geometry the language of space and form THE HISTORY OF MATHEMATICS THE HISTORY OF geometry the language of space and form John Tabak, Ph.D. GEOMETRY: The Language of Space and Form Copyright © 2004 by John Tabak, Ph.D. Permissions appear after relevant quoted material. 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. 132 West 31st Street New York NY 10001 Library of Congress Cataloging-in-Publication Data Tabak, John. Geometry: the language of space and form/John Tabak. p. cm. — (History of mathematics) Includes bibliographical references and index. ISBN 0-8160-4953-X (hardcover) 1. Geometry—History. I. Title QA443.5.T33 2004 516—dc222003017340 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.factsonfile.com Text design by David Strelecky Cover design by Kelly Parr Illustrations by Patricia Meschino 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. To Oliver Clifton Blaylock, friend and mentor. CONTENTS Acknowledgments xi Introduction xiii Part One: Geometry in Antiquity 1 1 Geometry Before the Greeks 3 2 Early Greek Geometry 10 Math without Numbers 12 The Pythagoreans 15 The Golden Section 17 Geometry in Athens 19 3 Major Mathematical Works of 24 Greek Geometry Elements by Euclid of Alexandria 24 Euclid Reexamined 31 The Method, On The Sphere and Cylinder, and Other Works by Archimedes 32 Conics by Apollonius of Perga 37 Investigating Conic Sections 42 Collection by Pappus of Alexandria 43 The End of the Greek Mathematical Tradition 48 Part Two: Projective Geometry 51 4 Mathematics and Art during the Renaissance 52 Leonardo da Vinci 57 Albrecht Dürer 61 5 The First Theorems 66 Marin Mersenne 73 6 Projective Geometry Rediscovered 75 Monge’s Students 78 Projective Geometry as a Mature Branch of Mathematics 84 Projective Geometry Today 87 Groups and Geometry 88 7 A Non-Euclidean Geometry 92 Is Our World Euclidean? 98 Part Three: Coordinate Geometry 101 8 The Beginnings of Analytic Geometry 102 Menaechmus and Apollonius of Perga 105 René Descartes 108 Algebraic Notation in Geometry 112 Pierre de Fermat 115 The Pythagorean Theorem and Cartesian Coordinates 118 9 Calculus and Analytic Geometry 120 Isaac Newton, the New Geometry, and the Old 123 Bipolar Coordinates 128 Leonhard Euler and Solid Geometry 130 Gaspard Monge 138 10 Differential Geometry 140 Georg Friedrich Bernhard Riemann 145 11 The Shape of Space and Time 153 Geometry and the Special Theory of Relativity 156 The Pythagorean Theorem and Special Relativity 159 The Geometry and Science of “Ordinary” Surfaces 160 Emmy Noether and Symmetry 163 12 Infinite-Dimensional Geometry 170 Chronology 177 Glossary 195 Further Reading 201 Index 211 ACKNOWLEDGMENTS The author is deeply appreciative of Frank Darmstadt, Executive Editor, for his many helpful suggestions and of Dorothy Cummings, Project Editor, for the insightful work that she did editing this volume. Special thanks to Penelope Pillsbury and the staff of the Brownell Library, Essex Junction, Vermont, for their extraordi- nary help with the many difficult research questions that arose during the preparation of this book. xi INTRODUCTION What is geometry? What is it that we learn when we learn geometry? Points, lines, and surfaces are some of the things that mathe- maticians study when they study geometry, but human beings have always been interested in questions of line and form. Cave paint- ings from Lascaux, France, made during the last ice age show remarkably sophisticated pictures of wild animals. These beautiful images were created about 15,000 years ago. In the history of humankind they are almost unimaginably old. Humans were hunt- ing mammoths when these pictures were painted. It was the Stone Age. There was no written language anywhere in the world until thousands of years after these images were completed, and yet these cave paintings show that 15,000 years ago cave artists were wonderfully sensitive in their use of line and form. Does this mean that they knew geometry? And if they did know some geometry, what part of the mathematical subject of geometry did they know? For centuries European, Middle Eastern, and North African mathematicians believed that they knew the answer to the ques- tion of what constitutes geometry. For them the answer was easy. The term geometry meant the geometry of the ancient Greeks. This type of geometry is called Euclidean geometry after one of the best known of all Greek mathematicians, Euclid of Alexandria. The answer to the question, What is it that we learn when we learn geometry? was (to them) equally obvious: We learn the the- orems and proofs in Euclid’s most famous work, Elements. These early mathematicians did not question whether other geometries might exist. It was their belief that most of the geometry that could be learned had already been learned elsewhere at an earlier time and that all that was left for them to do were to master the geometry of the Greeks and to clarify what they saw as those few points remaining in Greek geometry that still needed clarification. xiii xiv GEOMETRY Euclidean geometry was geometry. All of nature, they believed, was an exercise in Euclidean geometry. That the ancient Greeks made tremendous advances in geome- try is common knowledge. For almost a thousand years Greek cul- ture produced generation after generation of outstanding mathematicians. It is a record of mathematical excellence and longevity that no other culture has matched—either before or since. Beginning almost from scratch, the Greeks asked profound questions about the nature of mathematics and what it means to understand mathematics. They created the idea of proof, and they worked hard to place geometry on a firm logical foundation. Many Greek mathematical discoveries—the proofs as well as the state- ments of the results themselves—still sound modern to today’s reader. The Greeks invented mathematics in the sense that we now understand the term, but their work offers little insight into contemporary mathematicians’ current understanding of geometry. Today mathematicians recognize that Euclidean geometry is sim- ply one geometry among many. A more modern definition of geometry is that it is the study of geometric properties. That is the short, easy answer. But it is clearly an incomplete answer. It simply shifts our attention away from the word geometry and toward the phrase geometric properties. What are geometric properties? Points, lines, planes, angles, curves, surfaces, and bodies in three (and even more!) dimensions may all be worthy of geometric study, but not every property of an object is a geometric property. The shape of a triangle may be con- sidered a geometric property, but its color, its temperature, and its distance from the reader are all examples of properties that are not geometric. Perhaps that is obvious, but it is also important. The key to understanding what mathematicians study when they study geometry lies in recognizing which properties of an object are geometric. For example, imagine a triangle on a flat surface with the prop- erty that all three of its sides are the same length. (A triangle with three equal sides is called an equilateral triangle.) It is not difficult to prove that because our triangle has three equal sides it must also have three equal angles. That it has three equal angles is a conse- Introduction xv quence of the fact that it has three equal sides. We can go further: Because the three angles within our triangle are equal, the meas- ure of each angle must be 60°. So knowing that all three sides of our triangle are the same length allows us to deduce that all angles within our triangle are 60° angles. Now suppose that we tilt our triangle out of its original position. Is it still an equilateral triangle? Are all of its angles still of equal measure? Does each angle still measure 60°? Can we be sure, or do we have to reinvestigate our triangle each time we tilt it or move it a little off to the side? Most of us would simply assume that once we discover the meas- ures of the angles of a triangle we should not have to reestablish our discoveries just because we moved our triangle out of its original position. Euclid made the same assumption. Tilting or sliding the triangle from one position to the next should not change the length of the sides or the measures of the angles. Perhaps this is obvious to you. Less obvious, perhaps, is that this simple example also contains the key to understanding what it is that mathemati- cians study when they study geometry. Geometry is the study of just those properties of a figure that do not change under a particular set of motions. Euclidean geometry, for example, is the study of those properties of a figure that do not change when the figure is tilted (rotated) or when it is moved along a straight line (translated). When the ancient Greeks proved that some figure had a particular geometric property, the proof also applied to every figure, located anywhere, that could, through a series of translations and rotations, be made to coincide with the original figure.