Revised Con rming Pages The History of Mathematics AN INTRODUCTION Seventh Edition David M. Burton University of New Hampshire www.com bur83155 fm i-xii.tex i 01/13/2010 16:12 Revised Con rming Pages THE HISTORY OF MATHEMATICS: AN INTRODUCTION, SEVENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright c 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
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Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Burton, David M. The history of mathematics : an introduction / David M. Includes bibliographical references and index.com bur83155 fm i-xii.tex ii 01/13/2010 16:15 Revised Con rming Pages A ll these were honored in their generations, and were the glory of their times.
T here be of them, that have left a name behind them, that their praises might be reported. A nd some there be, which have no memorial; who are perished, as though they had never been; and are become as though they had never been born; and their children after them.com bur83155 fm i-xii.tex iii 01/13/2010 16:14 This page intentionally left blank www.com Revised Con rming Pages Contents Early Egyptian Multiplication 37 The Unit Fraction Table 40 Representing Rational Numbers 43 2.3 Four Problems from the Rhind Papyrus 46 The Method of False Position 46 A Curious Problem 49 Preface x–xii Egyptian Mathematics as Applied Arithmetic 50 2.4 Egyptian Geometry 53 Approximating the Area of a Circle 53 Chapter 1 The Volume of a Truncated Pyramid 56 Early Number Systems and Speculations About the Great Pyramid 57 Symbols 1 2.5 Babylonian Mathematics 62 A Tablet of Reciprocals 62 1.1 Primitive Counting 1 The Babylonian Treatment of Quadratic Equations 64 A Sense of Number 1 Two Characteristic Babylonian Problems 69 Notches as Tally Marks 2 2.6 Plimpton 322 72 The Peruvian Quipus: Knots as Numbers 6 A Tablet Concerning Number Triples 72 1.2 Number Recording of the Egyptians and Greeks 9 Babylonian Use of the Pythagorean Theorem 76 The History of Herodotus 9 The Cairo Mathematical Papyrus 77 Hieroglyphic Representation of Numbers 11 Egyptian Hieratic Numeration 15 Chapter 3 The Greek Alphabetic Numeral System 16 The Beginnings of Greek 1.3 Number Recording of the Babylonians 20 Mathematics 83 Babylonian Cuneiform Script 20 Deciphering Cuneiform: Grotefend and Rawlinson 21 3.1 The Geometrical Discoveries of Thales 83 The Babylonian Positional Number System 23 Greece and the Aegean Area 83 Writing in Ancient China 26 The Dawn of Demonstrative Geometry: Thales of Miletos 86 Chapter 2 Measurements Using Geometry 87 3.2 Pythagorean Mathematics 90 Mathematics in Early Pythagoras and His Followers 90 Civilizations 33 Nicomachus’s Introductio Arithmeticae 94 2.1 The Rhind Papyrus 33 The Theory of Figurative Numbers 97 Egyptian Mathematical Papyri 33 Zeno’s Paradox 101 A Key to Deciphering: The Rosetta Stone 35 3.3 The Pythagorean Problem 105 2.2 Egyptian Arithmetic 37 Geometric Proofs of the Pythagorean Theorem 105 v www.com bur83155 fm i-xii.tex v 01/13/2010 16:43 Revised Con rming Pages vi Contents Early Solutions of the Pythagorean Equation 107 The Almagest of Claudius Ptolemy 188 The Crisis of Incommensurable Quantities 109 Ptolemy’s Geographical Dictionary 190 Theon’s Side and Diagonal Numbers 111 4.5 Archimedes 193 Eudoxus of Cnidos 116 The Ancient World’s Genius 193 3.4 Three Construction Problems of Antiquity 120 Estimating the Value of ³ 197 Hippocrates and the Quadrature of the Circle 120 The Sand-Reckoner 202 The Duplication of the Cube 124 Quadrature of a Parabolic Segment 205 The Trisection of an Angle 126 Apollonius of Perga: The Conics 206 3.5 The Quadratrix of Hippias 130 Rise of the Sophists 130 Chapter 5 Hippias of Elis 131 The Twilight of Greek The Grove of Academia: Plato’s Academy 134 Mathematics: Diophantus 213 Chapter 4 5.1 The Decline of Alexandrian Mathematics 213 The Waning of the Golden Age 213 The Alexandrian School: The Spread of Christianity 215 Euclid 141 Constantinople, A Refuge for Greek Learning 217 4.1 Euclid and the Elements 141 5.2 The Arithmetica 217 A Center of Learning: The Museum 141 Diophantus’s Number Theory 217 Euclid’s Life and Writings 143 Problems from the Arithmetica 220 4.3 Diophantine Equations in Greece, India, Euclid’s Foundation for Geometry 144 and China 223 Postulates 146 The Cattle Problem of Archimedes 223 Common Notions 146 Early Mathematics in India 225 Book I of the Elements 148 The Chinese Hundred Fowls Problem 228 Euclid’s Proof of the Pythagorean Theorem 156 5.4 The Later Commentators 232 Book II on Geometric Algebra 159 The Mathematical Collection of Pappus 232 Construction of the Regular Pentagon 165 Hypatia, the First Woman Mathematician 233 4.3 Euclid’s Number Theory 170 Roman Mathematics: Boethius and Cassiodorus 235 Euclidean Divisibility Properties 170 5.5 Mathematics in the Near and Far East 238 The Algorithm of Euclid 173 The Algebra of al-Khowârizmı̂ 238 The Fundamental Theorem of Arithmetic 177 Abû Kâmil and Thâbit ibn Qurra 242 An Infinity of Primes 180 Omar Khayyam 247 4.4 Eratosthenes, the Wise Man of Alexandria 183 The Astronomers al-Tûsı̂ and al-Kashı̂ 249 The Sieve of Eratosthenes 183 The Ancient Chinese Nine Chapters 251 Measurement of the Earth 186 Later Chinese Mathematical Works 259 www.com bur83155 fm i-xii.tex vi 01/13/2010 16:43 Revised Con rming Pages Contents vii Chapter 6 Cardan’s Solution of the Cubic Equation 320 Bombelli and Imaginary Roots of the Cubic 324 The First Awakening: 7.4 Ferrari’s Solution of the Quartic Equation 328 Fibonacci 269 The Resolvant Cubic 328 6.1 The Decline and Revival of Learning 269 The Story of the Quintic Equation: The Carolingian Pre-Renaissance 269 Ruffini, Abel, and Galois 331 Transmission of Arabic Learning to the West 272 Chapter 8 The Pioneer Translators: Gerard and Adelard 274 6.2 The Liber Abaci and Liber Quadratorum 277 The Mechanical World: The Hindu-Arabic Numerals 277 Descartes and Newton 337 Fibonacci’s Liber Quadratorum 280 8.1 The Dawn of Modern Mathematics 337 The Works of Jordanus de Nemore 283 The Seventeenth Century Spread of Knowledge 337 6.3 The Fibonacci Sequence 287 Galileo’s Telescopic Observations 339 The Liber Abaci’s Rabbit Problem 287 The Beginning of Modern Notation: Some Properties of Fibonacci Numbers 289 François Vièta 345 6.4 Fibonacci and the Pythagorean Problem 293 The Decimal Fractions of Simon Stevin 348 Pythagorean Number Triples 293 Napier’s Invention of Logarithms 350 Fibonacci’s Tournament Problem 297 The Astronomical Discoveries of Brahe and Kepler 355 Chapter 7 8.2 Descartes: The Discours de la Méthode 362 The Writings of Descartes 362 The Renaissance of Mathematics: Inventing Cartesian Geometry 367 Cardan and Tartaglia 301 The Algebraic Aspect of La Géométrie 372 7.1 Europe in the Fourteenth and Fifteenth Descartes’s Principia Philosophiae 375 Centuries 301 Perspective Geometry: Desargues and Poncelet 377 The Italian Renaissance 301 8.3 Newton: The Principia Mathematica 381 Artificial Writing: The Invention of Printing 303 The Textbooks of Oughtred and Harriot 381 Founding of the Great Universities 306 Wallis’s Arithmetica Infinitorum 383 A Thirst for Classical Learning 310 The Lucasian Professorship: Barrow and Newton 386 7.2 The Battle of the Scholars 312 Newton’s Golden Years 392 Restoring the Algebraic Tradition: Robert Recorde 312 The Laws of Motion 398 The Italian Algebraists: Pacioli, del Ferro, and Later Years: Appointment to the Mint 404 Tartaglia 315 8.4 Gottfried Leibniz: The Calculus Controversy 409 Cardan, A Scoundrel Mathematician 319 The Early Work of Leibniz 409 7.3 Cardan’s Ars Magna 320 Leibniz’s Creation of the Calculus 413 www.com bur83155 fm i-xii.tex vii 01/13/2010 16:43 Revised Con rming Pages viii Contents Newton’s Fluxional Calculus 416 Scientific Societies 497 The Dispute over Priority 424 Marin Mersenne’s Mathematical Gathering 499 Maria Agnesi and Emilie du Châtelet 430 Numbers, Perfect and Not So Perfect 502 10.2 From Fermat to Euler 511 Chapter 9 Fermat’s Arithmetica 511 The Development of Probability The Famous Last Theorem of Fermat 516 Theory: Pascal, Bernoulli, The Eighteenth-Century Enlightenment 520 and Laplace 439 Maclaurin’s Treatise on Fluxions 524 Euler’s Life and Contributions 527 9.1 The Origins of Probability Theory 439 10.3 The Prince of Mathematicians: Carl Graunt’s Bills of Mortality 439 Friedrich Gauss 539 Games of Chance: Dice and Cards 443 The Period of the French Revolution: The Precocity of the Young Pascal 446 Lagrange, Monge, and Carnot 539 Pascal and the Cycloid 452 Gauss’s Disquisitiones Arithmeticae 546 De Méré’s Problem of Points 454 The Legacy of Gauss: Congruence Theory 551 9.2 Pascal’s Arithmetic Triangle 456 Dirichlet and Jacobi 558 The Traité du Triangle Arithmétique 456 Mathematical Induction 461 Chapter 11 Francesco Maurolico’s Use of Induction 463 9.3 The Bernoullis and Laplace 468 Nineteenth-Century Christiaan Huygens’s Pamphlet on Probability 468 Contributions: Lobachevsky to The Bernoulli Brothers: John and James 471 Hilbert 563 De Moivre’s Doctrine of Chances 477 11.1 Attempts to Prove the Parallel Postulate 563 The Mathematics of Celestial Phenomena: The Efforts of Proclus, Playfair, and Wallis 563 Laplace 478 Saccheri Quadrilaterals 566 Mary Fairfax Somerville 482 The Accomplishments of Legendre 571 Laplace’s Research in Probability Theory 483 Legendre’s Eléments de géométrie 574 Daniel Bernoulli, Poisson, and Chebyshev 489 11.2 The Founders of Non-Euclidean Geometry 584 Gauss’s Attempt at a New Geometry 584 Chapter 10 The Struggle of John Bolyai 588 Creation of Non-Euclidean Geometry: Lobachevsky 592 The Revival of Number Theory: Models of the New Geometry: Riemann, Fermat, Euler, and Gauss 497 Beltrami, and Klein 598 10.1 Marin Mersenne and the Search Grace Chisholm Young 603 for Perfect Numbers 497 11.3 The Age of Rigor 604 www.com bur83155 fm i-xii.tex viii 01/13/2010 16:43 Revised Con rming Pages Contents ix D’Alembert and Cauchy on Limits 604 Zermelo and the Axiom of Choice 701 Fourier’s Series 610 The Logistic School: Frege, Peano, and Russell 704 The Father of Modern Analysis, Weierstrass 614 Hilbert’s Formalistic Approach 708 Sonya Kovalevsky 616 Brouwer’s Institutionism 711 The Axiomatic Movement: Pasch and Hilbert 619 11.4 Arithmetic Generalized 626 Chapter 13 Babbage and the Analytical Engine 626 Extensions and Generalizations: Peacock’s Treatise on Algebra 629 Hardy, Hausdorff, The Representation of Complex Numbers 630 and Noether 721 Hamilton’s Discovery of Quaternions 633 Matrix Algebra: Cayley and Sylvester 639 13.1 Hardy and Ramanujan 721 Boole’s Algebra of Logic 646 The Tripos Examination 721 The Rejuvenation of English Mathematics 722 Chapter 12 A Unique Collaboration: Hardy and Littlewood 725 India’s Prodigy, Ramanujan 726 Transition to the Twentieth 13.2 The Beginnings of Point-Set Topology 729 Century: Cantor and Frechet’s Metric Spaces 729 Kronecker 657 The Neighborhood Spaces of Hausdorff 731 12.1 The Emergence of American Mathematics 657 Banach and Normed Linear Spaces 733 Ascendency of the German Universities 657 13.3 Some Twentieth-Century Developments 735 American Mathematics Takes Root: 1800–1900 659 Emmy Noether’s Theory of Rings 735 The Twentieth-Century Consolidation 669 Von Neumann and the Computer 741 12.2 Counting the Infinite 673 Women in Modern Mathematics 744 The Last Universalist: Poincaré 673 A Few Recent Advances 747 Cantor’s Theory of Infinite Sets 676 Kronecker’s View of Set Theory 681 Countable and Uncountable Sets 684 General Bibliography 755 Transcendental Numbers 689 Additional Reading 759 The Continuum Hypothesis 694 The Greek Alphabet 761 12.