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AN INTRODUCTION SECOND EDITION

9 L

^

F

l

^rL I /

PH

TIME LINE FOR THE HISTORY OF MATHEMATICS 3000

1000

B.C.E.

3000-2000

1000-500

B.C.E.

India:

hieroglyphic writing of numbers; Building of pyramids at Giza Iraq:

numbers Mesopotamia

2000-1000

in

Greece: Beginnings

India:

mathematics; Ptolemy and

techniques Iraq:

Jordan: Nicomachus and number theory

Nehemiah and

papyri

written; Ideas of linear

equations, volumes, areas

United States: Geometry used architectural designs

axiomatic mathematics;

Egypt: Diophantus and number

Discovery of incommensurabil-

theory; Hypatia and

Spain: Gerbert learns Arabic

commentaries

number system

Eudoxus and proportionality

equations, systems of equations

300-0

Egypt: Euclid and the Elements

Mississippian civilization

China: Liu Hui and mathematical

surveying techniques

Italy:

B.C.E.

400-800 Italy:

Boethius and elementary

mathematics

Archimedes and

Mexico: Development of Mayan numeration and astronomy

theoretical physics

Egypt: Apollonius and conic sections

India:

Aryabhata and

trigonometry; Brahmagupta

Turkey: Hipparchus and

and indeterminate analysis;

trigonometry

Development

of

Hindu-Arabic

decimal place-value number

system China: First tangent tables

1000

in

200-400

ity;

B.C.E.

Abu Kamil and advanced

algebraic techniques

practical

b.c.e.

China: Square and cube roots, systems of linear equations

3000

first

Greece: Plato, Aristotle, and

Cuneiform mathematical tablets written with Pythagorean theorem, quadratic Iraq:

of algebraic

AI-KhwarizmT and

Egypt: Israel:

geometry 500-300

Moscow

Development

algebra text

of

theoretical geometry

B.C.E.

Egypt: Rhind and

800-1000

C.E.

Egypt: Heron and practical

astronomy

China: Rod numerals, Pythagorean theorem

Beginnings of cuneiform

writing of

0-200

B.C.E.

Square root calculations, Pythagorean theorem

Egypt: Beginnings of

800

0

b.c.e.

B.C.E.

0

800

in

BARCODE ON NEXT PAGE i

1600

1200

1000

1200-1400

1000-1200 Iraq: AI-KarajT, induction,

and

Egypt: Ibn al-Haytham, of

sums

powers, and volumes

of

of

Kepler,

1800-1900

Newton, and

Algebraic number theory

celestial

physics

France: Jordanus and advanced

Descartes, Fermat, and analytic

algebra; Levi ben Gerson and

geometry

Galois theory

Groups and

Napier, Briggs, and logarithms

of

and the theory of equations Girard, Descartes,

England: Velocity, acceleration, and the mean speed theorem

cubic

equations

fields

Quaternions and the discovery

Oresme and

kinematics

Omar Khayyam and the

geometric solution

and

trigonometry

induction;

paraboloids Iran:

1600-1700

Iran: NasTr al-DTn al-TusT

Pascal triangle

1800

noncommutative algebra

Theory

of

matrices

The arithmetization

of analysis

Pascal, Fermat, and elementary

and spherical trigonometry; Bhaskara and the Pell equation

China: Chinese remainder

India: Al- Biruni

theorem; Solution

of

Peru: Quipus used for record

keeping

ibn Ezra

India:

combinatorics

Leonardo

of Pisa

Algebraic solution of the

Germany: Perspective and geometry

United States. Astronomical

Anasazi buildings

the Southwest

England:

New

algebra and

Projective geometry of

techniques for

Foundations

of

geometry

differential equations

1900-2000

Development

Set theory

Growth

Attempts to give

Algebraization of mathematics

logically

Lagrange and the analysis

heliocentric system \l\ete

Functions of several variables

of the

solution of polynomial equations

and algebraic

symbolism

1200

of

topology

Influence of computers

calculus

Poland: Copernicus and the

France:

of the calculus of

correct foundations to the

trigonometry texts

Zimbabwe: Construction of Great Zimbabwe structures

1000

geometry

Non-Euclidean geometry

cubic equation

mathematics

in

and the

solving ordinary and partial

introduction of Islamic

in

Differential

Leibniz,

Development

and arctangent

and Italy:

alignments

Newton,

1700-1800

Discovery of power series

for sine, cosine, Italy:

Vector analysis

1400-1600

and

complex

analysis projective geometry

invention of calculus

Spain: Arabic works translated

of

Pascal, Desargues, and

China: Pascal triangle used to

Abraham

probability

polynomial

equations

solve equations

in Latin;

Development

1600

1800

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31

A HISTORY OF

WITHDRAWN FROM BURLINGAME PUBLIC LIBRARY

MATHEMATICS An Introduction

'°'

A HISTORY OF

MATHEMATICS An Introduction Second Edition

VICTOR

J.

KATZ

University of the District of Columbia

Addison Wesley

Longman An

imprint of

Addison Wesley Longman,

Inc.

Reading, Massachusetts • Menlo Park, California • New York • Harlow, England Don Mills, Ontario • Sydney • Mexico City • Madrid • Amsterdam

To Phyllis, for long

talks,

long walks,

and afternoon naps

Reprinted with corrections, November 1998

Sponsoring Editor: Jennifer Albanese Production Supervisor: Rebecca Malone Project Manager: Barbara Pendergast

Prepress Services Buyer: Caroline Fell

Manufacturing Supervisor: Ralph Mattivello

Design Direction: Susan Carsten Text and Cover Design: Rebecca Lloyd

Cover Photo:

Lemna

© SuperStock

Art Editor: Sarah E. Mendelsohn

Composition and Prepress Services: Integre Technical Publishing Co.,

Inc.

Library of Congress Cataloging-in-Publication Data Katz, Victor

J.

A history of mathematics: 2nd

an introduction

/

Victor

J.

Katz.

ed. p.

cm.

Includes bibliographical references and index.

ISBN 0-321-01618-1 1.

Mathematics

QA21.K33 5 10'. 9

—

History.

I.

Title.

1998

—dc21

98-9273

CIP Copyright

©

1998 by Addison-Wesley Educational Publishers,

All rights reserved. in

No part

of this publication

may

Inc.

be reproduced, stored in a retrieval system, or transmitted,

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.

4

5

6

7

8

9

10— MA—00

Contents Preface

ix

PART ONE

Mathematics Before the Sixth Century chapter

i

Ancient Mathematics

1

1.1

Ancient Civilizations

1.2

Counting

2

4

1.3

Arithmetic Computations

1.4

Linear Equations

1.5

Elementary Geometry

1

.6

8

14

19

25

Astronomical Calculations

27

1.7

Square Roots

1.8

The Pythagorean Theorem

1.9

Quadratic Equations

30

35

CHAPTER 2 The Beginnings of Mathematics

CHAPTER

3

2.1

The

2.2

The Time of Plato

2.3

Aristotle

2.4

Euclid and the Elements

2.5

Euclid's Other

Earliest

in

Greek Mathematics

Greece 47

52

54

Works

58

95

Archimedes and Apollonius

102

3.1

Archimedes and Physics

3.2

Archimedes and Numerical Calculations

103

3.3

Archimedes and Geometry

3.4

Conics Before Apollonius

116

3.5

The Conics of Apollonius

117

111

46

VI

Contents

chapter 4 Mathematical Methods 4.

chapter

s

1

in Hellenistic

Astronomy Before Ptolemy

4.2

Ptolemy and the Almagest

4.3

Practical

The

135

145

156

Mathematics

Final Chapters of

Times

136

Greek Mathematics

168

5.1

Nicomachus and Elementary Number Theory

5.2

Diophantus and Greek Algebra

5.3

Pappus and Analysis

171

173

183

PART TWO Medieval Mathematics: 500-1400 chapter 6 Medieval China and India

192

Introduction to Medieval Chinese Mathematics

192

6.2

The Mathematics of Surveying and Astronomy

193

6.3

Indeterminate Analysis

6.4

Solving Equations

6.5

Introduction to the Mathematics of Medieval India

6.6

Indian Trigonometry

6.7

Indian Indeterminate Analysis

6.8

Algebra and Combinatorics

6.9

The Hindu-Arabic Decimal Place- Value System

6.

1

197

202

chapter 7 The Mathematics of Islam

chapter

s

7.1

Decimal Arithmetic

7.2

Algebra

7.3

Combinatorics

7.4

Geometry

7.5

Trigonometry

Mathematics

218

225

238

243

in

263

268 274

288

Medieval Europe

Geometry and Trigonometry

8.2

Combinatorics

292

300

307

8.3

Medieval Algebra

8.4

The Mathematics of Kinematics

interchapter Mathematics Around the World

1.2

230

240

8.1

1. 1

210

212

Mathematics

Mathematics

at the

in

314

327

Turn of the Fourteenth Century

America, Africa, and the

Pacific

327 332

Vll

Contents

PART THREE Early

Modern Mathematics: 1400-1700

chapter 9 Algebra 9.

chapter

chapter

io

it

The

1

343

Abacists

Italian

Germany, England, and Portugal

Algebra

9.3

The Solution of

9.4

The Work of Viete and Stevin

in France,

367

Renaissance

in the

Perspective

Geography and Navigation

10.3

Astronomy and Trigonometry

10.4

Logarithms

416

10.5

Kinematics

420

393 398

Geometry, Algebra, and Probability Analytic Geometry

.2

The Theory of Equations

1 1

.3

Elementary Probability

Number Theory Projective

445

448

460

468

The Beginnings of Calculus

469

12.1

Tangents and Extrema

12.2

Areas and Volumes

12.3

Power

12.4

Rectification of Curves and the

12.5

Isaac

12.6

Gottfried

12.7

First Calculus Texts

Newton

Seventeenth Century

458

Geometry

Series

in the

432

1 1

.5

385

389

10.1

10.2

1 1

348

358

Cubic Equation

the

Mathematical Methods

11.4

12

342

Renaissance

9.2

11.1

chapter

in the

475

492 Fundamental Theorem

503

Wilhelm Leibniz

522

532

PART FOUR

Modern Mathematics: 1700-2000 chapter

13

Analysis in the Eighteenth Century 13.1

Differential Equations

544

545

560

13.2

Calculus Texts

13.3

Multiple Integration

13.4

Partial Differential Equations:

13.5

The Foundations of Calculus

574

The Wave Equation 582

578

496

431

viii

Contents

chapter

chapter

14

is

Probability. Algebra,

chapter

chapter

16

Eighteenth Century

14.2

Algebra and Number Theory

14.3

Geometry

14.4

The French Revolution and Mathematics Education

14.5

Mathematics

Algebra

610

621

in the

Number Theory

650

652 662

15.2

Solving Algebraic Equations

15.3

Groups and Fields

15.4

Symbolic Algebra

15.5

Matrices and Systems of Linear Equations

Analysis

in the

—The Beginning of

687

706

16.2

The Arithmetization of Analysis

16.3

Complex Analysis

16.4

Vector Analysis

16.5

Probability and Statistics

in the

670

704

Nineteenth Century

Rigor

Analysis

Structure

677

16.1

in

637

640

Americas

Nineteenth Century

in the

n Geometry

is

in the

Probability

15.1

chapter

and Geometry

597

14.1

729

737 746 753

Nineteenth Century

Geometry

766

768

17.1

Differential

17.2

Non-Euclidean Geometry

17.3

Projective

Geometry

17.4

Geometry

in

17.5

The Foundations of Geometry

772

785

N Dimensions

Aspects of the Twentieth Century

792

797

805

18.1

Set Theory: Problems and Paradoxes

807

814

18.2

Topology

18.3

New

18.4

Computers and Applications

Ideas in Algebra

822

834

ANSWERS TO SELECTED PROBLEMS 857 GENERAL REFERENCES IN THE HISTORY OF MATHEMATICS INDEX AND PRONUNCIATION GUIDE l-l

863

596

Preface

APPROACH AND GUIDING PHILOSOPHY A Call For Change: Recommendations for the Mathematical Preparation of Teachers of Mathematics the Mathematical Association of America’s (MAA) Committee on the MathIn

,

ematical Education of Teachers in

recommends

that all prospective teachers of

mathematics

schools develop an appreciation of the contributions made by various cultures to the growth and devel-

opment 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

that mathematics

is

MAA,

knowledge of

an important

human

the history of mathematics

shows students

endeavor. Mathematics was not discovered in the

polished form of our textbooks, but often developed in intuitive and experimental fashion out of a need to solve problems. effectively used in exciting

This

new textbook

The

actual

development of mathematical ideas can be

and motivating students today.

in the history

of mathematics grew out of the conviction that not

only prospective school teachers of mathematics but also prospective college teachers of

mathematics need a background students.

It is

in history to

teach the subject

more

effectively to their

therefore designed for junior or senior mathematics majors

who

intend to

teach in college or high school and thus 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.

many of

the

problems ask the reader

to

develop a particular lesson.

student and prospective teacher will gain from this

from

there, a

knowledge

that will provide a

book

a

My

hope

is

In fact,

that the

knowledge of how we got here

deeper understanding of

many of the

important

concepts of mathematics.

IX

X

Preface

DISTINGUISHING FEATURES Flexible Organization Although the chief organization of the hook the material

By

organized topically.

is

is

by chronological period, within each period

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 in

methods of solution of higher degree polynomial equations, the work of Gauss

solving cyclotomic equations, and the work of Galois in using permutations to formulate

what

today called Galois theory.

is

Focus on Textbooks There is

is

an emphasis throughout the book on the important textbooks of various periods.

one thing to do mathematical research and discover new theorems and techniques.

quite another to elucidate these in a therefore, there

the

is

way

It

It is

can leam them. In nearly every chapter, more important texts of the time. These will be

that others

a discussion of one or

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.

Astronomy and Mathematics Two

chapters are devoted entirely to mathematical methods, that

mathematics was used

problems

to solve

in

is,

other areas of endeavor.

to the

A

ways

in

which

substantial part of

both of these chapters, one for the Greek period and one for the Renaissance, deals with

astronomy. In people.

It

is

fact, in

ancient times astronomers and mathematicians were usually the

crucial to the understanding of a substantial part of

understand the Greek model of the heavens and

model

to give predictions. Similarly,

heavens and see

how mathematicians

we

how mathematics was used

will discuss the

same

Greek mathematics in

applying

to

this

Copernicus-Kepler model of the

of the Renaissance applied mathematics to

its

study.

Non-Western Mathematics A

special effort has been

made

to consider

other than Europe. Thus, there

and the Islamic world. There

is

is

mathematics developed

substantial material

also an “interchapter" in

of the mathematics in the major civilizations

That comparison

is

at

in parts

on mathematics

in

of the world China, India,

which a comparison

is

made

about the turn of the fourteenth century.

followed by a discussion of the mathematics of various other societies

XI

Preface

around the world. The reader will see

many

how

certain mathematical ideas have occurred in

what we

places, although not perhaps in the context of

West

in the

call

“mathematics.”

Topical Exercises Each chapter contains many

exercises, collected by topic for easy access.

exercises are simple computational ones while others help to

fill

Some

of the

the gaps in the mathematical

in the text. For Discussion exercises are open-ended questions for which may involve some research to find answers. Many of these ask students think about how they would use historical material in the classroom. (Answers to most

arguments presented discussion, to

of the computational exercises are provided in the answer section.) Even attempt

many of the

exercises, they should at least read

them

if

readers do not

to gain a fuller understanding

of the material of the chapter.

Focus Essays For easy reference, many biographies of the mathematicians whose work

Biographies

discussed are in separate boxes. In particular, although participated in large

women

numbers

in

Special Topics

is

have for various reasons not

mathematical research, biographies of several important

mathematicians are included,

in contributing to the

women

women who

succeeded, usually against heavy odds,

mathematical enterprise.

There are also boxes on special topics scattered 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, and

a

comparison of various notions of continuity. There are also boxes containing important definitions collected together for easy reference.

Additional Pedagogy

Each chapter begins with a relevant quotation and

a description

of an important mathematical “event.” At the end of each chapter, a brief chronology of the mathematicians discussed will help students organize their knowledge.

also contains an annotated

list

which the students can obtain more information. of mathematics location of

in the inside front

some of the important

cover and a

Finally, there is a

map

—

in the inside

time line of the history

back cover indicating the

places mentioned in the text. Finally, given that students

may have difficulty pronouncing the names of some feature

Each chapter

of references to both primary and secondary sources from

mathematicians, the index has a special

a phonetic pronunciation guide.

PREREQUISITES A

working knowledge of one year of calculus

chapters of the

text.

demanding, but the

The mathematical titles

is

sufficient to understand the first twelve

prerequisites for the later chapters are

somewhat more

of the various sections indicate clearly what kind of mathematical

— Xll

Preface

knowledge

is

required. For example, a full understanding of Chapters

1

4 and

1

5 will require

that the student has studied abstract algebra.

NEW FOR THIS EDITION The generally

friendly reception to the

first

edition of this

the basic organization and content. Nevertheless,

improvements, both first

in clarity

edition as well as

new

and

in content,

book encouraged me

to maintain

have attempted to make a number of

based on the comments of many users of the

discoveries in the history of mathematics which have appeared

There are minor changes

in the recent literature.

I

changes include new material on combinatorics

in virtually

in the

every section, but the major

Islamic tradition, Newton’s derivation

of his system of the world, linear algebra in the nineteenth and twentieth centuries, and statistical ideas in the

without introducing

nineteenth century.

new

remaining errors. There are

the references to the literature have

new stamps

as illustrations.

or indeed elsewhere are fictitious.

—

have attempted to correct

I

errors of fact

all

would appreciate notes from anyone who discovers any new problems in every chapter, some of them easier ones, and

ones, but

been updated wherever possible. There are also a few

One should

note, however, that any portraits

on these stamps

purporting to represent mathematicians before the sixteenth century

There are no known representations of any of these people

that

have credible

evidence of being authentic.

COURSE FLEXIBILITY There

is far

more material

in the history first

in this text

of mathematics. In

than can be included in a typical one-semester course

fact, there is

adequate material for a

full

year course, the

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. For those instructors several

ways

to use this book. First,

who have

only one semester, however, there are

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.

Some

Equation Solving:

1.4,

possible themes with the appropriate section 1.8,

numbers

are

1.9, 2.4.3, 2.5, 5.2, 6.3, 6.4, 6.7, 6.8, 7.2, 8.3, 9.3, 9.4,

11.2, 14.2.4, 15.2

Ideas of Calculus: 2.3.2, 2.3.3, 2.4.9,

3.2, 3.3, 7.2.4, 7.4.4, 8.4, 10.5, 12, 13, 16.1,

16.2, 16.3, 16.4

Concepts of Geometry:

1.5, 1.8, 2.1.2, 2.2, 2.4, 3.3, 3.4, 3.5, 4.3, 5.3, 7.4, 8.1, 10.1,

11.1, 11.5, 14.3, 17, 18.2

Trigonometry, Astronomy, and Surveying:

1

.6, 4.

1

,

4.2, 6.2, 6.6, 7.5, 8.1,1 0.2, 10.3,

12.5.6, 13.1.3

Combinatorics, Probability, and

Linear Algebra:

Statistics: 6.8, 7.3, 8.2,

1

1.3, 14.1, 16.5

1.4, 14.2.2, 14.2.4, 15.5, 17.4, 18.3.3, 18.4.7

Number Theory: 2.1.1, 2.4.7, 5.1, 1.4, 14.2.3, 15.1 Modern Algebra: 6.8, 7.2, 8.3, 9.1, 9.2, 14.2, 15.2, 1

18.4.8

15.3, 15.4, 18.3, 18.4.4, 18.4.6,

Xlll

Preface

Third, one could cover in detail most of the

first

ten chapters

ideas from the later chapters, again following a particular theme.

and then pick selected

One could

also assign

various sections for individual or small-group reading assignments and reports.

ACKNOWLEDGMENTS Like any book,

this

one could not have been written without the help of many people. The

following people contributed to the

Marcia Ascher (Ithaca College), Kreiser (A.A.U.P.

),

J.

first

edition and their input continues to impact the text:

Lennart Berggren (Simon Fraser University), Robert

Robert Rosenfeld (Nassau Community College), and John Milcetich

(University of the District of Columbia).

Many

people made detailed suggestions for the second edition. Although

followed every one of them (and

may come

to regret that),

I

have not

1

sincerely appreciate the

thought they gave to improving the book. These people include Ivor Grattan-Guinness,

Kim Plofker, Eleanor Robson,

Richard Askey, William Anglin. Claudia Zaslavsky, Rebekka

William Ramaley, Joseph Albree, Calvin Jongsma, David Fowler, John

Struik,

Christian Thybo, Jim Tattersall, Judith Grabiner,

Stillwell,

Tony Gardiner, Ubi D’Ambrosio, Dirk

and David Rowe. My heartfelt thanks to all of them. The many reviewers of sections of the manuscript have also provided great help with their detailed critiques and have made this a much better book than it otherwise could have Struik,

been. First Edition Reviewers:

Duane Blumberg, University of Southwestern Louisiana;

Walter Czamec, Framingham State University; Joseph Dauben, Herbert

CUNY; sity;

Lehman College-

Harvey Davis, Michigan State University; Joy Easton, West Virginia Univer-

Carl FitzGerald, University of Califomia-San Diego; Basil Gordon, University of

Califomia-Los Angeles; Mary Gray, American University; Branko Grunbaum, 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

M. Petty, University of Missouri-Columbia; Howard Prouse, Mankato Helmut Rohrl, University of Califomia-San Diego; David Wilson, University of Florida; and Frederick Wright, University of North Carolina-Chapel Hill. Second Edition Reviewers: Salvatore Anastasio, State University of New York, New Platz; Bruce Crauder, Oklahoma State University; Walter Czamec, Framingham State College; William England. Mississippi State University; David Jabon, Eastern Washington University; Clinton State University;

University; Charles Jones, Ball State University; Michael Lacey, Indiana University; Harold

Martin, Northern Michigan University; James Murdock, Iowa State University;

Ken Shaw, Domina

Florida State University; Sverre Smalo, University of California, Santa Barbara;

Jimmy Woods, North Georgia College. I have also benefited greatly from conversations with many historians of mathematics various forums. In particular, those who have regularly attended the annual History

Eberle Spencer, University of Connecticut;

at

of Mathematics seminars, organized by Uta Merzbach, former curator of mathematics at the

National

Museum

of American History,

may

well recognize

some of

the ideas

discussed there. The book has also profited from discussions over the years with,

among

XIV

Preface

Rickey (Bowling Green State

others, Charles Jones (Ball State University), V. Frederick

(MAA), Israel Kleiner (York University), Abe Shenitzer Ubiratan D'Ambrosio (Univ. Estadual de Campinas), and Frank Swetz

University), Florence Fasanelli

(York University).

(Pennsylvania State University).

My students in History of Mathematics (and other) classes

Columbia have also helped me clarify many of my comments and correspondence from students and colleagues elsewhere in an effort to continue to improve this book. Special thanks are due to the librarians at the University of the District of Columbia and especially to Clement Goddard, who never failed to secure any of the obscure books I at

the University of the District of

ideas. Naturally,

I

welcome any

additional

requested on interlibrary loan. Leslie Overstreet of the Smithsonian Institution Libraries’ Special Collections Department was extremely helpful in finding sources for pictures.

Thanks

are

due

and George Duda, I

my

to

who

former editors

helped form the

at

first

also want to thank Jennifer Albanese.

Don Gecewicz,

HarperCollins, Steve Quigley, edition.

my new editor at Addison Wesley Longman, for

her suggestions and her patience as she pushed this book to completion, as well as Rebecca

Malone and Barbara Pendergast,

for their efforts in handling the production aspects,

and

Susan Holbert for preparation of the index.

My family has been very supportive during the many years of writing the book. my

Naomi last,

I

for help at various times

thank

being there

my

I

needed

her.

I

thank

my

children Sharon, Ari, and

and especially for allowing

me

to use our

wife Phyllis for long discussions

when

I

thank

parents for their patience and their faith in me.

owe

her

at

much more

I

computer.

And

any hour of the day or night and for than

I

can ever repay.

Victor

J.

Katz

)

PART ONE Mathematics Before the Sixth Century

h

a

p

r

t

Ancient

Mathematics Accurate reckoning. The entrance into the knowledge of all existing things and

all

obscure secrets. Introduction to Rhind Mathematical Papyrus'

M

in

some 3800

trying to

years ago, a teacher

is

develop mathematics problems to assign

to his students so they

can practice the ideas just

troduced on the relationship triangle.

Larsa

esopotamia: In a scribal school

among

the sides of a right

The teacher not only wants

to be difficult

enough

to

show who

the computations

really understands

the material but also w'ants the answers to

whole numbers so the students

in-

come

out as

will not be frustrated.

After playing for several hours with the few triples

of numbers he knows that satisfy the equation

( ct,b,c

cr

+

b

2

=

c

2 ,

a

new

idea occurs to him. With a few deft

strokes of his stylus, he quickly does

on a moist clay discovered

tablet

how

some

calculations

and convinces himself that he has

to generate as

many of these

triples as

necessary. After organizing his thoughts a bit longer,

he takes a fresh tablet and carefully records a table

list-

ing not only 15 such triples but also a brief indication

of some of the preliminary calculations.

He does

not,

however, record the details of his new method. Those will be

saved for his lecture to his colleagues. They will

then be forced to acknowledge his abilities, and his reputation as

one of the best teachers of mathematics

will

spread throughout the kingdom.

The opening quotation from one of the few documentary sources on Egyptian mathematics and the fictional story of the Babylonian scribe illustrate some of the difficulties in presenting an accurate picture of ancient mathematics. Mathematics certainly existed in virtually every ancient civilization of which there are records, but trained priests and scribes,

government

officials

mathematics for the benefit of the government

in

was always in the domain of specially whose job it was to develop and use

it

such areas as tax collection, measurement. 1

2

Chapter

1

Ancient Mathematics

building, trade, calendar making, and ritual practices. Yet, even though the origins of

many

mathematical concepts stem from their usefulness in these contexts, mathematicians have always exercised their curiosity by extending their ideas necessity. Nevertheless, because

on only

far

beyond

mathematics was a tool of power,

to the privileged few, often

much

detail.

In recent years, scholars have labored to reconstruct the

but

methods were passed

through an oral tradition. Hence the written records are

generally sparse and seldom provide

ilizations

the limits of practical

its

mathematics of ancient civ-

from whatever clues can be found. Naturally, they do not agree on every point,

enough consensus

India. In order to see

these civilizations,

we

we can

exists so that

mathematical knowledge

present a reasonable picture of the state of

in the ancient civilizations

most

clearly the similarities

of Egypt, Mesopotamia, China, and

and differences

mathematics of

in the

each civilization separately but will instead organize

will not treat

our discussion around the following key topics: counting, arithmetic computations, linear equations, elementary geometry, astronomical and calendrical computations, square roots, the “Pythagorean” theorem,

and quadratic equations. To place the story

in context,

we

begin with a brief description of the civilizations themselves and the sources from which

our knowledge of their mathematics

1

.

is

derived.

ANCIENT CIVILIZATIONS

1

Probably the oldest of the world's civilizations the Tigris to

and Euphrates

prominence

river valleys

in this area

is

that of

Mesopotamia, which emerged

sometime around 3500

b.c.e.

over the next 3000 years, including one based

Babylon, whose ruler Hammurapi conquered the entire area around 1700

By

this time, the civilization

had developed a national

in

Many kingdoms came

political loyalty

in the city

of

b.c.e. (Fig. 1.1).

and a pantheon of

gods, including Enlil, the god of Nippur, the traditional religious capital of Mesopotamia,

FIGURE

and Marduk. the city-god of Babylon.

1.1

A bureaucracy and a professional army had come into

existence, and a middle class of merchants and artisans had

Hammurapi on

a stamp

of peasants and the royal

of Iraq. tool,

officials. In addition, writing

which was then used

to aid the

government

1.2

a

stamp of Austria.

maintaining central control over a large

Writing was done by means of a stylus on clay

been excavated during the past 150 years (Fig.

FIGURE

the masses

tablets, thousands of which have was Henry Rawlinson (1810-1895) who, by the mid- 1850s, was first able to translate this cuneiform writing by comparing the Persian and Babylonian inscriptions of King Darius I of Persia (sixth century b.c.e.) on a rockface at Behistun (in modern Iran) describing a military victory. A large number of these tablets contain mathematical problems and solutions or mathematical tables. Hundreds of them have been copied, translated, and explained. They are generally rectangular but occasionally round in shape. They usually fit comfortably into one's hand and are an inch or so in thickness, although some are as small as a postage stamp and others are as large as an encyclopedia volume. We are fortunate that these tablets are virtually indestructible, because they are our only source for Mesopotamian mathematics. The written tradition that they represent died out under Greek domination in the last centuries b.c.e. and remained totally lost until the nineteenth century. The great majority of the excavated tablets date from the time of Hammurapi. although small collections date from the earliest beginnings of Mesopotamian civilization, from the centuries surrounding

area.

Babylonian clay tablet on

in

grown up between

had been invented as an accountancy

1.2). It

3

1

1000

b.c.e. ,

and from the Seleucid period around 300

ter will generally

Hammurapi),

but, as

to refer to the civilization

was

3

Ancient Civilizations

1

b.c.e.

Our discussion

in this

chap-

concern the mathematics of the “Old Babylonian” period (the time of

"Babylonian" itself

.

is

standard

of mathematics,

in the history

we

shall use the adjective

and culture of Mesopotamia, even though Babylon

the major city of the area for only a limited time.

Agriculture emerged in the Nile valley in Egypt about 7000 years ago, but the

first

dynasty to rule both Upper Egypt (the river valley) and Lower Egypt (the delta) dates

from about 3100 and

b.c.e.

The legacy of

priests, a rich court, and, for the

the

pharaohs included an

first

of officials

elite

kings themselves, a role as intermediary between

mortals and gods. This role fostered the development of Egypt's monumental architecture,

FIGURE

including the pyramids, built as royal tombs, and the great temples

1.3

(Fig. 1.3).

The pyramids

at

Gizeh.

The

scribes gradually developed the hieroglyphic writing

and temples. Jean Champollion of this writing early

in the

(

1790-1832)

is

at

Luxor and Karnak

which dots the tombs

chiefly responsible for the

of a multilingual inscription

—

the Rosetta Stone

—

in

hieroglyphics and Greek as well as

the later demotic writing, a form of the hieratic writing of the papyri (Fig.

Much of our knowledge of the mathematics of ancient Egypt, the hieroglyphs in the temples but

problems with

Papyrus purchased ,

FIGURE

1.4

Jean Champollion and a piece of the Rosetta Stone.

Museum

in

1

Mathematical Papyrus named ,

who purchased it at Luxor in

893 by

1

.4).

however, comes not from

from two papyri containing collections of mathematical

their solutions: the Rhincl

A. H. Rhind( 1833- 1863)

translation

first

nineteenth century. His work was accomplished through the help

V. S.

Golenishchev

(d.

for the

Scotsman

Moscow Mathematical who later sold to the Moscow

1858, and the

1947)

it

of Fine Arts. The former papyrus was copied about 1650 b.c.e. by the scribe

A'h-mose from an original about 200 years earlier and is approximately 8 feet long and inches high. The latter papyrus dates from roughly the same period and is over 5 feet long, 1

1

1

but only

some

3 inches high.

As

in the

Mesopotamian

we

case,

are fortunate that the dry

Egyptian climate allowed these mathematical papyri, along with hundreds of other papyri, to

be preserved, because again Greek domination of Egypt

in the

centuries surrounding the

beginning of our era was responsible for the disappearance of the native Egyptian hieratic script (Fig. 1.5).

Although there are legends dating Chinese evidence of such a civilization

earliest solid

Huang

near the there, the

River,

which are dated

Shang dynasty,

to

that the “oracle

is

civilization

back 5000 years or more, the

provided by the excavations

about 1600

b.c.e.

It

is

at

Anyang,

to the society centered

bones” belong, curious pieces of bone inscribed

with very ancient writing that were used for divination by the priests of the period. The

bones are the source of our knowledge of early Chinese number systems. Around the beginning of the

which

FIGURE

1.5

in turn

high official and scribe (15th century, b.c.e.).

millennium

b.c.e., the

Shang were replaced by

the

Zhou

dynasty,

there occured a great intellectual flowering, in

which the most famous philosopher was

Confucius. Academies of scholars were founded

in several

were hired by other feudal lords

by the coming of Amenhotep, an Egyptian

first

dissolved into numerous warring feudal states. In the sixth century b.c.e.,

The

them

in a

of the states. Individual scholars

time of technological growth caused

iron.

feudal period ended as the weaker states were gradually absorbed by the stronger

ones, until ultimately China

Under

to advise

his leadership.

He enforced

was

unified under the

China was transformed

Emperor Qin Shi Huangdi

in

221 b.c.e.

into a highly centralized bureaucratic state.

a severe legal code, levied taxes evenly, and

of weights, measures, money, and. especially, the written

demanded the standardization script. Legend holds that this

— 4

Chapter

1

Ancient Mathematics

emperor ordered the burning of all books from earlier periods to suppress dissent, but there is some reason to doubt that this decree was actually carried out. The emperor died in 210 B.c.r.. and his dynasty was soon overthrown and replaced by that of the Han, which was

400 years The Han completed

to last about

the establishment of a trained civil service, for

which a system of education was necessary. Among' the texts used for

two mathematical works, probably compiled early (Arithmetical Classic of the

Gnomon and

snan slut (Nine Chapters on the

Art).

It is

it

to

should be kept

in

and the Jiuzhang

impossible to date exactly the

it

is

at least several

first

fragmentary

generally believed that at least

China near the beginning of the Zhou period.

mind that even with this

be discussed took place

puipose were

in these texts, but since there are

records of older sources similar to the Nine Chapters, material was extant in

this

dynasty, the Zhoubi suanjing

the Circular Paths of Heaven)

Mathematical

discoveries of the mathematics contained

some of the

Han

in the

Of course,

dating, the Chinese mathematical developments

hundred years

and Egypt. Whether there was any transmission from

later than those in

Mesopotamia

these civilizations to

China

not

is

known.

A

civilization called the

Harappan arose

the third millennium B.c.r.. but there

Indian civilization for which there

by Aryan

tribes migrating

about the eighth century

had

to

is

is

in India

its

late in the

second millennium

monarchical states were established

manage complex systems, such

in the area,

had highly

stratified social

By

b.c.e.

and they

as fortifications, administrative centralization,

large-scale irrigation works. These states

and

systems headed by

kings and priests (brahmins). The literature of the brahmins was oral for

expressed

in

mathematics. The earliest

such evidence was formed along the Ganges River

from the Asian steppes

b.c.e..

on the banks of the Indus River

no direct evidence of

many

generations,

lengthy verses called Vedas. Although these verses probably achieved their

in

current form by

600

b.c.e., there are

no written records dating back beyond the current era

(Fig. 1.6).

Some priests.

It

of the material from the Vedic era describes the intricate sacrificial system of the is

in these

works, the Sulvasutras, that

we

find mathematical ideas. Curiously,

mathematics deals with the theoretical requirements for building altars out of bricks, as far as is known the early Vedic civilization did not have a tradition of brick technology, while the Harappan culture did. Thus there is a possibility that the mathematics although

FIGURE

in the

1.6

this

Sulvasutras was created in the Harappan period, although the

transmission to the later period

A Vedic

manuscript.

are the sources for our

.2

unknown. In any

civilizations in other parts of the

the data uncovered so

far give us

discussion, therefore, must await

1

currently

case,

it is

mechanism of

its

the Sulvasutras that

knowledge of ancient Indian mathematics.

Although there were b.c.e..

is

new

few clues

world before the

to their

first

millennium

mathematical knowledge.

Any

archaeological evidence.

COUNTING

—

The simplest mathematical idea and one that probably existed even before civilization is that of counting, in words and in more permanent form as written symbols. Although an interesting study can be made of number words in various languages (Sidebar 1.1), we restrict ourselves here to a discussion of number symbols. Several methods of organization can be distinguished in the w riting of these symbols. O ne method is called the grouping

1.2

SIDEBAR Number Words

5

Counting

1.1

Languages

in Various 18

10 (ten becomes teen)

English

eighteen

8,

Welsh

deu naw

2X9 (deu from dau =

Hebrew

shmona-eser

8,

Yoruba

eeji

Chinese

shih-pa

2, naw = 9) = 8, eser = 10) 20 less 2 (ogun = 20, eeji = 2) 10, 8 (shih = 10, pa = 8) 8, 10 (asta = 8, dasa = 10) 8, 10(uaxac = 8, lahun = 10) 2 from 20 (duo = 2, viginti = 20) 8 and 10 (okto = 8, deka = 10)

din logun

10 (shmona

Sanskrit

asta-dasa

Mayan

uaxac-lahun

Latin

duodeviginti

Greek

okto kai deka

English

forty

Welsh

de-ugeint

2

Hebrew

arba-im

4s (arba

40

4X10 (ten becomes ty) X 20

(de from dau

=

4,

im

is

=

2.

ugeint

=

20)

the plural ending)

Sanskrit

catvarim-sat

20 X 2 (from ogun = 20. eeji = 2) 4 X 10 (szu = 4, shih = 10) 4X10 (catvarah = 4, sat from dasa

Mayan

ca-ikal

2

Latin

quadraginta

Greek

tettarrakonto

4X10 (quad = 4, ginta from decern = 4X10 (tettara = 4, kunta from deka =

Yoruba

ogoji

Chinese

szu-shih

This table displays the words for 18 and 40 a linguistic analysis of the

method. In

made

to represent larger

such a representation of numbers

is

who

(ca

=

nine ancient and

2, kal is suffix for

modem

=

10)

20) 1

0)

10)

languages along with

2

/ is

used to represent the number

numbers. One of the

earliest

on a fossilized bone discovered

carbon-dated to some time around 20.000 represent, but one scholar

in

derivations.

simplest form, a stroke

its

repetitions are

word

X 20

b.c.e.

It is

1;

appropriate

dated occurrences of

at

Ishango

in

Zaire and

not clear what the strokes or notches

has studied the bone in detail believes they represent a

count of certain periods of the moon.

'

That early peoples used

this

elementary form of

numerical representation to deal with astronomical phenomena would confirm what will be seen

much

in

later

time periods

—

that the

development of mathematics goes hand

in

hand

with the development of astronomy. Artifacts similar to the Ishango bone and dating from

perhaps 8000

b.c.e.,

with regular groupings of notches perhaps representing astronomical

observations, have been found in central Europe as well.

A

more sophisticated example of

the grouping

developed by the Egyptians some 5000 years ago. the

first

several

powers of ten was represented by

familiar vertical stroke for

1.

method of number representation was In this hieroglyphic

a different

Thus 10 was represented by

Pi.

system each of ,

symbol, beginning with the 100 by 9, 1000 by

and

6

Chapter

1

Ancient Mathematics

V Arbitrary whole numbers were then represented by appropriate

10,000 by

the symbols. For example, to represent 12,643 the Egyptians

would write

(Note that the usual practice was to put the smaller digits on the

repetitions of

"|R

left.)

The hieroglyphic number system was the one used fbr writing on temple walls or when the scribes wrote on papyrus, they needed a form of

carving on columns. But

handwriting. For this purpose th ey developed the hieratic system, an example of a ciphered

system. Hcre each

number from

had a specific symbol, as did each multiple of 10

to 9

1

90 and each multiple of 100 from 100 to 900, and so on. A given number, for example 37, was written by putting the symbol for 7 next to that for 30. Since the symbol from 10 for 7

40

cipher

as

to

was £ and that and 200 as

necessary

30 was the

used

is

in

specific

r

Tk

i

—

ill

->'•

/Although a^ero symbol

is

HI,

not

,

CwS

'