Pi: A Source Book

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 falls into various classes. First and foremost there is a selection from the mathematical and computational literature of four millennia. There is also a variety of historical studies on the cultural significance of the number. Additionally, there is a selection of pieces that are anecdotal, fanciful, or simply amusing. 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, and new translations of works by Viete and Huygen. TOC:Preface.- Acknowledgements.- Introduction.- The Rhind Mathematical Papyrus-Problem 50.- Engles. Quadrature of the Circle in Ancient Egypt.- Archimedes. Measurement of a Circle.- Phillips. Archimedes the Numerical Analyst.- Lam & Ang. Circle Measurements in Ancient China.- The Banu Musa: The Measurement of Plane and Solid Figures.- Madhava's. The Power Series for Arctan and Pi.- Hope-Jones. Ludolph van Ceulen.- Viete. Variorum de Revus Mathematicis Reponsorum Liber VII.- Wallis. Computation of Pi by Successive Interpolations.- Wallis. Arithmetica Infinitorum.- Huygens. De Circuli Magnitudine Inventa.- Gregory. Correspondence with John Collins.- Jones. The First Use of Pi for the Circle Ratio.- Newton. Of The Method of Fluxions and Infinite Series.- Euler. Chapter 10 of Introduction to Analysis of the Infinite.- Lambert. Mémoire Sur Quelques Proprietés Remarquables Des Quantités Transcendentes Circulaires et Logarithmiques.- Lambert. Irrationality of Pi.- Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals.- Hermite. Sur La Fonction Exponentielle.- And much more.. Front Matter....Pages i-xix Extract from the Rhind Papyrus....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 [V.] The Ratio of the Diameter of any Circle to its Circumference is One [That is, is The Same for All Circles].....Pages 36-44 Appendix....Pages 45-50 Correspondence. Ludolph (or Ludolff or Lucius) van Ceulen....Pages 51-52 Capvt XVIII....Pages 53-67 Wallis. Computation of π by Successive Interpolations....Pages 68-77 Wallis. Arithmetica Infinitorum....Pages 78-80 Huygens. De Circuli Magnitudine Inventa (1654)....Pages 81-86 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....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 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 P I....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....Pages 400-401 In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes....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 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 The Deconstruction of Pi....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 677-797 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 falls into various classes. First and foremost there is a selection from the mathematical and computational literature of four millennia. There is also a variety of historical studies on the cultural significance of the number. Additionally, there is a selection of pieces that are anecdotal, fanciful, or simply amusing. 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, and new translations of works by Viete and Huygen. TOC:Preface.- Acknowledgements.- Introduction.- The Rhind Mathematical Papyrus-Problem 50.- Engles. Quadrature of the Circle in Ancient Egypt.- Archimedes. Measurement of a Circle.- Phillips. Archimedes the Numerical Analyst.- Lam & Ang. Circle Measurements in Ancient China.- The Banu Musa: The Measurement of Plane and Solid Figures.- Madhava's. The Power Series for Arctan and Pi.- Hope-Jones. Ludolph van Ceulen.- Viete. Variorum de Revus Mathematicis Reponsorum Liber VII.- Wallis. Computation of Pi by Successive Interpolations.- Wallis. Arithmetica Infinitorum.- Huygens. De Circuli Magnitudine Inventa.- Gregory. Correspondence with John Collins.- Jones. The First Use of Pi for the Circle Ratio.- Newton. Of The Method of Fluxions and Infinite Series.- Euler. Chapter 10 of Introduction to Analysis of the Infinite.- Lambert. Mémoire Sur Quelques Proprietés Remarquables Des Quantités Transcendentes Circulaires et Logarithmiques.- Lambert. Irrationality of Pi.- Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals.- Hermite. Sur La Fonction Exponentielle.- And much more..

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."

From the reviews of earlier editions:

"Few mathematics books serve a wider potential readership than does a source book and this particular one is admirably designed to cater for a broad spectrum of tastes: professional mathematicians with research interest in related subjects, historians of mathematics, teachers at all levels searching out material for individual talks and student projects, and amateurs who will find much to amuse and inform them in this leafy tome. The authors are to be congratulated on their good taste in preparing such a rich and varied banquet with which to celebrate pi."
- Roger Webster for the Bulletin of the LMS

"The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras."
- Ed Barbeau for MathSciNet

"Full of useful formulas and ideas, it is a vast source of inspiration to any mathematician, A level and upwards-a necessity in any maths library."
- New Scientist

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." From the reviews of earlier editions: "Few mathematics books serve a wider potential readership than does a source book and this particular one is admirably designed to cater for a broad spectrum of tastes: professional mathematicians with research interest in related subjects, historians of mathematics, teachers at all levels searching out material for individual talks and student projects, and amateurs who will find much to amuse and inform them in this leafy tome. The authors are to be congratulated on their good taste in preparing such a rich and varied banquet with which to celebrate pi."--Roger Webster for the Bulletin of the LMS "The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras."--Ed Barbeau for MathSciNet "Full of useful formulas and ideas, it is a vast source of inspiration to any mathematician, A level and upwards-a necessity in any maths library." - New Scientist 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. Overview: 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 ever been published. Mathematicians and historians of mathematics will find this book indispensable. Teachers from the seventh grade onward will find ample resources for anything from special topic courses to individual talks and special student projects.