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Undergraduate Texts in Mathematics
Editorial Board
S. Axler
K.A. Ribet
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John Stillwell
Mathematics
and Its History
Third Edition
123
John Stillwell
Department of Mathematics
University of San Francisco
San Francisco, CA 94117-1080
USA

Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA

K.A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720-3840
USA


several new sections in old chapters. The new sections fill gaps and update
areas where there has been recent progress, such as the Poincar´e conjec-
ture. The simple groups chapter includes some material on Lie groups,
thus redressing one of the omissions I regretted in the first edition of this
book. The coverage of group theory has now grown from 17 pages and 10
exercises in the first edition to 61 pages and 85 exercises in this one. As in
the second edition, exercises often amount to proofs of big theorems, bro-
ken down into small steps. In this way we are able to cover some famous
theorems, such as the Brouwer fixed point theorem and the simplicity of
A
5
, that would otherwise consume too much space.
Each chapter now begins with a “Preview” intended to orient the reader
with motivation, an outline of its contents and, where relevant, connections
to chapters that come before and after. I hope this will assist readers who
like to have an overview before plunging into the details, and also instruc-
tors looking for a path through the book that is short enough for a one-
semester course. Many different paths exist, at many different levels. Up
to Chapter 10, the level should be comfortable for most junior or senior
undergraduates; after that, the topics become more challenging, but also of
greater current interest.
vii
viii Preface to the Third Edition
All the figures have now been converted to electronic form, which has
enabled me to reduce some that were excessively large, and hence mitigate
the bloating that tends to occur in new editions.
Some of the new material on mechanics in Section 13.2 originally ap-
peared (in Italian) in a chapter I wrote for Volume II of
La Matematica
,

one in Section 2.2 comparing volume and surface area of the icosa-
hedron and dodecahedron, have now been broken into manageable
parts. Nevertheless, there are still a few challenging questions for
those who want them.
• Commentary has been added to the exercises to explain how they
relate to the preceding section, and also (when relevant) how they
foreshadow later topics.
• The index has been given extra structure to make searching easier.
To find Euler’s work on Fermat’s last theorem, for example, one no
longer has to look at 41 different pages under “Euler.” Instead, one
can find the entry “Euler, and Fermat’s last theorem” in the index.
• The bibliography has been redone, giving more complete publica-
tion data for many works previously listed with little or none. I have
found the online catalogue of the Burndy Library of the Dibner In-
stitute at MIT helpful in finding this information, particularly for
ix
x Preface to the Second Edition
early printed works. For recent works I have made extensive use of
MathSciNet, the online version of Mathematical Reviews.
There are also many small changes, some prompted by recent mathe-
matical events, such as the proof of Fermat’s last theorem. (Fortunately,
this one did not force a major rewrite, because the background theory of
elliptic curves was covered in the first edition.)
I thank the many friends, colleagues, and reviewers who drew my at-
tention to faults in the first edition, and helped me in the process of revision.
Special thanks go to the following people.
• My sons, Michael and Robert, who did most of the typing, and my
wife, Elaine, who did a great deal of the proofreading.
• My students in Math 310 at the University of San Francisco, who
tried out many of the exercises, and to Tristan Needham, who invited

by tracing their historical development.
In doing so, I have also tried to tie up some traditional loose ends. For
example, undergraduates can solve quadratic equations. Why not cubics?
They can integrate 1/
√
1 − x
2
but are told not to worry about 1/
√
1 − x
4
.
Why? Pursuing the history of these questions turns out to be very fruitful,
leading to a deeper understanding of complex analysis and algebraic ge-
ometry, among other things. Thus I hope that the book will be not only a
xi
xii Preface to the First Edition
bird’s-eye view of undergraduate mathematics but also a glimpse of wider
horizons.
Some historians of mathematics may object to my anachronistic use of
modern notation and (fairly) modern interpretations of classical mathemat-
ics. This has certain risks, such as making the mathematics look simpler
than it really was in its time, but the risk of obscuring ideas by cumber-
some, unfamiliar notation is greater, in my opinion. Indeed, it is practically
a truism that mathematical ideas generally arise before there is notation or
language to express them clearly, and that ideas are implicit before they
become explicit. Thus the historian, who is presumably trying to be both
clear and explicit, often has no choice but to be anachronistic when tracing
the origins of ideas.
Mathematicians may object to my choice of topics, since a book of

Jeremy Gray, George Odifreddi, and Abe Shenitzer. Their comments have
resulted in innumerable improvements, and any flaws remaining may be
due to my failure to follow all their advice. To them, and to Anne-Marie
Vandenberg for her usual excellent typing, I offer my sincere thanks.
John Stillwell
Monash University
Victoria, Australia
1989

Contents
Preface to the Third Edition vii
Preface to the Second Edition ix
Preface to the First Edition xi
1 The Theorem of Pythagoras 1
1.1 Arithmetic and Geometry 2
1.2 Pythagorean Triples 4
1.3 Rational Points on the Circle 6
1.4 Right-Angled Triangles 9
1.5 Irrational Numbers 11
1.6 The Definition of Distance 13
1.7 Biographical Notes: Pythagoras 15
2 Greek Geometry 17
2.1 The Deductive Method 18
2.2 The Regular Polyhedra 20
2.3 Ruler and Compass Constructions 25
2.4 Conic Sections 28
2.5 Higher-Degree Curves 31
2.6 Biographical Notes: Euclid 35
3 Greek Number Theory 37
3.1 The Role of Number Theory 38

7 Analytic Geometry 109
7.1 Steps Toward Analytic Geometry 110
7.2 Fermat and Descartes 111
7.3 Algebraic Curves 112
7.4 Newton’s Classification of Cubics 115
7.5 Construction of Equations, B´ezout’s Theorem 118
7.6 The Arithmetization of Geometry 120
7.7 Biographical Notes: Descartes 122
Contents xvii
8 Projective Geometry 127
8.1 Perspective 128
8.2 Anamorphosis 131
8.3 Desargues’s Projective Geometry 132
8.4 The Projective View of Curves 136
8.5 The Projective Plane 141
8.6 The Projective Line 144
8.7 Homogeneous Coordinates 147
8.8 Pascal’s Theorem 150
8.9 Biographical Notes: Desargues and Pascal 153
9Calculus 157
9.1 What Is Calculus? 158
9.2 Early Results on Areas and Volumes 159
9.3 Maxima, Minima, and Tangents 162
9.4 The Arithmetica Infinitorum of Wallis 164
9.5 Newton’s Calculus of Series 167
9.6 The Calculus of Leibniz 170
9.7 Biographical Notes: Wallis, Newton, and Leibniz 172
10 Infinite Series 181
10.1 Early Results 182
10.2 Power Series 185

13.6 The Vibrating String 261
13.7 Hydrodynamics 265
13.8 Biographical Notes: The Bernoullis 267
14 Complex Numbers in Algebra 275
14.1 Impossible Numbers 276
14.2 Quadratic Equations 276
14.3 Cubic Equations 277
14.4 Wallis’s Attempt at Geometric Representation 279
14.5 Angle Division 281
14.6 The Fundamental Theorem of Algebra 285
14.7 The Proofs of d’Alembert and Gauss 287
14.8 Biographical Notes: d’Alembert 291
15 Complex Numbers and Curves 295
15.1 Roots and Intersections 296
15.2 The Complex Projective Line 298
15.3 Branch Points 301
15.4 Topology of Complex Projective Curves 304
15.5 Biographical Notes: Riemann 308
Contents xix
16 Complex Numbers and Functions 313
16.1 Complex Functions 314
16.2 Conformal Mapping 318
16.3 Cauchy’s Theorem 319
16.4 Double Periodicity of Elliptic Functions 322
16.5 Elliptic Curves 325
16.6 Uniformization 329
16.7 Biographical Notes: Lagrange and Cauchy 331
17 Differential Geometry 335
17.1 Transcendental Curves 336
17.2 Curvature of Plane Curves 340

20.6 Octonions 428
20.7 Why C, H,andO Are Special 430
20.8 Biographical Notes: Hamilton 433
21 Algebraic Number Theory 439
21.1 Algebraic Numbers 440
21.2 Gaussian Integers 442
21.3 Algebraic Integers 445
21.4 Ideals 448
21.5 Ideal Factorization 452
21.6 Sums of Squares Revisited 454
21.7 Rings and Fields 457
21.8 Biographical Notes: Dedekind, Hilbert, and Noether 459
22 Topology 467
22.1 Geometry and Topology 468
22.2 Polyhedron Formulas of Descartes and Euler 469
22.3 The Classification of Surfaces 471
22.4 Descartes and Gauss–Bonnet 474
22.5 Euler Characteristic and Curvature 477
22.6 Surfaces and Planes 479
22.7 The Fundamental Group 484
22.8 The Poincar´e Conjecture 486
22.9 Biographical Notes: Poincar´e 492
23 Simple Groups 495
23.1 Finite Simple Groups and Finite Fields 496
23.2 The Mathieu Groups 498
23.3 Continuous Groups 501
23.4 Simplicity of SO(3) 505
23.5 Simple Lie Groups and Lie Algebras 509
23.6 Finite Simple Groups Revisited 513
23.7 The Monster 515

orem, but also the source of three great streams of mathematical thought:
numbers, geometry, and infinity.
The number stream begins with Pythagorean triples; triples of integers
(a, b, c) such that a
2
+ b
2
= c
2
. The geometry stream begins with the
interpretation of a
2
, b
2
,andc
2
as squares on the sides of a right-angled
triangle with sides a, b, and hypotenuse c. The infinity stream begins with
the discovery that
√
2, the hypotenuse of the right-angled triangle whose
other sides are of length 1, is an irrational number.
These three streams are followed separately through Greek mathemat-
ics in Chapters 2, 3, and 4. The geometry stream resurfaces in Chapter
7, where it takes an algebraic turn. The basis of algebraic geometry is
the possibility of describing points by numbers—their coordinates—and
describing each curve by an equation satisfied by the coordinates of its
points.
This fusion of numbers with geometry is briefly explored at the end of
this chapter, where we use the formula a

a
b
c
Figure 1.1: The Pythagorean theorem
Conversely, a solution of (1) by positive numbers a, b, c can be re-
alized by a right-angled triangle with sides a, b and hypotenuse c.Itis
clear that we can draw perpendicular sides a, b for any given positive num-
bers a, b, and then the hypotenuse c must be a solution of (1) to satisfy
the Pythagorean theorem. This converse view of the theorem becomes
interesting when we notice that (1) has some very simple solutions. For
example,
(a, b, c) = (3, 4, 5), (3
2
+ 4
2
= 9 + 16 = 25 = 5
2
),
(a, b, c) = (5, 12, 13), (5
2
+ 12
2
= 25 + 144 = 169 = 13
2
).
It is thought that in ancient times such solutions may have been used for
the construction of right angles. For example, by stretching a closed rope
with 12 equally spaced knots one can obtain a (3, 4, 5) triangle with right
angle between the sides 3, 4, as seen in Figure 1.2.