Two Millennia of Mathematics: From Archimedes to Gauss

235 Pages·2000·6.96 MB·English

by  George M. Phillips (auth.)

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
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Springer-Verlag New York,Inc. in 2000
Softcover reprint of the hardcover 1s t 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 under
graduate 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 log
arithms 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 mas
ter'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 emi
nent 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 great
est 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 continu
ing 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, litera
ture, 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 dimin
ished 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 be
ginnings 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