AHistory

ofMathematics

T H I R D E D I T I O N

Uta C. Merzbach and Carl B. Boyer

John Wiley & Sons, Inc.

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Library of Congress Cataloging-in-Publication Data:

Boyer, Carl B. (Carl Benjamin), 1906 1976.A history of mathematics / Carl B. Boyer and Uta Merzbach. — 3rd ed.

p. cm.Includes bibliographical references and index.ISBN 978-0-470-52548-7 (pbk.); ISBN 978-0-470-63039-6 (ebk.);ISBN 978-0-470-63054-9 (ebk.); ISBN 978-0-470-630563 (ebk.)

1. Mathematics—History. I. Merzbach, Uta C., 1933 II. Title.QA21.B767 2010510.9—dc22 2010003424

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

In memory of Carl B. Boyer(1906 1976)—U.C.M.

To the memory of my parents,Howard Franklin Boyer and

Rebecca Catherine (Eisenhart) Boyer—C.B.B.

Contents

Foreword by Isaac Asimov, xiPreface to the Third Edition, xiiiPreface to the Second Edition, xvPreface to the First Edition, xvii

1 Traces 1Concepts and Relationships, 1 Early Number Bases, 3Number Language and Counting, 5 Spatial Relationships, 6

2 Ancient Egypt 8The Era and the Sources, 8 Numbers and Fractions, 10Arithmetic Operations, 12 “Heap” Problems, 13 GeometricProblems, 14 Slope Problems, 18 Arithmetic Pragmatism, 19

3 Mesopotamia 21The Era and the Sources, 21 Cuneiform Writing, 22Numbers and Fractions: Sexagesimals, 23 PositionalNumeration, 23 Sexagesimal Fractions, 25 Approximations, 25Tables, 26 Equations, 28 Measurements: PythagoreanTriads, 31 Polygonal Areas, 35 Geometry as AppliedArithmetic, 36

4 Hellenic Traditions 40The Era and the Sources, 40 Thales and Pythagoras, 42Numeration, 52 Arithmetic and Logistic, 55

v

Fifth-Century Athens, 56 Three Classical Problems, 57Quadrature of Lunes, 58 Hippias of Elis, 61 Philolausand Archytas of Tarentum, 63 Incommensurability, 65Paradoxes of Zeno, 67 Deductive Reasoning, 70Democritus of Abdera, 72 Mathematics and the Liberal Arts, 74The Academy, 74 Aristotle, 88

5 Euclid of Alexandria 90Alexandria, 90 Lost Works, 91 Extant Works, 91The Elements, 93

6 Archimedes of Syracuse 109The Siege of Syracuse, 109 On the Equilibriums of Planes, 110On Floating Bodies, 111 The Sand-Reckoner, 112Measurement of the Circle, 113 On Spirals, 113Quadrature of the Parabola, 115 On Conoids and Spheroids, 116On the Sphere and Cylinder, 118 Book of Lemmas, 120Semiregular Solids and Trigonometry, 121 The Method, 122

7 Apollonius of Perge 127Works and Tradition, 127 Lost Works, 128 Cyclesand Epicycles, 129 The Conics, 130

8 Crosscurrents 142Changing Trends, 142 Eratosthenes, 143 Angles andChords, 144 Ptolemy’s Almagest, 149 Heron of Alexandria, 156The Decline of Greek Mathematics, 159 Nicomachus ofGerasa, 159 Diophantus of Alexandria, 160 Pappus ofAlexandria, 164 The End of Alexandrian Dominance, 170Proclus of Alexandria, 171 Boethius, 171Athenian Fragments, 172 Byzantine Mathematicians, 173

9 Ancient and Medieval China 175The Oldest Known Texts, 175 The Nine Chapters, 176Rod Numerals, 177 The Abacus and Decimal Fractions, 178Values of Pi, 180 Thirteenth-Century Mathematics, 182

10 Ancient and Medieval India 186Early Mathematics in India, 186 The Sulbasutras, 187The Siddhantas, 188 Aryabhata, 189 Numerals, 191Trigonometry, 193 Multiplication, 194 Long Division, 195Brahmagupta, 197 Indeterminate Equations, 199 Bhaskara, 200Madhava and the Keralese School, 202

vi Con t en t s

11 The Islamic Hegemony 203Arabic Conquests, 203 The House of Wisdom, 205Al-Khwarizmi, 206 ‘Abd Al-Hamid ibn-Turk, 212Thabit ibn-Qurra, 213 Numerals, 214 Trigonometry, 216Tenth- and Eleventh-Century Highlights, 216Omar Khayyam, 218 The Parallel Postulate, 220Nasir al-Din al-Tusi, 220 Al-Kashi, 221

12 The Latin West 223Introduction, 223 Compendia of the Dark Ages, 224Gerbert, 224 The Century of Translation, 226 Abacistsand Algorists, 227 Fibonacci, 229 Jordanus Nemorarius, 232Campanus of Novara, 233 Learning in the ThirteenthCentury, 235 Archimedes Revived, 235 Medieval Kinematics, 236Thomas Bradwardine, 236 Nicole Oresme, 238 The Latitudeof Forms, 239 Infinite Series, 241 Levi ben Gerson, 242Nicholas of Cusa, 243 The Decline of Medieval Learning, 243

13 The European Renaissance 245Overview, 245 Regiomontanus, 246 NicolasChuquet’s Triparty, 249 Luca Pacioli’s Summa, 251German Algebras and Arithmetics, 253 Cardan’s Ars Magna, 255Rafael Bombelli, 260 Robert Recorde, 262 Trigonometry, 263Geometry, 264 Renaissance Trends, 271 François Viète, 273

14 Early Modern Problem Solvers 282Accessibility of Computation, 282 Decimal Fractions, 283Notation, 285 Logarithms, 286 Mathematical Instruments, 290Infinitesimal Methods: Stevin, 296 Johannes Kepler, 296

15 Analysis, Synthesis, the Infinite, and Numbers 300Galileo’s Two New Sciences, 300 Bonaventura Cavalieri, 303Evangelista Torricelli, 306 Mersenne’s Communicants, 308René Descartes, 309 Fermat’s Loci, 320 Gregory ofSt. Vincent, 325 The Theory of Numbers, 326Gilles Persone de Roberval, 329 Girard Desargues andProjective Geometry, 330 Blaise Pascal, 332 Philippede Lahire, 337 Georg Mohr, 338 Pietro Mengoli, 338Frans van Schooten, 339 Jan de Witt, 340 Johann Hudde, 341René François de Sluse, 342 Christiaan Huygens, 342

16 British Techniques and Continental Methods 348John Wallis, 348 James Gregory, 353 Nicolaus Mercator andWilliam Brouncker, 355 Barrow’s Method of Tangents, 356

Con t en t s vii

Newton, 358 Abraham De Moivre, 372 Roger Cotes, 375James Stirling, 376 Colin Maclaurin, 376 Textbooks, 380Rigor and Progress, 381 Leibniz, 382 The BernoulliFamily, 390 Tschirnhaus Transformations, 398 SolidAnalytic Geometry, 399 Michel Rolle and Pierre Varignon, 400The Clairauts, 401 Mathematics in Italy, 402 The ParallelPostulate, 403 Divergent Series, 404

17 Euler 406The Life of Euler, 406 Notation, 408 Foundation ofAnalysis, 409 Logarithms and the Euler Identities, 413Differential Equations, 414 Probability, 416 The Theory ofNumbers, 417 Textbooks, 418 Analytic Geometry, 419The Parallel Postulate: Lambert, 420

18 Pre- to Postrevolutionary France 423Men and Institutions, 423 The Committee on Weightsand Measures, 424 D’Alembert, 425 Bézout, 427Condorcet, 429 Lagrange, 430 Monge, 433 Carnot, 438Laplace, 443 Legendre, 446 Aspects of Abstraction, 449Paris in the 1820s, 449 Fourier, 450 Cauchy, 452Diffusion, 460

19 Gauss 464Nineteenth-Century Overview, 464 Gauss: Early Work, 465Number Theory, 466 Reception of the DisquisitionesArithmeticae, 469 Astronomy, 470 Gauss’s Middle Years, 471Differential Geometry, 472 Gauss’s Later Work, 473Gauss’s Influence, 474

20 Geometry 483The School of Monge, 483 Projective Geometry: Poncelet andChasles, 485 Synthetic Metric Geometry: Steiner, 487Synthetic Nonmetric Geometry: von Staudt, 489 AnalyticGeometry, 489 Non-Euclidean Geometry, 494 RiemannianGeometry, 496 Spaces of Higher Dimensions, 498Felix Klein, 499 Post-Riemannian Algebraic Geometry, 501

21 Algebra 504Introduction, 504 British Algebra and the OperationalCalculus of Functions, 505 Boole and the Algebra of Logic, 506Augustus De Morgan, 509 William Rowan Hamilton, 510Grassmann and Ausdehnungslehre, 512 Cayley and Sylvester, 515Linear Associative Algebras, 519 Algebraic Geometry, 520Algebraic and Arithmetic Integers, 520 Axioms of Arithmetic, 522

viii Con t en t s

22 Analysis 526Berlin and Göttingen at Midcentury, 526 Riemann inGöttingen, 527 Mathematical Physics in Germany, 528Mathematical Physics in English-Speaking Countries, 529Weierstrass and Students, 531 The Arithmetization ofAnalysis, 533 Dedekind, 536 Cantor and Kronecker, 538Analysis in France, 543

23 Twentieth-Century Legacies 548Overview, 548 Henri Poincaré, 549 David Hilbert, 555Integration and Measure, 564 Functional Analysis andGeneral Topology, 568 Algebra, 570 Differential Geometryand Tensor Analysis, 572 Probability, 573 Bounds andApproximations, 575 The 1930s and World War II, 577Nicolas Bourbaki, 578 Homological Algebraand Category Theory, 580 Algebraic Geometry, 581Logic and Computing, 582 The Fields Medals, 584

24 Recent Trends 586Overview, 586 The Four-Color Conjecture, 587Classification of Finite Simple Groups, 591 Fermat’sLast Theorem, 593 Poincaré’s Query, 596 Future Outlook, 599

References, 601General Bibliography, 633Index, 647

Con t en t s ix

Foreword to the Second EditionBy Isaac Asimov

Mathematics is a unique aspect of human thought, and its history differsin essence from all other histories.As time goes on, nearly every field of human endeavor is marked

by changes which can be considered as correction and/or extension. Thus,the changes in the evolving history of political and military events arealways chaotic; there is no way to predict the rise of a Genghis Khan,for example, or the consequences of the short-lived Mongol Empire.Other changes are a matter of fashion and subjective opinion. The cave-paintings of 25,000 years ago are generally considered great art, and whileart has continuously—even chaotically—changed in the subsequent mil-lennia, there are elements of greatness in all the fashions. Similarly, eachsociety considers its own ways natural and rational, and finds the ways ofother societies to be odd, laughable, or repulsive.But only among the sciences is there true progress; only there is the

record one of continuous advance toward ever greater heights.And yet, among most branches of science, the process of progress is

one of both correction and extension. Aristotle, one of the greatest mindsever to contemplate physical laws, was quite wrong in his views onfalling bodies and had to be corrected by Galileo in the 1590s. Galen, thegreatest of ancient physicians, was not allowed to study human cadaversand was quite wrong in his anatomical and physiological conclusions. Hehad to be corrected by Vesalius in 1543 and Harvey in 1628. EvenNewton, the greatest of all scientists, was wrong in his view of the natureof light, of the achromaticity of lenses, and missed the existence of

xi

spectral lines. His masterpiece, the laws of motion and the theory ofuniversal gravitation, had to be modified by Einstein in 1916.Now we can see what makes mathematics unique. Only in mathematics

is there no significant correction—only extension. Once the Greeks haddeveloped the deductive method, they were correct in what they did,correct for all time. Euclid was incomplete and his work has beenextended enormously, but it has not had to be corrected. His theorems are,every one of them, valid to this day.Ptolemy may have developed an erroneous picture of the planetary

system, but the system of trigonometry he worked out to help him withhis calculations remains correct forever.Each great mathematician adds to what came previously, but nothing

needs to be uprooted. Consequently, when we read a book like A Historyof Mathematics, we get the picture of a mounting structure, ever taller andbroader and more beautiful and magnificent and with a foundation,moreover, that is as untainted and as functional now as it was when Thalesworked out the first geometrical theorems nearly 26 centuries ago.Nothing pertaining to humanity becomes us so well as mathematics.

There, and only there, do we touch the human mind at its peak.

xii Fo r ewo rd t o t h e Se cond Ed i t i o n

Preface to the Third Edition

During the two decades since the appearance of the second edition ofthis work, there have been substantial changes in the course of mathe-matics and the treatment of its history. Within mathematics, outstandingresults were achieved by a merging of techniques and concepts frompreviously distinct areas of specialization. The history of mathematicscontinued to grow quantitatively, as noted in the preface to the secondedition; but here, too, there were substantial studies that overcame thepolemics of “internal” versus “external” history and combined a freshapproach to the mathematics of the original texts with the appropriatelinguistic, sociological, and economic tools of the historian.In this third edition I have striven again to adhere to Boyer’s approach

to the history of mathematics. Although the revision this time includesthe entire work, changes have more to do with emphasis than originalcontent, the obvious exception being the inclusion of new findings sincethe appearance of the first edition. For example, the reader will findgreater stress placed on the fact that we deal with such a small number ofsources from antiquity; this is one of the reasons for condensing threeprevious chapters dealing with the Hellenic period into one. On the otherhand, the chapter dealing with China and India has been split, as contentdemands. There is greater emphasis on the recurring interplay betweenpure and applied mathematics as exemplified in chapter 14. Somereorganization is due to an attempt to underline the impact of institu-tional and personal transmission of ideas; this has affected most of thepre-nineteenth-century chapters. The chapters dealing with the nineteenthcentury have been altered the least, as I had made substantial changesfor some of this material in the second edition. The twentieth-century

xiii

material has been doubled, and a new final chapter deals with recenttrends, including solutions of some longstanding problems and the effectof computers on the nature of proofs.It is always pleasant to acknowledge those known to us for having had

an impact on our work. I am most grateful to Shirley Surrette Duffy forresponding judiciously to numerous requests for stylistic advice, even attimes when there were more immediate priorities. Peggy Aldrich Kid-well replied with unfailing precision to my inquiry concerning certainphotographs in the National Museum of American History. JeanneLaDuke cheerfully and promptly answered my appeals for help, espe-cially in confirming sources. Judy and Paul Green may not realize that acasual conversation last year led me to rethink some recent material. Ihave derived special pleasure and knowledge from several recent pub-lications, among them Klopfer 2009 and, in a more leisurely fashion,Szpiro 2007. Great thanks are due to the editors and production team ofJohn Wiley & Sons who worked with me to make this edition possible:Stephen Power, the senior editor, was unfailingly generous and diplo-matic in his counsel; the editorial assistant, Ellen Wright, facilitatedmy progress through the major steps of manuscript creation; the seniorproduction manager, Marcia Samuels, provided me with clear andconcise instructions, warnings, and examples; senior production editorsKimberly Monroe-Hill and John Simko and the copyeditor, PatriciaWaldygo, subjected the manuscript to painstakingly meticulous scrutiny.The professionalism of all concerned provides a special kind ofencouragement in troubled times.I should like to pay tribute to two scholars whose influence on others

should not be forgotten. The Renaissance historian Marjorie N. Boyer(Mrs. Carl B. Boyer) graciously and knowledgeably complimenteda young researcher at the beginning of her career on a talk presented at aLeibniz conference in 1966. The brief conversation with a total strangerdid much to influence me in pondering the choice between mathematicsand its history.More recently, the late historian of mathematics Wilbur Knorr set a

significant example to a generation of young scholars by refusing toaccept the notion that ancient authors had been studied definitively byothers. Setting aside the “magister dixit,” he showed us the wealth ofknowledge that emerges from seeking out the texts.

—Uta C. MerzbachMarch 2010

xiv P r e f a c e t o t he Th i r d Ed i t i on

Preface to the Second Edition

This edition brings to a new generation and a broader spectrum ofreaders a book that became a standard for its subject after its initialappearance in 1968. The years since then have been years of renewedinterest and vigorous activity in the history of mathematics. This hasbeen demonstrated by the appearance of numerous new publicationsdealing with topics in the field, by an increase in the number of courseson the history of mathematics, and by a steady growth over the years inthe number of popular books devoted to the subject. Lately, growinginterest in the history of mathematics has been reflected in other bran-ches of the popular press and in the electronic media. Boyer’s con-tribution to the history of mathematics has left its mark on all of theseendeavors.When one of the editors of John Wiley & Sons first approached me

concerning a revision of Boyer’s standard work, we quickly agreed thattextual modifications should be kept to a minimum and that the changesand additions should be made to conform as much as possible to Boyer’soriginal approach. Accordingly, the first twenty-two chapters have beenleft virtually unchanged. The chapters dealing with the nineteenth centuryhave been revised; the last chapter has been expanded and split into two.Throughout, an attempt has been made to retain a consistent approachwithin the volume and to adhere to Boyer’s stated aim of giving strongeremphasis on historical elements than is customary in similar works.The references and general bibliography have been substantially

revised. Since this work is aimed at English-speaking readers, many ofwhom are unable to utilize Boyer’s foreign-language chapter references,these have been replaced by recent works in English. Readers are urged

xv

to consult the General Bibliography as well, however. Immediately fol-lowing the chapter references at the end of the book, it contains additionalworks and further bibliographic references, with less regard to language.The introduction to that bibliography provides some overall guidance forfurther pleasurable reading and for solving problems.The initial revision, which appeared two years ago, was designed for

classroom use. The exercises found there, and in the original edition,have been dropped in this edition, which is aimed at readers outside thelecture room. Users of this book interested in supplementary exercisesare referred to the suggestions in the General Bibliography.I express my gratitude to Judith V. Grabiner and Albert Lewis for

numerous helpful criticisms and suggestions. I am pleased to acknowl-edge the fine cooperation and assistance of several members of theWiley editorial staff. I owe immeasurable thanks to Virginia Beets forlending her vision at a critical stage in the preparation of this manuscript.Finally, thanks are due to numerous colleagues and students who haveshared their thoughts about the first edition with me. I hope they will findbeneficial results in this revision.

—Uta C. MerzbachGeorgetown, Texas

March 1991

xvi P r e f a c e t o t he Se cond Ed i t i on

Preface to the First Edition

Numerous histories of mathematics have appeared during this century,many of them in the English language. Some are very recent, such asJ. F. Scott’s A History of Mathematics1; a new entry in the field,therefore, should have characteristics not already present in the availablebooks. Actually, few of the histories at hand are textbooks, at least not inthe American sense of the word, and Scott’s History is not one of them.It appeared, therefore, that there was room for a new book—one thatwould meet more satisfactorily my own preferences and possibly thoseof others.The two-volume History of Mathematics by David Eugene Smith2 was

indeed written “for the purpose of supplying teachers and students with ausable textbook on the history of elementary mathematics,” but it coverstoo wide an area on too low a mathematical level for most moderncollege courses, and it is lacking in problems of varied types. FlorianCajori’s History of Mathematics3 still is a very helpful reference work;but it is not adapted to classroom use, nor is E. T. Bell’s admirableThe Development of Mathematics.4 The most successful and app-ropriate textbook today appears to be Howard Eves, An Introduction tothe History of Mathematics,5 which I have used with considerablesatisfaction in at least a dozen classes since it first appeared in 1953.

1London: Taylor and Francis, 1958.2Boston: Ginn and Company, 1923 1925.3New York: Macmillan, 1931, 2nd edition.4New York: McGraw-Hill, 1945, 2nd edition.5New York: Holt, Rinehart and Winston, 1964, revised edition.

xvii

I have occasionally departed from the arrangement of topics in the bookin striving toward a heightened sense of historicalmindedness and havesupplemented the material by further reference to the contributionsof the eighteenth and nineteenth centuries especially by the use ofD. J. Struik, A Concise History of Mathematics.6

The reader of this book, whether layman, student, or teacher of acourse in the history of mathematics, will find that the level of mathe-matical background that is presupposed is approximately that of a col-lege junior or senior, but the material can be perused profitably also byreaders with either stronger or weaker mathematical preparation. Eachchapter ends with a set of exercises that are graded roughly into threecategories. Essay questions that are intended to indicate the reader’sability to organize and put into his own words the material discussed inthe chapter are listed first. Then follow relatively easy exercises thatrequire the proofs of some of the theorems mentioned in the chapter ortheir application to varied situations. Finally, there are a few starredexercises, which are either more difficult or require specialized methodsthat may not be familiar to all students or all readers. The exercises donot in any way form part of the general exposition and can be dis-regarded by the reader without loss of continuity.Here and there in the text are references to footnotes, generally bib-

liographical, and following each chapter there is a list of suggestedreadings. Included are some references to the vast periodical literature inthe field, for it is not too early for students at this level to be introducedto the wealth of material available in good libraries. Smaller collegelibraries may not be able to provide all of these sources, but it is well fora student to be aware of the larger realms of scholarship beyond theconfines of his own campus. There are references also to works inforeign languages, despite the fact that some students, hopefully notmany, may be unable to read any of these. Besides providing importantadditional sources for those who have a reading knowledge of a foreignlanguage, the inclusion of references in other languages may help tobreak down the linguistic provincialism which, ostrichlike, takes refugein the mistaken impression that everything worthwhile appeared in, orhas been translated into, the English language.The present work differs from the most successful presently available

textbook in a stricter adherence to the chronological arrangement and astronger emphasis on historical elements. There is always the temptationin a class in history of mathematics to assume that the fundamentalpurpose of the course is to teach mathematics. A departure frommathematical standards is then a mortal sin, whereas an error in historyis venial. I have striven to avoid such an attitude, and the purpose of the

6New York: Dover Publications, 1967, 3rd edition.

xviii P r e f a c e t o t he F i r s t E d i t i on

book is to present the history of mathematics with fidelity, not only tomathematical structure and exactitude, but also to historical perspectiveand detail. It would be folly, in a book of this scope, to expect that everydate, as well as every decimal point, is correct. It is hoped, however, thatsuch inadvertencies as may survive beyond the stage of page proof willnot do violence to the sense of history, broadly understood, or to a soundview of mathematical concepts. It cannot be too strongly emphasizedthat this single volume in no way purports to present the history ofmathematics in its entirety. Such an enterprise would call for the con-certed effort of a team, similar to that which produced the fourth volumeof Cantor’s Vorlesungen iiber Geschichte der Mathematik in 1908 andbrought the story down to 1799. In a work of modest scope the authormust exercise judgment in the selection of the materials to be included,reluctantly restraining the temptation to cite the work of every produc-tive mathematician; it will be an exceptional reader who will not notehere what he regards as unconscionable omissions. In particular, the lastchapter attempts merely to point out a few of the salient characteristicsof the twentieth century. In the field of the history of mathematicsperhaps nothing is more to be desired than that there should appear alatter-day Felix Klein who would complete for our century the type ofproject Klein essayed for the nineteenth century, but did not live tofinish.A published work is to some extent like an iceberg, for what is visible

constitutes only a small fraction of the whole. No book appears until theauthor has lavished time on it unstintingly and unless he has receivedencouragement and support from others too numerous to be namedindividually. Indebtedness in my case begins with the many eager stu-dents to whom I have taught the history of mathematics, primarilyat Brooklyn College, but also at Yeshiva University, the Universityof Michigan, the University of California (Berkeley), and the University ofKansas. At the University of Michigan, chiefly through the encourage-ment of Professor Phillip S. Jones, and at Brooklyn College through theassistance of Dean Walter H. Mais and Professors Samuel Borofsky andJames Singer, I have on occasion enjoyed a reduction in teaching load inorder to work on the manuscript of this book. Friends and colleaguesin the field of the history of mathematics, including Professor DirkJ. Struik of theMassachusetts Institute of Technology, Professor Kenneth O.May at the University of Toronto, Professor Howard Eves ofthe University of Maine, and Professor Morris Kline at New YorkUniversity, have made many helpful suggestions in the preparationof the book, and these have been greatly appreciated. Materials inthe books and articles of others have been expropriated freely, with littleacknowledgment beyond a cold bibliographical reference, and I take thisopportunity to express to these authors my warmest gratitude. Librariesand publishers have been very helpful in providing information and

P r e f a c e t o t he F i r s t E d i t i on xix

illustrations needed in the text; in particular it has been a pleasure to haveworked with the staff of John Wiley & Sons. The typing of the final copy,as well as of much of the difficult preliminary manuscript, was donecheerfully and with painstaking care by Mrs. Hazel Stanley of Lawrence,Kansas. Finally, I must express deep gratitude to a very understandingwife. Dr. Marjorie N. Boyer, for her patience in tolerating disruptionsoccasioned by the development of yet another book within the family.

—Carl B. BoyerBrooklyn, New York

January 1968

xx P r e f a c e t o t he F i r s t E d i t i on

1

Traces

Did you bring me a man who cannot number his fingers?—From the Egyptian Book of the Dead

Concepts and Relationships

Contemporary mathematicians formulate statements about abstract con-cepts that are subject to verification by proof. For centuries, mathematicswas considered to be the science of numbers, magnitudes, and forms. Forthat reason, those who seek early examples of mathematical activity willpoint to archaeological remnants that reflect human awareness of opera-tions on numbers, counting, or “geometric” patterns and shapes. Evenwhen these vestiges reflect mathematical activity, they rarely evidencemuch historical significance. Theymay be interestingwhen they show thatpeoples in different parts of the world conducted certain actions dealingwith concepts that have been considered mathematical. For such an actionto assume historical significance, however, we look for relationships thatindicate this actionwas known to another individual or group that engagedin a related action.Once such a connection has been established, the door isopen to more specifically historical studies, such as those dealing withtransmission, tradition, and conceptual change.

1

Mathematical vestiges are often found in the domain of nonliteratecultures, making the evaluation of their significance even more complex.Rules of operation may exist as part of an oral tradition, often in musicalor verse form, or they may be clad in the language of magic or ritual.Sometimes they are found in observations of animal behavior, removingthem even further from the realm of the historian. While studies ofcanine arithmetic or avian geometry belong to the zoologist, of theimpact of brain lesions on number sense to the neurologist, and ofnumerical healing incantations to the anthropologist, all of these studiesmay prove to be useful to the historian of mathematics without being anovert part of that history.At first, the notions of number, magnitude, and form may have been

related to contrasts rather than likenesses—the difference betweenone wolf and many, the inequality in size of a minnow and a whale, theunlikeness of the roundness of the moon and the straightness of a pinetree. Gradually, there may have arisen, out of the welter of chaoticexperiences, the realization that there are samenesses, and from thisawareness of similarities in number and form both science and mathe-matics were born. The differences themselves seem to point to likenesses,for the contrast between one wolf and many, between one sheep and aherd, between one tree and a forest suggests that one wolf, one sheep,and one tree have something in common—their uniqueness. In the sameway it would be noticed that certain other groups, such as pairs, can beput into one-to-one correspondence. The hands can be matched againstthe feet, the eyes, the ears, or the nostrils. This recognition of anabstract property that certain groups hold in common, and that we call“number,” represents a long step toward modern mathematics. It isunlikely to have been the discovery of any one individual or any singletribe; it was more probably a gradual awareness that may have devel-oped as early in man’s cultural development as the use of fire, possiblysome 300,000 years ago.That the development of the number concept was a long and gradual

process is suggested by the fact that some languages, including Greek,have preserved in their grammar a tripartite distinction between 1 and 2and more than 2, whereas most languages today make only the dualdistinction in “number” between singular and plural. Evidently, our veryearly ancestors at first counted only to 2, and any set beyond this levelwas designated as “many.” Even today, many people still count objectsby arranging them into sets of two each.The awareness of number ultimately became sufficiently extended

and vivid so that a need was felt to express the property in some way,presumably at first in sign language only. The fingers on a hand can bereadily used to indicate a set of two or three or four or five objects, thenumber 1 generally not being recognized at first as a true “number.” Bythe use of the fingers on both hands, collections containing up to ten

2 T r a c e s

elements could be represented; by combining fingers and toes, onecould count as high as 20. When the human digits were inadequate,heaps of stones or knotted strings could be used to represent a corre-spondence with the elements of another set. Where nonliterate peoplesused such a scheme of representation, they often piled the stones ingroups of five, for they had become familiar with quintuples throughobservation of the human hand and foot. As Aristotle noted long ago, thewidespread use today of the decimal system is but the result ofthe anatomical accident that most of us are born with ten fingers andten toes.Groups of stones are too ephemeral for the preservation of informa-

tion; hence, prehistoric man sometimes made a number record by cuttingnotches in a stick or a piece of bone. Few of these records remain today,but in Moravia a bone from a young wolf was found that is deeplyincised with fifty-five notches. These are arranged in two series, withtwenty-five in the first and thirty in the second: within each series, thenotches are arranged in groups of five. It has been dated as beingapproximately 30,000 years old. Two other prehistoric numerical arti-facts were found in Africa: a baboon fibula having twenty-nine notches,dated as being circa 35,000 years old, and the Ishango bone, with itsapparent examples of multiplicative entries, initially dated as approxi-mately 8,000 years old but now estimated to be as much as 30,000 yearsold as well. Such archaeological discoveries provide evidence that theidea of number is far older than previously acknowledged.

Early Number Bases

Historically, finger counting, or the practice of counting by fives andtens, seems to have come later than counter-casting by twos and threes,yet the quinary and decimal systems almost invariably displaced thebinary and ternary schemes. A study of several hundred tribes amongthe American Indians, for example, showed that almost one-third useda decimal base, and about another third had adopted a quinary or aquinary-decimal system; fewer than a third had a binary scheme, andthose using a ternary system constituted less than 1 percent of the group.The vigesimal system, with the number 20 as a base, occurred in about10 percent of the tribes.An interesting example of a vigesimal system is that used by the Maya

of Yucatan and Central America. This was deciphered some timebefore the rest of the Maya languages could be translated. In theirrepresentation of time intervals between dates in their calendar, theMaya used a place value numeration, generally with 20 as the primarybase and with 5 as an auxiliary. (See the following illustration.) Unitswere represented by dots and fives by horizontal bars, so that the number

Ea r l y Numbe r B a s e s 3

17, for example, would appear as (that is, as 3(5)+ 2). A verticalpositional arrangement was used, with the larger units of time above;hence, the notation denoted 352 (that is, 17(20)+ 12). Because thesystem was primarily for counting days within a calendar that had 360days in a year, the third position usually did not represent multiples of(20)(20), as in a pure vigesimal system, but (18)(20). Beyond this point,however, the base 20 again prevailed. Within this positional notation,the Maya indicated missing positions through the use of a symbol,which appeared in variant forms, somewhat resembling a half-open eye.

From the Dresden Codex of the Maya, displaying numbers. The secondcolumn on the left, reading down from above, displays the numbers 9, 9,16, 0, 0, which stand for 9 × 144,000+ 9 × 7,200+ 16 × 360+ 0+ 0= 1,366,560. In the third column are the numerals 9, 9, 9, 16, 0, representing1,364,360. The original appears in black and red. (Taken fromMorley 1915,p. 266.)

4 T r a c e s

In their scheme, then, the notation denoted 17(20 18 20)+0(18 20)+ 13(20)+ 0.

Number Language and Counting

It is generally believed that the development of language was essential tothe rise of abstract mathematical thinking. Yet words expressingnumerical ideas were slow in arising. Number signs probably precedednumber words, for it is easier to cut notches in a stick than it is toestablish a well-modulated phrase to identify a number. Had the problemof language not been so difficult, rivals to the decimal system mighthave made greater headway. The base 5, for example, was one of theearliest to leave behind some tangible written evidence, but by the timethat language became formalized, 10 had gained the upper hand.The modern languages of today are built almost without exceptionaround the base 10, so that the number 13, for example, is not describedas 3 and 5 and 5, but as 3 and 10. The tardiness in the development oflanguage to cover abstractions such as number is also seen in the factthat primitive numerical verbal expressions invariably refer to specificconcrete collections—such as “two fishes” or “two clubs”—and latersome such phrase would be adopted conventionally to indicate all sets oftwo objects. The tendency for language to develop from the concrete tothe abstract is seen in many of our present-day measures of length. Theheight of a horse is measured in “hands,” and the words “foot” and “ell”(or elbow) have similarly been derived from parts of the body.The thousands of years required for man to separate out the abstract

concepts from repeated concrete situations testify to the difficulties thatmust have been experienced in laying even a very primitive basis formathematics. Moreover, there are a great many unanswered questionsrelating to the origins ofmathematics. It is usually assumed that the subjectarose in answer to practical needs, but anthropological studies suggest thepossibility of an alternative origin. It has been suggested that the art ofcounting arose in connection with primitive religious ritual and that theordinal aspect preceded the quantitative concept. In ceremonial ritesdepicting creation myths, it was necessary to call the participants onto thescene in a specific order, and perhaps counting was invented to take care ofthis problem. If theories of the ritual origin of counting are correct, theconcept of the ordinal number may have preceded that of the cardinalnumber. Moreover, such an origin would tend to point to the possibilitythat counting stemmed from a unique origin, spreading subsequently toother areas of the world. This view, although far from established, wouldbe in harmonywith the ritual division of the integers into odd and even, theformer being regarded as male, the latter as female. Such distinctions

Numbe r L anguage and Coun t i ng 5

were known to civilizations in all corners of the earth, andmyths regardingthe male and female numbers have been remarkably persistent.The concept of the whole number is one of the oldest in mathematics,

and its origin is shrouded in the mists of prehistoric antiquity. The notionof a rational fraction, however, developed relatively late and was not ingeneral closely related to systems for the integers. Among nonliteratetribes, there seems to have been virtually no need for fractions. Forquantitative needs, the practical person can choose units that are suffi-ciently small to obviate the necessity of using fractions. Hence, therewas no orderly advance from binary to quinary to decimal fractions, andthe dominance of decimal fractions is essentially the product of themodern age.

Spatial Relationships

Statements about the origins of mathematics, whether of arithmetic orgeometry, are of necessity hazardous, for the beginnings of the subjectare older than the art of writing. It is only during the last half-dozenmillennia, in a passage that may have spanned thousands of millennia,that human beings have been able to put their records and thoughts intowritten form. For data about the prehistoric age, we must depend oninterpretations based on the few surviving artifacts, on evidence pro-vided by current anthropology, and on a conjectural backward extra-polation from surviving documents. Neolithic peoples may have hadlittle leisure and little need for surveying, yet their drawings and designssuggest a concern for spatial relationships that paved the way for geo-metry. Pottery, weaving, and basketry show instances of congruence andsymmetry, which are in essence parts of elementary geometry, and theyappear on every continent. Moreover, simple sequences in design, suchas that in Fig. 1.1, suggest a sort of applied group theory, as well as

FIG. 1.1

6 T r a c e s

propositions in geometry and arithmetic. The design makes it immedi-ately obvious that the areas of triangles are to one another as squares ona side, or, through counting, that the sums of consecutive odd numbers,beginning from unity, are perfect squares. For the prehistoric periodthere are no documents; hence, it is impossible to trace the evolution ofmathematics from a specific design to a familiar theorem. But ideas arelike hardy spores, and sometimes the presumed origin of a concept maybe only the reappearance of a much more ancient idea that had laindormant.The concern of prehistoric humans for spatial designs and relationships

may have stemmed from their aesthetic feeling and the enjoyment ofbeauty of form, motives that often actuate the mathematician of today.Wewould like to think that at least some of the early geometers pursued theirwork for the sheer joy of doing mathematics, rather than as a practical aidin mensuration, but there are alternative theories. One of these is thatgeometry, like counting, had an origin in primitive ritualistic practice. Yetthe theory of the origin of geometry in a secularization of ritualisticpractice is by no means established. The development of geometry mayjust as well have been stimulated by the practical needs of construction andsurveying or by an aesthetic feeling for design and order.We can make conjectures about what led people of the Stone Age to

count, to measure, and to draw. That the beginnings of mathematics areolder than the oldest civilizations is clear. To go further and categori-cally identify a specific origin in space or time, however, is to mistakeconjecture for history. It is best to suspend judgment on this matter andto move on to the safer ground of the history of mathematics as found inthe written documents that have come down to us.

Spa t i a l R e l a t i on sh i p s 7

2

Ancient Egypt

Sesostris . . . made a division of the soil of Egypt among theinhabitants. . . . If the river carried away any portion of a man’s lot, . . .the king sent persons to examine, and determine by measurement theexact extent of the loss. . . . From this practice, I think, geometry first

came to be known in Egypt, whence it passed into Greece.—Herodotus

The Era and the Sources

About 450 BCE, Herodotus, the inveterate Greek traveler and narrativehistorian, visited Egypt. He viewed ancient monuments, interviewedpriests, and observed the majesty of the Nile and the achievements of thoseworking along its banks. His resulting account would become a cornerstonefor the narrative of Egypt’s ancient history. When it came to mathematics,he held that geometry had originated in Egypt, for he believed thatthe subject had arisen there from the practical need for resurveying after theannual flooding of the river valley. A century later, the philosopher Aris-totle speculated on the same subject and attributed the Egyptians’ pursuit ofgeometry to the existence of a priestly leisure class. The debate, extending

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