The King of Infinite Space

A
LSO BY
D
AVID
B
ERLINSKI

One, Two, Three

The Deniable Darwin & Other Essays

The Devil's Delusion

Infinite Ascent

The Secrets of the Vaulted Sky

The Advent of the Algorithm

Newton's Gift

A Tour of the Calculus

Black Mischief: Language, Life, Logic, Luck

The Body Shop

Less Than Meets the Eye

A Clean Sweep

On Systems Analysis

EUCLID
AND HIS
ELEMENTS

DAVID BERLINSKI

BASIC BOOKS

A Member of the Perseus Books Group New York

Copyright © 2013 by David Berlinski

Published by Basic Books,
A Member of the Perseus Books Group

All rights reserved. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 250 West 57th Street, 15th Floor, New York, NY 10107.

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Designed by Trish Wilkinson
Set in 11.5 point Minion Pro

Library of Congress Cataloging-in-Publication Data
Berlinski, David, 1942–
The king of infinite space : Euclid and his Elements / David Berlinski.

pages cm

Includes index.

ISBN 978-0-465-03346-1 1.
Mathematics, Greek.

2.
Geometry—History. 3.
Euclid. 4.
Euclid. Elements. I.
Title.

II.
Title: Euclid and his Elements.

QA31.B47 2013

516.2—dc23

2012042492

10
9
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2
1

For Morris Salkoff

On peut avoir trois principaux objets dans l'étude de la vérité: l'un, de la découvrir quand on la cherche; l'autre, de la démontrer quand on la possède; le dernier, de la discerner d'avec le faux quand on l'examine.

—B
LAISE
P
ASCAL
,
D
E L
'
ESPRIT GÉOMÉTRIQUE

Preface

Teacher's Note

A Note on Sources

Appendix: Euclid's Definitions

Index

E
UCLID IS UNIVERSALLY
acclaimed great. His name is in no danger of being lost. He belongs in the company of men whose reputation defies revision. This is to establish Euclid's place, but hardly to say why so many years after his death, he continues to enjoy it.

Euclid is, of course, the author of the
Elements
, and the
Elements
is by far and away the most successful of mathematical textbooks. A textbook that has survived for more than two thousand years represents an uncommon achievement. Most textbooks have a short and ignominious life. They serve a purpose, but they do not inspire reverence.

Euclid's
Elements
is different.

No one has ever found a better way of presenting the elements of plane geometry; no sensitive teacher would think to use a substitute. There is none.

The
Elements
is not simply a great book in mathematics: it is a great book. The contemporary reader, eager for Euclid's personal revelations, will quit the
Elements
unsatisfied.
There is not a word about them. But in writing the
Elements
, Euclid found a way to impose his own powerful personality on the scattered propositions of geometry, and by imposing on them, created an immense structure, a logical space, a world in which there is growth, and form, and intimate dependencies among parts, something very large but not sprawling, the
Elements
itself the overflow in print, paper, or papyrus of a mind singular and determined.

Having revealed nothing of interest, Euclid, of course, has revealed everything of importance.

If this is not an artistic achievement, then nothing is.

Paris, 2012

T
HE
R
OMAN ARCHITECT
Marcus Vitruvius Pollio lived and worked in the first century
BC
. His treatise
Libri Decem
, or
The Ten Books,
was dedicated to Caesar Augustus some twenty years before the birth of Christ. Vitruvius was both an architect and a military engineer, and
The Ten Books
contain a remarkable account of classical architectural ideas and building methods. It is sophisticated. A building, he insists, must be durable, useful, and beautiful (
firmitas, utilitas, venustas
). These are simple but stringent standards. Very few buildings constructed in the past sixty years could meet them. Vitruvius writes as a critic as well as a commentator, a man prepared to judge men as well as buildings, and when he does, he takes pride in seeing things as they are.

In his sixth book,
De Architectura
, Vitruvius recounts a story, one told as well by Cicero, about Aristippus, a
fourth-century philosopher. Finding himself “shipwrecked and cast on the Rhodian shore,” he despaired.

Aristippus then happened to notice some geometrical figures scratched into the sand—triangles, perhaps, or circles, or straight lines suspended between points, the careless detritus of someone squatting by the seashore and thinking about shapes in space.

He said to his companions, “We can hope for the best, for I see signs of men.”

Aristippus was well known for his devotion to pleasure; he was notorious. When rebuked for sleeping with whores, he responded equably that a mansion does not become useless because it has already been used. We expect such men to be tested; we are disappointed if they are not. It is right that Aristippus found redemption in human solidarity—
the signs of men.

M
ATHEMATICS IS WHAT
mathematicians make of it. What other standard would apply? Still, mathematicians exhibit a very nice sense of what they should make of what they have made. They are, after all, border guards at their own frontiers. Is mathematical logic a part of mathematics? Or mathematical physics? Most mathematicians would say they are not. They never doubt the importance of these subjects. They are not blind. But mathematicians are as fussy as cats. And almost as conservative. Their deepest
commitment is to shapes and numbers, the seeing eye, the beating heart.

Counting would seem to come first, no? Every living creature makes the distinction between the thing that it
is
and the thing that it is
not.
Two numbers are required to express all of the imperatives of biology.

This
is me,
that
is not.

Long live the numbers then.

But then there is seeing. Shapes are metaphysically as compelling as numbers. A single point, after all, divides the universe into what is
at
the point and what is
not.

Long live the shapes too.

Shapes and numbers are in some sense coordinated. Points often have a numerical address. The latitude and longitude of Adelaide is 34 by 55S and 138 by 36E. The letters S and E may themselves be replaced by the numbers 19 and 5, their position in the alphabet. The result is a number marking Adelaide 345519138365. In just the same way, numbers often have a location. The number 345519138365 is notable in indicating the point where one finds
Adelaide.

Long live the shapes
and
the numbers.

S
OME CULTURES ARE
geometric in their sensibility, and others are not.

Prizing order, the Romans of the empire appreciated severity. They did not fool around.

A powerful visual orthodoxy dominated their landscape: amphitheaters, public monuments and squares, cities divided into blocks, senatorial villas arranged in a rectangle around an interior space, a great urban civilization spreading throughout southern Europe and the Mediterranean basin.

Strange in a people whose numerals (I, II, XXXII) left them unable elegantly to conduct their practical affairs.

Our own culture is very different. The historian Tony Judt has argued that in the nineteenth century, the railroad, by shrinking space, brought about a reorganization of time.
¹
A new standard of precision was first conceived and then enforced. Approximations that had long served the human race—sunrise, sunset, noon, midnight—were replaced by a complicated numerical apparatus, time divided into parts and parts of parts.

The result has been a culture that in comparison to the ancient world is numerically sophisticated but visually disgusting.

We
count,
they
saw.

It makes a difference—obviously so.

E
UCLID OF
A
LEXANDRIA
was born sometime in the fourth century
BC
, and he died sometime in the third. The year 300
BC
is very often designated as a time in which he flourished—Euclid of Alexandria, 300
fl.
, as historians sometimes write. Whatever the uncertainties of his birth and death, he was then at the height of his powers—alert, vibrant, and commanding. As a young man, Euclid is said to have been influenced by Plato's students, and he may well have attended the academy that Plato founded, mingling with the philosophers and inserting himself gregariously in their gathering gossip. Plato was devoted to geometry, even going so far as to assign to various deities a fondness for its study.