AHistoryof

MathematicsT 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, xi

Preface to the Third Edition, xiii

Preface to the Second Edition, xv

Preface to the First Edition, xvii

1 Traces 1Concepts and Relationships, 1 Early Number Bases, 3

Number Language and Counting, 5 Spatial Relationships, 6

2 Ancient Egypt 8The Era and the Sources, 8 Numbers and Fractions, 10

Arithmetic Operations, 12 “Heap” Problems, 13 Geometric

Problems, 14 Slope Problems, 18 Arithmetic Pragmatism, 19

3 Mesopotamia 21The Era and the Sources, 21 Cuneiform Writing, 22

Numbers and Fractions: Sexagesimals, 23 Positional

Numeration, 23 Sexagesimal Fractions, 25 Approximations, 25

Tables, 26 Equations, 28 Measurements: Pythagorean

Triads, 31 Polygonal Areas, 35 Geometry as Applied

Arithmetic, 36

4 Hellenic Traditions 40The Era and the Sources, 40 Thales and Pythagoras, 42

Numeration, 52 Arithmetic and Logistic, 55

v

Fifth-Century Athens, 56 Three Classical Problems, 57

Quadrature of Lunes, 58 Hippias of Elis, 61 Philolaus

and Archytas of Tarentum, 63 Incommensurability, 65

Paradoxes of Zeno, 67 Deductive Reasoning, 70

Democritus of Abdera, 72 Mathematics and the Liberal Arts, 74

The Academy, 74 Aristotle, 88

5 Euclid of Alexandria 90Alexandria, 90 Lost Works, 91 Extant Works, 91

The Elements, 93

6 Archimedes of Syracuse 109The Siege of Syracuse, 109 On the Equilibriums of Planes, 110

On Floating Bodies, 111 The Sand-Reckoner, 112

Measurement of the Circle, 113 On Spirals, 113

Quadrature of the Parabola, 115 On Conoids and Spheroids, 116

On the Sphere and Cylinder, 118 Book of Lemmas, 120

Semiregular Solids and Trigonometry, 121 The Method, 122

7 Apollonius of Perge 127Works and Tradition, 127 Lost Works, 128 Cycles

and Epicycles, 129 The Conics, 130

8 Crosscurrents 142Changing Trends, 142 Eratosthenes, 143 Angles and

Chords, 144 Ptolemy’s Almagest, 149 Heron of Alexandria, 156

The Decline of Greek Mathematics, 159 Nicomachus of

Gerasa, 159 Diophantus of Alexandria, 160 Pappus of

Alexandria, 164 The End of Alexandrian Dominance, 170

Proclus of Alexandria, 171 Boethius, 171

Athenian Fragments, 172 Byzantine Mathematicians, 173

9 Ancient and Medieval China 175The Oldest Known Texts, 175 The Nine Chapters, 176

Rod Numerals, 177 The Abacus and Decimal Fractions, 178

Values of Pi, 180 Thirteenth-Century Mathematics, 182

10 Ancient and Medieval India 186Early Mathematics in India, 186 The Sulbasutras, 187

The Siddhantas, 188 Aryabhata, 189 Numerals, 191

Trigonometry, 193 Multiplication, 194 Long Division, 195

Brahmagupta, 197 Indeterminate Equations, 199 Bhaskara, 200

Madhava and the Keralese School, 202

vi Cont en t s

11 The Islamic Hegemony 203Arabic Conquests, 203 The House of Wisdom, 205

Al-Khwarizmi, 206 ‘Abd Al-Hamid ibn-Turk, 212

Thabit ibn-Qurra, 213 Numerals, 214 Trigonometry, 216

Tenth- and Eleventh-Century Highlights, 216

Omar Khayyam, 218 The Parallel Postulate, 220

Nasir al-Din al-Tusi, 220 Al-Kashi, 221

12 The Latin West 223Introduction, 223 Compendia of the Dark Ages, 224

Gerbert, 224 The Century of Translation, 226 Abacists

and Algorists, 227 Fibonacci, 229 Jordanus Nemorarius, 232

Campanus of Novara, 233 Learning in the Thirteenth

Century, 235 Archimedes Revived, 235 Medieval Kinematics, 236

Thomas Bradwardine, 236 Nicole Oresme, 238 The Latitude

of Forms, 239 Infinite Series, 241 Levi ben Gerson, 242

Nicholas of Cusa, 243 The Decline of Medieval Learning, 243

13 The European Renaissance 245Overview, 245 Regiomontanus, 246 Nicolas

Chuquet’s Triparty, 249 Luca Pacioli’s Summa, 251

German Algebras and Arithmetics, 253 Cardan’s Ars Magna, 255

Rafael Bombelli, 260 Robert Recorde, 262 Trigonometry, 263

Geometry, 264 Renaissance Trends, 271 Francois Viete, 273

14 Early Modern Problem Solvers 282Accessibility of Computation, 282 Decimal Fractions, 283

Notation, 285 Logarithms, 286 Mathematical Instruments, 290

Infinitesimal Methods: Stevin, 296 Johannes Kepler, 296

15 Analysis, Synthesis, the Infinite, and Numbers 300Galileo’s Two New Sciences, 300 Bonaventura Cavalieri, 303

Evangelista Torricelli, 306 Mersenne’s Communicants, 308

Rene Descartes, 309 Fermat’s Loci, 320 Gregory of

St. Vincent, 325 The Theory of Numbers, 326

Gilles Persone de Roberval, 329 Girard Desargues and

Projective Geometry, 330 Blaise Pascal, 332 Philippe

de Lahire, 337 Georg Mohr, 338 Pietro Mengoli, 338

Frans van Schooten, 339 Jan de Witt, 340 Johann Hudde, 341

Rene Francois de Sluse, 342 Christiaan Huygens, 342

16 British Techniques and Continental Methods 348John Wallis, 348 James Gregory, 353 Nicolaus Mercator and

William Brouncker, 355 Barrow’s Method of Tangents, 356

Cont en t s vii

Newton, 358 Abraham De Moivre, 372 Roger Cotes, 375

James Stirling, 376 Colin Maclaurin, 376 Textbooks, 380

Rigor and Progress, 381 Leibniz, 382 The Bernoulli

Family, 390 Tschirnhaus Transformations, 398 Solid

Analytic Geometry, 399 Michel Rolle and Pierre Varignon, 400

The Clairauts, 401 Mathematics in Italy, 402 The Parallel

Postulate, 403 Divergent Series, 404

17 Euler 406The Life of Euler, 406 Notation, 408 Foundation of

Analysis, 409 Logarithms and the Euler Identities, 413

Differential Equations, 414 Probability, 416 The Theory of

Numbers, 417 Textbooks, 418 Analytic Geometry, 419

The Parallel Postulate: Lambert, 420

18 Pre- to Postrevolutionary France 423Men and Institutions, 423 The Committee on Weights

and Measures, 424 D’Alembert, 425 Bezout, 427

Condorcet, 429 Lagrange, 430 Monge, 433 Carnot, 438

Laplace, 443 Legendre, 446 Aspects of Abstraction, 449

Paris in the 1820s, 449 Fourier, 450 Cauchy, 452

Diffusion, 460

19 Gauss 464Nineteenth-Century Overview, 464 Gauss: Early Work, 465

Number Theory, 466 Reception of the Disquisitiones

Arithmeticae, 469 Astronomy, 470 Gauss’s Middle Years, 471

Differential Geometry, 472 Gauss’s Later Work, 473

Gauss’s Influence, 474

20 Geometry 483The School of Monge, 483 Projective Geometry: Poncelet and

Chasles, 485 Synthetic Metric Geometry: Steiner, 487

Synthetic Nonmetric Geometry: von Staudt, 489 Analytic

Geometry, 489 Non-Euclidean Geometry, 494 Riemannian

Geometry, 496 Spaces of Higher Dimensions, 498

Felix Klein, 499 Post-Riemannian Algebraic Geometry, 501

21 Algebra 504Introduction, 504 British Algebra and the Operational

Calculus of Functions, 505 Boole and the Algebra of Logic, 506

Augustus De Morgan, 509 William Rowan Hamilton, 510

Grassmann and Ausdehnungslehre, 512 Cayley and Sylvester, 515

Linear Associative Algebras, 519 Algebraic Geometry, 520

Algebraic and Arithmetic Integers, 520 Axioms of Arithmetic, 522

viii Cont en t s

22 Analysis 526Berlin and Gottingen at Midcentury, 526 Riemann in

Gottingen, 527 Mathematical Physics in Germany, 528

Mathematical Physics in English-Speaking Countries, 529

Weierstrass and Students, 531 The Arithmetization of

Analysis, 533 Dedekind, 536 Cantor and Kronecker, 538

Analysis in France, 543

23 Twentieth-Century Legacies 548Overview, 548 Henri Poincare, 549 David Hilbert, 555

Integration and Measure, 564 Functional Analysis and

General Topology, 568 Algebra, 570 Differential Geometry

and Tensor Analysis, 572 Probability, 573 Bounds and

Approximations, 575 The 1930s and World War II, 577

Nicolas Bourbaki, 578 Homological Algebra

and Category Theory, 580 Algebraic Geometry, 581

Logic and Computing, 582 The Fields Medals, 584

24 Recent Trends 586Overview, 586 The Four-Color Conjecture, 587

Classification of Finite Simple Groups, 591 Fermat’s

Last Theorem, 593 Poincare’s Query, 596 Future Outlook, 599

References, 601

General Bibliography, 633

Index, 647

Cont 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 subsequentmillennia, there are elements of greatness in all the fashions. Similarly,each society considers its own ways natural and rational, and finds theways of other 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.He had 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 mathe-

matics is there no significant correction—only extension. Once theGreeks had developed the deductive method, they were correct in whatthey did, correct for all time. Euclid was incomplete and his work hasbeen extended enormously, but it has not had to be corrected. His the-orems 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 whenThales worked 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 F o r ewo r d t o t he S e c ond Ed i t i on

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