THE ROLE OF MATHEMATICS IN SCIENCE NEW MATHEMATICAL LIBRARY PUBLISHED BY THEM ATHEMATICAASLS OCIATIOONF AMERICA Editorial Committee W. G. Chinn, Chairman (1983-85) Anneli Lax, Editor City College of San Francisco New York University Basil Gordon (1983-85) University of California, Los Angeles M. M. Schiffer (1983-85) Stanford University Ted Vessey (1983-85) St. Olaf College The New Mathematical Library (NML) was begun in 1961 by the School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the school syllabus. In a decade high the NML matured into a steadily growing series of some twenty titles of interest not only to the originally intended audience, but to college students and teachers at all levels. Previously published by Random House and L. W. Singer, the NML became a publication series of the Mathematical Association of America (MAA) in 1975. Under the auspices of the MA4 the NML will continue to grow and will remain dedicated to its original and expanded purposes. THE ROLE OF MATHEMATICS IN SCIENCE by M. M. Schiffer Stanford University and Leon Bowden University of Victoria, B. C. 30 THE MATHEMATICAL ASSOCIATION OF AMERICA ©Copyright 1984 by The Mathematical Association of America (Inc.) All rights reserved under International and Pan-American Copyright Conventions. Published in Washington, D.C. by The Mathematical Association of America Library of Congress Catalog Card Number: 84-60251 Print ISBN 978-0-88385-630-7 Electronic ISBN 978-0-88385-945-2 Manufactured in the United States of America Note to the Reader This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and under- standable to a large audience of school students and laymen. Most high of the volumes in the New Mathematical Library cover topics not usually included in the school curriculum; they vary in difficulty, and, even high within a single book, some parts require a greater degree of concentration than others. Thus, while the readers need little technical knowledge to understand most of these books, they will have to make an intellectual effort. If readers have encountered mathematics so far only in the classroom, they should keep in mind that a book on mathematics cannot be read quickly. Nor must they expect to understand all parts of the book on first reading. They should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark. On the other hand, sections containing thoroughly familiar material may be read very quickly. The best way to learn mathematics is to do mathematics. Readers are urged to acquire the habit of reading with paper and pencil in hand; in this way mathematics will become increasingly meaningful to them. The authors and editorial committee are interested in reactions to the books in this series, and hope that readers will write to: Anneli Lax, Editor, New Mathematical Library, NEW YORK UNIVERSITY, THE COURANT INSTITUTEO F MATHEMATICASCL IENCES2,5 1 Mercer Street, New York, N. Y. 10012. The Editors V NEW MATHEMATICAL LIBRARY 1 Numbers: Rational and Irrational by loan Nioen 2 What is Calculus About? by W. W. Sawyer 3 An Introduction to Inequalities by E. F. Beckenbach and R. Bellman 4 Geometric Inequalities by N. D. Kamrinoff 5 The Contest Problem Book I Annual High School Mathematics Examinations 1950-1960. Compiled and with solutions by Charles T. Salkind 6 The Lore of Large Numbers, by P. 1. Dauis 7 Uses of Infinity by Leo Zippin 8 Geometric Transformations I by 1. M. Yaglom, translated by A. Shields 9 Continued Fractions by Carl D. Okls 10 Graphs and Their Uses by Oystein Ore 11 Hungarian Problem Books I and 11, Based on the Eotvos 121 Competitions 1894-1905 and 1906-1928, translated by E. Rapaport 13 Episodes from the Early History of Mathematics by A. Aaboe 14 Groups and Their Graphs by I: Grossmun and W. Magnus 15 The Mathematics of Choice or How to Count Without Counting by loan Nicen 16 From Pythagoras to Einstein by K. 0. Friedrichs 17 The Contest Problem Book I1 Annual High School Mathematics Examinations 1961-1965. Compiled and with solutions by Churles T. Salkind 18 First Concepts of Topology by W. G. Chinn and N. E. Steenrod 19 Geometry Revisited by H. S. M. Cmeter and S. L. Creitzer 20 Invitation to Number Theory hy Oystein Ore 21 Geometric Transformations I1 by I. M. Yaglum, translated by A. Shields 22 Elementary Cryptandysis-A Mathematical Approach by A. Sinkou 23 Ingenuity in Mathematics by Ross Honsberger 24 Geometric Transformations I11 by I. M. Yagh, translated by A. Shenitzer 25 The Contest Problem Book I11 Annual High School Mathematics Examinations 1966-1972. Compiled and with solutions by C. T. Salkind and J. M. Earl 26 Mathematical Methods in Science by George Pdya 27 International Mathematical Olympiads- 1959-1977. Compiled and with solu- tions by S. L. Greitzer 28 The Mathematics of Games and Gambling by Edward W. Packel 29 The Contest Problem Book IV Annual High School Mathematics Examinations 1973-1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione and N. Shell 30 The Role of Mathematics in Science by M. M. Schifler and L. Bowden 31 International Mathematical Olympiads 1978- 1985 and forty supplementary problems. Compiled and with solutions by Murray S. Klamkin 32 Riddles of the Sphinx (and Other Mathematical Puzzle Tales) by Martin Cardner Other titles in preparation. Preface This little book is the outgrowth of a series of lectures given to a group of high school teachers and published as a mimeographed booklet by the School Mathematics Study Group. The aim of the booklet was to illustrate many ways in which mathematical methods have helped dis- covery in science. The present edition has the same objective. However, we now aim at a group of readers who, we assume, are interested in mathematics beyond the level of high school mathematics. We have added material, and we occasionally use some calculus and more intricate arguments than before. We hope that we will appeal to college students and general readers with some background in mathematics. This has also led to a change in style of exposition and choice of material. If we succeed in giving an impres- sion of the beauty and power of mathematical reasoning in science, the purpose of our work will have been achieved. We thank Professor R. Richtmyer for his comments on our treatment of relativity, in particular for his illuminating remark that the Lorentz transformation in space gives us more physical insight than that in one dimension. This leads to a simplification in deriving the conservaton laws of mechanics, which was elegantly done and woven into Section 7.10 by Professor P. D. Lax, whom we thank also for many other remarks that helped us clarify the exposition. We are deeply indebted to Dr. Anneli Lax, the editor of this series of books. Her editorial help has been most valuable and she has amply demonstrated that a good editor is an author’s best friend. M. M. Schifler L. Bowden 1984 vii Contents Preface vii Introduction 1 Chapter 1 The Beginnings of Mechanics 3 1.1 Archimedes' Law of the Lever 3 1.2 First Application: The Centroid of a Triangle 10 1.3 Second Application: The Area Under a Parabola 12 1.4 Third Application: The Law of the Crooked Lever 16 1.5 Galileo: The Law of the Inclined Plane 18 1.6 Stevin: The Law of the Inclined Plane 20 1.7 Insight and Outlook 24 Chapter 2 Growth Functions 25 2.1 The Exponential Law of Growth 25 2.2 Maxwell's Derivation of the Law of Errors 34 2.3 Differential and/or Functional Equations 44 2.4 The Problem of Predicting Population Growth 45 2.5 Cusanus' Recursive Formula for n 59 2.6 Arithmetic and Geometric Means 69 Chapter 3 The Role of Mathematics in Optics 15 3.1 Euclid's Optics 75 3.2 Heron: The Shortest Path Principle 76 3.3 Archimedes' Symmetry Proof 80 ix