| Preface |
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v | |
| Preface to the Second Edition |
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viii | |
| Acknowledgments |
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ix | |
| Introduction |
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xvii | |
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The Rhind Mathematical Papyrus-Problem 50 (∼ 1650 B.C.) |
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1 | (2) |
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Engels. Quadrature of the Circle in Ancient Egypt (1977) |
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3 | (4) |
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Archimedes. Measurement of a Circle (∼ 250 BC) |
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7 | (8) |
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Phillips. Archimedes the Numerical Analyst (1981) |
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15 | (5) |
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Lam and Ang. Circle Measurements in Ancient China (1986) |
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20 | (16) |
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The Banu Musa: The Measurement of Plane and Solid Figures (∼ 850) |
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36 | (9) |
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Madhava. The Power Series for Arctan and Pi (∼ 1400) |
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45 | (6) |
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Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) |
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51 | (2) |
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Viete. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) |
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53 | (15) |
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Wallis. Computation of π by Successive Interpolations (1655) |
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68 | (10) |
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Wallis. Arithmetica Infinitorum (1655) |
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78 | (3) |
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Huygens. De Circuli Magnitudine Inventa (1724) |
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81 | (6) |
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Gregory. Correspondence with John Collins (1671) |
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87 | (5) |
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Roy. The Discovery of the Series Formula for π by Leibniz, Gregory, and Nilakantha (1990) |
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92 | (16) |
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Jones. The First Use of π for the Circle Ratio (1706) |
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108 | (2) |
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Newton. Of the Method of Fluxions and Infinite Series (1737) |
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110 | (2) |
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Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of Discovered Fractions to Sum Infinite Series) (1748) |
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112 | (17) |
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Lambert. Memoire Sur Quelques Proprietes Remarquables Des Quantites Transcendentes Circularies et Logarithmiques (1761) |
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129 | (12) |
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Lambert. Irrationality of π (1969) |
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141 | (6) |
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Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853) |
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147 | (15) |
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Hermite. Sur La Fonction Exponentielle (1873) |
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162 | (32) |
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Lindemann. Ueber die Zahl π (1882) |
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194 | (13) |
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Weierstrass. Zu Lindemann's Abhandlung ``Uber die Ludolphsche Zahl'' (1885) |
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207 | (19) |
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Hilbert. Ueber die Trancendenz der Zahlen e und π (1893) |
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226 | (4) |
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Goodwin. Quadrature of the Circle (1894) |
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230 | (1) |
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Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) |
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231 | (5) |
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Singmaster. The Legal Values of Pi (1985) |
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236 | (4) |
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Ramanujan. Squaring the Circle (1913) |
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240 | (1) |
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Ramanujan. Modular Equations and Approximations to π (1914) |
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241 | (17) |
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Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933) |
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258 | (13) |
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Ballantine. The Best (?) Formula for Computing π to a Thousand Places (1939) |
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271 | (3) |
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Birch. An Algorithm for Construction of Arctangent Relations (1946) |
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274 | (2) |
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Niven. A Simple Proof that π Is Irrational (1947) |
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276 | (1) |
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Reitwiesner. An ENIAC Determination of π and e to 2000 Decimal Places (1950) |
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277 | (5) |
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Schepler. The Chronology of Pi (1950) |
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282 | (24) |
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Mahler. On the Approximation of π (1953) |
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306 | (13) |
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Wrench, Jr. The Evolution of Extended Decimal Approximations to π (1960) |
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319 | (7) |
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Shanks and Wrench, Jr. Calculation of π to 100,000 Decimals (1962) |
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326 | (24) |
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Sweeny. On the Computation of Euler's Constant (1963) |
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350 | (9) |
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Baker. Approximations to the Logarithms of Certain Rational Numbers (1964) |
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359 | (9) |
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Adams. Asymptotic Diophantine Approximations to E (1966) |
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368 | (4) |
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Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967) |
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372 | (28) |
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Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969) |
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400 | (2) |
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Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971) |
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402 | (10) |
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Todd. The Lemniscate Constants (1975) |
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412 | (6) |
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Salamin. Computation of π Using Arithmetic-Geometric Mean (1976) |
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418 | (6) |
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Brent. Fast Multiple-Precision Evaluation of Elementary Functions (1976) |
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424 | (10) |
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Beukers. A Note on the Irrationality of ζ(2) and ζ(3) (1979) |
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434 | (5) |
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van der Poorten. A Proof that Euler Missed ... Apery's Proof of the Irrationality of ζ(3) (1979) |
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439 | (9) |
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Brent and McMillan. Some New Algorithms for High-Precision Computation of Euler's Constant (1980) |
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448 | (8) |
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Apostol. A Proof that Euler Missed: Evaluating ζ(2) the Easy Way (1983) |
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456 | (2) |
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O'Shaughnessy. Putting God Back in Math (1983) |
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458 | (2) |
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Stern. A Remarkable Approximation to π (1985) |
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460 | (2) |
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Newman and Shanks. On a Sequence Arising in Series for π (1984) |
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462 | (19) |
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Cox. The Arithmetic-Geometric Mean of Gauss (1984) |
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481 | (56) |
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Borwein and Borwein. The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions (1984) |
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537 | (16) |
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Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984) |
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553 | (4) |
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Wagon. Is Pi Normal? (1985) |
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557 | (3) |
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Keith. Circle Digits: A Self-Referential Story (1986) |
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560 | (2) |
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Bailey. The Computation of π to 29,360,000 Decimal Digits Using Borweins' Quartically Convergent Algorithm (1988) |
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562 | (14) |
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Kanada. Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of π Calculation (1988) |
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576 | (12) |
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Borwein and Borwein. Ramanujan and Pi (1988) |
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588 | (8) |
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Chudnovsky and Chundnovsky. Approximations and Complex Multiplication According to Ramanujan (1988) |
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596 | (27) |
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Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989) |
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623 | (19) |
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Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989) |
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642 | (7) |
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Beukers, Bezivin, and Robba. An Alternative Proof of the Lindemann-Weierstrass Theorem (1990) |
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649 | (5) |
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Webster. The Tail of Pi (1991) |
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654 | (4) |
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Eco. An excerpt from Foucault's Pendulum (1993) |
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658 | (1) |
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Keith. Pi Mnemonics and the Art of Constrained Writing (1996) |
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659 | (4) |
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Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1997) |
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663 | (14) |
| Appendix I-On the Early History of Pi |
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677 | (6) |
| Appendix II-A Computational Chronology of Pi |
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683 | (3) |
| Appendix III-Selected Formulae for Pi |
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686 | (4) |
| Appendix IV-Translations of Viete and Huygens |
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690 | (21) |
| Bibliography |
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711 | (6) |
| Credits |
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717 | (4) |
| Index |
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721 | |
Other versions by this Author
