Ebook: Pi: A Source Book

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Content:
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 Banū Mūsā: The Measurement of Plane and Solid Figures ( ~ 850)....Pages 36-44
Mādhava. The Power Series for Arctan and Pi ( ~ 1400)....Pages 45-50
Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938)....Pages 51-52
Viète. Variorum de Rebus Mathematicis Reponsorum Liber VIII (1593)....Pages 53-67
Wallis. Computation of π by Successive Interpolations (1655)....Pages 68-77
Wallis. Arithmetica Infinitorum (1655)....Pages 78-80
Huygens. De Circuli Magnitudine Inventa (1724)....Pages 81-87
Gregory. 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 (1706)....Pages 108-109
Newton.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 Propriétiés Remarquables des Quantités Transcendentes Circulaires et Logarithmiques....Pages 129-140
Lambert. Irrationality of π....Pages 141-146
Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals....Pages 147-161
Sur la Fonction Exponentielle....Pages 162-193
Zu Lindemann’s, Abhandlung: „Über die Ludolph’sche Zahl“....Pages 194-206
Ueber die Transcendenz der Zahlen e und π....Pages 207-225
Quadrature of the Circle....Pages 226-229
House Bill No. 246, Indiana State Legislature, 1897....Pages 230-230
The Legal Values of Pi....Pages 231-235
Squaring the Circle....Pages 236-239
Modular Equations and Approximations to π....Pages 240-240
The Marquis and the Land-Agent; A Tale of the Eighteenth Century....Pages 241-257
The Best (?) Formula for Computing π to a Thousand Places....Pages 258-270
An Algorithm for the Construction of Arctangent Relations....Pages 271-273
A Simple Proof that π is Irrational....Pages 274-275
An ENIAC Determination of π and e to more than 2000 Decimal Places....Pages 276-276
The Chronology of Pi....Pages 277-281
The evolution of extended decimal approximations to π....Pages 282-305
Calculation of π to 100,000 Decimals....Pages 306-318
On the Computation of Euler’s Constant....Pages 319-325
Approximations to the logarithms of certain rational numbers....Pages 326-349
Asymptotic Diophantine Approximations to E....Pages 350-358
Applications of Some Formulae by Hermite to the Approximation of Exponentials and Logarithms....Pages 359-367
Mathematical Circles: A Selection of Mathematical Stories and Anecdotes....Pages 368-371
Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt)....Pages 372-399
The Lemniscate Constants....Pages 400-401
Computation of π Using Arithmetic-Geometric Mean....Pages 402-411
Fast Multiple-Precision Evaluation of Elementary Functions....Pages 412-417
A Note on the Irrationality of ζ(2) and ζ(3)....Pages 418-423
A Proof that Euler Missed …....Pages 424-433
Some New Algorithms for High-Precision Computation of Euler’s Constant....Pages 434-438
A Proof that Euler Missed: Evaluating ξ(2) the Easy Way....Pages 439-447
Putting God Back In Math....Pages 448-455
The Arithmetic-Geometric Mean of Gauss....Pages 456-457
The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions....Pages 458-459
A Simplified Version of the Fast Algorithms of Brent and Salamin....Pages 460-461
Is π Normal?....Pages 462-480
Circle Digits A Self-Referential Story....Pages 481-536
The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm....Pages 537-552
Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of π Calculation....Pages 553-556
Ramanujan and Pi....Pages 557-559
Approximations and complex multiplication according to Ramanujan....Pages 560-561
Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi....Pages 562-575
Pi, Euler Numbers, and Asymptotic Expansions....Pages 576-587
An Alternative Proof of the Lindemann-Weierstrass Theorem....Pages 588-595
The Tail of π....Pages 596-622
Eco. An excerpt from Foucault’s Pendulum (1993)....Pages 623-641
Pi Mnemonics and the Art of Constrained Writing....Pages 642-648
On the Rapid Computation of Various Polylogarithmic Constants....Pages 649-653
Back Matter....Pages 654-657
....Pages 658-658