The History of Mathematics The History of Mathematics A Brief Course Second Edition Roger Cooke University of Vermont @ WILEY-INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Copyright @ 2005 by John Wiley & Sons, Inc. AH 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 transmitted 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 Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc.. 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. 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For general information on our other products and services please contact our Customer Care Depart- ment within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Cooke, Roger, 1942 The history of mathematics: a brief course/Roger Cooke 2nd ed. p.cm. Includes bibliographical references and indexes. ISBN 978-0-471-44459-6 1. Mathematics-History. I. Title QA 21.C649 2005 510'.9-dc22 2004042299 Contents Preface xv Part 1. The World of Mathematics and the Mathematics of the World 1 Chapter 1. The Origin and Prehistory of Mathematics 3 1. Numbers 4 1.1. Animals'use of numbers 5 1.2. Young children's use of numbers 5 1.3. Archaeological evidence of counting 6 2. Continuous magnitudes 6 2.1. Perception of shape by animals 7 2.2. Children's concepts of space 8 2.3. Geometry in arts and crafts 9 3. Symbols 9 4. Mathematical inference 12 4.1. Visual reasoning 12 4.2. Chance and probability 13 Questions and problems 14 Chapter 2. Mathematical Cultures I 19 1. The motives for creating mathematics 19 1.1. Pure versus applied mathematics 19 2. India 21 2.1. The Sulva Sutras 22 2.2. Buddhist and Jaina mathematics 23 2.3. The Bakshali Manuscript 23 2.4. The siddhantas 23 2.5. Aryabhata I 24 2.6. Brahmagupta 25 2.7. Bhaskara II 25 2.8. Muslim India 26 2.9. Indian mathematics in the colonial period and after 26 3. China 27 3.1. Works and authors 29 3.2. China's encounter with Western mathematics 32 4. Ancient Egypt 34 5. Mesopotamia 35 6. The Maya 37 6.1. The Dresden Codex 37 vi CONTENTS Questions and problems 38 Chapter 3. Mathematical Cultures II 41 1. Greek and Roman mathematics 41 1.1. Sources 42 1.2. General features of Greek mathematics 43 1.3. Works and authors 45 2. Japan 50 2.1. Chinese influence and calculating devices 51 2.2. Japanese mathematicians and their works 51 3. The Muslims 54 3.1. Islamic science in general 54 3.2. Some Muslim mathematicians and their works 56 4. Europe 58 4.1. Monasteries, schools, and universities 58 4.2. The high Middle Ages 59 4.3. Authors and works 59 5. North America 62 5.1. The United States and Canada before 1867 63 5.2. The Canadian Federation and post Civil War United States 66 5.3. Mexico 69 6. Australia and New Zealand 70 6.1. Colonial mathematics 70 7. The modern era 72 7.1. Educational institutions 72 7.2. Mathematical societies 73 7.3. Journals 73 Questions and problems 73 Chapter 4. Women Mathematicians 75 1. Individual achievements and obstacles to achievement 76 1.1. Obstacles to mathematical careers for women 76 2. Ancient women mathematicians 80 3. Modern European women 81 3.1. Continental mathematicians 82 3.2. Nineteenth-century British women 85 3.3. Four modern pioneers 88 4. American women 100 5. The situation today 104 Questions and problems 105 Part 2. Numbers 109 Chapter 5. Counting 111 1. Number words 111 2. Bases for counting 113 2.1. Decimal systems 113 2.2. Nondecimal systems 114 3. Counting around the world 116 3.1. Egypt 116 CONTENTS vii 3.2. Mesopotamia 116 3.3. India 118 3.4. China 118 3.5. Greece and Rome 119 3.6. The Maya 121 4. What was counted? 122 4.1. Calendars 122 4.2. Weeks 125 Questions and problems 127 Chapter 6. Calculation 129 1. Egypt 129 1.1. Multiplication and division 130 1.2. "Parts" 131 1.3. Practical problems 134 2. China 135 2.1. Fractions and roots 136 2.2. The Jiu Zhang Suanshu 138 3. India 139 4. Mesopotamia 140 5. The ancient Greeks 142 6. The Islamic world 143 7. Europe 143 8. The value of calculation 145 9. Mechanical methods of computation 146 9.1. Software: prosthaphaeresis and logarithms 146 9.2. Hardware: slide rules and calculating machines 149 9.3. The effects of computing power 153 Questions and problems 154 Chapter 7. Ancient Number Theory 159 1. Plimpton 322 159 2. Ancient Greek number theory 164 2.1. The Arithmetica of Nicomachus 165 2.2. Euclid's number theory 168 2.3. The Arithmetica of Diophantus 170 3. China 172 4. India 175 4.1. Varahamihira's mystical square 175 4.2. Aryabhata I 175 4.3. Brahmagupta 175 4.4. Bhaskara II 178 5. The Muslims 179 6. Japan 180 7. Medieval Europe 181 Questions and problems 182 Chapter 8. Numbers and Number Theory in Modern Mathematics 187 1. Modern number theory 187 1.1. Fermat 187 viii CONTENTS 1.2. Euler 188 1.3. Lagrange 190 1.4. Legendre 191 1.5. Gauss 192 1.6. Dirichlet 193 1.7. Riemann 194 1.8. Fermat's last theorem 195 1.9. The prime number theorem 196 2. Number systems 197 2.1. Negative numbers and zero 197 2.2. Irrational and imaginary numbers 199 2.3. Imaginary and complex numbers 206 2.4. Infinite numbers 209 3. Combinatorics 210 3.1. Summation rules 210 Questions and problems 217 Part 3. Color Plates 221 Part 4. Space 231 Chapter 9. Measurement 233 1. Egypt 234 1.1. Areas 235 1.2. Volumes 239 2. Mesopotamia 241 2.1. The Pythagorean theorem 242 2.2. Plane figures 243 2.3. Volumes 244 3. China 244 3.1. The Zhou Bi Suan Jing 244 3.2. The Jiu Zhang Suanshu 247 3.3. The Sun Zi Suan Jing 248 3.4. Liu Hui 249 3.5. Zu Chongzhi . 250 4. Japan 252 4.1. The challenge problems 252 4.2. Beginnings of the calculus in Japan 253 5. India 257 5.1. Aryabhata I 257 5.2. Brahmagupta 262 Questions and problems 264 Chapter 10. Euclidean Geometry 269 1. The earliest Greek geometry 269 1.1. Thales 270 1.2. Pythagoras and the Pythagoreans 271 1.3. Pythagorean geometry 272 1.4. Challenges to Pythagoreanism: unsolved problems 274 CONTENTS ix 1.5. Challenges to Pythagoreanism: the paradoxes of Zeno of Elea 283 1.6. Challenges to Pythagoreanism: incommensurables 284 1.7. The influence of Plato 285 1.8. Eudoxan geometry 287 1.9. Aristotle 293 2. Euclid 296 2.1. The Elements 296 2.2. The Data 299 3. Archimedes 299 3.1. The area of a sphere 301 3.2. The Method 302 4. Apollonius 304 4.1. History of the Conies 305 4.2. Contents of the Conies 305 4.3. Apollonius' definition of the conic sections 306 4.4. Foci and the three- and four-line locus 308 Questions and problems 310 Chapter 11. Post-Euclidean Geometry 317 1. Hellenistic geometry 318 1.1. Zenodorus 318 1.2. The parallel postulate 319 1.3. Heron 320 1.4. Pappus 322 2. Roman geometry 325 2.1. Roman civil engineering 327 3. Medieval geometry 328 3.1. Late Medieval and Renaissance geometry 330 4. Geometry in the Muslim world 332 4.1. The parallel postulate 333 4.2. Thabit ibn-Qurra 333 4.3. Al-Kuhi 335 4.4. Al-Haytham 335 4.5. Omar Khayyam 336 4.6. Nasir al-Din al-Tusi 337 5. Non-Euclidean geometry 338 5.1. Girolamo Saccheri 339 5.2. Lambert and Legendre 341 5.3. Gauss 342 5.4. Lobachevskii and Janos Bolyai 343 5.5. The reception of non-Euclidean geometry 346 5.6. Foundations of geometry 348 6. Questions and problems 348 Chapter 12. Modern Geometries 351 1. Analytic and algebraic geometry 351 1.1. Fermat 351 1.2. Descartes 352 1.3. Newton's classification of curves 355