A HISTORY OF 7T (PI) Petr Beckmann Electrical Engineering Department, University of Colorado ST. MARTIN'S PRESS New York TO [;MUDLA, who never doubted the success of this book Copyright © 1971 by THE GOLEM PRESS All rights reserved. For information, write: St. Martin's Press, Inc., 175 Fifth Ave., New York, N.Y. 10010. Manufactured in the United States ofAmerica Library ofCongress Catalog Card Number: 74-32539 Preface The history of 17 is a quaint little mirror ofthe history ofman. It is the story of men like Archimedes of Syracuse, whose method of calculat­ ing 17 defied substantial improvement for some 1900 years; and it is also the story of a Cleveland businessman, who published a book in 1931 announcing the grand discovery that 17 was exactly equal to 256/81, a value that the Egyptians had used some 4,000 years ago. It is the story of human achievement at the University of Alexandria in the 3rd century B.C.; and it is also the story of human folly which made mediaeval bishops and crusaders set the torch to scientific libraries because they condemned their contents as works of the devil. Being neither an historian nor a mathematician, I felt eminently qualified to write that story. That remark is meant to be sarcastic, but there is a kernel oftruth in it. Not being an historian, I am not obliged to wear the mask of dispassionate aloofness. History relates ofcertain men and institutions that I admire, and others that I detest; and in neither case have I hesi­ tated to give vent to my opinions. However, I believe that facts and opinion are clearly separated in the following, so that the reader should run no risk of being overly influenced by my tastes and prejudices. Not being a mathematician, I am not obliged to complicate my explanations by excessive mathematical rigor. It is my hope that this little book might stimulate non-mathematical readers to become interested in mathematics, just as it is my hope that students of physics and engineering might become interested in the history ofthe tools they 4 PREFACE are using in their work. There are, however, two sure and all too well tried methods of how to make mathematics repugnant: One is to brutalize the reader by assertions without proof; the other is to -hit him over the head with epsilonics and proofs of existence and unicity. I have tried to steer a middle course between the two. A history of 17 containing only the bare facts and dates when who did what to 17 tends to be rather dull, and I thought it more interesting to mix in some ofthe background of the times in which 17 made progress. Sometimes I have strayed rather far afield, as in the case of the Roman Empire and the Middle Ages; but I thought it just as important to explore the times when 17 did not make any progress, and why it did not make any. The mathematical level ofthe book is flexible. The reader who finds the mathematics too difficult in some places is urged to do what the mathematician will do when he finds it too trivial: Skip it. This book, small as it is, would not have been possible without the wholehearted cooperation of the staff of Golem Press, and I take this opportunity to express my gratitude to every one of them. I am also indebted to the Archives Division of the Indiana State Library for making available photostats of Bill 246, Indiana House of Representa­ tives, 1897, and to the Cambridge University Press, Dover Publications and Litton Industries for granting permission to reproduce copyrighted materials without charge. Their courtesy is acknowledged in the notes accompanying the individual figures. I much enjoyed writing this book, and it is my sincere hope that the reader will enjoy reading it, too. Boulder, Colorado PetrBeckmann August 1970 PREFACE 5 PREFACE TO THE SECOND EDITION After all but calling Aristotle a dunce, spitting on the Roman Empire, and flipping my nose at some other highly esteemed institutions, I had braced myself for the reviews that would call this book the sick product ofan insolent ignoramus. My surprise was therefore all the more pleasant when the reviews were very favorable, and the first edition went out of print in less than a year. I am most grateful to the many readers who have written in to point out misprints and errors, particularly to those who took me to task (quite rightly) for ignoring the recent history of evaluating 17 by digital computers. I have attempted to remedy this shortcoming by adding a chapter on 17 in the computer age. Mr. D.S. Candelaber ofGolem Press had the bright idea ofimprinting the end sheets ofthe book with the first 10,000 decimal places of 17, and the American Mathematical Society kindly gave permission to reproduce the first two pages ofthe computer print-out as published by Shanks and Wrench in 1962. A reprint ofthis work was very kindly made available by one of the authors, Dr. John W. Wrench, Jr. To all of these, I would like to express my sincere thanks. I am also most grateful to all readers who have given me the benefit of their comments. I am particularly indebted to Mr. Craige Schensted of Ann Arbor, Michigan, and M. Jean Meeus of Erps-Kwerps, Belgium, for their detailed lists of misprints and errors in the first edition. Boulder, Colorado P.B. May 1971 PREFACE TO THE THIRD EDITION Some more errors have been corrected and the type has been re-set for the third edition. A Japanese translation of this book was published in 1973. Meanwhile, a disturbing trend away from science and toward the irrational has set in. The aerospace industry has been all but dismantled. CoIIege enroIIment in the hard sciences and engineering has significantly dropped. The disoriented and the guIlible flock in droves to the various Maharajas ofMumbo Jumbo. Ecology, once a respected scientific discipline, has become the buzzword of frustrated housewives on messianic ego-trips. Technology has wounded affluent intellectuals with the ultimate insult: They cannot understand it any more. Ignorance, anti-scientific and anti-technology sentiment have always provided the breeding ground for tyrannies in the past. The power of the ancient emperors, the mediaeval Church, the Sun Kings, the State with a capital S, was always rooted in the ignorance ofthe oppressed. Anti-scientific and anti-technology sentiment is providing a breeding ground for encroaching on the individual's freedoms now. A new tyranny is on the horizon. It masquerades under the meaningless name of "Society." Those who have not learned the lessons of history are destined to relive it. Must the rest of us relive it, too? Boulder, Colorado P.B. Christmas 1974 Contents 1. DAWN 9 2. THE BELT 20 3. THE EARLY GREEKS 36 4. EUCLID 45 5. THE ROMAN PEST 55 6. ARCHIMEDES OF SyRACUSE 62 7. DUSK 73 8. NIGHT 78 9. AWAKENING 87 10. THE DIGIT HUNTERS 99 11. THE LAST ARCHIMEDEANS 110 12. PRELUDE TO BREAKTHROUGH 121 13. NEWTON 134 14. EULER. " 147 15. THE MONTE CARLO METHOD 158 16. THE TRANSCENDENCE OF 17 •••••••••••••••••••••••166 17. THE MODERN CIRCLE SQUARERS 173 18. THE COMPUTER AGE 183 Notes 190 Bibliography 193 Chronological Table 196 Index 198 DAWN History records the names of royal bastards. but cannot tell us the origin of wheat. JEAN HENRI FABRE (1823-1915) million years or so have passed since the tool-wielding animal ~ called man made its appearance on this planet. During this 13 time it learned to recognize shapes and directions; to grasp the concepts of magnitude and number; to measure; and to realize that there exist relationships between certain magnitudes. The details of this process are unknown. The first dim flash in the darkness goes back to the stone age - the bone of a wolf with incisions to form a tally stick (see .figure on next page). The flashes become brighter and more numerous as time goes on, but not until about 2,000 B.C. do the hard facts start to emerge by direct documentation rather than by circumstantial evidence. And one of these facts is this: By 2,000 B.C., men had grasped the significance of the constant that is today denoted by TT, and that they had found a rough approximation of its value. How had they arrived at this point? To answer this question, we must return into the stone age and beyond, and into the realm of speculation. Long before the invention of the wheel, man must have learned to identify the peculiarly regular shape ofthe circle. He saw it in the pupils ofhis fellow men and fellow animals; he saw it bounding the disks of the Moon and Sun; he saw it, or something near it, in some flowers; and perhaps he was pleased by its infinite symmetry as he drew its shape in the sand with a stick. Then, one might speculate, men began to grasp the concept of magnitude - there were large circles and small circles, tall trees and small trees, heavy stones, heavier stones, very heavy stones. The transition from these qualitative statements to quantitative measure- 10 CHAPTER ONE ment was the dawn of mathematics. It must have been a long and arduous road, but it is a safe guess that it was first taken for quantities that assume only integral values - people, animals, trees, stones, sticks. For counting is a quantitative measurement: The measurement of the ... amount of a multitude of items. ­ Man first learned to count to two, and a long time elapsed before he learned to count to higher numbers.