geometry Revised Edition Math Chapter Title iii THE HISTORY OF geometry the language of space and form Revised Edition John Tabak, Ph.D. GEOMETRY: The Language of Space and Form, Revised Edition Copyright © 2011, 2004 by John Tabak, 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: Facts On File, Inc. An imprint of Infobase Learning 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tabak, John. Geometry, revised edition : the language of space and form / John Tabak.—Rev. ed. p. cm.—(The history of mathematics) Rev. ed. of Geometry: the language of space and form, c 2004. Includes bibliographical references and index. ISBN 978-0-8160-7942-1 ISBN 978-1-4381-3559-5 (e-book) 1. Geometry—History. I. Tabak, John. Geometry. II. Title. QA443.5.T33 2011 516.009—dc22 2010018627 Facts On File 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 Facts On File on the World Wide Web at http://www.infobaselearning.com Excerpts included herewith have been reprinted by permission of the copyright hold- ers; the author has made every effort to contact copyright holders. The publisher will be glad to rectify, in future editions, any errors or omissions brought to its notice. Text design by David Strelecky Composition by Hermitage Publishing Services Illustrations by Dale Williams Photo research by Elizabeth H. Oakes Cover printed by Yurchak Printing, Inc., Landisville, Pa. Book printed and bound by Yurchak Printing, Inc., Landisville, Pa. Date printed: May 2011 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 This book is printed on acid-free paper. To Oliver Clifton Blaylock, friend and mentor CONTENTS Preface x Acknowledgments xvi Introduction xvii Part One: Geometry in Antiquity 1 1 Geometry before the Greeks 3 2 Early Greek Geometry 8 Math without Numbers 10 The Pythagoreans 13 The Golden Section 15 Geometry in Athens 17 3 Major Mathematical Works of Greek Geometry 22 Elements by Euclid of Alexandria 22 Euclid Reexamined 27 The Method, On the Sphere and Cylinder, and Other Works by Archimedes 29 Conics by Apollonius of Perga 34 Investigating Conic Sections 38 Collection by Pappus of Alexandria 39 The End of the Greek Mathematical Tradition 44 Part Two: New Geometries 47 4 Mathematics and Art during the Renaissance 49 Leonardo da Vinci 53 Albrecht Dürer 57 5 The First Theorems 62 Marin Mersenne 69 6 Projective Geometry Rediscovered 70 Monge’s Students 72 Projective Geometry as a Mature Branch of Mathematics 79 Groups and Geometry 81 Projective Geometry, an Application 84 7 A Non-Euclidean Geometry 88 Finite Geometries 93 Is Our World Euclidean? 96 Part Three: Coordinate Geometry 99 8 The Beginnings of Analytic Geometry 101 Menaechmus and Apollonius of Perga 105 René Descartes 107 Algebraic Notation in Geometry 112 Pierre de Fermat 114 The Pythagorean Theorem and Cartesian Coordinates 117 9 Calculus and Analytic Geometry 119 Isaac Newton, the New Geometry, and the Old 122 Bipolar Coordinates 127 Leonhard Euler and Solid Geometry 128 10 Differential Geometry 137 Georg Friedrich Bernhard Riemann 143 Geometry on Spheres 148 11 The Shape of Space and Time 152 Geometry and the Special Theory of Relativity 158 The Pythagorean Theorem and Special Relativity 161 The Geometry and Science of “Ordinary” Surfaces 162 Emmy Noether and Symmetry 166 12 Infinite-Dimensional Geometries 174 David Hilbert 174 Stefan Banach 178 The Banach Tarski Paradox 183 Conclusion 190 Afterword: Interview with Professor Krystyna Kuperberg on Geometry, Intuition, and History 192 Chronology 203 Glossary 210 Further Resources 225 Index 235 PREFACE Of all human activities, mathematics is one of the oldest. Mathematics can be found on the cuneiform tablets of the Mesopotamians, on the papyri of the Egyptians, and in texts from ancient China, the Indian subcontinent, and the indigenous cul- tures of Central America. Sophisticated mathematical research was carried out in the Middle East for several centuries after the birth of Muhammad, and advanced mathematics has been a hallmark of European culture since the Renaissance. Today, mathematical research is carried out across the world, and it is a remarkable fact that there is no end in sight. The more we learn of mathematics, the faster the pace of discovery. Contemporary mathematics is often extremely abstract, and the important questions with which mathematicians concern them- selves can sometimes be difficult to describe to the interested nonspecialist. Perhaps this is one reason that so many histories of mathematics give so little attention to the last 100 years of dis- covery—this, despite the fact that the last 100 years have probably been the most productive period in the history of mathematics. One unique feature of this six-volume History of Mathematics is that it covers a significant portion of recent mathematical history as well as the origins. And with the help of in-depth interviews with prominent mathematicians—one for each volume—it is hoped that the reader will develop an appreciation for current work in mathematics as well as an interest in the future of this remarkable subject. Numbers details the evolution of the concept of number from the simplest counting schemes to the discovery of uncomputable numbers in the latter half of the 20th century. Divided into three parts, this volume first treats numbers from the point of view of computation. The second part details the evolution of the concept of number, a process that took thousands of years and culminated in what every student recognizes as “the real number line,” an x Preface xi extremely important and subtle mathematical idea. The third part of this volume concerns the evolution of the concept of the infi- nite. In particular, it covers Georg Cantor’s discovery (or creation, depending on one’s point of view) of transfinite numbers and his efforts to place set theory at the heart of modern mathematics. The most important ramifications of Cantor’s work, the attempt to axi- omatize mathematics carried out by David Hilbert and Bertrand Russell, and the discovery by Kurt Gödel and Alan Turing that there are limitations on what can be learned from the axiomatic method, are also described. The last chapter ends with the discov- ery of uncomputable numbers, a remarkable consequence of the work of Kurt Gödel and Alan Turing. The book concludes with an interview with Professor Karlis Podnieks, a mathematician of remarkable insights and a broad array of interests. Probability and Statistics describes subjects that have become cen- tral to modern thought. Statistics now lies at the heart of the way that most information is communicated and interpreted. Much of our understanding of economics, science, marketing, and a host of other subjects is expressed in the language of statistics. And for many of us statistical language has become part of everyday dis- course. Similarly, probability theory is used to predict everything from the weather to the success of space missions to the value of mortgage-backed securities. The first half of the volume treats probability beginning with the earliest ideas about chance and the foundational work of Blaise Pascal and Pierre Fermat. In addition to the development of the mathematics of probability, considerable attention is given to the application of probability theory to the study of smallpox and the misapplication of probability to modern finance. More than most branches of mathematics, probability is an applied dis- cipline, and its uses and misuses are important to us all. Statistics is the subject of the second half of the book. Beginning with the earliest examples of statistical thought, which are found in the writings of John Graunt and Edmund Halley, the volume gives special attention to two pioneers of statistical thinking, Karl Pearson and R. A. Fisher, and it describes some especially impor- tant uses and misuses of statistics, including the use of statistics xii GEOMETRY in the field of public health, an application of vital interest. The book concludes with an interview with Dr. Michael Stamatelatos, director of the Safety and Assurance Requirements Division in the Office of Safety and Mission Assurance at NASA, on the ways that probability theory, specifically the methodology of probabi- listic risk assessment, is used to assess risk and improve reliability. Geometry discusses one of the oldest of all branches of mathe- matics. Special attention is given to Greek geometry, which set the standard both for mathematical creativity and rigor for many cen- turies. So important was Euclidean geometry that it was not until the 19th century that mathematicians became willing to consider the existence of alternative and equally valid geometrical systems. This 19th-century revolution in mathematical, philosophical, and scientific thought is described in some detail, as are some alter- natives to Euclidean geometry, including projective geometry, the non-Euclidean geometry of Nikolay Ivanovich Lobachevsky and János Bolyai, the higher (but finite) dimensional geometry of Riemann, infinite-dimensional geometric ideas, and some of the geometrical implications of the theory of relativity. The volume concludes with an interview with Professor Krystyna Kuperberg of Auburn University about her work in geometry and dynamical systems, a branch of mathematics heavily dependent on ideas from geometry. A successful and highly insightful mathematician, she also discusses the role of intuition in her research. Mathematics is also the language of science, and mathematical methods are an important tool of discovery for scientists in many disciplines.