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A History of

MATHEMATICS

An Introduction

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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

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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

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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