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THE AGE OF GENIUS 1300 to 1800 Michael J. Bradley, Ph.D. The Age of Genius: 1300 to 1800 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. (Michael John), 1956– The age of genius : 1300 to 1800 / Michael J. Bradley. p. cm.—(Pioneers in mathematics) Includes bibliographical references and index. ISBN 0-8160-5424-X 1. Mathematicians—Biography. 2. Mathematics, Medieval. I. Title. QA28.B728 2006 510.92'2—dc22 2005032354 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 vii Acknowledgments ix Introduction xi 1 CHAPTER Ghiya¯th al-D¯ın Jamsh¯ıd Mas’u¯d al-Ka¯sh¯ı (ca. 1380–1429): Accurate Decimal Approximations 1 Early Astronomical Writings 2 Determining the Value of π 5 Roots, Decimals, and Domes 8 Estimating sin(1°) 10 Minor Works 12 Conclusion 13 Further Reading 13 2 CHAPTER François Viète (1540–1603): Father of Modern Algebra 15 Lawyer, Tutor, Government Official, and Code-Breaker 16 Early Writings on Mathematics and Science 17 Modern Algebra Introduced as the Analytic Art 18 Theory of Equations Provides Diverse Methods of Solution 21 Further Advances in Geometry, Trigonometry, and Algebra 23 Conclusion 28 Further Reading 29 3 CHAPTER John Napier (1550–1617): Coinventor of Logarithms 31 Inventor and Theologian 32 Rumored to Be a Magician 34 Napier’s Bones Aid in Multiplication 35 Logarithms Simplify Computation 36 Logarithms Receive International Acclaim 39 Additional Mathematical Contributions 41 Conclusion 42 Further Reading 43 4 CHAPTER Pierre de Fermat (1601–1665): Father of Modern Number Theory 45 Professional Life Leaves Time for Mathematical Investigations 46 Origins of Analytic Geometry 47 Essential Ideas in Calculus 49 Fundamentals of the Theory of Probability 52 Questions about Primes and Divisibility Define Modern Number Theory 52 Writing Numbers as Sums of Powers 54 Conclusion 57 Further Reading 57 5 CHAPTER Blaise Pascal (1623–1662): Coinventor of Probability Theory 59 Discoveries in Projective Geometry 60 Calculating Machine Performs Addition and Subtraction 62 Experiments on Vacuums and Air Pressure 63 Foundations of Probability Theory and the Arithmetic Triangle 65 Study of the Cycloid Reinvigorates Pascal’s Interest in Mathematics 67 Conclusion 69 Further Reading 70 6 CHAPTER Sir Isaac Newton (1642–1727): Calculus, Optics, and Gravity 71 Education and Early Life 72 Infinite Series and the General Binomial Theorem 73 Method of Fluxions Introduces the Formal Theory of Calculus 75 Additional Mathematical Treatises 79 A New Theory of Light 80 Laws of Motion and the Principle of Universal Gravitation 84 Activities beyond Mathematics and Physics 86 Conclusion 87 Further Reading 87 7 CHAPTER Gottfried Leibniz (1646–1716): Coinventor of Calculus 89 Family and Education 90 Service to Royal Patrons 91 General Theory of Calculus 93 Additional Mathematical Discoveries 97 Philosophy, Dynamics, and Theology 100 Conclusion 101 Further Reading 101 8 CHAPTER Leonhard Euler (1707–1783): Leading Mathematician of the 18th Century 103 Student Years, 1707–1726 104 Early Years at St. Petersburg Academy, 1727–1741 105 Middle Years at Berlin Academy, 1741–1766 110 Return to St. Petersburg Academy, 1766–1783 113 Conclusion 114 Further Reading 115 9 CHAPTER Maria Agnesi (1718–1799): Mathematical Linguist 117 Early Family Life 118 Instituzioni Analitiche (Analytical Institutions) 119 Reactions to the Book 121 “The Witch of Agnesi” 122 Second Career after Mathematics 124 Conclusion 125 Further Reading 125 10 CHAPTER Benjamin Banneker (1731–1806): Early African-American Scientist 127 Tobacco Farmer 128 Wooden Clock 128 Diverse Interests 129 Astronomer 130 Surveying the District of Columbia 132 1792 Almanac 134 Professional Almanac-Maker 135 Honors and Memorials 138 Conclusion 139 Further Reading 140 Glossary 141 Further Reading 153 Associations 157 Index 158 PREFACE M athematics is a human endeavor. Behind its numbers, equations, formulas, and theorems are the stories of the people who expanded the frontiers of humanity's mathematical knowledge. Some were child prodigies while others developed their aptitudes for mathematics later in life. They were rich and poor, male and female, well educated and self-taught. They worked as professors, clerks, farmers, engineers, astronomers, nurses, and philosophers. The diversity of their backgrounds testifies that mathematical tal- ent is independent of nationality, ethnicity, religion, class, gender, or disability. Pioneers in Mathematics is a five-volume set that profiles the lives of 50 individuals, each of whom played a role in the development and the advancement of mathematics. The overall profiles do not represent the 50 most notable mathematicians; rather, they are a collection of individuals whose life stories and significant contributions to mathematics will interest and inform middle school and high school students. Collectively, they represent the diverse talents of the millions of people, both anonymous and well known, who developed new techniques, discovered innovative ideas, and extended known mathematical theories while facing challenges and overcoming obstacles. Each book in the set presents the lives and accomplishments of 10 mathematicians who lived during an historical period. The Birth of Mathematics profiles individuals from ancient Greece, India, Arabia, and medieval Italy who lived from 700 b.c.e. to 1300 c.e. The Age of Genius features mathematicians from Iran, France, England, Germany, Switzerland, and America who lived between vii viii The Age of Genius the 14th and 18th centuries. The Foundations of Mathematics presents 19th-century mathematicians from various European countries. Modern Mathematics and Mathematics Frontiers profile a variety of international mathematicians who worked in the early 20th and the late 20th century, respectively. The 50 chapters of Pioneers in Mathematics tell pieces of the story of humankind's attempt to understand the world in terms of numbers, patterns, and equations. Some of the individuals profiled contributed innovative ideas that gave birth to new branches of mathematics. Others solved problems that had puzzled mathematicians for centuries. Some wrote books that influenced the teaching of mathematics for hundreds of years. Still others were among the first of their race, gender, or nationality to achieve recognition for their mathematical accomplishments. Each one was an innovator who broke new ground and enabled their successors to progress even further. From the introduction of the base-10 number system to the development of logarithms, calculus, and computers, most significant ideas in mathematics developed gradually, with countless individuals making important contributions. Many mathematical ideas developed independently in different civilizations separated by geography and time. Within the same civilization, the name of the scholar who developed a particular innovation often became lost as his or her idea was incorporated into the writings of a later mathematician. For these reasons, it is not always possible to identify accurately any one individual as the first person to have discovered a particular theorem or to have introduced a certain idea. But then mathematics was not created by one person or for one person; it is a human endeavor. ACKNOWLEDGMENTS A n author does not write in isolation. I owe a debt of thanks to so many people who helped in a myriad of ways during the creation of this work: To Jim Tanton, who introduced me to this fascinating project. To Jodie Rhodes, my agent, who put me in touch with Facts On File and handled the contractual paperwork. To Frank K. Darmstadt, my editor, who kept me on track throughout the course of this project. To Karen Harrington, who thoroughly researched the material for the chapter on Pierre de Fermat. To Warren Kay and Charles Kay, who generously allowed me to use a photograph of their collection of slide rules, and to Kevin Salemme, who took the photograph. To Larry Gillooly, George Heffernan, Sylvie Pressman, Suzanne Scholz, and Ernie Montella, who all assisted with the translations of Latin, Italian, French, and German titles. To Steve Scherwatzky, who helped me to become a better writer by critiquing early drafts of many chapters. To Melissa Cullen-DuPont, who provided valuable assistance with the artwork. To Amy L. Conver, for her copyediting. To my wife, Arleen, who provided constant love and support. To many relatives, colleagues, students, and friends, who inquired and really cared about my progress on this project. To Joyce Sullivan, Donna Katzman, and their students at Sacred Heart School in Lawrence, Massachusetts, who created poster pre- sentations for a math fair based on some of these chapters. ix