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A History of
MATHEMATICS
An Introduction
This page intentionally left blank
A History of
MATHEMATICS
An Introduction
Third Edition
Victor J. Katz
University of the District of Columbia
Addison-Wesley
Boston
San Francisco
New York
London Toronto Sydney Tokyo Singapore Madrid
Mexico City Munich Paris Cape Town Hong Kong Montreal
To Phyllis, for her patience, encouragement, and love
Editor in Chief: Deirdre Lynch
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Cover photo: Tycho Brahe and Others with Astronomical Instruments, 1587, “Le Quadran Mural”
1663. Blaeu, Joan (1596–1673 Dutch). Newberry Library, Chicago, Illinois, USA © Newberry
Library/SuperStock.
Many of the designations used by manufacturers and sellers to distinguish their products are claimed
as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a
trademark claim, the designations have been printed in initial caps or all caps.
Library of Congress Cataloging-in-Publication Data
Katz, Victor J.
A history of mathematics / Victor Katz.—3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-321-38700-7
1. Mathematics—History. I. Title.
QA21.K.33 2009
510.9—dc22
2006049619
Copyright © 2009 by Pearson Education, Inc.
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 the prior written permission of the publisher. Printed in the United States of
America. For information on obtaining permission for use of material in this work, please submit a
written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street,
Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned
.com/legal/permissions.htm.
1 2 3 4 5 6 7 8 9 10—CRW—12 11 10 09 08
Contents
Preface
PART ONE
. . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Ancient Mathematics
Chapter 1
Egypt and Mesopotamia
1.1 Egypt . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Mesopotamia . . . . . . . . . . . . . . . . . . . . . .
1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
1
2
10
27
28
30
Chapter 2
The Beginnings of Mathematics in Greece
32
2.1
2.2
2.3
The Earliest Greek Mathematics . . . . . . . . . . . . . .
The Time of Plato . . . . . . . . . . . . . . . . . . . .
Aristotle . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
33
41
43
47
48
Euclid
3.1 Introduction to the Elements . . . . . . . . . . . . . . . .
3.2 Book I and the Pythagorean Theorem . . . . . . . . . . . .
3.3 Book II and Geometric Algebra . . . . . . . . . . . . . .
3.4 Circles and the Pentagon Construction . . . . . . . . . . . .
3.5 Ratio and Proportion . . . . . . . . . . . . . . . . . . .
3.6 Number Theory . . . . . . . . . . . . . . . . . . . . .
3.7 Irrational Magnitudes . . . . . . . . . . . . . . . . . .
3.8 Solid Geometry and the Method of Exhaustion . . . . . . . .
3.9 Euclid’s Data . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
50
51
53
60
66
71
77
81
83
88
90
92
Chapter 3
vi
Contents
Chapter 4
Archimedes and Apollonius
4.1 Archimedes and Physics . . . . . . . . . . . . . . . . .
4.2 Archimedes and Numerical Calculations . . . . . . . . . . .
4.3 Archimedes and Geometry . . . . . . . . . . . . . . . .
4.4 Conic Sections before Apollonius . . . . . . . . . . . . . .
4.5 The Conics of Apollonius . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
94
96
101
103
112
115
127
131
Chapter 5
Mathematical Methods in Hellenistic Times
133
5.1
5.2
5.3
Astronomy before Ptolemy . . . . . . . . . . . . . . . .
Ptolemy and the Almagest . . . . . . . . . . . . . . . . .
Practical Mathematics . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
134
145
157
168
170
The Final Chapters of Greek Mathematics
6.1 Nicomachus and Elementary Number Theory . . . . . . . . .
6.2 Diophantus and Greek Algebra . . . . . . . . . . . . . . .
6.3 Pappus and Analysis . . . . . . . . . . . . . . . . . . .
6.4 Hypatia and the End of Greek Mathematics . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
172
173
176
185
189
191
192
Chapter 6
PART TWO
Medieval Mathematics
Chapter 7
Ancient and Medieval China
195
7.1
7.2
7.3
7.4
7.5
7.6
196
197
201
209
222
225
226
228
Chapter 8
Introduction to Mathematics in China . . . . . . . . . . . .
Calculations . . . . . . . . . . . . . . . . . . . . . .
Geometry . . . . . . . . . . . . . . . . . . . . . . .
Solving Equations . . . . . . . . . . . . . . . . . . . .
Indeterminate Analysis . . . . . . . . . . . . . . . . . .
Transmission To and From China . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
Ancient and Medieval India
230
8.1
8.2
8.3
231
233
237
Introduction to Mathematics in India . . . . . . . . . . . .
Calculations . . . . . . . . . . . . . . . . . . . . . .
Geometry . . . . . . . . . . . . . . . . . . . . . . .
Contents
vii
Equation Solving . . . . . . . . . . . . . . . . . . . .
Indeterminate Analysis . . . . . . . . . . . . . . . . . .
Combinatorics . . . . . . . . . . . . . . . . . . . . .
Trigonometry . . . . . . . . . . . . . . . . . . . . . .
Transmission To and From India . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
242
244
250
252
259
260
263
Chapter 9
The Mathematics of Islam
9.1 Introduction to Mathematics in Islam . . . . . . . . . . . .
9.2 Decimal Arithmetic . . . . . . . . . . . . . . . . . . .
9.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Combinatorics . . . . . . . . . . . . . . . . . . . . .
9.5 Geometry . . . . . . . . . . . . . . . . . . . . . . .
9.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . .
9.7 Transmission of Islamic Mathematics . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
265
266
267
271
292
296
306
317
318
321
Chapter 10
Mathematics in Medieval Europe
10.1 Introduction to the Mathematics of Medieval Europe . . . . . .
10.2 Geometry and Trigonometry . . . . . . . . . . . . . . . .
10.3 Combinatorics . . . . . . . . . . . . . . . . . . . . .
10.4 Medieval Algebra . . . . . . . . . . . . . . . . . . . .
10.5 The Mathematics of Kinematics . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
324
325
328
337
342
351
359
362
Chapter 11
Mathematics around the World
11.1 Mathematics at the Turn of the Fourteenth Century . . . . . . .
11.2 Mathematics in America, Africa, and the Pacific . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
364
365
370
379
380
8.4
8.5
8.6
8.7
8.8
PART THREE
Chapter 12
Early Modern Mathematics
Algebra in the Renaissance
383
12.1 The Italian Abacists . . . . . . . . . . . . . . . . . . .
12.2 Algebra in France, Germany, England, and Portugal . . . . . .
12.3 The Solution of the Cubic Equation . . . . . . . . . . . . .
385
389
399
viii
Contents
12.4 Viète, Algebraic Symbolism, and Analysis . . . . . . . . . .
12.5 Simon Stevin and Decimal Fractions . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . .
407
414
418
420
Chapter 13
Mathematical Methods in the Renaissance
13.1 Perspective . . . . . . . . . . . . . . . . . . . . . . .
13.2 Navigation and Geography . . . . . . . . . . . . . . . .
13.3 Astronomy and Trigonometry . . . . . . . . . . . . . . .
13.4 Logarithms . . . . . . . . . . . . . . . . . . . . . .
13.5 Kinematics . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
423
427
432
435
453
457
462
464
Chapter 14
Algebra, Geometry, and Probability in the Seventeenth Century
14.1 The Theory of Equations . . . . . . . . . . . . . . . . .
14.2 Analytic Geometry . . . . . . . . . . . . . . . . . . .
14.3 Elementary Probability . . . . . . . . . . . . . . . . . .
14.4 Number Theory . . . . . . . . . . . . . . . . . . . . .
14.5 Projective Geometry . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
467
468
473
487
497
499
501
504
Chapter 15
The Beginnings of Calculus
15.1 Tangents and Extrema . . . . . . . . . . . . . . . . . .
15.2 Areas and Volumes . . . . . . . . . . . . . . . . . . .
15.3 Rectification of Curves and the Fundamental Theorem . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
507
509
514
532
539
541
Chapter 16
Newton and Leibniz
16.1 Isaac Newton . . . . . . . . . . . . . . . . . . . . . .
16.2 Gottfried Wilhelm Leibniz . . . . . . . . . . . . . . . .
16.3 First Calculus Texts . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
543
544
565
575
579
580
PART FOUR
Chapter 17
Modern Mathematics
Analysis in the Eighteenth Century
583
17.1 Differential Equations . . . . . . . . . . . . . . . . . .
17.2 The Calculus of Several Variables . . . . . . . . . . . . . .
584
601
Contents
ix
17.3 Calculus Texts . . . . . . . . . . . . . . . . . . . . .
17.4 The Foundations of Calculus . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
611
628
636
639
Chapter 18
Probability and Statistics in the Eighteenth Century
18.1 Theoretical Probability . . . . . . . . . . . . . . . . . .
18.2 Statistical Inference . . . . . . . . . . . . . . . . . . .
18.3 Applications of Probability . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
642
643
651
655
661
663
Chapter 19
Algebra and Number Theory in the Eighteenth Century
19.1 Algebra Texts . . . . . . . . . . . . . . . . . . . . .
19.2 Advances in the Theory of Equations . . . . . . . . . . . .
19.3 Number Theory . . . . . . . . . . . . . . . . . . . . .
19.4 Mathematics in the Americas . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
665
666
671
677
680
683
684
Chapter 20
Geometry in the Eighteenth Century
20.1 Clairaut and the Elements of Geometry . . . . . . . . . . . .
20.2 The Parallel Postulate . . . . . . . . . . . . . . . . . .
20.3 Analytic and Differential Geometry . . . . . . . . . . . . .
20.4 The Beginnings of Topology . . . . . . . . . . . . . . . .
20.5 The French Revolution and Mathematics Education . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
686
687
689
695
701
702
706
707
Chapter 21
Algebra and Number Theory in the Nineteenth Century
21.1 Number Theory . . . . . . . . . . . . . . . . . . . . .
21.2 Solving Algebraic Equations . . . . . . . . . . . . . . . .
21.3 Symbolic Algebra . . . . . . . . . . . . . . . . . . . .
21.4 Matrices and Systems of Linear Equations . . . . . . . . . .
21.5 Groups and Fields—The Beginning of Structure . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
709
711
721
730
740
750
759
761
Chapter 22
Analysis in the Nineteenth Century
764
22.1 Rigor in Analysis . . . . . . . . . . . . . . . . . . . .
22.2 The Arithmetization of Analysis . . . . . . . . . . . . . .
22.3 Complex Analysis . . . . . . . . . . . . . . . . . . . .
766
788
795
x
Contents
22.4 Vector Analysis . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
807
813
815
Chapter 23
Probability and Statistics in the Nineteenth Century
23.1 The Method of Least Squares and Probability Distributions . . .
23.2 Statistics and the Social Sciences . . . . . . . . . . . . . .
23.3 Statistical Graphs . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
818
819
824
828
831
831
Chapter 24
Geometry in the Nineteenth Century
24.1 Differential Geometry . . . . . . . . . . . . . . . . . .
24.2 Non-Euclidean Geometry . . . . . . . . . . . . . . . . .
24.3 Projective Geometry . . . . . . . . . . . . . . . . . . .
24.4 Graph Theory and the Four-Color Problem . . . . . . . . . .
24.5 Geometry in N Dimensions . . . . . . . . . . . . . . . .
24.6 The Foundations of Geometry . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
833
835
839
852
858
862
867
870
872
Chapter 25
Aspects of the Twentieth Century and Beyond
25.1 Set Theory: Problems and Paradoxes . . . . . . . . . . . .
25.2 Topology . . . . . . . . . . . . . . . . . . . . . . .
25.3 New Ideas in Algebra . . . . . . . . . . . . . . . . . .
25.4 The Statistical Revolution . . . . . . . . . . . . . . . . .
25.5 Computers and Applications . . . . . . . . . . . . . . . .
25.6 Old Questions Answered . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . .
874
876
882
890
903
907
919
926
928
Appendix A
Using This Textbook in Teaching Mathematics
A.1 Courses and Topics . . . . . . . . . . . . . . . . . . .
A.2 Sample Lesson Ideas to Incorporate History . . . . . . . . . .
A.3 Time Line . . . . . . . . . . . . . . . . . . . . . . .
931
931
935
939
General References in the History of Mathematics
. . . . . . . . .
945
Answers to Selected Exercises . . . . . . . . . . . . . . . . .
949
Index and Pronunciation Guide . . . . . . . . . . . . . . . . .
961
Preface
In A Call For Change: Recommendations for the Mathematical Preparation of Teachers of
Mathematics, the Mathematical Association of America’s (MAA) Committee on the Mathematical Education of Teachers recommends that all prospective teachers of mathematics in
schools
. . . develop an appreciation of the contributions made by various cultures to the growth and
development of mathematical ideas; investigate the contributions made by individuals, both female
and male, and from a variety of cultures, in the development of ancient, modern, and current
mathematical topics; [and] gain an understanding of the historical development of major school
mathematics concepts.
According to the MAA, knowledge of the history of mathematics shows students that
mathematics is an important human endeavor. Mathematics was not discovered in the polished
form of our textbooks, but was often developed in an intuitive and experimental fashion in
order to solve problems. The actual development of mathematical ideas can be effectively
used in exciting and motivating students today.
This textbook grew out of the conviction that both prospective school teachers of mathematics and prospective college teachers of mathematics need a background in history to teach
the subject more effectively. It is therefore designed for junior or senior mathematics majors
who intend to teach in college or high school, and it concentrates on the history of those topics
typically covered in an undergraduate curriculum or in elementary or high school. Because
the history of any given mathematical topic often provides excellent ideas for teaching the
topic, there is sufficient detail in each explanation of a new concept for the future (or present)
teacher of mathematics to develop a classroom lesson or series of lessons based on history.
In fact, many of the problems ask readers to develop a particular lesson. My hope is that
students and prospective teachers will gain from this book a knowledge of how we got here
from there, a knowledge that will provide a deeper understanding of many of the important
concepts of mathematics.
Distinguishing Features
FLEXIBLE ORGANIZATION
Although the text’s chief organization is by chronological period, the material is organized
topically within each period. By consulting the detailed subsection headings, the reader can
choose to follow a particular theme throughout history. For example, to study equation solving
one could consider ancient Egyptian and Babylonian methods, the geometrical solution
methods of the Greeks, the numerical methods of the Chinese, the Islamic solution methods
for cubic equations by use of conic sections, the Italian discovery of an algorithmic solution
of cubic and quartic equations, the work of Lagrange in developing criteria for methods of
xii
Preface
solution of higher degree polynomial equations, the work of Gauss in solving cyclotomic
equations, and the work of Galois in using permutations to formulate what is today called
Galois theory.
FOCUS ON TEXTBOOKS
It is one thing to do mathematical research and discover new theorems and techniques. It
is quite another to elucidate these in such a way that others can learn them. Thus, in many
chapters there is a discussion of one or more important texts of the time. These are the works
from which students learned the important ideas of the great mathematicians. Today’s students
will see how certain topics were treated and will be able to compare these treatments to those
in current texts and see the kinds of problems students of years ago were expected to solve.
APPLICATIONS OF MATHEMATICS
Two chapters, one for the Greek period and one for the Renaissance, are devoted entirely to
mathematical methods, the ways in which mathematics was used to solve problems in other
areas of study. A major part of both chapters deals with astronomy since in ancient times
astronomers and mathematicians were usually the same people. To understand a substantial
part of Greek mathematics, it is crucial also to understand the Greek model of the heavens
and how mathematics was used in applying this model to give predictions. Similarly, I
discuss the Copernicus-Kepler model of the heavens and consider how mathematicians of the
Renaissance applied mathematics to its study. I also look at the applications of mathematics
to geography during these two time periods.
NON-WESTERN MATHEMATICS
A special effort has been made to consider mathematics developed in parts of the world other
than Europe. Thus, there is substantial material on mathematics in China, India, and the
Islamic world. In addition, Chapter 11 discusses the mathematics of various other societies
around the world. Readers will see how certain mathematical ideas have occurred in many
places, although not perhaps in the context of what we in the West call “mathematics.”
TOPICAL EXERCISES
Each chapter contains many exercises, organized in order of the chapter’s topics. Some
exercises are simple computational ones, while others help to fill gaps in the mathematical
arguments presented in the text. For Discussion exercises are open-ended questions, which
may involve some research to find answers. Many of these ask students to think about how
they would use historical material in the classroom. Even if readers do not attempt many of
the exercises, they should at least read them to gain a fuller understanding of the material
of the chapter. (Answers to the odd numbered computational problems as well as some odd
numbered “proof” problems are included at the end of the book.)
FOCUS ESSAYS
Biographies For easy reference, many biographies of the mathematicians whose work is
discussed are in separate boxes. Although women have for various reasons not participated
in large numbers in mathematical research, biographies of several important women mathematicians are included, women who succeeded, usually against heavy odds, in contributing
to the mathematical enterprise.
Preface
xiii
Special Topics Sidebars on special topics also appear throughout the book. These include
such items as a treatment of the question of the Egyptian influence on Greek mathematics, a
discussion of the idea of a function in the work of Ptolemy, a comparison of various notions of
continuity, and several containing important definitions collected together for easy reference.
ADDITIONAL PEDAGOGY
At the start of each chapter is a relevant quotation and a description of an important mathematical “event.” Each chapter also contains an annotated list of references to both primary
and secondary sources from which students can obtain more information. Given that a major
audience for this text is prospective teachers of secondary or college-level mathematics, I
have provided an appendix giving suggestions for using the text material in teaching mathematics. It contains a detailed list to correlate the history of various topics in the secondary
and college curriculum to sections in the text; there are suggestions for organizing some of
this material for classroom use; and there is a detailed time line that helps to relate the mathematical discoveries to other events happening in the world. On the back inside cover there
is a chronological listing of most of the mathematicians discussed in the book. Finally, given
that students may have difficulty pronouncing the names of some mathematicians, the index
has a special feature: a phonetic pronunciation guide.
Prerequisites
A knowledge of calculus is sufficient to understand the first 16 chapters of the text. The
mathematical prerequisites for later chapters are somewhat more demanding, but the various
section titles indicate clearly what kind of mathematical knowledge is required. For example,
a full understanding of chapters 19 and 21 will require that students have studied abstract
algebra.
Course Flexibility
The text contains more material than can be included in a typical one-semester course in
the history of mathematics. In fact, it includes adequate material for a full year course, the
first half being devoted to the period through the invention of calculus in the late seventeenth
century and the second half covering the mathematics of the eighteenth, nineteenth, and twentieth centuries. However, for those instructors who have only one semester, there are several
ways to use this book. First, one could cover most of the first twelve chapters and simply
conclude with calculus. Second, one could choose to follow one or two particular themes
through history. (The table in the appendix will direct one to the appropriate sections to include when dealing with a particular theme.) Among the themes that could be followed are
equation solving; ideas of calculus; concepts of geometry; trigonometry and its applications
to astronomy and surveying; combinatorics, probability, and statistics; and modern algebra
and number theory. For a thematic approach, I would suggest making every effort to include
material on mathematics in the twentieth century, to help students realize that new mathematics is continually being discovered. Finally, one could combine the two approaches and
cover ancient times chronologically, and then pick a theme for the modern era.
xiv
Preface
New for this Edition
The generally friendly reception of this text’s first two editions encouraged me to maintain
the basic organization and content. Nevertheless, I have attempted to make a number of
improvements, both in clarity and in content, based on comments from many users of
those editions as well as new discoveries in the history of mathematics that have appeared
in the recent literature. To make the book somewhat easier to use, I have reorganized
some material into shorter chapters. There are minor changes in virtually every section,
but the major changes from the second edition include: new material about Archimedes
discovered in analyzing the palimpsest of the Method; a new section on Ptolemy’s Geography;
more material in the Chinese, Indian, and Islamic chapters based on my work on the new
Sourcebook dealing with the mathematics of these civilizations, as well as the ancient
Egyptian and Babylonian ones; new material on statistics in the nineteenth and twentieth
centuries; and a description of the eighteenth-century translation into the differential calculus
of some of Newton’s work in the Principia. The text concludes with a brief description of
the solution to the first Clay Institute problem, the Poincarè conjecture. I have attempted to
correct all factual errors from the earlier editions without introducing new ones, yet would
appreciate notes from anyone who discovers any remaining errors. New problems appear in
every chapter, some of them easier ones, and references to the literature have been updated
wherever possible. Also, a few new stamps were added as illustrations. One should note,
however, that any portraits on these stamps—or indeed elsewhere—purporting to represent
mathematicians before the sixteenth century are fictitious. There are no known representations
of any of these people that have credible evidence of being authentic.
Acknowledgments
Like any book, this one could not have been written without the help of many people.
The following people have read large sections of the book at my request and have offered
many valuable suggestions: Marcia Ascher (Ithaca College); J. Lennart Berggren (Simon
Fraser University); Robert Kreiser (A.A.U.P.); Robert Rosenfeld (Nassau Community College); John Milcetich (University of the District of Columbia); Eleanor Robson (Cambridge
University); and Kim Plofker (Brown University). In addition, many people made detailed
suggestions for the second and third editions. Although I have not followed every suggestion,
I sincerely appreciate the thought they gave toward improving the book. These people include
Ivor Grattan-Guinness, Richard Askey, William Anglin, Claudia Zaslavsky, Rebekka Struik,
William Ramaley, Joseph Albree, Calvin Jongsma, David Fowler, John Stillwell, Christian
Thybo, Jim Tattersall, Judith Grabiner, Tony Gardiner, Ubi D’Ambrosio, Dirk Struik, and
David Rowe. My heartfelt thanks to all of them.
The many reviewers of sections of the manuscript for each of the editions have also provided great help with their detailed critiques and have made this a much better book than
it otherwise could have been. For the first edition, they were Duane Blumberg (University
of Southwestern Louisiana); Walter Czarnec (Framington State University); Joseph Dauben
(Lehman College–CUNY); Harvey Davis (Michigan State University); Joy Easton (West
Virginia University); Carl FitzGerald (University of California, San Diego); Basil Gordon
(University of California, Los Angeles); Mary Gray (American University); Branko Grun-
Preface
xv
baum (University of Washington); William Hintzman (San Diego State University); Barnabas
Hughes (California State University, Northridge); Israel Kleiner (York University); David E.
Kullman (Miami University); Robert L. Hall (University of Wisconsin, Milwaukee); Richard
Marshall (Eastern Michigan University); Jerold Mathews (Iowa State University); Willard
Parker (Kansas State University); Clinton M. Petty (University of Missouri, Columbia);
Howard Prouse (Mankato State University); Helmut Rohrl (University of California, San
Diego); David Wilson (University of Florida); and Frederick Wright (University of North
Carolina at Chapel Hill).
For the second edition, the reviewers were Salvatore Anastasio (State University of New
York, New Paltz); Bruce Crauder (Oklahoma State University); Walter Czarnec (Framingham State College); William England (Mississippi State University); David Jabon (Eastern
Washington University); Charles Jones (Ball State University); Michael Lacey (Indiana University); Harold Martin (Northern Michigan University); James Murdock (Iowa State University); Ken Shaw (Florida State University); Svere Smalo (University of California, Santa
Barbara); Domina Eberle Spencer (University of Connecticut); and Jimmy Woods (North
Georgia College).
For the third edition, the reviewers were Edward Boamah (Blackburn College); Douglas
Cashing (St. Bonaventure University); Morley Davidson (Kent State University); Martin J.
Erickson (Truman State University); Jian-Guo Liu (University of Maryland); Warren William
McGovern (Bowling Green State University); Daniel E. Otero (Xavier University); Talmage
James Reid (University of Mississippi); Angelo Segalla (California State University, Long
Beach); Lawrence Shirley (Towson University); Agnes Tuska (California State University at
Fresno); Jeffrey X. Watt (Indiana University–Purdue University Indianapolis).
I have also benefited greatly from conversations with many historians of mathematics
at various forums, including numerous history sessions at the annual joint meetings of the
Mathematical Association of America and the American Mathematical Society. Among
those who have helped at various stages (and who have not been mentioned earlier) are
V. Frederick Rickey, United States Military Academy; Florence Fasanelli, AAAS; Israel
Kleiner, York University; Abe Shenitzer, York University; Frank Swetz, Pennsylvania State
University, and Janet Beery, University of Redlands. In addition, I want to thank Karen Dee
Michalowicz, of the Langley School, who showed me how to reach current and prospective
high school teachers, and whose untimely death in 2006 was such a tragedy. In addition,
I learned a lot from all the people who attended various sessions of the Institute in the
History of Mathematics and Its Use in Teaching, as well as members of the 2007 PREP
workshop on Asian mathematics. My students in History of Mathematics classes (and others)
at the University of the District of Columbia have also helped me clarify many of my
ideas. Naturally, I welcome any additional comments and correspondence from students and
colleagues elsewhere in an effort to continue to improve this book.
My former editors at Harper Collins, Steve Quigley, Don Gecewicz, and George Duda,
who helped form the first edition, and Jennifer Albanese, who was the editor for the second
edition, were very helpful. And I want to particularly thank my new editor, Bill Hoffman, for
all his suggestions and his support during the creation of both the brief edition and this new
third edition. Elizabeth Bernardi at Pearson Addison-Wesley has worked hard to keep me
on deadline, and Jean-Marie Magnier has caught several errors in the answers to problems,
xvi
Preface
for which I thank her. The production staff of Paul C. Anagnostopoulos, Jennifer McClain,
Laurel Muller, Yonie Overton, and Joe Snowden have cheerfully and efficiently handled their
tasks to make this book a reality.
Lastly, I want to thank my wife Phyllis for all her love and support over the years, during
the very many hours of working on this book and, of course, during the other hours as well.
Victor J. Katz
Silver Spring, MD
May 2008
1
PART ONE
Ancient Mathematics
chapter
Egypt and Mesopotamia
Accurate reckoning. The entrance into the
knowledge of all existing things and all
obscure secrets.
—Introduction to Rhind
Mathematical Papyrus1
M
esopotamia: In a scribal school in Larsa some 3800 years
ago, a teacher is trying to develop mathematics problems
to assign to his students so they can practice the ideas just
introduced on the relationship among the sides of a right triangle.
The teacher not only wants the computations to be difficult enough
to show him who really understands the material but also wants the
answers to come out as whole numbers so the students will not b
