The King of Infinite Space Read online

  THE KING OF INFINITE SPACE

  ALSO BY DAVID BERLINSKI

  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

  THE

  KING OF INFINITE

  SPACE

  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 8 7 6 5 4 3 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.

  —BLAISE PASCAL, DE L’ESPRIT GÉOMÉTRIQUE

  Contents

  Preface

  I

  Signs of Men

  II

  An Abstraction from the Gabble

  III

  Common Beliefs

  IV

  Darker by Definition

  V

  The Axioms

  VI

  The Greater Euclid

  VII

  Visible and Invisible Proof

  VIII

  The Devil’s Offer

  IX

  The Euclidean Joint Stock Company

  X

  Euclid the Great

  Teacher’s Note

  A Note on Sources

  Appendix: Euclid’s Definitions

  Index

  Preface

  EUCLID 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

  THE KING OF INFINITE SPACE

  Chapter I

  SIGNS OF MEN

  L’homme c’est rien; l’oeuvre c’est tout

  (The man is nothing; the work, everything).

  —GUSTAVE FLAUBERT

  THE ROMAN 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.

  MATHEMATICS 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.

  Shape
s 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.

  SOME 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.

  EUCLID OF ALEXANDRIA 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.

  The circumstances under which Euclid composed his masterpiece, the Elements, remain, like the details of his life, largely unknown. There is some evidence that Euclid taught at the great library in Alexandria founded by Ptolemy I. The Euclid of the Elements is stern, logical, unrelenting, a man able to concentrate the powers of his mind on what is abstract and remote. It would be fascinating to know the details of his life in Alexandria, to see Euclid toddling off to the baths or with a sense that he has let things get out of hand, submitting to having his eyebrows trimmed. There are suggestions here and there that as a teacher, Euclid was urbane, helpful, and kind. Among its other virtues, the Elements is a great textbook; perhaps Euclid read aloud from his masterpiece as the warm sunlit air slitted through the dancing motes of dust, his students unaware that they were the first to hear a lesson that others would hear so many times and from so many other voices.

  As a mathematician, Euclid took from his predecessors, men such as Eudoxus and Theaetetus, and gave to his successors, Apollonius and Archimedes. He summarized; he adjusted and refined; he was a living synthetic force and in very short order a monument—all this we know from what we can guess and from later commentaries, but the man himself remains invisible, his influence conveyed by his industry, a magnificent mole in the history of thought, a great digger of tunnels.

  He must have been a man of heavy architecture, and at some point in his concourse with those endlessly gabbling philosophers, he gathered up his robes and, with a dawning sense of his powers, determined that he had something to offer that they had not seen and could not express.

  FOR MORE THAN two thousand years, geometry has meant Euclidean geometry, and Euclidean geometry, Euclid’s Elements. It is the oldest complete text in the Western mathematical tradition and the most influential of its textbooks.

  The first book of Euclid’s Elements contains 48 propositions, the second, 14. There are in all thirteen books comprising 467 propositions, and two more books of uncertain authorship, which are donkey-tailed onto older editions of the Elements.

  The propositions in Books I through IV are concerned with points, straight lines, circles, squares, right angles, triangles, and rectangles, the stable shapes of art and architecture. Books V through IX develop a theory of magnitudes, proportions, and numbers. The remaining books are devoted to solid geometry. Every book of the Elements is compelling, but the Euclid of myth and memory is the Euclid of the first four books of his treatise.

  In every generation, a few students have found themselves ravished by the Elements. “At the age of eleven,” Bertrand Russell recalls in his autobiography, “I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.”

  A course in Euclidean geometry has long been a part of the universal curriculum of mankind. Those not ravished by its study have very often remarked that the Euclidean discipline did them good nevertheless. It improved their mental hygiene. Students study algebra at roughly the same time that they study geometry, and curiously enough, they rarely remark on the improvement that it confers.

  Algebra, students complain, is just nasty.

  NOTHING FROM EUCLID’S hand survives into the twenty-first century. We know Euclid only from copies of copies, these passing through the mangler of translation from Greek to Latin and then to Arabic and back to Greek and finally to medieval Latin. Modern versions of Euclid are based on a tenth-century Greek manuscript identified in the eighteenth century by the French scholar François Peyrard. There is a poignant distinction between the solidity of Euclid’s thoughts and the perishable papyrus he used to express them. Long before Euclid, the Babylonians wrote laboriously on tablets. Plop went the wet clay on a long table. Inscriptions by means of a curved stylus—chiff, chiff, chiff. The oven of the sun. And thereafter, immortality. We can see their words as well as their works. Euclid himself we cannot see at all.

  If Euclid imposed order on his subject by making it a system, it was an order so severe as to force geometry into a fixed shape until at least the Italian Renaissance in the sixteenth century. Thereafter, a long and confusing process followed in which the Euclidean monument was variously chipped away, until in the nineteenth century, mathematicians discovered non-Euclidean geometries, Euclidean geometry becoming one among many, mathematicians half-mad with possibilities absorbing themselves with spaces that bulged like basketballs, or curved like saddlebacks, or that went on forever without getting anywhere.

  Euclid’s Elements represents the great achievement of the Greek mathematical tradition. Archimedes was a more brilliant mathematician than Euclid. He gave to the world what great mathematicians always give, and that is a record of his genius, but in the idea of an axiomatic system, Euclid gave to mathematics something even more enduring, and that was a way of life. It was a way of life invisible to the people who preceded the Greeks, and it was invisible as well to the Chinese, the masters of a subtle technological culture.

  And as one might expect, it is invisible to everyone else as well—now, today, even so—and must as a result be taught like any other artifact of civilization.

  1. Tony Judt, “The Glory of the Rails,” New York Review of Books, December 23, 2010.

  Chapter II

  AN ABSTRACTION FROM THE GABBLE

  As all suns smolder in a single sun The word is many but the word is one.

  —G. K. CHESTERTON

  AN AXIOMATIC SYSTEM is a stylized organization of intellectual life, an abstraction from the gabble. Euclid conceived of an axiomatic system in order to fulfill an ambition that had before Euclid gone unconceived and so unexpressed: to derive all of the propositions about geometry fro
m a handful of assumptions. The Egyptians who built the pyramids surely knew something about pyramids. They were not unsophisticated. They had a feel for measurements and mensuration. But what they knew, they knew incompletely. They took what they needed; they had no grasp of the whole. Euclid believed that there is a form of unity beneath the diversity of experience, and it is this that marks the difference between Euclid and the Egyptian mathematicians, men of the lash.

  Euclid required a double insight before he could strike for immortality. The first: that the various propositions of geometry could be organized into a single structure; and the second: that the principle of organization binding geometric propositions must be logical, and so alien to geometry itself.

  These are radically counterintuitive ideas, Pharaonic in their audacity.

  Euclid’s assumptions are commonly called axioms, and sometimes postulates; his conclusions, theorems. A proof is a chain connecting the axioms to the theorems in logically unassailable links. Euclid assumed five axioms, and from these he derived 467 theorems.

  A sense of this intellectual power, its grandeur—this is the Euclidean gift. The Pythagoreans before Euclid were men consumed by the rapture of mathematics. They communed with the numbers, and they were often tempted by gross intellectual follies. They took pleasure in mumbo jumbo. Euclid is by comparison imperturbable. There is no rapture in the Elements, but neither is there anything insane. The structure that Euclid created is intellectually accessible to anyone capable of following an argument.