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THE BIRTH OF MATHEMATICS Ancient Times to 1300 Michael J. Bradley, Ph.D. The Birth of Mathematics: Ancient Times to 1300 Copyright © 2006 by Michael J. Bradley, Ph.D. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval systems, without permission in writing from the publisher. For information contact: Chelsea House An imprint of Infobase Publishing 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Bradley, Michael J. The birth of mathematics : ancient times to 1300 / Michael J. Bradley. p. cm.—(Pioneers in mathematics) Includes bibliographical references and index. ISBN 0-8160-5423-1 (acid-free paper) 1. Mathematicians—Biography. 2. Mathematics, Ancient. 3. Mathematics— History—To 1300. I. Title. QA28.B73 2006 510.9—dc22 2005030563 Chelsea House books are available at special discounts when purchased in bulk quantities for businesses, associations, institutions, or sales promotions. Please call our Special Sales Department in New York at (212) 967-8800 or (800) 322-8755. You can find Chelsea House on the World Wide Web at http://www.chelseahouse.com Text design by Mary Susan Ryan-Flynn Cover design by Dorothy Preston Illustrations by Dale Williams Printed in the United States of America MP FOF 10 9 8 7 6 5 4 3 2 1 This book is printed on acid-free paper. CONTENTS Preface Acknowledgments Introduction CHAPTER 1 Thales of Miletus (ca. 625–ca. 547 B.C.E.): Earliest Proofs of Geometrical Theorems Early Years Natural Philosopher First Proofs of Theorems in Mathematics Discoveries in Astronomy Ingenious Solutions to Practical Problems Legends about Thales Conclusion Further Reading CHAPTER vii ix xi 1 2 2 3 6 8 10 11 12 2 Pythagoras of Samos (ca. 560–ca. 480 B.C.E.): Ancient Greek Proves Theorem about Right Triangles First Student is Paid to Learn Pythagorean Society Blends Mysticism with Mathematics Early Research in Number Theory Ratios in Music and Astronomy Pythagorean Theorem Irrational Numbers 14 15 16 17 20 21 23 Five Regular Solids Conclusion Further Reading CHAPTER 3 Euclid of Alexandria (ca. 325–ca. 270 B.C.E.): Geometer Who Organized Mathematics Professor of Mathematics Elements Original Results in Elements Criticisms of Euclid’s Methods Parallel Postulate Euclid’s Additional Writings Conclusion Further Reading CHAPTER 29 30 31 33 35 36 38 40 41 4 Archimedes of Syracuse (ca. 287–212 B.C.E.): Innovator of Techniques in Geometry Inventor of Practical Machines Approximation of Pi Using Inscribed and Circumscribed Polygons Method of Exhaustion to Estimate Areas and Volumes Creative Problem-Solver Investigations of Large Numbers Conclusion Further Reading CHAPTER 25 27 28 43 44 46 48 51 53 55 56 5 Hypatia of Alexandria (ca. 370–415 C.E.): First Woman of Mathematics The “Perfect” Human Being Commentaries on Classical Mathematics Books Famous Teacher, Philosopher, and Scientist Brutally Murdered 57 58 59 62 63 Conclusion Further Reading CHAPTER 6 A¯ ryabhata I (476–550 C.E.): From Alphabetical Numbers to the Rotation of the Earth ¯ ¯ Aryabhatı ¯ya (Aryabhata’s Treatise) Arithmetical Methods Geometric Techniques Tables of Sines Algebraic Advances Astronomical Theories Second Astronomical Treatise Conclusion Further Reading CHAPTER 65 65 67 68 69 71 73 75 75 77 77 78 7 Brahmagupta (598–668 C.E.): Father of Numerical Analysis 79 Br¯ahmasphutasiddh¯anta (Improved Astronomical System of Br¯ahma) Arithmetical Innovations New Geometrical Techniques Algebrai