Pi: A Source Book

Our intention in this collection is to provide, largely through original writings, an ex­ tended account 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 most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a signifi­ cant number of the most important mathematicians have contributed to its unfolding­ directly or otherwise. Pi is one of the few mathematical concepts whose mention evokes a response of recog­ nition and interest in those not concerned professionally with the subject. It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of pi is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathe­ maticians with examples of many current mathematical techniques as weIl as a palpable sense of their historical development. Why a Source Book? Few books serve wider potential audiences than does a source book. To our knowledge, there is at present no easy access to the bulk of the material we have collected. 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 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 Mathematical Circles: A Selection of Mathematical Stories and Anecdotes....Pages 400-401 Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt)....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 ξ(2) the Easy Way....Pages 456-457 Putting God Back In Math....Pages 458-459 69.30 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 Eco. 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 677-736 "This book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on pi collected herein include selections from the mathematical and computational literature over four millennia, a variety of historical studies on the cultural significance of the number, and an assortment of anecdotal, fanciful, and simply amusing pieces. For this new edition, the authors have updated the original material while adding new material of historical and cultural interest. There is a substantial exposition of the recent history of the computation of digits of pi, a discussion of the normality of the distribution of the digits, new translations of works by Viete and Huygen, as well as Kaplansky's never-before-published 'Song of Pi.'"--Publisher's website "This book documents the 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 most whimsical aspects of mathematics. Much significant mathematics originates with pi, and many great mathematicians have contributed to this story's unfolding." "Mathematicians and historians of mathematics will find this book indispensable. Teachers at every level can find here ample resources for anything from individual talks and student projects to special topics courses."--Jacket