Canadian Mathematical Society Societe mathematique du Canada

Editors-in-Chief Redacteurs-en-chef Jonathan M. Borwein Peter Borwein

Springer Science+Business Media, LLC

CMS Books in MQthemQtics OUVfQges de mQthemQtiques de /Q SMC

1 HERMAN/KuCERAlSIMSA Equations and Inequalities 2 .ARNOLD Abelian Groups and Representations of Finite Partially Ordered Sets 3 BORWEIN/LEWIS Convex Analysis and Nonlinear Optimization 4 LEVIN/LuBINSKY Orthogonal Polynomials for Exponential Weights 5 KANE Reflection Groups and Invariant Theory 6 PHILLIPS Two Millennia of Mathematics 7 DEUTSCH/BEST Approximation in Inner Product Spaces

George M. Phillips

Two Millennia of Mathematics From Archimedes to Gauss

, Springer

George M. Phillips Mathematical Institute University of St. Andrews St. Andrews KY16 9SS Scotland

Editors-in-Chie! Redacteurs-en-che! Jonathan M. Borwein Peter Borwein Centre for Experimental and Constructive Mathematics Department of Mathematics and Statistics Simon Fraser University Burnaby, British Columbia VSA IS6 Canada

Mathematics Subject Classification (2000): 00A05, 0lA05

Library of Congress Cataloging-in-Publication Data PhilIips, G.M. (George McArtney)

Two millennia of mathematics : from Archimedes to Gauss / George M. Phillips. p. cm. - (CMS books in mathematics ; 6)

Includes bibliographical references and index. ISBN 978-1-4612-7035-5 ISBN 978-1-4612-1180-8 (eBook) DOI 10.1007/978-1-4612-1180-8 1. Mathematics-Miscellanea. 2. Mathematics-History. I. Title. 11. Serles.

QA99 .P48 2000 51O-dc21 00-023807

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© 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York,Inc. in 2000 Softcover reprint of the hardcover 1 st edition 2000

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ISBN 978-1-4612-7035-5 SPIN 10762921

Preface

This book is intended for those who love mathematics, including under­graduate students of mathematics, more experienced students, and the vast number of amateurs, in the literal sense of those who do something for the love of it. I hope it will also be a useful source of material for those who teach mathematics. It is a collection of loosely connected topics in areas of mathematics that particularly interest me, ranging over the two millennia from the work of Archimedes, who died in the year 212 Be, to the Werke of Gauss, who was born in 1777, although there are some references outside this period. In view of its title, I must emphasize that this book is certainly not pretending to be a comprehensive history of the mathematics of this period, or even a complete account of the topics discussed. However, every chapter is written with the history of its topic in mind. It is fascinating, for example, to follow how both Napier and Briggs constructed their log­arithms before many of the most relevant mathematical ideas had been discovered. Do I really mean "discovered"? There is an old question, "Is mathematics created or discovered?" Sometimes it seems a shame not to use the word "create" in praise of the first mathematician to write down some outstanding result. Yet the inner harmony that sings out from the best of mathematics seems to demand the word "discover." Patterns emerge that are sometimes reinterpreted later in a new context. For example, the relation

showing that the product of two numbers that are the sums of two squares is itself the sum of two squares, was known long before it was reinterpreted

vi Preface

as a property of complex numbers. It is equivalent to the fact that the modulus of the product of two complex numbers is equal to the product of their moduli. Other examples of the inner harmony of mathematics occur again and again when generalizations of known results lead to exciting new developments.

There is one matter that troubles me, on which I must make my peace with the reader. I need to get at you before you find that I have cited my own name as author or coauthor of 11 out of the 55 items in the Bibliography at the end of this book. You might infer from this either that I must be a mathematician of monumental importance or that I believe I am. Neither of these statements is true. As a measure of my worth as a mathematician, I would not merit even one citation if the Bibliography contained 10,000 items. (Alas, the number 10,000 could be increased, but let us not dwell on that.) However, this is one mathematician's account (mine!) of some of the mathematics that has given him much pleasure. Thus references to some of the work in which I have shared demonstrates the depth of my interest and commitment to my subject, and I hope that doesn't sound pompous. I think it may surprise most readers to know that many interesting and exciting results in mathematics, although usually not the most original and substantial, have been obtained (discovered!) by ordinary mortals, and not only by towering geniuses such as Archimedes and Gauss, Newton and Euler, Fermat and the hundreds of other well-known names. This gives us a feel for the scale and the grandeur of mathematics, and allows us to admire all the more its greatest explorers and discoverers. It is only by asking questions ourselves and by making our own little discoveries that we gain a real understanding of our subject. We should certainly not be disappointed if we later find that some well-known mathematician found "our" result before us, but should be proud of finding it independently and of being in such exalted company. One of the most impressive facts about mathematics is that it talks about absolute truths, which are not dependent on opinion or fashion. Any theorem that was proved two thousand years ago, or at any time in the past, is still true today.

No two persons' tastes are exactly the same, and perhaps no one else could or would have made the same selection of material as I have here. I was extremely fortunate to begin my mathematical career with a mas­ter's degree in number theory at the University of Aberdeen, under the supervision of E. M. Wright, who is best known for his long-lived text An Introduction to the Theory of Numbers, written jointly with the emi­nent mathematician G. H. Hardy. I then switched to approximation theory and numerical analysis, while never losing my love for number theory, and the topics discussed in this book reflect these interests. I have had the good fortune to collaborate in mathematical research with several very able mathematicians, valued friends from whom I have learned a great deal while sharing the excitement of research and the joys of our discoveries. Although the results that most mathematical researchers obtain, including

Preface vii

mine, are of minuscule importance compared to the mathematics of great­est significance, their discoveries give enormous pleasure to the researchers involved.

I have often been asked, "How can one do research in mathematics? Surely it is all known already!" If this is your opinion of mathematics, this book may influence you towards a different view, that mathematics was not brought down from Mount Sinai on stone tablets by some mathematical Moses, all ready-made and complete. It is the result of the work of a very large number of persons over thousands of years, work that is still continu­ing vigorously to the present day, and with no end in sight. A rather smaller number of individuals, including Archimedes and Gauss, have made such disproportionately large individual contributions that they stand out from the crowd.

The year 2000 marks the 250th anniversary of the death of J. S. Bach. By a happy chance I read today an article in The Guardian (December 17, 1999) by the distinguished pianist Andras Schiff, who writes, "A musician's life without Bach is like an actor's life without Shakespeare." There is an essential difference between Bach and Shakespeare, on the one hand, and Gauss, a figure of comparable standing in mathematics, on the other. For the music of Bach and the literature of Shakespeare bear the individual stamp of their creators. And although Bach, Shakespeare, and Gauss have all greatly influenced the development, at least in Europe, of music, litera­ture, and mathematics, respectively, the work of Gauss does not retain his individual identity, as does the work of Bach or Shakespeare, being rather like a major tributary that discharges its waters modestly and anonymously into the great river of mathematics. While we cannot imagine anyone but Bach creating his Mass in B Minor or his Cello Suites, or anyone but Shakespeare writing King Lear or the Sonnets, we must concede that all the achievements of the equally mighty Gauss would, sooner or later, have been discovered by someone else. This is the price that even a prince of mathematics, as Gauss has been described, must pay for the eternal worth of mathematics, as encapsulated in the striking quotation of G. H. Hardy at the beginning of Chapter 1.

Mathematics has an inherent charm and beauty that cannot be dimin­ished by anything I write. In these pages I can pursue my craft of seeking to express sometimes difficult ideas as simply as I can. But only you can find mathematics interesting. As Samuel Johnson said, "Sir, I have found you an argument; but I am not obliged to find you an understanding." I find this a most comforting thought.

The reader should be warned that this author likes to use the word "we." This is not the royal ''we'' but the mathematical "we," which is used to emphasize that author and reader are in this together, sometimes up to our necks. And on the many occasions when I write words such as "We can easily see," I hope there are not too many times when you respond with "Speak for yourself!"

viii Preface

If you are like me, you will probably wish to browse through this book, omitting much of the detailed discussion at a first reading. But then I hope some of the detail will seize your attention and imagination, or some of the Problems at the end of each section will tempt you to reach for pencil and paper to pursue your own mathematical research. Whatever your mathematical experience has been to date, I hope you will enjoy reading this book even half as much as I have enjoyed writing it. And I hope you learn much while reading it, as I indeed have from writing it.

George M. Phillips Crail, Scotland

Acknowledgments

Thanks to the wonders of MEX, an author of a mathematics text can produce a book that is at least excellent in its appearance. Therefore, I am extremely grateful not only to the creators of ~TEX but also to those friends and colleagues who have helped this not so old dog learn some new tricks. Two publications have been constantly on my desk, Starting YTfj(, by C. D. Kemp and A. W. Kemp, published by the Mathematical Institute, University of St Andrews, and Learning YTfj(, by David F. Griffiths and Desmond J. Higham, published by SIAM. I am further indebted to David Kemp for ad hoc personal tutorials on ~TEX, and to other St Andrews colleagues John Howie and Michael Wolfe for sharing their know-how on this topic. It is also a pleasure to record my thanks to John O'Connor for his guidance on using the symbolic mathematics program Maple, which I used to pursue those calculations in the book that require many decimal places of accuracy. My colleagues John O'Connor and Edmund Robertson are the creators of the celebrated website on the History of Mathematics, which I have found very helpful in preparing this text.

I am also very grateful to my friend and coauthor Halil Oruc; for his help in producing the diagrams, and to Tricia Heggie for her cheerful and unstinting technical assistance.

My mathematical debts are, of course, considerably greater than those already recorded above. In the Preface I have mentioned my fortunate be­ginnings in Aberdeen, and it is appropriate to give thanks for the goodness of my early teachers there, notably Miss Margaret Cassie, Mr John Flett, and Professor H. S. A. Potter. In my first lecturing appointment, at the University of Southampton, I was equally fortunate to meet Peter Tay-

x Acknowledgments

lor, my long-time friend and coauthor from whom I learned much about numerical analysis.

Several persons have kindly read all or part of the manuscript, and their comments and suggestions have been very helpful to me. My thanks thus go to my good friends Cleonice Bracciali in Brazil, Dorothy Foster and Peter Taylor in Scotland, Herta Freitag and Charles J. A. Halberg in the U.S.A., and Zeynep Ko~ak and Halil Oru~ in Turkey. Of course, any errors that remain are my sole responsibility. In addition to those already mentioned I would like to acknowledge the encouragement and friendship, over the years, of Bruce Chalmers, Ward Cheney, Philip Davis, Frank Deutsch, and Ted Rivlin in the U.S.A.; Peter Lancaster, A. Sharma, Bruce Shawyer, and Sankatha Singh in Canada; A. Sri Ranga and Dimitar Dimitrov in Brazil; Colin Campbell, Tim Goodman, and Ron Mitchell in Scotland; Gracinda Gomes in Portugal; Wolfgang Dahmen in Germany; Zdenek Kosina and Jaroslav Nadrchal in the Czech Republic; Blagovest Sendov in Bulgaria; Didi Stancu in Romania; Lev Brutman in Israel; Kamal Mirnia in Iran; B. H. Ong, H. B. Said, W.-S. Tang, and Daud Yahaya in Malaysia; Lee Seng Luan in Singapore; Feng Shun-xi, Hou Guo-rong, L. C. Hsu, Shen Zuhe, You Zhao-yong, Huang Chang-bin, and Xiong Xi-wen in China; and David Elliott in Australia. I must also thank Lee Seng Luan for introducing me to the wonderful book of Piet Hein, Grooks, published by Narayana Press, and in particular to the "Grook" that I have quoted at the beginning of Chapter 4. This was often mentioned as we worked together, since it so cleverly sums up the tantalizing nature of mathematical research.

Mathematics has been very kind to me, allowing me to travel widely and meet many interesting people. I learned at first hand what my dear parents knew without ever leaving their native land, that we are all the same in the things that matter most. I have felt at home in all the countries I have visited. It pleases me very much that this book appears in a Canadian Mathematical Society series, because my mathematical travels began with a visit to Canada. It was on one of my later visits to Canada that I met the editors, Peter and Jon Borwein. I am grateful to them for their support for this project. Their constructive and kind comments encouraged me to add some further material that, I believe, has had a most beneficial influence on the final form of this book.

I wish to acknowledge the fine work of those members of the staff of Springer, New York who have been involved with the production of this book. There are perhaps only two persons who will ever scrutinize every letter and punctuation mark in this book, the author and the copyeditor. Therefore, I am particularly grateful to the copyeditor, David Kramer, who has carried out this most exacting task with admirable precision.

George M. Phillips Crail, Scotland

Contents

Preface v

Acknowledgments ix

1 From Archimedes to Gauss 1 1.1 Archimedes and Pi . . . 2 1.2 Variations on a Theme. 8 1.3 Playing a Mean Game 18 1.4 Gauss and the AGM 33

2 Logarithms 45 2.1 Exponential Functions 45 2.2 Logarithmic Functions 49 2.3 Napier and Briggs .. 60 2.4 The Logarithm as an Area . 72 2.5 Further Historical Notes 76

3 Interpolation 81 3.1 The Interpolating Polynomial 81 3.2 Newton's Divided Differences 88 3.3 Finite Differences . . . . . 93 3.4 Other Differences . . . . . . . 98 3.5 Multivariate Interpolation . . 105 3.6 The Neville-Aitken Algorithm. 115

xii Contents

3.7 Historical Notes. 119

4 Continued Fractions 121 4.1 The Euclidean Algorithm 121 4.2 Linear Recurrence Relations . 131 4.3 Fibonacci Numbers. 138 4.4 Continued Fractions 147 4.5 Historical Notes ... 161

5 More Number Theory 165 5.1 The Prime Numbers 166 5.2 Congruences. . . . . 172 5.3 Quadratic Residues. 181 5.4 Diophantine Equations . 188 5.5 Algebraic Integers ... 194 5.6 The equation x3 + y3 = z3 . 204 5.7 Euler and Sums of Cubes 209

References 215

Index 219