Pi : A Source Book

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

• ISBN13: 9780387989464
• ISBN10: 0387989463
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-10-01
• Publisher: Springer Verlag

Summary

Documents the history of Pi from the dawn of mathematical time to the present. Reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. DLC: Pi.

Table of Contents

Preface	v
Preface to the Second Edition	viii
Acknowledgments	ix
Introduction	xvii
The Rhind Mathematical Papyrus-Problem 50 (∼ 1650 B.C.)	1	(2)
Engels. Quadrature of the Circle in Ancient Egypt (1977)	3	(4)
Archimedes. Measurement of a Circle (∼ 250 BC)	7	(8)
Phillips. Archimedes the Numerical Analyst (1981)	15	(5)
Lam and Ang. Circle Measurements in Ancient China (1986)	20	(16)
The Banu Musa: The Measurement of Plane and Solid Figures (∼ 850)	36	(9)
Madhava. The Power Series for Arctan and Pi (∼ 1400)	45	(6)
Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938)	51	(2)
Viete. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)	53	(15)
Wallis. Computation of π by Successive Interpolations (1655)	68	(10)
Wallis. Arithmetica Infinitorum (1655)	78	(3)
Huygens. De Circuli Magnitudine Inventa (1724)	81	(6)
Gregory. Correspondence with John Collins (1671)	87	(5)
Roy. The Discovery of the Series Formula for π by Leibniz, Gregory, and Nilakantha (1990)	92	(16)
Jones. The First Use of π for the Circle Ratio (1706)	108	(2)
Newton. Of the Method of Fluxions and Infinite Series (1737)	110	(2)
Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of Discovered Fractions to Sum Infinite Series) (1748)	112	(17)
Lambert. Memoire Sur Quelques Proprietes Remarquables Des Quantites Transcendentes Circularies et Logarithmiques (1761)	129	(12)
Lambert. Irrationality of π (1969)	141	(6)
Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853)	147	(15)
Hermite. Sur La Fonction Exponentielle (1873)	162	(32)
Lindemann. Ueber die Zahl π (1882)	194	(13)
Weierstrass. Zu Lindemann's Abhandlung ``Uber die Ludolphsche Zahl'' (1885)	207	(19)
Hilbert. Ueber die Trancendenz der Zahlen e und π (1893)	226	(4)
Goodwin. Quadrature of the Circle (1894)	230	(1)
Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935)	231	(5)
Singmaster. The Legal Values of Pi (1985)	236	(4)
Ramanujan. Squaring the Circle (1913)	240	(1)
Ramanujan. Modular Equations and Approximations to π (1914)	241	(17)
Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933)	258	(13)
Ballantine. The Best (?) Formula for Computing π to a Thousand Places (1939)	271	(3)
Birch. An Algorithm for Construction of Arctangent Relations (1946)	274	(2)
Niven. A Simple Proof that π Is Irrational (1947)	276	(1)
Reitwiesner. An ENIAC Determination of π and e to 2000 Decimal Places (1950)	277	(5)
Schepler. The Chronology of Pi (1950)	282	(24)
Mahler. On the Approximation of π (1953)	306	(13)
Wrench, Jr. The Evolution of Extended Decimal Approximations to π (1960)	319	(7)
Shanks and Wrench, Jr. Calculation of π to 100,000 Decimals (1962)	326	(24)
Sweeny. On the Computation of Euler's Constant (1963)	350	(9)
Baker. Approximations to the Logarithms of Certain Rational Numbers (1964)	359	(9)
Adams. Asymptotic Diophantine Approximations to E (1966)	368	(4)
Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967)	372	(28)
Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)	400	(2)
Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971)	402	(10)
Todd. The Lemniscate Constants (1975)	412	(6)
Salamin. Computation of π Using Arithmetic-Geometric Mean (1976)	418	(6)
Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976)	424	(10)
Beukers. A Note on the Irrationality of ζ(2) and ζ(3) (1979)	434	(5)
van der Poorten. A Proof that Euler Missed ... Apery's Proof of the Irrationality of ζ(3) (1979)	439	(9)
Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980)	448	(8)
Apostol. A Proof that Euler Missed: Evaluating ζ(2) the Easy Way (1983)	456	(2)
O'Shaughnessy. Putting God Back in Math (1983)	458	(2)
Stern. A Remarkable Approximation to π (1985)	460	(2)
Newman and Shanks. On a Sequence Arising in Series for π (1984)	462	(19)
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	(16)
Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984)	553	(4)
Wagon. Is Pi Normal? (1985)	557	(3)
Keith. Circle Digits: A Self-Referential Story (1986)	560	(2)
Bailey. The Computation of π to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988)	562	(14)
Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of π Calculation (1988)	576	(12)
Borwein and Borwein. Ramanujan and Pi (1988)	588	(8)
Chudnovsky and Chundnovsky. Approximations and Complex Multiplication According to Ramanujan (1988)	596	(27)
Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989)	623	(19)
Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989)	642	(7)
Beukers, Bezivin, and Robba. An Alternative Proof of the Lindemann-Weierstrass Theorem (1990)	649	(5)
Webster. The Tail of Pi (1991)	654	(4)
Eco. An excerpt from Foucault's Pendulum (1993)	658	(1)
Keith. Pi Mnemonics and the Art of Constrained Writing (1996)	659	(4)
Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1997)	663	(14)
Appendix I-On the Early History of Pi	677	(6)
Appendix II-A Computational Chronology of Pi	683	(3)
Appendix III-Selected Formulae for Pi	686	(4)
Appendix IV-Translations of Viete and Huygens	690	(21)
Bibliography	711	(6)
Credits	717	(4)
Index	721