A Smoother Pebble This page intentionally left blank A Smoother Pebble Mathematical Explorations DONALD C. BENSON OXFORD UNIVERSITY PRESS 2003 OXPORD UNIVERSITY PRESS Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Copyright © 2003 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. 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, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Benson, Donald C. A smoother pebble : mathematical explorations / Donald C. Benson. p. cm. Includes bibliographical references and index. ISBN 0-19-514436-8 1. Mathematics—History—Popular works. I. Title. QA21 .B46 2003 510—dc21 2002042515 98765432 Printed in the United States of America on acid-free paper Acknowledgments First, I would like to thank my wife, Dorothy, for reading the entire manu- script many times and making many valuable suggestions. Thanks to Ned Black and Donald Chakerian for reading parts of the manuscript and making valuable comments. I would also like to acknowledge the assistance and encouragement of Kirk Jensen at Oxford University Press. This book was typeset by the author using WtjiX Many thanks to the authors of the ETgX packages used here and all who have contributed to CTAN, the Comprehensive TgX Archive Network. The line drawings were done by the author using MetaPost and Gnu- Plot. This page intentionally left blank Contents Introduction I Bridging the Gap 1 Ancient Fractions 5 The Egyptian Unit Fractions 6 Egyptian arithmetic 8 The greedy algorithm 13 The Babylonians and the Sexagesimal System 15 Sexagesimal fractions 16 2 Greek Gifts 18 The Heresy 20 Magnitudes, Ratio, and Proportion 22 Method 1—proportion according to Eudoxus 24 Method 2—Attributed to Theaetetus 26 3 The Music of the Ratios 33 Acoustics 35 The rotating circle 36 Waveforms and spectra 39 Psychoacoustics 44 Consonance versus dissonance 45 Critical bandwidth 46 Intervals, Scales, and Tuning 49 Pythagorean tuning 49 Approximating m octaves with n fifths 51 Equal-tempered tuning 54 1 viii Contents II The Shape of Things 4 Tubeland 61 Curvature of Smooth Curves 62 The inner world of the curve-bound inchworm 62 Curves embedded in two dimensions 63 Curves embedded in three dimensions 64 Curvature of Smooth Surfaces 65 Gaussian curvature — Extrinsic definition 66 Tubeland—A fantasy 68 Triangular excess 71 Euclidean Geometry 73 The parallels axiom 75 Non-Euclidean Geometry 75 Models of non-Euclidean geometries 77 5 The Calculating Eye 82 Graphs 84 The need for graphs 85 "Materials" for graphs 86 Clever people invented graphs 89 Coordinate Geometry 93 Synthetic versus analytic 94 Synthetic and analytic proofs 95 Straight lines 99 Conic sections 101 III The Great Art 6 Algebra Rules 111 Algebra Anxiety 112 Arithmetic by Other Means 115 Symbolic algebra 116 Algebra and Geometry 120 Al-jabr 121 Square root algorithms 122 Contents ix 7 The Root of the Problem 128 Graphical Solutions 129 Quadratic Equations 130 Secrecy, Jealousy, Rivalry, Pugnacity, and Guile 135 Solving a cubic equation 138 8 Symmetry Without Fear 142 Symmetries of a Square 145 The Group Axioms 148 Isometrics of the Plane 150 Patterns for Plane Ornaments 151 Catalog of border and wallpaper patterns 151 Wallpaper watching 158 9 The Magic Mirror 160 Undecidability 160 The Magic Writing 162 IV A Smoother Pebble 10 On the Shoulders of Giants 167 Integration Before Newton and Leibniz 168 Archimedes' method for estimating pi 168 Circular reasoning 170 Completing the estimate of pi 171 Differentiation Before Newton and Leibniz 172 Descartes's discriminant method 173 Fermat's difference quotient method 176 Galileo's Lute 177 Falling bodies 177 The inclined plane 179 11 Six-Minute Calculus 184 Preliminaries 185 Functions 186 Limits 188 Continuity 189 The Damaged Dashboard 191 The broken speedometer 193 The derivative 194 The broken odometer 199 The definite integral 201 Roller Coasters 206 The length of a curve 206 Time of descent 208 x Contents 12 Roller-Coaster Science 212 The Simplest Extremum Problems 213 The rectangle of maximum area with fixed perimeter 213 The lifeguard's calculation 215 A faster track 217 A road-building project for three towns 219 Inequalities 220 The inequality of the arithmetic and geometric means 221 Cauchy's inequality 223 The Brachistochrone 223 The geometry of the cycloid 227 A differential equation 228 The restricted brachistochrone 230 The unrestricted brachistochrone 235 Glossary 243 Notes 249 References 257 Index 261 [...]... possible than with the decimal system Babylonian cuneiform tablets contain many tables of numerical calculations There are tables that imply a knowledge of the Pythagorean Theorem long before Pythagoras To facilitate calculations with fractions, the Babylonians used a table (e.g., Table 1) of reciprocals of ordinary sexagesimal numbers An important use of Table 1 is to replace division by a natural number... mathematical seashore Paths are T the accumulated footprints of those who came before There are many paths to choose from—some leading to minor curiosities and others leading to important mathematical goals In this book, I intend to point out a few paths that I believe are both curious and important—paths with mainstream destinations I intend to show mathematics as a human endeavor, not a cold unapproachable... 1800, a promising innovation in 1900, and a universal commonplace in mathematics, science, business, and everyday life in 2000? Answer: Graphs 1 2 Introduction Part III is concerned with algebra, the language of mathematics Solving equations was a passionate undertaking for five Italian mathematicians of the sixteenth century For them, algebraic knowledge was booty of great value, the object of quarrels,... to have such a record of ancient Greek mathematics; however, the beginnings of Greek mathematics are more shadowy Thales of Miletus (6257-546? BCE) and Pythagoras of Samos (580?-500? BCE) were the first Greek mathematicians Miletus was a Greek coastal city of Asia Minor (now Turkey), and Samos is a Greek island—both are on the Aegean Sea Both men are said to have brought back knowledge from travels... four diagonal lines is twice the area of the small shaded square because the diagonal square consists of four triangles and the shaded square 21 Greek Gifts consists of two triangles The side of the diagonal square is the diagonal of the shaded square Thus, if a and c denote the side and diagonal, respectively, of the shaded square, we have The numerical values of a and c depend on the unit of measurement—for... incommensurable magnitudes created a crisis in the foundations of their mathematics At any rate, they resolved this "logical scandal" by making a distinction between geometric and arithmetic magnitudes and by developing a theory of ratio and proportion In ordinary usage, ratio and proportion are sometimes used interchangeably, but here we will make a more careful distinction Specifically, an equality of ratios... greedy algorithm (Proposition 1.1) for Egyptian unit fractions, was also acquainted with decimal and sexagesimal numbers He used sexagesimal numbers to give a solution, accurate to the equivalent of nine decimal places, of a certain cubic equation Egyptian unit fractions have long ago ceased to be a tool for serious computation Today they are merely a source of curious problems On the other hand, we are... (1.1) and (1.3)) are all that are needed The fact—show in equations (1.1), (1.3), and (1.4)—that 5/7 has more than one representation as a sum of unit fractions indicates a serious flaw in the Egyptian system for fractions How is it possible that such an awkward system remained in use for thousands of years? There are several possible answers: 1 The system was adequate for simple needs 2 The system was... Eudoxus Define what it means for one ratio to be equal to or greater than another ratio.4 As we will see in the next section, this can be done even if we have not defined what a ratio A : B actually is Ratios can be left undefined just as lines and points in geometry are undefined 2 Anthyphairesis Define a ratio A : B in terms of the natural numbers even when A and B are not numerical magnitudes Further... numbers All the rest is the work of man." There is essentially one way to understand the natural numbers However, there are several different ways to define fractional numbers—also known as rational numbers The fractions in current use a numerator and denominator separated by a bar, for example, 5/7—we call common fractions This notation originated in India in the twelfth century and soon spread to Europe, . York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei. Press. Library of Congress Cataloging-in-Publication Data Benson, Donald C. A smoother pebble : mathematical explorations / Donald C. Benson. p. cm. Includes bibliographical references and . 2000? Answer: Graphs. 1 T Part III is concerned with algebra, the language of mathematics. Solv- ing equations was a passionate undertaking for five Italian mathemati- cians of