Pi: A Source Book

1. Front Matter

   Pages i-xix

   PDF

2. The Rhind Mathematical Papyrus-Problem 50 ( ~ 1650 B.C.)

   • Lennart Berggren, Jonathan Borwein, Peter Borwein

   Pages 1-2

3. Quadrature of the Circle in Ancient Egypt

   • Hermann Engels

   Pages 3-6

4. Measurement of a Circle

   • Lennart Berggren, Jonathan Borwein, Peter Borwein

   Pages 7-14

5. Archimedes the Numerical Analyst

   • G. M. Phillips

   Pages 15-19

6. Circle Measurements in Ancient China

   • Lam Lay-Yong, Ang Tian-Se

   Pages 20-35

7. The Banū Mūsā: The Measurement of Plane and Solid Figures ( ~ 850)

   • Lennart Berggren, Jonathan Borwein, Peter Borwein

   Pages 36-44

8. Mādhava. The Power Series for Arctan and Pi ( ~ 1400)

   • K. Balagangadharan

   Pages 45-50

9. Hope-Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938)

   • Ludolph van Ceulen, Mathematical Gazette

   Pages 51-52

10. Viète. Variorum de Rebus Mathematicis Reponsorum Liber VIII (1593)

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 53-67

11. Wallis. Computation of π by Successive Interpolations (1655)

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 68-77

12. Wallis. Arithmetica Infinitorum (1655)

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 78-80

13. Huygens. De Circuli Magnitudine Inventa (1724)

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 81-87

14. Gregory. Correspondence with John Collins (1671)

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 87-91

15. The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha

    • Ranjan Roy

    Pages 92-107

16. The First Use of π for the Circle Ratio (1706)

    • William Jones

    Pages 108-109

17. Newton.Of the Method of Fluxions and Infinite Series (1737)

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 110-111

18. On the Use of the Discovered Factors to Sum Infinite Series

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 112-128

19. Mémoire Sur Quelques Propriétiés Remarquables des Quantités Transcendentes Circulaires et Logarithmiques

    • M. Lambert

    Pages 129-140

20. Lambert. Irrationality of π

    • Lennart Berggren, Jonathan Borwein, Peter Borwein

    Pages 141-146

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.

• Microsoft Access
• algebra
• arithmetic
• boundary element method
• calculus
• development
• function
• functions
• geometry
• history of mathematics
• interpolation
• knowledge
• logarithm
• number theory
• special function
• combinatorics

From the reviews:

"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

"Should be on every mathematician’s coffee-table! … The seventy articles comprising the source-book proper range from historical articles and classic by such players as Wallis, Huyghens, Newton, and Euler, to the articles on irrationality and transcendence … . Pi: A source Book is truly an amazing book, irresistible in its own way, and filled with gems. And once it’s on your coffee-table, feel free to do more than just browse: it’s pretty well-suited for more in-depth study … . Obviously the book is highly recommended." (Michael Berg, MathDL, January, 2001)

From the reviews of the third edition:

"This is the third edition of the by now classical Pi: a source book. … contains some notes on the computation of individual (binary) digits of p, some considerations on the normality of the decimal expansion of p, and two more sections on the history of p. This book is still a classic work of reference for anyone with an interest in fascinating Pi.” (F. Beukers, Mathematical Reviews, 2005h)

"The 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." (Zentralblatt für Didaktik der Mathematik, November, 2004)

"This is the third edition of a comprehensive selection of about 70 articles on the number pi and related constants. … This edition contains a new supplement on the recent history of the computation of digits of pi … . Furthermore, new translations of articles by Viète and Huygens have been added. Altogether, this volume provides a fascinating overview of many hundred years of research and will delight and enlighten amateur lovers of pi and professional mathematicians alike." (Ch. Baxa, Monatshefte für Mathematik, Vol. 148 (1), 2006)

"This is a fascinating reference book, which consists almost entirely of facsimiles of 70 articles about pi, followed by appendices which look at its early history as well as much computational information. … it should be something to appear in any mathematical library, as it does contain so much material. … There is something for all levels of readers in this book, from the 14-year old who may wish to know a little of the history, to the professional mathematician seeking information … ." (Anthony C. Robin, The Mathematical Gazette, Vol. 90 (518), 2006)

• Department of Mathematics and Statistics, Simon Fraser University, Burnaby, Canada

  Lennart Berggren, Jonathan Borwein, Peter Borwein

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