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  Contents

  Title Page

  Copyright Notice

  Dedication

  Preface

  1. DAWN

  2. THE BELT

  3. THE EARLY GREEKS

  4. EUCLID

  5. THE ROMAN PEST

  6. ARCHIMEDES OF SYRACUSE

  7. DUSK

  8. NIGHT

  9. AWAKENING

  10. THE DIGIT HUNTERS

  11. THE LAST ARCHIMEDEANS

  12. PRELUDE TO BREAKTHROUGH

  13. NEWTON

  14. EULER

  15. THE MONTE CARLO METHOD

  16. THE TRANSCENDENCE OF π

  17. THE MODERN CIRCLE SQUARERS

  18. THE COMPUTER AGE

  Notes

  Bibliography

  Chronological Table

  Index

  Copyright

  TO SMUDLA,

  who never doubted the success of this book

  Preface

  The history of π is a quaint little mirror of the history of man. It is the story of men like Archimedes of Syracuse, whose method of calculating π 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 π 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 of truth in it. Not being an historian, I am not obliged to wear the mask of dispassionate aloofness. History relates of certain men and institutions that I admire, and others that I detest; and in neither case have I hesitated 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 of the tools they 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 π containing only the bare facts and dates when who did what to π tends to be rather dull, and I thought it more interesting to mix in some of the background of the times in which π 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 π did not make any progress, and why it did not make any.

  The mathematical level of the 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 Representatives, 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.

  Petr Beckmann

  Boulder, Colorado

  August 1970

  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 of an 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 π by digital computers. I have attempted to remedy this shortcoming by adding a chapter on π in the computer age.

  Mr. D.S. Candelaber of Golem Press had the bright idea of imprinting the end sheets of the book with the first 10,000 decimal places of π, and the American Mathematical Society kindly gave permission to reproduce the first two pages of the computer print-out as published by Shanks and Wrench in 1962. A reprint of this 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.

  P.B.

  Boulder, Colorado

  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. College enrollment in the hard sciences and engineering has significantly dropped. The disoriented and the gullible flock in droves to the various Maharajas of Mumbo 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 of the 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?

  P.B.

  Boulder, Colorado

  Christmas 1974

  1

  DAWN

  History records the names of royal bastards, but cannot tell us the origin of wheat.

  JEAN HENRI FABRE

  (1823-1915)

  A million years or so have passed since the tool-wielding animal called man made its appearance on this planet. During this 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 here). 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 π, 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 of the circle. He saw it in the pupils of his 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 measurement 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. There is a fair amount of evidence for this,1 perhaps none of it more fascinating than that preserved in man’s languages: In Czech, until the Middle Ages, there used to be two kinds of plural — one for two items, another for many (more than two) items, and apparently in Finnish this is so to this day. There is evidently no connection between the (Germanic) words two and half; there is none in the Romance languages (French: deux and moitié) nor in the Slavic languages (Russian: dva and pol), and in Hungarian, which is not an Indo-European language, the words are kettö and fél. Yet in all European languages, the words for 3 and 1/3, 4 and 1/4, etc., are related. This suggests that men grasped the concept of a ratio, and the idea of a relation between a number and its reciprocal, only after they had learned to count beyond two.

  A stone age tally stick. The tibia (shin) of a wolf with two long incisions in the center, and two series of 25 and 30 marks. Found in Vèstonice, Moravia (Czechoslovakia) in 1937.2

  The next step was to discover relations between various magnitudes. Again, it seems certain that such relations were first expressed qualitatively. It must have been noticed that bigger stones are heavier, or to put it into more complicated words, that there is a relation between the volume and the weight of a stone. It must have been observed that an older tree is taller, that a faster runner covers a longer distance, that more prey gives more food, that larger fields yield bigger crops. Among all these kinds of relationships, there was one which could hardly have escaped notice, and which, moreover, had no exceptions:

  The wider a circle is “across,” the longer it is “around.”

  And again, this line of qualitative reasoning must have been followed by quantitative considerations. If the volume of a stone is doubled, the weight is doubled; if you run twice as fast, you cover double the distance; if you treble the fields, you treble the crop; if you double the diameter of a circle, you double its circumference. Of course, the rule does not always work: A tree twice as old is not twice as tall. The reason is that “the more … the more” does not always imply proportionality; or in more snobbish words, not every monotonic function is linear.

  Neolithic man was hardly concerned with monotonic functions; but it is certain that men learned to recognize, consciously or unconsciously, by experience, instinct, reasoning, or all of these, the concept of proportionality; that is, they learned to recognize pairs of magnitude such that if the one was doubled, trebled, quadrupled, halved or left alone, then the other would also double, treble, quadruple, halve or show no change.

  And then came the great discovery. By recognizing certain specific properties, and by defining them, little is accomplished. (That is why the old type of descriptive biology was so barren.) But a great scientific discovery has been made when the observations are generalized in such a way that a generally valid rule can be stated. The greater its range of validity, the greater its significance. To say that one field will feed half the tribe, two fields will field the whole tribe, three fields will feed one and a half tribes, all this applies only to certain fields and tribes. To say that one bee has six legs, three bees have eighteen legs, etc., is a statement that applies, at best, to the class of insects. But somewhere along the line some inquisitive and smart individuals must have seen something in common in the behavior of the magnitudes in these and similar statements:

  No matter how the two proportional quantities are varied, their ratio remains constant.

  For the fields, this constant is 1 : ½ = 2 : 1 = 3 : 1½ = 2. For the bees, this constant is 1 : 6 = 3 : 18 = 1/6. And thus, man had discovered a general, not a specific, truth.

  This constant ratio was not obtained by numerical division (and certainly not by the use of Arabic numerals, as above); more likely, the ratio was expressed geometrically, for geometry was the first mathematical discipline to make substantial progress. But the actual technique of arriving at the constancy of the ratio of two proportional quantities makes little difference to the argument.

  There were of course many intermediate steps, such as the discovery of sums, differences, products and ratios; and the step of abstraction, exemplified by the transition from the statement “two birds and two birds make four birds” to the statement “two and two is four.” But the decisive and great step on the road to π was the discovery that proportional quantities have a constant ratio.

  From here it was but a dwarf’s step to the constant π: If the “around” (circumference) and the “across” (diameter) of a circle were recognized as proportional quantities, as they easily must have been, then it immediately follows that the ratio

  circumference: diameter = constant for all circles.

  This constant circle ratio was not denoted by the symbol π until the 18th century (A.D.), nor, for that matter, did the equal sign ( = ) come into general use before the 16th century A.D. (The twin lines as an equal sign were used by the English physician and mathematician Robert Recorde in 1557 with the charming explanation that “noe .2. thynges, can be moare equalle.”) However, we shall use modern notation from the outset, so that the definition of the number π reads

  where C is the circumference, and D the diameter of any circle.

  And with this, our speculative road has reached, about 2,000 B.C., the dawn of the documented history of mathematics. From the documents of that time it is evident that by then the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π as given by (1).

  * * *

  BUT the Babylonians and the Egyptians knew more about π than its mere existence. They had also found its approximate value. By about 2,000 B.C., the Babylonians had arrived at the value

  and the Egyptians at the value

  How to measure π in the sands of the Nile

  How did these ancient people arrive at these values? Nobody knows for certain, but this time the guessing is fairly easy.

  Obviously, the easiest way is to take a circle, to measure its circumference and diameter, and to find π as the ratio of the two. Let us try to do just that, imagining that we are in Egypt in 3,000 B.C. There is no National Bureau of Standards; no calibrated measuring tapes. We are not allowed to use the decimal system or numerical division of any kind. No compasses, no pencil, no paper; all we have is stakes, ropes and sand.

  So we find a fairly flat patch of wet sand along the Nile, drive in a stake, attach a pie
ce of rope to it by loop and knot, tie the other end to another stake with a sharp point, and keeping the rope taut, we draw a circle in the sand. We pull out the central stake, leaving a hole O (see figure above). Now we take a longer piece of rope, choose any point A on the circle and stretch the rope from A across the hole O until it intersects the circle at B. We mark the length AB on the rope (with charcoal); this is the diameter of the circle and our unit of length. Now we take the rope and lay it into the circular groove in the sand, starting at A. The charcoal mark is at C; we have laid off the diameter along the circumference once. Then we lay it off a second time from C to D, and a third time from D to A, so that the diameter goes into the circumference three (plus a little bit) times.

  If, to start with, we neglect the little bit, we have, to the nearest integer,

  To improve our approximation, we next measure the little left-over bit EA as a fraction of our unit distance AB. We measure the curved length EA and mark it on a piece of rope. Then we straighten the rope and lay it off along AB as many times as it will go. It will go into our unit distance AB between 7 and 8 times. (Actually, if we swindle a little and check by 20th century arithmetic, we find that 7 is much nearer the right value than 8, i.e., that E7 in the figure here is nearer to B than E8, for 1/7 = 0.142857…, 1/8 = 0.125, and the former value is nearer π – 3 = 0.141592 … However, that would be difficult to ascertain by our measurement using thick, elastic ropes with coarse charcoal marks for the roughly circular curve in the sand whose surface was judged “flat” by arbitrary opinion.)

  We have thus measured the length of the arc EA to be between 1/7 and 1/8 of the unit distance AB; and our second approximation is therefore

  for this, to the nearest simple fractions, is how often the unit rope length AB goes into the circumference ABCD.

  And indeed, the values