Pi: A Source Book / Edition 2

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Overview

"This book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on pi collected herein include selections from the mathematical and computational literature over four millennia, a variety of historical studies on the cultural significance of the number, and an assortment of anecdotal, fanciful, and simply amusing pieces." For this new edition, the authors have updated the original material while adding new material of historical and cultural interest. There is a substantial exposition of the recent history of the computation of digits of pi, a discussion of the normality of the distribution of the digits, new translations of works by Viete and Huygens, as well as Kaplansky's never-before-published "A Song about Pi."

"...this new edition of a definitive source book on pi contains a surprising amount of the most important mathematics that contribute to the unfolding of pi, plus a complete history."

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Editorial Reviews

From the Publisher
From the reviews:

"Few mathematics books serve a wider potential readership than does a source book and this particular one is admirably designed to cater for a broad spectrum of tastes: professional mathematicians with research interest in related subjects, historians of mathematics, teachers at all levels searching out material for individual talks and student projects, and amateurs who will find much to amuse and inform them in this leafy tome. The authors are to be congratulated on their good taste in preparing such a rich and varied banquet with which to celebrate pi."

Roger Webster for the Bulletin of the LMS

"The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras."

Ed Barbeau for MathSciNet

"Full of useful formulas and ideas, it is a vast source of inspiration to any mathematician, A level and upwards—a necessity in any maths library."

New Scientist

"Should be on every mathematician’s coffee-table! … The seventy articles comprising the source-book proper range from historical articles and classic by such players as Wallis, Huyghens, Newton, and Euler, to the articles on irrationality and transcendence … . Pi: A source Book is truly an amazing book, irresistible in its own way, and filled with gems. And once it’s on your coffee-table, feel free to do more than just browse: it’s pretty well-suited for more in-depth study … . Obviously the book is highly recommended." (Michael Berg, MathDL, January, 2001)

From the reviews of the third edition:

"This is the third edition of the by now classical Pi: a source book. … contains some notes on the computation of individual (binary) digits of p, some considerations on the normality of the decimal expansion of p, and two more sections on the history of p. This book is still a classic work of reference for anyone with an interest in fascinating Pi.” (F. Beukers, Mathematical Reviews, 2005h)

"The book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on pi collected herein include selections from the mathematical and computational literature over four millennia, a variety of historical studies on the cultural significance of the number, and an assortment of anecdotal, fanciful, and simply amusing pieces." (Zentralblatt für Didaktik der Mathematik, November, 2004)

"This is the third edition of a comprehensive selection of about 70 articles on the number pi and related constants. … This edition contains a new supplement on the recent history of the computation of digits of pi … . Furthermore, new translations of articles by Viète and Huygens have been added. Altogether, this volume provides a fascinating overview of many hundred years of research and will delight and enlighten amateur lovers of pi and professional mathematicians alike." (Ch. Baxa, Monatshefte für Mathematik, Vol. 148 (1), 2006)

"This is a fascinating reference book, which consists almost entirely of facsimiles of 70 articles about pi, followed by appendices which look at its early history as well as much computational information. … it should be something to appear in any mathematical library, as it does contain so much material. … There is something for all levels of readers in this book, from the 14-year old who may wish to know a little of the history, to the professional mathematician seeking information … ." (Anthony C. Robin, The Mathematical Gazette, Vol. 90 (518), 2006)

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Product Details

  • ISBN-13: 9780387989464
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 10/28/1999
  • Edition description: 2ND
  • Edition number: 2
  • Pages: 752
  • Product dimensions: 7.28 (w) x 10.22 (h) x 1.72 (d)

Table of Contents

1 The Rhind mathematical papyrus-problem 50 (ca. 1650 B.C.) 1
2 Engels : Quadrature of the circle in ancient Egypt (1977) 3
3 Archimedes : Measurement of a circle (ca. 250 B.C.) 7
4 Phillips : Archimedes the numerical analyst (1981) 15
5 Lam and Ang : Circle measurements in ancient China (1986) 20
6 The Banu Musa : the measurement of plane and solid figures (ca. 850) 36
7 Madhava : The power series for Arctan and Pi (ca. 1400) 45
8 Hope-Jones : Ludolph (or Ludolff or Lucius) van Ceulen (1938) 51
9 Viete : Variorum de Rebus mathematicis reponsorum liber VII (1593) 53
10 Wallis : Computation of [pi] by successive interpolations (1655) 68
11 Wallis : Arithmetica infinitorum (1655) 78
12 Huygens : De Circuli magnitudine inventa (1654) 81
13 Gregory : correspondence with John Collins (1671) 87
14 Roy : The discovery of the series formula for [pi] by Leibniz, Gregory, and Nilakantha (1990) 92
15 Jones : The first use of [pi] for the circle ratio (1706) 108
16 Newton : Of the method of fluxions and infinite series (1737) 110
17 Euler : chapter 10 of Introduction to analysis of the infinite (on the use of the discovered fractions to sum infinite series) (1748) 112
18 Lambert : Memoire Sur Quelques Proprietes Remarquables Des Quantites Transcendentes Circulaires et Logarithmiques (1761) 129
19 Lambert : Irrationality of [pi] (1969) 141
20 Shanks : Contributions to mathematics comprising chiefly of the rectification of the circle to 607 places of decimals (1853) 147
21 Hermite : Sur La Fonction Exponentielle (1873) 162
22 Lindemann : Ueber die Zahl [pi] (1882) 194
23 Weiserstrass : Zu Lindemann's Abhandlung "Uber die Ludolphsche Zahl" (1885) 207
24 Hilbert : Ueber die Transzendenz der Zahlen e und [pi] (1893) 226
25 Goodwin : Quadrature of the circle (1894) 230
26 Edington : House bill no. 246, Indiana State Legislature, 1897 (1935) 231
27 Singmaster : The legal values of Pi (1985) 236
28 Ramanujan : Squaring the circle (1913) 240
29 Ramanujan : Modular equations and approximations to [pi] (1914) 241
30 Watson. The Aarquis and the land agent : a tale of the eighteenth century (1933)
31 Ballantine. The best (?) formula for computing [pi] to a thousand places (1939) 271
32 Birch. An algorithm for construction of arctangent relations (1946) 274
33 Niven. A simple proof that [pi] is irrational (1947) 276
34 Reitwiesner. An ENLAC determination of [pi] and e to 2000 decimal places (1950) 277
35 Schepler. The chronology of Pi (1950) 282
36 Mahler. On the approximation of [pi] (1953) 306
37 Wrench, Jr. the evolution of extended decimal approximations to [pi] (1960) 319
38 Shanks and Wrench, Jr. calculation of [pi] to 100,000 decimals (1962) 326
39 Sweeny. On the computation of Euler's constant (1963) 350
40 Baker. Approximations to the logarithms of certain rational numbers (1964) 359
41 Adams. Asymptotic diophantine approximations to e (1966) 368
42 Mahler. Applications of some formulae by hermite to the approximations of exponentials of logarithms (1967) 372
43 Eves. In mathematical circles; a selection of mathematical stories and anecdotes (excerpt) (1969) 400
44 Eves. Mathematical circles revisited; a second collection of mathematical stories and anecdotes (excerpt) (1971) 402
45 Todd. The lemniscate constants (1975) 412
46 Salamin. Computation of [pi] using arithmetic-geometric mean (1976) 418
47 Brent. Fast multiple-precision evaluation of elementary functions (1976) 424
48 Beukers. A note on the irrationality of [Zeta](2) and [Zeta](3) (1979) 434
49 Van der Poorten. A proof that Euler missed ... Apery's proof of the irrationality of [Zeta](3) (1979) 439
50 Brent and McMillan. Some new algorithms for high-precision computation of Euler's constant (1980) 448
51 Apostol. A proof that Euler missed : evaluating [zeta](2) the easy way (1983) 456
52 O'Shaughnessy. Putting God back in math (1983) 458
53 Stern. A remarkable approximation to [pi] (1985) 460
54 Newman and Shanks. On a sequence arising in series for [pi] (1984) 462
55 Cox. The arithmetic-geometric mean of gauss (1984) 481
56 Borwein and Borwein. The arithmetic-geometric mean and fast computation of elementary functions (1984) 537
57 Newman. A simplified version of the fast algorithms of Brent and Salamin (1984) 553
58 Wagon. Is Pi normal? (1985) 557
59 Keith. Circle digits : a self-referential story (1986) 560
60 Bailey. The computation of [pi] to 29,360,000 decimal digits using Borwein's quartically convergent algorithm (1988) 562
61 Kanada. Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of [pi] calculation (1988) 576
62 Borwein and Borwein. Ramanujan and Pi (1988) 588
63 Chudnovsky and Chudnovsky. Approximations and complex multiplication according to Ramanujan (1988) 596
64 Borwein, Borwein and Bailey. Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi (1989) 623
65 Borwein, Borwein and Dilcher. Pi, Euler numbers, and asymptotic expansions (1989) 642
66 Beukers, Bezivin, and Robba. An alternative proof of the Lindemann-Weierstrass theorem (1990) 649
67 Webster. The tale of Pi (1991) 654
68 Eco. An except from Foucault's pendulum (1993) 658
69 Keith. Pi mnemonics and the art of constrained writing (1996) 659
70 Bailey, Borwein, and Plouffe. On the rapid computation of various polylogarithmic constraints (1997) 663
App. I On the early history of Pi 677
App. II A computational chronology of Pi 683
App. III Selected formulae for Pi 686
App. IV Translations of Viete and Huygens 690
A pamphlet on Pi 721
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