Pi: A Source Book Third Edition Springer Science+B usiness Media, LLC Lennart Berggren Jonathan Borwein Peter Borwein Pi: A Source Book Third Edition Springer Lennart Berggren Jonathan Borwein Department of Mathematics Faculty of Computer Science Simon Fraser University Dalhousie University Burnaby, BC V5A 1S6 6050 University Avenue Canada Halifax, Nova Scotia Canada, B3H 1W 5 Peter Borwein Department of Mathematics Simon Fraser University Burnaby, BC V5A 1S6 Canada Mathematics Subject Classification (2000): 01-00, 01A05, 01A 75, 11-00, 11-03, 26-03, 65-03, 68-03 Library of Congress Cataloging-in-Publication Data Pi, a source book 1 [edited by 1 Lennart Berggren, Jonathan Borwein, Peter Borwein.-3rd ed. p. cm. Includes bibliographical references and index. l. Pi. I. Title: Pi. II. Berggren, Lennart. m. Borwein, Jonathan M. IV. Borwein, Peter B. QA484.P5 2003 516.22-dc22 2003066023 Printed on acid-free paper. ISBN 978-1-4419-1915-1 ISBN 978-1-4757-4217-6 (eBook) DOI 10.1007/978-1-4757-4217-6 © 2004, 2000, 1997 Springer Science+Business Media New York Originally published by Springer-Verlag New York, LLC in 2004. Softcover reprint of the hardcover I st edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 SPIN 10953327 springeronline.com Preface to the Third Edition Our aim in preparing this edition is to bring the material in the collection of papers in the second edition of this source book up to date. Moreover, several delightful pieces became available and are added. This substantial supplement to the third edition serves as a stand-alone exposition of the recent history of the computation of digits of pi. It also includes a discussion of the thorny old question of normality of the distribution of the digits. Additional material of historical and cultural interest is included, the most notable being new translations of the two Latin pieces of Viete (translation of article 9 (Excerpt 1): Various Responses on Mathematical Matters: Book VII (1593) and (Excerpt 2): Defense for the New Cyclometry or ''Anti-Axe''), and a thorough revision of the translation of Huygens's piece (article 12) published in the second edition. We should like to thank Professor Marinus Taisbak of Copenhagen for grappling with Viete's idiosyncratic style to produce the new translation of his work. We should like to thank Karen Aardal for permission to use her photograph of Ludolph's new tombstone in the Pieterskerk in Leiden, the Smithsonian Institution for permission to reproduce a fine photo of ENIAC, and David and Gregory Chudnovsky for providing a "Walk on the digits of pi." We should also like to thank Irving Kaplansky for his gracious permission to include his "A Song about Pi." Finally, our thanks go to our colleagues whose continued interest in pi has encouraged our publisher to produce this third edition, as well as for the comments and corrections to earlier editions that some of them have sent us. L. Berggren J. Borwein P. Borwein Simon Fraser University December 2003 Preface to the Second Edition We are gratified that the first edition was sufficiently well received so as to merit a second. In addition to correcting a few minor infelicities, we have taken the opportunity to add an Appendix in which articles 9 and 12 by Viete and Huygens respectively are translated into English. While modem European languages are accessible to our full community-at least through colleagues-this is no longer true of Latin. Thus, following the suggestions of a reviewer of the first edition we have opted to provide a serviceable if fairly literal translation of three extended Latin excerpts. And in particular to make Viete's opinions and style known to a broader community. We also record that in the last two years distributed computations have been made of the binary digits of 7r using an enhancement due to Fabrice Bellard of the identity made in article 70. In particular the binary digits of 7r starting at the 40 trillionth place are 00000 111110011111. Details of such ongoing computations, led by Colin Percival, are to be found at www.cecm.sfu.ca/projects/pihex. Corresponding details of a billion (230) digit computation on a single Pentium II PC, by Dominique Delande using Carey Bloodworth's desktop 7r program and taking under nine days, are lodged at www.cecm.sfu.ca/personalljborweinlpLcover.html. Here also are details of the computation of 236 digits by Kanada et al. in April 1999. We are grateful for the opportunity to thank Jen Chang for all her assistance with the cover design of the book. We also wish to thank Annie Marquis and Judith Borwein for their substantial help with the translated material. Lennart Berggren Jonathan Borwein Peter Borwein Simon Fraser University July 5, 1999 Preface Our intention in this collection is to provide, largely through original writings, an extended account of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a significant number of the most important mathematicians have contributed to its unfolding--directly or otherwise. Pi is one of the few mathematical concepts whose mention evokes a response of recog nition and interest in those not concerned professionally with the subject. It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of pi is virtually the only topic from the most ancient stratum of mathe matics that is still of serious interest to modem mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of math ematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathematicians with exam ples of many current mathematical techniques as well as a palpable sense of their historical development. Why a Source Book? Few books serve wider potential audiences than does a source book. To our knowledge, there is at present no easy access to the bulk of the material we have collected. Both professional and amateur mathematicians, whether budding, blooming, or begin ning to wilt, can find in it a source of instruction, study, and inspiration. Pi yields wonderful examples of how the best of our mathematical progenitors have struggled with a problem worthy of their mettle. One of the great attractions of the literature on pi is that it allows for the inclusion of very modem, yet still highly accessible, mathematics. Indeed, we have included several prize winning twentieth century expository papers, and at least half of the collected material dates from the last half of the twentieth century. While this book is definitely a collection of literature on, and not a history of, pi, we anticipate that historians of mathematics will find the collection useful. As authors we viii Preface believe that one legitimate way of exhibiting the history of a concept is in gathering a coherent collection of original and secondary sources, and then to let the documents largely tell their own stories when placed in an appropriate historical and intellectual context. Equally, teachers at every level will find herein ample supplementary resources: for many purposes from material for special topic courses to preparatory information for semi nars and colloquia and guidance for student projects. What Is Included? We have chosen to include roughly 70 representatives of the accumulated literature on pi. In the Contents each piece is accorded a very brief but hopefully illuminating description. This is followed by an Introduction in which we highlight some further issues raised by the collection. Finally, since the pre-Newtonian study of pi presents many more problems for the reader than does the material after the time of Huygens, we have included an Appendix On the Early History of Pi. We have also provided two other Appendices. Computational Chronology of Pi offers a concise tabular accounting of computational records, and Selected Formulae for Pi presents a brief compendium of some of the most historically or computa tionally significant formulas for pi. The pieces in the collection fall into three broad classes. The core of the material is the accumulated mathematical research literature of four millennia. Although most of this comes from the last 150 years, there is much of interest from ancient Egypt, Greece, India, China, and medieval Islam. We trust that readers will appreciate the ingenuity of our earliest mathematicians in their valiant attempts to under stand this number. The reader may well find this material as engrossing as the later work of Newton, Euler, or Ramanujan. Seminal papers by Lambert, Hermite, Lindemann, Hilbert and Mahler, to name but a few, are included in this category. Some of the more important papers on the number e, on zeta functions, and on Euler's constant have also been included as they are inextricably interwoven with the story of pi. The second stratum of the literature comprises historical studies of pi, based on the above core sources, and of writings on the cultural meaning and significance of the number. Some of these are present here only in the bibliography such as Petr Beckmann's some what idiosyncratic monograph, A History of Pi. Other works on the subject are provided in extenso. These include Schepler's chronology of pi, some of Eves's anecdotes about the history of the number, and Engels' conjecture about how the ancient Egyptians may have computed pi. Finally, the third level comprises the treatments of pi that are fanciful, satirical or whimsical, or just wrongheaded. Although these abound, we have exercised considerable restraint in this category and have included only a few representative pieces such as Keith's elaborate mnemonic for the digits of pi based on the poem "The Raven," a recent offering by Umberto Eco, and the notorious 1897 attempt by the state of Indiana 1 to legislate the value of pi. Lennart Berggren Jonathan Borwein Peter Borwein Simon Fraser University September 6, 1996 1 Oddly enough, the third page of this bill is apparently missing from the Indiana State Library and thus may now exist only in facsimile! Preface ix Some Points of Entry For the reader looking for accessible points of introduction to the collection we make the following suggestions: • As general introduction: 35. Schepler. The Chronology of Pi (1950) 282 64. Borwein, Borwein, and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) 588 • As an introduction to irrationality and transcendence: 33. Niven. A Simple Proof that 7r Is Irrational (1947) 276 49. van der Poorten. A Proof that Euler Missed. .. Apery's Proof of the Irrationality of ~(3) (1979) 439 24. Hilbert. Ueber die Transzendenz der Zahlen e und 7r (1893) 226 • As an introduction to elliptic integrals and related subjects: 30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) 258 55. Cox. The Arithmetic-Geometric Means of Gauss (1984) 481 • As an introduction to the computational issues: 37. Wrench, Jr. The Evolution of Extended Decimal Approximations to 7r (1960) 319 47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) 424 70. Bailey, Borwein, and Plouffe. On The Rapid Computation of Various Polylogarithmic Constants (1997) 663 • For a concise synopsis, the final "Pamphlet" makes an excellent self-contained entry. 721