Table Of Content3.1416
And All That
PHILIP J. DAVIS
WILLIAM G. CHINN
Birkhauser Boston. Basel. Stuttgart
TO OUR FAMILIES
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 prior permission of
the copyright owner.
Second edition. First edition published in 1%9 by Simon & Schuster, New York.
© Birkhauser Boston, 1985
ABCDEFGHIJ
ISBN-13: 978-0-8176-3304-2 e-ISBN-13: 978-1-4615-8519-0
DOl: 10.1007/978-1-4615-8519-0
Library of Congress Cataloging in Publication Data
Davis, Philip J., 1923-
3.1416 and all that.
Bibliography: p.
I. Mathematics-Popular works. I. Chinn, William G.
II. Title.
QA93.D3 1985 510 85-1314
CIP-Kurztitelaufnahme der Deutschen Bibliothek
3.1416 [Three onejour one six] and all that /
Philip J. Davis and William G. Chinn.-2. ed.
Boston; Basel; Stuttgart: Birkhiiuser, 1985.
1. AnI!. im VerI. Simon and Schuster, New York
HE: Davis, PhilipJ. [Mitverf.]; Chinn, William
G. [Mitverf.]
CONTENTS
Foreword vii
Introduction ix
1. The Problem That Saved a Man's Life 1
2. The Code of the Primes 7
3. Pompeiu's Magic Seven 14
4. What Is an Abstraction? 20
5. Postulates-The Bylaws of Mathematics 27
6. The Logical Lie Detector 33
7. Number 40
8. The Philadelphia Story 63
9. Poinsot's Points and Lines 65
10. Chaos and Polygons 71
11. Numbers, Point and Counterpoint 79
12. The Mathematical Beauty Contest 88
13. The House That Geometry Built 94
14. Explorers of the Nth Dimension 101
15. The Band-Aid Principle 108
16. The Spider and the Fly 117
17. A Walk in the Neighborhood 123
18. Division in the Cellar 131
19. The Art of Squeezing 137
20. The Business of Inequalities 144
21. The Abacus and the Slipstick 152
22. Of Maps and Mathematics 159
23. "Mr. Milton, Mr. Bradley-Meet Andrey
Andreyevich Markov" 164
24. 3.1416 and All That 172
Bibliography 177
Ancient and Honorable Society ofP i
Watchers:
1984 Report 177
Bibliography 181
ACKNOWLEDGMENTS
Most of the articles in this book originally appeared in Science
World. Grateful acknowledgment is made to Science World for per
mission to use them here. We thank the Scientific American for
permission to reprint the article entitled "Number."
The authorship of the individual articles is as follows:
Philip J. Davis: 1,3,4,7,8,9, 12, 14, 15, 18,22,23, 24.
William G. Chinn: 2, 5, 6, 10, 11, 13, 16, 17, 19,20,21.
The first author would like to acknowledge his indebtedness to
Professor Alexander Ostrowski and to Drs. Barry Bernstein and
John Wrench for information incorporated in some of his articles.
The second author would like to acknowledge his indebted
ness to Professor G. Baley Price for permission to use materials
from one of his articles.
FOREWORD
LYTTON STRACHEY tells the following story. In intervals of
relaxation from his art, the painter Degas used to try his hand at
writing sonnets. One day, while so engaged, he found that his in
spiration had run dry. In desperation he ran to his friend Mallarme,
who was a poet. "My poem won't come out," he said, "and yet I'm
full of excellent ideas." "My dear Degas," Mallarme retorted,
"poetry is not written with ideas, it is written with words."
If we seek an application of Mallarme's words to mathematics
we find that we shall want to turn his paradox around. We are led
to say that mathematics does not consist of formulas, it consists of
ideas. What is platitudinous about this statement is that mathe
matics, of course, consists of ideas. Who but the most unregenerate
formalist, asserting that mathematics is a meaningless game played
with symbols, would deny it? What is paradoxical about the state
ment is that symbols and formulas dominate the mathematical page,
and so one is naturally led to equate mathematics with its formulas.
+ =
Is not Pythagoras' Theorem a2 b2 c2? Does not the Binomial
+ = + +
Theorem say that (a b) 2 a2 2ab b2? What more need be
said-indeed, what more can be said? And yet, as every devotee
who has tried to be creative in mathematics knows, formulas are
not enough; for new formulas can be produced by the yard, while
original thought remains as remote as hummingbirds in January.
To appreciate mathematics at its deeper level we must pass
from naked formulas to the ideas that lie behind them.
In the present volume, we have selected for reprinting a num
ber of pieces that appeared principally in Science World, a periodi
cal with a wide circulation among students and teachers. In writing
these articles it was our aim to deal with a number of diverse areas
of current mathematical interest and, by concentrating on a limited
aspect of each topic, to expose in a modest way the mathematical
ideas that underlie it. It has not been possible, in the few pages
allotted to each essay, to present the topics in the conventional text-
vII
viii Foreword
book sense; our goal has been rather to provide a series of ap
petizers or previews of coming attractions which might catch the
reader's imagination and attract him to the thoroughgoing treat
ments suggested in the bibliographies. Each article is essentially
self-contained.
What reason can we put forward for the study of mathematics
by the educated man? Every generation has felt obliged to say a
word about this. Some of the reasons given for this study are that
mathematics makes one think logically, that mathematics is the
Queen of the Sciences, that God is a geometer who runs His uni
verse mathematically, that mathematics is useful in surveying fields,
building pyramids, launching satellites. Other reasons are that life
has become increasingly concerned with the manipulation of sym
bols, and mathematics is the natural language of symbols; that
"Euclid alone has looked upon beauty bare"; that mathematics can
be fun. Each of these has its nugget of truth and must not be
denied. Each undoubtedly can be made the basis of a course of
instruction.
But we should like to suggest a different reason. Mathematics
has been cultivated for more than four thousand years. It was
studied long before there were Democrats and Republicans or any
of our present concerns. It has flourished in many lands, and the
genius of many peoples has added to its stock of ideas. Mathe
matics dreams of an order which does not exist. This is the source
of its power; and in this dream it has exhibited a lasting quality that
resists the crash of empire and the pettiness of small minds. Mathe
matical thought is one of the great human achievements. The study
of its ideas, its past and its present, can enable the individual to
free himself from the tyranny of time and place and circumstance.
Is not this what liberal education is about?
Summer, 1968
INTRODUCTION
3.1416 AND ALL THAT is a beautiful book written by two masters
of popularization. Philip J. Davis and William G. Chinn are math
ematicians of many dimensions, who are interested in much more
than mathematics. In the following twenty-four essays, you obtain
pleasant introductions to powerful ideas of mathematics. In fact,
you will have fun reading this book of mathematics. Essayists Davis
and Chinn write with style, vigor, and a sense of the relationship
of mathematics to other fields, including biology, history, music,
philosophy, and even golf and woodworking.
Professors Chinn and Davis are much aware of mathematics as a
human activity. In their essays, they provide precious insights into
how mathematicians think. They also show us that mathematics is
alive and growing.
If you like mathematics, this book will cause you to like it more.
If you are wary of mathematics, then reading this book is likely to
make you a mathematical convert.
Donald J. Albers
Second Vice-President
The Mathematical Association of America
ix
1
THE PROBLEM THAT
SAVED A MAN'S LIFE
TRUTH IS STRANGER THAN FICTION-even in the world
of mathematics. It is hard to believe that the mathematical ideas
of a small religious sect of ancient Greece were able to save-some
twenty-five hundred years later-the life of a young man who lived
in Germany. The story is related to some very important mathe
matics and has a number of strange twists. *
The sect was that of the Pythagoreans, whose founder, Pythag
oras, lived ,in the town of Crotona in southern Italy around 500
B.C. The young man was Paul Wop· >kehl, who was professor of
mathematics in Darmstadt, Germany, in the early 1900's. Between
the two men lay a hundred generations, a thousand miles, and
many dead civilizations. But between the two men stretched the
binding cord of a mathematical idea-a thin cord, perhaps, but a
strong and enduring one.
Pythagoras was both a religious prophet and a mathematician.
He founded a school of mathematics and a religion based upon the
notion of the transmigration of souls. "All is number," said Pythag
oras, and each generation of scientists discovers new reasons for
• I have the story about the young man-Paul Wolfskehl-from the
renowned mathematician Alexander Ostrowski. Professor Ostrowski himself
heard the story many years ago and claims there is more to 'it than mere
legend.-P.l.D.
1
2 3.1416 and All That
thinking this might be true. "Do not eat beans, and do not touch
white roosters," said Pythagoras, and succeeding generations won
der whatever was in the man's head. "The square on the hypotenuse
is equal to the sum of the squares on the sides," said Pythagoras,
and each generation of geometry students has faithfully learned
this relation-a relation that has retained its importance over the
intervening years.
Paul Wolfskehl was a modern man, somewhat of a romantic.
He is as obscure to the world as Pythagoras is famous. Wolfskehl
was a good mathematician, but not an extremely original one. The
one elegant idea he discovered was recently printed in a textbook
without so much as a credit line to its forgotten author. He is prin
cipally remembered today as the donor of a prize of 100,000 marks
-long since made worthless by inflation-for the solution of the
famous unsolved "Fermat's Last Problem." But this prize is an im
portant part of the story, and to begin it properly, we must return
to the square on the hypotenuse.
The Theorem of Pythagoras tells us that if a and b are the
lengths of the legs of a right-angled triangle and if c is the length
of its hypotenuse, then
+
a2 b2 = c2•
The cases when a = 3, b = 4, c = 5 and when-a'= 5, b = 12,
c = 13 are very familiar to us, and probably were familiar to the
mathematicians of antiquity long before Pythagoras formulated his
general theorem.
The Greeks sought a formula for generating all the integer
(whole number) solutions to Pythagoras' equation, and by the year
250 A.D. they had found it. At that time, the mathematician Dio
phantus wrote a famous book on arithmetic (today we might prefer
to call it the "theory of numbers"), and in his book he indicated
the following: If m and n are any two integers, then the three in
+
tegers m2 - n2, 2mn, and m2 n2 are the two legs and the hypote
nuse of a right triangle. For instance, if we select m = 2, n = 1, we
obtain 3,4, and 5; if we select m = 3, n = 2, we obtain 5, 12, 13;
and so on.
Our story now skips about fourteen hundred years-while the
The Problem That Saved a Man's Life 3
world of science slept soundly-to the year 1637, a time when
mathematics was experiencing a rebirth.
There was in France a king's counselor by the name of Pierre
Fermat. Fermat's hobby was mathematics, but to call him an ama
teur would be an understatement, for he was as skilled as any
mathematician then alive. Fermat happened to own an edition of
Diophantus, and it is clear that he must have read it from cover to
cover many times, for he discovered and proved many wonderful
things about numbers. On the very page where Diophantus talks
+
about the numbers m2 - n2, 2mn, and m2 n2, Fermat wrote in
the margin: "It is impossible to separate a cube into two cubes, a
fourth power into two fourth powers, or, generally, any power
above the second into two powers of the same degree. I have dis
covered a trulv marvelous demonstration which this margin is too
narrow to contain."
Let's see what Fermat meant by this. By Pythagoras' Theorem,
a square, c2, is "separated" into two other squares, a2 and b2• If
+
we could do likewise for cubes, we would have c3 = a3 b3• If
+
we could do it for fourth powers, we would have c4 = a4 b4• But
Fermat claims that this is impossible. Moreover, if n is any integer
+
greater than 2, Fermat claims, it is impossible to have CD = aD bD,
where a, b, and c are positive integers. He left no indication of what
his demonstration was.
From 1637 to the present time, the most renowned mathe
maticians of each century have attempted to supply a proof for
Fermat's Last Theorem. Euler, Legendre, Gauss, Abel, Cauchy,
Dirichlet, Lame, Kummer, Frobenius, and hosts of others have
tackled it. In the United States, and closer to our own time,
L. E. Dickson, who was professor at the University of Chicago,
and H. S. Vandiver, a professor at the University of Texas, have
gone after the problem with renewed vigor and cunning. But all
have failed. However, many experts feel that the solution to the
problem is now within the scope of methods that have been
developed in the last few decades.
Perhaps the most ingenious of all the contributions to the
