 175–176), noting structural and paleographic resemblances between the Brāhmī and the earliest Chinese (Chapter 8) numerical notations, argues for a Chinese origin. However, there is little evidence of contact between the two regions at this period, and the only paleographic similarity between the systems is the common use of horizontal strokes for 1, 2, and 3. It has occasionally been proposed that the Greek alphabetic numerals inspired the Brāhmī numerals, given their appearance following the Alexandrine period, the strong trade ties with the Greco-Iranian kingdoms of Parthia and Bactria, and the structural similarities between the two systems. However, the evidence for the “alphabeticity” of the Brāhmī numerals is weak at best (see the previous discussion), and there is no paleographic correspondence between the Greek and Brāhmī numerals. It is most plausible that the Brāhmī numerals are derived from the Egyptian hieratic or demotic numerals. Burnell (1968 [1874]) argued for a demotic origin, while Bühler’s (1963 [1895]) much more prominent analysis argued for a hieratic origin. Th e three systems are structurally similar: they are all decimal, hybrid ciphered- additive/multiplicative-additive systems, and represent 200, 300, 2000, and 3000 by adding quasi-multiplicative strokes to the signs for 100 or 1000. Th ere are resemblances in around one-third of the sign-forms, and very close resemblances for a few, such as 9 (hieratic = i; demotic = i; early Brāhmī = i) (Bühler 1963 [1895]: 115–119; Salomon 1995, 1998). While there was not tremendous Egypto-Indic cultural contact, Ptolemaic traders reached as far as the city of Muziris (modern Cranganore) on the Malabar Coast, and Aśoka is known to have sent Buddhist missionaries to Alex- andria (Basham 1980: 187). Of the two Egyptian systems, I believe the demotic to be a more likely ancestor, because in the Ptolemaic period the use of hieratic numerals was very limited. Th us, although the demotic and Brāhmī systems diff er in both the power at which multiplication is used and the direction of writing, I believe that a demotic origin should be adopted as a working hypothesis. Th e Brāhmī numerals spread throughout the Indian subcontinent during the Mauryan period. Only in the northwest, where Kharoṣṭhī numerals predominated, South Asian Systems 193 did the Brāhmī numerals fail to penetrate until around the fourth century ad. Th ey were used primarily for writing dates on stone inscriptions and copper land grants. Th us, a full set of numeral-signs up to at least 1000 is attested, and the numeral-signs can be assigned exact dates. Other functions for which Brāhmī numerals were used include stichometry and the recording of fi nancial transactions. While it is interesting to speculate on the use of Brāhmī numerals on other materials than stone and copper, the Indian climate and geography are unsuitable for the survival of perishable materials. In Central Asia, manuscripts in the Tocharian language were written using a variant of Brāhmī script and numerals from the sixth to ninth centuries, and in this drier climate, Brāhmī numerals are attested to have been used on wood tablets, palm leaves, and paper (Sander 1968). Th ere is no surviving evidence that the Brāhmī numerals were used for arithmetic or accounting; the rather substantial body of medieval Indian mathematical works, sometimes attributed to very early dates, use either lexical numerals only, or employ one of the alphasyllabic notations to be described here. After the Kharoṣṭhī script died out in the fourth century ad, variants of the Brāhmī numerals were the only ones used in India until the late sixth or early seventh century. Th ey spread not only throughout the Indian subcontinent, but also into Central and Southeast Asia, regions that were heavily infl uenced by India during this period. In some Central Asian manuscripts, numeral-phrases were written from top to bottom rather than from left to right (Renou and Filliozat 1953: 702). Th ere was enormous variation in the shapes of the numeral-signs from location to location, suggesting that, as with the Indian scripts, no pressure existed toward the formation of interregional standards. Th e primary regional division, between northern and southern systems, began as early as the second century ad, and these two basic variants diverged further in later centuries. Th e end of the traditional Brāhmī numerals and the later local additive variants was a gradual process, instigated not by external infl uences but by the invention of ciphered-positional notation beginning in the late sixth or early seventh century ad. Over the next couple of centuries, the older ciphered-additive forms became increasingly rare, and by the ninth century ad, the Brāhmī numerals had been replaced by the modern ciphered-positional system throughout India and Southeast Asia. Only on the southern tip of India and in Sri Lanka were additive systems retained (though in an altered form) until signifi cantly later.

From Addition to Position No numerical notation systems have been as widely studied and discussed as the ciphered-positional systems that originated in India, primarily because, through the intermediary of the Arabs, these systems are ancestral to Western numerals. 194 Numerical Notation

While it would be teleological to portray the history of numerals in a linear fashion leading to our own system, the present near-universality of the ciphered- positional, decimal structure originating in India and spreading westward throughout the Islamic and Western spheres requires explanation. Around ad 600, a change began in the writing of dates in the Brāhmī- derived scripts of India and Southeast Asia. Instead of writing smaller numbers with ciphered-additive notation and larger numbers with multiplicative-additive notation, all numbers were written using paleographic variations of the nine Brāhmī numeral-signs and a dot to indicate zero – a purely ciphered-positional system. Th e spread of the older additive systems throughout South and Southeast Asia between the third century bc and the seventh century ad was followed by a second wave of diff usion of the positional principle and zero (seventh century ad onward), wherein the additive systems changed into positional ones. Th e change is actually a very simple one. Th e unit-signs were retained, but the power-signs were replaced with a single sign for zero. Th is process is confi rmed by comparing the signs for 1 through 9 in the nonpositional (Brāhmī-derived) systems with the very similar unit-signs used with a zero in later periods. Th is was not the fi rst zero; the Babylonians (Chapter 7) and the Maya (Chapter 9) had already invented the positional principle and zero-signs well before this time. Some scholars have claimed that the Babylonian zero diff used eastward to India just as it diff used westward to Greece (Février 1948: 585; Menninger 1969: 398–9). Yet the Babylonian and Maya systems were both cumulative-positional, and used a sub-base in addition to a base (Babylonian 10 and 60, Maya 5 and 20). Similarly, while Greek and Arab astronomers used positional fractions with a zero, these had a sexagesimal (Babylonian-derived) base and never used positional notation for whole numbers. Th us, there is no apparent historical relation between these systems and the later Indian one. Th e Indian positional system rstfi combined ciphering, a zero, and a single, decimal base. It is frequently claimed that the earliest example of ciphered-positional numerals is found on the Sankheda or Mankani copper plate bearing the date 346 in the Kalacuri era, which translates to 595 ad (Bühler 1896: 78; Smith and Karpinski 1911: 46; Das 1927a: 118; Kak 1990: 199). Th is plate is a donation charter of Dadda III, used to certify a land grant. When discussing any land grant, the issue of a later forgery always arises, as attempts to claim land by producing such evidence were common in India (as elsewhere). Since much of the paleographic evidence for early positional numerals comes from such land grants, we must be cautious to avoid dating inscriptions simply by the date as inscribed, and also take paleography and historical context into account. We must also remember that texts containing positional notation that are transcriptions or translations of earlier works must not be assigned an early date based simply on their putative earlier authors. South Asian Systems 195

While we need not go so far as Kaye (1919: 346), who claimed that all positional numerals in India prior to the ninth century ad were forgeries, most Indologists are very wary of the Sankheda plate (Salomon 1998: 61). Th e fact that it is ninety years older than any other positional numeral-phrase suggests that we need to question carefully any sixth-century ad evidence for ciphered-positional numerals in India. Th e earliest surviving and unquestionable examples of ciphered-positional numerals with a zero derive, not from India itself, but from Southeast Asia, in Khmer, Old Malay, and Cham inscriptions from the late seventh century ad. A calendrical inscription found at Sambor (in Cambodia) and written in a mixture of Old Khmer and Sanskrit is dated 605 in the Śaka dating system, or 683 ad; the zero appears as a small dot (Coedès 1931: 327). As the Sambor inscription is a calendrical passage rather than a land grant or fi nancial document, it is unlikely to be a forgery (Diller 1996: 126). Similar inscriptions from the Old Malay kingdom of Sriwijaya have been found at Palembang and at Kotakapur on the nearby island of Bangka dating to 683, 684, and 686 ad, or 605, 606, and 608 Śaka, respectively; in these cases, the zero was written as a circle rather than as a closed dot (Diller 1995). It is intriguing that the Old Khmer and Old Malay inscriptions appear in the same year (Diller 1995: 66). Nevertheless, there is no reason to believe, as Kaye (1907) did, that the existence of these inscriptions must mean that ciphered-positional numerals actually originated in Southeast Asia and diff used from there to India. Th e existence of intermediate additive-positional forms from the sixth and early seventh centuries ad, which I will discuss later, coupled with the probability that some of the earlier copper grant plates are authentic, make it likely that the invention of ciphered-positional numerals occurred around ad 600. However, there is compelling evidence for something akin to the positional principle and zero in certain earlier literary and religious texts. In texts using the bhutasa khya or word-numeral system, special cryptic words for one through nine could be combined in a sort of positional fashion with a word for zero in order to represent dates lexically in a fashion quite diff erent from that of ordinary Sanskrit number-words (Datta and Singh 1962 [1935]: 53–63; Mukherjee 1977). Th e earliest attested text to use this system is the Yavanajātaka of Sphujidhava (ad 269), a Sanskrit version of a Greek astronomical text (Yano 2006: 15). In it, the words ‘moon’ and ‘earth’ mean one, ‘eye’ and ‘twin’ two, and so on, with ‘sky’ and ‘dot’ meaning ‘zero’, suggesting a sort of positionality of words if not of graphemes, while also allowing multiple words to represent the number, a key to producing numeral-phrases that are unambiguous and yet fl exible enough to fi t into strict Sanskrit verse (Plofker 2007: 395). Th e bhutasa khya system is thus suggestive of positionality but does not constitute a system of graphic numerical signs, nor should its use be taken to imply the widespread use of decimal positional numerals 196 Numerical Notation in Indian manuscripts. Notably, the regular name for Indian numeral symbols, anka ‘mark’, is also a numerical word for ‘nine’ in this system, which suggests that there were originally only nine numerical graphemes – that is, that the zero, and place-value, were latecomers to Indian numerical notation even though they existed conceptually in the word-numerals (Clark 1929: 229–230). Th e earliest Sanksrit word for the zero-sign, śûnya-bindu (literally, ‘void-dot’), is fi rst used in Subhandu’s Vasavadatta, written around 600 ad (Sen 1971: 175; Salomon 1998: 63). Th is evidence suggests a correspondence between the early use of numeral-words and the structurally identical later use of the ten numeral-signs. Th us, the literary concept of a “zero-space” in Hindu thought, the use of chronograms, and the term śûnya-bindu in the fi fth and sixth centuries ad may have prefi gured the eventual development of ciphered-positional numerals. Within the in- terlinked tradition of Indian religious and mathematical thought, the invention of the zero is as much a metaphysical concept as it is an arithmetical tool, if not more so. Almost all the attested early ciphered-positional numerals are nonarithmetical, and are simply used to register dates and other numbers on inscriptions and copper plates. It would be erroneous to assume that the positional numerals originally had an arithmetical function, and then to use this assumption to hypothesize an ancient mathematical tradition of ciphered-positional numerals. Th ere is no evidence that true ciphered-positional numerals were used prior to the middle of the sixth century ad, and any claim prior to the middle of the seventh century ad requires careful examination. Th e Indian climate and topog- raphy are not particularly suitable for the survival of materials other than stone and metal, and we certainly do not have as much evidence as we would like. Nevertheless, there is plenty of inscriptional evidence for continued use of the old additive numerals from the sixth through the eighth centuries ad, they decline signifi cantly in frequency only in the ninth century. It is unlikely that all the evidence for ciphered-positional numerals was lost where so much survives for the additive system – though not impossible, if positional numerals were used only on perishable media at fi rst. Further evidence for the chronology of the shift comes from inscriptions dating from the late sixth to the middle of the eighth century ad from the Orissa region, which are written with unusual mixed structures combining the features of the older additive and newer positional notations (Datta and Singh 1962 [1935]; Acharya 1993; Salomon 1998: 62–63). Th e earliest of these is from the Urlam copper plates of the Eastern Gangâ king Hastivarman, dated to 578 ad, in which the Gangâ era year 80 is written as the additive sign for 80 followed by a zero, but this date is questionable (Salomon 1998: 62). Acharya (1993) describes many Oris- san inscriptions dating from 635 to 690 ad in which dates such as 137 are written as 100 3 7 rather than 137. Th is series of dates leads directly into the first fully South Asian Systems 197 positional date found in India, on the Siddhantam grant of Devendravarman (195 Gangâ = 693 ad), just ten years after the Southeast Asian examples mentioned earlier (Salomon 1998: 62).3 Datta and Singh (1962 [1935]: 52) mention additional eighth-century ad examples combining additive and positional notation, which they characterize as representing the gradual forgetting of the older system. Th eir argument rests on the claim that these are quite late examples of additive notation and that the positional principle was well established by this time. In fact, it is far more likely that they represent incomplete attempts to incorporate the novel positional principle into inscriptions. Th ese mixed numeral-phrases confi rm the hypothesis of an early seventh-century origin of positional numerical notation in India. In the eighth century, the positional system gained signifi cant ground, and was preferred by 800 ad. Around this time, the spread of scientifi c knowledge from India to China (primarily through the medium of Buddhist scholarship) led to awareness of the ciphered-positional numerals in China. In the Kaiyuan period (713 to 741 ad), the Indian astronomer Qutan Xida (Gautama Siddhartha) translated an Indian calendar into Chinese, using positional numerals (with a dot for zero), and commented on their ease of use (Gupta 1983: 24). In the ninth century ad, the additive system became much scarcer. Salomon (1998: 62) notes a striking late example from the Ahar stone inscription in north central India, which is a composite record of documents of diff erent dates; those up to 865 ad are all dated using additive numerals, and those from 867 ad using positional numerals, providing precise information on the date of replacement. Th e plate of Vināyakapalā (931 ad) is an extremely late northern (Nagari) inscription containing the additive system (Singh 1991: 170). By the tenth century ad, only the far south of India (Tamil and Malayalam-speaking areas) consistently used additive systems, but even there, the old system was replaced by a purely multiplicative-additive structure. Of all the descendants of the Brāhmī system, only the Sinhalese numerical notation system preserved the old structure until comparatively recently.

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

3 I am uncertain what to make of Mukherjee’s (1993) assertion that the copper-plate inscription of Devakhadga expresses the date 73 in the Harsha era (starting 606 ad) using positional numerals, which would thus be dated to 678 ad. 198 Numerical Notation

Table 6.2. North Indian numerical notation systems

Script0123456789 Bengali ؘؙؐؑؒؓؔؕؖؗ Devanagari Gujarati Marathi Oriya Punjabi Nepali to present the paleographic data concerning the development of Indian numeral- signs from 800 ad to the present day (cf. Salomon 1998; Ifrah 1998: 367–385 for more complete analyses of this issue). Nevertheless, a look at some of the more important variations on this common pattern of ciphered-positional decimal systems is warranted. Many of these systems (or very close descendants thereof ) have been employed for well over one thousand years and continue to be used. Most major South Asian languages have their own alphasyllabaries and numeral-signs. Th eir numerical notation systems are structurally identical to one another and to Western numerals. Today, all these indigenous systems are in competition with Western numerals, especially for commercial and scientifi c purposes. In religious and formal contexts, the traditional numerals are still frequently preferred.

North India Th e ancestor of the northern Indian numerical notation systems is the Brāhmī system used in the Gupta Empire, which ruled most of northern India from the Indus to the Ganges from the fourth to sixth centuries ad, and also signifi cantly infl uenced central India. Th ese earliest Gupta Brāhmī numerals were nonpositional, but the idea of positionality and the zero sign spread quickly through the systems of the region. Th e most common modern varieties of this subgroup are the Bengali, Devanagari, Gujarati, Marathi, Oriya, Nepali, and Punjabi; they are thus used in a swath across Pakistan, northern and central India, Nepal, and Bang- ladesh. Th e northern Indian systems are also directly ancestral to both the modern Arabic and “Maghribi” numerals associated with the Arabic script, and thus, indirectly, to Western numerals. Th e similarities between Western numeral-signs and many of the north Indian numerals, especially those for 0, 2, and 3, are quite evident in Table 6.2. South Asian Systems 199

Table 6.3. Central Asian numerical notation systems

Script0123456789 Tibetan

Mongolian

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

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

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

4 Despite Ifrah’s assertion (1998: 382) that each of the Agnean, Kutchean, and Khotanese scripts of Chinese Turkestan would have used a set of ten positional numerals, I know of no evidence that this was the case. 200 Numerical Notation

Table 6.4. South Indian numerical notation systems

Script0123456789 Telugu Kannada as Kawi (the ancient script of Java) and Cham (used in Vietnam until the thirteenth century) originally used hybrid additive numerical notation systems on the Brāhmī model, but these transformed into ciphered-additive positional systems in the seventh century. Th e modern descendants of these systems include Khmer, Th ai, Burmese, Lao, Balinese, and Javanese. Of these, Balinese and Javanese are closely related to one another but paleographically distant from any other South Asian systems. Th ey use Javanese letters to represent certain numbers, while retaining older signs derived from Kawi for the others (0, 4, 5, and 6). Th e Southeast Asian systems are shown in Table 6.5.

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

Table 6.5. Southeast Asian numerical notation systems

Script0123456789 Khmer Th ai Burmese Lao Balinese Javanese South Asian Systems 201

Table 6.6. Tamil numerals

123456789 Units ABCDEFGHI 10 J 100 K 1000 L 6408 = FLDKH

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

5 Curiously, this system is structurally identical to the Armenian alphabetic notation of Anania Shirakatsi (Chapter 5), but it would be an error to make too much of this resemblance. 202 Numerical Notation one. (Sinhalese retained the Brāhmī structure, while all other Brāhmī-derived systems became ciphered-positional.) Th e Chinese classical numerals (Chapter 8) are multiplicative-additive, so contact with Chinese Buddhists might have stimulated the development of the unique notations characteristic of areas of southern India that were also Buddhist, or, more likely, made their retention more appealing. However, there is no paleographic similarity between the Chinese and Tamil numerals, making diff usion from China improbable. Moreover, none of this evidence explains why the Tamils retained their system even after they ceased to practice Buddhism. Another possibility is that the users of the Tamil and Malayalam systems did not switch to positional notation as an eff ort to maintain their cultural distinctness from the north. Late in the system’s history, an abbreviated form of the Tamil numerals developed that, for some numbers, adds an element of positional notation by omitting the power-signs for 10, 100, and 1000. For instance, Pihan notes that while 21 was traditionally written BJA (2 × 10 + 1), it could also be written BA, abbreviating the phrase without any loss of information (Pihan 1860: 117). Such numeral-phrases are purely ciphered-positional. While this presents no problem for numerals that lack any empty positions, a zero sign is needed in other cases; however, no zero appears in any pre–twentieth-century Tamil numeral-phrases. Sometimes, rather than using a sign for zero, Tamil writers used the power-signs for 10, 100, and 1000 to eliminate ambiguity. In one case, 2205 is written as BBKE (2, 2, 100, 5), which indicates that the second 2 is to be understood as a hundreds value rather than as a tens value, and that therefore the fi rst 2 must be understood as 2000 (Guitel 1975: 614–615). Such phrases combine multiplicative-additive and ciphered-positional notation. Th ese mixed multiplicative and positional phrases are no longer used, and appear to be a transitional product of the colonial period, when contact with the West began in earnest. Apparently, in the nineteenth century some Tamil astronomers were computing using a mixed decimal and sexagesimal computation system by manipulating pebbles or shells on a fl at surface in a manner reminiscent of Greek or Babylonian techniques, but there was no corresponding numerical notation (Neugebauer 1952). Today, some formal Tamil writings use the traditional numerals, while for most commercial and informal purposes an ordinary zero sign is used, making the system ciphered-positional. Most literate Tamils today are familiar with and use the Western numerals.

Malayalam Th e Malayalam script, like Tamil, is derived from the Grantha script of southern India. It is used to write the Dravidian language of the same name used in Kerala at India’s southwestern tip. It fi rst emerged as a distinct script around 700 ad, South Asian Systems 203

Table 6.7. Malayalam numerals

123456789 Units MNOPQR S TU 10 W 100 X 1000 Y 6408 = RYPXT although its letters and numeral-signs are closely related to those of the other Brāhmī-derived scripts. Th e Malayalam numerical notation system, like Tamil and Sinhalese, was unaff ected by the cultural and political infl uence that rendered the northern Indian numerical notation systems structurally identical and paleographically similar. Th e traditional Malayalam numeral-signs are indicated in Table 6.7 (Pihan 1860: 122–125; Ifrah 1998: 335). Th e similarities between the Tamil and Malayalam systems are striking. Both systems are decimal and multiplicative-additive, and written from left to right. Th ere are many paleographic similarities between the numeral-signs of the two systems, thus refuting the claim that the Tamil numerals are phonetic in origin. As in Tamil, there is no power-sign for the units, and the ‘1’ is understood in any numeral-phrase with a units value. Furthermore, Malayalam numbers above 10,000 are also expressed using multiplicative combinations of the sign for 1000 with those for 10 and 100, as necessary. None of these similarities is particularly surprising, given the close cultural and geographic proximity of these two Dravidian peoples. Th e only structural diff erence between Tamil and Malayalam numeration is that Malayalam numeral-phrases were never expressed using the hybrid additive and positional notation that was occasionally used later in the Tamil system’s history. Th ere are few distinctly Malayalam inscriptions that date before ad 1000, by which time it had already acquired its multiplicative-additive structure. Th e numeral-signs are derived from those of the Grantha script, showing that the Ma- layalam numerals are native to South Asia, although, as with Tamil, we cannot exclude the possibility of some infl uence from Buddhist China. Th e fact that Buddhism was maintained longer in the south than in northern India may be a partial explanation for the diff erence in structure. A millennium of trade with and domination by other peoples of South Asia, most of whom used ciphered-positional notation, did not aff ect the structure of the Malayalam system. Malayalam writers employed this system regularly until the middle of the nineteenth century, at which time European contact introduced the zero and the idea of positionality. A new sign for zero was introduced (V), which, when combined with the nine regular unit- signs, produced a regular ciphered-positional system. Today, the older Malayalam 204 Numerical Notation

Table 6.8. Sinhalese numerals

123456789 1s abcdefghi 10s jkl mno pqr 100 s 1000 t 3684 = ctfsqd system is used rarely if at all (largely by those who need to understand old texts), and is quickly becoming a historical curiosity.

Sinhalese Th e Sinhalese (or Singhalese) script developed from the model of the southern Brāhmī scripts for use among the speakers of the Sinhalese language, and was infl uenced by the Grantha script that is ancestral to the Tamil and Malayalam scripts used for writing the Dravidian languages of southern India and northern Sri Lanka (although Sinhalese is an Indo-European language). It is an alphasyllabary, written from left to right, and is used today in Sri Lanka and the Maldives. Th e traditional Sinhalese numeral-signs are indicated in Table 6.8 (Pihan 1860: 140–141; Ifrah 1998: 332). Sinhalese has distinct signs for each of the units, each of the decades, 100, and 1000. Th e numeral-signs for the units resemble many of those used in other South Asian numerical notation systems, and many of the signs also resemble, but are not derived from, the phonetic signs of the Sinhalese script. Numeral-phrases are written from left to right. Th e system is ciphered-additive below 100, but it is multiplicative for the hundreds and thousands, combining the unit-signs with the appropriate power-signs. It is unknown how 10,000 and higher numbers were written, though Pihan (1860: 141) speculates that it may have been through multiplicative forms such as those used in Tamil and Malayalam. Th e Sinhalese numerals are thus structurally identical to the old Brāhmī system. Th e Sinhalese did not adopt the positional system when the peoples of In- dia (except Tamil and Malayalam speakers) and Southeast Asia did, between the seventh and ninth centuries ad. Sinhalese inscriptions and texts used this system throughout the medieval period, seemingly unaff ected by the shift to positionality occurring in other Indian numerical notation systems. Pihan (1860) shows no awareness of any structural changes in the Sinhalese numerals in use at the time South Asian Systems 205 he was writing; although his knowledge of the numerals was limited, there is no reason to believe that they were in signifi cant decline in the mid nineteenth century. Modern Sinhalese writings normally use the Western numerals, although the traditional numerals are retained for certain formal purposes.

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

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

6 Th e number of consonant-signs and vowel diacritics varies from script to script, and there are also signs for V syllables (isolated vowels) and CCV syllables. 206 Numerical Notation

Table 6.9. Âryabhata’s numerals

% &G % &G ka kha ga gha a ki khi gi ghi i 1 2 3 4 5 100 200 300 400 500

70 80 90 100 7000 8000 9000 10,000 named Ardubarius in the seventh-century Byzantine text, Chronicon Paschale. His work focused on astronomy (he is regarded as the fi rst great Indian astronomer) but also contains much pure mathematics, in addition to the cosmological hypotheses from which the exact sciences in ancient India cannot be divorced. His numerals are concise and readable with little training, and while not infi nitely extendable, are capable of expressing very high numbers. Table 6.9 shows the signs of Âryabhata’s numerals, using modern Nagari signs for convenience (Fleet 1911a; Guitel 1975: 582–583). Due to the structure of Indian alphasyllabaries, while there are many hundreds of possible syllables, to learn the signs of the system one need only learn the two sets of signs, which then can be combined with one another. Th e thirty-three unmarked signs, in their assigned order and divided into groups on the basis of similar phonet- ics, assume the numerical values 1–25, 30–90, and 100, as shown in the leftmost fi ve columns of Table 6.9. Th e ingenious principle involved in this system is that changing South Asian Systems 207

Table 6.10. Order of power diacritics

ka ki ku kri kli ke kai ko kau 100 (1) 102 104 106 108 1010 1012 1014 1016

the vowel attached to one of the basic signs alters its numerical value. When combined with the vowel i, the signs take on the numerical values 100–2500, 3000–9000, and 10,000, as shown in the rightmost fi ve columns. While this means that there are two signs with the value 100 – ha () and ki () – this has little potential to cause confusion. Each successive vowel diacritic multiplies the value of the sign by 100 with respect to its predecessor, as shown in Table 6.10 (indicating only the combinations of k + vowels). Using these signs in combination, any number up to 1018 could be expressed, and Âryabhata’s system does not exhaust the available diacritics. Numeral-phrases were written with the lowest powers on the left, which refl ects the order of powers of the Sanskrit lexical numerals, but which is opposed to the Indian numerical notation systems, in which the highest power was on the left. No sign for zero was needed, and none used. Th e signs for 11 through 19 and 21 through 25 were not strictly necessary; 15 could be written as GA instead of B without any ambiguity, but the latter was more concise. Th ese extraneous signs normally were not used in numbers such as 85, which was written as G (5 + 80) rather than B( (15 + 70). In some cases, these rules were violated (we do not know why), so that in one astronomical table, 106 is written as 16 + 90 and 37 as 16 + 21 (Guitel 1975: 587). Table 6.11 indicates several numeral-phrases written alphasyllabically.

Table 6.11. Alphasyllabic numeral-phrases

Alphasyllabic Value Representation Transcription and Sign-values kha va 62 % 2 60 ta ha ta ki 116 OR or 16 100 16 100 ra chu 70,040 7E 40 70,000 kha ya ghi li cu śu 765,432 %'&#<( 2 30 400 5000 60,000 700,000 ma vi nu jri sri 98,206,025 3;H 25 6000 200,000 8,000,000 90,000,000 phu ghe 40,000,220,000 & 220,000 40,000,000,000 208 Numerical Notation

Th e best way to conceive of this system is as a base-100, or centesimal, multiplicative-additive system with a decimal and ciphered sub-base. Unlike most multiplicative-additive systems, however, there can be up to two unit-signs within each power of 100, each of which combines with its own power-sign. For instance, in the representation of 9800 in Table 6.11, the signs for 8 (;) and 90 (H) each combine separately with the diacritic sign for 100 ( ). Th e system is slightly irregular below 100 in that the basic thirty-three signs include signs for 11 through 25. Âryabhata recognized that the system was centesimal rather than decimal, as he distinguished the set of centesimal powers, or varga ‘classes’, from the intermediate powers, or avarga (Das 1927a: 110; Jha 1988: 80). Âryabhata’s system is not positional, since placing an unmodifi ed consonant-sign in the middle of a numeral-phrase would render it meaningless. If Âryabhata was unfamiliar with positional numeration, then he may have developed alphasyllabic numerals because of the insuffi ciency of the Brāhmī ciphered- additive system for writing large numbers, a task that could be done very concisely using his own system. However, there is no evidence that the calculations that Âryabhata undertook were done directly with these numerals. Instead, their function was primarily to effi ciently express very large numbers in verse, in the manner in which traditional Indian mathematics was written (Clark 1929: 232–233). Th e versifi cation of mathematical expressions, while seemingly constraining and complicating the expression of numbers, assisted the user mnemonically while retaining the brevity of other numerical systems. It is possible that Âryabhata was familiar with the Greek alphabetic numerals in addition to the Brāhmī numerals. Âryabhata’s work was inspired in part by Greek astronomical writings, and Fleet (1911a), among others, has argued that both Ârya- bhata’s astronomy and his numerals are derived from Greek sources. However, even if he borrowed the general idea of using script-signs as numerals from the Greeks – and there is no defi nite evidence either way – this does not tell us very much, because the two systems are radically diff erent (Das 1927a: 111–114). Even if he did know the Greek numerals, they played little role in the invention of his own system, whose idiosyncratic features, such as a base of 100, are not found in other systems. Another possibility is that even though his system was not positional and lacked a sign for zero, Âryabhata had a complete knowledge of ciphered- positional numeration when inventing his alphasyllabic system (Ganguli 1927). As I discussed earlier, the concept of śûnya or “emptiness” existed in the fi fth century ad and may have prefi gured the use of positional numerals in India, but no good evidence survives for an actual ciphered-positional numeration system prior to the seventh century ad, long after Âryabhata’s death. Ifrah’s (1998: 450–451) statement that the use and adoption of Âryabhata’s system “caused the Indian discoveries of the place-value system and zero, which took place before Âryabhata’s time, to be South Asian Systems 209 irretrievably lost to history” is typical of the confused anti-empiricism of recent research on this system. Th e nonpositionality and relative complexity of Âryabhata’s system argue against his having been familiar with positionality. While Âryabhata’s numerals were known to Indian astronomers and mathematicians long after his death, they were used solely in the context of commentaries on his work. Th e system’s lack of facility for arithmetic, coupled with the diffi culty in pronouncing the resulting numeral-phrases in Sanskrit, led to its abandonment even among most of Âryabhata’s proponents (Jha 1988: 85–86; Yano 2006: 149). It was replaced, in part, by the regular numerical notation systems of India, but it also gave rise to a variety of successor systems for correlating phonetic signs with numerical values, most notably the katapayâdi system. While these successors were not as unusual as Âryabhata’s system, they were far more successful.

Katapayâdi Numerals When later scholars experimented with alphasyllabic numeration starting in the ninth century ad, they immediately saw that an alphasyllabary could also be turned into a ciphered-positional system. Known as katapayâdi, the signs of this system are shown in Table 6.12 (Fleet 1911b; Datta and Singh 1962 [1935]: 70). In this system, each V and CV syllable is given a value from 0 to 9. Unlike Âryabhata’s system, changing the vowel of the syllable does not change its numerical value, so that ka = ki = ku = 1. Two of the signs (ña and na) take on the value of zero, as did isolated vowel-signs (those representing a V syllable alone, without any consonantal component), which did not have a numerical value in Âryabhata’s system. CCV syllables do not have their own numerical values, but are considered to have the value of the consonant to the left of the vowel, so that tva = va = 4 and ntya = ya = 1. As a result, any sequence of syllables can be assigned a numerical value, and any number has a wide variety of possible phonetic transcriptions. Katapayâdi numerals were read with the lowest power on the left, as in Âryabhata’s numerals. Th us, the word bhavati or F ‘is’ had the numerical value 644. Th e name katapayâdi itself is taken from the four syllables (ka, ta, pa, ya) that are assigned the value 1 in this system. Although it is unusual in that each digit from 0 to 9 has several alphasyllabic values that represent it, structurally this system is an ordinary ciphered-positional and decimal system. Th e earliest example of the katapayâdi numerals is from the Grahacāranibandhana by the astronomer Haridatta, written in ad 683 (Sarma 1999: 274). Datta and Singh (1962 [1935]: 71) place its invention around ad 500 and claim that it must have been known to Âryabhata himself, but there is no textual evidence to support this assertion. Haridatta was a direct intellectual descendant of Âryabhata, and used his predecessor’s system as the basis for his own. Nevertheless, as the katapayâdi 210 Numerical Notation

Table 6.12. Katapayâdi numerals

% &G ka kha ga gha ua 1 2 3 4 5

5 6 7 8 is ciphered-positional, like the general Indian positional numerals in ascendance at the time, the existing positional numerals must certainly have infl uenced him. Th us, this system is very likely a blend of Âryabhata’s numerals and the ordinary Indian positional numerals. Aside from being able to express any number, it gave every word a numerical value, and gave every number many corresponding words. Th is would have allowed for the construction of various mnemonic devices to aid scholars and students, and would have served a prosodic function (for astronomical texts were written in Sanskrit verse, which had strict metrical rules). Th e kata- payâdi numerals were also important in the Hindu traditions of number-magic, divination, and chronograms in which the sum of the numerical values of the signs of a word or verse produced a meaningful date. Th e katapayâdi numerals, as well as related systems that are identical except for the use of local script-signs and the assigning of diff erent digit-values to various signs, were used continuously throughout much of India for many centuries. South Asian Systems 211

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

Aksharapallî A third variety of alphasyllabic numerals, sometimes confused with the ciphered- positional katapayâdi in the scholarly literature, is known as aksharapallî numeration (after akshara, the word for the CV syllable-clusters that comprise the basic unit of the Indian alphasyllabaries). Whereas Âryabhata’s system was multiplicative-additive, and the katapayâdi system was ciphered-positional, the aksharapallî systems are ciphered-additive and decimal, assigning the numerical values 1–9, 10–90, and sometimes also the low hundreds (but never as high as 1000, to my knowledge) to a set of phonetic signs. It was used very widely for paginating books, and was written in the margins from top to bottom with the highest power at the top. Unlike the fi rst two alphasyllabic numerical notation systems, there was never a single regular system for correlating signs with numerical values in the akshara- pallî. Datta and Singh’s (1962 [1935]: 73) search through old manuscripts revealed no fewer than three signs for 1, twelve diff erent signs for 4, and nine signs for 60. Th is is less complex than it seems, because within specifi c traditions, there were set sequences of signs that would be understood by anyone working within them. In some instances, parts of these sequences may be comprehensible; for instance, in Nepali manuscripts from the eleventh to the fourteenth centuries, the numbers 1 through 3 were represented by the syllables e, dvi, and tri, which correspond to the Nepali lexical numerals (Burnell 1968 [1874]: 66). In other cases, the signs used appear to have been assigned almost randomly. Datta and Singh (1962 [1935]: 73) list many signs for which they cannot even attach a plausible syllabic value. Dialect 212 Numerical Notation diff erences in pronunciation or paleographic variation among scripts may account for the irregularity of the aksharapallî systems. Th e aksharapallî numerals are ciphered-additive, but the lack of a regular correlation between the signs and numerical values suggests no obvious origin. Th ey may have developed directly from the Brāhmī ciphered-additive numerals, with only the use of phonetic rather than abstract symbols to distinguish them. Th is matter is made even more complex by the fact that many modern scholars still maintain that the origin of the Brāhmī numeral-signs was as a modifi cation of phonetic signs (as mentioned earlier). Indian scholars considered the aksharapallî to be part of the varnasankhya tradition of alphasyllabic numeration, so I believe they are related to the other alphasyllabic Indian systems, although possibly with some infl uence from the Brāhmī system. Th e primary use of aksharapallî numerals was for the pagination of manuscripts. Th is may explain why there was apparently no need for any such system to express numbers in the high hundreds and thousands. Aksharapallî numerals had the greatest and most consistent level of use of any of the alphasyllabic numerals of India. Th ey were used with great frequency in the manuscripts of the Jains until the sixteenth century, although it is not clear why this system would appeal specifi cally to Jains (Datta and Singh 1962 [1935]: 74). Th ey also survived for a very long time in Nepal (Burnell 1968 [1874]: 65). Temple (1891) goes into great detail concerning a ciphered-additive numerical notation system used for arithmetic by Hindu astrologers in Burma in the late nineteenth century. While he does not indicate whether the signs were alphasyllabic, nor does he use the name aksharapallî (or any other name) to describe this notation, no other ciphered-additive notation is likely to have been used in Burma at that time. Aksharapallî numerals appear to have thrived along the Malabar Coast; they were still common enough in Malayalam-speaking regions in the middle of the nineteenth century to be included in some grammars (Bendall 1896). Th ese groups are all relatively distant from the central political and religious movements of India, so their survival may refl ect the marginal status of such places in Indian history. Ak- sharapallî systems continue to be used occasionally throughout India, Bangladesh, Nepal, Tibet, Burma, Cambodia, Th ailand, and Java (Ifrah 1998: 484). Th e tradition of Indian alphasyllabic numeration, while lasting well over a millennium and playing a signifi cant role in Indian astronomy, astrology, and mathematics, did not infl uence numeration practices outside of South and Southeast Asia. Th e diff erent varnasankhya systems have in common a suitability for fl exibly representing numerical phrases in verse, but in mathematical traditions that did not involve versifi cation, alphasyllabic numeration was undesirable. Moreover, the Arabic and Western scripts were unsuitable for modifi cation to suit the unusual structure of the alphasyllabic numerals, and both the Arabic abjad and the Greek South Asian Systems 213 alphabet had their own alphabetic numerical traditions. Th us, while the Indian tradition of ciphered-positional numerals spread fairly readily into the Arab world, alphasyllabic numeration remained a strictly South Asian phenomenon.

Arabic Positional Th e Arabic script is written from right to left, and is basically consonantal, though with some representation of vowel sounds. Th e earliest Arabic speakers used the hybrid cumulative-additive/multiplicative-additive Nabataean numerals (Chapter 3); this was replaced after the Islamic conquest of Greek-speaking regions by an alphabetic system (the abjad numerals) akin to the Greek, Hebrew, and Syriac systems (Chapter 5). Shortly after the introduction of abjad numerals, however, users of the Arabic script became aware of Indian ciphered-positional numerals, and developed their own system on this basis, whose modern numeral-signs are shown in Table 6.13. Th is system is written with the higher powers on the left, rather than from right to left following the direction of the Arabic script. Th us, 26,049 would be written . Burnett (2000b) discusses this phenomenon in light of the fact that Western numerals are read from left to right but in this case in accordance with the direction of the script, noting that this may have led to some confusion when the Arabic numerals were borrowed into the West as to the direction in which the numerals should be read. In the modern Arabic numeral-signs, there are alternate signs used for 4 ( ) and 5 ( ), and the modern zero sign is written with a dot instead of with a circle because the circle was already assigned the value of fi ve. Th ere are undeniable paleographic resemblances between the Arabic positional numeral-signs and those used in medieval north India. Table 6.14 compares the Arabic positional numerals found in eleventh-century mathematical and astronomical treatises with the inscription found at Gwalior, India, dated to 876 ad, containing the Nagari numerals used in medieval India. Th ese signs are very similar, and it is thus safe to assume that the Arabic numerals have an Indian origin. In some cases, as for 2, 3, 7, 8, and 9, the Nagari numeral-sign became rotated or inverted, which may have resulted from the scribal practice of writing from top to bottom, then rotating the manuscript to read it (Ifrah 1998: 532–533). Th e fact that medieval and modern Arabic scholars are unanimous in attributing an Indian origin to these signs, and call them ḥisāb al-hindi (Indian numerals), abundantly confi rms the paleographic evidence. Th e social context of the transmission appears to have been limited to the exact sciences initially, specifi cally to astronomy. In 662 ad, a Syrian Christian bishop, Severus Sebokht, noted the Hindu profi ciency in astronomy, commenting that “as for their skilful methods of calculation and their computing which belies 214 Numerical Notation

Table 6.13. Arabic positional numerals

0123456789

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

Table 6.14. Early Arabic and Nagari positional numerals

0123456789 Arabic Nagari 0123456789 South Asian Systems 215

From their origins in the late eighth and early ninth centuries ad, the numerals spread throughout the Islamic world, though not without resistance or confusion. Many conservative scribes and bookkeepers resisted the new numerals in favor of older calculation on the fi ngers and with numeral-words. In his Kitāb al-mu’allimīn, the ninth-century scholar al-Jāhiz recommended fi nger reckoning above the ḥisāb al-hindi because it needed neither speech nor writing, a position echoed a century later by the historian al-Sūlī in his Adab al-kuttāb (Kunitzsch 2003: 4–5). Whether this functional explanation is complete, or whether other, ideological considerations came into play, is unknown. Lemay (1977: 440–444) questions the extent to which the Indian numerals were known to the Arabs before the tenth century, and shows that there was confusion among some Arabic thinkers over how they worked. While positional numerals began to dominate in both mathematical and nonmathematical contexts starting in the eleventh century, astrological texts remained far more conservative, retaining the abjad numerals solely until the fourteenth century (Lemay 1982: 385–386). Contrary to the diff usion of most numerical notation systems, scientifi c functions rather than commerce or religion provided a signifi cant impetus for the transmission of the positional numerals from India westward to the Arabs. It is nonetheless generally the case that medieval Arabic arithmetic did not distinguish commercial, astronomical, divinatory, and other arithmetical practices unambiguously. Th e Arabs borrowed not only the Indian numerals, but also a host of computational techniques and devices, including the dust-board, a fl at tablet strewn with sand into which fi gures could be written for undertaking computations (Bag 1990: 290–293; Folkerts 2001: 14). Other techniques available included a complex Greek-derived system of fi nger reckoning and the use of counters or shells; accordingly, the use of written pen-and-paper arithmetic was apparently not part of the initial practice of Indian-derived numeration. Th e earliest major Arabic arithmetical text to advocate Indian numeration instead was the Kitāb al-fusūl fī al-hisāb al-hindī of al-Uqlīdisī ‘the Euclidean’, written in Damascus in ad 952/3, the earliest extant copy of which dates to ad 1186 (Saidan 1966; Burnett 2006: 16). Yet al-Uqlīdisī recommended concealing the Indian origin of the technique, using the fi rst nine letters of the Greek alphabet or the Arabic abjad instead of the numeral-signs for 1 through 9 (Saidan 1966: 478–479). Many of the earliest Arabic arithmetic texts had names such as kitāb al-takht ‘book of the board’, confi rming the association of the new numerals with dust-board computation (Smith and Mourad 1927). Yet once the ḥisāb al-hindi had been fi rmly established by the eleventh and twelfth centuries, Arabic mathematical texts began to advocate doing computations with paper and ink instead of the dust-board, which of necessity involved rubbing out intermediate steps of 216 Numerical Notation computations and thus could lead to error. Clearly, Arab mathematicians’ sentiments toward the innovation were complex – both recognizing it as highly useful in comparison to earlier techniques, yet desiring to improve it to suit their own purposes and prejudices. Despite their prominence, the Arabic positional numerals have given rise to few structurally distinct descendants. Some of the cryptographic systems of the Ottoman Empire (Chapter 10) may have been inspired by the Arabic positional system, although most of these systems are additive and may be more closely related to the abjad numerals. A number of sub-Saharan African ciphered-positional systems developed in the twentieth century (Chapter 10) were created by inventors who knew the Arabic and/or Western positional numerals. While the debt of Western numerals to Arabic positional notation is undeniable, they are directly related to the ciphered-positional Maghribi numerals (known sometimes as ghubar numerals, and in medieval Latin as fi gure toletane ‘Toledan fi gures’) described in the following section, and are thus at best phylogenetic “cousins” of the signs commonly used in the modern Islamic world. Burnett (2000c) notes, however, that routes of transmission were complex, and the standard Arabic numerals were described in several Western European texts, through the medium of the Crusader state at Antioch in the twelfth-century Latin translation Liber Mamonis, and through the city of Pisa in the Latin translation of the Hebrew works of Abraham ibn Ezra (see Chapter 5). Th ese “Eastern gures”fi (known in Latin as fi gure indice ‘Indian fi gures’) were, if never widespread, at least known to Western mathematicians, but by the early thirteenth century, the migration of Toledan translators to northern Italy marked the decline of “Indian fi gures” in Western Europe (Burnett 2002a). Th e Arabic numerals enjoy a degree of currency and use in the modern world second only to the Western numerals. Th ey are used regularly in most contexts throughout all regions that employ the Arabic script, and are thus found regularly from Morocco to Indonesia. While global commerce and the eff ects of mass media have introduced Western numerals into the Arabic-speaking world, and most literate users of the Arabic script know them, it is unlikely that this will have any long-term eff ect on the use of Arabic numerals.

Maghribi (“GHUBAR”) Numerals A set of ciphered-positional numerals quite distinct from the regular Arabic system was used in North Africa and southern Spain during the medieval era and sporadically thereafter. Th ese numerals, sometimes known in Arabic as ḥisāb al-ghubar ‘dust-reckoning’ and commonly known in the scholarly literature as ghubar ‘dust, sand’ numerals, are particularly important for this study because they are the South Asian Systems 217

Table 6.15. Maghribi numerals

0123456789 ^ U V Wc X Y e Z [g \ h ] immediate ancestor of the Western numerals.7 While the paleographic forms of the Maghribi numeral-signs vary, representative examples are indicated in Table 6.15 (Gandz 1931, Souissi 1971, Labarta and Barceló 1988). In Chapter 5, I showed that North Africa and Spain were quite distinct from the rest of the Arabic world, both in their use of a diff erent ordering of the abjad numerals and in their use of special “Fez numerals.” Likewise, comparing the Maghribi numerals to the standard Arabic positional numerals (either the medieval or modern forms), while they are both decimal, ciphered-positional numerical notation systems written with the highest powers on the left, the two systems diff er paleographically. Potential early examples of the Maghribi numerals come from two documents dated from 874 and 888 ad, respectively, in texts from the Maghreb (Gandz 1931: 394). It is perhaps notable that the fi rst textual example of the Maghribi numerals comes only one year after the appearance of the fi rst regular Arabic positional numerals known in Egypt, but this may simply refl ect accidents of survival and discovery. If these ninth-century texts actually contain Maghribi numerals, this would accord well with their tenth-century diff usion into the Latin manuscript tradition in Spain. However, Kunitzsch (2003: 11) argues, contrarily, that the earliest unambiguous Western Arabic/Maghribi numeral-signs in Arabic texts are from MS Florence, Or. 152, dating to the middle of the thirteenth century (!). A good part of this discrepancy arises from the fact that it is not a simple matter to distinguish the Maghribi numerals from the Eastern forms in early texts; clarifying the origin of the divergence will help to resolve the problem. Smith and Karpinski (1911: 98), Das (1927b: 359), and Datta and Singh (1962 [1935]) argue that the Maghribi numerals are closer to the original Indian forms, and thus represent an earlier transmission, than the later Eastern Arab forms. Th ey claim from this that the Maghribi numerals were the ones used by al-Khwārizmī and other early mathematicians; however, such a conclusion is overly speculative. Another theory, popular from about 1915 until 1935, holds that the Maghribi numerals (and hence Western numerals) came from India to Spain via Neo- Pythagoreans in Byzantium, while the standard Arabic numerals came from In- dia via the caliphate of Baghdad (Carra de Vaux 1917; Cajori 1919; Gandz 1931:

7 Following Kunitzsch (2003: 10), who argues persuasively that the term “ghubar” has been misapplied, I use the label Maghribi, refl ecting the system’s geographical origin. 218 Numerical Notation

395; Miller 1933; Lattin 1933: 184–185). Yet there is no evidence for ciphered-positional numerals in Byzantine Greece prior to the twelfth century (Wilson 1981). Th e “rejectionist” view is further refuted by the fact that medieval Arabs, Western Europeans, and Byzantines were in accord that the numerals were of Indian origin (Lemay 1982: 382). It is best now regarded as an ethnocentric relic of the misguided notion that Indians and Arabs were uncreative. Th e term ghubar, with its unusual meaning of “dust” or “sand,” has prompted some comment as to the function of the number. Das (1927b: 358) and Gandz (1931) assert that the ghubar tradition represented a sort of Arabic abacus, but Kunitzsch (2003) shows that texts discussing the system refer to a takht ‘board’ on which writings were made and from which items could be erased – that is, boards covered with dust or sand that were used as calculating boards by drawing fi gures on them. Some of the variation between the Arabic positional and Maghribi numerals may be explained by their use on diff ering media, the former in permanent media and the latter for arithmetical calculations on sand-boards. Th eir forms, thus xedfi by the separation of contexts, might have become entrenched through centuries of use in disparate parts of the Islamic world. However, no actual “dust-numerals” in that medium survive, of course. Kunitzsch (2003: 9–10) argues that despite the terms ḥisāb al-hindi and ḥisāb al-ghubar being references to diff erent media and computational techniques, they do not necessarily imply two distinct sets of graphemes associated with each. While the term ghubar seems to have originated in Tunisia and is associated with Maghribi scholars such as ibn Khaldun (whose fourteenth- century Muqaddimah mentions only ghubar, abjad, and zimām [Coptic] numerals, not hindi) (Lemay 1982: 384–387), the distinction in sign-forms is thus better understood as a geographical rather than a functional one. In fact, the ordinary Arabic numerals and the Maghribi numerals were quite similar until the twelfth century; their numeral-signs for almost all values are similar enough to be explained as graphic variations of a common system of Indian derivation (the medieval Nagari ciphered-positional system). In a tenth-century manuscript written by the Persian astronomer Sijzi, the form of numerals used is intermediate between the Arabic and Maghribi forms (Mazaheri 1974). Maghribi numerals are thus a subset of the larger class of Indian-derived positional, decimal numerals (Lemay 1977: 437), both of which stand in contrast to the abjad numerals described in Chapter 5. Th is does not mean that the transmission of positional numeration was a singular event, but it does suggest that it was not a matter of two distinct “waves” of diff usion, one into the Maghreb and the other into the rest of the Arab world, but rather a signifi cantly more complex series of episodes that resulted in two parallel systems. While the Maghribi numerals began as a paleographic variant of the Indian numerals, they eventually took on a distinct cultural meaning among the scribes, South Asian Systems 219 astronomers, and mathematicians of the western Islamic world. Th is is partly because of the relative independence of polities such as the caliphate of Cordoba from the Baghdad-based Abbasid caliphate. Th e traditionalism of the Maghrebi and Andalusians may partly explain why the numerals persisted even after the rest of the Islamic world had adopted the signs now used throughout the modern world (Ifrah 1998: 539). Th ey were still regularly used in Spain and North Africa in the fi fteenth and sixteenth centuries, and sporadically thereafter (Labarta and Barceló 1988). Th e Maghribi numerals would be little more than a paleographic curiosity, merely one of many ways of writing Arabic numerals, if not for the fact that through them, Western Europe adopted ciphered-positional numerals. For the past seventy-fi ve years, all major scholars have agreed that the resemblances between Maghribi and Western numerals, coupled with the fi rst appearance of the latter system in medieval Spain, demonstrate this origin. Yet even as the Western numerals developed along their own trajectory within the Christian European context, the Maghribi numerals survived for almost a full millennium. Ifrah (1998: 535) provides examples of arithmetical texts written using the system from as late as the eighteenth century, and suggests that the system may have survived into the nineteenth century before being completely replaced by the standard Arabic numerals.

Western Numerals From their origin as a foreign and suspicious novelty during the medieval period, the ten Western numerals, structured by the use of the positional principle, have become so familiar that it is easy for the nonspecialist to forget that there are other numerical notation systems. Th e ubiquity and universality of the Western numerals make understanding their origin and diff usion all the more important. Unfortunately, no monograph has dealt systematically with the topic since Hill (1915), whose work is rather outdated as a result of advances in paleography. Th e fi rst example of Western numerals is generally held to be the Codex Vigi- lanus, written in 976 in the monastery of Albelda near the town of Logroño in northern Spain, in which the numerals are described (in Latin) as “Indian fi gures” (Hill 1915: 29; Burnett 2002b: 241). Th e nine units are listed, in descending order, but no zero-sign, probably because the signs were intended for use with a counting board. Th ese signs are shown in Table 6.16. Th ese fi gures are very similar to the Maghribi numerals shown in Table 6.11, and in fact there is no reason to consider them a separate system, except that they are used in a Latin and Christian text from northern Spain rather than in an Ara- bic one from Andalusia. Toledo was a major center for the transmission of Arabic knowledge to the Christian West in the tenth and eleventh centuries, and Lemay 220 Numerical Notation

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

123456789 lmnopqrst

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

8 A remaining puzzle is why the Roman numerals for 1 through 9 were never employed on the counters; although Burnett (1997: 11) argues that this would have been impractical, I do not see any reason why even a long numeral-phrase like VIIII for 9 would have been too long to have been used in this fashion. South Asian Systems 221

Th e spread of Western numerals into the tradition of manuscript writing (in both mathematical and other texts) began in earnest in 1202, at which time the mathematician Leonardo of Pisa, better known as Fibonacci, promoted their use in his Liber Abaci (Book of the Abacus). Despite its name, the purpose of Fibon- acci’s text was to promote not the use of the abacus, but rather the use of written numerals for computation, with nine unit-signs and a zero-sign (k), following Arabic practices advocated by scholars such as al-Uqlīdisī and al-Khwārizmī. Later thirteenth-century texts following in Fibonacci’s wake, such as Alexander de Villa Dei’s Carmen de algorismo and John de Sacrobosco’s Algorismus vulgaris, used the term algorismus, a corruption of the name al-Khwārizmī, to refer to this new art (Burnett 2006: 19). While he did not use the term himself, Fibonacci was thus the forerunner of the algorithmists who, in direct confl ict with the abacists, promoted the use of written numerals for computation rather than the use of counting boards (cf. Evans 1977, Murray 1978). Th is technique is the precursor to modern computational techniques with pen and paper. Despite the unquestionable importance of Gerbert, Fibonacci, and other mathematicians in introducing ciphered-positional numerals to the West and promot- ing their use, their eventual adoption is not a vindication of a “great man” theory of history. Among the names given to the zero-sign in a late twelfth-century Latin manuscript (Cambridge, Trinity College R.15.16, Fol. Av) was “chimaera,” suggesting that it was assimilated only with diffi culty into the conceptual system of medieval mathematical thought. Western numerals were not initially given the same conceptual status as letters of the alphabet, or even as Roman numerals. Th ey were, instead, seen as characteres, signs to be made on physical artifacts, and only gradually assimilated into texts as written signs (Burnett 2006: 29). Th e diffusion of the Western numerals from Andalusia and North Africa to the West occurred slowly, numerous times and by several diff erent routes, some of which were more fruitful than others (Gibson and Newton 1995: 316). In fact, Greek and Ital- ian mathematical manuscripts contain the standard Arabic “Eastern” positional numerals, not ghubar-derived ones, through the twelfth century (Burnett 2002b). A late twelfth-century Latin manuscript from Bavaria contrasts the “Toledan fi gures” and the “Indian fi gures” (Burnett 2002b: 241). Only in the thirteenth century, when many of the important Toledan astronomers moved to northern Italy, were the standard Arabic forms fully abandoned. Contact between the Arab and Western cultural spheres followed several paths in the Middle Ages: through Spain, to be sure, but also through Norman Sicily, along main trade routes from African cities such as Tunis and Tripoli to Venice and Genoa, through the Cru- sader states such as Antioch, and through Byzantine Arabs (Burnett 2006). Far from being an instantaneous adoption, then, the Western numerals were used only by a small number of Western European scholars in the Middle Ages. 222 Numerical Notation

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

9 Th e quasi-alphabetic “Fez numerals” used by notaries in the Maghrib (Chapter 5) served similar functions, but in that case the nonpositional system was the obscure one and the positional system the more commonplace. South Asian Systems 223

Table 6.17. Early Western numerals in Europe

Location First Attested Example Common Use 976: Codex Vigilanus (Hill 1915: Spain 1490:a dating pages in texts 29) c. 1325: banking records and c. 1050–1075: Pandulf of Capua’s account books in major cities Italy De Calculatione (Struik 1968; Menninger (Gibson and Newton 1995) 1969: 428) mid to late 11th century: abacus France c. 1400: dating, accounting, etc. treatises (Hill 1915: 29) c. 1130: Adelard of Bath’s transla- 1525–1550: archival records, tion of al-Khwārizmī accounting books (Jenkinson England 13th century: Wells Cathedral (fi rst 1926) epigraphic use) 1600–1630: probate inventories (Morley 1947: 81) (Wardley and White 2003) 1143: translation of al-Khwārizmī Germany/ 1525–1550 (Smith and into Latin at Vienna Austria Karpinski 1911: 133) (Menninger 1969: 411) 12th century: commentaries on c. 1400: Ottoman conquest of Greece Euclid’s Elements (Wilson 1981) most Greek-speaking areas c. 1275–1300: Valdemar’s year- c. 1550 (books, manuscripts, Scandinavia book (Kroman 1974: 120) records) c. 1310–1330: Haukr Erlends- son’s translation of Carmen de c. 1550 (books, manuscripts, Iceland algorismo (Benedict 1914: 17; records) Bekken 2001) 1490–1510: travelogues, scien- 1415: Livro da Virtuosa tifi c documents (Barrados de Portugal Bemfeitoria (Barrados de Carvalho 1957: 125) Carvalho 1957: 124–125) 16th–17th century: general use (de Oliveira Marques 1996) 17th century: sundials early 18th century (reforms of Russia (Ryan 1991) Peter the Great) a Arabic documents from Spain used the ghubar numerals extensively from the tenth century ad onward; this date refers only to their common use in Christian Spain. the numerals expanded from monks and astronomers did Western numerals begin to replace Roman numerals more generally, however. Th e common use of Western numerals in Europe was surely aided by the transmission of double-entry bookkeeping from Italy in the fourteenth and fi f- teenth centuries; this technique greatly benefi ts from ciphered-positional numerical notation (Jenkinson 1926: 267). Yet the Italian merchant Francesco Datini (c. 1335–1410), an early adopter of double-entry bookkeeping, switched his 224 Numerical Notation accounts to Western numerals in 1366 but adopted double-entry accounts only in 1383, suggesting that the numerals may in fact have been the cause rather than the eff ect (Crosby 1997: 205–206). By 1600, however, Roman numerals were essentially absent in most accounting traditions. Continental Europeans started to date coins with Western numerals in the fi f- teenth century, the fi rst being a Swiss coin from 1424; Austria followed in 1456, and France, Germany, and the Low Countries in the fi nal quarter of the fteenthfi century (Hill 1915: 94–105).10 Th e earliest Western numerals on English coins are those of an issue of Henry VIII (undated, but with the regnal name “Henric 8”) from 1526–44 (Wardley and White 2003: 15–16). Th e increasing complexity of the alloying of coinage, particularly the need for complex divisions involving fractions, between the thirteenth and seventeenth centuries made the adoption of Western numerals by goldsmiths and mint workers highly useful (Williams 1995). Once coins began to be minted and records kept using the new numerals, their spread to a large segment of the populace was inevitable. Th e rise in frequency of the Western numerals corresponds well with the birth of printing in the middle of the fi fteenth century. Th e rise of literacy after the invention of the printing press, and the consequent expansion of the use of numeration to a broader range of people, correlated with a new willingness on the part of the middle class to use the new invention for a variety of purposes, including bookkeeping, inscriptions on coins and seals, foliation, and stichometry. Th roop (2004) emphasizes that the integration of Western numerals into Western typography was both a technical and a conceptual issue; typographers had to determine not only how to distinguish 0 / o / O or 1 / l (a problem not yet fully resolved today) but also how to produce numerals that accorded with the aesthetic canons of the rest of the typeface, a task fi rst accomplished by Claude Garamond (1480–1561). Doing so further helped in the standardization of numeral-forms, which had been extremely fl exible prior to the fi fteenth century (Hill 1915). Printed books were fi rst paginated in Western numerals in 1470: an edition of Chrysostomus’s Ho- miliae in Rome (McPharlin 1942: 20–21) and Werner Rolewinck’s Sermo in festo praesentationis in Cologne (Archibald 1921: 423). Bibles began being printed using Western numerals in the mid sixteenth century (Williams 1997). In all of these cases, micro-scale processes and eff ects relating to users’ social status and proximity to major centers of learning aff ected the exact dates at which the transition from Roman to Western numerals occurred. Wardley and White (2003) have demonstrated that even within a single country (England) and considering

10 A copper coin of Norman Sicily dated to 533 ah (1138 ad) is the earliest positionally dated coin in Europe, but it is inscribed and dated using the Arabic script and numerals (Hill 1915: 16; Menninger 1969: 439). South Asian Systems 225 only a single document type (probate inventories), regional diff erences in the dates of adoption of Western numerals were as much as a century – in Newcastle and some regions near the Th ames, Roman numerals were still used commonly as late as 1700. Nevertheless, the general trend in all regions was always in favor of Western over Roman notation. By the eighteenth century, Roman numerals in Western Europe served primarily archaic and formal functions, ending two millennia of their eff ective domination. It is remarkable that a system that had not yet been established in its heartland in the fi fteenth century could almost entirely replace not only the Roman numerals but also many other ancient systems in less than three centuries. Th e spread of the Western numerals throughout the world, and their eventual replacement of large numbers of indigenous numerical notation systems, occurred, however, only when European countries had become politically powerful, as the modern world-system began to form with the Western European nations at the core. Th e replacement of non-Western numerical notation systems began en masse in the sixteenth century.11 Within a few decades of Europeans reaching the New World, the Aztec and Maya systems had become obsolete and the Inka khipu greatly restricted in scope. At around the same time, the ciphered-additive systems of Eastern Europe and the Caucasus (Cyrillic, Glagolitic, Armenian, and Geor- gian) began to be replaced by Western numerals or Arabic positional numerals. Th e modern era of colonialism brought about the replacement of further systems starting in the nineteenth century. Th e Hebrew, Coptic, and Syriac alphabetic numerals all continue to be used for religious and formal purposes, but Western numerals are used in most other contexts. Th e indigenous numerical notation systems of South and East Asia have not been completely replaced, but they too have been supplanted for many purposes by the Western numerals. While there is no functional reason for the replacement of one ciphered-positional system by another, the dominance of the European nations, coupled with the desire to have a single, universally intelligible symbol system, have made the Western numerals an attractive option. Even in places like Japan and Th ailand, which were never under direct political control by a European power, the Western numerals are usually preferred. By the late nineteenth century, the history of the Western numerals had been suffi ciently obscured that considerable disagreement arose regarding their purported origins. While the correct notion that they originated in India and were

11 Curiously, the sixteenth century also marked the development of the attribution “Arabic” to the Western numerals, whereas earlier they had always been seen (correctly) as an Indian invention (Clark 1929: 217). It is possible that this occurred due to growing awareness of Arabic learning among early modern scholars. 226 Numerical Notation transmitted through an Arabic intermediary was always prominent, Kaye (1919) and Carra de Vaux (1917), among others, espoused the theory that their origin was among Greek Neo-Pythagoreans, who transmitted the knowledge of positionality to the Persians and ultimately to India. Th is imperialistic theory has no redeeming virtues and no evidence in its favor; there is no evidence that the Neo-Pythagoreans knew of place value. Th is theory rests chiefl y on the misguided notion that India has produced nothing of real scientifi c value. At the same time, the spread of Western numerals in the nineteenth and twentieth centuries has spawned a host of descendants among North American and African peoples, such as the Iñupiaq, Cherokee, Oberi Okaime, and Mende systems (Chapter 10). Many of these are structurally diff erent from the Western numerals, and are not simply the ordinary system recast with new numeral-signs. Modern octal, binary, and hexadecimal numbers used for electronics make use of the Western numeral-signs and ciphered-positional structure, merely substituting diff erent numerical bases. As well, while not structurally signifi cant, graphic variations of many of the numeral-signs are quite common (e.g., ! vs. #, $ vs. %, & vs. *, ( vs. )). Despite the near-universality of the Western numerals, the history of numerical notation is thus far from complete.

Summary We end in the modern era with a host of local numerical notation systems and two (the Western and Arabic) that spread enormously on the heels of political conquests, but we should not forget the origins of these systems in the South Asian regional tradition originating with Brāhmī. Th e common feature of the South Asian systems is the set of nine Brāhmī unit-signs – which persist, though greatly altered, in the surviving numerical notation systems – and a decimal structure. Only the alphasyllabic systems use distinct signs, the letters of the Indian alphasyllabaries. It is common practice to end studies of numerical notation with the Western numerals (e.g., Guitel 1975, Ifrah 1998). Yet to do so portrays the spread of West- ern numerals throughout the world as the inevitable replacement of worse with better systems, in continuous progress from primitive beginnings to the perfection of the Western decimal positional system, an achievement that can never be surpassed. Most surviving modern systems are ciphered-positional, which indicates that they are useful, but this does not demonstrate that they are the inevitable conclusion of a teleological historical process. If technology truly spread only through the diff usion of what is functional and the replacement of what is not, one would expect that structurally identical systems should expand with equal rapidity and geographical reach. Th e South Asian phylogeny, with so many decimal ciphered- positional systems surviving and in use, provides a good testing ground for this South Asian Systems 227 theory. Because only the Arabic and Western numerals (and, to a lesser extent, Nagari) have spread extensively, their diff usion must be due mainly to sociopolitical factors. Furthermore, the geographic distribution of the surviving nonpositional systems is no less than that of many positional ones. Why would the additive Tamil system be as widespread as, say, the Khmer system, if functionality is of supreme importance? Do we really expect Tibetan ciphered-positional numerals to survive and Chinese multiplicative-additive ones to decline? I do not mean to suggest that functionality has nothing to do with the spread of numerical notation systems, especially ones such as the Arabic and Western numerals that have been used extensively for accounting, arithmetic, and mathematics. Yet to proclaim the Western numerals’ spread as the triumph of functionality and reason over illogic and unwieldiness is to ignore the many ciphered-positional systems that have failed to spread – or failed to survive. While, owing to the political might of nations that use them, the Western numerals are very important, they are merely one branch of one phylogeny. In placing them in the middle of my study, I choose to emphasize that their present triumph is neither inevitable nor eternal. chapter 7

Mesopotamian Systems

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

228 Table 7.1. Mesopotamian numerical notation systems

System 1 10 60 100 120 600 1000 1200 3600 7200 10,000 36,000 Archaic Systems Sexagesimal ABCDEF Bisexagesimal AB C M NO Bisexagesimal 2 SZ P Q R Proto-Elamite decimal AB T [ ]

229 Cuneiform Systems Sumerian fgf h j k Assyro- Babylonian fgfi gi Mari fg f fm x Hittite fgfi Old Persian fg w Babylonian positional fgf g f g 230 Numerical Notation

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

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

1 However, Price and Pospisil (1966) claim that the Kapauku of Papua New Guinea derived their sexagesimal lexical numerals from the comparable Babylonian numerical notation. 2 Over the past twenty years of research, the chronology of protohistoric Mesopotamia has been shifted backward; older sources tend to regard the Uruk IV period as representing the early third rather than the late fourth millennium bc. 3 My discussion of the systems (including their functions) is derived almost entirely from the work of Nissen, Damerow, and Englund (1993: 25–29). Mesopotamian Systems 231 highest to lowest and regularly replace lower power-signs with higher ones wherever possible. A diffi culty is that a given numeral-sign may be found in several of the proto-cuneiform systems, but its value often varies from system to system. Th us, B is equal to 10 A in some systems, but to 6 A in others. Yet while we can identify the numerical ratio between the values of any two signs within a system, we often cannot identify the specifi c quantity represented by any one sign. For systems used for counting discrete objects, it is easy to identify the basic sign for 1, since fractions of humans do not normally occur in texts, but for systems that measure area or capacity, we can never ascertain with certainty which sign (if any) has the basic value of one unit. Following Nissen, Damerow, and Englund (1993), I present the values for these metrological systems as ratios, since we can only tell the value of a sign relative to the other signs of the system. Despite having diff erent numeral-signs and diff erent numerical values, all the proto-cuneiform numerical systems have much in common. All are cumulative- additive, although some individual numeral-signs are formed multiplicatively – e.g., D (600) = C (60) × B\(10). Groups of identical signs were sometimes sorted into two or three rows for easy reading, but this is not a universal rule, and some tablets contain long strings of signs. Numerals were most often grouped with signs arranged from highest to lowest (although there are some rare exceptions, which may be scribal errors). A single numeral-phrase, together with one or more ideograms, was enclosed in a box in a section of the text.4 Th e proto-cuneiform texts are mainly accounting documents, often written on both sides – the obverse with a series of amounts of commodities, the reverse with a single total. Th e greater-than-expected prevalence of round or nearly round numerals in proto-cuneiform texts allows the identifi cation of hypothetical problems that were used as training exercises and thus were not actual economic texts (Friberg 1998). Learning the various signs, the ratios among them, and how to construct texts would have required considerable scribal training, including “school mathematics” associated with the temple economy at Uruk (Robson 2007: 63–64). For instance, the late fourth-millennium tablet W20044,20 is an exercise in calculating the area of an irregular quadrilateral fi eld (Robson 2008: 30). Th roughout the history of the various systems’ use, only a very small portion of the populace would have had access to the training necessary to master the proto- cuneiform notations.

4 Note that the proto-cuneiform script was written vertically in columns reading from top to bottom, but I follow Assyriological convention (and that used by Nissen, Damerow, and Englund) in showing the signs rotated ninety degrees counterclockwise and thus read horizontally from left to right. Th is convention refl ects a similar change in the direction of writing cuneiform signs around the middle of the third millennium bc. 232 Numerical Notation

Table 7.2. Sexagesimal numerals

36,000 3600 600 60 10 1 1/2 Sexagesimal (S)a F =10 E = 6 D =10 C =6 B =10 A =2 L Sexagesimal (S’) ( =6 & =10 ^ a Th e letters in parentheses in this table and the following ones are those assigned to each system by Nissen, Damerow, and Englund (1993) in their research.

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

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

Table 7.3. Bisexagesimal numerals

7200 1200 120 60 10 1 1/2 Bisexagesimal (B) O =6 N =10 M =2 C =6 B =10 A =2 L Bisexagesimal (B*) R =10 Q =2 P =6 Z =10 S Mesopotamian Systems 233

Table 7.4. GAN2 numerals

E =6 F =10 B =3 W =6 A =10? L

GAN2 System

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

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

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

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

Table 7.5. EN numerals

B =10 A =2 Ä =2 L 234 Numerical Notation

Table 7.6. ŠE numerals

System Š D =10 C =3 E =10 B =6 A =5 J System Š’ Æ =10 ô =6 ö =5 ò =5 û System Š* ë =3 è =10 ï =6 î =5 ì

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

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

Table 7.7. U4 numerals

\\+ =10 , =3 - =10 . * =12 10 months / 1 month 10 days 1 day 1 year Mesopotamian Systems 235 signs resembling later archaic numerals and contain the correct total of tokens, suggesting that the systems are connected (Nissen, Damerow, and Englund 1993: 127–129). Furthermore, there are some similarities between the three-dimensional tokens and the proto-cuneiform ideograms, suggesting that the tokens developed into writing through the recognition that, if the total of a transaction is written on clay, one need not actually use the clay tokens but need only record their values. Schmandt-Besserat’s conclusions have been received with some skepticism (see especially Lieberman 1980, Zimansky 1993).5 Firstly, the scope in time and space of the token system is far greater than that of the proto-cuneiform numerals; it is implausible that such a widespread phenomenon represents a uniform system. Moreover, in Schmandt-Besserat’s study of tokens from the Uruk-Jemdet Nasr period (c. 3000 bc), around two-thirds of the tokens come from Susa in Iran, while only 10 percent come from the very thoroughly excavated site at Uruk (Lieberman 1980: 353). Th is suggests that the token system is unlikely to have given rise to numerals and writing at Uruk. Finally, some of the most common tokens are correlated with proto-cuneiform signs for rare objects such as nails and days of labor, whereas given the accounting function established for the tokens, we would expect livestock, people, and grain to be the most common tokens, as is the case in proto-cuneiform texts (Zimansky 1993: 316). Th is discrepancy points to a further problem. Th e archaic numeral systems always place a numeral-phrase in front of an ideographic sign; “16” + “sheep” = “16 sheep,” and so on. While the proto-cuneiform numerals partly fuse quantity and quality, because diff erent systems represent diff erent commodities, they do not do so completely, because one always needs a further sign to indicate exactly what is being counted. With the tokens, however, there is no separation of numerals and the objects being counted; to show sixteen sheep, one simply uses sixteen tokens for “sheep.” Th us, there is no correspondence between the archaic numeral- signs and the shapes of tokens. Th e use of tokens sealed within bullae appears to have been an accounting technology that pre-dated, but then later coexisted with, the proto-cuneiform numerals. While some early proto-cuneiform numerals are found on clay bullae, this is insuffi cient evidence that tokens led to numerals. Conversely, numerical signs resembling the proto-cuneiform ones have been found, not on bullae, but on ordinary clay tablets in late preliterate contexts at Uruk as well as at Jebel Aruda, Susa, and elsewhere (Nissen, Damerow, and Englund 1993: 127–130; see especially Figures 113, 114). Th ese tablets have numerical signs only (no ideograms), and disobey the ordinary rule that once a certain

5 I cannot hope to address her claim that the tokens are ancestral to the proto-cuneiform script, and will restrict myself to the similarities and diff erences between the token system and the proto-cuneiform numerals. 236 Numerical Notation number of lower-valued signs have been written, they are replaced with a single higher-valued sign. For instance, one tablet from Jebel Aruda contains twenty-two B signs, among others (Nissen, Damerow, and Englund 1993: 130). In any of the later systems, twenty-two signs would have to be replaced by a smaller number of higher-valued signs. Because these inscriptions are found in late preliterate contexts and are similar but not identical to the proto-cuneiform numerals, they are immediately ancestral to them and date from a period when the system was still being developed, weakening the hypothesis that they derived from tokens. While we do not know the language in which proto-cuneiform numerals were read, the Sumerian lexical numeral system is mainly sexagesimal, and furthermore, 10 is a sub-base in the lexical numerals just as it is in the proto-cuneiform numerals (Powell 1971, 1972a, 1972b). On this basis, Powell (1972b: 172) has correctly discerned that “the presence of a sexagesimal system of notation in the archaic texts from Uruk and Jemdet Nasr constitute [sic] the best – indeed irrefutable – evidence that Sumerian is the language of those texts.” Yet, as Høyrup (2006: 79–82) points out, proto-cuneiform notation was not meant to record any spoken language, but was an artifi cial recording system designed for limited purposes, and only very gradually became assimilated to Sumerian.6 Th e multiplicity of proto-cuneiform numeral systems and bases is not lexical in origin, but is likely based on Sumerian metrological systems in the Uruk period, for which we have minimal nontextual evidence. We do, however, have substantial textual evidence for the metrological systems of the Early Dynastic and later periods. Th e ratios between various signs in the proto-cuneiform numeral systems dealing with measures of capacity, area, and weight are similar to the ratios found in later Sumerian metrological systems. Th is supports the contention that the odd ratios of some of the older systems are due to unattested metrological systems that continued into better-documented periods. Th e increasing administrative demands associated with the rise of the Uruk city-state in the late fourth millennium bc created a new need for record keeping, metrology, and accounting, of which the numerals and the clay bullae are two distinct consequences.

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

6 Høyrup attributes this purpose as being the administration of a multiethnic society of slaveholders and immigrant slaves, which we need not accept to recognize that the general principle is true. Mesopotamian Systems 237

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

7 Liverani’s (1983) intriguing conclusion that a fragmentary Uruk IV-period clay tablet indented with holes may have served as a counting board has not been confi rmed and must remain tentative unless further fi nds are made. 8 Schmandt-Besserat (1984, 1992) has made much of the parallel between the many proto- cuneiform numerical notation systems and the use of “numeral classifi ers” in Japanese, the Mayan languages, and others, where the set of numerals is modifi ed depending on the class of object being counted. Numeral classifi ers are not a feature of Sumerian. If taken to its logical conclusion, this would imply that the modern Japanese do not have a concept of abstract number. 238 Numerical Notation prevent confusion as to the type of thing being counted, or to correspond to metrological systems. Th ere is no qualitative diff erence between the Uruk systems and the modern use of Roman numerals to distinguish the foreword of a book from its main text, or the use of hexadecimal numerals for computing purposes. Ironically, one of the principles behind the “new mathematics” movement in North America in the 1960s was the claim that teaching students to calculate using numerical systems of diff erent bases would improve their understanding of abstract number concepts. I believe the Uruk scribes had an abstract number concept, but realized that abstract written numerals were not the most effi cient solution to the problems they were facing. Th e theoretical importance attributed to the proto-cuneiform numerals as evidence of an evolutionary stage of cognition is entirely unwarranted.

Convergence and Decline While they are an interesting early example of numerical notation, the proto- cuneiform numerals did not diff use extensively or last for an extended period. Th ere are no signifi cant resemblances between the proto-cuneiform numerals and the Egyptian hieroglyphic numerals (Chapter 2), which may precede the proto- cuneiform systems in any case. Th eir only descendants were the proto-Elamite systems used from about 3000 bc at the site of Susa and elsewhere in modern Iran, and the later Sumerian numerals. Th e start of the Early Dynastic period in Mesopotamian history marked a turning point in the history of its numerals. Beginning around 2900 bc, there was a marked decline in the frequency of almost all the proto-cuneiform numerical systems, while the sexagesimal system rapidly assumed the functions of the other systems. While the system for measuring area

(GAN2) continued to be used as late as the Fara period (c. 2500 bc), it was in decline and considered archaic by that point (Nissen, Englund, and Damerow 1993: 137–138). While each metrological system had its own numerical notation system in the archaic period, eventually offi cials decided it was better to express all numbers, regardless of function, using a single notation. Th ere are three plausible explanations for this convergence. Th e simplest is that the use of so many systems for so many diff erent functions was cumbersome for administration, potentially confusing, and open to abuse. Th is may simply be a modern prejudice attributable to the Western use of only one set of numerals. While 200 years is a short time in the context of world history, it is a long time for a truly ineffi cient set of systems to persist. Secondly, while the archaic texts were used at only a very few locales (mainly at Uruk), the later numerals were used throughout Mesopotamia. If the Early Dynastic period marks the fi rst era when Mesopotamian numerals were employed for long-distance communication, the use of a single system to facilitate communication among many individuals Mesopotamian Systems 239 would be advantageous. Finally, changes in Sumerian metrological systems may have reduced the usefulness of the proto-cuneiform systems by eliminating the fi t between metrology and numeration.

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

9 As with the proto-cuneiform numerals, I have represented the numerals as they would be read horizontally (following Assyriological convention; cf. Damerow and Englund 1989) rather than vertically (cf. Englund 1996). 240 Numerical Notation

Table 7.8. Proto-Elamite numerals (discrete objects)

1 10 60 100 120 600 1000 1200 3600 10000 Function A B C D E Inanimate Sexagesimal objects A B C MN Grain Bisexagesimal products A B T []Animate Decimal objects

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

Table 7.9. Proto-Elamite metrological numerals Capacity X =6 D =10 C =3 E =10 B =6 A =5 J =2 Ä (ŠE) K Area E =10? B =3 W =6 A (GAN2) Mesopotamian Systems 241

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

Sumerian Th e only system among the multitude of proto-cuneiform systems to survive into the Early Dynastic period (2900 to 2350 bc) was the sexagesimal (or, more accurately, the decimal-sexagesimal) system. While it was originally used only for counting discrete objects, it began to be used for all numerical functions as the older metrological systems were abandoned. At the beginning of the Early Dynas- tic, signifi cant changes were taking place in the script of the region. Th e older ideographic and curviform proto-cuneiform symbol system slowly transformed into a writing system that used wedge-shaped (cuneiform) signs and expressed phonetic as well as conceptual information. From this, we can tell that Sumerian was the language in which the script was read. Yet, despite these alterations to the script, the numeral-signs remained identical to the archaic sexagesimal ones. One important change occurred around the twenty-seventh century bc, when the numerals, like the entire script, underwent a ninety-degree rotation, so that they were written and read horizontally from left to right rather than vertically from top to bottom. Th e Sumerian numerals are shown in Table 7.10 (Nissen, Damerow, and Englund 1993: 28). Th ese six numeral-signs were combined to make a cumulative-additive numerical notation system. Normally, groups of four or more signs were arranged in two rows to facilitate rapid reading. Because this system has signs for both 60 and 3600 (= 602), it has a sexagesimal component. In a purely sexagesimal system, one would need to repeat each sign up to fi fty-nine times, which is impractical, but the Sumer- ian system also has sub-base signs for 10, 60 × 10 (600), and 3600 × 10 (36,000). Th e latter two signs are multiplicative combinations of the small circle for 10 with the sexagesimal signs for 60 and 3600. Th is decimal sub-base is similar, but not identical, to the use of the sub-base of 5 in the Roman numerals. While the fi gures of the 242 Numerical Notation

Table 7.10. Sumerian archaic numerals

1 10 60 600 3600 36,000 Vertical GB HIEF Horizontal AB C DE F 14,254 = EEE\DDD CCCC\BBB\AA DD CCC\ AA (3 × 3600) + (5 × 600) + (7 × 60) + (3 × 10) + (4 × 1)

19 = BB \Ay(20 LAL 1)

Roman sub-base (V, L, D) could occur only once in any numeral-phrase because 5, 50, and 500 are half of 10, 100, and 1000, respectively, the decimal signs in the archaic Mesopotamian numerals could be repeated up to fi ve times. Th e sign for 60 is a large version of the sign for 1, just as the sign for 3600 is a large version of the sign for 10. Because the “big 1” is 60 times greater than the regular 1, but the “big 10” (3600) is 360 times greater than its counterpart, I cannot agree with Lieberman’s (1980: 343) suggestion that these signs represent the use of “size-value,” which then evolved into “place-value,” over time. Th is feature simply derives from the fact that two styli, one twice as large as the other, were used to impress numerical signs on clay tablets (Powell 1972a: 11–12). Th e Sumerian numerals provide the fi rst evidence for the use of subtractive notation to express certain numbers, especially those that end in 8 or 9 in the Western numerals, using a Sumerian ideogram that corresponded with the phonetic value LAL. Th us, instead of writing 19 as one sign for 10 plus nine signs for 1, it could be written as 20 – 1, as seen in Table 7.10. A sign or signs placed inside the LAL sign indicated an amount to be subtracted from the signs preceding it. Th is technique was used at Fara (ancient Šuruppak), perhaps as early as 2650 bc (Jestin 1937: Pl. LXXXIV). Th ere is no evidence of subtractive lexical numerals in Sumerian comparable to the Latin duodeviginti and undeviginti; the Sumerian words for 18 and 19 are etymologically ‘10 + 5 + 3’ and ‘10 + 5 + 4’, respectively (Powell 1971: 47). Rather, this innovation had its origin strictly in numerical notation and the desire to express numbers more concisely. As in the archaic period, Early Dynastic numerals are found overwhelmingly in economic or administrative texts, or in scribal exercises related to these functions. In the archaic period, there was no indication how calculations were being done (though calculations must have been made). At Fara, however, Sumerian “tables of squares,” geometrical and arithmetical exercises, and other arithmetical aids Mesopotamian Systems 243

Table 7.11. Sumerian cuneiform numerals

1 10 60 600 3600 36,000 216,000 fgfh n k onm

LAL: u

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

GAL ‘big’, and is quite rare (Powell 1972a: 7). Powell also describes an even more complex phrase for 12,960,000 (216,000 × 60), šargal šunutaga, ‘big everything which hand cannot touch’. For such lexical phrases, we need to ask at what point a phrase ceases to become part of a numerical notation system. Th e subtractive ideogram LAL is used in this system, as in the archaic one, but it is depicted using two cuneiform wedges. Th e replacement of the curviform by cuneiform numerals was a gradual process. While the older system required additional styli to write numerals, and its numeral-signs generally took up more space, it stood out more clearly in a text of cuneiform characters, making totaling easier (Powell 1972a: 12). Additionally, a norm developed by which the two sets of numerals could be used side by side to indicate diff erent functions. Possibly the older numerals indicated quantities directly counted, possibly using clay counters, while the cuneiform numerals enumerated quantities of objects not actually present to be counted (Lieberman 1980: 344–345). Alternately, Damerow (1996: 238) notes that some Early Dynastic economic texts from Girsu use the older numerals for amounts of grain and the cuneiform numerals for amounts of animals, and hypothesizes that this may have been done to avoid confusing the two diff erent categories when taking sums. isTh use of multiple numerical notation systems is analogous to the modern use of Roman and Western numerals side by side. Th e round Sumerian numerals had been abandoned by the Ur III period (when Sumerian rulers regained control of Mesopotamia), and are not attested later than 2050 bc (Powell 1972a: 13). Th e Akkadian conquest, the most important political event of third-millennium bc Mesopotamia, had a minimal eff ect on numeration. Th e Akkadian kings and offi cials (c. 2350–2150 bc) were content to use the cuneiform and even the archaic numerals for most of the same purposes for which they had been used in the Early Dynastic period. More change in the numerals is visible in the Neo-Sumerian Ur III period (2150 to 2000 bc), during which the archaic numerals disappeared entirely. One slight modifi cation that was tried in some Akkadian texts was to write multiples of 60 using units followed by the Akkadian lexical numeral for 60, šu-ši (:), using multiplicative notation (Labat 1952: 244–247). Th us, instead of writing 120 as ff, it would be written as ff:. Th is is a much more cumbersome representation, and probably was used in part to distinguish 120 (ff) from 2 (ff). For the higher decades – 70, 80, and 90 – the regular Sumerian forms were always used by the Akkadians (fg, fgg, and fggg, respectively). Regardless, many Akkadian inscriptions where šu-ši could have been used are written in the ordinary Sumerian fashion. Th e Sumerian cuneiform system is ancestral to all the later systems of Mesopo- tamia. Th e Semitic cuneiform decimal systems (Eblaite and Assyro-Babylonian) were directly derived from a Sumerian ancestor. Th e decimal structure of these Mesopotamian Systems 245 systems refl ected the lexical numerals of the Semitic languages of its users. While Th ureau-Dangin (1939: 107) believed this tradition to have been developed in the Old Akkadian period (starting c. 2350 bc), it is now clear from the library at Ebla that it developed as early as 2500 bc (Pettinato 1981). Th e sexagesimal cumulative- positional system used in Babylonian mathematics and astronomy was also modeled on the Sumerian cuneiform system. It may have arisen in the Ur III period, and was used by the twentieth century bc at the very latest (cf. Powell 1976, Whiting 1984). Th e Sumerian cuneiform system continued to be used for most purposes until the Old Babylonian period (c. 2000–1595 bc). Around that time, the Assyro-Baby- lonian decimal system began to be used for most administrative, commercial, and literary functions, while the sexagesimal positional system was used for mathematics and astronomy – once again perpetuating the tradition of using multiple numerical notation systems for multiple purposes. Several Old Babylonian tablets provide translations from the old Sumerian additive numerals to the new positional system (Nissen, Damerow, and Englund 1993: 146–147), indicating either a need to learn the new positional system or, alternatively, that the older cuneiform system was already being forgotten. By the fi fteenth century bc, it had disappeared from regular use. However, a peculiar vestige of the base-60 Sumer- ian system survived in certain late inscriptions, particularly those indicating the sizes of buildings (De Odorico 1995: 4). One such example is the Nameninschrift of the Assyrian king Sargon II (722–705 bc), which describes the dimensions of the fortress at Khorsabad as “16,283 cubits, the numeral of my name,” notated using Sumerian numerals rather than the Assyro-Babylonian system that would be expected (Fouts 1994: 207). In this case, the notation served literally to indicate a numeral corresponding to the royal name of Sargon by correlating the signs for 3600, 600, and 60 with the phonetic values šar, nêr, and šûš, which could not have been done using the Assyro-Babylonian decimal system. Th is inscription has been compared to later Greek, Hebrew, and Arabic gematria using alphabetic numerals (Chapter 5), but it seems to be an earlier and independent development.

Eblaite Th e inhabitants of the city-state of Ebla (in the western part of modern Syria) spoke a West Semitic language but were strongly infl uenced by Sumerian culture. A great library of thousands of Eblaite cuneiform texts dating certainly to the period prior to 2350 bc (the Akkadian conquest), and possibly as early as 2500 bc, provide us with ample evidence regarding the numerals used by the Eblaites (Pettinato 1981). Th is system is almost identical to that used by the Babylonians some centuries later, and refl ects the shared Semitic language and culture of these 246 Numerical Notation

Table 7.12. Eblaite numerals

1 10 60 100 1000 10,000 100,000 AB C mi-at li-im rí-bax ma-i-at OR ma-i-hu fgf

AA\rí-bax AA li-im AAA mi-at CBBAA AA\\\\\\\ AAA 2 10,000 4 1000 6 100 60 10 10 1 1 = 24,682 two groups in contrast to those of the Sumerians. As indicated in Table 7.12, the Eblaite numerical notation system consisted of two sets of numeral-signs for numbers below 100, one curviform and the other cuneiform (corresponding with the Sumerian archaic and cuneiform systems), but only one set of expressions for the powers above 100 (Pettinato 1981: 183–184). Th e Eblaite system is cumulative-additive for values less than 100, and multiplicative-additive above that point. Th e signs for 1, 10, and 60 are ideographic signs identical to those used in the two Sumerian sets of numerals. Th e two sets of numeral-signs served quite separate functions: the curviform numerals were used for basic enumeration and counting discrete objects, while the cuneiform numerals were used only for capacity measures such as the mina and gubar, as well as for regnal years of kings (Pettinato 1981: 183–184). Th e sign for 60 was used to express the tens values in numbers between 60 and 99; its presence, a holdover from Sumerian, is the major irregularity in an otherwise perfectly decimal system. Th e “signs” for numbers above 100 are in fact the Eblaite lexical numerals and were combined multiplicatively with the unit-signs as necessary. Because it is decimal and multiplicative-additive above 100, this system required only one ideographic sign (the crescent or vertical wedge) for the higher powers; however, the repetition of intraexponential signs for the units, coupled with the use of complex two- and three-syllable words, meant that numerals were fairly long and cumbersome. To reduce this length, two features were often used. Firstly, just as in the Sumerian system, subtractive numerals were sometimes used for certain numbers to eliminate the need to write seven, eight, or nine unit-signs by placing the subtrahend after the syllable lal or lá. Secondly, the words mi-at for 100 and li-im for 1000 were often shortened to the single syllables mi and li, respectively. Th us, in one text, 7879 (expressing a number of gubar measures of barley) is written in cuneiform numerals as 7 li 8 mi 60 10 10 lá-1 (7 × 1000 + 8 × 100 + 60 + 10 + 10 – 1) (Pettinato 1981: 134). Such syllabic abbreviations are reminiscent of the Greek acrophonic numeral-signs (Chapter 4). Mesopotamian Systems 247

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

Assyro-Babylonian Common Because historians of mathematics are especially interested in the origins of our base-60 units of time and the division of the circle, enormous attention has been paid to the Babylonian positional numerals – the cumulative-positional, base-60 system used for astronomy and mathematics. Th e far more common decimal and additive numerals (which I call the Assyro-Babylonian common system), which were used for most economic, monumental, and literary purposes throughout Mesopotamia, are almost forgotten. Th is system came into common use in the Old Babylonian period (starting c. 2000 bc), a position it would maintain for over 1,500 years. Th e numeral-signs of this system are shown in Table 7.13 (De Odorico 1995: 4). Th e system is cumulative-additive below 100, multiplicative-additive above 100, and is always written from left to right. For the most part, it is purely decimal. Th e units were expressed cumulatively, except that 9 could be written using three overlaid vertical strokes ($) as an alternative to writing it with nine strokes (9) (De Odorico 1995: 4n). Th e tens values were usually expressed decimally using one through nine Winkelhaken corner wedges for 10. Th e vertical wedge for 60 is identical to that for 1, but unlike the Sumerian system, f was not normally used to represent 60 alone, which would have created ambiguity, but only in combination with signs for 10 and 1 to write numbers from 70 to 99. As in the Akkadian variety of the Sumerian cuneiform system, the lexical numeral šu-ši (:) was used to indicate 60 or multiples thereof; however, this phonetic form was not used to write 70, 80, or 90, and even for 60 its use declined over time. Th erefore, there were as many as three diff erent numeral-signs for 60, one decimal ü( ) and two sexagesimal (: or f). Above 100, the multiplicative principle was used quite freely and could be combined in various ways to express very high numbers. Th e sign for 100 was the syllabic sign ME, an abbreviation of the Babylonian word for 100, me’at, while that for 1000 was a multiplicative combination of the signs for 10 and 100. For instance, a scribe from the period of Sargon II wrote 305,412 as shown in 248 Numerical Notation

Table 7.13. Assyro-Babylonian common numerals

1 10 60 100 1000 fg f or : i gi

305,412 = 3i5gi4ia2 ((3 × 100 + 5) × 1000) + (4 × 100) + 10 + 2

Table 7.13 (Ifrah 1998: 139). In theory, this system could be extended as far as one wished by juxtaposing signs for 100 and 1000 repeatedly, even though there was no sign for zero. Furthermore, unlike the Sumerian system, in which the signs for 1 and 60 were identical, this system presented no ambiguities to the reader. While this system was sometimes called “Akkadian” (Th ureau-Dangin 1939), it was rarely used during the period of Akkadian control of Mesopotamia and began to predominate only during the Old Babylonian period. It originated in response to the increased power of Semitic peoples in Mesopotamia in the latter part of the third millennium bc: Akkadians, to be sure, but also Eblaites, Babylonians, and others. Its structure refl ects the decimal lexical numerals of the Semitic languages rather than Sumerian lexical numerals, although the continued use of a special sign for 60 gives testament to its descent from the Sumerian numerals. All of the administrative, commercial, literary, and religious texts of the Babylonians and the Assyrians were written using this set of numerals. Perhaps the greatest signifi cance of the Assyro-Babylonian common system is the large number of descendant systems it produced. Earliest among these is the system used at the city-state of Mari around 1800 bc, which blends features of this system and the Babylonian positional system. In the middle of the second millennium bc, both the Ugaritic and the Hittite cuneiform scripts began using numerals based on the Assyro-Babylonian ones, in the context of Mesopotamian trade with these polities. Th e Ugaritic texts written between the fi fteenth and twelfth centuries bc at Ugarit on the Mediterranean coast use the cuneiform ideograms for 1 (f) and 10 (g); however, numerals in Ugaritic were normally written lexically (Gordon 1965: 42). Th is was likely borrowed from the Assyro-Babylonian system, but we have no idea whether higher numbers could be expressed through numerical notation. Th e Old Persian cuneiform numerical notation system, developed in the sixth century bc (by which time Mesopotamia was under Persian rule), also derived from the Assyro-Babylonian system rather than from any of the numerous other systems then used in the region. Finally, as I argued in Chapter 3, the earliest Levantine systems (Phoenician and Aramaic) developed around 800 bc as a blend of Egyptian hieroglyphic (or perhaps Hittite) and Assyro-Babylonian infl uences. Mesopotamian Systems 249

Table 7.14. Graphic changes in numeral-phrases

478940 Sumerian p qsrt Babylonian 4789d

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

Babylonian Positional Th e Babylonian positional numeral system is assigned such great importance by many historians of mathematics that one could easily get the impression that it was the only noteworthy form of Mesopotamian numeration. Despite Neugebauer’s (1957: 17) warning that the positional numerals are a relatively minor part of the body of Babylonian numerals, these sexagesimal positional numerals, used for mathematics, have been assigned priority over much more widespread systems (Sumerian and Assyro-Babylonian). In fact, positional numerals were used in only a limited set of mathematical and astronomical contexts and over a much shorter period, serving primarily as a means of easing certain computations (Robson 2008: 75–6). Th e system uses only two basic numeral-signs, the vertical wedge f for 1 and the corner-wedge or Winkelhaken g for 10, to write any number between 1 and 59. Th us, small numeral-phrases were usually identical to those of the Sumerian cuneiform system. Nevertheless, certain graphic changes (shown in Table 7.14) were made to the numeral-phrases for 4, 7, 8, 9, and 40, so that, instead of grouping signs in at most two rows of up to fi ve signs, three rows of no more than three signs were used. Th is shift eliminated any phrases that placed four or fi ve signs side by side, and may have increased the system’s legibility (Powell 1972a: 16). Unlike the earlier cumulative-additive Mesopotamian systems, this system was cumulative-positional, combining the two basic signs in multiple positions to 250 Numerical Notation express powers of 60. It was thus a base-60 system with a sub-base of 10. It had an additive structure within each power, because of the way that 10-signs and 1-signs combine together, but a positional structure among diff erent powers. Just as in the Sumerian and Assyro-Babylonian systems, subtractive notation was frequently used to write numbers such as 9 (10 lal 1) and 19 (20 lal 1) (Th ureau-Dangin 1939: 106). According to the rules of the system, 4,252,914 would be written as a9 d4 b a4 = (19 × 603) + (44 × 602) + (20 × 60) + 14. In addition to expressing integers, positional numerals could be used to express fractional powers of 60: 1/60, 1/3600, 1/216,000, and so on. During the Old Babylonian period, the positional numerals did not have any sign for zero to indicate an empty position within a numeral-phrase, nor was there any way to distinguish an integer from a fraction (i.e., there was no “sexagesimal point”). However, many texts list numbers in columns in which the positional values of all the numbers are lined up with one another, making misinterpretation less likely (Powell 1976: 421). When numbers are embedded in the middle of a text or occur alone, the lack of a zero leads to ambiguity; there is no way, except through contextual information, to determine which positional value expressed which power, and thus a single numeral-phrase could have an infi nite number of readings. Th e simple phrase a2 3 could mean 723 (12 × 60 + 3), 43,380 (12 × 3600 + 3 × 60), 12.05 (12 + 3/60), and so on, depending on which positional values we assume are indicated. When the empty position was both preceded and followed by numerals, this diffi culty was sometimes obviated by using a large empty space to indicate the empty position (Neugebauer 1957: 20). Th us, 1 b (80) could be distinguished from 1 b (3620). Yet this technique was not used universally, and in some texts what looks to be a large space does not bear any numerical signifi cance. Unless numeral-signs were arranged in columns, there was no way during the Old Babylonian period to distinguish numbers where the empty position came at the end or beginning of the numeral-phrase. Nevertheless, by organizing numbers in columns, and through commonsense interpretations of texts, Babylonian mathematicians would not have experienced insurmountable diffi culties in reading numbers despite these ambiguities. Th e Babylonian positional notation probably developed, in fact, in the twenty- fi rst century bc, during the Ur III (Neo-Sumerian) period. Th e late Sumerian system of weight units is purely sexagesimal and notated in a way that could be ancestral to positional notation (Powell 1972a: 14). Powell (1976: 420) also found positional numerals on several early texts, which led him to assert that the development of positional numerals occurred in the twenty-fi rst century bc at the very latest. Robson (2007: 78–79) discusses twenty-fi rst-century texts from the cities of Umma and Girsu that clearly depict sexagesimal place-value numerals. Th e development of the notation may have resulted from Ur III administrative reforms, Mesopotamian Systems 251 which followed from managing much larger amounts of goods than had previously been the case (Powell 1976: 422; Høyrup 1985: 9). Nissen, Damerow, and Englund (1993: 142), however, remain agnostic regarding Ur III positional numerals, because most texts can be dated only paleographically, and the numerals do not show much variation throughout time. I am unconvinced by Whiting’s (1984) assertion that the positional numerals developed as early as the Old Akkadian period (i.e., the twenty-fourth or twenty-third century bc). Most of the early texts containing the positional numerals are mathematical texts of the Old Babylonian tradition, and thus date between 2000 and 1600 bc, with the majority from the latter part of that period (Powell 1976: 419). Th ese range from simple multiplication tables and arithmetical exercises to complex problems that can legitimately be called algebra. Education in the positional notation and in “pure” mathematics was a signifi cant component of scribal education, although of course only a small proportion of the total populace would have had access to such training. Figure 7.1 is a small square clay tablet on which a squaring exercise has been written in Babylonian positional numerals; the two numbers on the left are each 2, 30 (150), and their product (22,500) is depicted on the right as 6, 15 – the fi nal position, whose value would be 0, must be understood from context (Nemet-Nejat 2002: 262–263). Many arithmetical exercises and texts for translating numerals into the new positional system date from the Old Babylonian period, indicating the existence of a vigorous process for teaching the system to scribes (Nissen, Damerow, and Englund 1993: 142–147). Yet, because nonmathematical texts did not contain positional numerals, scribes who did not write mathematical texts probably would not have been familiar with the positional numerals. Th e converse is not the case, however; mathematical texts containing positional numerals are often dated in Assyro-Babylonian numerals, showing that mathematicians also knew the common system (Neugebauer 1957: 17). Th e numerals were rarely attested to have been used to perform arithmetical calculations directly, as opposed to writing down results. However, it is possible that calculations were made on clay “scratch pads” that could be moistened and rewritten to record results, after erasing the preliminary work (Powell 1976: 420–421). It is equally probable that some sort of counting board or wax writing tablet was used, on which the intermediate steps of calculations were not preserved (Høyrup 2002). We do not know much about the precise steps by which arithmetical computations were performed. After the end of the Old Babylonian period around 1600 bc, attested texts containing sexagesimal positional numerals are extremely sparse for over a millennium. Th e degree to which there was an actual decline in the system’s use is unclear, however. Th e small number of post–Old Babylonian texts, in combination with the relatively small number of place-value mathematical texts regardless 252 Numerical Notation

Figure 7.1. An Old Babylonian clay tablet (YBC 7294); the numerals at left each signify 150 (2 × 60 + 3 × 10) and the product, 22,500, is at right. Courtesy Yale Babylonian Collection. of period, may suffi ce to explain the decline, or it may have actually waned in use. Robson (2007: 154–156) has published a small number of mathematical texts from late second-millennium bc Babylonia that denote numbers using this notation. In Neo-Assyrian texts such as lnam ğišḫur ankia, an omen text dating to 712 bc, a technique called “downwards upwards, upwards downwards,” playfully altered sexagesimal numerals by swapping positional registers, so that, for instance, 140 (2 × 60 + 20) became 22 (20 + 2 × 1) (Robson 2008: 148). Th e positional numerals reemerged during the Seleucid period (the beginning of which is dated from the Alexandrine conquest of 332 bc) in new contexts and with several structural diff erences from the Old Babylonian system. First, subtractive expressions such as “20 lal 1” for 19 were no longer used in Seleucid texts (Neugebauer 1957: 5). Second, while in the Old Babylonian period the positional system was used exclusively in mathematical texts, by the Seleucid period it was used in astronomical texts as well (Neugebauer 1957: 14). Th e most important change was the introduction, in certain circumstances, of a sign serving some of the functions of zero, usually written as 0 or %, to fi ll in an empty position within a numeral-phrase. Th is sign could be used at the beginning of a numeral-phrase to indicate that the ones place was empty (i.e., to distinguish a fraction from an integer) or in a medial position to prevent misreading 3620 Mesopotamian Systems 253 as 80, as in the earlier example, but never phrase-fi nally (Neugebauer 1957: 20). Neugebauer (1941) emphasized that the primary role of the zero-sign was as much epigraphical as it was mathematical. He demonstrated that in a small number of texts, a “zero-sign” was inserted, apparently superfl uously, in numeral-phrases preceded by an amount in tens and followed by an amount in units. Th is was done in order to preclude misreading b 7 (20 × 60 + 7, or 1207) as b7 (27); by writing the former as b07, the latter interpretation is prohibited. In such numeral- phrases, the “zero-sign” does not indicate an empty position, but simply sepa- rates two consecutive positions (Neugebauer 1941: 213). In fact, the zero-sign was originally used to separate sentences, which may indicate that it originally merely indicated separation or space. It is unlikely that the Babylonians conceived of zero as an abstract number. No cuneiform positional text contains the bare numeral- phrase 0; it always occurs in phrases along with other signs. Th us, 0 was not equivalent to 0 in the same way that b was equivalent to 20. Th e abstract concept of zero accompanied by a special sign for that concept developed independently among the Greeks and Indians, but probably never among the Babylonians. Despite its use of the positional principle, the system was restricted to an extremely small group of Babylonian mathematicians and astronomers in the Old Babylonian and Seleucid periods. It does not appear to have been known by merchants or most administrators, and certainly did not diff use to other peoples of the Middle East, such as the Aramaeans, Phoenicians, and Persians. Th ere is no reason to believe, as was formerly held by some, that the Babylonian positional numerals survived long enough to be ancestral to the Indian positional numerals (Février 1948: 585; Menninger 1969: 398–399). Th e only direct descendant of the Babylo- nian positional numerals is the sexagesimal Greek positional system used by classical mathematicians and astronomers to represent fractions. Around the second century bc, the Greeks combined the Babylonian system with their alphabetic numerals to produce a positional, base-60 numerical notation system (Chapter 5). Neugebauer (1975: 590) states that the use of a sexagesimal division of the circle into sixty parts by Eratosthenes (ca. 250 bc) is the earliest evidence of this borrowing, although Eratosthenes did not use the sexagesimal fractions. Th is Greek system was used only in mathematical and astronomical texts, and only for fractions. Th e quasi-positional cuneiform system used in a few texts in the city-state of Mari also appears to derive in part from the Babylonian positional system. While the Seleucid astronomical texts are important from the perspective of the history of science, they represent the work of a limited group of scholars whose knowledge was being surpassed by Greek mathematics and astronomy even in the fourth century bc. Th e Greek alphabetic numerals were those used for everyday purposes as well as for mathematics, and so, slowly, the Babylonian system fell into decay. Th e last example of positional cuneiform numerals dates from the first 254 Numerical Notation

Table 7.15. Mari numerals

1 10 100 1000 10,000 fgimx century ad (Powell 1972: 6a). Th at this system, the fi rst positional system ever and one much lauded by modern scholars, should be replaced, after having nearly been abandoned once before by its own inventors, suggests that the advantages of positional systems do not correlate closely with their survival.

Mari Th e city of Mari, located on the Euphrates River at the border of modern Syria and Iraq, was an independent city-state between the twentieth and eighteenth centuries bc. During this time, it was engaged in extensive trade relations with Canaan and Babylonia. A large number of cuneiform tablets have been recovered from Mari, mainly dating to the eighteenth century bc, upon which a very unusual numerical notation system has been found. Th is system is shown in Table 7.15 (Durand 1987; Ifrah 1998: 142–146). Below 100, the system is purely cumulative-additive. For the units, it is identical to the Assyro-Babylonian system, but in the tens, there is no special sign for 60; rather, the higher decades are written as ü, é, â, and å, respectively. For the hundreds, the multiplicative ideogram i (ME) was often used (preceded by unit-signs for multiples). However, some texts omit the i sign entirely, turning the system into a quasi-positional one. Th us, 476 is written in one inscription as p\é\6 (Soubeyran 1984: 34). Th e sign for 1000 is a syllabic representation of “LI-IM,” while the sign for 10,000 is created by superposing the signs for 10 and 1000. For powers above 1000, the system is multiplicative-additive; there are no instances where higher power signs are omitted. Th e omission of the multiplicative sign for 100 is very interesting, but it does not represent a fully positional system. If it did, we would expect 476 to be written as p\7\6 (4 7 6), not as p\é\6 (4 70 6). Th e Mari system resembles the experiments with positionality in India in the seventh century ad, before the positional principle was fully adopted (Chapter 6). Most of the administrative and commercial texts from Mari use the Assyro- Babylonian common system, while mathematical texts mostly use the Babylo- nian positional numerals. One text (M7857) uses both systems side by side (Rob- son 2008: 130). We thus know that the Mari scribes understood these systems Mesopotamian Systems 255

Table 7.16. Hittite cuneiform numerals

1 10 60 100 1000 10,000

fgf or :i; \<

7169= 7;\fi:9 (7 × 1000 + 1 × 100 + 60 + 9) perfectly well. What may have happened, at least in the case of mathematical texts, is that the aberrant system was used unoffi cially (perhaps for calculation) and then retranscribed for offi cial purposes using the sexagesimal positional numerals (Soubeyran 1984: 34). It is possible that the decimal structure of the Mari numerals was borrowed from the Assyro-Babylonian ones, and that the idea of using positional notation for the hundreds was taken from the Babylonian positional numerals. Th is system should be considered as an aberrant and short-lived experiment with positionality, as the conquest of Mari by Hammurabi in 1755 bc ended its use.

Hittite Cuneiform In Chapter 2, I discussed the Hittite hieroglyphic system, which was probably borrowed from the Egyptian hieroglyphic system or the Linear B (Mycenaean) system. A separate Hittite script, written in cuneiform characters and related to the various Mesopotamian scripts, was used at the Hittite capital of Hattusha between the seventeenth and thirteenth centuries bc. Th e numerals used alongside this script, shown in Table 7.16, are similar to those of the Assyro-Babylonian common system (Rüster and Neu 1989: Table 7). Th e system is decimal and cumulative-additive below 100, while the multiplicative principle is used for higher powers. Numeral-phrases are written from left to right. To write 60 or multiples of 60, the Akkadian loanword šu-ši (:) was employed, just as it could be in other cuneiform systems. Yet, when writing numeral-phrases between 70 and 99, a simple vertical wedge represented 60 (Rüster and Neu 1989: 271). Th e sign for 100 is simply the ME syllable of Assyro-Babylonian numerals borrowed into Hittite, while the complex and apparently lexical expressions for 1000 and 10,000 appear to be indigenous. Th us, 7169 was written as shown in Table 7.16 (Rüster and Neu 1989: Table 7). Th ere is no evidence for the use of subtractive notation in the Hittite cuneiform numerical notation system. 256 Numerical Notation

Table 7.17. Old Persian cuneiform numerals

1 10 100 fgw

604= 6wp

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

Old Persian Th e Old Persian script was invented early in the domination of the Achaemenid Empire over Mesopotamia, probably near the beginning of the reign of Darius I (522 to 486 bc) (Testen 1996). It is an alphasyllabary; thus, while its letter-signs are cuneiform, it represents a distinct break from the older Assyrian and Babylo- nian scripts. Th e Old Persian numerals, shown in Table 7.17, closely resemble the Assyro-Babylonian common numerals (Testen 1996: 136). Th e system is decimal and cumulative-additive, with numeral-phrases written from left to right. Th e sign for 100 combines multiplicatively with preceding unit- signs in the one attested text containing a large numeral-phrase. No known Old Persian texts have numerals of 1000 or higher. Whereas the Assyro-Babylonian units could be grouped in sets of two or three, in up to three rows, Old Persian units and tens were arranged in at most two rows, with odd units represented at twice the size of paired ones. Th ere is no trace of sexagesimal notation in Old Persian; 60 is expressed with six signs for 10 and 600 as 6 × 100. Although the Old Persian script has traditionally been regarded as strictly a display-oriented script used for prestige purposes related to the Achaemenid monarchy, the recent translation and interpretation of an Old Persian administrative record from the Persepolis Fortifi cation Archive has cast some doubt on this notion (Stolper and Tavernier 2007). Figure 7.2 shows an Old Persian Mesopotamian Systems 257

Figure 7.2. Old Persian tablet from the Persepolis Fortifi cation Archive (Fort. 1208–101, obverse), which begins with the number 604, the only evidence for multiplicative notation for the Old Persian hundreds. Courtesy of Matthew Stolper. cuneiform text (Fort. 1208–101, obverse) originally excavated in 1933 from Per- sepolis but published only in 2007, which begins with the numeral 604, probably denoting a quantity of a dry capacity measure (Stolper and Tavernier 2007: 12–13). Th is numeral is the highest found to date in any Old Persian inscription; it is one of only two to contain the sign for 100, and the only one to denote higher hundreds, thus attesting the hybrid multiplicative structuring. As an isolated document among many thousands of Elamite and Aramaic economic and other texts found at Persepolis, Fort. 1208–101 does not prove widespread Old Persian literacy or numeracy in the Achaemenid period, but it does provide tantalizing evidence that the script’s use was not as limited-purpose as was once believed. Th e invention of the Old Persian numerical notation system occurred in a context of cultural contact between Persians and Assyrians/Babylonians in the late sixth century. Th ere are a number of Babylonian/Persian bilingual inscriptions, and the two systems existed side by side at that time (as evidenced by the thousands of cuneiform documents at Persepolis). Th e absence of any attested trace of sexagesimal notation is a unique feature that clearly diff erentiates this system from its Assyro-Babylonian ancestor, however. While Fort. 1208–101 shows that Old 258 Numerical Notation

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

Summary Th e commonality among the Mesopotamian numerical notation systems is their use of cumulative notation with signs for 1 and 10. In other respects, there is considerable variation among these systems, whether due to linguistic (Sumerian vs. Semitic) or functional (administrative vs. mathematical) factors. Th e survival of the sexagesimal base over a period of nearly 3,500 years is testament to the Baby- lonians’ preservation of Sumerian traditions, but most of the numerals used after 2500 bc were primarily decimal. Th e mathematical functions of the various systems, while interesting to historians of mathematics, are minimal in comparison to their administrative and literary functions. Th e conquest of Mesopotamia by the Achaemenids and later the Seleu- cids sounded the death knell for native Mesopotamian traditions, after which Aramaic and then Greek numerals were used for most purposes. Even the positional numerals, the hallmark of Babylonian arithmetical achievement, quickly disappeared under conditions of cultural and political domination. Th e Assyro- Babylonian common system was borrowed and modifi ed in regions of the Middle East where Mesopotamian infl uence was strong, but the history of Mesopotamian numerals is largely linear rather than branching, with each system giving rise to its successor but not giving rise to many systems outside Mesopotamia. Multiple systems were often employed at the same time within Mesopotamia, the use of which was determined by their contexts in ways that remain unclear. With the sole exception of the Levantine systems, which derive from both Mesopotamian and Hieroglyphic ancestors, the Mesopotamian systems did not give rise to a large number of descendants either within Mesopotamia or without. chapter 8

East Asian Systems

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

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

259 Table 8.1. East Asian numerical notation systems

1 2 3 4 5 6 7 8 9 10 100 1000 10,000 0 Shang / Zhou fghijl mno J:(~ Chinese classical ₏ℛₘ⥪℣⏼ₒ⏺⃬◐䤍◒ₖ榅

Rod-numerals JKLMNOPQRfJfJ

260 Late rod- numerals JKLSTOPQVfJfJ 0 Chinese commercial ሌልሎሏሐሑሒሓሔ◐x ◒ₖ 0 Chinese positional ₏ℛₘ⥪℣⏼ₒ⏺⃬◐ 0 Kitan 1234567890/ Jurchin !@#$%^&*()Ñ¢£ East Asian Systems 261

Table 8.2. Shang numerals

123456 789 1 fghijklmno 10 q <>?$% & 100 : 1000 ( 10,000 ~ 4539 = ) * > o numerical indications of tribute received, animals hunted, numbers of sacrifi cial victims, or counts of days, months, or other miscellaneous quantities relating to divinations (Takashima 1985: 45). Th e attested numeral-signs used on oracle-bone inscriptions are shown in Table 8.2 (Needham 1959: Tables 22, 23; Djamouri 1994: 39). Th e Shang numerical notation system combines the nine unit-signs with signs for the powers of 10; it is thus multiplicative-additive and decimal. Th e numeral- signs 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. Th e numbers 20, 30, and 40 were never expressed using multiplicative expressions involving the signs for 2, 3, and 4. As in most multiplicative- additive systems, there was no sign for zero; if a particular power was not needed, no sign indicated its absence in the numeral-phrase. Th ese 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. Th ere was also a symbol for ‘and’ which was placed between the hundreds and tens, or the tens and ones (Martzloff 1997: 182). Th e 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

Th e 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). eTh 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 quantifi ed. On this basis, Djamouri (1994: 33) regards Shang numeral-phrases as determinatives of noun- phrases, 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. Th is 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 diff usion 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 confi rms 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. Th e 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. Th e function of the Shang numerals is quite clear, however, in the context of royal divinatory inscriptions.1 Th ere 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, fi rst 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 exemplifi ed in the Yijing. East Asian Systems 263

Table 8.3. Zhou numerals

1234 56 7 89 1 fghi jl m np j l n p 10 rÄÅæ r r Unattested r r 100 û 1000 ù 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). Th e 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). Th e 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 numeral- signs 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 diff erent 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). Th e power-signs immediately ancestral to the classical Chinese ones were among the variants used in the late Zhou period. Th ese signs, shown in Table 8.4, diff er 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. Th e 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 unifi cation 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). Th is 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 fl at 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. Th e system is quite simple to learn and use; vertical and horizontal lines are suffi cient to write any number. For the units position, vertical strokes signifi ed 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). Th us, Chinese scholars learned the following rhyme (Li and Du 1987: 10):

Units are vertical, tens are horizontal, Hundreds stand, thousands lie down; Th us 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)

123456789 1s JKLMNOPQR 10s ABCDEFGHI 100s JKLMNOPQR 1000s ABCDEFGHI

762 P F K 7008 G Q 905,920 I E R B 6.49 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). Th e 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 refl ected 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 fi ve 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. Th e rod-numeral system was infi nitely 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 numeral- phrases containing both whole and fractional positions, the ones position could be

2 Martzloff (1997: 205) coins the term “dispositional” to refl ect this irregularity. 266 Numerical Notation identifi ed 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 dif- ferent-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). Th e 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). Tex- tual 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. Th e Daodejing (Tao Te Ching), written in the early third century bc, advises that “[g]ood mathematicians do not use counting-rods,” confi rming 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 (Th e Math- ematical Classic of Master Sun) in the fourth or early fi fth 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 infl uenced their development. While the systems diff er 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. Th e 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 zero- sign (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 infl uence in Japan was enormous. Th e 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). Th ere is no evidence of their use outside China, Japan, and Korea. Th e last coins to use rod-numerals are the fi ve chu coins of the Liang Dynasty (502–557 ad), but these numerals are highly irregular (de Lacouperie 1883: 316–317). Th ey 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 signifi cant 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 scientifi c 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 fi ve cumulative strokes had previously been required, this was probably done to simplify the signs, though it meant that the written system diff ered from that used with physical rods. Th ese 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. Th e fi rst 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 fi rst to use a circular sign for zero in conjunction with the classical numerals. 268 Numerical Notation

Table 8.6. Late rod-numerals

1234567890 Zong JK LM N OPQR 0 (1, 100, ...) S T V Heng ABCD E FGHI 0 (10, 1000, ...) S U W

Old Style New Style 5804 E Q 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. Th e 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. Th e earliest strictly mathematical Chinese text, the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), which dates no later than the fi rst 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). Th ereafter 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 propo- sitional mathematics in the Song Dynasty. Yet the modifi cations to the system, including the addition of a zero-sign, suggest that the rod-numerals, as an infi nitely 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 rod- numerals, with their vertical lines, to be insuffi ciently literary, and replaced them with the classical numerals (Needham 1959: 8). East Asian Systems 269

Th e introduction of the bead-abacus (suan pan) in the fourteenth or fi fteenth century brought this novelty into direct competition with the rod-numerals. Th e earliest surviving suan pan dates from the sixteenth century, but the text Dui- xiang siyan zazi of 1337 indisputably depicts one (Martzloff 1997: 213–215). Textual sources indicate that the suan pan was perceived as being more effi cient for computational purposes than the rod-numerals. Th e divorcing of rod-numerals from the physical manipulation of rods made their use in written form rather archaic. At fi rst, the suan pan was an instrument for popular arithmetic, while computing rods were retained by mathematicians and the elite (Jami 1998: 4). Th roughout 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 fi rst 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 Japa- nese 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 infl u- ence of the system of numbered rods developed by the English mathematician John Napier (Needham 1959: 72). Th is 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. Th e 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). Th e manipulation of rod-numerals on boards appears to have been nearly as important to ancient and medieval Chinese scientifi c and commercial calculation as the bead-abacus would later be. Th eir origin and persistence must have had a great deal to do with their effi ciency 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. Th e rod-numerals

5 Wang Ling (1955: 91) reports that Chinese logarithmic tables were still written with rod- numerals 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 rod- numerals to replace the multiplicative-additive classical numerals entirely, or at least to facilitate their transformation into a ciphered-positional system.

Chinese Classical Th e 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 diff erences between that system and the way the numerals are normally used today, ancient numeral-phrases are still easy to read. Th e 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. Th e 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. Th ere 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 numeral- phrases, 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. Th ere 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). Th ese signs were most often used in literary contexts, for paginating certain texts, and when denoting days of the month. In the fi fth-century mathematical manuscripts found at Dunhuang in Central Asia, however, they were also used for mathematical purposes (Martzloff 1997: 185). Th ey 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

12 3456789 ₏ℛor₳ₘ⥪℣⏼ₒ⏺⃬ yi er san si wu liu qi ba jiu

10 100 1000 10,000 100,000,000 ◐䤍 ◒ 嚻or ₖⅎor ⎓or嚻嚻 shi bai qian wan yi or wan wan

20 30 40 ◓◔ ◛ nian sa xi

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. Th e system did not stop there, however. By the fi rst century ad, multiples of 100,000,000 could be written by placing a multiplicative numeral- phrase 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 5 118 䤍◐⏺RU₏䤍₏◐⏺ 100 10 8 1 100 1 10 8 74,002 ₒ嚻⥪◒ℛ 7 10,000 4 1000 2 1,072,419 b7d2c4ba9 100 7 10,000 2 1000 4 100 10 9 4,703,600,854 4a7ƒ3b6ad8b5a4 4 10 7 100 mil. 3 100 6 10 10,000 8 100 5 10 4 272 Numerical Notation

Table 8.8. Chinese higher power-signs

Lower Series Middle Series Upper Series Sign Phonetic Value xia deng zhong deng shang deng d wan 104 104 104 ƒ yi 105 108 108 á zhao 106 1016 1016 í jing 107 1024 1032 £ gai 108 1032 1064 ó zi 109 1040 10128 Ñ rang 1010 1048 10256 ñ gou 1011 1056 10512 ¿ jian 1012 1064 101024 ¬ zheng 1013 1072 102048 ½ zai 1014 1080 104096 system of power-signs that was assigned three diff erent series of values, as shown in Table 8.8 (Needham 1959: 87; Martzloff 1997: 99).6 Th ese power-signs combine multiplicatively with the nine basic unit signs, and thus extend the basic system. Needham (1959: 87) asserts that these signs fi rst 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 fi fth 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 diff erent 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. Th e fi rst 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 Th e middle series (zhong deng) values in Martzloff and Needham diff er 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 fi rst millennium ad. None of these systems was ever in common use. Th e 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 unifi ed administrative apparatus under the Qin and Han Dynasties, they spread throughout areas under direct and indirect imperial control. Th e unifi cation of China led to many eff orts to standardize the forms of Chinese script and numeral-signs, although this was not accomplished to any signifi cant 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 codifi ed, however, Chinese writers began to use calligraphic variants and other modifi cations of the basic system for specifi c functions. Th ese variants used diff erent numeral-signs (ranging from mild paleographic variations to radically diff erent signs), but their structure is identical to that of the basic system (decimal and multiplicative-additive). Th ese 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 fi rst century bc (Needham 1959: 5, Table 22). Struc- turally, they are identical to the classical numerals, but while the classical numeral- signs are quite simple, the accountant’s numerals were intentionally made very complex; thus they were considered more elegant and less susceptible to falsifi cation. Th e 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 falsifi cation. 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). Th ese 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. Th ese signs are shown in Table 8.9. Th e diff usion of the Chinese classical numerals was associated with the spread of Chinese political infl uence 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. Th e 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

123456789 Çüéâäàåçê

10 100 1000 10000 ëèïì

North Korea). Th e Chinese numerals were borrowed directly (without any transformation) by the Japanese as part of the kanji characters starting in the third century ad. Th e Korean hangul script developed in the fi fteenth century has no corresponding numerical notation system, but Koreans often used the Chinese classical system. Th e 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). Th e 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 modifi - cation. 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. Th e classical Chinese numerals were nonpositional and used no zero-sign for over a millennium after their development. Th e 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, fi rst 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). Th is transmission refl ects the enormous scientifi c 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 Th is name is the Sinicization of the author’s original name, Gautama Siddharta. 276 Numerical Notation

In the mid thirteenth century, a period of scientifi c vigor during the late Song Dynasty, the fi rst zero-signs appeared alongside classical Chinese numerals in mathematical texts. Th e fi rst such text was the Shu shu jiu zhang of 1247, the same document in which the zero-sign is fi rst found with rod-numerals (Libbrecht 1973: 69). Th is modifi cation 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 ciphered- positional. Yet, during the Song Dynasty zero was used only to fi ll 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 scientifi c 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.” Th e fi rst 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 fi rst text to describe the Chinese commercial numerals or ma zi, and additionally contains the fi rst complete description of the bead-abacus orsuan 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). Th e 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) Ciphered-positional Western 2d4b6 2d[4b[6 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. Th e 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 scientifi c contexts, the nine unit-signs along with a sign for zero are used positionally, as in the seventeenth- century logarithm tables. In contexts where there is concern with forgery, the “accountant’s numerals” can be used. Th e fi nal two options are to use the commercial or Hangzhou numerals, which I will describe later, or Western numerals. Th e Chinese classical numerals are ancestral to several numerical notation systems. Th e ciphered-positional variant Chinese numerals used in modern mathematics are, of course, one such descendant, as are, more indirectly, the Hangzhou numerals. Th e 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 scientifi c 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 fl avor. 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 Th e Chinese commercial numerals (often known as “Hangzhou numerals”)8 arose in the sixteenth century. Th e 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. Th e 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 fi ve. Th e most common versions of the power-signs for 10 through 10,000 are obvious variants of the classical system’s power-signs. Th e circular sign for zero was in use in both the rod-numerals and the classical system. Th is evidence strongly suggests that the commercial numerals originated as a blend of the late rod-numerals and the Chinese classical numerals. Th e system is multiplicative-additive, with the zero used only to fi ll 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). Numeral- phrases 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. Th is 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

123456789 JKLSsFGHt or u

10 100 1000 10,000 0 vx or w y or 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). Th e 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. Th e 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 S\\ G 4 7 40,709 z0x0t 10,000 0 100 0 9 K 2 26 vF OR ,F 10 6 OR 20 6 JF JF 1 6 162 xaK OR xag 100 10 2 tJ t 9 1 9 917 xvG OR xvG 100 10 7 OR 100 10 7 L0 3 0 3008 y0H 1000 0 8 5000 sy 5 1000 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. Th ey 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. Th ey are most common in regions where Cantonese is spoken, including Hong Kong.

Kitan Th e 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 infl uence 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. Th e 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. Th e 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 diffi cult 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. Th e Kitan script and numerals did not outlast the period of Kitan independence, East Asian Systems 281

Table 8.13. Kitan numerals

123456789 1 123456789 10 0,. 100 / 473 = 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 Th e 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. Th e 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 fi fteenth centuries, but may have developed somewhat earlier (Kiyose 1977). Th e 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. Th e 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 Man- chu 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 282 Numerical Notation

Table 8.14. Jurchin numerals

123 4 5 6 7 8 9 10 !@# $ % ^ & * ( ) HPX ƲXZH LODQ GXZLQ ģXQƲD QLQJX QDGDQ ƲDNXQ X\XQ ƲXZD

11 12 13 14 15 16 17 18 19 20 ÅÉæ Æ ô ö ò û ù ÿ DPģR ƲLUKRQ JRUKRQ GXUKRQ WRERKRQ QLOKXQ GDUKRQ QL\XKXQ RQL\RKRQ RULQ

30 40 50 60 70 80 90 100 1000 10,000 ÖÜá í ó ú ñ Ñ ¢ £ JXģLQ WHKL VXVDL QLQMKX QDGDQMX MKDNXQMKX X\XQMX WDQJX PLQJDQ WXPDQ

100, the multiplicative principle was always employed. Th us, 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 signifi cant Chinese infl uences; the script was ciallyoffi introduced in 1145 by Emperor Xizong, with a number of “small script” characters added (Kara 1996: 235). Th e Jurchin numerals are found on many monuments of the Jin Dynasty and some manuscript fragments. Th e 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). Th e earlier signs diff er 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. Th e Jurchin were one of the major constituent groups of the Man- chu 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. Th is 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. Th e increasing rarity of the Chinese numerals in Japan and Korea represents not only the functional rejection of an “ineffi cient” system, but also resistance to a Chinese cultural feature in favor of the more international Western numerals. Th e 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. Th e Chinese classical numerals are well suited to being read because they account for the irregularities in spoken Mandarin. Th e 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. Th e system’s fl exibility and its correspondence with language are thus advantages rather than hindrances. Th e 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 multiplicative- additive (Shang/Zhou, Chinese classical and commercial, Kitan). Yet there can hardly be any doubt that these systems comprise a cultural phylogeny. Th e 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 fi rst 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. Th ese two interrelated traditions are associated most closely with the Maya and Aztec civilizations, respectively.1 Both these traditions fl ourished until the Span- ish 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. Th e numeral-signs of the major Mesoamerican systems are shown in Table 9.1.

Bar-and-Dot Th e 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). Th is 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 5 20 400 8000 0 Bar-and-dot (monumental) V e 2 3\9 Bar-and-dot (codices) V E 1 0\] Aztec V X v Y x y Texcocan line-and-dot TU V 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, refl ecting both the strong interest in dating and calendrics and the practice of incorporating numerical values into the names of deities and individuals. Th e 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 fi rst 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 diffi cult. Ornamental crescents were often employed in order to “fi ll in” a numeral that would otherwise have an empty space, and these can easily be confused with dots (Th ompson 1971: 130). Similarly, decorative lines were sometimes added to bars for aesthetic purposes, which can make it diffi cult to distinguish one from two bars. In addition, in Maya monumental inscriptions and also (with a diff erent 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. 286 Numerical Notation

Table 9.2. Bar-and-dot numerals r R m z 18: vertical vs. horizontal orientation 13 vs. 11 (with ornamental crescents)

21( 20/‘moon’ 20/‘moon’ 20/‘moon’ (?) (monumental) (codices) (Isthmian)

3\ 9 0\] 0/‘completion’ 0/‘completion’ (monumental) (codices)

Th e 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. Th e accompanying bar-and-dot numerals could be placed above, below, or to either side of a 20-glyph (Kelley 1976: 23). Th e 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) (Th ompson 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. Th ere is considerable paleographic variation in the signs used, but a “shell” sign was commonly used in the codices, while diff erent signs were used in monumental writing. eTh 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 (Th ompson 1971: 137), which I will discuss later in greater detail, I do not see any reason to deny the Maya their zero. Th e Maya zero-sign is defi nitely 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 fi rst identifi able signs of Mesoamerican writing systems, and occur in all three of the major Forma- tive 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 signifi cant. Understanding the early history of bar-and-dot numeration, however, does not give clear priority to any of the three. Th e 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. Th e Cascajal block, which appears to date to the early fi rst 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). Th ere is apparently no connection between the Cascajal block and Isthmian or indeed any other Mesoamerican script. Th e 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 fi rst 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 fi rst time in vertical columns to indicate periods of time and specifi c dates in the famous “Long Count” system, the calendar expressing dates as a series of numerals indicating fi ve time periods. We can thus assign absolute dates to these artifacts from the calendrical evidence alone. Th e 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 Th e longer second-century Isthmian inscriptions, such as the La Mojarra stela 1 (156 ad) and the Tuxtla statuette (162 ad), contain bar-and- dot numeral-phrases including Long Count dates (Justeson and Kaufman 1993). Th e 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. Th e stela also contains a sign identifi ed 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. Th e 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. Th e 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 fi rst 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 defi nitively 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 bar- and-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). Th is 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. Th e fi rst 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

Th is 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 diff usion 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 diff erent bases, diff erent directions of writing, diff erent media, and serve extraordinarily diff erent sets of functions. Th e 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. Th ere are many thousands of identifi ed Maya inscriptions from the Classic period (150–900 ad), the vast majority of which contain at least some bar-and-dot numerals. Th ere is no identifi able regional variation in the form or ornamentation of the numerals within the Maya sphere of infl uence. 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 Oa- xaca region, and continued to use bar-and-dot numeration throughout the Classic period) or from the more infl uential 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). Th ere is some evidence that the bar-and-dot numerals survived in central Mexico until the Spanish conquest. In Mixtecan-Pueblan texts such as Codex Fejervary- Mayer and Codex Cospi, sets of bars and dots arranged vertically or horizontally could represent counted bundles of off erings (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. Th ese 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. Th e 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 290 Numerical Notation

Table 9.3. Maya period glyphs !@#$% kin uinal tun katun baktun 1 day 20 kins 18 uinals 20 tuns 20 katuns

2001: 3). Th e 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 (Th ompson 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). Th is 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 fi ve 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 specifi c fi xed Long Count date, fi ve numeral-glyph combinations were required (one for each period, written from longest to shortest). Th ese were normally written in pairs of columns, most commonly but not

5 Th e terms katun and baktun mean, literally, ‘20 tuns’ and ‘400 tuns’. Th e 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. Th ere are several extremely rare glyphs for longer periods, again with coined names: pictun (8000 tuns), calabtun (160,000 tuns), and kin- chiltun (3,200,000 tuns), which presume a purely vigesimal progression of dates (Closs 1986: 303). 6 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 refl ect 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) i% n$ (14 katuns) (10 tuns) j# 9@ (0 uinals) (12 kins) l! exclusively read from left to right and top to bottom. Th ese 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-Martinez- Th ompson 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 fi ve numerical coeffi cients 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 coeffi cient of a time period was zero (e.g., “0 uinals” in Table 9.4), the Maya included both a zero coeffi cient 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 coeffi cient of zero was suppressed (Th ompson 1971: 139). In a few texts, period-glyphs were omitted entirely, and dates were written simply by placing the fi ve coeffi cients in a single vertical column. As mentioned already, the technique was present in the Isthmian and Zapotec inscriptions by the fi rst 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 period- glyphs, 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 fi ve numbers without period glyphs is found on the fteenth-centuryfi Madrid Codex that may qualify, but otherwise no other codices have them. 292 Numerical Notation

Table 9.5. Initial Series date without period glyphs

I 9 baktuns

N 14 katuns

J 10 tuns

] 0 uinals

L 12 kins other. Table 9.5 shows the Long Count date 9.14.10.0.12 as it would be written in this manner. Th is system requires that all the relevant numerical coeffi cients be included, even for periods for which there is a zero coeffi cient, to ensure that the correct quantity of time is counted. Th e 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 cumulative- positional 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. Po- sitional numerical notation systems do this by having each successive position represent the next higher power of a base. Th us, 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 fi ve 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 fi ve 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 9 × 144,000 days 1,296,000 days

N 14 × 7200 days 100,800 days

J 10 × 360 days 3600 days

] 0 × 20 days 0 days

L 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 quasi- numerical expression for a count of 40,000 cacao beans (Houston 1997). Th e Yucatecan, Ch’olan, and Tzeltalan languages all use numeral classifi ers – 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 classifi ers.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). Th ese diff erent 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 multiplicative- additive system into a ciphered-positional one by adding a zero-sign and delet- ing the power-signs for 10, 100, 1000, and so on, so that ₒ◒榅⥪◐⃬ (7 × 1000 + (0) + 4 × 10 + 9) becomes ₒ2◐⃬(7049). Th ere are similarities between this transformation and the removal of the Maya period-glyphs, but in the Chi- nese case, the removed power-signs are numerical (representing the powers of 10), whereas in the Maya case the period-glyphs are calendrical. Th e assumption that the fourth position of the Maya numerals means “7200” is wrong. Even so, it could still have been read as “7200 days.” Th is 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 rec- ognizes that Long Count dates (with or without period-glyphs) are capable of being read positionally or nonpositionally. Teeple (1931) and Th ompson (1971) claimed that such dates should be considered as a count of tuns (years), in which the fi nal 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 fi rst 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, fi rst, 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. Th ese 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 affi liation 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 diff erently 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 fi ve separate nonpositional counts of fi ve diff erent time periods. Th e 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 fi ve diff erent 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-Th ompson 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. Th e Dresden Codex is a repository of calendrical data, including what appear to be multiplication tables, but there are no calculations on paper. Th ere is specifi c ethnohistorical evidence concerning Maya computation, from Landa’s Relación de las cosas de Yucatan, which suggests that the sixteenth- century Maya did not calculate directly using bar-and-dot numerals:

Th eir count is by fi ves 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. Th ey have other very long counts and they extend them in infi nitum, 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. Th ey make their counts on the ground or on something smooth. (Tozzer 1941: 98)

Computation was done on some sort of fl at 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 (Th ompson 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 fi ve and twenty, supporting the idea that the ancient Maya may have done similarly (Th ompson 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. Th e 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. Th e manipulation of counters is identical, but the reading of the results is very diff erent. 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 cumulative- positional numerical notation system. Th e most parsimonious explanation is that the omission of period-glyphs was abbreviatory but did not entail a radical reread- ing of the numerical coeffi cients. In his analysis of Maya arithmetic, Fulton noted that “it is possible to have a strictly positional notation, not altogether diff erent 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. Invert- ing this insight, I believe that the Maya bar-and-dot system had a zero, but did not use the positional principle. Th is 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 fi rst 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 diff erent 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. Th e 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 quasi- positional 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 Th ese 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 Th ompson 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 bar- and-dot numerals, head-variant glyphs are associated with the fi ve calendrical coeffi cients (baktun, katun, tun, uinal, kin), and thus assume elements of a vigesimal structure. Th e head-variant numerals from 1 through 12 are written with elementary signs. Th e 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. Th ere 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 signifi cance 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). Th e 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 specifi c deities and other symbolism associated with each glyph, see Th ompson (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. Th e 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. Th e head-variant numerals are relatively common on Maya inscriptions, though less common than the bar-and-dot numerals. Th ey also appear occasionally in the Dresden Codex, though not in the other Postclassic codices (Th ompson 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 Isth- mian inscriptions contains head-variant numerals, and many centuries lie between the decline of the Olmec civilization and the appearance of head-variant glyphs. Th e head variants are extremely diff erent 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 diffi cult 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). Th e 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 fi ve 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 dot- only 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. Th e infl uence 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. Th us, unless the use of dots for units developed independently in the two diff erent parts of Mesoamerica, the dot-numerals must be descended from the bar-and-dot system. Th e 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 diff erence being that with the Aztec numerals, the dots were more regularly grouped in fi ves, and higher numbers were written using diff erent 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 dot- numerals in some areas outside their tributary area, and were used in many of the post-Conquest Mixtec codices (Terraciano 2001).

Aztec Th e 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. Th e Aztec tributary network, which embraced numerous small states, produced a large number of manuscripts, using a combination of ideographic and phonographic signs. Th e 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. Th e Aztecs most defi nitely possessed a vigesimal numerical notation system, whose signs are shown in Table 9.8. Th e sign for 1 is the dot that was commonly used for units throughout Meso- america. Th e signs for the vigesimal powers are depictions of objects: for 20, a agfl (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 ttttt xxx vvvvv tttt vvv 27,469 = (3 × 8000) + (8 × 400) + (13 × 20) + (9 × 1) hold copal incense (xiquipilli) (Harvey 1982: 190). Th ese 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 fi ve identical signs were arranged in sets of fi ve for easier reading. Groups of fi ve signs were sometimes joined to one another with a horizontal line underneath the set. Th e purely vigesimal structure of the Aztec numerical notation system and the shapes of its numeral-signs are quite diff erent from those of the lowland Meso- american 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 infl uenced directly by the lowland Mesoamerican systems of the Maya. Th e Mexican dot numerals do not constitute a numerical notation system according to my defi nition, because they lack a base, meaning that the Aztec system was invented relatively independently. Th e 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 diff erent 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 Telleri- ano-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. Th e 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. Th e 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 fi ve, 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 diff erent 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 post- conquest manuscripts, fractions could be depicted by segments of 1/4, 1/2, and 3/4 of a dot, low multiples of fi ve by fi lling in quarters of the pantli fl ag 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 fi rst 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 fl ag for 20 is placed. Th is 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.” Th is 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 Th e 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 diff erent system, which I will call the “Texcocan line-and-dot” system. Th ese documents have been studied extensively by Herbert Harvey and Barbara Williams and are identifi ed collectively as the “Tepetlaoztoc Group” (Harvey and Williams 1980, 1981, 1986; Harvey 1982; Williams and Harvey 1988, 1997). Th e numeral-signs of this system are shown in Table 9.9 (Harvey and Williams 1980: 500). Th is system is cumulative-additive, with a base of 20 and a sub-base of 5. Th e sign for 5 consists of fi ve 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 fi ve 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). Th is 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. Th ese 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). Th e 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). Th e 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. Th e Códice de Santa María Asunción used a modifi ed 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 diff erent sections, but served very diff erent 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 diff erent individuals to indicate their lengths. In another section, known as tlahuelmatli, the plots of land 304 Numerical Notation

Table 9.9. Texcocan line-and-dot numerals

1 5 20 100 TUVW from the milcocoli section were redrawn as rectangles (regardless of their original shape). Th is 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 fi ve 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). Th is 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. Th ese 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 fi nd 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 offi cials by the inhabitants of the Tepetlaoztoc region in an eff ort to convince colonial offi cials that the populace was overworked; extensive description in Spanish confi rms the numerical values (Harvey 1982: 193). Curiously, this text combines Aztec numerals and Texcocan line-and-dot notation. Lines and chunked groups of fi ve lines indicate 1 and 5, respectively. To write larger numbers, dots organized in lines of fi ve were placed beside the signs for 20, 400, and 8000. Th e dots were placed in a single row, with the signs for 20 and 400 above them and the 8000 sign below them. Th us, where the regular Aztec numerals use these three signs cumulatively, the Kingsbor- ough 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. Th e 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 fl ag 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). Th e erratic nature of the system suggests that whoever wrote it was extremely inventive and was in the process of experimenting with diff erent means of representation. Th e 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. Th us, 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 diff erent registers around a rectangle is quite diff erent 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 diff erent from Western and Roman numerals that the Spanish could not have introduced them. Nevertheless, these may be instances of stimulus diff usion, which the Texcocan scribes developed with an awareness of Western and/or Roman numerals but without adopting the form and structure of those systems. Th at 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 line- and-dot numerals probably existed in the pre-Conquest period. After 1545, epi- demic 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 dif- ferent 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 fi xed value but could mean 5, 10, or 30, depending on their confi guration (Langley 1986: 141). Th e 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 confi rm 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” 260- day 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. Th e 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 diff erences 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 Th e 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. Th e Maya head-variant glyphs, a sort of ciphered symbolic code that expresses only units up to 19, constitute a partial exception to this rule. Th e 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 fi t neatly into the phylogenetic classifi cation 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. Th e majority of this chapter, however, deals with systems that emerged in colonial settings under the infl uence 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 affi liations remain inscrutable enough that no defi nite conclusions can be reached.

Inka Th e Inka civilization was an enormous state on the Pacifi c coast of South Amer- ica 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 diff erent 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 fi rst 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. Th ese 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). Th e designation that pendant cords hang “down” and top cords hang “up” is an artifi ce; 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. Th e 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. Th ere is no sign for zero; instead, a space was left on the cord in an empty position. Th e 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, fi ve-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 confl ated 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. Th ree diff erent 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 Th e spellings ‘Inka’ and ‘khipu’ currently enjoy favor with Andeanists over the older ‘Inca’ and ‘quipu.’ Miscellaneous Systems 311

Single knot Long knot Figure-8 knot 10s, 100s, etc. Units: 2-9 Units: 1

Figure 10.1. Khipu knots. diff erent 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 diff erent knot, a fi gure-8. Th e use of diff erent 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). Th e use of long or fi gure-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. Th e 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 fi gure-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 fi gures 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 sub- sidiaries 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. Th e fi rst well- substantiated evidence for khipu use comes from Middle Horizon sites (c. 600– 1000 ad) associated with the Wari civilization in coastal Peru (Conklin 1982). 312 Numerical Notation

T1 S3 776 26

P1 P2 S1 P3 P4 S2 360 23 6 102 291 20

Figure 10.2. Khipu structure. Miscellaneous Systems 313

Th ese khipus cannot be deciphered numerically because of their deteriorated condition (although they may have used nondecimal bases), and they use color in very diff erent 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). Th e 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-Pacifi c region (Birket-Smith 1966). Th ere is no evidence of any connection between the khipu notation and any other numerical notation system, and thus it is defi nitive 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 unconfi rmably) that such khipus may have been part of a double-entry accounting system. Th e 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. Th e 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 force- fully 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). Th e 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. Th e 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 diff erent 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 topo- nymic 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 arrange- ments of three fi gure-8 knots (which normally represent the numerical value 1), suggesting that this served as an identifi er 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 identifi ed three-term number sets that occur on some of the Puru- chuco khipus that do not fi t 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). Th e 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 diffi cult 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 defi nitively 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 unconfi rmed. 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-fl edged 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. Th e 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 sixteenth- century 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 confi rmed 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 fi ve rows by four columns, in each square of which is found a number of circles: fi ve dots in the fi rst column, three in the second, two in the third, and a single dot in the fourth. Moreover, some of the dots have been fi lled 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) eff ort 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 diff usion 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. Th is shift involved changes in how texts were organized, written, and read, and also forced pre-existing Quechua ideas about numbering and counting into confl ict 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 refl ected 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. Th e 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 diff erent from the pre-colonial khipu, but confi rm that the practice continued in vary- ing forms well after the colonial period (Hyland 2003). Th ese were not simply tallying systems, however, but were cumulative-positional and decimal, and thus constitute a survival of the Inka numerical notation system. A signifi cant 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). Th e modern “episteme of numbers” rests on diff erent 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-fi rst centuries, signifi cant 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 pre- modern 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 numeral- signs and the structure of the system. Th e Bambara numeral-signs are shown in Table 10.1. Th e 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. Th e value of a set of vertical strokes is doubled if a horizontal line is crossed through it (eff ectively 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. Th e 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). Th is 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. 318 Numerical Notation

Table 10.1. Bambara numeral-signs

12345 a aa aaa aaaa aaaaa 678910 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 efghi 80 90 100 110 120 jklmn 130 140 150 160 170 op q r s 180 190 tu

Th e 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 diff erent in most respects – its frequent use of the cumulative principle, the presence of a vigesimal component, and its numeral- signs. I have no idea whether this system continues to be used, though I suspect that it does not.

Berber Th e 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. Th e 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 Tifi nigh script (still used by the modern Tuareg for love letters, domestic ornamentation, and games) is descended from it (O’Connor 1996). Th ere 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. Th e 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). Th e 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 diff erent contexts or at diff erent 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. Th e question of the Berber system’s ancestor (if any) is still open. It is possible that it was an entirely independent development. Th e 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. Th e Phoenician/Punic numerical notation system is quite diff erent in its structure, lacking a sign for 5, and employing a special sign for 20 and a hybrid multiplicative-additive structure above 100. Th e use of | for 1 and > for 5 is superfi cially similar to the Roman system; Ghadames was an important 320 Numerical Notation

Table 10.2. Berber numerals

1 5 10 50 100 500 1000 Hanoteau 1íóúñ Ñ Si Mohammed Serif 1íóúñªº Rohlfs 1íóúšªº 1/4: E 1/2: J 3/4: 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 diff erent directions and have diff erent signs for the higher values. Vycichl (1952: 83) suggests that the system derives from the South Arabian numerals. Th e 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 fi rst century bc and was never used in Africa, so to accept this theory requires that we believe in a two-thousand- year unattested history for this system. Th e 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). Th is 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. Th ough 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 diff usion. Ultimately, more data are needed for this system to be assigned unambiguously to any phylogeny. Th e 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 Ghad- ames region, but only ascertained the meanings of the signs through great eff ort and negotiation. He thus believed that the system was semi-cryptographic, restricting the fl ow of information concerning prices to a limited group of Berber traders in order to give them an advantage over Arab traders. Th e system is not especially diffi cult 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

12345678910 1234567890 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-Efi k, 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). Th e script represented an arcane revealed liturgical language of the sect, but was not used to write Ibibio. Th e Oberi Okaime numeral-signs are shown in Table 10.3 (Hau 1961: 295). Th e system is ciphered-positional and vigesimal; it is the only known ciphered- positional base-20 system with no sub-base, with the partial exception of the Maya head-variant glyphs. Th e 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. Th e 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 infl uenced 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. Th e numerals were used in a relatively small number of liturgical texts and personal letters among the members of the Oberi Okaime sect. Th e 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 322 Numerical Notation

Table 10.4. Bamum numerals (original)

123456789 ¶áíóúñÑ¿¬

1 10 100 1000 10000 ½¼ ¡ « » 76 = Ñ¼ñ½ or Ñ¼½ñ to have been unsuccessful (Abasiattai 1989: 506). Western and sometimes Arabic positional numerals are used in the region today.

Bamum Th e 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. Th e earliest Bamum numerals are shown in Table 10.4 (Dugast and Jeff reys 1950: 6). Th is 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 Jeff reys 1950: 30). Th e 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 Jeff reys 1950: 98). At this point in the system’s history, we could well consider it to be a set of lexical numerals. Th is is the same problem we encountered with the Shang/Zhou and Chinese classical systems (Chapter 8), which, not coincidentally, also are multiplicative- additive 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 Jeff reys (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␧)

1234567890 öòûù ÿÖ Ü ¢£¥ ciphered-positional by removing the power-signs (Dugast and Jeff reys 1950: 30). Th e 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. Th e mf␧mf␧ numerals are shown in Table 10.5 (Dugast and Jeff reys 1950: 31). During its heyday in the fi rst 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). Neverthe- less, 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 infl uenced 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. Th e Mende numeral-signs are shown in Table 10.6 (Tuchscherer 1996: 71–75). Th e 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. Th ere are two signs for 10. The fi rst, 10 (+) in Table 10.6, combines additively with the units for 1 through 9 in order to write 11 through 19, 324 Numerical Notation

Table 10.6. Mende (Kikakui) numerals

1 23456 789 a bcdeF ghi 10 (+) 10 (×) 100 1000 10,000 100,000 1,000,000 kjmnoP ß d 14 k \\\\ba 128 hjm \F 60,009 io \\e\e\e\e\e\e 5,555,555 e\j\m\n\o\p\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). Th e 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. Th is 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 infi nitely 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 fi rst 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 fi rst 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 infl uenced 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). Th e 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 diff erent signs for 10 (one additive, one multiplicative) and the use of repeated strokes to indicate exponents are features that are not attested elsewhere. Th us, 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 fi rst decades of the twentieth century, it, rather than a foreign numerical notation system, could have inspired the decimal base of the system. Th e 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. Th e Bagam syllabary was invented early in the twentieth century in western Cam- eroon and used briefl y by the Eghap (known in scholarly literature as the Bagam) of that region (Tuchscherer 1999). Th e only text to preserve Bagam writing and numerals is a recently discovered 1917 description of the system by a British colonial military offi cer, Captain L. W. G. Malcolm. Th e 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. Th e 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 326 Numerical Notation

Table 10.7. Decimal systems of sub-Saharan Africa

123456789010 Bagam qrstuVwxy z Bété LMNOPQRSTK U Fula (Dita) Çüéâäàåçêë Fula (Adama Ba) èïîìÄÅÉæÆô Kpelle ABCDEFGHI J Manding VWXYZ,./<> Osmaniya +,-./:;<=> Wolof ; :[{]}-_=+ suggests that the Bagam system may also have had this structure. Th e 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). Th e Bété numerals were invented in late 1957 or early 1958 by Frédéric Bruly- Bouabré, 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 infl uence from the Western numerals (or perhaps the Arabic numerals, although it is not clear whether Bruly- Bouabré knew Arabic at all). Th e unusual sign for 10 may have been used multiplicatively or additively in conjunction with the unit-signs. Th ere 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 eff orts 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 ciphered- positional decimal numerical notation systems. Th e fi rst 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. Th e 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 infl uence from Western numerals (Dalby 1969: 173–174). Neither system was ever used except by its inventor. Th e 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 defi nite historical ancestry can be assigned to them. Th e script was used traditionally for tax records as well as for offi cial 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 Man- dinka, and was intended to provide a means of communication accessible without the need for formal schooling. Th e numerals are ciphered-positional and decimal, and perhaps are related graphically to the Western numerals; however, numeral- phrases 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). Th e Osmaniya decimal ciphered-positional numerals, like the script, were written from left to right. Th e fact that the script fully represented vowel sounds and was written from left to right shows infl uence 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 offi cial 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 ciphered- positional numerals (Dalby 1969: 165–168). Faye, who was literate in both French and Arabic, presumably drew more infl uence 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 Afri- can 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. Th is 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. Th e 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 fi ve 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). Th ere 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). Th e 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. Th e 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 cumulative- additive numerical notation systems. Regardless, even tallying systems that use specifi c 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:

Th e numbers are continually used as follows: a perpendicular stands for a unit, fi ve 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 fi fties. It will be seen that two of these symbols are not dissimilar to Arabic numerals, namely, the circle and the symbol for fi fty. Th e Arabic symbol for fi ve 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 fi fty. ... 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.

Th is 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 cumulative- additive numeral-phrase with a base of 10, a sub-base of 5, and a special sign for 20. Th is sort of notation is qualitatively diff erent 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 Th e writing system of the Harappan civilization, centered in the Indus River valley, is one of the great remaining mysteries in the fi eld of script decipherment. It was used from around 2500 bc to 1900 bc on several thousand very short inscriptions (averaging fi ve 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. Th e 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 confi rm the existence of a numerical notation system in the ancient Indus Valley, much less determine its origin, history, or function. Th ere 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

12345678910 111 111 1111 1111 11111 11111 1 11 111 1111 Short strokes \11 111 \111 1111 \1111 11111 ? ? 151 70 38 38 70 7 2 1

Long strokes a aa aaa aaaa aaaaa aaaaaa aaaaaaa 149 365 314 64 22 3 6

Th ese signs probably represent low numbers in a cumulative fashion; the short strokes are grouped into sets of three, four, or fi ve, just as the signs of most other cumulative systems. Th e 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 ‘fi sh’ sign + is attested in combination with three, four, six, and seven strokes), we are relatively confi dent 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). Th is 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’. Th e 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. Th is 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). Th is 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, Can- hujo-daro, and Kalibangan, inscribed with sets of vertical strokes and crescents/ hooks (and sometimes other script-signs). Th ese 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 Mohenjo- daro, 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 specifi c 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 insuffi cient 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 Th ere is some confusion over the identifi cation of this object, which is assigned diff erent artifact numbers by Parpola (1994) and Pettersson (1999). Miscellaneous Systems 333 must be 8. Th e limited linguistic reconstructions regarding Proto-Dravidian (even granting the controversial hypothesis that the Harappan language was a Dravid- ian 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). Th e 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. Th e 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. Th e 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. Th ere 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. Th e 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. Th e Harappan civilization declined precipitously after 1900 bc, although it may have survived in certain regions for a century or two longer. Th ere is no evidence that the Indus numerals had any infl uence 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. Th e 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 “prompt- book” to assist priests in reciting memorized texts (Bockman 1989: 155). It was reputedly invented in the twelfth century ad (Coulmas 1996: 353). Th e earliest dat- able dongba texts, however, are from the middle of the eighteenth century (Bock- man 1989: 153). Th e 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 fi rst made available to Western scholarship by F. H. Nichols in 1904, three repeated “South Asian” swastika-like signs precede six vertical strokes. Th ese 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. Th is 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 specifi c 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 fi ve strokes (3 + 3, 4 + 3, 4 + 4, 3 + 3 + 3); 5 is depicted with a single row of fi ve strokes, but 15 is depicted with X (10) followed by 5 indicated in two lines of three and two strokes respectively. Th e 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 numeral- signs, and all appear to be cumulative-additive and decimal (i.e., signs are repeated, but no sign is repeated more than nine times). Th e 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

123456789 d efg hijk l 10 20 30 40 50 60 70 80 90 mnopqrstu 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. Th ese numerals are shown in Table 10.9 (Pinnow 1972: 828). Th e system is ciphered-additive, as it has signs for 1 through 9 and 10 through 90. Th e 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. Th e separate signs for the decades from 10 to 90 may have obviated the need for a zero-sign. Th ere may also have been signs for 100 and 1000, which presumably would combine multiplicatively with the unit-signs, but this cannot be confi rmed (Pinnow 1972: 831). Th e 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 fi rst 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). Th e Varang Kshiti numeral-signs resemble those of various South Asian systems, but none of these resemblances proves a specifi c 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 suffi cient antiquity to have been infl uenced by Brāhmī. Th e Varang Kshiti script and numerals are still used in both primary and adult education, and eff orts 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 Th e 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. Th e earliest (Source Version) of the Pahawh Hmong numerals are shown in Table 10.10 (Smalley 1990: 79). Th is 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. Th e 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. Th e 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. Th e use of two signs for 10 (one additive and one multiplicative) is analogous to the multiplicative-additive Mende system described earlier. Th ere 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. Th e 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 infl uence on its invention. eTh standard Chinese numerals are multiplicative-additive, so they may have infl uenced the development of the Pahawh Hmong system. Th is 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. Th e Pahawh Hmong numeral-signs are entirely dif- ferent from the Chinese ones, so we should not presume any infl uence 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). Th is 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. Th e addition of a sign for zero and the abandonment of the power-signs change the system’s structure radically. Th is 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

123456789 abcdefghi

10(+) 20 10(×) 100 1000 jklmn

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 power- signs. 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.). Th is 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

12 3 4 5 6 7890 pqrs t u vwxo

10 100 1000 10,000 100,000 1,000,000 10,000,000 100,000,000 < > >< ? ?< + +< _ 1,000,000,000 10,000,000,000 100,000,000,000 1,000,000,000,000 _< { {< } 338 Numerical Notation immigrated to the West (especially Australia), Hmong numerals are used not only in Laos and Hmong-speaking parts of Vietnam, but also in Western countries. At present, both the Lao and Western ciphered-positional numerals challenge the Hmong numerals, so it is not clear how long they will continue to be used. Regardless of the eventual success of any of these systems, it is noteworthy that three variants of the Pahawh Hmong system were developed with such rapidity, each with a diff erent structure.

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

6 While the Ryukyu Islands have been under Japanese control since the seventeenth century, the cultural infl uences in the archipelago have been at least as much Chinese as Japanese, given its location in the East China Sea. 7 I have corrected a couple of errors in Chamberlain’s tables where numeral-signs were assigned incorrect values. Miscellaneous Systems 339

Table 10.12. Ryukyu numerals (money)

123456789 10 mung Y 100 mung A AA AAA AAAA KÇüéâ 1 kwang LMNOBPQRS 10 kwang CTUVZ[îì 100 kwang E\G y I\z ] 1000 kwang :| 10,000 kwang ; - = 352 kwang, 250 mung grounds. Both systems are cumulative-additive and decimal, with a sub-base of 5. Th e multiples of each power from 1 to 4 are mainly cumulative (exceptions include the “100 kwang” money count and the hundreds value in the fi rewood count), and the multiples from 5 through 9 combine the appropriate sign for 5 with the required number of additional units. Th e numeral-signs are largely abstract. In some cases, the sign for fi ve of a power is derived by halving the sign for one of the next higher power, (e.g., 5000 kwang vs. 10,000 kwang in Table 10.12, or 500 vs. 1000 in Table 10.13). Numeral-phrases were written in a roughly vertical fashion. Figure 10.5 depicts one of the sho-chu-ma studied by Chamberlain. We do not know when or how the sho-chu-ma were fi rst used. Chamberlain states, “Th e custom may be traced to a hearsay knowledge of the Chinese written character among the Luchuan [Ryukyu] peasantry, who, not possessing suffi cient learning to employ this character itself, and not being encouraged by their rulers to acquire the elements of an education deemed unsuitable to their lowly station, developed a make-shift of their own” (1898: 383). Th e sign for 10 bundles / 10 kwang, +, was probably borrowed from the identical Chinese sign for 10 (Chamberlain 1898: 384).

Table 10.13. Ryukyu numerals (fi rewood bundles)

123456789 1 A AA AAA AAAA B= 10 CêëèDïîìÄ 100 EHIJFäàåç 1000 G 340 Numerical Notation

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

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

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

8 It is highly improbable that the Roman numerals are an ancestor of the Ryukyu system. Miscellaneous Systems 341

Figure 10.6. Samoyed tallying stick showing diff erent means of notation. eTh fi rst two instances are primarily sequential/ordinal, while the third is a cumulative-additive notation for 333. Source: Jackson and Montefi ore 1895: 403. but underspecifi ed. While Samoyedic peoples traditionally were nonliterate, there is evidence for an indigenous numerical notation system of notched sticks among at least some Samoyedic peoples. Figure 10.6 depicts a selection of tallying sticks found by the British explorer Frederick George Jackson (1860–1938) during his 1893 expedition by sledge (Jackson and Montefi ore 1895: 403). Th ese tallies refl ect a diversity of numerical practices. The fi rst two do not appear to be base-structured numerical notation, but rather to represent counting through one-to-one correspondence using notches across the width of a stick; the V-like signs may represent 5, 10, or some other value. Th e third stick, however, apparently indicates ones, tens, and hundreds with I, X, and ᅷ,if the annotations are correct. Th e fourth artifact may in fact be non-numerical (e.g., an ownership mark). Th e use of I and X for 1 and 10 suggests a possible link with Roman numerals, but the sign for 100 does not, and is instead identical to the “Tuscan tallies” used in nineteenth-century Chioggia (Chapter 4). Th is system of signs might thus be in some sense a pan-Eurasian cultural phenomenon (Ifrah 1998) or a borrowing from Russian users of Roman numerals; alternately, it may simply be that when using wood or bone as a tallying medium, signs of one, two, and three straight notches are an evident way to indicate successive decimal powers. 342 Numerical Notation

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

9 See Fischer (1997) for a remarkably complete summary of dozens of decipherment attempts from the 1860s to the present. Miscellaneous Systems 343

Table 10.14. Putative Easter Island numerals

12345678910 ! @ # $ % #\# #\#\# & ^ 6 (?) ) 35 = 3 × 10 + 5 (?)

In combination, these diffi culties lead to the conclusion the rongorongo script had no numerical notation. Unlike other undeciphered scripts, such as Linear A, for which the numeral-signs are obvious and frequent, rongorongo texts lack any of the markers that would help identify such signs. Despite Bianco’s (1990: 39) statement that “[i]l est normal de trouver un système représentatif des nombres dans une écriture ancienne, les tablettes pascuanes ne pouvaient échapper à cette règle générale,” there exists no iron law that every script must have its own numerals. It is quite possible that the texts were not used for functions in which numerical notation was necessary or useful. Alternately, it may have been a ciphered system, in which case we would be unable to identify numeral-signs unless the script were deciphered more fully. A fi nal possibility is that so few rongorongo texts survive (about two dozen) that any former system is now lost. Cherokee One of the most famous instances of stimulus diff usion, in the form of the indigenous invention of a script by a nonliterate person on the basis of hearsay knowledge, was the creation of a syllabary for the Cherokee (Tsalagi) language around 1820 by Sequoyah. It is less commonly known that several years after inventing his syllabary, probably around 1830, Sequoyah also developed a decimal numerical notation system. Th is system is preserved only in two manuscripts now held at the Gilcrease Museum in Tulsa, Oklahoma, written by Sequoyah for John Howard Payne, the American dramatist and poet, who spent much time among the Chero- kee and particularly with Sequoyah in the 1830s (Walker and Sarbaugh 1993: 77). Th e numerals of this system are shown in Table 10.15 (Holmes and Smith 1977: Appendixes II and III). Th e system is ciphered-additive for numbers from 1 to 99. Th ere are distinct signs for 1 through 19 and each decade from 20 through 90. Th e signs for 1 through 20 are grouped graphically into sets of 5 (1–5, 6–10, 11–15, 16–20), but there is no structural relation among the signs in each subgrouping. Th is vigesimal element is very curious, since the Cherokee lexical numerals are purely decimal. Presumably, the signs for the tens between 20 and 90 combine additively with the unit-signs for 1 through 9, while the signs for 10 through 19 are 344 Numerical Notation

Table 10.15. Cherokee numerals

12345678910 ABCDEFGHIJ 11 12 13 14 15 16 17 18 19 20 KLMNOPQRST 30 40 50 60 70 80 90 100 ×10 UVWXYZ[]{ used only on their own. Th e documentary evidence neither confi rms nor refutes this supposition, but if the signs for 10 through 19 were combined with the signs for the tens, the signs for 30, 50, 70, and 90 would have been redundant. For writing numbers above 100, the system is not ciphered-additive but multiplicative-additive.10 Th e sign indicated as “x 10” always combines multiplicatively with the sign for 100, and multiplies the value of the phrase by ten. Perhaps I am overinterpreting the fi rst element of this sign, but it strikes me as being similar to the cursive English word ‘times’. While Sequoyah had hoped that his numerical notation system would be adopted, just as the syllabary had, when he laid it before the Cherokee tribal council, they voted against it and in favor of the Western numerals (Holmes and Smith 1977: 293). As a result, we know of the Cherokee system only from the two surviving Payne documents, only one of which transliterates the numeral- signs into Western numerals. One of these is dated 1839 (in Western numerals) by Payne, suggesting that Sequoyah may still at that time have been attempting to resuscitate his system’s fortunes (Holmes and Smith 1977: Appendix III). After Sequoyah’s death in 1843, the Cherokee numerals ceased to be used for the most part. However, some modern Cherokee are certainly aware of them; Figure 10.7 depicts a modern clock, purchased online, in which the Cherokee numerals for 1 through 12 indicate the hours.

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

10 Strikingly, the Cherokee system is structurally identical to the Jurchin system (Chapter 8), including the use of distinct signs for 10 through 19, and is very similar to the Ethiopic (Chapter 5) and Sinhalese (Chapter 6) systems, both of which are ciphered-additive below 100 and multiplicative-additive above. Miscellaneous Systems 345

Figure 10.7. Cherokee numeral clock. Author’s photo. project, and has been adopted more widely among the Inupiat.11 Th e numeral- signs of this system are shown in Table 10.16. Th e system is cumulative-positional with a base of 20 and a sub-base of 5. Th e numeral-signs are written using slightly diagonal vertical strokes with a value of 1, above which slightly diagonal horizontal strokes are placed, each with a value of 5. When the numerals are handwritten, the vertical and horizontal strokes are of the same width, but sometimes in print the horizontal strokes are shown somewhat thicker than the vertical strokes. Th e zero-sign is reported to be symbolic of a human fi gure’s arms crossed over the chest, but is also similar to the Western zero-sign. Fortunately, we have enormous detail regarding the context of the system’s invention. Th e students, having completed work on binary notation, realized that the lexical numerals of the Iñupiaq language were base-20, and took it upon themselves to develop a vigesimal numerical notation system that would better

11 My information on this system is based entirely on very fruitful discussions with W. Clark Bartley, the non-Inupiat instructor of the mathematics class in which the system was developed. 346 Numerical Notation

Table 10.16. Iñupiaq numerals

12345678910 Çüéâäàåçêë 11 12 13 14 15 16 17 18 19 0 èïîìÄÅÉæÆô correspond to their lexical numerals.12 At fi rst, an attempt was made to develop ciphered signs for 10 through 19, but this was found to be taxing on the memory of users. Th e students instead developed a cumulative-positional system that requires only two diff erent strokes (vertical for ones, horizontal for fi ves) and a zero. At the time, neither they nor their teacher were familiar with other cumulative-positional systems such as the Chinese rod-numerals (Chapter 8) or the quasi-positional Mesoamerican bar-and-dot numerals (Chapter 9). Th e students knew the Western numerals, and had had a brief introduction to Chisanbop fi nger computation, a quinary-decimal calculating technology that also inspired them. Th e Iñupiaq system is unusual in that its invention was specifi cally in the context of mathematical education; it was always meant to aid students in working with arithmetic. Although the choice of cumulative-positional notation with a sub-base was stimulated by the diffi culty entailed in memorizing the twenty separate symbols that a ciphered system would have required, it had the added eff ect of facilitating arithmetic using physical counters. Techniques were quickly developed to manipulate numbers using popsicle sticks to represent the vertical and horizontal strokes of the written numerals, thus producing a computational device whose results could easily be written on paper thereafter. In some cases, the students found it more convenient to use this device in a purely base-5 fashion (i.e., with up to four horizontal sticks for 5 instead of only three, as in the numerical notation system). While the system is understood by a number of youth of northern Alaska, as well as by some educators, its eventual success is still very uncertain, as Western numerals are strongly preferred by many educators. Although Inupiat children trained in this system have had considerable success in their mathematics education, the very small number of users of this system limits its present value as a communication tool. It is too early to say whether the offi cial adoption of this

12 While the Iñupiaq lexical numerals are vigesimal with a sub-base of 5, they deviate from the numerical notation system described here in the use of subtractive formations for 9 (10 − 1), 14 (15 − 1), and 19 (20 − 1), as well as in the use of a word for 6 that is not derived from that for 5. Miscellaneous Systems 347

Figure 10.8. Zuni irrigation stick; the right side of the stick is a sequential-ordinal tally in which every fi fth notch is diagonal and every tenth notch an X; the sum (24) is indicated in cumulative-additive fashion at left. Source: Cushing 1892: 298. system by the Commission on Inupiat History, Language, and Culture will help its chances of survival.

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

13 To be precise, the two slanted notches diff er slightly, as do the two X marks, but it is not clear from Cushing’s drawing exactly what the distinctions are. 348 Numerical Notation

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

Siyaq A very unusual set of numerical notation systems was employed by Arabic, Per- sian, Islamic Indian, and Ottoman administrators between the tenth and nineteenth centuries for representing numbers in fi nancial transactions. While they are known by many names (dewani by the Arabs, siyaq by the Persians and Turks, and rokoum in India) and exhibit enormous paleographic variability, they all share a common origin and structure. Recognizing that it is slightly inappropriate to refer to all variants of the numerals as “siyaq,” I will nevertheless group them all here under this single term. Th e Persian siyaq numeral-signs are shown in Table 10.17 (Kazem-zadeh 1915: Plates I–III). Miscellaneous Systems 349

Table 10.17. Siyaq numerals

123 4 5 6 7 8 9 1s ab c d e f g h i 10s jklmnop qs 100s tuvwxy z A B 1000s CDE F GHI JK 10000s LMNOPQRST

Th e siyaq numerals have nine distinct signs for each power of 10, and thus the system is basically ciphered-additive and decimal. Numeral-phrases are written from right to left, although in numeral-phrases containing both units and tens, the unit-sign is found to the right of the tens-sign (i.e., before it rather than after it). Often, individual signs are ligatured together, making it diffi cult in some cases to distinguish the individual components of a numeral-phrase. Th ere are similarities among the signs for diff erent multiples of the same power. For instance, the signs for the tens all have a short diagonal stroke connected to a long hooked horizontal stroke followed by some additional component, while the signs for the hundreds all have a small curved stroke at the left followed by a separate additional component. Moreover, there are similarities among the signs for the same multiples of diff erent powers. Th us, 9, 90, 9000, and 90,000 all have a common graphic element roughly resembling an obliquely slanted Western numeral 3. Th ese similarities suggest that the system is multiplicative-additive and that each sign is composed of a unit-sign on the right and a power-sign on the left. Yet this classifi cation would be overly simplistic, since there are many imperfections in the numeral-signs that defy a simple multiplicative explanation; for instance, the numeral 900 does not conform to the pattern just established. Th e solution to this taxonomic conundrum is that the siyaq numeral-signs were not originally abstract signs. Rather, the signs in Table 10.17 are extremely reduced cursive versions of the corresponding Arabic lexical numerals (Kazem- zadeh 1915). Because the Arabic lexical numerals are multiplicative-additive (like those of English and most other languages), when they were cursively reduced into abstract and nonphonetic siyaq signs, they retained a visual vestige of their original multiplicative nature. Th us, the Arabic word for ‘thousand’, alf, which is written phonetically as , is reduced but still visible in the siyaq numeral 1000. Th is unusual origin also explains the odd structural features of the siyaq numerals, such as the placing of the units before the tens in numeral-phrases. Yet siyaq numeral-phrases could not be read phonetically; they are all too reduced to be understandable except to those trained in the system’s use. Especially in 350 Numerical Notation non-Arabic-speaking areas, the association between numeral-words and numeral- signs was limited. Th us, the siyaq system is numerical notation, not a set of lexical numerals. Bagheri’s (1998) derivation of the siyaq numerals from the Pahlavi numerals used in early medieval Persia (Chapter 3), fails to account for the siyaq system’s irregular structure. Th e earliest document containing siyaq numerals (in fact, the dewani variant used by the Arabs) is a list of expenses and receipts presented to the Abbasid caliph Al-Moktadir Billâh by his minister, Ali ibn ‘Isa, dating to 306 a.h. (919 ad) (Kazem-zadeh 1915: 14; Bagheri 1998). Th e numeral-signs from this text are already quite impossible to read as lexical numerals, so these signs may have been used even earlier. Th e numbers expressed in this text and later ones normally represent monetary amounts, but in some cases expressed weights or discrete quantities of objects (Kazem-zadeh 1915: 31–32). Th e term siyaqat ‘style, arrangement, method’ was fi rst applied to this system in the Kitab al-Fihrist (Catalogue Book) of Ibn al- Nadim, dating to 377 ah / 977–978 ad (Bagheri 1998). Th e siyaq numerals were likely chosen over the Arabic positional numerals or some other system in order to control who could read a particular document. Th ey also prevented fi nancial corruption by making forgery, falsifi cation, and alteration more diffi cult (Bagheri 1998). Th e siyaq numerals were used for many centuries in all of the major successor states to the Abbasids. Th e system was especially popular in the Ottoman Empire from 1300 onward, and in Persia from the time of the Safavid Dynasty (1501–1736) onward. Th e Ottomans apparently stopped using siyaq numerals in the nineteenth century. However, it was still taught in Persian elementary schools until the 1930s (Bagheri 1998: 297).

Cistercian For most purposes, medieval European scribes used Roman numerals (in Western Europe) or Greek alphabetic numerals (in Eastern Europe), with the use of West- ern numerals becoming increasingly frequent from the eleventh century onward. Yet, beginning in the early thirteenth century, an unusual system began to be used in a limited number of manuscripts and marked on objects, primarily in contexts associated with the scribal tradition of the Cistercian monks. I therefore call this system “Cistercian numerals,” even though neither its earliest nor its latest users were Cistercians. While it has been ignored in synthetic works on numerical notation, thanks to the recent work of David King (1995, 2001), which supersedes all earlier research completely, we now have extensive information about it. Th e precursor of the full-fl edged Cistercian numerals was a set of eighteen symbols introduced by John of Basingstoke (John of Basing), archdeacon of Miscellaneous Systems 351

Table 10.18. Numerals of John of Basingstoke

123456789 1s ABCDEFGHI 10s JKLMNOPQR 75 = :

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

Table 10.19. Alphabetic and numerical values of the ars notaria/Basingstoke’s system

abcdefghil mnopqrstu a BEHKNQCFILOR ADGJMP - 258205080369306090147104070 352 Numerical Notation

Table 10.20. Cistercian horizontal numeral-signs

123456789 1s abcde!ghi 10s jklmn#pqr 100s stuvw$yz0 1000s 12345%789 alphabetic order and correlated with their numerical values in Basingstoke’s system, a clear pattern emerges, as seen in Table 10.19 (King 2001: 68–69). It is remotely possible that the graphic similarities and patterning of the ars notaria and Basingstoke’s system were developed independently. Nevertheless, since we know the former to have been invented in the twelfth century, the most parsimonious theory is that Basingstoke learned the ars notaria and then hit on the idea of assigning numerical values to these alphabetic signs. Th is raises the possibility that perhaps Basingstoke’s travels in Greece taught him the ciphered-additive Greek alphabetic numerals (Chapter 5), leading him to hit on the idea of using ars notaria letters as numerals. Th e use of alphabetic numerals was infrequent in Western Europe, but would have been common in early thirteenth-century Athens. While Basingstoke’s numerals appear in only two texts other than the Chronica maiora, one of these is a late thirteenth-century manuscript from a Cistercian monastery, Whalley Abbey in Cheshire. Th is is relevant because the next place we fi nd a system like Basingstoke’s numerals is in late thirteenth-century Cistercian manuscripts from France and Belgium. While these signs were slightly diff erent from his numerals, they were derived from the earlier English signs. Th e most common variant of this system is shown in Table 10.20 (King 2001: 102).14 Whereas Basingstoke’s numerals had signs only for the units and tens, this more developed system included signs for the hundreds and thousands as well, and used a horizontal base stroke rather than a vertical one. Nevertheless, the structure of the system is essentially the same, only with four positions instead of two, with the units in the top left, the tens in the bottom left, the hundreds in the top right, and the thousands in the bottom right. Th us, we may classify it as a ciphered-positional system based on orientation, or as a ciphered-additive system for which signs for the same multiple of diff erent powers happened to resemble one another. Table 10.21 shows how several numbers would have been expressed using this system.

14 King (2001: 39) provides a chart illustrating the enormous variation in this system within what was, after all, a very limited manuscript tradition. Miscellaneous Systems 353

Table 10.21. Horizontal Cistercian numeral-phrases

157 < 2345 . 6666 ; 9002 ,

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

Table 10.22. Vertical Cistercian numerals

123456789 STUVWXYZ[ (4 + 1) (6 + 1) (6 + 2) (8 + 1)

5107 = > 354 Numerical Notation as frequently as the horizontal numerals, they were employed in a wider set of contexts, including mercantile and scientifi c ones. Th ey are inscribed on an astrolabe from Picardy, the only example we have for their use on an object rather than in a text (King 2001: 131–151). A fi fteenth-century arithmetical text from Normandy describes a technique for writing numbers higher than 10,000 by placing a sort of bracket around a lower number using the multiplicative principle, so that 126,000 would be written as ? (King 2001: 159). Th ey also occur outside of northern France on a late fi fteenth-century astronomical table from Segovia, which belongs to a set of eclipse computations by the Jewish Spaniard, Abraham Zacuto (Chabás and Goldstein 1998). Unusually, a few manuscripts from Bruges describe their use as markings on wine barrels and wine gauges, as part of the mercantile practices of vintners starting in the late fourteenth century and used as late as 1720 (King 2001: 164–171, 239–242). Unfortunately, no marked wine barrels or related artifacts exist to complement the textual evidence. Th e advent of printing in the middle of the fi fteenth century, and the decline in the Cistercians' fortunes that accompanied the Reformation, were disastrous for this system. Th e numerals ceased to be used regularly in the sixteenth century. After this point, interest in the numerals from an academic and mystical perspective increased. Th ey appear as historical curiosities in many sixteenth-century texts, most notably De occulta philosophia (1531–33) by Agrippa of Nettesheim (King 2001: 190–202). Th ey are also found in De numeris (1539) by Johannes Noviomagus and De subtilitate libri XXI (1550) by Girolamo Cardano, both of which cite Agrippa as an authority.15 In these texts, and frequently thereafter, the numerals were mistakenly thought to be “Chaldean,” an appellation often used in the Renaissance to refer to mystical learning supposedly diff used from the Near East, especially Babylonia. Even Cajori (1928: 68–69) cites the Chaldean theory of Agrippa, although skeptically. Th e use of the Cistercian numerals in well-known mystical and mathematical texts ensured that they were never completely forgotten, even though knowledge of their true origin was lost. Th ey were described in various works on magic, the occult, and astrology, as well as in a variety of early works on numerical notation (King 2001: 210–238). Yet, apart from their use in wine gauging in Bruges mentioned earlier, they were used only rarely after 1550. A group of Parisian Free- masons used the numerals in some of their private correspondence with fellow members in the 1780s (King 2001: 243–246). Th e last nonscholarly mention of the numerals was by a number of German nationalistic authors in the early twentieth century, who saw the Cistercian numerals as a sort of proto-Aryan runic numeration (King 2001: 251–261).

15 Curiously, however, Noviomagus lists the horizontal rather than the vertical numerals, and some of Cardano’s vertical numeral-signs are more similar to Basingstoke’s thirteenth- century numerals than they are to the later vertical signs. Miscellaneous Systems 355

Table 10.23. Keutuklu numerals

123456789 1s abcdefghi 10s jklmnopqr 100s stuvwxyz0 1000s 123456789

Ottoman Cryptographic Th roughout the period of Ottoman dominance in the Middle East, between roughly 1450 and 1900, the standard Arabic positional numerals (Chapter 6) were by far the most common system in use, while various ciphered-additive systems, most notably the Arabic abjad numerals (Chapter 5), were used in certain contexts. In addition, a number of quasi-cryptographic systems that bear little to no resemblance to the Arabic alphabetic or positional systems were used by Ottoman administrators (particularly military clerks). Four of these were reported in West- ern scholarly literature by M. J. A. Decourdemanche (1899). While their historical importance is not great, they were structurally ingenious, and three of them have unusual structural properties. Th e fi rst system, known as keutuklu, was used by clerks to record data concerning the recruitment of Christian youth into the Ottoman army (Decourdemanche 1899: 261). Th e signs of this system are shown in Table 10.23 (Decourdemanche 1899: 260). Th is system has the appearance of a decimal ciphered-additive system with unique signs for each multiple of each power of 10. Yet the numeral-signs are not arbitrary, but are constructed by adding small circles to the set of nine basic unit-signs (one circle for the tens, two for the hundreds, and three for the thousands). An alternate way of looking at this system, then, would be to regard it as multiplicative-additive, with the unit-signs being the basic linear frames for 1 through 9 and the power-signs being one, two, or three circles. Additionally, there are graphic resemblances among the signs for 2 through 5 (which are based on the sign b and transpositions thereof ) and among the signs for 6 through 9 (based on f). However one chooses to look at it, there is a clear graphic resemblance between the signs in each column (e.g., 6, 60, 600, and 6000), which dispels one of the objections often leveled at ciphered-additive systems, namely the ardu- ousness of memorizing many diff erent signs. Because Decourdemanche does not describe how these signs combined with one another to produce numeral-phrases, we do not know whether numerals were written from left to right as in the Arabic 356 Numerical Notation

Table 10.24. Ordouï cheïlu numerals

123456789 1s ABCDEFGHI 10s JKLNNOPQR 100s STUVWXYZ) 1000s !@#$%^&*(

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

16 While the total number of diagonals (left and right) equals the relevant number in the ones column, this pattern does not hold for the higher powers. Miscellaneous Systems 357

Table 10.25. Ordoui numerals for personnel

123456789 1s BCDEKLMTU 10s V@#$%:;<= 100s >?[]+,-./ 1000s _ are grouped erratically in sets of three or four (1–4: one right stroke; 5–7: two right strokes; 8–10: three right strokes ... 800–1000: eight right strokes). Moreover, there is no common feature among the multiples of diff erent powers, so that 4, 40, and 400 have no inherent similarity. Finally, whereas the ordouï cheïlu could be used to write any number up to 10,000, the highest sign in this system was 1000. Th is variant originated from the structure of the Arabic abjad (Chapter 5). Th e twenty-eight-sign abjad was divided into eight mnemonic groups of three or four signs apiece, and the numerical values assigned to the abjad correlate perfectly with the divisions of this system. Moreover, the numeral-signs could be used not only in their numerical sense, but also to represent the appropriate Arabic letter. Th e third notable Ottoman cryptographic system, known as damgalu ‘inspec- tion’, was used for marking numerals on military equipment, and also could be used as a cryptographic script (Decourdemanche 1899: 264–265). Th e signs of this system are shown in Table 10.26 (Decourdemanche 1899: 265). Th e damgalu numerals have twenty-eight signs, like the Arabic abjad numerals, corresponding to 1 through 9, 10 through 90, 100 through 900, and 1000, organized

Table 10.26. Damgalu numerals

123456789

1s Çüéâäàåçê Ç ü 10s ëèïîìÄÅ æ É é â ä à å ç ê ë è 100s æ æ É É É æ æ æ É ï 1000s æ 358 Numerical Notation into a decimal ciphered-additive system. Instead of simply using alphabetic signs, however, each sign has four registers, in each of which a line or a dot is placed. Th is allows for only sixteen (24) combinations, so two additional signs were used to represent the last twelve signs. Th e fi rst, æ, was placed under any sign whose bottom register was occupied by a line, while the second, É, was placed under signs whose bottom register was a dot. Th e signs shown in Table 10.26 are the variety of damgalu used by the Ottoman navy, while a separate system was used in the army, which was identical except that it used the sixteen combinations in a dif- ferent order. Th ere is no correlation between the sequence of dots and lines and the numerical values in question, so the damgalu is particularly cryptographic. Th e additional signs for the last twelve numerical values are not structurally signifi cant; their use was simply a necessity imposed by the lack of adequate signs available with the sixteen basic combinations. Th e damgalu numerals were each correlated with one of the twenty-eight signs of the Arabic abjad, and could be used to stand for phonetic as well as numerical values. We know remarkably little about these systems’ origin or history, save that they were employed in the nineteenth century, when Decourdemanche reported on their use. A system nearly identical to the keutuklu system is described in a sixteenth- century Moroccan manuscript, where it is called, strikingly, qalam hindī ‘Indian fi gures’ (Sanchez Perez 1935). Th e inventors of the ordoui and damgalu systems were surely familiar with the Arabic abjad numerals, since they were organized according to the structure of the Arabic abjad and could stand either for a numeral or for the corresponding letter. Th e unusual structure of the keutuklu system may derive from the Arabic positional numerals. Th ey are treated by Decourdemanche as already being obsolete at the turn of the twentieth century, and do not appear to have survived past the end of the Ottoman Empire.

Summary Because the systems described in this chapter are not part of a single phylogeny, they share little in common. Few have been studied in any histories of numeration. While systems such as the Inka khipu have been ignored or belittled because they are not associated with a script, a formal analysis of their properties shows them to be ordinary numerical notation systems. Th e failure to recognize other systems, such as the Cherokee, Pahawh Hmong, and various African systems, derives from the marginalization of these societies in the modern world-system. Such systems may also be thought to be unimportant because they are derived from Western numerals and because they typically have failed to be adopted on a widespread basis. Th is is unfortunate, because they are often structurally distinct from Western numerals, contradicting the assumption that African and other Miscellaneous Systems 359 cultures infl uenced by Western imperialism are devoid of mathematical achievements, or that such achievements are at best purely derivative. Moreover, these newly invented systems help us understand the social contexts that lead to the development of new systems. In many of the cases described here, Western, Arabic, or other positional systems were available to be adopted. Th e fact that indigenous systems were invented indicates that the desire to resist imperialistic institutions or to produce local alternatives to foreign inventions often moti- vates the development of these and other systems. Even where systems are not in widespread use, they can remain important sources of cultural capital upon which script users can draw when necessary. Th is practice diff ers little from the use and retention of Roman numerals for prestige functions in Western societies, despite the near-ubiquity of Western numerals otherwise. Several ethnographically attested systems (such as Naxi, Oberi Okaime, Varang Kshiti, and Bambara) were invented and used specifi cally for ritual, liturgical, or divinatory purposes. Th is rationale parallels both the earliest Mesoamerican numerals (apparently used for calendrics and deity names) and the earliest East Asian numerals (used in divinatory contexts on oracle bone inscriptions). Sev- eral of the inventors of scripts and numerical systems were religious fi gures, and the association of script invention with religious visions or divine revelation has been noted elsewhere (Houston 2004: 235). Postgate, Wang, and Wilkinson (1995) argue that all early writing and numeration served bookkeeping/administrative functions, regardless of the attested evidence, claiming that the oracle-bones and Mesoamerican inscriptions are not reliable evidence of the initial functions of the scripts, and that the equivalent of proto-cuneiform tablets, long perished, once existed in those societies. Yet if colonial-period scripts and numerical systems can be invented for nonaccounting functions, why not ancient ones also? I have not discussed modern artifi cial numerical notation systems, such as the binary and hexadecimal numeration used in computing or the color-coded system of numeration used on resistors (International Electrotechnical Commission 2004). In these cases, there is a technological impetus toward the invention of particular forms of numeration.17 Yet these systems, in combination with the ethnographic evidence from the systems described here, conform to a pattern suggesting that the creation of numerical notation systems remains an ongoing project. More signifi cantly, despite the predominance of Western and Arabic positional numerals worldwide, there exist technological, social, and cultural reasons for the continued invention of new numerical notation systems in the twenty-fi rst century and beyond.

17 Th e same might be said of the use of I, V, and X shapes in tally-type systems! chapter 11

Cognitive and Structural Analysis

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

360 Cognitive and Structural Analysis 361 true universals are interesting, statistical regularities are also important, and may in fact be caused by cognitive factors similar to those that produce universals. Exceptions can help explain why the regularity exists in the fi rst place, by clarifying the conditions under which it does not apply. One must also be aware that a universal may be only contingent or apparent – for instance, an exceptional case may not be well attested, or may simply not have been invented yet. Whenever there are exceptions, I have as an expositionary device stated the regularity in its universal (exceptionless) form and then discussed the exceptions in the text. Statistical regularities (general patterns that have exceptions) are quite dif- ferent from implicational or conditional regularities, which take the form “If system A exhibits feature X, then it will also exhibit feature Y” and that apply to only a subsection of the universe of numerical notation systems. Frequently, systems to which an implicational regularity does not apply do not actually violate it; rather, the feature does not exist in outlying systems. Th ey can be either universal (within that context), “If A, then B,” or statistically probable, “If A, then usually B.” Th e systems included in this study are those that are attested in the ethnographic or historical record. Th e clever skeptic can imagine systems that violate any of the regularities to be presented here. Such systems have already been invented by scholars (Dwornik 1980–81), cryptographers (Wrixon 1989: 103), and science fi ction writers (Pohl 1966: 179–192). Th is does not demonstrate that these generalizations are not “true” regularities, or even that they are not universals, but merely proves that they are not logical necessities. Because existing numerical notation systems satisfy these constraints, even though it is not logically required that they do so, we must look instead to psychological and utilitarian constraints as the source of both universals and statistical regularities. Th at these constraints are apparently so great as to produce universals among 100 or more structurally distinct numerical notation systems confi rms the constraining power of the mind, working in conjunction with the perceived environment. Greenberg’s (1978) examination of regularities in lexical numerals has been of particular use in formulating my list of regularities. Where appropriate, I have indicated the correlations between my regularities and those he found for lexical numerals, without confi rming or denying the validity of the latter set. ereTh are, however, many regularities for lexical numerals that do not apply to numerical notation systems, and vice versa. For every instance in which there is a parallel between lexical numerals and numerical notation, there is another in which there are signifi cant diff erences between the two domains. Because of these diff erences, the regularities of numerical notation systems cannot possibly be derived from a biologically hard-wired “universal grammar.” 362 Numerical Notation

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

Axioms

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

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

General Regularities

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

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

1 Greenberg (1978: 254–255) off ers a similar principle concerning lexical numerals, which he calls the “thesis of continuity.” Cognitive and Structural Analysis 363 conceivably be developed for the sole purpose of recording a set of nonsequential numbers with religious signifi cance, or a group of users might use one system for representing odd numbers and an entirely diff erent one for even numbers. Such unusual systems have never been implemented. One of the crucial functions of numerals is ordinal enumeration, for which only a continuous set of integers will suffi ce.

G2. All systems use a base of 10 or a multiple of 10 for representing natural numbers.

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

G3. All systems form numeral-phrases through addition. G4. No system forms numeral-phrases through division.

Th ese two regularities are universals. Addition will always be found among the arithmetical steps by which a system is used to derive the values of numeral-phrases, whether it is the only operation (as in cumulative-additive and ciphered-additive systems) or not (multiplicative-additive, cumulative-positional, and ciphered- positional systems). It is possible to imagine a purely multiplicative system – for instance, one that expresses all numbers as prime numbers or as the product of prime numbers – but this has never occurred. Th is does not imply, however, that every numeral-phrase in a system uses addition; the units and the powers of the base are expressed with single signs in many systems, and thus do not involve addition. Addition is frequently combined with multiplication, a very eff ective means of expressing large numbers. While addition and multiplication are common, subtraction is extremely rare (being found only in the Roman numerals and a few Mesopotamian systems), and division is absent entirely from the operations used to form numeral-phrases for integers. It is certainly possible to imagine 50 and 10 being expressed as “2 100” or “2 20,” 364 Numerical Notation but this is not attested. Lexical numerals use division only in the form of multiplication by ½ or ¼, and this is very rare (Greenberg 1978: 261). Even this operation is never found in numerical notation. Th ere is the physical division of Etruscan, Roman, and Ryukyu tallying-based signs (e.g., Roman V is the top half of X), but this is a nonarithmetical graphic technique governing the formation of signs, and can be interpreted as doubling or halving with equal validity. Th is regularity does not deny that fractions can be expressed in numerical notation. Th e absence of division (and the rarity of subtraction) may be a matter of representational convenience (since such numeral-phrases would need to use large divisors to express smaller numbers), or a consequence of the rarity of such operations in lexical numeral systems.

G5. All numerical notation systems are ordered and read from the highest to the lowest power of the base.

Th is is a near-universal, which I call the Ordering Principle. Positional systems could be read from the lowest power to the highest, but this never occurs; in other words, interexponential ordering is respected throughout a numeral-phrase. While the powers of additive systems could be placed in any order (e.g., in classical Roman numerals, which do not normally use subtraction, 217 could be written as IICVCX), this occurs only when the writer has made an error. Subtractive forms such as the modern Roman numeral IX for 9 do not violate this principle, because they involve intraexponential structuring only. Many lexical numeral systems disobey this principle, including those of many of the major European languages (e.g., Italian sedici = 6 + 10 although diciasette = 10 + 7). However, numerical notation systems almost always reserve “low-high” forms for intraexponential subtraction or multiplication. Users of a system know, upon encountering a lower numeral-sign followed by a higher one, to be alert for the need to use an operation other than addition. Th e Ordering Principle also applies to sub-bases, which always precede signs for the next-lower full power. Th ere are a couple of minor exceptions to the Ordering Principle; in certain alphabetic systems, including the Greek, Glagolitic, and Cyrillic alphabetic numerals, the numbers 11 through 19 are often written with the sign for 10 at the end of the numeral-phrase (e.g., Cyrillic 12 = bj (2 + 10), not jb (10 + 2)). Th ese exceptions refl ect the word order of the lexical numerals of the languages of these systems’ users.

G6. No system uses signs for the arithmetical operations used to derive the value of a numeral-phrase.

Th is is a near-universal. Even though all systems form numeral-phrases through addition, and many of them also use multiplication, this is almost never directly represented with a sign. Th is is in contrast to lexical numerals, in which it is very Cognitive and Structural Analysis 365 common to express at least some operations with words, such as the German ‘sechshundert-fünf-und-vierzig’ ‘six hundred-fi ve-and-forty’, Latin ‘duodeviginti’ ‘two-from-twenty’, and other phrases that are even more complex. In fact, lexical numerals almost never express subtraction without some indication of the operation (Greenberg 1978: 258–259). Such signs in numerical notation would render numeral-phrases less concise without providing any additional clarity because the system’s principle dictates the operations used. Th e (near-) universality of the Ordering Principle means that the operation to be used can be inferred easily from the context. When lexical numerals show arithmetical operations explicitly, it is often because unusual ordering is being employed, as in the two numerals just mentioned. Th ere are, of course, systems that use graphemes that implicitly refer to a particular operation (e.g., the use of a hasta in Greek alphabetic numerals (Chapter 5) to indicate multiplication by 1000, or the ‘times 10’ sign in the Cherokee numerals (Chapter 10). Th ese signs always primarily bear numerical values, and simply happen to be used only in the context of particular arithmetical operations. Th e only actual exception to this regularity is the use of the additive grapheme ‘and’ in the Shang Chinese numerals (Chapter 8). Th is may refl ect the highly lexical nature of Chinese numerical notation and the linguistic origins of the system.

G7. Th e only visual features used to determine the numerical value of fi gures in numerical notation systems are shape, quantity, and position.

Th is is universal or perhaps nearly universal. Th e relevant features for determining the value of a numeral-phrase are the shapes of the particular signs used, the quantity of those signs, and their position. Th e color of the signs, their relative size, and other extraneous graphic features do not aff ect the value of the phrase. It is possible to conceive of a system where diff erent registers of sign sizes, rather than their position, would determine the value of signs within numeral-phrases, thereby eliminating the need for a zero-sign. We could, for instance, use West- ern signs from 1 through 9 to write 462 as 462 and 402 as 42 – no zero being required because the size of the 4 indicates its value. A similar system could apply diff erent colors to diff erent powers of the base. An exception to this pattern is that in the proto-cuneiform and archaic Sumerian systems (Chapter 7) the sign for 60 is a large version of the sign for 1. A second partial exception is the use of red- and black-colored rods in the Chinese rod-numerals (Chapter 8), but this was only done occasionally and only distinguished positive from negative numbers. Features whose diff erent values are easily diff erentiated visually and that are easily represented in writing are highly desirable for numerical notation. Using size or color would be extremely diffi cult, requiring users to employ many diff erent- colored inks or to distinguish between diff erent-sized registers of signs. 366 Numerical Notation

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

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

2 Note, however, that the modern Greek lexical numerals for 13 through 19 follow the inverse order: dekatreis ‘10 + 3’... dekaennia ‘10 + 9’. Cognitive and Structural Analysis 367

G10. No system uses an identical representation for two diff erent numbers.

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

Implicational Regularities

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

Th is is a universal. While it is easy to imagine a system with a base of 10 and a sub-base of 3, and while such a system would be able to express every number uniquely (and, in fact, somewhat more concisely than with a sub-base of 5), this and similar nondivisor sub-bases are never attested in numerical notation systems. Greenberg (1978: 270) notes that at least two lexical numeral systems, Coahuilt- eco and Sora, have this feature, but it is extremely rare in lexical numerals as well. Numerical notation systems are sometimes structured by means of additional numbers that are not divisors of their primary bases (such as the use of a sign for 4 in the base-10 Nabataean and Kharoṣṭhī systems), but these additional numbers are not sub-bases, because they do not recur throughout the system (i.e., 40, 400, and 4000 do not have their own special signs). 368 Numerical Notation

I2. No ciphered system has a sub-base.

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

I3. If a system is cumulative, it will group intraexponential signs in groups of between three and fi ve signs.

Th is regularity, which I call the Rule of Four, is nearly universal but has a few minor exceptions. Humans are limited in their cognitive capacities, and work most effi ciently when information is packaged in groups of three to fi ve elements. Cumu- lative systems cope with this limit either by using sub-bases (e.g., Roman, Maya bar- and-dot), by using spacing to distinguish groups (e.g., Egyptian hieroglyphs, Aztec), or both (Babylonian sexagesimal). Probably the limit of fi ve is even a bit too high for the human mind to grasp. Two full exceptions to the Rule of Four are the Inka and the Bambara numerals (Chapter 10), which always use groups of up to nine unit- signs. Partial exceptions include the Hittite hieroglyphs (Chapter 2) and the South Arabian numerals (Chapter 4), in which chunking in groups of three to fi ve signs was an option, but in other cases groups of up to nine signs were used. I will discuss the Rule of Four further in the section “Subitizing and Chunking.”

I4. If a system is multiplicative-additive for a particular power of its base, it will also be multiplicative-additive for all higher powers of the base.

Th is exceptionless regularity applies to hybrid systems – those that are cumulative- additive or ciphered-additive for some powers, but multiplicative-additive for Cognitive and Structural Analysis 369 others. In such systems, it is always the higher rather than the lower powers that are multiplicative, and once the “switch” to the multiplicative principle has been made, it applies for all higher powers. No system is multiplicative-additive for lower powers while following some other principle for higher powers, or alternates multiplicative and non-multiplicative powers. Hybrid multiplication is primarily useful for extending a system further without developing increasingly large inventories of signs, but results in somewhat longer numeral-phrases. It is thus more useful for large powers than for small ones. Moreover, in many lexical numeral systems, multiples of lower powers are expressed with a single word, but multiples of higher powers separate the units and power components (e.g., Latin sex, sexa- ginta, sescenti vs. sex milia). Th e point at which this shift in lexical numerals occurs sometimes may aff ect whether or not a certain power is expressed multiplicatively in the corresponding numerical notation system, but it cannot be the only factor because many systems do not use multiplication at all.

I5. Whenever the multiplicative principle is used in a system, the unit-sign or signs (multiplier) will precede the power-sign (multiplicand).

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

I6. No multiplicative system uses 1 as a power-sign.

Th is regularity is virtually exceptionless. Multiplicative-additive systems combine unit-sign multipliers with power-sign multiplicands, but there is never a separate power-sign for the units. Rather, the numbers from 1 up to the base of the system are expressed through unit-signs alone. While a power-sign for 1 would be consistent with the principle of combining unit-signs with power-signs, it would be completely extraneous and provide no additional information. While most lexical numeral systems are multiplicative-additive, they similarly do not 370 Numerical Notation use power-signs for 1. Th e only exception to this rule is that the earliest Bamum numerals (Chapter 10) had a separate power-sign for 1, which was entirely distinct from the unit-sign for 1. Th is sign was used only for a brief time, however, before the Bamum system became ciphered-positional. In any event, the use of distinct graphemes for the unit-sign and the power-sign eliminated any ambiguity. In all other multiplicative-additive systems, there is no power-sign for 1.

I7. All multiplicative expressions involve only bases or their powers as multiplicands.

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

I8. All composite multiplicands are strictly multiplicative.

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

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

3 In a few medieval French manuscripts, XX (20) is used as a multiplicand in emulation of this lexical formation, however. Cognitive and Structural Analysis 371 following regularities are nonuniversals whose interest lies not in their regularity but rather in the frequent assumption that they are universal, despite numerous signifi cant exceptions. I once held many of these propositions to be universal, based on my intuition or preliminary reading, but under more careful scrutiny they have proved to be less regular than they fi rst appeared. Th e existence of these exceptions does not make the generalizations irrelevant, but it does require an accounting of the processes that lead to exceptions. In all other respects, they are ordinary statistical regularities.

N1. Some additive numerical notation systems are infi nitely extendable without the need to invent new signs.

One of the primary benefi ts cited for positional notation is that any number, no matter how large, can be written without needing to develop new signs. How- ever, several additive systems that use multiplication, such as the Ethiopic numerals (Chapter 5), the Armenian numerals of Shirakatsi (Chapter 5), the classical Tamil and Malayalam numerals (Chapter 6), the Babylonian common numerals (Chapter 7), the Chinese classical numerals (Chapter 8), and the Mende numerals (Chapter 10), are also infi nitely extendable by using composite power-signs as multiplicands (see the previous section, I8). Th ese techniques obviate the need to develop new signs for higher powers of the base, and thus produce an infi nitely extendable system. It is sometimes far more concise to use these systems for writing large numbers than it is to use ciphered-positional systems; for instance, the Ethiopic expression for 100,000,000, , requires only two signs where nine Western numerals are needed. It is fair to say that all systems that are purely cumulative-additive or ciphered-additive are fi nite in scope.

N2. Some positional systems are not infi nitely extendable.

Th is is not a universal, even though it borders on being a logical necessity. Th e Cistercian system (Chapter 10), which is best understood as one based on orientational or rotational position, is not infi nitely extendable; once the four positions are occupied (used for writing ones, tens, hundreds, and thousands), the system has reached its end. Th e same can be said for the Texcocan numerals (Chapter 9), which also use orientational position in the form of vertical and horizontal registers. Positional systems that are linear are all infi nitely extendable, as one can keep adding new positions in front of the highest power. In practice, even in positional systems, artifi cial forms of notation, or expressions that combine lexical and nonlexical numerals, are used for the highest numbers. While my editorial style guide for this book tells me to write 900,000,000,000,000,000,000, I am much more 372 Numerical Notation likely to write 900 quintillion or, most likely of all, 9.0 × 1020, both for reasons of conciseness and because it is very diffi cult for humans to readily read such long strings of numbers (see the section, “Subitizing and Chunking”).

N3. Some additive systems use a sign for zero.

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

N4. Some systems are not written or read in a one-directional straight line.

While most numerical notation systems are purely linear, several systems are read in a more convoluted manner. Th e Chinese commercial system (Chapter 8) and the Texcocan numerals of the Kingsborough Codex (Chapter 9), both of which are multiplicative-additive, place unit-signs in a row beneath the corresponding power-signs, so that the phrase is written and read in a zigzag fashion. Th e same is true for the multiplicative component of the Greek alphabetic numerals (Chapter 5) above 10,000, except that the unit-signs are placed above the multiplicative sign for 10,000 (M). Th e most extreme nonlinear systems are the Cistercian numerals (Chapter 10) and the ordinary Texcocan numerals (Chapter 9), both of which use orientational rather than linear position, the former through four rotational orientations, the latter through horizontal and vertical registers.

N5. Not all independently invented systems are cumulative-additive.

Based on limited evidence from the circum-Mediterranean region, or even based on nonempirical theoretical reasoning, some authors claim that cumulative- additive systems are the most ancient or basic form of numerical notation (Hallpike 1979, Damerow 1996, Dehaene 1997). However, the evidence presented in this study refutes that assumption. Th e Shang numerals (Chapter 8) are multiplicative-additive, Cognitive and Structural Analysis 373 the Inka khipu numerals (Chapter 10) are cumulative-positional, and the Bam- bara numerals (Chapter 10) are ciphered-additive (with a cumulative component). If the Brāhmī numerals (Chapter 6) were developed independently of Egyptian or Greek infl uence, this would also be an independently invented ciphered-additive system. Th ere is no evidence that any of these systems had cumulative-additive antecedents. It is probably not coincidental that several independently invented systems (Egyptian hieroglyphs, proto-cuneiform, Indus) and others that may have been independently invented (Etruscan) are cumulative-additive, because many nonliterate societies use unstructured tally marks, an obvious antecedent to cumulative-additive numerical notation. Yet this tendency does not reach the status of a universal evolutionary law or even a solid generalization.

Cognitive Explanations of Synchronic Regularities Synchronic regularities are features or patterns that hold true for all or most systems, or, in the case of implicational regularities, are true of systems that possess a certain feature. I have attempted to outline commonsensical or functional explanations for these regularities. Th ey refl ect common cognitive patterns among the inventors and users of systems, and thus have the potential to tell us a great deal about pan-human numerical cognition. Because some of these regularities have exceptions, we must exercise caution in attributing these regularities to neurological “hard-wiring.” Conversely, because these are not logical requirements – it is possible to conceive of systems that fulfi ll all the axioms A1 through A4 and yet violate one or more of the regularities just listed – we need to understand why the universe of attested systems is so much smaller than the universe of possible systems. Ideally, we would like to have more information about the decision-making processes and behavior that produced these regularities. However, since individual inventors were very rarely considerate enough to have left detailed records concerning their choice of a specifi c principle or base, and because very few systems were invented within living memory, indirect techniques are necessary. Moreover, since individuals may have been unaware of the advantages of diff erent means of representation, it may be more productive to analyze representational effi ciency without reference to individual decision making. Th e following cognitive factors are always relevant to some unknown degree in the processes relating to the development and use of numerical notation systems. It is simply impossible that so many specifi c regularities in the structure of numerical notation systems – ones that are not strictly determined by the logic of those systems – would emerge unless the various representational techniques were constrained by certain cognitive considerations. 374 Numerical Notation

Phrase Ordering One of the central principles of the cognitive sciences (of which cognitive anthropology is an important branch) is that information is often most useful when it is structured. Since numerical notation is a representational system used to help record, remember, and use numerical information, several synchronic regularities relate directly to the ordering of signs within numeral-phrases. Most obvious among these is the Ordering Principle (G5), which orients all systems in a highest- to-lowest direction of powers. Yet a number of other rules, including G6 (absence of signs for operations), G4 (absence of division as an operation), I4 (governing the switch from addition to multiplication in hybrid systems), I5 (unit-signs precede power-signs), and I6 (1 is not a power-sign), relate directly to the arrangement of numeral-signs within numeral-phrases. Something important is going on, from a cognitive perspective, to constrain the ordering of signs in numerical notation systems. Th e logical place to start in the analysis of the Ordering Principle is to look at similar attempts by linguists to explain ordering in lexical numerals (Salzmann 1950; Hurford 1975, 1987; Stampe 1976). Yet, while this research is important as a description of systems, it fails to explain the phenomena it describes, unless the assumption is made that descriptive rules map perfectly onto cognitive processes. While I have called the regularities that I have described “rules,” I do not mean that they are applied by individuals, either consciously or unconsciously. Rather, they are the outcomes of broader cognitive principles that must be examined in order to explain those regularities. In positional systems, signs must be put in their proper order to ensure that a numeral-phrase is interpreted correctly. A positional system that did not do this would be unworkable, since every numeral-phrase would have many equally valid readings. In many other systems, however, no logical requirement prohibits irregular ordering. In cumulative-additive and ciphered-additive systems, for example, the signs could be placed in any order without any ambiguity, since the values of the signs are simply added. In multiplicative-additive systems, as long as each unit-sign is associated with a specifi c power-sign, the resulting sign pairs could be placed in any order. Yet the Ordering Principle is nearly exceptionless, and such irregular phrases usually occur only where the writer has made an error. In cumulative systems, ordering ensures that identical signs are grouped together. If 327 could be written in Roman numerals as ICIXCVCX, the advantage of cumulation would be greatly reduced by the fact that identical signs are far apart from one another. Even requiring that the signs for each power be grouped together, one could still write CCCVIIXX, XXCCCVII, XXVIICCC, VIICCCXX, or VIIXXCCC instead of CCCXXVII, the only acceptable form. Cognitive and Structural Analysis 375

Th e Ordering Principle applies to all systems, and regulates intraexponential ordering in cumulative systems as well as interexponential ordering in both additive and positional systems. Greenberg (1978: 274) suggests that one cognitive principle favoring “larger + smaller” formations in lexical numerals is that, by beginning with the largest power, the fi rst element closely approximates the fi nal result, producing an expectation of the eventual size of the phrase in the listener/hearer. I think that explanations involving this factor of “successive approximation” also apply to numerical notation, and that the desire rapidly to approximate a value is in part responsible for the Ordering Principle. In particular, it explains why numeral-phrases that are in order, but read from the lowest to highest power, never occur. Note, however, that because numerical notation is a visual medium, numeral-phrases can be read in any order, regardless of the manner in which they are written, whereas spoken lexical numerals must be heard sequentially. Yet, while numerical notation systems are nearly always ordered from highest to lowest powers, lexical numeral systems are not. Th e few exceptions to the Ordering Principle in numerical notation (e.g., Greek, Cyrillic, and Glagolitic alphabetic numerals for 11 through 19) result directly from comparable irregular ordering in the corresponding lexical numerals. We then might expect to fi nd viola- tions of the Ordering Principle wherever the lexical numerals of a system’s users also do so. Since 17 in Latin is septendecim, we should expect the Roman numeral for 17 to be the unattested VIIX. Upon reading the number 16 aloud as sixteen, English speakers rapidly transform the “high-low” order numeral-phrase into a “low-high” lexical numeral. Th ese diff erences between lexical and graphic numeration suggest that additional factors must be involved. In many circumstances, ordering constraints in numerical notation result from the omnipresent concern with avoiding ambiguity, coupled with the relative infl exibility of numeral-signs. Almost all numerical notation systems are designed to minimize the possibility that a reader will misinterpret signs; that is, each specifi c numeral- phrase has only one numerical meaning (rule G10). Lexical numeral systems use a variety of techniques other than ordering to eliminate ambiguity. For instance, modern German (among many other European languages) uses stem alteration to distinguish 16 (sechzehn = 6 + 10) from 60 (sechzig = 6 × 10), and Classical Sanskrit uses pitch accent alone to distinguish 108 (aštáçatam = 8 + 100) from 800 (aštaçatám = 8 × 100), even though the numerical value and ordering of the two elements in each word are identical. Numerical notation systems lack this fl exibility; one of their conveniences is that they use a relatively limited set of discrete signs. Th erefore, the strict ordering of numeral-phrases is far more important. Ordering is also essential for unambiguously indicating which arithmetical operations are used to derive the values of numeral-phrases, because numerical notation systems do not explicitly express signs for the operations being used (rule G6). 376 Numerical Notation

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

Subitizing and Chunking Th e Rule of Four (rule I4) is the nearly exceptionless regularity that cumulative systems group signs in three to fi ve (4 ± 1) units, tending signifi cantly toward the lower end of this range. Th is feature developed independently at least seven times (Egyptian hieroglyphic, Etruscan, proto-cuneiform, Chinese rod-numerals, Maya bar-and-dot, Indus Valley, Iñupiaq) and is very widespread (although not universal) in cumulative systems throughout history. Th e Rule of Four is explained most parsimoniously by reference to the process of subitizing, or the ability to enumerate rapidly small quantities of discrete objects without having to count them explicitly. Humans have been shown experimentally to be able to enumerate groups of between one and three dots rapidly and with almost no error, and groups of four dots with some error and slightly less quickly, but most individuals cannot count groups of fi ve or more dots at a glance without signifi cant error or considerable delay (Mandler and Shebo 1982). Subitizing probably results from the physiological constraints of the mechanism by which our visual system local- izes objects in space (Dehaene 1997: 68). Th e implications of this principle for numerical notation are obvious: long groups of undiff erentiated cumulative signs (e.g., |||||||, ||||||||, |||||||||) will take longer to read and result in more errors than if some technique is used to avoid them. A common way in which systems conform to the Rule of Four is by dividing long sets of signs into smaller groups. Cumulative systems that do not use 5 as a sub-base, including most of the systems of the Hieroglyphic, Levantine, and Cognitive and Structural Analysis 377

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

4 I have listed the zong forms only, but the heng forms are simply ninety-degree rotations of the former, and thus the same principle applies. 378 Numerical Notation cumulative forms were sometimes replaced with ciphered signs: S, T, and V, so that no phrase required more than three repeated signs. Th ese three independent reductions strongly suggest that fi ve signs is cognitively too many, and that even four signs may be diffi cult to perceive. In noncumulative systems, signs are not repeated intraexponentially and do not need to be counted, and thus subitizing is mostly irrelevant to them. A separate cognitive principle, called chunking, applies to ciphered and multiplicative systems. As fi rst described by Miller (1956), humans have a limited ability to memorize and recall long sets of bits of information, with the maximum being the “magical number” 7 ± 2. In order to deal with long lists of information, it is much easier to recode input into a series of chunks, each of which contains a small number of bits. In practice, chunks of three or four bits eff ectively balance the limits of the memory, which restricts the maximum size of chunks, and the desire to minimize the number of chunks necessary. Chunking strongly aff ects the structure of noncumulative numerical notation systems. For instance, in Western numerals and many other ciphered-positional systems, it is typical to divide long numbers up into sets of three numbers (e.g., 123,456,789). Four-digit numbers are sometimes but not always grouped in this way (1000 vs. 1,000), but it is normal for all fi ve-digit and longer numbers to be subdivided. Doing so not only groups large series of numbers into manage- able chunks, it also accords precisely with the millesimal (base-1000) structure of American English lexical numerals (thousand, million, billion, trillion, etc.), and to a lesser degree with the mixed base-1000/base-1,000,000 lexical numerals of British English and many other European languages. Similarly, in the multiplicative-additive Chinese classical system, 10,000 and 100 million are specially structured to divide numeral-phrases into four pairs of unit-signs plus power-signs. Finally, one reason why most hybrid ciphered-additive/multiplicative-additive systems switch to multiplication at 1000 or 10,000 is that doing so groups signs into chunks of no more than three or four signs. For instance, in the Fez numerals (Chapter 5), which use a multiplier at 1000 by placing a horizontal line under a sign or signs, 658,379 is written (reading from right to left) as ipu-h\-n\-x , or (9 + 70 + 300) + (8 + 50 + 600) × 1000, thus dividing the numeral-phrase into two chunks of three bits on the basis of the subscript multiplier used. Chunking in numerical notation in all of these cases is at least partly correlated with the base- structure of the lexical numerals of associated languages, however. Whether there is a connection between subitizing and chunking cannot be resolved here. Subitizing may in fact be a specifi c example of how chunking aff ects humans' ability to perceive and encode patterns of discrete visual objects. Chunk- ing has a much broader range of applications, as it is not restricted to visual information and applies to tasks other than simple enumeration. Subitizing, as the Cognitive and Structural Analysis 379 direct cause of the Rule of Four, has far more signifi cant eff ects on numerical notation systems than does chunking, in general.

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

5 Th is raises the interesting question, beyond the scope of this book, why apparently no speakers of languages whose lexical numerals violate the Rule of Ten have invented numerical notation systems. 380 Numerical Notation

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

Diachronic Regularities In contrast to synchronic regularities, which constrain the possible structural outcomes of numerical notation systems, diachronic regularities apply not to individual systems, but to temporal trends among systems. Th ey exemplify change rather than stasis in numerical notation systems. To analyze diachronic regularities requires that we shift from the numerical notation system as the unit of analysis and toward the event of change (cf. Mace and Pagel 1994). Two processes of change, which I will simply call transformation and replacement, exhibit Cognitive and Structural Analysis 381 diachronic regularities. Transformation occurs when an older system gives rise to a structurally distinct descendant. It presumes a direct phylogenetic relationship between the ancestral and descendant systems, but does not tell us what happens to the ancestor after it gives rise to the descendant. We must establish the structures of both the ancestor and its descendant, correctly identify that the latter is derived from the former, and ideally understand the process by which the new system arose from the old. For my purposes, transformation does not include cases where the ancestral and descendant systems have the same basic structure. Replacement occurs when one system becomes extinct and is supplanted by another. It does not matter whether the system being replaced is directly related, indirectly related, or unrelated to its successor. For my purposes, a system that is essentially moribund but is understood by a very small group of specialists is considered to be extinct. Both transformation and replacement are severely constrained in their possible outcomes, and thus, while there are far fewer diachronic regularities than synchronic ones, their eﬀ ects on the pattern of historical change in numerical notation over time are considerable. Diachronic regularities are nonrandom patterns of cultural change that can be meaningfully called evolutionary. To admit that the cultural evolution of numerical notation is real is not to concede that it is linear, however, nor does it require that such changes be regarded as adaptive.

Transformation of Systems Table 11.1 summarizes all the cases where a system uses a diff erent intraexponential or interexponential principle than its ancestor. Th ese comprise all cases of transformation of principle for which adequate evidence exists, considering only the fi ve basic principles, but omitting other features (base, use of multiplication for higher powers, and other structuring signs). I have omitted cases, such as the Ryukyu sho-chu-ma tallies (Chapter 10), whose ancestor cannot be identifi ed with confi dence. In a few instances where a system is a blend of two ancestors, I simplify the relationship between ancestor and descendant, but this does not signifi - cantly aff ect the data because such cases usually have one clearly identifi able main ancestor, with the second system contributing far less. In all, twenty-two systems use a diff erent principle than their ancestor.6 Th ere is considerable but nevertheless constrained variability in the possible transformations of systems. Table 11.2 quantifi es the frequencies of these structural transformations, fi rst by graphing the changes according to both intraexponential

6 In the analysis that follows, I will treat the quasi-positional Maya numerals (with zero) as cumulative-positional even though, strictly speaking, this is not the case, as I have shown (Chapter 9). Ciphered-positional Ciphered-positional Principle Cumulative-positional Cumulative-positional Ciphered-additive Ciphered-additive Ciphered-positional Ciphered-positional Multiplicative-additive Multiplicative-additive Cumulative-positional Ciphered-additive Ciphered-additive Multiplicative-additive Multiplicative-additive Ciphered-additive Ciphered-positional Descendant positionalBabylonian “positional” Maya positional Roman hieratic Egyptian Cumulative-positional Pahlavi CodexKingsborough positional Greek positional Indian Armenian (Shirakatsi) Multiplicative-additive Âryabhata Tamil/Malayalam Iñupiaq Multiplicative-additive Cherokee Kshiti Varang Bamum Mende Multiplicative-additiveMultiplicative-additiveMultiplicative-additive Jurchin Multiplicative-additiveMultiplicative-additive Chinese positional Hmong (second stage) Katapayadi (modern) Malayalam Ciphered-positional Ciphered-positional Principle Cumulative-additive Cumulative-additive Cumulative-additive Cumulative-additive Ciphered-additive Ciphered-additive Ciphered-additive Ciphered-additive Ciphered-positional Ciphered-positional Ciphered-positional Ciphered-positional Ciphered-positional Multiplicative-additive (mfemfe) Bamum 66 Âryabhata Malayalam 794 Sumerian 2 bar-and-dot Maya 3 Roman 9 hieroglyphic Egyptian 5 Persian Middle Cumulative-additive 6 Aztec Cumulative-additive 5 alphabetic Greek 6 Brāhmī 6 Armenian alphabetic Brāhmī Brāhmī Ciphered-additive 88 Chinese classical Chinese classical 10 Hmong Pahawh 101010 Western 10 Western 10 positional Indian Western/Arabic 10 Western/Arabic Bamum Chapter Ancestor(s) Table 11.1. case) of systems (by Table Transformation

382 Cognitive and Structural Analysis 383

Table 11.2. Transformation of systems

Descendant’s Structure Ancestor’s Structure Cu-Ad Cu-Po Ci-Ad Ci-Po Mu-Ad Total Cu-Ad x3 2 016 Cu-Po 0x 0 000 Ci-Ad 00 x 235 Ci-Po 01 2 x25 Mu-Ad 00 1 5x6 Total 04 5 7622 Intraexponential Changes Interexponential Changes Cu-Ci 2 Ad-Po 10 Ci-Cu 1 Po-Ad 4 Cu-Mu 1 Mu-Cu 0 Ci-Mu 5 Mu-Ci 6 and interexponential dimensions, and then by considering each dimension of change separately. Alternatively, these changes can be represented graphically as in Figure 11.1. Ver- tical arrows indicate intraexponential transformations; horizontal arrows indicate interexponential transformations; and diagonal lines involve both types of change,

3 Cumulative-Additive Cumulative-Positional

2 1

1 Ciphered-Additive 2 Ciphered-Positional

1 3 2

5 Multiplicative-Additive

Figure 11.1. Transformation of systems (graphic representation) 384 Numerical Notation with the numbers indicating the frequency of each change. Dotted lines indicate changes that are only attested in modern contexts (1800–present). Th ree important regularities can be extrapolated from these data:

T1. Additive systems do not develop from positional ancestors.

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

T2. Cumulative systems do not develop from noncumulative ancestors.

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

T3. Th e only transformation that involves both intra- and interexponential change is the invention of multiplicative-additive systems from ciphered-positional ones, and vice versa.

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

7 If the Ryukyu cumulative-additive numerals were developed on the basis of the Chinese rod-numerals (Chapter 8), this would constitute a fi fth exception. Cognitive and Structural Analysis 385 change.8 Th ese changes are presumably too radical alterations of principle to occur. Yet the rise of ciphered-positional systems based on multiplicative-additive antecedents occurs fi ve times (albeit sometimes in conjunction with some externally introduced knowledge of positionality). While this transformation involves both intra- and interexponential change, it is nonetheless relatively simple, involving only the elimination of power-signs and the addition of a sign for zero. In two other cases (the Bamum and Mende numerals, already mentioned), the reverse change occurred, with ciphered-positional systems giving rise to multiplicative- additive descendants. Th ese three regularities tend over time to increase the frequency of noncumulative systems over cumulative ones, and of positional over additive systems. Th e reverse changes occur only in modern contexts, and the resulting systems have not been extremely successful. Among noncumulative systems, there is no trend favoring ciphered over multiplicative systems, or vice versa. Of the twenty possible transformations, only eleven are attested (only eight in pre-modern contexts), three of which (encompassing eight of the twenty-two examples of change in Table 11.2) result in ciphered-positional systems. Th e nine unattested transformations either result in cumulative-additive systems or involve both intra- and interexponential change. Th e trend toward ciphered-positional notation is partly explained by this transformational pattern. A second type of transformation does not involve changes to either the intraexponential or the interexponential principle, but only to the use of multiplication in higher powers of a system’s base. While only a limited number of systems use such a feature, it produces an important diachronic regularity.

T4. When one system that uses the multiplicative principle gives rise to another, the power above which the descendant is multiplicative is not higher than that of the antecedent.

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

8 Th e other three are all changes in principle resulting in cumulative-additive systems, which as I have already stated, is never known to have occurred. Hieratic 100,000

Meroitic Demotic 1000 10,000

Brahmi Greek 100/1000 Alphabetic 1000/10,000

Sinhalese Tamil/ Aryabhata Hebrew Arabic abjad Coptic Glagolitic Ethiopic

386 100 Malayalam 1 1000 1000 1000 Cyrillic 100 1 Fez 1000 Syriac Zimam 1000/10,000 1000/10,000

Chinese Assyro- 1 Babylonian 100

Jurchin Mari 100 1000 Figure 11.2. Changes in hybrid multiplicative power. Cognitive and Structural Analysis 387

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

Replacement of Systems Th e second diachronic process concerns the extinction of systems and their replacement by other systems, regardless of any phylogenetic relation between the two. I use the term “replacement” to refer only to systems that are supplanted more or less completely by other systems, even though doing so obscures cases where a system is replaced for many functions while continuing to be used regularly for others. Th e replacement of systems is far more frequent than the transformation of systems, because one system may replace many systems, but rarely does one system give rise to multiple systems that use a diff erent principle (cf. Table 11.1). Moreover, a system may be replaced by one that has the same basic structure. Table 11.3 compares the structures of extinct systems with those of the systems that replace them (including cases where the two systems have the same structure). Two clear trends emerge from the examination of patterns of replacement:

R1. Positional systems are not replaced by additive systems.

Th ere are only two partial exceptions to this rule, both of which involve the replacement of cumulative-positional systems. Th e quasi-positional system used in the Mesopotamian city-state of Mari in the eighteenth century bc (Chap- ter 7), which was occasionally used in place of the Assyro-Babylonian system, was eventually replaced by that system after the Babylonian conquest of Mari. Positional numeral-phrases in the Mari system were used only rarely and only in the hundreds position, however; this system is actually best regarded as a short- lived experimental combination of the Babylonian positional (mathematical) and additive (scribal) systems. Th e second exception involves the replacement of the 388 Numerical Notation

Table 11.3. Replacement of systems

Extinct Replaced by System Cu-Ad Cu-Po Ci-Ad Ci-Po Mu-Ad Total Cu-Ad 11 0 15 9 1 36 Cu-Po 1 0 1 4 0 6 Ci-Ad 1 0 3 19 1 24 Ci-Po 0 0 0 12 0 12 Mu-Ad 0 0 1 6 2 9 Total 13 0 20 50 4 87

Intraexponential Replacement Interexponential Replacement Cu Ci Mu Ad Po Cu 12 29 1 Ad 35 34 Ci 1341Po 216 Mu 0 72

Babylonian cumulative-positional numerals used in mathematics and astronomy by the Greek alphabetic numerals following Alexander the Great’s conquest of Mesopotamia and the gradual domination of Greek over Mesopotamian learning in the exact sciences. Again, this is only a partial exception, because the Greeks borrowed and adopted Babylonian sexagesimal positional numerals in their own mathematics and astronomy, producing the sexagesimal ciphered-positional fractions (Chapter 5). All other positional systems were replaced by other positional systems (in fact, by ciphered-positional systems) or survive to the present day.

R2. Noncumulative systems are not replaced by cumulative systems.

Th ere is one exception to this rule. Th e Gothic numerals of Wulfi la’s script (Chapter 5) were ciphered-additive and were based on the Greek alphabetic numerals. Yet, because they were used primarily in Western and Central Europe, they were replaced by the cumulative-additive Roman numerals. Gothic numerals were used in only a limited number of texts, and so, while other alphabetic systems such as the Greek alphabetic numerals survived and thrived in competition with Roman numerals, the Gothic numerals were overwhelmed. Despite the importance of Roman numerals as an instrument of Roman imperialism, they never totally displaced any of the ciphered-additive systems of Eastern Europe or the Middle East, although they replaced other cumulative-additive systems, such as the Etruscan numerals (Chapter 4) and various Levantine systems. Despite the historical importance of cumulative-positional systems, such as the Babylonian Cognitive and Structural Analysis 389 positional numerals and the Chinese rod-numerals, no cumulative-positional system has ever totally replaced any other system. Th e comparison of patterns of replacement with patterns of transformation is instructive, as the eff ects of the two processes overlap. Positional systems are rarely ancestral to additive systems (except in modern colonial contexts) and tend to replace additive systems over time (but not vice versa). Th e obvious eff ect is that positional systems become more frequent over time, while additive ones become less frequent. A similar eff ect is seen in the intraexponential dimension, where noncumulative systems are rarely ancestral to cumulative ones, and tend to replace cumulative systems over time, gradually decreasing the frequency of cumulative systems. Yet there are also considerable diff erences between patterns of invention and patterns of replacement. While the intraexponential transformation of cumulative systems into noncumulative ones is comparatively rare, the intraexponential replacement of cumulative systems by ciphered or multiplicative ones is very frequent. To understand why systems transform and are replaced in these patterned yet complex ways requires attention to the cognitive eff ects on individuals of changing the manner in which numbers are represented.

Cognitive Explanations of Diachronic Regularities In explaining synchronic regularities, it was necessary only to show that the presence of a feature was correlated with some cognitive factor that, if the feature were absent, would be inconvenient to a system’s users. Because these regularities were universal or near-universal, these explanations largely involved considerations of hypothetical exceptions. Th eir universality also means that social context is less relevant to explanations of their existence. In explaining diachronic regularities, we are considering patterned variability among systems, so we may also ask whether a descendant system is more or less convenient than its ancestor, or whether a successor is more or less convenient than the system it supersedes, with respect to a number of cognitive criteria. To explain diachronic patterns, I compare the observed trends with the cognitive advantages or disadvantages of particular features of systems. Where trends correspond to increased effi ciency in some respect, theories can be developed connecting individual decision making to the observed patterns. Evaluations of effi ciency can be derived from abstract principles of economy in some cases (e.g., a short numeral-phrase is better than a long one) or from principles derived from cognitive science. As with explanations of synchronic regularities, this is an indirect means of reconstructing cognitive processes, one made necessary by the limitations of the data. Th ese explanations will be more complex than those of synchronic regularities simply because they must explain change rather than 390 Numerical Notation stasis. Cognitive explanations for diachronic regularities explain not only why a feature came into existence, but also how one system compares to another in some respect. Th is also raises the possibility that advantages as well as disadvantages may be involved in the choice between any particular pair of systems.

Conciseness Th e conciseness of a numeral-phrase is simply the number of signs needed to write that particular number. It is thus a property of a numeral-phrase, not of a numerical notation system. All other things being equal, a system that requires many signs to write numbers is more cumbersome than one that requires few signs. Because many systems are infi nitely extendable, it is impossible to state exactly the average number of signs needed to express numbers, and in any case, this would not necessarily be useful, since very large numbers are quite rare. Yet, because a system that regularly requires long numeral-phrases is going to be quite cumbersome to use, we wish to evaluate in general whether a system’s numeral-phrases are long or short. I consider the average length of numeral-phrases for all numbers from 1 to 999 to be a good rough measure of a system’s conciseness. Table 11.4 shows the conciseness of each principle (presuming a base-10 system with no sub-base for each case) for a variety of numbers. In general, ciphered systems are the most concise, requiring only one sign per power. For any natural number, no system is ever more concise than a purely ciphered-additive system. While ciphered-positional systems are usually more concise than nonciphered ones, for round numbers they are sometimes less concise because they require zero-signs in the empty positions (e.g., Roman numeral C = 100). Nevertheless, cumulative systems are almost always less concise than their ciphered and multiplicative counterparts, even for small and/or round numbers. Multiplicative-additive systems are slightly less concise than ciphered systems, because they often require two signs (a unit-sign and a power-sign) where the latter need only one. Yet, because they are additive, and thus do not require a zero-sign, they are more concise than ciphered-positional systems for round and nearly round numbers. Additive systems are only slightly more concise than positional systems that use the same intraexponential principle; the diff erence in their conciseness is equal to the number of empty positions present, for which positional systems require zero-signs. Th e comparative eff ect of this diff erence is far less than that between cumulative and noncumulative systems. Th e use of bases higher than 10 has variable eff ects on a system’s conciseness, depending on which principle it uses. Cumulative systems become far less concise through the use of higher bases; in a pure base-20 cumulative-additive system, such as the Aztec numerals, each sign may be repeated up to nineteen times, so Cognitive and Structural Analysis 391

Table 11.4. Conciseness of systems

Number of Signs Required (base-10 system, no sub-base, no hybrid multiplication) Ciphered- Ciphered- Multiplicative- Cumulative- Cumulative- additive positional additive additive positional (Chinese (Egyptian Number (Georgian) (Western) Classical) Hieroglyphic) (Inka) 6 11 1 6 6 27 22 3 9 9 70 12 2 7 8 100 13 1 1 3 400 13 2 4 6 649 33 5 1919 870 23 4 1516 2003 24 3 5 7 4268 44 7 2020 9080 24 4 1719 Average 2.70 2.89 4.49 13.59 13.78 (1-999) that 399 requires thirty-eight signs instead of only twenty in a base-10 system (average 22.82 signs/numeral-phrase from 1 to 999). Yet, for a noncumulative system, using a higher base makes numeral-phrases slightly more concise! In a ciphered- positional system like the Oberi Okaime numerals (Chapter 10), all numbers from 1 to 19 require only a single sign each, from 20 to 399 only two signs, and from 400 to 8000 only three signs (average 2.58 signs/numeral-phrase from 1 to 999). Th is advantage in conciseness, however, is off set by an increase in the sign-counts of these systems (see the following discussion). Th e use of sub-bases in cumulative systems improves their conciseness. By using a sub-base of 5 in a decimal cumulative-additive system, a number such as 870 requires only seven signs instead of fi fteen (DCCCLXX vs. CCCCCCCCXXXXXXX), and for all numbers less than 1000 the average conciseness is reduced from 13.59 signs per numeral-phrase to 7.45 signs per numeral-phrase. While this reduces the disadvantage of cumulative systems as compared to noncumulative ones, it never eliminates it. A cumulative system with a sub-base has additional round numbers, which are often expressed more concisely; whereas only fourteen numbers less than 1000 are expressed as or more concisely in a cumulative-additive than in a ciphered-positional system, the introduction of a sub-base of 5 into the cumulative- additive system raises that number to fi fty-four. Th e absence of sub-bases in most 392 Numerical Notation noncumulative systems (rule I2), and their relative frequency in cumulative ones, is due to the enormous advantage in conciseness that a sub-base provides to the latter but not to the former. Th e use of subtractive notation is rare in numerical notation systems, being found only in Roman numerals and some of the Mesopotamian cuneiform systems. It does carry a considerable advantage in conciseness, since 1999 in additive Roman numerals is MDCCCCLXXXXVIIII but MCMXCIX (or even MIM) when subtractive notation is used. However, because subtractive numeral-phrases do not group similar signs together, parsing and reading them may be more diffi cult. Th e relative paucity of subtractive numeral-phrases in Roman numerals in the classical period, coupled with the fact that subtractive notation is cross-culturally rare, suggests that its advantages were not perceived as being great. Th e use of a hybrid multiplicative component for higher powers of some additive systems has a slightly negative eff ect on the conciseness of systems that possess this feature. For example, a ciphered-additive system that has no multiplicative component, such as the Georgian alphabetic numerals (Chapter 5), expresses 4000 with one sign, while a similar system with a multiplicative component for higher powers, such as Sinhalese (Chapter 6), requires two signs. Similarly, where a purely cumulative-additive system, such as the Greek acrophonic numerals, requires four signs to write 4000 (XXXX), a hybrid multiplicative system, like the Phoenician system, requires fi ve (μ\a\aaa). Using multiplicative expressions for higher powers obviates the need to develop distinct signs for each multiple of each higher power, but such expressions will contain both unit-signs and power-signs and thus be somewhat longer. Th e diachronic trend toward noncumulative systems over cumulative ones strongly accords with their far greater conciseness. Th e trend in favor of positional systems over additive ones, by contrast, does not have a basis in conciseness, since additive systems are slightly more concise than their positional counterparts.

Sign-count A system’s sign-count is the total number of signs its users must know in order to read and write numbers. A system with a smaller sign-count is generally easier to learn and use than one with a larger sign-count because of the decreased mnemonic eff ort involved. For systems such as the Western numerals, the sign-count is ten. Yet this seemingly simple defi nition produces many complexities when attempting to enumerate how many distinct signs a user of a system requires. For systems such as the Phoenician numerals (Chapter 3) and the republican Roman numerals (Chapter 4), there are multiple signs for many numbers, some of which represent regional or diachronic variability that would be irrelevant to individual Cognitive and Structural Analysis 393 users, while others may be multiple signs that every user needed to learn. Certain signs (normally for very large numbers) may have developed late in a system’s history or have been used by only a few writers. In other cases, in which a sign is composed of two largely undisguised other signs, a rather arbitrary decision must be made whether to count it as a separate sign. Should the Sumerian sign for 36,000, k, be counted as a sign separately from its constituent parts, j (3600) and g (10)? Finally, and perhaps most importantly, the issue of sign-count cannot be considered properly without also considering the numerical limit of a set of signs; the Indus numerals may have only two signs, but these can express only numbers from 1 to 99, whereas the Western numerals have ten signs but can express any number. Keeping these reservations in mind, it is nevertheless possible and useful to compare the sign-counts of diff erent types of systems. Cumulative systems, which rely on the repetition of a small number of identical signs, have much lower sign-counts than ciphered ones, which use a wider variety of unrepeated signs. Positional systems, which do not require additional signs to be invented for successive powers, are more economical in sign-count than additive ones, although a sign for zero is highly useful. Cumulative-positional systems, which combine both of these advantages, have extremely small sign-counts (one to three distinct signs). Th e sign-counts of cumulative-additive systems are very low, but depend on their extendability; a decimal cumulative-additive system without a sub-base requires only one sign for each power of 10 that can be expressed (usually four to seven signs, with more if a sub-base is used). Multiplicative-additive systems have slightly larger sign-counts than ciphered-positional ones because, while ciphered- positional systems need only signs from 1 up to the system’s base, and 0, multiplicative-additive ones need signs for each power of the base. Th e sign-count for a ciphered-positional system is normally equal to its base, while that of a multiplicative-additive system equals its base plus one sign per power. Ciphered-additive systems, which require one sign for each multiple of each power of the base, have extremely large sign-counts, normally twenty or more, although using script-signs as numeral-signs can mitigate the inconvenience. From lowest to highest, the sign- counts of systems of diff erent principles are: cumulative-positional Æ cumulative- additive Æ ciphered-positional Æ multiplicative-additive Æ ciphered-additive. Table 11.5 lists some systems whose sign-counts are unambiguous, thus allowing them to be compared. Base structure has variable eff ects on sign-count. For cumulative systems, a system’s primary base does not aff ect its sign-count, since a single sign per power will suffi ce, as long as it can be repeated as often as is necessary. For noncumulative systems, however, using a higher base than 10 is extremely detrimental, requiring many more signs to be developed. Th e only noncumulative systems with bases 394 Numerical Notation

Table 11.5. Sign-counts (selected systems)

Sign- System Chapter Structure Sign Inventory count Babylonian Cumulative- 7 2 positional positional f g Egyptian Cumulative- hiero- 2 7 additive q r s t u v w glyphic Ciphered- Western 6 10 positional 1234567890

Multiplicative- A\B\C\D\E\F\G\H\I Tamil 6 12 additive J\K\L a b c d e f g h i Ciphered- Gothic 5 27 additive j k l m n o p q r s t u v w x y z {

higher than 10 are the Maya head-glyphs (Chapter 9; base-20, but uses a sub- base of 10 to decrease mnemonic eff ort), the Oberi Okaime numerals (Chapter 10; base-20, used only briefl y and by few individuals), and Âryabhata’s numerals (Chapter 10; base-100, uses script-signs to decrease eff ort and has a sub-base of 10, and was used by only a single school of mathematical thought). Th e use of a sub-base also has variable eff ects on a system’s sign-count, depending on the system’s structure. For a cumulative system, introducing a sub-base increases its sign-count slightly. A cumulative-positional system requires only one extra sign (for the sub-base), while a cumulative-additive system requires one extra sign per power (e.g., the Roman numerals V, L, and D). In either case, this increase in sign-count is off set by an enormous saving in conciseness; it is safe to say that a base-60 cumulative-additive system, such as the Sumerian cuneiform numerals (Chapter 7), could not exist without a sub-base. Yet, in the one ciphered system that has a sub-base (see rule I3), the Maya head-variant glyphs (Chapter 9), introducing a sub-base actually decreases the sign-count; instead of signs for 0 through 19, it requires only fourteen signs (for 0 through 13) with 14 through 19 (and sometimes also 13) written with glyphs combining 10 with 4 through 9. Finally, the use of multiplication for higher powers (hybrid systems) greatly reduces the sign-count of ciphered systems, but has minimal benefi t in cumulative systems. A single multiplicative power-sign can be combined with a set of existing ciphered unit-signs (1 through 9 in a decimal system) to avoid needing Cognitive and Structural Analysis 395 new signs for each multiple of each power. Most ciphered-additive systems of the Hieroglyphic (Chapter 2), Alphabetic (Chapter 5), and South Asian (Chap- ter 6) families, as well as systems such as the Jurchin (Chapter 8) and Cherokee (Chapter 10), use hybrid multiplication to express large numbers. By contrast, the use of hybrid multiplication in cumulative systems, such as the Levantine (Chap- ter 3) and many of the later Mesopotamian systems (Chapter 7), has a minimal eff ect on sign-count. Where each successive power has its own power-sign, and power-signs are combined only with signs from 1 up to the base (e.g., Aramaic F = 100, G = 1000, ± = 10,000, each of which is combined with up to nine cumulative strokes), there is no economy of sign-count; all that multiplication does is avoid repeating signs other than the unit strokes (e.g., Faaaa instead of FFFF for 400). Rarely, in systems such as the Assyro-Babylonian common system, whole cumulative-additive numeral-phrases, including both unit-signs and signs for higher powers, combine multiplicatively with large power-signs, so that 10,000 was written as 10 (g) times 1000 (gi), 100,000 as 100 (h) times 1000 (gi), and so on. Doing so eliminates the need for new signs for higher powers of 10. In such cases, there is a moderate savings in sign-count. Other than the reversal of the positions of ciphered-positional and multiplicative- additive systems, there is an inverse correlation between a system’s conciseness and its sign-count. Th us, the observed diachronic trend from cumulative to noncumulative systems is unexpected if sign-count is an overwhelmingly important factor. However, the trend toward positional over additive systems may have such a basis, although ciphered-positional systems have larger sign-counts than most cumulative-additive systems and only slightly smaller ones than multiplicative- additive systems.

Extendability A system’s extendability is measured by the largest number that can be written with it. Unlike conciseness and sign-count, both of which are relevant to the writing even of low numbers, infi nite extendability, which is characteristic of most positional systems, becomes particularly important only when there is a strong societal inclination to express very large numbers (especially where numerals are used commonly for mathematics). However, any increase in extendability – even the addition of a new power-sign to an additive system – can be considered an increase in the capabilities of a system to represent numbers, regardless of the specifi c functions for which such developments are used. While some multiplicative-additive systems are infi nitely extendable (rule N1), and some orientational positional systems are not (rule N2), the general rule that positional systems are infi nite in scope while additive ones are not is largely 396 Numerical Notation correct. Even so, some additive systems are much more easily extended than others. Additive systems that use multiplication, whether throughout the system (fully multiplicative-additive systems) or only for larger powers (hybrids), are generally able to express larger numbers than non-multiplicative ones. Th is is because in many such systems, power-signs may be multiplied by entire numeral-phrases rather than single signs, and/or because multiple power-signs placed side by side can be used to indicate repeated multiplications. Most pure multiplicative- additive systems can express numbers as large as 100,000, and many have limits as high as 10 million. In the abstract, there is no reason why ciphered systems should be more extendable than cumulative ones, but in actuality, they are slightly more extendable, usually having limits of 10,000 or higher whereas many cumulative- additive systems are used only for numbers up to 1000 or 10,000 (such as the modern Roman numerals). Th is may be related to conciseness – beyond 1000 many cumulative-additive numeral-phrases become so long that the best alternative is to transform the system’s structure, or to adopt a new system entirely. Th e use of a particular base constrains but does not dictate the numerical limit of a system, because one must also take into account how many powers of the base the system can represent. All other things being equal in terms of sign-count and conciseness, a system with a base higher than 10 can represent larger numbers than a base-10 system (e.g., 203 = 8000, so the Aztec numerals can represent any quantity up to 160,000 using only four diff erent symbols, whereas a similar base- 10 system could represent only numbers below 10,000). Th e use of sub-bases has no eff ect on extendability. Th e greater extendability of positional systems correlates with the trend over time toward positionality over addition. It should be noted, however, that practically any numerical notation system can be extended without great diffi culty should the need arise, either by developing new numeral-signs or by introducing a structural change such as hybrid multiplication. Where such changes have not been made, there was no overwhelming need for them. A great preponderance of the numbers used in both pre-modern and modern contexts are below 1000, and nearly any numerical notation system can deal with such small quantities. Infi nite extendability is really relevant only in mathematical contexts.

Eff ect of Cognitive Factors Table 11.6 summarizes the conciseness, sign-count, and extendability of systems using each of the fi ve basic combinations of principle (presuming all other factors to be identical). Each principle is ranked on the three criteria I have discussed (1 being best, 5 being worst). Cognitive and Structural Analysis 397

Table 11.6. Ranking of systems by cognitive factors

Conciseness Sign-count Extendability 154 Ciphered-additive Normally 10,000 – 2.70 18–30 1 million 231 Ciphered-positional 2.89 10–11 Normally infi nite 343 Multiplicative-additive 4.49 12–14 Normally 100,000 + 425 Cumulative-additive 13.59 4–7 Normally 1000 – 100,000 511 Cumulative-positional 13.78 1–3 Normally infi nite

Th e conciseness and sign-count of a system are inversely correlated, except that ciphered-positional systems have a slightly smaller sign-count and are also slightly more concise than multiplicative-additive systems. Th is correlation is not a coincidence, because one of the most eff ective ways to increase conciseness is to reduce many signs (one-to-one correspondence) to one, which must involve inventing new signs. Yet there is no correlation between conciseness and extendability or between sign-count and extendability. Very concise systems may be highly extendable (ciphered-positional) or limited (ciphered-additive), just as systems with small sign-counts may be highly (cumulative-positional) or less (cumulative-additive) extendable. Th e reason for this is that conciseness, sign-count, and extendability are properties of dimensions of systems (intraexponential or interexponential), not of the systems themselves: ciphered systems are the most concise, cumulative systems have the smallest sign-counts, and positional systems are the most extendable. Because each system is structured both intra- and interexponentially, any system will be less than optimal in at least one of these dimensions. Th e Western numerals are less concise than the ciphered-additive Greek alphabetic numerals and has more signs than the cumulative-additive Roman numerals, both of which they replaced. It is a good compromise between maximum conciseness and minimum sign-count, but it is maximally effi cient in neither respect. A major problem arises when we attempt to extend the analysis of cognitive- structural motivations of specifi c instances of invention or replacement to produce general rules. Th e diachronic trend toward positionality over addition, and toward 398 Numerical Notation ciphering and multiplication over cumulation, suggests that addition and cumulation should be seen as negative or inferior principles. To do so neglects an important consideration, which is that cumulative systems are more common than ciphered ones and additive systems more common than positional ones, and that for many millennia, cumulative-additive systems were the most common type. Moreover, if we explain the trend toward ciphering as a desire to maximize conciseness, we must deal with the fact that the trend toward positionality is in opposition to this desire, since positional systems are less concise than their additive counterparts. To explain long-term diachronic trends, we must acknowledge that the weight- ing of the cognitive advantages and disadvantages of diff erent principles was not equal in all time periods or in all social contexts. Where there are diachronic trends – such as that favoring noncumulative systems over cumulative ones – they result from changing evaluations of the importance of various merits and defects of diff erent principles. If we want to understand why those evaluations might have changed – for instance, why ciphering (and thus conciseness) came over time on a worldwide basis to be preferred over the small sign-counts of cumulative systems – we must understand the historical conditions under which such evaluations were made. Th e question of diachronic trends becomes even trickier when we examine the eff ects of features other than intraexponential and interexponential principle. Table 11.7 summarizes these eff ects. Th e presence of any of these features may mitigate any negative eff ects or reduce the advantages of the use of a principle. Moreover, their eff ects on conciseness and sign-count vary depending on a system’s intraexponential principle, adding an additional layer of complexity to the analysis of its merits and disadvantages. Although we can speak of the cognitive merits and disadvantages of a system’s base and/or sub-base, we would not expect diachronic trends to exist for this feature, because these are often a consequence of the lexical numerals of its users’ language(s) rather than the result of a conscious decision to alter a system. Simi- larly, hybrid multiplication is a fl exible way of achieving greater extendability at little extra cost in sign-count or conciseness. Yet the only diachronic regularity concerning hybrid multiplication, rule T4, concerns the power above which multiplication is used, not its simple presence or absence. While the effi ciency of systems is relevant to the diachronic patterns I have described, potential improvements in a system are not necessarily perceived auto- matically and regarded as relevant by its users. Th ere are levels of diff erence too small to be relevant, and perhaps too small to be perceived. For instance, the minimal diff erence in conciseness between ciphered-additive and ciphered-positional systems, while recognizable, does not appear to exceed a minimum threshold level (above which, presumably, the additive would be preferred over the positional), Cognitive and Structural Analysis 399

Table 11.7. Overall eff ects of other features

Conciseness Sign-count Extendability Cumulative Other Cumulative Other Much Base > 10 Much less Much more No eff ect Higher higher Sub-base More N/A Higher N/A No eff ect Hybrid multi- Slightly less Slightly less Usually Much Higher plication none lower

while more salient features such as the much smaller sign-count of ciphered- positional 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 benefi t. Where signifi cant social factors, such as political hegemony, are involved in the transformation and replacement of systems, otherwise important considerations of effi ciency may be irrelevant to users. Th ere 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 fi ve 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. Th us, 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 diff erent, 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 specifi c 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 specifi c 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. Th e 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 (specifi cally, 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 diff erence in structure must have been perceived as important, is erroneous. Th e analysis of the structure of numerical notation systems is insuffi cient 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 specifi c situations in the history of numerical notation. chapter 12

Social and Historical Analysis

Th e primary function of numerical notation is to communicate numerical values. One cannot even lie eff ectively 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 effi ciency 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. Th e role of various social factors in explaining the history and development of numerical notation systems diff ers 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 identifi ed seventeen factors that infl uenced 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. Th ey complement the transformations and replacements of systems that I discussed in Chapter 11, and also help to explain the cultural diff u- sion 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 eff ects on the history of numerical notation systematically, as I did with the dif- ferential eff ects of structural patterns. Th ere 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 eff ects that are directly opposite to others. Th ere 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 disadvan- tageous for new functions for which numerical notation is required.

Although numerical notation systems possess effi ciency-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. Th e analysis of utility must therefore always be linked to the analysis of social context. No system is absolutely “effi - cient” 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. Th e development of Texcocan variants of the Aztec numerical notation system (Chapter 9) was probably motivated by new demands relating to land mensuration and sur- veying in highland Mexico in the early colonial period. Th e changes involved need not be so drastic as to produce a system that employs entirely diff erent principles. Th ey 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 suffi ce, 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. Th e 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). Th is 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. Th us, 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 modifi ed Chinese positional numerals are normally used in scientifi c 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. Th e 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. Th us, even though other systems known to Europe- ans 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. Th us, 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 diffi cult to displace. Th e property of cultural systems that the current popularity of a cultural phenomenon aff ects 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 diffi cult to displace once it had achieved a critical mass of popularity (Shermer 1995: 74–75). Th is inertia is due in part to the diffi culties 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 signifi cant disadvantage. Numerical notation systems are not diffi cult 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.

Th e 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. Th e 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. Th e most notable example is the prevalence of Western numerals throughout the world, accompanying Western Euro- pean 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. Th e eff ect of imperialism can be seen most clearly when the systems of the dominant and subordinate powers are structurally identical, thereby eliminating diff erential effi ciency for specifi c functions as an explanation. Th us, 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. Th is factor in combination with the previous two is an extremely powerful explanatory tool. Th e transformation and replacement of numerical notation systems often depends on social needs relating to administration, bookkeeping, and the exact sciences. Th ese functions also allow large and complex societies to dominate less complex ones. Th us, 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. Th is potentially explains why cumulative and additive systems tend to be replaced over time – their inferiority for some administrative and scientifi c 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 long- distance trade, since the latter circumstance might make it advantageous simply to adopt one’s partner’s system wholesale. For instance, several ancient eastern Medi- terranean 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. Th e 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. Th is was one of the factors behind the development of the Armenian and Georgian numerical notation systems (Chapter 5), under the infl uence of Greek-speaking missionaries during the fourth and fi fth centuries ad. Th e 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.

Th roughout this study, I have downplayed the role of computational effi ciency 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. Th e 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. Th e 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. Th e 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 signifi cantly 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 diff erent 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 subcate- gories), 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 diff erent 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 diff erent 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 diff erent types of object, possibly to avoid confusing the diff erent categories.

7. At the time of the diff usion of a numerical notation system into a region, the principle of the ancestral system may be adopted, but using an indigenous set of numeral- signs.

Th e 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 diff erent cultural identity than that held by those who transmitted the system, possibly in the process obscuring the new system’s origin. Th e 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 diff erent 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. Th is factor must be diff erentiated from the paleographic divergence of systems that once were unifi ed, 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 unifi ed (see factor #14). Social and Historical Analysis 409

8. A descendant system may be structurally distinct from its ancestor because of diff erences in the lexical numerals associated with them.

Systemic transformations sometimes result from eff orts 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 fi t 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. Th e shift from sexagesimal to decimal numerical notation in Meso- potamia corresponds well with the shift in political dominance from Sumerian to Semitic speakers (although sexagesimal elements were retained in Assyro- Babylonian numeration for millennia thereafter). In some cases, the additional signs of a system rather than its major features are aff ected. Th e 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 fl uid situation in which, depending on social position, personality, or other factors, individuals and groups adopt the novelty at diff erential 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. Th e eff ect 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) eff ectively 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 suffi ces 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 effi ciency, 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. Th ese new numerals were a foreign invention and could be seen as undesirable by xeno- phobic administrators. Th ey were also associated with merchants and moneylend- ers, 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 fl ow 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. Th e siyaq numerals and Turkish cryptographic numerals (Chapter 10) appear to have their origins in the desire of certain categories of individuals to control the fl ow of information. Th e Cistercian numerals (Chapter 10) also may have occasionally been used cryptographically, particularly in the latter part of their history. Th e Fez numerals (Chapter 5), originally used quite widely, were eventually used only in contracts in order to conceal values and thus prevent forgery and modifi cation. 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 eff ect, 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.

Th e retention of Roman numerals in the West is the best-known example of such a situation. Th ey are often used today, in contexts such as clock faces, Social and Historical Analysis 411 monumental inscriptions, copyright dates of fi lms, 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. Th ey 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 oversimplifi ed 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. Th e 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 Assyro- Babylonian 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. Th e 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. Th e 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. Th is 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 ciphered- positional 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 refl ect 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.

Th e 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. Th ese 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). Th e development of a cursive script tradition and a media on which distinct cumulative signs could be reduced gradually to ligatured ciphered ones was a signifi cant 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). Th e paleographic diff erences 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 diff erent 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. Th e adoption of Western numerals was somewhat infl uenced by the increasing use that the burgeoning middle classes of Western Europe were making of books and coins in the fi fteenth 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 unifi ed region becomes politically fragmented or because of geographical separation caused by migration. Th is 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. Th e 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. Th e numerals’ spread into the Arab world and eventually to Western Europe continued the process of paleographic divergence, so that today it is diffi cult to see any resemblance among the numeral-signs of Europe, the Middle East, and South Asia. Th is 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 Mid- dle 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. Th ese 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.

Th is 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, akshar- apallî) 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.

Th is rare circumstance has nevertheless in two cases resulted in important structural changes. Th e 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). Th e complexity of the head-variant glyphs and the consequent diffi culty in inscribing them on monuments refutes any simple functional explanation for their development. Th e development of positionality in India and the shift from ciphered- additive notation to ciphered-positional numerals with a zero (Chapter 6) similarly related to metaphysical and literary rather than practical concerns. Positional- ity 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 prefi guring 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 eff ects of various social circumstances. None of these factors refutes the fi ndings 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. Th e 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 diff erent 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). Th us, 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). Th e 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. Th e 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 eff ect of such vestigial survivals is small, and under any calculation the Egyptian hieroglyphic and hieratic numerals have the longest period of use.1 Th ere is no correlation between a system’s principle and its longevity; all fi ve combinations of principle are found multiple times among long-lived systems. Th e fact that thirteen of the twenty- eight 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. Th e 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 off er 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 specifi c 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 Shang/Zhou Mu-Ad −1300 −250 1050 Siyaq Ci-Ad 900 1925 1025 may seem more effi cient 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 insuffi - cient 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 ineffi - cient, 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 fi nger arithmetic and the abacus could be used. If the Roman numeral-phrases were insuffi ciently concise, subtractive phrases could be used. Th e 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 suffi ciently 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 diffi cult 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 diffi cult 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 eff ective for communication until some critical mass is reached. Th is is unlikely to occur unless there is a signifi cant 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. Th is 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. Th e 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 numeral- signs 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). Th ese changes, while seemingly inconsequential, sometimes can have great eff ects (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. Th ese 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. Th ese 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. More- over, some phylogenies are far more unifi ed than others. While the Italic systems share many structural features, others (such as the East Asian and Mesoamerican

2 Th is 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 diffi cult to learn. Social and Historical Analysis 419 systems) can be identifi ed as being descended from a common ancestor only through historical context. Even so, the fact that such traditions can be identifi ed at all highlights the remarkable stability of numerical notation systems.

Civilization and Systemic Invention Th ere is no reason to postulate a qualitative gulf between cases of independent invention and “secondary” developments that have an ancestor. Th e 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 Stew- ard (1955: 182) maintained, every borrowing must be construed as an independent recurrence of cause and eff ect. 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 diff erent from that by which systems are modeled on a specifi c ancestor. Seven numerical notation systems were almost certainly invented independently of any specifi c infl uence from other systems: the Egyptian hieroglyphic (Chapter 2), Mesopotamian proto-cuneiform (Chapter 7), Shang Chinese (Chap- ter 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 rod- numerals (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. Th e development of independently invented numerical notation systems coin- cides 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 Th e 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. Th ere is likewise no evidence of the employment of numerical notation systems in many other African or early New World civilizations. Th ere is thus a strong but imperfect correlation between the origin of numerical notation and the emergence of complex civilizations. Th is 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. Th is could also help to account for the development of numerical notation systems in colonial situations. A diffi culty 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 fi rst used for administration and record keeping. In Shang China, however, the fi rst 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 specifi c 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 specifi c 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. Th e earliest Mesoamerican inscription with a sound provenience and clear dating (San Jose Mogoté, Monument 3) contains only the day-name

3 Th e 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 sacrifi ces to be made). Finally, the Mesopotamian proto-cuneiform tablets are simply numerical systems combined with pictorial signs for commodities. Th us, 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 diff erent functions in diff erent civilizations, and since not all civilizations developed either scripts or numerical notation systems. Th e pursuit of answers to this question thus requires the accumulation of new data by scholars of individual civilizations.

The Macrohistory of Numerals Th e phylogenetic study of structural transformations of systems and the non- phylogenetic 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 aff ected 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 ciphered- positional 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. Th ere 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 fi gures I derive the total number of systems in use in every century from 3000 bc to 2000 ad. Th is 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. Th ere 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 fl uctuations. When these fi gures 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 interrelation- ships among cultural contact, imperialism, and the invention and diff usion of diff erent sorts of numerical notation systems. 3000–800 bc: Th is period of slow growth and relative stability saw the invention of the earliest systems of the Old World civilizations, fi rst in Egypt and Meso- potamia, 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 oppor- tunities for cross-cultural contact were not as great as they would later become. Th e 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. Th e 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 eff ect 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 eff ect 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. Th e most signifi cant 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: Th is 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 defi nition 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. Th e 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. Th is period also saw the development of the fi rst 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: Th is 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 ciphered- additive, were invented in the Alphabetic and South Asian families, derived from the Greek alphabetic and Brāhmī systems. Th e fi rst ciphered-positional system appeared in India around ad 600. In this period, ciphered systems fi rst 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 eff ect 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: Th is 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. Th is fi nding 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: Th is was the only period in history when there was a pro- longed decline in the number of systems in use. Th is 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 fi ve 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 fi rst 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. Th e 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. Th e 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. Th e 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. Th e 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 hierarchi- cal networks of interregional socioeconomic interaction) before 1500, their ability to overwhelm older knowledge systems was not nearly as great (Abu-Lughod 1989). Th e capitalist world-system was and is an agent of an entirely diff erent 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. Th e 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. Th e 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. Th e invention of the printing press in the middle of the fi fteenth century encouraged a signifi cant 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 fi rst 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. Th e 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. Th is 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. Th e rise of the capitalist world-system was the most signifi cant 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 insignifi cant. 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? Th is 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. Th e 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 eff ort 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 diff usion). 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, scientifi c (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 fi ve 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 signifi cant advantage over ciphered-positional numeration for a set of specifi c functions, however, it will not Social and Historical Analysis 429 be widely adopted. Th e Western numerals are so prevalent as a representational system that they would have to be practically useless for such specifi c 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. Th ere is no such class of individuals today, since one can fi nd 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 fi rst 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, Th e Railroad; other numbers were Luis Melián Lafi nur, Olimar, sulphur, the reins, the whale, the gas, the caldron, Napoleon, Agustín de Vedia. In place of fi ve 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, fi ve ones, an analysis which is not found in the ‘numbers’ Th e 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-defi ned 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 eff ectively. Over the past 5,500 years, more than 100 diff erent 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. Th ere 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 fi t into the typology I have constructed, because no attested system is so aberrant that it cannot be described within it. A multidimensional typology better refl ects 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 classifi cation. 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. Th ese patterns thus refute radically relativistic notions concerning the way in which concepts are determined by culture. Recognizing the signifi cant diff erences in how societies think about numbers (especially number symbolism), the core of comparable features common to all numerical notation systems demonstrates that these diff erences are surmountable. 432 Numerical Notation

Th e 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. Th erefore, 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. Th ese 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 diff usionary relationships among systems. Th e 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 suffi ce. 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 identifi ed, and various features of systems can be evaluated as to how well they refl ect them. eTh existence of diachronic regularities and the commonalities among independent events of systemic transformation and replacement refute, or at least redefi ne, the commonly held anthropological dichotomy between independent invention (analogy) and diff usion (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 specifi c 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 specifi c contexts of use. Even so, considerations other than purely structural or cognitive ones are often very important. Th e evaluation of a system also requires that we take into account the medium on which it is used, Conclusion 433 the linguistic affi liation 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 fi rst and foremost representational systems used to communicate. Th ey 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 diff ers signifi cantly 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 diff usion, 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. Th e fact that number has two very diff erent 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, fi n- ger 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 effi - ciency as the prime mover behind the evolution of numerical notation is fatally fl awed. Th e 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 fi gures 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 diffi culty, 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 modifi ed signifi cantly except for changes in the forms of numeral- signs. 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. Th is is not simply a coincidence, but is the outcome of broad social changes related to the rise of capitalism in West- ern Europe and throughout the modern world-system: mercantilism, printing, double-entry bookkeeping, dated coinage, and vernacular religious literature. Th e 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 diff usion 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 eff ectiveness. Th e 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. Th e 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 eff ects of social factors on individual choices. As previously lost systems come to light – and as new systems are invented – the fi ndings of this study will no doubt be revised and refi ned. Yet, because human minds work with numbers in similar ways – notwithstanding the memorious Funes – and because similar social factors aff ect decisions, there is no reason to think that the future of numeration will be greatly diff erent 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’]: Th e use of phonetic signs that correspond to the fi rst 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: Th e 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. Th e Arabic and Hebrew scripts are consonantaries. Also known as abjads. Counting board: Any of various fl at 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: Th e 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: Th e principle governing how the values of diff erent powers in a numerical notation system are arranged and combined to derive the total value of numeral- phrases. Intraexponential: Th e 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 specifi c language. Logogram: A grapheme that represents a specifi c word in a language but is essentially nonphonetic. Mathematics: Th e science that deals with the logic of quantity, shape, and arrangement. Metrology: Th e 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 specifi c 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: Th e 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., fi rst, second, third …). Cf. cardinal number. Ostracon (pl. ostraca): A potsherd (pottery fragment) containing text, usually written with ink. Paleography: Th e study of pre-modern handwriting, usually on perishable materials, including the study of changes in the forms of graphemes. Cf. epigraphy. Phylogeny: Th e 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 aff ected by their position or place within the numeral-phrase. Also known as place-value. Power: Th e 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’]: Th e enumeration of lines of prose or poetry in texts; the measurement of the length of texts by lines of fi xed or average length. Stimulus diff usion: Th e transmission and adoption of an innovation where the basic principle of the innovation diff uses, 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: Th e 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: Th e practice of making a series of relatively permanent marks to indicate an ongoing total of some objects. Cf. one-to-one correspondence. Translinguistic: Of graphic symbols, not linked inextricably to a particular language; able to be read in multiple languages. Uncial [Latin unciales ‘inch-long’]: Writing in Greek, Roman, or sometimes other writing systems in late antiquity, characterized by large, rounded majuscule letters. Unit fraction: A fraction of the form 1/N, whose numerator is 1. Vigesimal: Having a numerical base or sub-base of 20.

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Index

Numerical notation systems are indexed alphabetically by name of system (e.g., “Aramaic numerals”). Number words are indexed under “lexical numerals, Language-name” (e.g., “lexical numerals, English”). Writing systems are indexed under “writing system, Script-name” (e.g., “writing system, Greek”). abacists 123, 147, 220 Aegean numerical notation systems abacus 97, 104, 115–6, 123–4, 144, 147, 182, 218, See Linear A / Linear B numerals 220, 259, 264, 266, 269, 315 Afro-Asiatic languages 319 Gerbert’s abacus 220 Agabo 152 ghubar 218 Agora (Athens) 100, 142 Greco-Roman 266 Agrippa of Nettesheim 354 Inka 315 Ahar stone inscriptions 197 pebble-board 97, 104, 144 Akinidad stela 52, 53 and Roman numerals 115–16, 124 Akkadians 71, 243–4, 248 schety 182 Akko 76 soroban 264 aksharapallî numerals 205, 210–13 suan pan 259, 264, 269 Aksumites 153 Abaj Tabalik 288 Albelda (Logrono), Spain 219 Abbasid caliphate 219, 403 al-Biruni 170 Abraham ibn Ezra 159, 184, 216, 237 Alexander the Great 143, 388 abstract numbers 18, 237 Alexandria 153, 192 abugida See alphasyllabary Alexandrine conquest 252, 258, 413 Abusir papyri 43, 47 Alexandrine period 104, 143, 192 Achaeans 64 Algorismus vulgaris 221 Achaemenid Empire 73, 76, 86, 83, 92, 142, 249, algorithmists 123, 147, 221 256–8, 413 al-Jāhiz 215 acrophonic principle 102–3, 107–8, 128, 435 al-Khwārizmī, Muhammad ibn Mūsā 128, 166, Adab al-kuttāb 215 214, 217 Adair, James 348 al-Mansur (Abbasid caliph) 214 additive principle 12, 371–2, 384, 387, 389, 392, al-Moktadir Billâh (Abbasid caliph) 350 395–6, 405, 425, 432, 435 alphabet 75, 92, 157, 187, 435 See also writing defi nition 12, 435 system extendability 371–2, 395–6 alphabetic systems 35, 67, 132, 133–87, 222, 387, Adelard of Bath 120, 223 395, 413, 424–5, 428

471 472 Index alphasyllabary 133, 152, 189, 256, 435 with astronomical fractions 170 alphasyllabic numeration See Indian with Egyptian demotic numerals 55 alphasyllabic numerals with Egyptian hieratic numerals 47–9 Alphonsine tables 169 with Etruscan numerals 97 al-Sijzi 214, 218 with Greek acrophonic numerals 104 al-Sūlī 215 with Greek alphabetic numerals 138, 144 Altaic languages 280 with Hebrew alphabetic numerals 159 Altintepe 64 with Levantine systems 92 al-Uqlīdisī 215 with Maghribi numerals 165 al-Ya’qūbī 214 with Meroitic numerals 53 Amenhotep III 41 with Roman numerals 115, 117 Amharic language 152 with Western numerals 220–2 analogy vs. homology 23–4, 432 with zimām numerals 150 Anatolia 63–4, 72 See also abacus; computational technologies; Ancient Letters (Manichaean) 87–8 fi nger-reckoning; tallying Andalusia 219 Arjabhad See Âryabhata Antioch 216, 221 Armenia 145 Anyang (Shang capital) 259 Armenian numerals 173–5, 185–6, 225, 406, 413, Anza stela 152 416, 426 Apollinius 144 decline and replacement 175, 225, 426 Aqaba 78 ars notaria 351 Araba 80 Arte del Cambio 123 Arabia 68, 78 Âryabhata 177, 205–9 Arabic abjad numerals 91, 118, 127, 133, 146, Âryabhata’s numerals 205–9, 379, 381–2, 394, 411 149–51, 162–7, 169, 171–2, 213, 215, 217–18, Âryabhatiya 205 318, 328, 357–8, 403, 411, 414 Asia Minor 64–6, 68 decline and replacement 165–6 Aśoka 83, 188–9, 192 diff usion and transmission 149–51 Assyrians 65, 70–1, 76, 248, 257 fractions 165–6 Assyro-Babylonian common numerals 44, 63, functions 165–7 65, 71–2, 91, 229, 245, 247–51, 254–6, 258, origins 162–5 371, 377, 387, 395, 409, 411, 416, 422 Arabic positional numerals 91, 118, 147–8, 166, astrolabe 150, 183 185–6, 213, 215, 227, 309, 324–8, 384, 403, Astronomia 183 408, 412, 416, 425, 428 astronomical fractions 167–70, 185 infl uence on African systems 325–8 Athens 99, 102, 104, 142–3, 105–6 Arabico-Hispanic numerals 94, 127–9, 426 Augsburg 125 Arad 50, 52, 73 Augustan period 111–12 Aramaeans 70–1, 73, 78, 158 Aurelian (Roman emperor) 78 Aramaic language 71, 73, 83 Austria 223–4 Aramaic numerals 50, 65, 67–8, 70–4, 83, 86, Axial Age 424 91–2, 96, 156, 248–9, 258, 377, 406, Aymara 316 411, 413 Aztec, Aztecs 284, 289, 300 diff usion and transmission 72, 74, 413 Aztec numerals 225, 285, 300–2, 305, 368, 381–2, functions 73–4 384, 390, 402, 405, 419 origins 71–3, 406, 411 Archimedes 144 Babylonia, Babylonians 194, 248, 252, 254, 257, Ardubarius See Âryabhata 289 Argos acrophonic numerals 94, 102, 132 Babylonian positional numerals 11, 167, 229, Arifmetika 182 248–55, 367–8, 377, 379, 381–2, 387–9, Aristarchus 138 402, 416 arithmetic 2, 30–1, 47–9, 53, 55, 74, 92, 97, 104, origins 250–1, 402 115, 117, 138, 144, 150–1, 159, 165–6, 170, use in mathematics 251–2, 388 214–15, 220–2, 251, 268, 295, 315, 324, 337, Bactria 192 346, 431 Bagam numerals 325–6 with Arabic abjad numerals 165 Baghdad 166, 214, 217, 219 with Arabic positional numerals 166, 214–15 Balinese numerals 200 with Aramaic numerals 74 Bambara numerals 309, 317–19, 368, 373, 418–20 Index 473

Bamum numerals 322–4, 325, 370, 381–2, 384–5 European 126–7, 129–31, 353 mf␧mf␧ variant 323, 381–2 Hebrew 159–60 Bangka 195 Indian 195 Bangladesh 198, 212 Indus (Harappan) 331–2 bar-and-dot numerals (Mesoamerican) 225, Inka 313 284–97, 301, 368, 376, 380–22, 401, 416, 419 Kitan 280 decline and replacement 290, 401 Mesoamerican 285–99, 302, 307–8 origins 287–8, 419 Mesopotamian 233–4 quasi-positional 290–7, 381–2 calendar numerals 94, 129–31 Rule of Four 368, 376 Cambodia 212 use in mathematics 295–6 Cameroon 322, 325 base (numerical) 4, 435 Canaan 254 Basingstoke numerals 350–2 Canhujo-daro 332 Bede 146 Cantonese language 280 Belize 284 capacity system submultiples See Horus-eye ben Gerson, Levi 169 fractions Bengali numerals 198 capitalism 126, 225, 426–7, 433–4 Benin civilization 328 Cardano, Girolamo 354 Berber numerals 319–21 cardinal numbers 4, 435–6 Berlin Papyrus 49 Caria 55, 134, 140 Bété numerals 326 Carmen de algorismo 221 bhutasaṃkya 195 Carolingian Renaissance 118 Bianco, Jean 342–3 Carthage 76, 319 bisexagesimal numeration 229 Cascajal block 287 blended systems 222, 411, 435 Cecil, William (Lord Burghley) 124 Bodra, Lako 335 centesimal numeration 435 Bogazkoy 64 Central Asia 68 bolorgir 173 Central Asian systems 199 Bonampak mural 293 Chamberlain, Basil Hall 338–40 Borges, Jorge Luis 430 Changan (China) 275 Borough, Christopher 182 Char Bakr 167 boustrophedon 67, 99, 107, 134, 435 Cherokee numerals 27, 226, 309, Brāhmī numerals 56, 67, 84, 115, 146, 189–97, 343–5, 358, 365, 381–2, 384–5, 204, 208, 212, 226, 267, 335, 373, 381–2, 395, 403–4 406, 408, 412, 416, 419, 424 Chickasaw numerals 348 diff usion and transmission 192–3, 204, 412 chikusaku 267 origins 191, 406 Chilam Balam 290 transition to ciphered-positional 193–7 Childe, Gordon 14 Bronze Age 58 China 197, 202, 262, 273, 275, 281–2, 333, 403, Bruges 354 409, 422, 426 Buddhism 88, 197, 202–3, 268, 273, 275 Chinese classical numerals 177, 191, 202, Bulgaria 180–1 227, 260, 269–78, 281, 283, 336, 340, bullae 234–5 370–2, 378, 380–2, 403, 409, 411, Bungus, Petrus 125 415–6, 428 Bürgermeisterbuch (Frankfurt) 124 diff usion and transmission 273, 277, 281, 336, Burma 212 340, 403 Burmese numerals 200 modern persistence 409, 415–16, 428 Byzantine Empire 118, 147, 165, 172, 181, 185 positional variants 260, 270, 283, 381–2, Byzantium 86 403 variant forms 273–8 Cairo 150 Chinese commercial numerals 260, 269, caldéron 129 277–80, 283, 372, 411 calendars 36, 46, 126–7, 129–31, 159–60, 195, Chinese counting-rod numerals 264–70, 269, 283, 233–4, 270, 273, 280, 285–99, 302, 307–8, 365, 376–7, 379, 389, 411, 416, 419, 425 313, 331–2, 353 diff usion and transmission 267, 269 Chinese 270, 273 origins 266, 419 Egyptian 36, 46 Chinese language 262 474 Index

Chioggia 98 electronic 407 Chisanbop fi nger computation 380 fi nger reckoning 215, 379–80, 417, 433 Ch’olan languages 293 ketsujo 338 Christianity 81, 321, 355–8, 405 kitāb al-takht 215 Chronica maiora 351 LoDagaa cowrie shells 18 Chronicon Paschale 206 sangi 267 chronograms 119, 159, 166–7, 196, 210, 224, 413, tables of squares 242 435 takht 218 Chrysostomus, Johannes 224 See also abacus, Chinese counting rods, chunking 15, 376–9, 435 tallying cifre chioggiotte 98 conciseness 395, 397–9, 402, 432 ciphered principle 11, 48, 133, 142, 168, 368, 378, concrete counting 18, 234, 237 389, 393, 396, 421, 425, 432, 435–6 consonantaries 70, 81, 89, 319, 435 ciphered-additive systems 12–13, 75, 146, 165, conspicuous computation 112 185–7, 374, 382–3, 384, 388, 390–1, 393–5, Constantinople 147, 182 397, 424 constraints 6, 404, 431 ciphered-positional systems 12–13, 193–5, 378, Continuity Principle 362 382–4, 388, 390–1, 393–4, 397, 424, 425, 428 Coptic language 148, 151 Cistercian numerals 350–4, 371–2, 410 Coptic numerals 44, 54, 56, 145, 148–9, 185–6, Codex Cospi 289 225, 369, 405, 411, 416 Codex Fejervary-Mayer 289 Copts 149–50, 153 Codex Kingsborough See Kingsborough Codex Cordoba 219 numerals counting board See abacus Codex Mariano Jimenez 307 cowrie shells 18 Codex Puteanus 119 Cretan hieroglyphic numerals 34, 43, Codex Reginensis 119 58–61 Codex Selden 289 Crete 56, 98 Codex Sierra 302 Croatia 178 Codex Telleriano-Remensis 301 cryptography 39–40, 216, 410 Codex Urbinas Latinus 146 cryptoquantum numeration, Maya 307 Codex Vigilanus 219, 223 cultural evolution 401, 419–20 Codex Vindobonenisis 795 155 cultural identity 187, 408–9 Códice de Santa María Asunción 303 cumulative principle 11, 22, 142, 368, 374, 376–8, Códice Vergara 303 384, 388–9, 391, 393, 394, 405, 425, 432, cognitive psychology 14 435–6 coins See money defi nition 11, 435–6 Cologne 224 cumulative-additive systems 12–13, 144–6, 372–4, colonialism 225, 309, 316, 347, 384, 389, 402, 382–4, 388–97, 422, 424 405, 420, 425–6 cumulative-positional systems 12–13, 382–3, Columna rostrata 112 387–8, 390–7, 394 Commentary on the Sentences of Peter Lombard cursive 44, 47, 90–1, 111, 118, 145, 149, 172–3, 122 349, 412, 418, 436 Commission on Inupiat History, Language and defi nition 436 Culture 347 eff ect on system development 91, 412 comparative ethology 15 Cushing, Frank 347–8 comparativism 6–8 Cypriote syllabic numerals 34, 65–6, 72, 96 composite multiplicands 370–1 Cypriote tallies 329–30 computational technologies 18, 124–6, 215–16, Cyprus 58, 63, 65–6, 329–31 218, 220, 234–5, 242, 264, 266–7, 268–9, Cyrenaic numerals 100–1 338, 346–8, 348, 379–80, 315, 406–7, 412, Cyrene 100–1 417, 433 Cyrillic numerals 146, 180–2, 185–6, 225, 364, apices 220 366, 369, 375, 416, 426 body counting 18 decline and replacement 182, 225, 426 bullae 234–5 divergence from Ordering Principle 364, 375 chikusaku 267 Chisanbop 346, 380 da xie shu mu zi numerals 277, 410 dust-board 215–16, 218, 412 Dadda III 194 Index 475

Damascus 76, 78–9 Egypt 49, 55, 64, 68, 71–3, 79, 149–51, 163, 333, Damerow, Peter 236–40, 243, 408 405–6, 417, 419, 422, 424 damgalu numerals 357–8 Egyptian demotic numerals 34, 40, 49, 54–6, 73, Daodejing (Tao Te Ching) 266 140–1, 192, 405–6, 413, 416 Dark Age (Greece) 63 decline and replacement 56, 405 Datini, Francesco 223 diff usion and transmission 55–6, 140–1 Dawei, Cheng 276 functions 54, 55 De calculatione 223 origins 49, 54 De inventione litterarum 184 use in mathematics 40, 55 De numeris 354 Egyptian hieratic numerals 34, 40–9, 66, 73, 79, De occulta philosophia 354 91, 156, 192, 381–2, 384, 403, 412–13, de Sacrobosco, John 221 415–16, 418 De subtilitate libri XXI 354 diff usion and transmission 49, 413 De temporum ratione 146 fractions 46 de Villa Dei, Alexander 221 functions 41, 43–4, 48 Dead Sea Scrolls 156 multiplicative 42, 46–7, 403 decimal numeration 436 origins 41, 44, 46–8 Decourdemanche, M. J. A. 355–8 relationship with hieroglyphs 39–42, 46–8 Dehaene, Stanislas 17, 29 use in mathematics 40, 46–7, 49 Deir el Medina 47 Egyptian hieroglyphic numerals 34–44, 66, Dengfeng County (China) 266 71–4, 91, 96, 109, 238, 248, 333, 368, 373, denotation 2, 30–1 376, 381–2, 384, 408, 412, 415–19 determinism 6 comparison to hieratic 39–42 Devanagari numerals 198 cryptographic ciphered numerals 39, 40 Devendravarman 197 decline and replacement 44, 47, 417 dewani See Siyaq numerals diff usion and transmission 39, 40, 43–4, 71–4 Dhyāna text 88 fractions 42–3 diachronic analysis 7, 32, 360–1, 380–401, functions 40–2 418, 432 multiplicative 41–2 diacritic 436 origins 35–8, 419 Diela 280 Egyptian language 35, 37, 148 Distance Numbers (Maya) 291 Egyptian Mathematical Leather Roll 49 divination 210 Elagabalus (Heliogabalus) 115 Djaga people 329 Elamite language 258 Djamouri, Redouane 262 Elements 147 Dogon people 328 Elephantine 71, 73 Dravidian languages 201–2, 204, 330 Elizabeth I (English monarch) 124 Dresden Almagest 183 El Portón 288 Dresden Codex 291, 295, 297–9, 307 emporion 141 Duixiang siyan zazi 269 England 124, 130, 133, 223–5, 351 Dunhuang (Gansu province) 270, 273 Englund, Robert 239–40, 408 Durandus of Saint-Pourcain 122 Epidaurus numerals 94, 132 dust-boards 166, 215, 412 epigraphy 436–7 episemons 134 Early Dynastic period (Mesopotamia) 236, 238, Eratosthenes 253 242, 244, 407 erkat’agir 173 East Asian systems 259–83, 359, 366, 418, Ethiopia 133, 148, 152–4, 169 425 Ethiopic numerals 54, 145, 148, 152–4, 186, 282, East Germanic languages 154 371, 416 Easter Island 342 Etruria, Etruscans 93, 96–7 Easter Island numerals 342–3 Etruscan “abacus-gem” cameo 97 Eastern Roman Empire 145–6, 185 See also Etruscan numerals 62, 67, 76, 94–8, 100, 103–4, Byzantine Empire 113–14, 132, 364, 373, 376, 379, 380, 388, Ebla 245, 248 405, 419–20 Eblaite numerals 244–7, 422 decline and replacement 97–8, 388, 405 Edessa 80–1, 160 origins 95–8, 103, 419–20 Eghap people 325 Euclid 147 476 Index

Euphrates River 254 Gansu corridor 273 Evans, Arthur 58 Gaon, Saadia 158 Exchequer 126 Garamond, Claude 224 extendability 31, 371, 387, 395–9, 402, 432 Garni 174–5 Gbili (Kpelle chief) 327 Fairservis, Walter 331–2 Geertz, Cliff ord 9 Faliscan language 96 Ge’ez language 103, 108, 152 Fara (Ṧuruppak) 242 gematria 159 Fara period 238 Genoa 221 Farsi language 86, 91 Georgian numerals 177–8, 185–6, 225, 392, 406, Fasti Danici 131 413, 416, 426 Faye, Assane 327 Gerbert of Aurillac (Pope Sylvester II) 123, 220 Fenghuangshan, Hubei province 266 Germany 124, 127, 130, 133, 223–4, 354 Fez numerals 151, 165, 171–3, 185–6, 378, 410 Ghadames 319–20 Fibonacci (Leonardo of Pisa) 221 ghubar numerals See Maghribi numerals fi gure indice, “Indian fi gures” 216, 219, 221 Gilcrease Museum (Oklahoma) 343 fi gure toletane, “Toledan fi gures” 216, 221 Girsu 244, 250 fi ngers, fi nger-reckoning 116, 215, 346, 379–80, Glagolitic numerals 146, 178–80, 185–6, 225, 364, 417, 433 375, 426 Flanders 184 decline and replacement 180, 426 Florence 123, 222, 410 divergence from Ordering Principle 364, 375 fractions 362–4 Goldenweiser, Alexander 6 Arabic abjad numerals 165–6 Goody, Jack 18 Aramaic numerals 71 Gothic language 154 astronomical (sexagesimal) 167–70, 253, 388 Gothic numerals 145, 154–6, 185–6, 388 Aztec numerals 302 Grahacāranibandhana 209 Babylonian numerals 388 Grantha numerals 199 Berber numerals 319 graphemes 3, 19, 195, 218, 365, 436 Chinese counting-rod numerals 265 Greco-Roman period (Egypt) 44 Cretan hieroglyphic numerals 59 Greece, Greeks 76, 93, 98–9, 102–4, 108, 142–5, Egyptian hieratic numerals 46 223, 351–2, 407–8 Egyptian hieroglyphic numerals 42 Greek acrophonic numerals 3, 62, 94, 96, Fez numerals 171 99–106, 128, 132, 134, 139, 141–2, 144–5, Greek acrophonic numerals 104, 139 246, 392, 407–8 Greek alphabetic numerals 139, 141 archaic variants 94, 100–2 Horus-eye fractions 42–3, 363 arithmetic with 144–5, 407 Linear A numerals 57 decline and replacement 104–5 Phoenician numerals 75 diff usion and transmission 103 proto-cuneiform numerals 231 epichoric variants 100–2 Roman numerals 115–16, 125 fractions 104, 139, 141 Sogdian numerals 87–8 functions 100, 103, 105, 144, 408 Syriac alphabetic numerals 160 origins 103 Texcocan line-and-dot numerals 303 use with abacus 104, 144 unit-fractions 42, 139, 437 Greek alphabetic numerals 54–5, 62, 67, 76–7, Western numerals 224 79, 82, 86, 115, 117–18, 126–7, 134, 138–47, France 121, 223–4, 353–4 151–7, 164, 171–2, 178–9, 181, 185–6, 192, Frankfurt 222 208, 253, 258, 317, 352, 364–6, 369, 372, 375, frequency dependent bias 404, 408 381–2, 388, 397, 405, 411, 413, 416–17, 424 Fula (Adama Ba) numerals 326–7 abacus 144 Fula (Dita) numerals 326 arithmetic with 144–5 Funes, the Memorious 430 decline and replacement 142, 147 diff usion and transmission 142, 145–6, 151, Galton’s problem 5 155–7, 172, 178–9, 181, 424 Ganda 329 episemons 134, 142 Gandhara 83 fractions 139 Gangâ dynasty 189, 196 functions 144–5 Ganges River 198 hasta 138, 365 Index 477

modern persistence 147, 411 Hittite cuneiform numerals 229, 255–6 Ordering Principle in 364, 369, 375, 413 Hittite hieroglyphic numerals 34, 43, 63–4, 71, origins 55, 134, 140–1 96, 255–6, 368, 406, 422, 424 positional variants 253, 381–2 Hittite language 63 Greek language 65 Hittites 63–4 Greenberg, Joseph 7, 361, 375 Hmong See Pahawh Hmong numerals Griffi th, F. Ll 52, 53 Ho (Bihar province) 335 Guaman Poma de Ayala, Don Felipe 315–16 Homiliae 224 Guatemala 284, 289, 296 homology 23–4 Guitel, Geneviève 10–11, 29 Honduras 284 Gujarati numerals 198 Hong Kong 269, 280 Gupta Empire 84, 86, 198, 205, 413–14 Horus-eye fractions 42–3, 363 ḥurūf al-zimām 149 Hadramauti language 107 hybrid systems 12–13, 70, 113, 132, 181, Halicarnassus 140, 142 257, 368, 378, 392, 394, 396, 431, Hammurabi 255 435–6 Han Dynasty 264, 273 Hypsicles 168 Hangzhou numerals See Chinese commercial numerals Iberian peninsula 128 Hao (China) 262 Ibibio-Efi k people 321 Harappan civilization 189, 330, 333 Ibn al-Banna 173 Harappan numerals See Indus numerals Ibn al-Nadim 350 Haridatta 209 Ibn ‘Isa, Ali 350 Harris Papyrus 44, 45, 47 Ibn Khaldun 151, 173, 218 Harris, Marvin 24, 28 Iceland 121, 223 Harvard University 126 iconicity 18 Harvey, Herbert 303, 307 ideograms 63, 88, 235, 436 hasta 138, 148, 365 Ifrah, Georges 29–30 Hastivarman 196 Imazighen See Berber numerals Hatra 80 imperialism 112, 359, 388, 404–5, 425–6 See also Hatran numerals 68, 69, 80–1, 86, 377, 413 colonialism Hattusha 255–6 implicational regularities 367–70, 373 Hau, Kathleen 321–2 Inam gišhur ankia 252 Hebrew alphabetic numerals 133, 145, 149, 156–9, India 84, 166, 193–4, 197–8, 202, 212, 254, 420, 185–6, 225, 409, 411, 413, 416 424, 426 origins 156–7 Indian alphasyllabic numerals 205–13, positional variants 158–9 413 Hebrew hieratic numerals 50–2 Indian numerals 166, 196–200, 253, 266–7, Helcep Sarracenicum 120 381–2, 384, 408, 411–2, 414, 416 Hellenistic period 132, 144 Indo-European languages 63 heng and zong registers 264, 278 Indonesia 216 Henry VIII (English monarch) 224 Indus numerals 191, 309, 330–3, 373, 376–7, 393, heqat 46 418–20 Hermann of Carinthia 183 Indus Valley 166, 191, 198, 330, 333, 363, 419, 422 Herodianic numerals See Greek acrophonic Inka Empire 312 numerals Inka numerals 8, 225, 309–17, 358, 368, 373, 405, Herodotos 144 418–19 Hibeh papyrus 142–3 abacus 315 hieroglyph 34, 436 prohibition of use 316 Hieroglyphic systems 34–67, 132, 376, 387, 395 status as numerical notation 312–15 ḥisāb abjad 162 See also khipu ḥisāb al-djummal 166 interexponential structure 11–12, 364, 368, 375, ḥisāb al-ghubar 216–18 381–4, 388, 397, 411, 431, 436 ḥisāb al-hindī 166, 213–15, 218 defi nition 11–12, 436 ḥisāb al-qalam al-Fāsī 171 Iñupiaq language 345 ḥisāb al-ūqūd 165 Iñupiaq numerals 226, 309, 344–7, 376–7, Histoire comparée des numérations écrites 10 380–2, 384, 409 478 Index

Ionia 55, 140–1 ketsujo 338 Ionian numerals See Greek alphabetic keutuklu numerals 355 numerals Khafaji 230 Iran 68, 74, 86, 161 Kharoṣṭhī numerals 68–9, 83–4, 377 Iranian languages 87 Khasekhem 38 Iraq 86, 161, 254 khipu 21, 310–12, 315, 338 Iron Age 52 See also Inka numerals Islam 88, 92, 146, 166, 215, 218–19, 403 See also khipukamayuq 312, 315 Muslims Khirbet el-Qôm ostracon 157 Israel 52, 159 Khmer numerals 200, 227 Italic systems 35, 66–7, 93–132, 377, 407, 418 Khorsabad 245 Italy 93, 95–6, 102–3, 105, 109, 118, 129, 147, 216, Kingsborough Codex numerals 302, 305–6, 372, 221, 223–4, 403, 419, 420 381–2, 384 Kitāb al-Fihrist 350 Jackson, George 341 Kitāb al-fusūl fi al-ḥisāb al-hindī 215 Jacobites 161 Kitāb al-mu’allimin 215 Jains 212 Kitāb al-takht 215 Janneus, Alexander 157 Kitan language 280 Japan 225, 267, 269, 338, 407, 409 Kitan numerals 260, 275, 277, 280–1, 283 Japanese numerals 340 Kluge, Th eodor 7 Java 212 Kobel, Jacob 124 Javanese numerals 200 Korea 409 Jebel Aruda 235 Kotakapur 195 Jemdet Nasr 230 Kpelle numerals 326–7 Jesuits 276, 426 Krakow 182 Jews, Judaism 50–2, 149–50, 156–9 See also Kroeber, Alfred 27–8 Hebrew alphabetic numerals Kuhn, Th omas 29 jiaguwen 259 Kululu lead strips 64 Jiahu, Henan province 262 Kunitzsch, Paul 218 Jin Dynasty 281–2 Kusumapura (Bihar) 205 Jiushao, Qin 267 Jiuzhang suanshu (Nine Chapters on the La Mojarra stela 287 Mathematical Art) 268 La Venta 287 John of Basingstoke (John of Basing) 350–2 Lachish 50 Johnston, Alan 100–3 Lake Van 174 Judaean weights 51 Lam Lay-Yong 266 Judah 52, 73, 157 Lamasba tablet 115 Jurchin people 267, 275, 281 Landa, Diego de 295 Jurchin numerals 260, 281–3, 277, 381–2, 387, Landnámabók 121 395, 409 Lanfranco 222 Lao numerals 200, 338 Kadesh-barnea 50–2 Late Period (Egypt) 54, 413 Kahun Papyrus 44–5, 47, 49 Latin alphabetic numerals 146, 183–6 Kaiyuan period 197 Latin language 105, 216, 369, 427 Kaiyuan zhanjing 275 Layard, A. H. 71 Kaktovik, Alaska 344 Lebanon 161 Kalacuri era 194 Leigh, Howard 307 Kalibangan 332 Lelang (Pyongyang), North Korea 273 Kamara, Kisimi 323 Leonardo of Pisa (Fibonacci) 123, 221 Kannada numerals 199 Letoon 105 Kantè, Souleymane 327 Levant 52, 68, 70–1, 73–6, 78, 157, 405, Karatepe inscription 75 411 Kartir 86 Levantine systems 35, 44, 52, 65–92, 106, 132, katapayâdi numerals 205, 209–11, 381–2, 411 248, 258, 376–7, 388 Kenadiid, Ismaan Yuusuf 327 diff usion and transmission 68, 70, 91–2 Kensington Rune Stone 131 origins 35, 44, 52, 68, 258 Kerala, India 202 Lex de Gallia Cisalpina 112 Index 479 lexical numerals 3–4, 7–8, 10, 17, 20, 22, 361, Luwian language 63–4, 105 364–7, 369, 374–5, 378–80, 409, 414, 436 Luwian numerals See Hittite hieroglyphic Arabic 349 numerals Aramaic 70–1, 74, 86 Lycian numerals 65, 76, 103, 105–7 Armenian 174 auditory aspect 20, 366, 375 Macedonia 178 Bamum 322 macrohistory 25, 422–9 Cherokee 343 Madrid Codex 307 Chinese 262, 273, 277 Maghreb 172, 217–9, 318 Coahuilteco 367 Maghribi numerals 151, 198, 216–18, 412, 416 defi nition 3, 436 magic number 7±2 14, 378 Eblaite 246 magic squares 175 Egyptian 35–7 Magna Graecia 98 English 22, 378 Magnitskii 182 Etruscan 94, 96 Mahrnâmag 89–90 Georgian 177 Malabar Coast 212 German 10, 375 Malayalam language 212 Gothic 155 Malayalam numerals 201–4, 371, 381–2, 409, 416 Greek 100, 138–9, 366, 375 Malcolm, Captain L. W. G. 325 Hebrew 50 Maldives 204 Iñupiaq 345 Mali 317 Jurchin 409 Mama, Nji 322 Latin 109, 111, 242 Manchu people 280–2 Maya 288, 298–9, 414 Mandarin language 283 Mende 325 Manding language 327 Middle Persian 86 Manding numerals 326–7 Nabataean 79 Mani 89 Nepali 211 Manichaean numerals 68–9, 89–90, 92 proto-Dravidian 333 Mankani copper plate 194–5 quasi-lexical numerical notation 262, 283 Mao Zedong 277 relation to Ordering Principle 365, 369, 374 Marathi numerals 198 Sanskrit 196, 207, 273 Mari 248, 253–4, 387 Slavic languages 180 Mari numerals 229, 254–5, 387 Slavonic 366 Mari Schad Ormizd 89 Sogdian 87 Maronite Christians 161 Sora 367 mathematics 4, 29–30, 402, 417, 436 See also Sumerian 236–7, 242, 244, 248 arithmetic, computational technologies Turkish 88 Mauryan Empire 191–2, 406 Ugaritic 248 Maximus Planudes 147 Li Shou 259 Maya civilization 194, 284–5, 299–300 Liao Dynasty 280 Maya head-variant numerals 297–9, 308, 368, Liber Abaci 221 394, 414 Liber Mamonis 183, 216 Maya numerals See bar-and-dot numerals Linear A numerals 34, 43, 56–8, 321, 406, 408, 422 Mende language 323, 366 Linear B numerals 34, 43, 58, 61–3, 95–6, 255, Mende numerals 226, 323–5, 366, 371, 381–2, 406, 422 384–5 linen lists (Egypt) 37, 39 Mercedarian friars 316 ling sign 276 Meroe 52, 54 literacy 224, 226, 424, 427 Meroitic numerals 52–4, 66 Livro de Virtuosa Bemfeitoria 223 Mesoamerica 284, 401, 419 LoDagaa people 18 Mesoamerican calendar 290–7, 299, 307 logogram 35, 56, 61, 281, 436 Mesoamerican systems 31, 284–308, 359, 377, logosyllabary 322 379, 410, 418 Long Count (Maya) 287, 290, 292, 294 Mesopotamia 68, 72, 73, 86, 163, 169, 228, 237, lost-letter theory 101, 114 241, 243, 247–8, 256, 258, 313, 333, 388, Lower Egypt 49, 55, 73 406–9, 412, 419, 422, 424, 436 Luoyang 263 Mesopotamian systems 228–58 480 Index

Mesrop Mashtots 173–4 Moscow School of Mathematics and Navigation Methen 40 182 Methodius 146, 178–81 Moso See Naxi numerals Metonic cycle 131 Mozarabs 171–2 metrology 239–40, 436 multiplication 18 Mexican dot-numerals 299–300 multiplicative principle 11, 73, 84, 435–6 Mexico 288, 402 defi nition 11, 436 Mexico City 129 multiplicative-additive systems 12–3, 368, 371–2, mf␧mf␧ numerals See Bamum numerals 374, 378, 382–97, 425 Miao people 275 cognitive factors 390–7 Microcosmographia 122 composite multiplicands 371 Middle Formative Period (Mesoamerica) decline and replacement 388–9 287–9 phylogenetic analysis 385–7 Middle Kingdom (Egypt) 42, 49 structure 368, 374 Middle Persian language 86, 89 transformation of 384–7 Middle Persian numerals 68–9, 80, 86–7, 381–2, Munda languages 334–5 384, 412 Munshi numerals 328 cursive reduction 87, 412 Muqaddimah 151, 218 Middle Persian period 90 Muslims 118, 149–50, 172, 318 See also Islam Midrash 159–60 Muziris 192 Milesian numerals See Greek alphabetic Mycenae, Myceneans 63–4, 95 numerals Mycenean numerals See Linear B numerals Miletus 63, 134, 140 Mysticae numerorum signifi cationis 125 Miller, George 14, 378 mina 71 Nabataean numerals 68, 69, 78–80, 92, 107, 158, Minaea, Minaeans 107–8 213, 377, 413 Ming Dynasty 269, 276, 278, 282 Nagari numerals 213, 218, 227 Minoan numerals See Linear A numerals Nahuatl language 300, 302 Minoans 57, 408 Nana Ghat 190 Mixe-Zoquean languages 299, 380 Napier, John 269 Mixtecs 289, 299–300, 302 Narmer mace-head 38, 112, 420 Mochica 311 Nāsik Cave 190 Mohenjo-daro 332 Natural History 110 Mommsen, Th eodor 101, 114 natural numbers 10 money 30–1, 412, 427, 434 Naukratis 55, 141 alphabetic numerals 142–3 Naxi language 333 Arabic abjad numerals 165 Naxi numerals 309, 333–4, 419 Arabico-Hispanic numerals 129 Negev 73 Brahmi numerals 189 Nemea acrophonic numerals 94, 102, 132 Chinese numerals 266–7, 273, 279 Neo-Babylonian empire 76 Cyrenaic numerals 101 Neo-Hittite kingdoms 63, 65–6, 71–2, 105 Etruscan numerals 95–6 Nepal 198, 212 Greek acrophonic numerals 100–1, 104 Nepali numerals 198 Greek alphabetic numerals 147, 158, 164 Nestorian Christians 87, 161 Hebrew alphabetic numerals 157–8 Neugebauer, Otto 253 Nabataean numerals 79 New Kingdom (Egypt) 41 Passamaquody numerals 348 New Mathematics 238 Phoenician numerals 76 Newcastle (England) 225 Roman numerals 114, 116, 118 Nichols, F. H. 334 Ryukyu numerals 338–9 Nickerson, Raymond 19 Shang and Zhou numerals 263 Nigeria 322, 328 Western numerals 224 Nimrud 71 Mongols 267 Ninni, A. P. 98 Monte Albán 288 Njoya, Ibrahim 322–3 Moravians 178 noncumulative systems 378–9, 384–5, 388–9, 391 Morocco 133, 165, 171–3, 216 nonuniversal regularities 370–2 Moscow Papyrus 49 Norman, Donald 32 Index 481

Normandy 354 Oaxaca 288 North Africa 217 Oberi Okaime numerals 226, 321–2, 391, 394, North America 347, 426 409 North Indian numerals 198 Ocreatus 120 Noviomagus, Johannes 354 Ogowe River, Gabon 329 Novum Testamentum in Linguam Amharicam 154 Old Akkadian period 245, 251 Nueva corónica y buen gobierno 315–1 6 Old Babylonian period 245, 248, 250–3 number 4, 20, 436 Old Chinese language 262 numeral classifi ers 293 Old Church Slavonic language 178, 180 numeral phrases 4, 8, 11, 22, 66, 111, 133, 363–7, Old Kingdom (Egypt) 37, 39, 42, 46 370–2, 374–6, 390–2, 396, 436 Old Persian numerals 86, 229, 248–9, 256–8 conciseness 371–2, 390–1 Old Syriac language 81 defi nition 4, 436 Old Syriac numerals 68, 69, 78, 81–3, 92, 146, interexponential ordering 364 377 multiplicative 363–4, 370 Olmec 287–8, 299 structure 11, 22, 363, 367, 375–6, 390, 364 Olynthus numerals 94, 101 subtractive 363, 392 one-to-one correspondence 3, 15, 23, 115, 299, numeral words See lexical numerals 328–9, 397, 436–7 numeral-signs 2–3, 11, 19, 39, 393, 396–7, 418, open vs. closed notation 20 436 oracle bone inscriptions 259 defi nition 2–3, 436 Ordering Principle 364–6, 369, 374–5 numerical notation 2–3, 5, 7, 9–33, 38, 236–8, ordinal numbers 2, 4, 435–6 245, 309, 316, 359–434, 436 ordouï cheïlu numerals 356 biological prerequisites 16–17 ordoui numerals 356–7 and capitalism 426–7, 433–4 Orissa 196 cognitive analysis 14–16, 236–8, 360–401, Oriya numerals 198 432 Oscan language 96 and colonialism/imperialism 309, 316, 384, Osmaniya numerals 326–7 389, 402, 404–6, 409, 420, 425–7 ostraca 43, 49, 73, 437 computation with 2, 29–33, 402–3, 406–7, Otlazpan 307 433 Ottoman cryptographic numerals 355–8, 410 decline and replacement 387, 405, 417, 432 Ottoman Empire 151, 216, 350, 358 defi nition 3, 436 Oztoticpac Lands Map 303 diachronic analysis 32, 360, 380–9, 399–400, 421, 431 Padua 123, 222 diff usion and transmission 23–8, 402, 405, pagination 124, 127, 147–8, 150, 161, 167, 178, 408, 414, 432–3 182, 210, 270, 427 functions 2, 29–33, 38, 245, 402, 404, 407, Pahawh Hmong numerals 277, 309, 336–8, 358, 411, 413–14, 433 379, 381–2, 410 longevity 415–19 Pahlavi numerals 68, 69, 90–2, 350, 381–2, 384, medium of writing 412, 432 412 origins 18, 23–8, 359, 419–21, 432–3 Pakistan 74, 83, 198 perceived effi ciency 29–33 Palembang 195 phylogenetic analysis 7, 26–8, 408, 411–12, 433 paleography 436–7 relationship to lexical numerals 19–22, 361, Palermo Stone 39 366, 379, 409, 433 Palestine 164 sociopolitical factors 17, 24, 359, 366, 398, Palmyra 78, 158 401–30, 432 Palmyrene numerals 68–9, 76–8, 92, 366–7, structure 3, 360–401, 431 377, 413 synchronic analysis 32, 360, 362–7, 399–400, Palsgrave, John 123 431 Panajachel 296 transformation of 384, 405, 414, 432 Pandulf of Capua 223 typology 9–14 Panini 205 universals 5, 360–1 papyrus 43 and writing systems 19–23, 372, 413, 417–8, Paris 354 420–1, 433 Paris Codex 307 numerology 151, 159, 210, 414 Paris, Matthew 351 482 Index

Parnavaz (Georgian monarch) 177 GAN2 system 233, 240 Parthian Empire 80, 86, 192 origins 234–6 Passamaquoddy numerals 348 Rule of Four 376 Payne, John Howard 343–4 Rule of Ten 379 peasant numerals See calendar numerals ŠE systems 233–4, 240 period-glyphs (Maya) 291–7 U4 system 233–4 Persepolis Fortifi cation Archive 256 proto-Elamite numerals 57, 229, 238–41 Persia 86, 166, 249, 257, 350 Ptolemaic era 39, 41, 48, 54–6, 104, 141–2, 145, Peter the Great 182 157, 192, 405 Petra 78–9 Ptolemy (mathematician) 168 Philae 44, 56 Ptolemy II Soter (Egyptian pharaoh) 142 Phoenician numerals 43–4, 65–6, 68–9, 71, Ptolemy Philadelphos (Egyptian pharaoh) 39 74–6, 81, 91, 96, 102–3, 140, 248, 319, 377, Punjabi numerals 198 392, 406, 411 Puruchuco 314 diff usion and transmission 65–6, 76, 103 Pylos 62–3 multiplicative 74–5, 96, 103 origins 43–4, 74–6, 406, 411 Qa’ba inscription 86 phoneticism 20, 133 qalam hindī 358 phonograms 35, 280 Qatabanian language 107 phrase ordering 374–6, 431 Qin Dynasty 264, 270, 273, 277 Phrygians 64 qoppa 134 phylogeny 25, 437 Quechua language 8, 312, 315–16 Piaget, Jean 237 quinary numeration 380, 437 Picardy 354 Qutan Xida 197, 275 pictography 58, 114, 437 QWERTY principle 404 Pisa 216 Pithom 39 Rapa Nui See Easter Island place value See positional principle Rechenbiechlin 124 Pliny the Elder 110 Reconquista 127 Pompeii 112 Regulae de numerorum abaci rationibus 220 Ponce de Leon papers 129 Reisner Papyrus 49 Portugal 124, 223 Relación de las cosas de Yucatan 295 positional principle 10, 11–12, 122, 124, 167, relativism 5 193–7, 253–5, 266–7, 365, 384–5, 387–9, 421, replacement 380–1, 387–9, 401, 402, 405, 407, 432, 434–7 413, 421 increase in frequency 122, 124, 226, 385, 389, Restivo, Sal 9 425 Rhind Mathematical Papyrus 49 modern additive descendants 384 rod-numerals See Chinese counting-rod nonlinear orientation 306, 351–2, 371–2 numerals quasi-positionality 39, 77, 108, 290–7, 372, rokoum See Siyaq numerals 387 Rolewinck, Werner 224 Postclassic period (Maya) 289 Roman Empire 54, 56, 93, 98, 118, 132, 143, 154, post-positional systems 428 388, 402–3, 405, 424 power (mathematical) 4, 437 Roman numerals 10, 31, 56, 67, 78, 80, 82, 93–6, Predynastic era, Egyptian 42 101–2, 104, 109–32, 146, 184, 222–5, 241–2, Presargonic period 243 283, 290, 302, 306, 320, 341, 347, 364, 368, principle of limited possibilities 6 374, 377, 381–2, 385, 388, 392, 394, 397, printing press 124, 224, 354, 427 402–3, 405–7, 410–12, 415–17, 424, 426–7, Problemata 379 429 proto-cuneiform numerals 230–8, 240, 365, 367, and abaci 115–16, 124 373, 376, 379, 419, 422 Arabico-Hispanic numerals 127–9 bisexagesimal systems 232 arithmetic with 31, 115–18 cognitive correlates 236–8 calendar numerals 129–31 computer-aided decipherment 230–1 cursive variants 111, 118 concrete counting 234 decline and replacement 116, 120, 122–7, double documents 234 222–5, 412, 426–7, 429 EN system 233 diff usion and transmission 115, 132, 403, 405 Index 483

fractions 115–16, 124 Secret of Secrets 182 longevity 415–6 Sefer ha-Mispar 159, 184 lost-letter theory 101, 114 Segovia 354 medieval 118–19, 120–21 Seleucid Empire 81, 86, 92, 104, 145, 157, 252–3, modern persistence 126–7, 410–1 258, 367, 413 multiplicative 94, 111–13, 116, 121, 132, 385 Semitic languages 73, 245, 248, 409 Ordering Principle in 364, 374 Sequoyah 27, 343–4 origins 113–15 Serbia 178, 181 positional variants 120, 122, 381–2 Sermo in festo praesentationis 224 prestige functions 126–7, 410–1, 428 sexagesimal systems 167–8, 185, 229–30, 232, Rule of Four 368 379, 437 sub-base 110–11, 241–2 Shalmaneser V 71 subtractive principle in 109, 111, 377, 392 Shang and Zhou numerals 31, 259, 260–2, 264, variant forms 109, 116, 128–9, 320 270, 283, 365, 372, 415–16, 419 Roman Republic 94, 96–7 quasi-lexical 262, 365 Rome 86, 93, 96–7, 224, 319, 405, 417 Shang Dynasty 259, 262 Rule of Four 368, 376, 380, 407 shang fang da zhuan numerals 276, 411 Rule of Ten 363, 379–80 Shapur I (Middle Persian monarch) 80 rūmī 171 Shirakatsi, Anania 174–7, 402 runic numerals See calendar numerals Shirakatsi’s numerals 174–7, 185–6, 317, 381–2, Rus 182 402 Russia 133, 179, 182, 223 sho-chu-ma numerals See Ryukyu numerals Russian language 180 Shu shu ji yi 272 Ryukyu numerals 269, 277, 338–40, 364, Shu shu jiu zhang (Mathematical Treatise in 384, 407 Nine Sections) 267 Shuxue wenda 277 Saba 107 Siberia 340 Sabaean language 107–8 Sicily 62, 95, 102–3, 221, 224 Sacred Round 307 Siddhantam grant 197 sade 134 Sidon 74–6 Safavid Dynasty 350 Sierra Leone 323, 325 Saint Cyril 146, 178–81 sign-count 31, 392, 394–5, 397–9, 402, 432 Saint Frumentius 153 Sinhalese language 204 Saka calendar 195 Sinhalese numerals 204, 282, 392, 409, 416 Salamis tablet 104 siyaq numerals 348–50, 410, 416, 419 Salisbury Cathedral 222 size-value 365 Samaria ostraca 50, 79 Sogdian language 87 Samos 140 Sogdian numerals 68, 69, 87–8 Samoyed numerals 340–1 Somali language 327 san (sampi) 134 Song Dynasty 267–8, 276, 377 San Andrés cylinder seal 287–8 South Arabian numerals 94, 103, 107–9, 132, 154, San José Mogote 288 320, 368, 385 sangi 267 divergence from Rule of Four 368 Sankheda copper plate 194–5 quasi-positional 108 Sanskrit language 195, 375 South Asian systems 67, 92, 132–3, 188–227, 335, Saqqara 71, 73, 103 387, 395, 413, 424, 428 Šargal Šunutaga 244 South Semitic languages 78, 107 Sargon II (Assyrian king) 245, 248 Southeast Asian numerals 199–200 Sasanian Empire 86, 160 Spain 76, 116, 118, 127–9, 147, 151, 171–3, 216–23 Saussure, Ferdinand de 8 Spanish New World conquests 284, 289, 299– Scandinavia 129–31, 223 300, 316, 401, 425 schety 182 Sphujidhava 195 Schitanie udobnoe 182 Sri Lanka 193, 200, 204 Schmandt-Besserat, Denise 234–5, 421 Sriwijaya 195 scripts See writing systems statistical regularities 361, 370 Sebokht, Severus 213–14 Stein, Aurel 84 Second Intermediate Period 49 stela 437 484 Index

Stephen of Pisa 183 Talmud 159 Steward, Julian 419 tamgas 179 stichometry 105, 148, 178, 437 Tamil numerals 200–3, 227, 370, 381–2, 409, 416 stimulus diff usion 27, 437 composite multiplicands 371 Suan fa tong zong 276 connection to Malayalam 203, 381–2 sub-base 4, 437 origins 200–1 Subhandu 196 Tang Dynasty 270 subitizing 14–15, 376–9, 435, 437 Tangut people 275 sub-Saharan decimal positional numerals Tannery, Paul 145 325–8 Tarikh 214 subtractive principle 106, 111–12, 119, 128, 242, Tax, Sol 296 244, 250, 391–2, 418 taxograms 61 successive approximation 375 Tell Qasile 71 Sumerian language 241, 409 Tell Uqair 230 Sumerian numerals 229, 232, 238, 240–50, 333, Telugu numerals 199 365, 367, 377, 379, 381–2, 394, 407–8, Tenochtitlan 300 411–12, 416, 422 Teotihuacan 289, 299, 307 cuneiform vs. curviform 243, 408 Teotihuacani numerals 307 use in mathematics 245, 407 Tepe Yahya 239 śûnya-bindu 196, 208, 414 Tepetlaoztoc 303, 305 Sunzi suan jing 266 Texcocan line-and-dot numerals 285, 303–6, survival of the mediocre 417 371–2, 402 Susa (Iran) 234–5, 238–41 fractions 303 Susa III period 239 non-infi nite 371 Suzhou (China) 280 nonlinear 372 Switzerland 127, 224 origins 306, 402 syllabary 64, 437 Th ai numerals 200 syllabograms 56, 64, 281 Th ailand 212, 225 synchronic regularities 7, 32, 360–80, 399–401, Th ames River 225 432 Th eon of Alexandria 168 axioms 362 Th espiae 100 cognitive analysis 373–80 Tibet 212 Syntaxis 168 Tibetan numerals 227, 405 Syria 70, 76, 78, 82, 163–4, 254 Tikal 288 Syriac alphabetic numerals 82, 87, 146, 160–1, Tocharian language 193, 199 164, 185–6, 225, 387, 411, 416 Tod, Marcus Niebuhr 99–101 arithmetic with 161 Toledo (Spain) 171, 219 blended system 411 Toltecs 300 expression of large numbers 160 Tomb U-j (Abydos) 37, 420 fractions 160 Tongwen suanzi qianban 273 longevity 416 transformation 380–7, 401–2, 405, 407, 409, multiplicative 160, 387 411, 421 origins 160–1 constraints 383–4 defi nition 380–1 tables of squares 242 transformational grammar 117 Tajikistan 87 translinguistic notation 22, 418, 433, 437 tallying 15, 21, 58, 62, 95, 97–8, 102–3, 130, 132, Trigger, Bruce 6 310–11, 316, 329–32, 347–8, 419, 436–7 Tripoli 221 and Roman numerals 115 Tsenhor papyrus 55 connection to cumulative-additive systems T’uabant’iwn 176 373 Tuareg people 319 defi nition 437 Tunis 221 Rule of Four 407 Tunisia 218 Samoyed numerals 341 Turay, Mohamed 323 sho-chu-ma (Ryukyu) 277, 381, 407 Turkey 161 while inebriated 15 Turkish language 88 Zuni 347 Tuscany 93 Index 485

Tuxtla statuette 287 and Christianity 219–20 Tyre 74–6 diff usion and transmission 22, 219, 221–3, Tzeltalan languages 293 224–7, 321, 409, 426–7 near-universality of 2, 219, 226, 428–9 Ugaritic numerals 248, 424 origins 146–48, 219, 222 Umayyad caliphate 87 positive characterizations of 29, 31, 346 Umbrian language 96 prohibition of use 123–4, 222 Umma 250 Whalley Abbey (Cheshire) 352 uncial 148, 437 Wiener, Charles 316 unilinear evolution 2, 29, 421 Williams, Barbara 303, 307 unit-fractions 42, 437 Winkelhaken 243, 247 universal grammar 361 Wolof numerals 326–8 universalism 5, 361 Worm, Ole 131 universals 6, 360 writing, scripts 19–22, 413, 417–18 Upper Paleolithic 16, 23 writing system Ur III period 244, 250–1, 367, 402 Arabic 89, 133, 149, 198, 435–6 Urartian language 64 Aramaic 71, 73, 76, 80, 84 Urartians 64 Armenian 173 Urton, Gary 8, 312–13 asomtavruli 177 Uruk 230, 235, 240 Bamum 322–3 Uruk period 37, 230, 235–6, 407 Berber 319 Uto-Aztecan languages 300 Bhattiprolu 200 Uygurs 88 Book Pahlavi 87, 90, 91 Uzbekistan 87 Brāhmī 84, 188, 200 Canaanite 74 Vâkâtaka grants 190 Cham 195, 200 Valera, Blas 316 Chinese 270, 417 Valley of Mexico 302 chu’ nom (Vietnam) 275 Varang Kshiti numerals 309, 334–335, 381–2, cismaanya 327 384, 410 Coptic 44, 148–9 varnasankhya systems 205, 212 Cretan hieroglyphic 57–9 See also Indian alphasyllabic numerals cuneiform 73, 167, 228, 248–9, 436 Vasavadatta 196 Cypriote syllabary 65 Vatican Codex 302 Cypro-Minoan 65 Venice 221 dongba (Naxi) 333–4 vernacular languages 427 Egyptian demotic 35, 54, 148 Vietnam 273 Egyptian hieratic 35, 49, 55 vigesimal numeration 379, 437 Egyptian hieroglyphic 35, 44 Vināyakapalā 197 Elamite 73 vinculum 402 Ethiopic 133, 152 Vygotsky, Lev 237 Etruscan 94–5, 109 far soomaali 327 Walid I (Umayyad caliph) 164 geba (Naxi) 333 Wallerstein, Immanuel 426 Georgian (asomtavruli) 177 Wari civilization 311 Georgian (mxedruli) 177 Warring States Period 264, 266, 273 Glagolitic 178 Wells Cathedral 223 Gothic 154, 388 Wen Di 266 Grantha 199, 202, 204 West African systems 328, 404, 408 Greek 44, 63, 96, 99, 101, 105, 134, 148, 437 Western numerals 2, 10, 22, 29–32, 67, 118, Greek (Chalcidic) 114 123–4, 146–8, 159, 182, 185, 188, 193, 198, Greek (Euboean) 94 213, 216–27, 271, 277, 283, 290, 302, 306, Greek (Ionic) 140–1 309, 321–3, 325–8, 335, 338, 344, 346, 365, Greek epichoric scripts 99, 101, 102, 105 381–2, 384, 397, 401, 403, 408–12, 418, Gupta 199 426–9 hangul (Korean) 275 arithmetical use 29–30, 222 Hasmonean 157 in Bibles 224, 427 Hatran 80 486 Index writing system (cont.) Vai 323, 327 Hebrew 50, 156, 435–36 Varang Kshiti 334–5 Hittite cuneiform 63, 248, 255 Visigothic 118 Indus (Harappan) 330–1, 333 Wolof 327 Isthmian 287, 291 Zapotec 287, 291 Jurchin 281–2 Wulfi la 154, 388 kanji (Japanese) 275 Kannada 199 Xanthus 105 Kawi 200 Xcalumkin 294 Kharoṣṭhī 83, 84, 188, 193 Xiyin, Wanyan 282 Kikakui (Mende) 323–4 Xizong (Chinese emperor) 282 Kitan 280–1 Kpelle 327 yakâ-ne talápha 348 Latin 22, 95, 109, 115, 183, 327, 417, 437 Yang, Shong Lue 336 Linear A 56–7 Yavanajātaka 195 Linear B 64 Yellow Emperor 260 Luwian 63 Yezdigird III (Persian monarch) 87 Lycian 105 Yoruba civilization 328, 420 Malayalam 199–200, 202 Yucatan 284, 289 malimasa (Naxi) 333 Yucatecan language 293 Manichaean 87, 89 Yue, Xu 272 Maya 285, 287 Yunnan province (China) 333 Meroitic cursive 49, 52 Meroitic hieroglyphic 35, 52 Zacuto, Abraham 354 Mesoamerican 255, 289 zā’irajah technique of divination 151 Middle Persian 86 Zapotec civilization 288, 299 Minaeo-Sabaean 152 Zapotec language 380 mxedruli 177 Zapotec numerals 288 Nabataean 79 Zenobia 78 Nestorian 160 zero 22, 362, 365, 371–2, 390–2, 414 N’ko 327 alphabetic numerals 147, 159, 169, 186–7 North Semitic 108 Arabic numerals 169, 213–14 Oberi Okaime 321 Âryabhata’s numerals 208–9 Ol Cemet’ 334–5 astronomical fractions 169 Old Khmer 195 Babylonian numerals 250–3 Old Malay 195 Bété numerals 326 Old Persian 73, 256 Chinese classical numerals 275–7 Old Syriac 80 Chinese commercial numerals 278 Pahawh Hmong 336 Chinese counting-rod numerals 267–8 Phoenician 70, 74–5, 99, 156 Indian positional numerals 194–7, 427 proto-cuneiform 230 Inka numerals 310 rongorongo 342 Iñupiaq numerals 345–6 runic 129–31 katapayâdi numerals 209 Sanskrit 195 Linear A numerals 57 Semitic 108, 140 Malayalam numerals 203 Serto 160 Maya numerals 286, 292, 294, 296–7 Sinhalese 199, 200, 204 Oberi Okaime numerals 321 Sogdian 87–8 Pahawh Hmong numerals 336–7 Sorang Sompeng 334–5 Roman numerals 120 South Arabian 107–8 Tamil numerals 202 Sumerian 230, 241, 243 Texcocan numerals 304, 306 Syriac 81–3, 87, 89 Western numerals 123–4, 221, 224 Tamil 199–200 Zhang, Jiajie 32 Telugu 199 Zhou Dynasty 262–3 Teotihuacan 307 Zhou numerals See Shang and Zhou numerals Tifi nigh 319 zimām numerals 149–52, 171–2, 186, 218, 387, Tocharian 199 411 Ugaritic 248 Zoroastrianism 89, 91 Uygur 280 Zuñi numerals 347–8