com A History of Mathematics THIRD EDITION Uta C. Merzbach and Carl B. Boyer John Wiley & Sons, Inc.com Copyright r 1968, 1989, 1991, 2011 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or trans mitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Pub lisher, or authorization through payment of the appropriate per copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750 8400, fax (978) 646 8600, or on the web at www.
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Boyer and Uta Merzbach. Includes bibliographical references and index.9 dc22 2010003424 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 www.com In memory of Carl B. To the memory of my parents, Howard Franklin Boyer and Rebecca Catherine (Eisenhart) Boyer C.com 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 1 Concepts and Relationships, 1 Early Number Bases, 3 Number Language and Counting, 5 Spatial Relationships, 6 2 Ancient Egypt 8 The 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 21 The 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 40 The Era and the Sources, 40 Thales and Pythagoras, 42 Numeration, 52 Arithmetic and Logistic, 55 v www.com vi Content s 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 90 Alexandria, 90 Lost Works, 91 Extant Works, 91 The Elements, 93 6 Archimedes of Syracuse 109 The 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 127 Works and Tradition, 127 Lost Works, 128 Cycles and Epicycles, 129 The Conics, 130 8 Crosscurrents 142 Changing 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 175 The 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 186 Early 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 www.com Content s vii 11 The Islamic Hegemony 203 Arabic 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 223 Introduction, 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 245 Overview, 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 François Viète, 273 14 Early Modern Problem Solvers 282 Accessibility 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 300 Galileo’s Two New Sciences, 300 Bonaventura Cavalieri, 303 Evangelista Torricelli, 306 Mersenne’s Communicants, 308 René 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 René François de Sluse, 342 Christiaan Huygens, 342 16 British Techniques and Continental Methods 348 John Wallis, 348 James Gregory, 353 Nicolaus Mercator and William Brouncker, 355 Barrow’s Method of Tangents, 356 www.com viii Content s 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 406 The 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 423 Men and Institutions, 423 The Committee on Weights and Measures, 424 D’Alembert, 425 Bézout, 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 464 Nineteenth-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 483 The 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 504 Introduction, 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 www.com Content s ix 22 Analysis 526 Berlin and Göttingen at Midcentury, 526 Riemann in Göttingen, 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 548 Overview, 548 Henri Poincaré, 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 586 Overview, 586 The Four-Color Conjecture, 587 Classification of Finite Simple Groups, 591 Fermat’s Last Theorem, 593 Poincaré’s Query, 596 Future Outlook, 599 References, 601 General Bibliography, 633 Index, 647 www.com Foreword to the Second Edition By Isaac Asimov Mathematics is a unique aspect of human thought, and its history differs in 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 are always 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 while art has continuously—even chaotically—changed in the subsequent millennia, there are elements of greatness in all the fashions.
Similarly, each society considers its own ways natural and rational, and finds the ways 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 minds ever to contemplate physical laws, was quite wrong in his views on falling bodies and had to be corrected by Galileo in the 1590s.
Galen, the greatest of ancient physicians, was not allowed to study human cadavers and was quite wrong in his anatomical and physiological conclusions. He had to be corrected by Vesalius in 1543 and Harvey in 1628. Even Newton, the greatest of all scientists, was wrong in his view of the nature of light, of the achromaticity of lenses, and missed the existence of xi www.com xii F ore word to the Sec ond Edition spectral lines. His masterpiece, the laws of motion and the theory of universal 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 the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been 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 with his calculations remains correct forever. Each great mathematician adds to what came previously, but nothing needs to be uprooted.