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“historymaster” — 2009/7/1 — 14:16 — page i — #1

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Mathematics in Historical Context

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“historymaster” — 2011/4/5 — 10:19 — page ii — #2

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Images on Cover from Left to Right:

Issac Newton

Felix Klein

James Joseph Sylvester

Joseph-Louis Lagrange

Solomon Lefschetz

Hypatia

Brahmagupta

c 2009 by

The Mathematical Association of America (Incorporated)

Library of Congress Catalog Card Number 2009928078

Print ISBN: 978-0-88385-570-6

Electronic ISBN: 978-1-61444-502-9

Printed in the United States of America

Current Printing (last digit):

10 9 8 7 6 5 4 3 2 1

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“historymaster” — 2009/7/1 — 14:16 — page iii — #3

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Mathematics in Historical Context

Jeff Suzuki

Brooklyn College

®

Published and Distributed by

The Mathematical Association of America

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“historymaster” — 2009/7/1 — 14:16 — page iv — #4

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Committee on Books

Paul Zorn, Chair

Spectrum Editorial Board

Gerald L. Alexanderson, Editor

Robert E. Bradley

Richard K. Guy

Keith M. Kendig

Edward W. Packel

Sanford Segal

Amy E. Shell-Gellasch

Robert S. Wolf

William W. Dunham

Michael A. Jones

Steven W. Morics

Kenneth A. Ross

Franklin F. Sheehan

Robin Wilson

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“historymaster” — 2009/7/1 — 14:16 — page v — #5

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SPECTRUM SERIES

The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new

mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other

monographs of high interest that will appeal to a broad range of readers, including students and

teachers of mathematics, mathematical amateurs, and researchers.

777 Mathematical Conversation Starters, by John de Pillis

99 Points of Intersection: Examples—Pictures—Proofs, by Hans Walser. Translated from the original

German by Peter Hilton and Jean Pedersen

Aha Gotcha and Aha Insight, by Martin Gardner

All the Math That’s Fit to Print, by Keith Devlin

Calculus Gems: Brief Lives and Memorable Mathematics, by George F. Simmons

Carl Friedrich Gauss: Titan of Science, by G. Waldo Dunnington, with additional material by Jeremy

Gray and Fritz-Egbert Dohse

The Changing Space of Geometry, edited by Chris Pritchard

Circles: A Mathematical View, by Dan Pedoe

Complex Numbers and Geometry, by Liang-shin Hahn

Cryptology, by Albrecht Beutelspacher

The Early Mathematics of Leonhard Euler, by C. Edward Sandifer

The Edge of the Universe: Celebrating 10 Years of Math Horizons, edited by Deanna Haunsperger

and Stephen Kennedy

Euler and Modern Science, edited by N. N. Bogolyubov, G. K. Mikhailov, and A. P. Yushkevich.

Translated from Russian by Robert Burns.

Euler at 300: An Appreciation, edited by Robert E. Bradley, Lawrence A. D’Antonio, and C. Edward

Sandifer

Five Hundred Mathematical Challenges, by Edward J. Barbeau, Murray S. Klamkin, and William O.

J. Moser

The Genius of Euler: Reflections on his Life and Work, edited by William Dunham

The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the

assistance of Jean Pedersen.

The Harmony of the World: 75 Years of Mathematics Magazine, edited by Gerald L. Alexanderson

with the assistance of Peter Ross

How Euler Did It, by C. Edward Sandifer

Is Mathematics Inevitable? A Miscellany, edited by Underwood Dudley

I Want to Be a Mathematician, by Paul R. Halmos

Journey into Geometries, by Marta Sved

JULIA: a life in mathematics, by Constance Reid

The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow

Lure of the Integers, by Joe Roberts

Magic Numbers of the Professor, by Owen O’Shea and Underwood Dudley

Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect

Shuffle, by S. Brent Morris

Martin Gardner’s Mathematical Games: The entire collection of his Scientific American columns

The Math Chat Book, by Frank Morgan

Mathematical Adventures for Students and Amateurs, edited by David Hayes and Tatiana Shubin.

With the assistance of Gerald L. Alexanderson and Peter Ross

Mathematical Apocrypha, by Steven G. Krantz

Mathematical Apocrypha Redux, by Steven G. Krantz

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Mathematical Carnival, by Martin Gardner

Mathematical Circles Vol I: In Mathematical Circles Quadrants I, II, III, IV, by Howard W. Eves

Mathematical Circles Vol II: Mathematical Circles Revisited and Mathematical Circles Squared, by

Howard W. Eves

Mathematical Circles Vol III: Mathematical Circles Adieu and Return to Mathematical Circles, by

Howard W. Eves

Mathematical Circus, by Martin Gardner

Mathematical Cranks, by Underwood Dudley

Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell

Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau

Mathematical Magic Show, by Martin Gardner

Mathematical Reminiscences, by Howard Eves

Mathematical Treks: From Surreal Numbers to Magic Circles, by Ivars Peterson

Mathematics: Queen and Servant of Science, by E.T. Bell

Mathematics in Historical Context,, by Jeff Suzuki

Memorabilia Mathematica, by Robert Edouard Moritz

Musings of the Masters: An Anthology of Mathematical Reflections, edited by Raymond G. Ayoub

New Mathematical Diversions, by Martin Gardner

Non-Euclidean Geometry, by H. S. M. Coxeter

Numerical Methods That Work, by Forman Acton

Numerology or What Pythagoras Wrought, by Underwood Dudley

Out of the Mouths of Mathematicians, by Rosemary Schmalz

Penrose Tiles to Trapdoor Ciphers . . . and the Return of Dr. Matrix, by Martin Gardner

Polyominoes, by George Martin

Power Play, by Edward J. Barbeau

Proof and Other Dilemmas: Mathematics and Philosophy, edited by Bonnie Gold and Roger Simons

The Random Walks of George Pólya, by Gerald L. Alexanderson

Remarkable Mathematicians, from Euler to von Neumann, by Ioan James

The Search for E.T. Bell, also known as John Taine, by Constance Reid

Shaping Space, edited by Marjorie Senechal and George Fleck

Sherlock Holmes in Babylon and Other Tales of Mathematical History, edited by Marlow Anderson,

Victor Katz, and Robin Wilson

Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan,

Douglas S. Kurtz, and David Pengelley

Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance

of Jean Pedersen.

The Trisectors, by Underwood Dudley

Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford

Who Gave You the Epsilon? and Other Tales of Mathematical History, edited by Marlow Anderson,

Victor Katz, and Robin Wilson

The Words of Mathematics, by Steven Schwartzman

MAA Service Center

P.O. Box 91112

Washington, DC 20090-1112

800-331-1622

FAX 301-206-9789

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Introduction

This book emerged from a discussion I had with Don Albers in the Spring of 2002, when

he suggested the idea of a book that would describe the world of the great mathematicians:

“What would Newton see, if he looked out his window?” I really liked the idea, and planned

to pick out a dozen or so mathematicians and write a hundred or so pages of history discussing how mathematics, society, and mathematicians interacted with one another. Before

I knew it, I had written six hundred pages. I’ve cut those down to the present text, which is

part mathematics, part mathematical biography, and part history.

My hope is to convey some of the fascinating and complex relationships between mathematicians, mathematics, and society. In these pages, you will find how world events shaped

the lives of mathematicians like Archimedes and de Moivre; how artistic conventions inspired some of the mathematical investigations of Abu’l-Wafā and al-Khayyāmı̄; how Newton and Poincaré affected

the political events of their time; and how mathematical concepts

p

7

like the irrationality of 2 drove cultural development.

Finally, some thanks and dedication. First, to Rob Bradley of Adelphi University and

Ross Gingrich of Southern Connecticut State University for suggesting the title of the

book (my title was a completely uninspired “Mathematics and History”). Next, to my wife

Jacqui, who listened with great forbearance as I tried to reconcile information from different sources, which often spelled and indexed the same name in different ways, or gave

different dates for the same historical event, or completely different interpretations of the

significance of an event. Last but not least, I’d like to dedicate this book to my children,

William and Dorothy: May the past be the guide to the future.

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“historymaster” — 2009/7/1 — 14:16 — page ix — #9

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Contents

Introduction

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1 The Ancient World

1.1 Prehistory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Egypt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Mesopotamia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

2

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2 The Classical World

15

2.1 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The Romans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 China and India

3.1 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 The Importance of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 The Islamic World

4.1 Early Islam . . . . . . . . . .

4.2 The Rise and Fall of Dynasties

4.3 The Age of Invasion . . . . .

4.4 Andalusia . . . . . . . . . . .

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5 The Middle Ages

119

5.1 Norman Sicily . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Early Medieval France . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 High Medieval France . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6 Renaissance and Reformation

157

6.1 The Italian Peninsula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2 Central Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7 Early Modern Europe

7.1 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2 The Low Countries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Great Britain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

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8 The Eighteenth Century

231

8.1 Great Britain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.2 Central and Eastern Europe . . . . . . . . . . . . . . . . . . . . . . . . . . 242

8.3 France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

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9 The Nineteenth Century

9.1 France . . . . . . .

9.2 Great Britain . . .

9.3 Italy . . . . . . . .

9.4 Germany . . . . .

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277

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10 The United States

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10.1 From Colony to Country . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

10.2 The Gilded Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

11 The Modern World

11.1 Before the Great War . .

11.2 The Era of the Great War

11.3 The Depression Era . . .

11.4 World War Two . . . . .

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337

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357

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Epilog

385

Bibliography

387

Figure Citations

395

Index

397

About the Author

409

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1

The Ancient World

1.1

Prehistory

When anthropologists speak of a group’s culture, they mean the sum of all the human

activity of the group: how they talk, think, eat, play, and do mathematics. Every culture in

the world creates some sort of mathematics, though it may be on a very basic level. The

most ancient evidence of mathematical activity comes from a wolf’s bone on which fiftyfive notches are carved, grouped in sets of five. The bone was found in the Czech Republic

and is about 35,000 years old. A similar artifact was found at Ishango, on the shores of

Lake Edward in Zaire. The Ishango bone (from a baboon) dates to around 18,000 B.C., and

is particularly intriguing since along one side, the notches are grouped into sets of 11, 13,

17, and 19, suggesting an interest in prime numbers.

At that time, mankind obtained food by hunting animals and gathering plants. But about

ten thousand years ago, an unknown group of people invented agriculture. This forced

people to establish permanent settlements so the plants could be cared for until harvest: the

first cities.

The culture of city-dwellers is called civilization (from civilis, “city-dweller” in Latin).

Since cities were originally farming communities, they were established in regions that

provided the two fundamental needs of agriculture: fertile soil and a reliable water supply.

River flood-plains provide both, and the most ancient civilizations developed around them.

Figure 1.1. The Ishango Bone. Photograph courtesy of Science Museum of Brussels.

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1. The Ancient World

Egypt

In Egypt, civilization developed around the Nile river, whose annual flooding deposited silt

that fertilized the fields of the Egyptian farmers. As early as 4000 B.C. the Egyptians may

have noticed that it took approximately 365 days from one Nile flood to the next. The Egyptian calendar divided the year into 12 months of 30 days, with 5 extra days celebrated as

the birthdays of the main gods of the Egyptian pantheon. These days of religious festivities

were the original holidays (“Holy days”).

Egypt is divided into two main parts, Lower Egypt and Upper Egypt. Lower Egypt

consists of the marshy areas where the Nile empties into the Mediterranean, known as the

Delta (because of its resemblance to the Greek letter , pointing southwards); the term

would later be applied to any river’s outlet into a sea or lake. Upper Egypt is formed by the

region between the Delta and the First Cataract, or set of waterfalls, at Aswan. According

to tradition, Upper and Lower Egypt were united around 3100 B.C. by Narmer, whom the

Greeks called Menes. Archaeological evidence points to an even earlier King Scorpion who

ruled over a united kingdom, and recently the tombs of King Scorpion and his successors

have been identified.

The kings of Egypt were known as pharaohs (“Great House,” referring to the Royal

Palace). The pharaohs were revered as gods, though this did not spare them from being

criticized, plotted against, and deposed. Though the rulers changed, the fundamental in-

Figure 1.2. Egypt and the Fertile Crescent

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1.2. Egypt

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stitutions of pharaonic Egypt would last for nearly three thousand years, a period of time

usually referred to as Dynastic Egypt.

Dynastic Egypt is traditionally divided into thirty dynasties, according to a division first

described by the Greek-Egyptian scholar Manetho around 280 B.C. These thirty dynasties

are grouped into five periods: the Old Kingdom (roughly 3100 B.C. to 2200 B.C.); a First

Interregnum (2200 B.C. to 2100 B.C.); a Middle Kingdom (2100 B.C. to 1788 B.C.); a Second Interregnum (1788 B.C. to 1580 B.C.); and a New Kingdom (1580 B.C. to 1090 B.C.).1

Manetho’s division of the reigns has been retained, though his dates are no longer considered accurate; indeed, the dates of ancient Egypt are very uncertain and Egyptologists

themselves differ by up to a hundred years on the dates of the reigns of early pharaohs.

The best-known features of Egyptian civilization are the pyramids. The step pyramid

at Sakkara, near the Old Kingdom capital of Memphis, is the most ancient; it was completed under the direction of the architect Imhotep around 2650 B.C. The largest and best

known pyramid is the Great Pyramid at Gizeh, finished around 2500 B.C. and the oldest of

the Seven Wonders of the World. Moreover, it is the only one of the Seven Wonders still

standing, a tribute to the skill of the Egyptian builders and the dry climate of Egypt.

By 2700 B.C., a form of writing had been invented. Since many examples were found

adorning the walls of Egyptian temples, it was erroneously believed that the writings were

religious in nature. Hence this form of Egyptian writing became known as hieroglyphic

(“sacred writing” in Greek).

Hieroglyphs may have originally been pictograms (small pictures of the object being

represented, such as a set of wavy lines to represent a river) or ideograms (stylized figures

that represent an abstract concept, such as the emoticon 😉 to indicate “just kidding”).

However, this form of writing would require the knowledge of hundreds, if not thousands,

of symbols to represent the common words in a language, so hieroglyphs soon took on new

meanings as sounds. For example, we might draw to represent the word “river” or to

refer to a man named Rivers, but in time might come to represent the sound “riv-” or

even the initial sound “r-.”

1.2.1

Egyptian Mathematics

To write numbers, we could spell them out: two hundred forty-one. However, writing out

number words is tedious and all civilizations have developed special symbols for numbers.

The use of a vertical or horizontal stroke to represent “one” is nearly universal, as is the use

of multiple strokes to represent larger numbers: thus, “three” would be . Of course, such

from

.

notation rapidly gets out of hand: try to distinguish

The next logical step would be to make a new symbol for a larger unit. In hieroglyphic,

represents ten,represents one hundred, represents one thousand, and other symbols were

used for ten thousand, one hundred thousand, and one million. To indicate a large number,

the symbols would be repeated as many times as necessary: two hundred forty-one would

be

(note that the Egyptians wrote from right to left, and placed the greatest

values first). Since the value of the number is found by adding the values of the symbols,

this type of numeration is called additive notation.

H

H

@

@

@

1 Menes is the first pharaoh of the First Dynasty. King Scorpion and his immediate successors, who predate

Menes, are collectively grouped into Dynasty Zero.

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1. The Ancient World

Figure 1.3. From the Rosetta Stone, line 10. Inscription reads, “On the 17th day of Paophi, he [Ptolemy

V] brought….”

Fractions were generally expressed as sums of unit fractions. Thus 52 would be written

1

as 15

and 13 . These unit fractions were written by placing a symbol, _

^ , read as ro (an

open mouth), above the hieroglyphic representation of the denominator. Thus 52 would be

_

^ _

^ .

Some common (non-unit) fractions had special symbols, though the only one used with

any consistency was a symbol for two-thirds. The immense length of Egyptian history can

be illustrated by the two thousand year history of this symbol: in the Old Kingdom, two^ but a thousand years later, this form would change into _

^ . It

thirds was written as _

_

^

would be another thousand years before this form was written .

By 2600 B.C., a cursive form of hieroglyphic, suitable for writing on softer materials,

came into existence: hieratic. All Egyptian mathematical treatises are written in hieratic,

on papyrus, cloth, or leather. Unfortunately, these materials disintegrate rapidly when exposed to water, insects, and sunlight, so our knowledge of Old Kingdom mathematics is

limited to indirect evidence (like the existence of the pyramids or an accurate calendar) and

hieroglyphic numerals.

The oldest mathematical texts date to the Middle Kingdom, six hundred years after the

construction of the Great Pyramid of Gizeh. The Reisner papyrus, named after its discoverer, dates to the reign of the Twelfth Dynasty pharaoh Sesostris (around 1900 B.C.) who

established a chain of forts south of Aswan to keep Egypt’s borders secure against the Nubians. Sesostris also sponsored a number of building projects, including additions to the

great temple complex at Karnak (near Thebes, the capital of the Middle Kingdom).

The Reisner papyrus is a set of four worm-eaten rolls, and appears to be a set of books

for a construction site. The readable portions include a list of employees, as well as calculations of volumes and areas. A typical computation is determining the number of workmen

needed to excavate a tomb, given the dimensions of the pit and the expected volume of dirt

to be removed by a workman each day (apparently 10 cubic cubits, or roughly 30 cubic

feet).

Sesostris’s dynasty ended with the reign of Sebeknefru (around 1760 B.C.), the earliest

female ruler whose existence is definitively established. Her reign was peaceful, but her

death touched off a civil war and over the next century, nearly seventy pharaohs would rule

in quick succession.

The Moscow or Golenishchev papyrus (named after its current location or discoverer)

was written during this era. It includes a number of geometric problems. For example,

Problem 7 asks to find the dimensions of a triangle with area 20 setat, where the ratio of

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the height to the base is 2 1=2 to 1. The scribe’s solution is to double the area (making

a rectangle with sides in ratio 2 1=2 to 1) then multiply by 2 1=2: this produces a square

(of area 100), whose side can be found by taking the square root; this is also the triangle’s

height. Dividing by 2 1=2 gives the base. Problem 9 finds the surface area of a basket

(possibly a hemisphere or a half-cylinder).

Problem 14 contains what is perhaps the greatest mathematical discovery of the ancient

Egyptians: the computation of the exact volume of the frustum of a pyramid. The idea of a

mathematical formula was as yet non-existent; the student was expected to generalize from

the given example. Thus the Moscow papyrus gives:

Problem 1.1. Find the volume for a frustum 6 cubits high with [square] base 4 cubits and

[square] top 2 cubits. Square the base to get 16; multiply [top side] 2 by [bottom side] 4 to

get 8; square the top 2 to get 4; add these together to get 28. Multiply the height 6 by 1=3,

to get 2; multiply 2 by 28 to get 56, the volume.

Thus if the pyramid has a square base of side a, square top of side b, and height h then

its volume will be given by V D 31 h a2 C ab C b 2 .

Shortly after the Moscow papyrus was written, Egypt was invaded by a mixed group

of tribes, consisting primarily of Semites from Palestine and Hurrians from Asia Minor

(modern-day Turkey). These invaders swept away all opposition by using a new weapon

of war: the horse and chariot. They conquered Egypt but kept its political and religious

institutions intact, establishing themselves around 1680 B.C. as the Fifteenth Dynasty. The

Egyptians named them the Hyksos (“Rulers of Foreign Lands”).

Our first complete text of Egyptian mathematics dates to this period, a thousand years

after the building of the first pyramids. The Rhind papyrus (named after its discoverer)

was, according to the preface, written down in the fourth month of the Inundation Season

in the thirty-third year of the reign of A-user-Re by the scribe A’ H – MOS È (fl. 1650 B.C.?).

The introduction to the Rhind papyrus highlights an important problem of history: precise

dating of an event. Often we have a better idea of the day and month than for the year

itself: because of the reference to the fourth month of the Inundation Season, we know

that the Rhind was copied around September. But we do not know A-user-Re’s dates with

any certainty, so the best guess for the thirty-third year of the reign of A-user-Re is around

1650 B.C.

A’h-mosè claimed the Rhind was based on an older work, dating back to the Thirteenth

Dynasty pharaoh Ne-ma’et-Re and thus contemporaneous with the Moscow papyrus, a

claim that might or might not be true. Why would someone deny that their work was original? In the days before printing, all texts had to be copied by hand: they were literally

manuscripts (“hand written” in Latin). Since it takes just as much effort to copy a worthless text as a worthwhile one, this meant that the only texts that were copied and recopied

were the most important. One way to give your work a veneer of importance is to attribute

it to some great author of the past or some time period that people looked back to with nostalgia: hence, the first five books of the Old Testament are attributed to Moses, even though

they were not written down until a thousand years after the events they claim to describe.

The first part of the Rhind papyrus consists of the quotient of 2 and the odd numbers

from 3 to 101. From this, we can discern many details of how the Egyptians performed

1

1

calculations. A typical computation is 2 divided by 13, whose quotient is 18 C 52

C 104

.

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The computation proceeds as follows:

n

n

n

1

1=2

1=4

1=8

4

8

13

6

3

1

52

104

1=2

1=4

1=2

1=4

1=8

1=8

The second, third, and fourth lines seem to represent the product of 13 and 1=2, 1=4, and

1

D 14 .

1=8, respectively. The fifth line seems to refer to the fact that 134 D 52 implies 13 52

1

Likewise, the last line obtains 13 104

D 18 from 13 8 D 104. The lines indicated with the

n can then be interpreted to read:

1

1

1

1

1

1

1

13

C

C

D1C C C C

8

52

104

2

8

4

8

1

1

1

1

C 104

D 2, and thus 2 divided by 13 is 18 C 52

C 104

.

Hence 13 18 C 52

In the Rhind papyrus, A’h-mosè shows how to solve simple linear problems using a

variety of methods, including the method of false position. The student was expected to

generalize from the examples, which began with:

Problem 1.2. A number and its seventh make 19 [i.e., x C 71 x D 19]. Suppose 7; its seventh

is 1, and together 8. Divide 19 by 8, then multiply by 7. Solution: 16 and 12 and 18 .

To find the area of a circle, A’h-mosè gave the example:

Problem 1.3. Find the area of a circular plot of land with a diameter of 9 khet. Take away

one-ninth of the diameter, leaving 8; multiply 8 by itself to get the area, 64 setat.

The volume of a cylinder was computed by multiplying the area of the base (using the

above method) by the height.

The problems in the Rhind are all linear, though the contemporary Kahun papyrus

seems to suggest the Egyptians could also solve non-linear equations. Unfortunately the

problem statement is missing, and must be reconstructed from the method of solution.

1.2.2

The New Kingdom

The Hyksos conquest of Egypt depended on advanced technology: the horse, chariot, and

compound bow. But all of these were soon duplicated by native Egyptians, who made ready

to throw out their masters. A revolt began in Thebes, and by 1521 B.C., the Hyksos were

expelled and a new dynasty, the eighteenth, was formed. Mathematicians might smile at the

name of the first pharaoh of the new dynasty: Ahmose.2 Ahmose went on to conquer Nubia,

to the south, and his successors established a vast Egyptian Empire that ultimately extended

well into Palestine. The Kingdom of Egypt underwent a great cultural renaissance.

One of the critical events of the Eighteenth Dynasty concerned religion. For millennia,

the Egyptians were polytheists, revering many gods, though chief among them was Amon.

Around 1350 B.C., the pharaoh Amenhotep IV (“Amon is satisfied”) became a monotheist,

2 The

name was a popular one during and immediately after the Hyksos era.

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worshipping a single god. In Amenhotep’s case, the single god was the sun god Aton,

and he renamed himself Akhenaton (“One Useful to Aton”). Akhenaton established a new

capital at Tell-el-Amarna, where Egyptian art and sculpture flourished.

Akhenaton attempted to eliminate the worship of other gods besides Aton, and in this

he failed. His appointed successor Smenkhkare had a very short reign, and was succeeded

by an eight-year-old boy named Tutankhaton. With the accession of a minor, the polytheists

saw the chance to return Egypt to the old religion. They forced the young pharaoh to restore

Amon to the position of chief among many gods, and to rename himself Tutankhamun

(“Living Image of Amon”). Tutankhamun’s unplundered tomb was discovered in 1922 by

Howard Carter and the fifth Earl of Carnarvon, George Edward Stanhope. A few months

later Stanhope died suddenly, giving rise to a belief in a “curse of the pharaohs,” though

Carter, the first to actually enter the tomb, survived another seventeen years.

The internal religious warfare weakened the Eighteenth Dynasty and led to the rise of

the most famous Egyptian dynasty of all, thanks to its pharaoh Rameses the Great. Rameses, the third pharaoh of the Nineteenth Dynasty, built enormous monuments to himself

all over Egypt. The most spectacular was at Abu Simbel, where four 67-foot-high statues

of a seated Rameses loom beside the entrance to a temple. In the 1960s, the damming of

the Nile at Aswan formed a lake that threatened to submerge the temples, so they were cut

apart and reassembled on higher ground.

The Berlin papyrus was written during the Nineteenth Dynasty and gives the first clear

example of a non-linear problem solved by the ancient Egyptians. The problem is:

Problem 1.4. A square with an area of 100 square cubits is to be divided into two squares

whose sides are in ratio 1 to 21 14 [i.e., 1 to 21 C 14 ].

The scribe’s solution is by means of false position: supposing the side of the bigger

9

square is 1, making the side of the smaller square 34 , the total area will be 1 16

, which is

1

1

the area of a square of side 1 4 . Divide 10 by 1 4 to get 8, then multiply the initial guess

(1) by 8 to yield the actual side of the larger square; the side of the smaller square is thus 34

of 8 or 6.

After Rameses, the Egyptian Empire underwent a long and final decline. The reasons

are varied, but one factor may have contributed more than anything else. The Egyptian

Empire was conquered and controlled by warriors using bronze. But around the reign of

Rameses the Great and the writing of the Berlin papyrus, the Hittites, an obscure tribe living

in the foothills of the Caucasus mountains, learned how to refine iron from its ores. An iron

equipped army could easily destroy one outfitted with bronze. The Egyptians adopted the

horses, chariots, and bows of the Hyksos, and might have adopted the iron technology of

the Hittites as well—but there are few sources of iron ore in Egypt.

1.3

Mesopotamia

Egypt is the westernmost of the ancient river civilizations, and is just south of one end

of the Fertile Crescent, a region that includes modern Iraq, southeastern Turkey, Syria,

Israel, Lebanon and Jordan. A second great river civilization arose at the eastern end of

the Fertile Crescent, in modern Iraq. Located between the Tigris and Euphrates rivers, it

became known as Mesopotamia (“between the rivers” in Greek). Today, the Tigris and

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Euphrates merge about a hundred miles inland of the Persian Gulf, though in ancient times,

the two rivers flowed separately into the sea.

Most historians believe that agriculture was invented in Mesopotamia, and thus civilization began in this land between the rivers. The most ancient civilization in this region was

the Sumerian, and by 3000 B.C., a group of Sumerian cities existed in Lower Mesopotamia,

the region closer to the Persian Gulf. Unlike the contemporary Egyptians, the Sumerians

never united into a single kingdom, and each city was a separate political entity with its

own customs, army, and foreign policy; hence they are generally referred to as city-states.

Like the Nile, the Tigris and Euphrates flood periodically. But while the flooding of

the Nile is a gentle, welcome event that signals the beginning of the year and a renewal of

life, Mesopotamian floods are often catastrophic events. One flood tale was included in the

story of Gilgamesh, the King of Uruk (a city along the Euphrates). When a close friend

dies, Gilgamesh searches the world for the secret of immortality, and comes across the one

man who has found it: Utnapishtim. He tells Gilgamesh the story of how the world became

so populated that the gods, tired of hearing the racket made by so many people, decided to

exterminate mankind with a great flood. One of the gods, Ea, warned Utnapishtim of what

was to happen, and had him build a boat in which to save himself and his family. To find

when the waters had receded enough to land, Utnapishtim released a dove, which could find

no resting place and thus returned to the boat, and later a raven, which found a roost and

never returned. The legend of Gilgamesh is almost certainly one of the inspirations of the

Biblical story of Noah. As for immortality, Utnapishtim explained to Gilgamesh how to find

the plant that confers immortality, but the plant is eaten by a snake while Gilgamesh sleeps.

Again, the parallels to the Biblical story of how mankind was deprived of immortality

through the actions of a serpent suggest that the ancient Hebrews drew heavily upon these

legends.

1.3.1

Positional Notation

The Egyptians wrote on papyrus, leather, cloth, or stone. The Mesopotamians had none of

these in abundance; instead, they had mud. Fortunately, mud could be written on with a

stylus while still wet, and then baked into a brick-hard clay tablet. Such tablets can last

for millennia, virtually unchanged. In contrast to a handful of Egyptian papyri, we have

thousands of clay tablets.

Mesopotamian script consists of cuneiform (“wedge-shaped” in Greek) symbols, distinguished by the number of marks and their orientation. Deciphering an arbitrary text with

no knowledge of the content would be virtually impossible, but if the text is mathematical,

we can make considerable headway. Consider the table text reproduced in Figure 1.4. As

you read down the rows of the first and second vertical columns, the number of symbols

increases from one to nine, after which a new symbol, , appears. It seems reasonable to

suppose that indicates a unit, indicates two units, and so on, up until indicates ten units;

then

indicates eleven and so on. Mesopotamian numeration appears to be additive.

However, consider the third column. The first entry is obliterated, but if our assumption

that represents one and represents ten is correct, then the second through seventh rows

read: four, nine, sixteen, twenty-five, thirty-six, and forty-nine. These are the squares of

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1.3. Mesopotamia

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Figure 1.4. A Table Text from Nippur.

the numbers in the second column, so the eighth entry ought to represent sixty-four. So

how shall we interpret the fact that the number is written using five s? If there is any

consistency in the table text, then one of the s must represent sixty and the remaining four

must represent four. This suggests that Mesopotamian notation was positional, base 60.

There are some drawbacks in the positional notation used in Mesopotamia. First, there

was no way to indicate the order of magnitude: by itself might mean one unit, or it might

mean one sixty, or it might even mean one sixtieth. Context was the only way to decide

between the possible interpretations.

Another problem was that there was no way to indicate the lack of an order of magnitude. For example, consider the eleventh line of the table text. The number in the third

column should be the square of eleven, or one hundred twenty-one. This is two sixties and

one unit, and the scribe has properly recorded the number, leaving a wide space between

the s representing the sixties and the representing the unit. But a careless or inexperienced scribe might not leave enough space between the orders of magnitude, making it

possible to read this as three s: three units or possibly three sixties.

Our own system uses a number of devices to solve these problems. First, we have

nine different quantity symbols (1 through 9), so that adjacent symbols actually represent

different orders of magnitude: 123 represents 3 units, 2 tens, and 1 hundred. Moreover,

we use a decimal point to separate units from fractions of a unit, while a tenth symbol

(0) indicates the lack of an order of magnitude. Thus 10:1 is 1 ten, no units, and 1 tenth.

In modern transcriptions of Babylonian sexagesimals, the comma is used to separate the

orders of magnitude and the semicolon (;) separates the units from the fractions; the zero

is used freely. Thus 1 sixty and 1 sixtieth would be written 1; 0I 1. However (and this is

important to emphasize), the Mesopotamians never indicated the sexagesimal point, and it

was not until very late in their history that a symbol for an empty space was used.

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In 1898 an American expedition began excavating the city of Nippur, the center of

worship of the Sumerian god Enlil (“Lord Wind”), and thus one of the main religious

centers in Mesopotamia. There they found thousands of mathematical cuneiform texts in

the eastern section of the city (known as the scribal quarter). Originally attributed to being

from a “temple library”, though later expeditions cast doubt on this claim, these tablets

are of the type known as table texts, and may have been written as early as 2200 B.C.

These are essentially multiplication tables, and provide many examples of Mesopotamian

numeration; there are enough careless errors on the tablets to suggest they were written by

apprentice scribes.

One might expect the multiplication tables to include the multiples of 1, 2, 3, and so

on, and most of the tablets do. But tables exist for the products of (in sexagesimal) 3; 20

or even 44; 26; 40. Such tables might be explained as “make work” assigned by teachers

to keep students busy, but a more likely explanation is found by noting that dividing by a

1

1

number like 18 is the same as multiplying by 18

, and in sexagesimal, 18

D 0I 3; 20. For

1

44; 26; 40 we note that 0I 0; 44; 26; 40 D 81 . Thus these tablets were probably used for

division problems.

1.3.2

Babylon

The disunity of the Sumerican city-states made them easy prey for any unified conqueror.

Under Sargon of Agade (or Akkad), a group of tribes speaking a Semitic language conquered the Sumerians of Mesopotamia around 2300 B.C. Since Sargon and the Akkadians

ruled over a diverse population of which they themselves were only a small minority, we

speak of the Akkadian Empire. It was the first empire in history. It lasted less than a century,

and by 2200 B.C., the empire was destroyed and the capital city of Akkad was so devastated

that even today its exact location is unknown. But the brief existence of the Akkadian Empire had some lasting consequences. One of the main results was that Sumerian (a language

unrelated to any others) gradually disappeared as a spoken language, to be replaced with

Semitic languages (such as Akkadian). This linguistic shift affected place names, and an

obscure Sumerian city, Ka-dingr (“The Gate of God” in Sumerian) was translated literally

into Akkadian as Bab-ilu, a name that eventually became Babylon.

Around 1800 B.C. the Babylonians conquered all of Mesopotamia. Since most of the

problem texts seem to date to around this time, Mesopotamian mathematics is frequently

referred to as Babylonian mathematics.

The most famous Babylonian king was Hammurabi, who reigned around 1750 B.C.

Hammurabi achieved lasting fame for having inscribed on a stele (stone pillar) a complex

and sophisticated legal code. This was not the first written code of laws, though it is one of

the earliest that we have in its entirety.

The stele was originally in Sippar, at the temple of the sun god Shamash. At the top

is a depiction of Hammurabi receiving the laws from Shamash. Flames emerge from the

god’s shoulders, which is suggestive of the much later Biblical myth of Moses receiving

the ten commandments from a flaming god. The code of Hammurabi divides people into

several classes, including noblemen, commoners, and slaves. The law is primarily punitive

and consists of prescriptions: if X occurs, the penalty is Y . Many crimes are punishable by

death or mutilation, but monetary fines are also common. The legal theory of Hammurabi’s

empire gave the classes distinct responsibilities and punishments: in general, harm to a

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noble warranted the greatest punishment, and harm to a slave the least. But nobles were

also held to a higher standard of behavior, and misconduct by a noble was more severely

punished: if a noble stole livestock (as opposed to other types of property), they had to

pay 30 times the value, but a commoner only 10 times. The code also regulates prices and

wages. The law code must have made a great impression on the people of the time, for

shortly after Hammurabi’s death, an invading army carried the stele away to Susa, where it

was discovered in 1901 by French archaeologists.

We cannot compare the legal codes of Babylon and Egypt, for the surviving Egyptian

law codes date to a time much later than Hammurabi. We may, however, compare their

mathematics, for the Rhind papyrus and the Babylonian problem texts are contemporaneous. Many Babylonian problems dealt with canals, a necessity of life in Mesopotamia,

required both for irrigation and flood control. A typical problem was:

Problem 1.5. A canal 5 GAR long, 1 12 GAR wide, and 12 GAR deep is to be dug. Each

worker is assigned to dig 10 GIN, and is paid 6 SE. Find the area, volume, number of

workers, and total cost.

where GAR, GIN, and SE are units of quantity (the GIN is equal to a cubic GAR, for

example).

The Babylonians frequently ventured beyond simple linear equations. Many canal problems resulted in quadratic equations, though a few “pure math” problems were posed:

Problem 1.6. The igibum exceeded the igum by 7, and the product of the two is 1; 0. Find

the igibum and the igum.

The problem is equivalent to solving x.x C 7/ D 60, and the scribe’s solution relied on an

identity we would write as

aCb

2

2

a

b

2

2

D ab

Thus given the product ab and a ˙ b, we can find a b. Finally, the igibum and the igum

can be found using

aCb

a b

C

Da

2

2

aCb a b

Db

2

2

There is some evidence that degenerate cubics with rational solutions were solved by

transformation into a standard form and referring to tables of values. Even approximate

solutions to exponential equations were found using linear interpolation (also known as the

method of double false position or the method of secants).

Much of our knowledge of Babylonian geometry comes from a French expedition

to Susa, about 200 miles east of Babylon. In 1936, the expedition uncovered a horde of

cuneiform texts, including some on geometrical subjects, which seem to be contemporaneous with the arrival of Hammurabi’s stele. One tablet shows a circle with the cuneiform

numbers 3 and 9 on the circumference and 45 on the inside. This suggests that the area

of a circle was found by taking the circumference, 3, squaring it to get 9, then dividing

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by 12 to get 0I 45. This gives an even worse approximation for the area of a circle than

the one used by the Egyptian scribes. This might suggest the Mesopotamians were poor

geometers, though other tablets of the Babylonian era show they knew of and used the

Pythagorean Theorem; the most remarkable is a tablet that seems to indicate the diagonal

of a square is 1I 24; 51; 10 of its side which, if converted to its decimal equivalent, gives

p

2 1:414213 : : :.

In addition, Plimpton 322, believed to have been written around Hammurabi’s time,

gives a list of 15 rational numbers that satisfy a2 C b 2 D c 2 , which suggests an early

interest in number theory, and might point to the knowledge of the Pythagorean Theorem

as well. Another tablet gives relationships between the area An of a regular n-gon with a

side length of Sn as:

A5 D 1I 40S52

A6 D 2I 37; 30S62

A7 D 3I 41S72

These values are accurate to within about 3%. Moreover, if C is the circumference of

the circle circumscribed about a hexagon, then the tablet also suggests S6 D 0I 57; 36C ,

accurate to within 1%.

1.3.3

The Iron Empires

Around 1400 B.C., the Hittites of Asia Minor learned how to smelt iron. Before they could

turn this to their advantage, they were conquered by the Assyrians, who were the first to

field an army of iron.

In the ancient world, a conquered population could expect death or enslavement. The

Assyrians added a new horror to war, and routinely tortured and mutilated their captives in

what can only be called a calculated attempt to terrify potential adversaries into submission.

The end result of this policy was that the Assyrian empire, established by force, had to be

maintained by force and was ultimately destroyed by force.

In 612 B.C., the Assyrian capital of Nineveh was utterly destroyed by an alliance that included the Medes under Cyaxeres and the Chaldeans under Nabu-apal-usur (“Nabu guards

the prince”), usually known as Nabopolassar. Under him the Chaldeans rose to become a

mighty empire. His son, Nabu-kudurri-usur (“Nabu protects my boundaries”) rebuilt Babylon and turned it into the capital of a new empire, properly called the Chaldean Empire but

also referred to as the Babylonian or Neo-Babylonian Empire.

Nabu-kudurri-usur is better known as the Nebuchadnezzar of the Bible (though we

will use the more correct spelling Nebuchdrezzar). According to tradition, Nebuchadrezzar

married a Medean princess, but she became so homesick for her homeland in the foothills

of Asia Minor that he had built for her a fabulous tiered garden. The “Hanging Gardens of

Babylon” became one of the Seven Wonders of the World.

In 598 B.C., Judea revolted against Babylonian rule. Nebuchadrezzar suppressed the

revolt, deported some of the leading citizens to Babylonia, but left the kingdom, temple,

and local government intact. A second revolt began. Nebuchadrezzar captured Jerusalem

in 586 B.C., destroyed the Temple of Solomon, and arranged for a second deportation. This

began the era of Jewish history known as the Babylonian Captivity.

Meanwhile, the Medes were moving into Asia Minor, where they met the expanding

Lydian Empire. Five years of inconclusive warfare between the Lydians and the Medes

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were brought to an abrupt end when, during a battle, “day turned into night:” a solar eclipse

occurred. The opposing commanders took this as a sign of the displeasure of the gods, and

signed a hasty peace. We can calculate that a total eclipse was visible from Asia Minor on

May 28, 585 B.C., so this battle is the earliest historical event that can be given an exact

date.

According to one story, the eclipse was predicted by the first mathematician we know

by name: Thales of Miletus.

For Further Reading

For the history of Egypt and the Fertile Crescent, see [1, 18, 60, 68, 106, 130]. For the

mathematics and science of the region, see [22, 41, 52, 84, 85].

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The Classical World

2.1

The Greeks

Greece is a mountainous land of limited fertility, so settlements tended to cluster in the narrow valleys and on the various peninsulas that jut out into the Mediterranean. The relative

isolation of the settlements encouraged them to form independent city-states. Like Sumeria, the individual Greek city-states were no match for a united conqueror, so by the time

of A’h-mosè, mainland Greece was part of the Minoan Empire, centered on the island of

Crete. The Minoan royal palace at Knossos was an impressive structure. Spread over six

acres of land, it contained hundreds of rooms—and flush plumbing. Around 1500 B.C., the

volcanic island of Thera exploded, causing a tsunami so devastating that the Minoans never

recovered; this was probably the origin of the legend of Atlantis, retold by Plato a thousand

years later.

The tsunami weakened the Minoans and the Greeks revolted. They invaded Crete,

burned the palace and other centers of civilization, and lost the secret of flush plumbing

for thousands of years. The Greeks established the Mycenaean Empire, named after one of

their main cities. Around 1250 B.C., the Mycenaens besieged and destroyed Troy, a city on

the coast of Asia Minor. But shortly after, the bronze-equipped conquerors of Troy were

overcome by iron-wielding Dorians from the north. The Dorian dialect of Greek was difficult for the Mycenaeans to understand; they lampooned Dorian speech as “bar-bar” (much

as we might describe someone’s ramblings as “yadda-yadda”) and called them barbarians,

a term that has since been applied to cultures that have no permanent cities. Since it is

easy (and usually incorrect) to associate “uncivilized” with “simple-minded,” a particularly

plain type of architectural column was later called Doric in the (mistaken) belief that they

were products of Dorian builders.

By 700 B.C., Greece had recovered from the Dorian invasions, and the population was

booming. To relieve crowding, a city-state would establish colonies around the Mediterranean. The “mother city,” or metropolis in Greek, would equip a colony with people,

ships, and government. As often as not, the colonies became independent of the mother

city within a few generations, though some kept close ties.

These colonies went out in two directions, which can lead to confusion among students

of classical geography. The Greek settlers who went west, towards the Italian peninsula and

the island of Sicily, were the first Greeks encountered by the Romans. Thus the Romans

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referred to southern Italy as Magna Graecia, Latin for “Greater Greece” (where in this

context, “Greater” means “Larger”).

Other Greek colonists went east, to Asia Minor. Many who settled there came from the

shores of the Ionian Sea, on the west coast of modern Greece. In commemoration of their

original homeland, the Greek colonies in Asia Minor were collectively known as Ionia,

even though they nowhere border the Ionian Sea.

2.1.1

Thales and Pythagoras

T HALES (fl. 6th cent. B.C.) lived in Miletus, in Ionia. He is credited with having “discovered” five geometrical propositions (which suggests that he did not prove them):

1. A circle is bisected by its diameter.

2. The base angles of an isosceles triangle are equal.

3. Vertical angles are equal.

4. If, in a triangle, a side and its two adjacent angles are equal to the side and two adjacent

angles of another triangle, then the two triangles are equal.1

5. An angle inscribed in a semicircle is a right angle.

There is a story that Thales sacrificed an ox upon his discovery of the last theorem.

The life of Thales is well documented in the Histories of Herodotus, who lived in the

fifth century B.C. Herodotus has been called the “father of history,” because he was the first

to apply recognizably modern historical methods to the problem of the past, as well as the

“father of falsehoods,” because so much of what he concludes was fantastical and hard to

believe. However, the more fantastical statements were usually Herodotus quoting what

someone else said, so a proper title for Herodotus might be the “father of journalism.”

According to Herodotus, Thales predicted the eclipse that ended the wars between the

Lydians and the Medes. The peace was sealed by a marriage between Astyages, the King

of the Medes, and Aryenis, the sister of Croesus, the King of the Lydians. Croesus then

turned his attention south, towards the Greeks, so by Thales’s time, most of Ionia had been

absorbed by Lydia. Miletus, recognizing the futility of resistance, allied itself with Lydia

and retained a measure of independence.

Part of the secret of the success of the Lydian Empire was a remarkable invention made

around 650 B.C.: coinage. The value of money is often underrated and frequently misunderstood. Economic activity relies on trade, but trade requires two people, each having what

the other person wants. Money serves as a universally desirable commodity. The problem

is making theory and practice coincide. One way is to make money out of silver or gold,

both rare, durable metals.

However, silver or gold by themselves are merely valuable; they are not money. Imagine

the problems that would arise from trying to pay for a purchase using silver or gold. First,

1 Greek geometers used the term “equal” to mean one of three things. First, two figures are equal if one can

be superimposed on the other through a sequence of rigid transformations; today we use the term “congruent.”

Alternatively, two figures were equal if one could be dissected and reconstituted to form the other; today we would

say the two figures have the same area. Finally, two figures were equal if their lengths, areas, or volumes had a

ratio of one to one.

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the purchaser would have to cut off a sliver of metal and measure its size (weighing would

be the easiest method). But before accepting it the prudent seller would verify its weight

and purity. Discrepancies over perceived weight or purity would lead to endless argument,

and trade would grind to a halt. In the seventh century B.C., Croesus’s predecessors had

a marvelous idea: a trusted authority could stamp a disk of metal guaranteeing its weight

and purity. Thus, coinage was invented. The commerce flowing through Lydia made her

kings immensely wealthy, so the Greeks used “rich as Croesus” to describe anyone of great

wealth.

Around 553 B.C., a usurper named Kurush from Fars rebelled against Astyages, and deposed him by 550 B.C. in a relatively bloodless civil war. Kurush, his homeland Fars (east of

Mesopotamia), and his family, the Hakhamani, are better known to us by the Latinization

of the Greek forms of their names: Cyrus, Persia, and the Achaemenids. Croesus saw an

opportunity to invade and conquer the Medes (which he could do under the guise of liberating them from a foreign conqueror). Before invading, he consulted his advisors, and got an

outside opinion from the Oracle of Delphi, supposedly possessed of the gift of prophecy.

She told him that if he invaded, he would bring down a mighty empire. This sounded like

good news to Croesus, and he sent in his army.

The border of the Lydian Empire was the river Halys. Herodotus noted (with doubt) a

belief among the Greeks that Thales diverted the river so a bridge could be built across it. If

this story is true, then Thales, like so many later mathematicians, was a military engineer.

Croesus’s army invaded Persia, and promptly fulfilled the Oracle’s prophecy. Unfortunately

the mighty empire that Croesus brought down was his own, which Cyrus conquered in

547 B.C. Ionia, as part of the Lydian Empire, became part of the Persian Empire. Again,

Miletus remained independent.

Cyrus employed a novel strategy: tolerance. For example, the worship of Marduk had

been suppressed by the Chaldean king Nabonidus; Cyrus promised to restore the worship

of Marduk. With the worshippers and priests of Marduk on his side, Cyrus conquered the

Chaldean Empire easily in 538 B.C. After the conquest, Cyrus kept his promise.

Another group disaffected by the Chaldeans were the Jews, many of whom had been

living in Babylon since the time of Nebuchadrezzar. Cyrus allowed the Jews to return to

Palestine to rebuild the temple. Few bothered: most had made comfortable lives for themselves in Babylon, though they did send monetary gifts with the returnees.

While in Babylon, the Jews maintained their cultural identity by a rigid adherence to

the old customs. The Palestinian Jews, on the other hand, let their customs evolve naturally.

As a result, the Babylonian Jews, returning to Palestine, felt the indigenous Jews were

practicing the “wrong” form of Judaism. Since the chief city of the Palestinian Jews was

Samaria, this was the beginning of a great schism in Judaism and as a result, the Babylonian

Jews (who became dominant) learned to hate and detest the Samaritans. Half a millennium

later, this hatred would be used to good effect in a story that argued for the fundamental

unity of all mankind.

Egypt was added to the Persian Empire in 525 B.C. by Cyrus’s son Cambyses, and by

the time of his death in 521 B.C., the Persian Empire included Egypt and most of the Fertile

Crescent. It was the largest empire the West had yet seen.

During the conquest of Egypt, a number of Greeks were captured by the Persians and

transported to Babylon. There is a tradition that P YTHAGORAS of Samos (580–500 B.C.)

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was among this group. Pythagoras had been in Egypt since 535 B.C., and spent five years

in Babylon before making his way back to Greece. In 518 B.C., Pythagoras left the eastern

Mediterranean forever and settled in Croton, at the heel of the “boot” of modern Italy.

Croton was the home of Milon (or Milo) the wrestler, the most famous athlete in antiquity. Milon’s strength was legendary. During an Olympic procession around 540 B.C., he

is said to have carried a fully grown ox across the stadium, a distance of about 600 feet.

Milon, as the local sports hero, was greatly honored in Croton and in 510 B.C. led an expedition that destroyed the neighboring town of Sybaris, whose inhabitants were well known

for their rich lifestyle (hence the word sybaritic).

At Croton, Pythagoras established a secretive, mystical school that lasted about a century. At least one story suggests that Milon was Pythagoras’s patron, and that Milon’s

daughter became one of the first Pythagoreans. None of Pythagoras’s own work has survived; we have only what his followers claim he said. Some of the rules of the school, such

as always wearing white and not eating beans, have parallels with ancient Egyptian practices. Pythagoras and his followers were struck by the many relationships among numbers;

they sought to analyze the physical world in terms of these number relations. For example,

they apparently discovered that the sum of the first n odd numbers is n2 . Since the Greek

word for number is arithmos, this study of number properties became known as arithmetic.

Perhaps the best evidence of the beauty of mathematics comes from the Pythagorean

study of music. According to one rather dubious story, Pythagoras happened to be passing

by a smithy, and noticed that the noise of the falling hammers sounded pleasant. Upon

investigation, he discovered that the weights of the hammers had a whole number ratio to

one another. This began a study of music from a mathematical perspective.

Rather than using hammers, the Pythagoreans used a monochord: a one stringed instrument with a movable bridge. If the bridge divided the string into two equal parts, the two

parts could be plucked one after another (melodically) or simultaneously (harmonically). In

both cases, the sounds went together well: the string ratio of 1 to 1 produced a consonance

(now known as unison, from the Latin “one sound”).

Suppose the bridge was moved so the string was divided in a 2 to 1 ratio. The shorter

part would produce a higher pitched version of the tone produced by the longer part, and

the 2 W 1 ratio produces another consonance.2

Another consonant ratio corresponds to string lengths in a ratio of 3 W 2. Finally, an

“inversion” takes the lower note of a set and replaces it with a higher pitched version of

the same note (in this case, by halving the string length). Inverting the 3 W 2 ratio produces

a 2 W 3=2 ratio, which we can simplify to 4 W 3. The evidence of the senses, as well as

the elegance of the numerical ratios, make us regard this ratio as consonant. Moreover, the

three lengths together give us a 6 W 4 W 3 ratio, which has a remarkable property: the ratio of

the difference between the first and second to the difference between the second and third is

equal to the ratio between the first and third. In this case, the ratio of 6 4 to 4 3 is equal

to the ratio of 6 to 3. This type of ratio is now called a harmonic ratio (and the numbers

are said to form a harmonic progression). On the other hand, if we began with the 4 W 3

ratio and inerted it, we would obtain a 3 W 2 ratio, giving us three numbers in an arithmetic

progression.

2 Changing the tension or thickness of the string also affects the tone. This fact makes the hammer story extremely improbable, since several factors would influence the fundamental tones produced by dropping hammers.

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Suppose we tuned an instrument so the strings had a 4 W 3 W 2 or a 6 W 4 W 3 ratio. The

instrument would produce three notes, with the property that any combination of the notes

would form a consonance. However, a three note repertoire is rather limited, so additional

notes were added. Various schemes were tried, but eventually the division of the 2 W 1

interval into eight notes became standardized (Euclid was the first to note this division,

though it certainly predated him). Consequently, the 2 W 1 ratio is said to correspond to

an interval of an octave. From lowest to highest, these notes are now designated as C-DE-F-G-A-B, with the eighth note also called C and beginning the pattern again.3 The C-G

interval, which spans five notes, corresponds to the 3 W 2 ratio; hence this ratio is referred

to as a fifth. Likewise the C-F ratio, which spans four notes, corresponds to the 4 W 3 ratio,

and is designated a fourth. This means that the F-G interval must correspond to a ratio of

9 W 8. We can designate this ratio as defining a tone. But the internote ratios cannot all be

9 W 8, since 97 W 87 ¤ 2 W 1. In fact, the problem is worse than that: there are no whole

numbers p, q for which p 7 W q 7 D 2 W 1. Thus it is impossible to divide the octave into

equal intervals. This is the root of what is called the tuning problem: how can we make

a product of the power of one rational number equal another rational number? If the two

rational numbers have different prime factors, then the problem is unsolvable; the best we

can hope for is an approximation.

Any solution to the tuning problem must sacrifice some of the intervals. In Pythagorean

tuning, the octave and fifths are retained, while the fourths and tones are sacrificed where

necessary. For example, if C-D and D-E both correspond to a 9 W 8 ratio, then to make

C-F correspond to a fourth (and thus the ratio 4 W 3), then E-F must correspond to the

ratio 256 W 243. Since 2562 W 2432 is approximately equal to the ratio 9 W 8 that defines

a tone, this new and inelegant ratio is designated a semitone. In order for all fourths to be

perfect, it is necessary that any sequence of four notes consist of two tones and a semitone.

Likewise, for all fifths to be perfect, any sequence of five notes must contain three tones

and a semitone. The octave, which consists of a fourth and a fifth together, thus consists

of five tones and two semitones, with the semitones five notes apart. Reconciling all these

factors produces the Pythagorean tuning:

C to D

9W8

D to E

9W8

E to F

256 W 243

F to G

9W8

G to A

9W8

A to B

9W8

B to c

256 W 243

The F-B fourth is slightly sharp (the upper note is slightly high in pitch), but the remaining

fourths and fifths are perfect. This solution is uniquely determined up to the starting point.

In the above, the lowest note of the scale is C (so we have two tones followed by a semitone,

followed by three tones and a semitone), but if the lowest note is E (and thus we begin with

a semitone, followed by three tones and a semitone, then two tones), you are playing in the

“Dorian mode.”

In addition to studying music, Pythagoras began a tradition of examining mathematical results in an “immaterial” and “intellectual” manner, thereby turning it into a “liberal

art.” This probably means that Pythagoras introduced the deductive method to mathematics, while restricting its domain to the theoretical properties of abstract objects. We can

3 To

distinguish the two Cs, the German physicist Hermann von Helmholtz used a system of marks. Thus if

our first note is C, then the same note one octave higher is designated c, and the same note one octave higher still

is designated c0. This last note corresponds to “middle C.”

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draw an interesting parallel: the ancient Olympic games highlighted the skills of the warrior, presented in an abstract form divorced from an actual siege or battle. Although the

term itself is Latin, the idea of liberal arts originated with the Greeks—and slavery. Slavery was a key element of almost every culture before the present day, though generally

men (and women, and children) were slaves because of military conquest, rather than race.

Pythagoras himself was probably a slave during his time in Babylon.

Slaves worked with their hands: hence, the work done by slaves came to be known as

manual labor, from manus (“hand” in Latin). Free men, on the other hand, were expected

to pursue the liberal arts, from the Latin libera (“free man” in Latin). Free women, incidentally, were expected to bear children; raising the children was a task for slaves.

The mathematics of the Egyptians and the Babylonians concerned itself with practical

matters, whether it was computing the height of a pyramid or the cost of digging a canal.

Because of its association with manual labor, this type of mathematics was deemed fit only

for slaves. The Greeks gave the name logistics to this practical, computational mathematics.

The difference between logistics and a liberal art like geometry can be illustrated in the

following way. The procedure for finding the area of a parallelogram:

Rule 2.1. The area of a parallelogram is the base times the height.

is logistics, while the theorem:

Theorem 2.1. If two parallelograms have equal bases, and equal heights, then either can

be dissected and rearranged to form the other.

is geometry.

The Pythagoreans seem to have been the first to make proof an essential part of mathematics. They are credited with discovering and proving at least five theorems. Four of these

are:

1. The angles in a triangle are together equal to two right angles.

2. The angles in an n-gon are together equal to the angles in n

2 triangles.

3. The exterior angles in a polygon are together equal to four right angles.

4. The space about a point can be filled with regular triangles, squares, or hexagons.

The last may have led to Pythagorean discovery of three of the five regular solids: the

tetrahedron, formed by equilateral triangles; the cube, formed by squares; and the dodecahedron, formed by regular pentagons.

The fifth Pythagorean discovery was the Pythagorean Theorem. A few myths about the

Pythagorean Theorem ought to be discussed. The most often repeated myth comes to us

from Proclus (writing in the fifth century A.D.): “Those who like to record antiquities”

claim Pythagoras sacrificed an ox (a hundred oxen in some accounts) upon the discovery of the relationship between the sides of a right triangle. Proclus sounds dubious: the

Pythagoreans believed in transmigration of souls, and were adamantly opposed to animal

sacrifices. The story is also suspiciously like one told about Thales. Another story is that

Pythagoras learned the theorem in Egypt, from “rope stretchers” who routinely formed 34-5 right triangles using a knotted chord. There is no evidence of any Egyptian knowledge

of the Pythagorean Theorem in any form.

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If Pythagoras did not discover the theorem independently, then he may have learned

about it in Babylon. More concretely, Pythagoras may have learned how to construct what

are now called Pythagorean triplets: three numbers, a, b, and c, that satisfy a2 C b 2 D c 2 .

Pythagoras constructed triplets in the following way: if a is any odd number, b half of one

less than the square of a, and c one more than b, then a2 C b 2 D c 2 . For example, if a D 5,

b D 21 52 1 D 12, and c D 12 C 1 D 13, and 52 C 122 D 132 .

Both the Pythagorean Theorem and the tuning problem lead to incommensurable quantities, discovered by the Pythagoreans during the fifth century B.C. Today we would say

that two quantities are incommensurable if the ratio between them corresponds to an irrational number. We do not know how incommensurable quantities were discovered or who

discovered them, and even the identity of the first pair of incommensurable quantities is unknown. One of the better candidates is the side and diagonal of a regular pentagon inscribed

in a circle. This suggests that the discoverer was H IPPASUS (fl. ca. 430 B.C.) from Metapontum, who was apparently expelled from the order for revealing to outsiders the Pythagorean

methods of inscribing a regular pentagon in a circle and a regular dodecahedron in a sphere.

The school did not long survive the discovery of incommensurable quantities, for it

began to interfere in local politics. By the middle of the fifth century B.C., it was suppressed

by the authorities. Real progress in mathematics came from the Greek heartland—which

was at that point in a struggle for its very existence.

2.1.2

The Wars of Greece

The greatest of the Persian kings was Cambyses’s successor Darius, from a collateral

branch of the Achamaenids. Darius became king in 521 B.C., and concentrated on internal

improvements. He established a vast system of roads, complete with military patrols to deter bandits, and an efficient postal system. Regarding the latter, Herodotus wrote: “Neither

rain, nor sleet, nor dark of night stays [prevents] these couriers from the swift completion

of their appointed rounds.”4

In 500 B.C. the Ionian city-states, led by independent Miletus, revolted against Persian

rule. They pled for help from the Greek mainland, but only Athens and Eretria sent more

than token assistance. Miletus finally lost its independence when the Persians conquered it

in 494 B.C., and Darius organized a punitive expedition to deal with the interfering Greeks.

Darius, a shrewd diplomat, encouraged the neutrality of the other Greek city-states by announcing that his battle was with Athens and Eretria alone.

In 490 B.C., twenty thousand Persian troops laid siege to Eretria. After six days, a traitor

opened the city gates to the Persians, who destroyed the city and carried its population off as

slaves. With Eretria successfully reduced, the Persians crossed the narrow strait separating

Euboea from mainland Greece, and made ready to crush Athens. The Athenians sent a

runner, Pheidippides, to seek help from Sparta.

The Spartans conquered their neighbors the Messinians around 600 B.C., and reduced

the inhabitants to the status of serfs, or helots. The helots outnumbered the Spartans ten

to one, so to keep the helots subjugated, the Spartans made their society increasingly militaristic. Children were subject to a rigorous examination upon birth, and any who seemed

4 The architect William

Mitchell Kendall thought the phrase appropriate for the New York General Post Office,

and when the building was completed in 1912, the inscription appeared along its façade. It is not the motto of the

U.S. Postal Service.

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weak or sickly were exposed and left to die. Education emphasized physical fitness, poetry,

and music.

At the age of 20, a Spartan male joined one of several military units. The members of

the unit ate together, lived together, fought together—and if necessary, died together. One

Greek visitor, after eating at the communal barracks, found the food so unpalatable that he

is said to have remarked, “Now I know why the Spartans are unafraid of death.” Silence was

encouraged, and if one had to speak, one should be brief about it and issue short statements

that cut right to the point. Since Sparta is located in the region of the Peloponnesus known

as Laconia, the other Greeks commented on their laconic way of speaking.

The Spartans agreed to help Athens, but for political reasons, they could not send assistance until after the full Moon. Pheidippides ran back to Athens to report the bad news.

Boldly, the Athenian commanders made the decision to attack before the Spartans arrived.

This turned out to be the correct decision, for the lightly armed Persians were no match for

the heavily armed and armored Greek hoplite (which referred to their armor). The Persians

were forced back to their ships, losing over six thousand men in the process. The Athenians lost less than two hundred. Pheidippides ran the 26 miles from the battle site to the

marketplace in Athens, where he announced victory (nike in Greek) over the Persians on

the beaches of Marathon. Then he collapsed and died of exhaustion, having run about 150

miles in two days.

Egypt revolted against the Persians at the same time, and keeping the rich province of

Egypt was far more important than punishing the Greeks. Thus the next great invasion of

Greece did not occur until 480 B.C., under Darius’s successor Xerxes. This time it was clear

that the intent was not to punish Athens, but instead to conquer all of Greece.

At the pass of Thermopylae, 1400 Greeks, including 300 Spartans, faced the entire

Persian army. The exact size of the Persian army is unknown, but it may have been around

100,000 soldiers, which included 10,000 elite troops, known as “Immortals.” According to

Herodotus, the Spartans were warned by a native of Trachis (a village near Thermopylae)

that when the Persians fired their arrows, they would be so numerous they would blot out

the sun. A Spartan named Dienices remarked, “Good! We will have shade to fight in.”

Defeat was inevitable, but the Spartans stopped the Persian advance for three days (dying

to the last man), buying enough time for other parts of Greece to improvise hasty defenses.

It was not enough: Xerxes laid siege to Athens and burned it to the ground. But at the naval

battle of Salamis, the Persians lost nearly half their fleet; Greek losses were insignificant.

Xerxes withdrew—for one year. The next year, Xerxes returned with another army, suffered

another defeat (at Plataea), and withdrew again.

To fight Persia, Athens organized an alliance, now known as the Delian League, because

its headquarters and the treasury were located on the island of Delos. Member states of the

alliance could either contribute ships and men, or the equivalent in money. Most members

chose to supply money, letting Athens build and man the ships of the fleet. Eventually the

Delian League included all of Ionia, and the Persian threat began to recede.

In theory the alliance, with no enemy to fight, ought to have been disbanded, but Athens

argued that it was still necessary to defend Greece from Persia. Member states grudgingly

agreed to maintain what was rapidly becoming the Athenian Navy, but after a while, the

threat of Persian invasion seemed remote. Tired of paying for a navy that did it no good, the

inhabitants of the island of Naxos attempted to withdraw from the alliance. The Athenians

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Figure 2.1. The Athenian Empire

refused to accept the withdrawal, laid siege to Naxos, and in 470 B.C., captured the city,

destroyed its fortifications, and sold the inhabitants into slavery. Other member states were

cowed into submission, their annual dues being converted into tribute, and the Athenian

Empire was established.

Pericles was an ambitious aristocrat who came to power in 461 B.C. and presided over

the Golden Age of Athens. The great playwrights Aeschylus and Sophocles lived and

worked in Athens in this period; indeed, one of Pericles’s first recorded actions was paying

for the production of one of Aeschylus’s plays in 472 B.C. The Parthenon, considered one of

the most elegant buildings in the world, was also constructed in the time of Pericles, using

funds technically belonging to the Delian League. When Thucydides of Melesias objected

to this misappropriation of funds, Pericles arranged to have him ostracized (a political maneuver that forced Thucydides into exile for ten years).5

The Golden Age of Athens was a great era of building, democracy, literature—and

slavery. At the height of the Golden Age, there were perhaps fifty thousand (male) citizens

with the franchise—and over a hundred thousand slaves. Many worked at the state-owned

silver mines of Laurion, which were so profitable that by 483 B.C. each Athenian citizen

received a share of the revenues. Even today, mining is an extremely dangerous profession;

in the fifth century B.C., the death toll among the slaves must have been horrific, and some

estimate that slaves only survived an average of two years in the mines.

Only three states of the Delian League maintained a measure of independence, contributing ships and men to the fleet rather than money to the treasury. Geography may have

played a role: all were large islands off the shore of Asia Minor, so of necessity they were

forced to have strong navies. From north to the south, the islands were Lesbos, Chios, and

Samos.

5 The term

comes from ostraka (“pottery shard”) onto which the name of an individual might be written: those

who received more than a specified percentage of the votes were sent into exile, with the understanding that their

property and person were to remain undisturbed in their absence.

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Mathematicians of the Golden Age

Chios joined the league in 479 B.C. There may have been a school there established by

Pythagoras, for O ENOPIDES of Chios (fl. 450 B.C.) was one of the more celebrated mathematicians of the era. We know nothing about Oenopides’s life, though there is some evidence he visited Athens. He would not have been the first: as the vibrant center of a growing empire, Athens attracted the best minds of the age. For example, Herodotus moved to

Athens to write his Histories about the recent Persian wars. Oenopides is credited with being the first to construct a perpendicular and to construct an angle equal to a given angle.

As the Parthenon and other large buildings were under construction by the time of Oenopides, it seems unlikely that he was actually the first to do either of these things. More likely,

Oenopides was the first to construct the figures using only compass and straightedge, the

tools of the liberal arts, not those of the manual ones.

Around 462 B.C., A NAXAGORAS (500–428 B.C.) came to Athens from Clazomenae, in

Ionia. Anaxagoras may have been the first to bring the Ionian tradition of rational inquiry to

Athens. Anaxagoras suggested that the sun was not a god, but instead a hot rock larger than

the Peloponnese. This suggestion might have been based on the following logic: First, the

Moon eclipses the sun, which implies the sun is more distant. Second, the sun and Moon

appear to be the same size, so the sun must be larger than the Moon. Hence the region of

totality during an eclipse must be smaller than the Moon. From accounts of the total eclipse

of April 30, 463 B.C., Anaxagoras might have learned that the shadow of the Moon covered

most of the Peloponnese; this would allow him to give a lower bound for the size of the

Moon and sun.

The other Athenians were not as impressed with the Ionian traditions, and Anaxagoras was charged with heresy. In an account written five hundred years later, Plutarch said

Anaxagoras “wrote on” the squaring of the circle while awaiting trial, though Plutarch gave

no details. The events surrounding the trial itself are somewhat hazy; Anaxagoras was apparently condemned to death, though in fact he lived out the remaining years of his life in

Lampsacus in Ionia. Pericles may have secured Anaxagoras’s acquittal (there is a tradition

that Pericles was a student of Anaxagoras), but other accounts suggest that Anaxagoras had

already left Athens by the time charges were brought, and that the trial and condemnation

occurred in absentia.

If Plutarch’s account can be taken at face value, Anaxagoras may have been the first to

examine one of what became known as the Three Classical Problems of antiquity:

1. Trisection of an angle: given an angle, divide it into three equal angles.

2. Duplication of the cube: given a cube, to construct another with twice the volume.

3. Squaring the circle: given a circle, to construct a square equal to it in area.

The greatest mathematician of the era was Oenopides’s countryman, H IPPOCRATES

(470–410 B.C.). Hippocrates of Chios should not be confused with his contemporary, Hippocrates of Cos, the physician. Hippocrates of Chios was originally a merchant who came

to Athens to retrieve a cargo lost to piracy. The legal proceedings against the pirates took so

long that, to support himself, Hippocrates became the first known professional teacher of

mathematics. Moreover, he wrote a textbook, called the Elements of Geometry (now lost),

for his students.

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B

F

E

D

A

C

Figure 2.2. Hippocrates Lunes

Hippocrates was apparently aware that:

Theorem 2.2. The area of a circle is proportional to the area of the square on its diameter.

However, he probably did not have a proof.

While attempting to square the circle, Hippocrates was able to show that certain lunes

(regions bound by circular arcs) were equal in area to certain rectilineal figures (see Figure

2.2). If ABC is an isosceles right triangle inscribed in a semicircle, and ADC is a segment

of a circle similar to the segments AEB, BF C (two segments are similar when their central

angles are equal), then the lune ABCD is equal to the triangle ABC . Hippocrates found

several other results, all equating the area of a lune with the area of a certain rectilineal

figure.

Hippocrates also took steps towards duplicating the cube. Given two quantities, a and

d , then the problem of inserting one mean proportional between them is the problem of

finding b so a W b D b W d . We could insert two mean proportionals, b and c, if a W b D b W

c D c W d . Hippocrates showed that in this case, a3 W b 3 D a W d . Thus if d D 2a, then the

cube with a side of b will have twice the volume of a cube with a side of a. Hippocrates

himself was unable to solve the problem of finding two mean proportionals.

The most enigmatic mathematician of the era was D EMOCRITUS (ca. 460–370 B.C.),

who came from Abdera at the northern edge of the Athenian Empire. Democritus traveled

around the Mediterranean, and made at least one visit to Athens, where he was snubbed

by Anaxagoras. Although none of Democritus’s writings have survived, references to his

work in other sources indicate that he was the first to give correct formulas for the volume

of a pyramid and cone, though he apparently did not give a proof. A work of his known

only by its title, Two Books on Irrational Lines and Solids, suggests that the existence of

incommensurable figures was well known by his time.

H IPPIAS (b. ca. 460 B.C.) also came to Athens, from Elis, an independent city-state on

the western shore of the Peloponnesus. He was a member of a new school of philosophy,

the Sophists. Unlike the Pythagoreans, who were secretive and would teach only those who

wished to become Pythagoreans, the Sophists would teach anyone who paid them. Teaching

seemed too much like manual labor to other philosophers, who reviled the Sophists. It might

seem that there was a small market for philosophy lessons, but one of the skills taught by

the Sophists was the skill of debate, critical for success in the Athenian legal system.

Hippias might have taught in Sparta briefly, but found the Spartans uninterested in his

primary subjects (astronomy, geometry, and logistics). Thus he made his way to Athens,

where his skills were in demand. He was the first to invent a curve that could not be constructed using compass and straightedge. Since the curve could be used to square the circle,

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D

C

K

G

F

E

L

A

N

H

M

B

Figure 2.3. Trisectrix or Quadratrix

it is known as the quadratrix, from the Latin quadratus, “square” (see Figure 2.3); it is not

known if Hippias knew that the curve could be used in this fashion.

To construct the quadratrix, take the square ABCD, and let DKB be the quadrant of

a circle. Let the side DC drop parallel to itself towards AB while at the same time the

radius AD rotates until it coincides with AB. The intersection of the radius and the side of

the square at points G, N , etc. form the quadratrix. If angle KAB is to be trisected, then

draw EF parallel to AB and divide EA into thirds at L (which can be done using compass

and straightedge); draw ML which intersects the curve at N . Then †NAB is one-third

†KAB. Squaring the circle relies on the theorem that arc DB is to DA as DA is to AH ;

thus locating the point H on the quadratrix allows us to square the circle.

A NTIPHON (480 B.C.–411 B.C.) and B RYSON (b. ca. 450 B.C.) were two other important

Sophists. Antiphon was Athenian, and Bryson may have come to Athens to study under

Socrates. Antiphon worked on the problem of squaring the circle, and apparently suggested

that its area could be found by considering the area of an inscribed polygon; this suggests an

early use of the method of exhaustion. Bryson took Antiphon’s method further and bounded

the area between the area of inscribed and circumscribed polygons.

The problem of squaring the circle attracted enough popular attention by this time to

warrant a reference in The Birds (414 B.C.) by Aristophanes. The main characters, Euelpides and Pisthetairos, leave Athens to found the utopian “Cloud Cuckoo Land,” and are

subsequently bombarded by unsolicited and impractical advice. The astronomer M ETON

(b. ca. 440 B.C.), whose observation that 235 lunar months very nearly equals 19 solar years

led to the development of a very accurate luni-solar calendar, appears as a character with a

plan to design the city: “With this straight ruler here I measure this, so that your circle here

becomes a square.” Pisthetairos replies, “This man’s a Thales,” then drives him away with

a beating.

Beating geometers played only a minor role in Aristophanes’s work. A more constant

theme was the stupidity and pointlessness of war. Athens’s quest to build an empire led

to the Great Peloponnesian War, which began in 431 B.C. In response, Aristophanes wrote

The Acharnians (425 B.C.), The Knights (424 B.C.), and The Peace (421 B.C.), criticizing the

leadership and conduct of the war, and calling for its end. The plays were well-received, but

war proved more popular. The Athenians also passed a law limiting political satire, though

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this did not prevent Aristophanes from writing his greatest work, Lysistrata (411 B.C.). In

it, Lysistrata convinces the wives and mistresses of the Athenian and Spartan soldiers to

withhold sex until they cease their endless war.

Lysistrata ends happily. The Great Peloponnesian War ended in 404 B.C. with the destruction of the Athenian Empire. Sparta emerged the dominant power in the Greek world.

Athens, having lost her worldly empire, was about to establish one far greater, and far more

important. Built on ideas, not force, it would prove far more lasting, and to this day we are

very much a part of the Athenian intellectual empire.

2.1.4

Mathematicians and the Academy

Plato, a student of Socrates, fought in the last five years of the Peloponnesian War, but was

born to an aristocratic family and had political ambitions. But in 399 B.C. Socrates was

ordered to commit suicide on charges of having “corrupted youth.” This convinced Plato

that politics was no place for a man with a conscience, and he left Athens. Plato spent the

next dozen years traveling about the Mediterranean. He visited Egypt and southern Italy,

where he met Pythagoreans and taught Dion, the brother-in-law of the Tyrant of Syracuse,

Dionysius.6

According to one story, Dionysius grew angry at Plato and arranged to have the philosopher sold into slavery. Plato was saved from this unhappy fate by A RCHYTAS (428–347

B.C.), the leading citizen of Taras. Taras (Tarentum in Latin, and now Taranto, Italy) was

the only colony ever successfully founded by Sparta, populated by bastard children born

during the Messinian Wars. Spartan culture valued military prowess; Archytas had been

chosen as strategos (general) seven times, and was never defeated.7 Spartan culture also

valued music; Archytas is believed to have invented the ˛˛ , which Aristotle derided

as a rattle, useful for keeping children occupied so they do not break things (although it has

been suggested that the ˛˛ is the same instrument that adorns a number of vases

of the fourth century B.C. alongside a number of other religious symbols). Archytas is also

credited with associating the arithmetic, geometric, and harmonic means with musical intervals, and was the first to solve the problem of duplicating the cube, using a complex

method involving the intersection of two space curves.

When Plato returned to Athens in 387 B.C., he founded the most famous school in

history: the Academy. Plato may have been the first to use the term “mathematics”, which

comes from the Greek mathema (“that which is learned”). He emphasized mathematics

as a way to train the mind in deductive thinking, so the Academy became a center for

mathematical teaching and research. Archytas joined the faculty almost immediately.

The name of the school commemorates a rather unsavory event in the history of Athens.

According to one story Theseus, a king of Athens and great hero, abducted Helen, a

princess of Sparta (later to become Helen of Troy). An Athenian named Academus revealed

where Theseus hid Helen, and she was rescued by her brothers Castor and Polydeuces (Latinized as Pollux). Tradition placed Academus’s estate just outside the city walls of Athens;

the site was purchased by a wealthy admirer of Plato and donated to him.

6 In

Greek political theory, a “tyrant” is the term used for any non-hereditary ruler.

is not clear which campaigns he fought in, but southern Italy was a constant battleground during the time

period, and since, by tradition, a strategos could not succeed himself, the continued election of Archytas had to

have some basis.

7 It

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Other stories are told about the school. Johannes Tzetzes, a Byzantine author, claimed

that a plaque above its entrance read, “Let No One Unversed in Geometry Come Under My

Roof.” Since Tzetzes wrote six hundred years after the Academy closed its doors forever,

the claim is dubious. Another story highlights the difference between the manual and the

liberal arts: a student asked Plato the value of knowledge, at which point Plato told a servant

to give the student a coin, “since he must have value for what he learns.” Then the student

was expelled from the Academy.

The greatest mathematician associated with the Academy was E UDOXUS of Cnidus

(408–355 B.C.), a student of Archytas in Tarentum. The life of Eudoxus coincided with a

resurgence of Persian interest in Greece. Athens, Thebes, Corinth, and Argo all had historic

grudges against Sparta, and the Persians were well aware of this. Late in 396 B.C., Pharnabazus, a satrap (provincial governor) of the Persian Empire, let these cities know that if

they attacked Sparta, they would receive his support. Thus, less than a century after the

Greeks united to fend off the Persian Empire, they united again—this time at the urging of

the very same Persian Empire against one of their former allies. In 395 B.C., the Corinthian

War began.

A Spartan fleet was destroyed just offshore Cnidus in 394 B.C., when Eudoxus was fourteen. Since the destruction of Sparta would leave expansionist Athens supreme among the

Greek city-states, Persian policy reversed itself and in 387 B.C., the Spartans and Persians

negotiated the “King’s Peace,” binding on all of Greece, though no Greek city-state besides

Sparta was consulted.

Eudoxus visited Athens shortly afterwards. He was so poor he had to stay at the Piraeus

(the port section), and walked ten kilometers uphill each day to the Academy. Eudoxus only

stayed in Athens a few months, before going to Egypt. There a sacred bull licked his cloak,

which meant (according to the priests) that he would die young, but famous. After a year in

Egypt, he went to Cyzicus, where he founded a school before returning to Athens around

368 B.C. While Plato was away in Sicily, attempting to tutor King Dionysius the Younger

of Syracuse, he left the Academy in the hands of Eudoxus. Dionysius, like his father, grew

angry with the philosopher and eventually Plato returned to Athens. This allowed Eudoxus

to return to Cnidus where he stayed the rest of his life.

Eudoxus was responsible for developing the theory of proportion and ratio, a crucial

step in the development of the idea of a real number. Two magnitudes were said to have a

ratio if either could be multiplied to exceed the other (thus, a circle and a square could have

a ratio, but not a circle and a line). To compare two ratios, Eudoxus used the definition:

Definition. Two ratios are equal if, given any equimultiples of the first and third, and

any equimultiples of the second and fourth, the equimultiples of the first and second, and

the equimultiples of the third and fourth, alike equal, exceed, or fall short of the latter

equimultiples.

A

C

What this complicated definition means is that given any two ratios, B

and D

, and any

whole numbers m and n, then if mA D nB whenever mC D nD, and mA > nB whenever

mC > nD, and mA < nB whenever mC < nD, then the ratios are equal. Another way to
A
C
interpret Eudoxus’s complex definition is that if two ratios B
and D
are equal, then given
n
any rational number m , the ratios are both greater than, both less than, or both equal to the
rational number.
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With the theory of proportions (which Euclid preserved as Book V of the Elements),
Eudoxus was apparently able to prove the propositions of Democritus and Hippocrates
regarding the volumes of cones and pyramids, and the area of a circle. However, we have
no copies of his proofs.
M ENAECHMUS (fl. 350 B.C.) was a student of Eudoxus in Cyzicus, and was one of the
first to study the conic sections in a systematic fashion.8 He discovered specific properties,
or symptoms of the conic sections, that translate into the modern algebraic equations ky D
x 2 or kx D y 2 for the parabola and xy D k 2 for a hyperbola. Using these symptoms,
Menaechmus presented a new and simple method of duplicating the cube. Hippocrates
had shown that this problem reduced to inserting two mean proportionals between two
quantities; Menaechmus showed that these two mean proportionals could be found using
the intersection of a parabola and a hyperbola. In particular, suppose we wish to insert two
mean proportionals between a and b. We seek to find x, y that satisfy the proportionality
a W y D y W x D x W b. From the first and second terms of the proportionality, we have
a W y D y W x; hence ax D y 2 , so x, y, are on a parabola. From the first and third terms,
we have a W y D x W b; hence ab D xy and x, y are on a hyperbola. The intersection of
the parabola ax D y 2 and ab D xy can be used to find the desired quantities x and y.
One of the stories told about Menaechmus is that he came into the service of the household of Philip, the king of Macedon as a tutor for the young Prince Alexander. After a
particularly difficult lesson, Alexander asked whether there was an easier way to learn
mathematics. “Sire,” Menaechmus replied, “there is no royal road to geometry.”
2.1.5
The Hellenistic Kingdoms
By the time of Philip of Macedon, it was obvious that the Persian Empire was nothing but
an empty shell, ready to topple at the slightest push. Philip intended to supply that push.
After uniting the Macedonian tribes and modernizing their army, he conquered the Greeks
and prepared to march against the Persian Empire. Unfortunately he was assassinated in
336 B.C.by a conspiracy that included his ex-wife Olympias.
The conspiracy placed their son Alexander on the throne. Over the next ten years, he
embarked on a career of conquest never before seen in the ancient world, conquering an
empire that stretched from Greece to the borders of India. He would have gone further, but
his soldiers, who had faithfully followed him across the width of what was once the Persian
Empire, rebelled at the thought of getting even farther from home. Thus, Alexander turned
back. There is a story that he wept at the thought that there were “no more worlds to
conquer,” though this is probably apocryphal: Alexander knew that India lay just beyond
the borders of his e...
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