Contents

"Pi : A Source Book"

L B Berggren, J M Borwein, P B Borwein (Eds.)



CONTENTS


Preface

Acknowledgments

Introduction

1. The Rhind Mathematical Papyrus - Problem 50 (~1650 B.C.)

2. Engels. Quadrature of the Circle in Ancient Egypt (1977)

3. Archimedes. Measurement of a Circle (~250 BC)

4. Phillips. Archimedes the Numerical Analyst (1981)

5. Lam and Ang. Circle Measurements in Ancient China (1986)

6. The Banu Musa: The Measurement of Plane and Solid Figures (~850)

7. Madhava. The Power Series for Arctan and Pi (~1400)

8. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938)

9. Viète. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)

10. Wallis. Computation of by Successive Interpolations (1655)

11. Wallis. Arithmetica Infinitorum (1655)

12. Huygens. De Circuli Magnitudine Inventa (1724)

13. Gregory. Correspondence with John Collins (1671)

14. Roy. The Discovery of the Series Formula for ; by Leibniz, Gregory, and Nilakantha (1990)

15. Jones. The First Use of for the Circle Ratio (1706)

16. Newton. Of the Method of Fluxions and Infinite Series (1737)

17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748)

18. Lambert. Mémoire Sur Quelques Propriétés Remarquables Des Quantités Transcendentes Circulaires et Logarithmiques (1761)

19. Lambert. Irrationality of (1969)

20. Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853)

21. Hermite. Sur La Foncion Exponentielle (1873)

22. Lindemann. Ueber die Zahl (1882)

23. Weierstrass. Zu Lindemann's Abhandlung "Über die Ludolphsche Zahl" (1885)

24. Hilbert. Ueber die Trancendenz der Zahlen e und (1893)

25. Goodwin. Quadrature of the Circle (1894)

26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935)

27. Singmaster. The Legal Values of Pi (1985)

28. Ramanujan. Squaring the Circle (1913)

29. Ramanujan. Modular Equations and Approximations to (1914)

30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933)

31. Ballantine. The Best (?) Formula for Computing to a Thousand Places (1939)

32. Birch. An Algorithm for Construction of Arctangent Relations (1946)

33. Niven. A Simple Proof that is Irrational (1947)

34. Reitwiesner. An ENIAC Determination of and e to 2000 Decimal Places (1950)

35. Schepler. The Chronology of Pi (1950)

36. Mahler. On the Approximation of (1953)

37. Wrench, Jr. The Evolution of Extended Decimal Approximations to (1960)

38. Shanks and Wrench, Jr. Calculation of to 100,000 Decimals (1962)

39. Sweeny. On the Computation of Euler's Constant (1963)

40. Baker. Approximations to the Logarithms of Certain Rational Numbers (1964)

41. Adams. Asymptotic Diophantine Approximations to E (1966)

42. Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967)

43. Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)

44. Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971)

45. Todd. The Lemniscate Constants (1975)

46. Salamin. Computation of using Arithmetic-Geometric Mean (1976)

47. Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976)

48. Beukers. A Note on the Irrationality of (2) and (3) (1979)

49. van der Poorten. A Proof that Euler Missed...Apéry's Proof of the Irrationality of (3) (1979)

50. Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980)

51. Apostol. A Proof that Euler Missed: Evaluating (2) the Easy Way (1983)

52. O'Shaughnessy. Putting God Back in Math (1983)

53. Stern. A Remarkable Approximation to (1985)

54. Newman and Shanks. On a Sequence Arising in Series for (1984)

55. Cox. The Arithmetic-Geometric Mean of Gauss (1984)

56. Borwein and Borwein. The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions (1984)

57. Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984)

58. Wagon. Is Pi Normal? (1985)

59. Keith. Circle Digits: A Self-Referential Story (1986)

60. Bailey. The Computation of to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988)

61. Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of Calculation (1988)

62. Borwein and Borwein. Ramanujan and Pi (1988)

63. Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988)

64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989)

65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989)

66. Beukers, Bézivin, and Robba. An Alternative Proof of the Lindemann-Weierstrass Theorem (1990)

67. Webster. The Tail of Pi (1991)

68. Eco. An excerpt from Foucault's Pendulum (1993)

69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996)

70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996)

Appendix I - On the Early History of Pi

Appendix II - A Computational Chronology of Pi

Appendix III - Selected Formulae for Pi

Bibliography

Credits

Index