Pi: A Source Book

The aim of this book is to provide a complete history of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics, and a suprising amount of the most important mathematics and mathematicians have contributed to its unfolding. Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. Yet, despite this, no source book on pi has been published. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet still accessible, mathematics. Mathematicians and historians of mathematics will find this book indespensable. Teachers at every level from the seventh grade onward will find here ample resources for anything from special topic courses to individual talks and special student projects. The literature on pi included in this source book falls into three classes: first a selection of the mathematical literature of four millennia, second a variety of historial studies or writings on the cultural meaning and significance of the number, and third, a number of treatments on pi that are fanciful, satirical and/or whimsical. 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 A complete history of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious and sometimes the silliest aspects of mathematics. Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. Yet, despite this, no source book on pi has been published until now. One of the beauties of this subject is that it allows for the inclusion of very modern, yet still accessible, mathematics.Mathematicians and historians of mathematics will find this book indispensable, while teachers at every level from the seventh grade onward will find ample resources for anything from special topic courses to individual talks and special student projects. Following a selection of the mathematical literature over four millennia, the book covers a variety of historical writings on the cultural meaning and significance of the number, and the whole is rounded off by a number of treatments on pi that are fanciful, satirical and/or whimsical.