Pi: A Source Book

Abstract

3rd ed. "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 Huygen, as well as Kaplansky's never-before-published 'Song of Pi.'"--Publisher's website. Preface. -- Acknowledgments. -- Introduction. -- The Rhind mathematical papyrus-problem 50 (̃1650 B.C.). -- Engles. Quadrature of the circle in ancient Egypt (1977). -- Archimedes. Measurement of a circle (̃250 B.C.). -- Phillips. Archimedes the numerical analyst (1981). -- Lam and Ang. Circle measurements in ancient China (1986). -- The Banū Mūsā: the measurement of plane and solid figures (̃850). -- Mādhava. The power series for Arctan and Pi (̃1400). -- Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938). -- Viète. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593). -- Wallis. Computation of Pi by successive interpolations (1655). -- Wallis. Arithmetica Infinitorum (1655).- Huygens. De Circuli Magnitudine Inventa (1654). -- Gregory. Correspondence with John Collins (1671). -- Roy. The discovery of the series formula for Pi by Leibniz, Gregory, and Nilakantha (1990). -- Jones. The first use of Pi for the circle ratio (1706). -- Newton. Of the method of fluxions and infinite series (1737). -- Euler. Chapter 10 of "Introduction to analysis of the infinite" (On the use of the discovered fractions to sum infinite series) (1748). -- Lambert. Mémoire Sur Quelques Proprietés Remarquables Des Quantités Transcendentes Circulaires et Logarithmiques (1761).- Lambert. Irrationality of Pi (1969). -- Shanks. Contributions to mathematics comprising chiefly of the rectification of the circle to 607 places of decimals (1853). -- Hermite. Sur La Fonction Exponentielle (1873). -- Lindemann. Ueber die Zahl Pi (1882). -- Weierstrass. Zu Lindemann's Abhandlung "Über die Ludolphsche Zahl" (1885). -- Hilbert. Ueber die Transzendenz der Zahlen e und Pi (1893). -- Goodwin. Quadrature of the circle (1894). -- Edington. House bill no. 246, Indiana State Legislature, 1897 (1935). -- Singmaster. The legal values of Pi (1985). -- Ramanujan. Squaring the circle (1913). -- Ramanujan. Modular equations and approximations to Pi (1914). -- Watson. The Marquis and the land agent: a tale of the Eighteenth Century (1933). -- Ballantine. The best formula for computing Pi to a thousand places (1939). -- Birch. An algorithm for construction of arctangent relations (1946). -- Niven. A simple proof that Pi is irrational (1947). -- Reitwiesner. An ENIAC determination of Pi and e to 2000 decimal places (1950). -- Schepler. The chronology of Pi (1950). -- Mahler. On the approximation of Pi (1953). -- Wrench, Jr. The evolution of extended decimal approximations to Pi (1960). -- Shanks and Wrench, Jr. Calculation of Pi to 100,000 decimals (1962). -- Sweeny. On the computation of Euler's constant (1963). -- Baker. Approximations to the logarithms of certain rational numbers (1964). Adams. Asymptotic diophantine approximations to e (1966). -- Mahler. Applications of some formulae by Hermite to the approximations of exponentials of logarithms (1967). -- Eves. In mathematical circles; a selection of mathematical stories and anecdotes (excerpt) (1969). -- Eves. Mathematical circles revisited; a second collection of mathematical stories and anecdotes (excerpt) (1971). -- Todd. The Lemniscate constants (1975). -- Salamin. Computation of Pi using arithmetic-geometric mean (1976). --Brent. Fast multiple-precision evaluation of elementary functions (1976). -- Beukers. A note on the irrationality of [Apéry's constant](2) and [Apéry's constant](3) (1979). -- van der Poorten. A proof that Euler missed ... Apéry's proof of the irrationality of [Apéry's constant](3) (1979). -- Brent and McMillan. Some new algorithms for high-precision computation of Euler's constant (1980). -- Apostol. A proof that Euler missed: evaluating [Apéry's constant](2) the easy way (1983). -- O'Shaughnessy. Putting God back in math (1983). -- Stern. A remarkable approximation to Pi (1985). -- Newman and Shanks. On a sequence arising in series for Pi (1984). -- Cox. The arithmetic-geometric mean of Gauss (1984). -- Borwein and Borwein. The arithmetic-geometric mean and fast computation of elementary functions (1984). -- Newman. A simplified version of the fast algorithms of Brent and Salamin (1984). -- Wagon. Is Pi normal? (1985). -- Keith. Circle digits: a self-referential story (1986). -- Bailey. The computation of Pi to 29,360,000 decimal digits using Borwein's quartically convergent algorithm (1988). -- Kanada. Vectorization of multiple-precision arithmetic program and 201,326, 000 decimal digits of Pi calculation (1988). -- Borwein and Borwein. Ramanujan and Pi (1988). -- Chudnovsky and Chudnovsky. Approximations and complex multiplication according to Ramanujan (1988). -- Borwein, Borwein and Bailey. Ramanujan, modular equations, and approximations to Pi or How to compute one billion digits of Pi (1989). -- Borwein, Borwein and Dilcher. Pi, Euler numbers and asymptotic expansions (1989). -- Beukers, Bezivin, and Robba. An alternative proof of the Lindemann-Weierstrass theorem (1990). -- Webster. The tale of Pi (1991). -- Eco. An excerpt from Foucault's Pendulum (1993). -- Keith. Pi mnemonics and the art of constrained writing (1996). -- Bailey, Borwein, and Plouffe. On the rapid computation of various polylogarithmic constants (1997).