Pi: A Source Book / Edition 2

Pi: A Source Book / Edition 2

by Lennart Berggren, Peter B. Borwein, Jonathan M. Borwein, Peter Borwein
     
 

"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… See more details below

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."

Product Details

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

Table of Contents

1The Rhind mathematical papyrus-problem 50 (ca. 1650 B.C.)1
2Engels : Quadrature of the circle in ancient Egypt (1977)3
3Archimedes : Measurement of a circle (ca. 250 B.C.)7
4Phillips : Archimedes the numerical analyst (1981)15
5Lam and Ang : Circle measurements in ancient China (1986)20
6The Banu Musa : the measurement of plane and solid figures (ca. 850)36
7Madhava : The power series for Arctan and Pi (ca. 1400)45
8Hope-Jones : Ludolph (or Ludolff or Lucius) van Ceulen (1938)51
9Viete : Variorum de Rebus mathematicis reponsorum liber VII (1593)53
10Wallis : Computation of [pi] by successive interpolations (1655)68
11Wallis : Arithmetica infinitorum (1655)78
12Huygens : De Circuli magnitudine inventa (1654)81
13Gregory : correspondence with John Collins (1671)87
14Roy : The discovery of the series formula for [pi] by Leibniz, Gregory, and Nilakantha (1990)92
15Jones : The first use of [pi] for the circle ratio (1706)108
16Newton : Of the method of fluxions and infinite series (1737)110
17Euler : chapter 10 of Introduction to analysis of the infinite (on the use of the discovered fractions to sum infinite series) (1748)112
18Lambert : Memoire Sur Quelques Proprietes Remarquables Des Quantites Transcendentes Circulaires et Logarithmiques (1761)129
19Lambert : Irrationality of [pi] (1969)141
20Shanks : Contributions to mathematics comprising chiefly of the rectification of the circle to 607 places of decimals (1853)147
21Hermite : Sur La Fonction Exponentielle (1873)162
22Lindemann : Ueber die Zahl [pi] (1882)194
23Weiserstrass : Zu Lindemann's Abhandlung "Uber die Ludolphsche Zahl" (1885)207
24Hilbert : Ueber die Transzendenz der Zahlen e und [pi] (1893)226
25Goodwin : Quadrature of the circle (1894)230
26Edington : House bill no. 246, Indiana State Legislature, 1897 (1935)231
27Singmaster : The legal values of Pi (1985)236
28Ramanujan : Squaring the circle (1913)240
29Ramanujan : Modular equations and approximations to [pi] (1914)241
30Watson. The Aarquis and the land agent : a tale of the eighteenth century (1933)
31Ballantine. The best (?) formula for computing [pi] to a thousand places (1939)271
32Birch. An algorithm for construction of arctangent relations (1946)274
33Niven. A simple proof that [pi] is irrational (1947)276
34Reitwiesner. An ENLAC determination of [pi] and e to 2000 decimal places (1950)277
35Schepler. The chronology of Pi (1950)282
36Mahler. On the approximation of [pi] (1953)306
37Wrench, Jr. the evolution of extended decimal approximations to [pi] (1960)319
38Shanks and Wrench, Jr. calculation of [pi] to 100,000 decimals (1962)326
39Sweeny. On the computation of Euler's constant (1963)350
40Baker. Approximations to the logarithms of certain rational numbers (1964)359
41Adams. Asymptotic diophantine approximations to e (1966)368
42Mahler. Applications of some formulae by hermite to the approximations of exponentials of logarithms (1967)372
43Eves. In mathematical circles; a selection of mathematical stories and anecdotes (excerpt) (1969)400
44Eves. Mathematical circles revisited; a second collection of mathematical stories and anecdotes (excerpt) (1971)402
45Todd. The lemniscate constants (1975)412
46Salamin. Computation of [pi] using arithmetic-geometric mean (1976)418
47Brent. Fast multiple-precision evaluation of elementary functions (1976)424
48Beukers. A note on the irrationality of [Zeta](2) and [Zeta](3) (1979)434
49Van der Poorten. A proof that Euler missed ... Apery's proof of the irrationality of [Zeta](3) (1979)439
50Brent and McMillan. Some new algorithms for high-precision computation of Euler's constant (1980)448
51Apostol. A proof that Euler missed : evaluating [zeta](2) the easy way (1983)456
52O'Shaughnessy. Putting God back in math (1983)458
53Stern. A remarkable approximation to [pi] (1985)460
54Newman and Shanks. On a sequence arising in series for [pi] (1984)462
55Cox. The arithmetic-geometric mean of gauss (1984)481
56Borwein and Borwein. The arithmetic-geometric mean and fast computation of elementary functions (1984)537
57Newman. A simplified version of the fast algorithms of Brent and Salamin (1984)553
58Wagon. Is Pi normal? (1985)557
59Keith. Circle digits : a self-referential story (1986)560
60Bailey. The computation of [pi] to 29,360,000 decimal digits using Borwein's quartically convergent algorithm (1988)562
61Kanada. Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of [pi] calculation (1988)576
62Borwein and Borwein. Ramanujan and Pi (1988)588
63Chudnovsky and Chudnovsky. Approximations and complex multiplication according to Ramanujan (1988)596
64Borwein, Borwein and Bailey. Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi (1989)623
65Borwein, Borwein and Dilcher. Pi, Euler numbers, and asymptotic expansions (1989)642
66Beukers, Bezivin, and Robba. An alternative proof of the Lindemann-Weierstrass theorem (1990)649
67Webster. The tale of Pi (1991)654
68Eco. An except from Foucault's pendulum (1993)658
69Keith. Pi mnemonics and the art of constrained writing (1996)659
70Bailey, Borwein, and Plouffe. On the rapid computation of various polylogarithmic constraints (1997)663
App. IOn the early history of Pi677
App. IIA computational chronology of Pi683
App. IIISelected formulae for Pi686
App. IVTranslations of Viete and Huygens690
A pamphlet on Pi721

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