Pi: A Source Book

Description

Front Matter....Pages i-xix The Rhind Mathematical Papyrus-Problem 50 (~ 1650 B.C.)....Pages 1-2 Quadrature of the Circle in Ancient Egypt....Pages 3-6 Measurement of a Circle....Pages 7-14 Archimedes the Numerical Analyst....Pages 15-19 Circle Measurements in Ancient China....Pages 20-35 The Measurement of Plane and Solid Figures (~850)....Pages 36-44 The Power Series of Arctan and Pi (~1400)....Pages 45-50 Ludolph (or Ludolff or Lucius) van Ceulen....Pages 51-52 Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)....Pages 53-67 Computation of π by Successive Interpolations....Pages 68-77 Arithmetica Infinitorum (1655)....Pages 78-80 De Circuli Magnitudine Inventa....Pages 81-86 Correspondence with John Collins (1671)....Pages 87-91 The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha....Pages 92-107 The First Use of π for the Circle Ratio....Pages 108-109 Of the Method of Fluxions and Infinite Series (1737)....Pages 110-111 On the Use of the Discovered Factors to Sum Infinite Series....Pages 112-128 Mémoire Sur Quelques Propriétés Remarquables des Quantités Transcendentes Circulaires et Logarithmiques....Pages 129-140 Lambert. Irrationality of π ....Pages 141-146 Contributions to Mathematics Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals....Pages 147-161 Sur La Fonction Exponentielle....Pages 162-193 Ueber die Zahl π ....Pages 194-206 Zu Lindemann’s Abhandlung: „Über die Ludolph’sche Zahl“....Pages 207-225 Ueber die Transcendenz der Zahlen e und π....Pages 226-229 Quadrature of the Circle....Pages 230-230 House Bill No. 246, Indiana State Legislature, 1897....Pages 231-235 The Legal Values of Pi....Pages 236-239 Squaring the Circle....Pages 240-240 Modular Equations and Approximations to π ....Pages 241-257 The Marquis and the Land-Agent; A Tale of the Eighteenth Century....Pages 258-270 The Best (?) Formula for Computing π to a Thousand Places....Pages 271-273 An Algorithm for the Construction of Arctangent Relations....Pages 274-275 A Simple Proof that π is Irrational....Pages 276-276 An ENIAC Determination of π and e to more than 2000 Decimal Places....Pages 277-281 The Chronology of Pi....Pages 282-305 On the Approximation of π ....Pages 306-318 The evolution of extended decimal approximations to π ....Pages 319-325 Calculation of π to 100,000 Decimals....Pages 326-349 On the Computation of Euler’s Constant....Pages 350-358 Approximations to the logarithms of certain rational numbers....Pages 359-367 Asymptotic Diophantine Approximations to E....Pages 368-371 Applications of Some Formulae by Hermite to the Approximation of Exponentials and Logarithms....Pages 372-399 In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)....Pages 400-401 Mathematical Circles Revisited ; A Selection Collection of Mathematical Stories and Anecdotes (excerpt) (1971)....Pages 402-411 The Lemniscate Constants....Pages 412-417 Computation of π Using Arithmetic-Geometric Mean....Pages 418-423 Fast Multiple-Precision Evaluation of Elementary Functions....Pages 424-433 A Note on the Irrationality of ζ(2) and ζ(3)....Pages 434-438 A Proof that Euler Missed.......Pages 439-447 Some New Algorithms for High-Precision Computation of Euler’s Constant....Pages 448-455 A Proof that Euler Missed: Evaluating ζ(2) the Easy Way....Pages 456-457 Putting God Back In Math....Pages 458-459 A remarkable approximation to π ....Pages 460-461 On a Sequence Arising in Series for π ....Pages 462-480 The Arithmetic-Geometric Mean of Gauss....Pages 481-536 The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions....Pages 537-552 A Simplified Version of the Fast Algorithms of Brent and Salamin....Pages 553-556 Is π Normal?....Pages 557-559 Circle Digits A Self-Referential Story....Pages 560-561 The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm....Pages 562-575 Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of π Calculation....Pages 576-587 Ramanujan and Pi....Pages 588-595 Approximations and complex multiplication according to Ramanujan....Pages 596-622 Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi....Pages 623-641 Pi, Euler Numbers, and Asymptotic Expansions....Pages 642-648 An Alternative Proof of the Lindemann-Weierstrass Theorem....Pages 649-653 The Tail of π....Pages 654-657 An excerpt from Foucault’s Pendulum (1993)....Pages 658-658 Pi Mnemonics and the Art of Constrained Writing....Pages 659-662 On the Rapid Computation of Various Polylogarithmic Constants....Pages 663-676 Back Matter....Pages 677-716