**Uh-oh, it looks like your Internet Explorer is out of date.**

For a better shopping experience, please upgrade now.

# A PI: A Source Book

## 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.

## Product Details

ISBN-13: | 9780387949246 |
---|---|

Publisher: | Springer-Verlag New York, LLC |

Publication date: | 06/01/1997 |

Pages: | 760 |

Product dimensions: | 7.31(w) x 10.27(h) x 1.72(d) |

## Read an Excerpt

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.

## First Chapter

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.

## Table of Contents

reface

Acknowledgments

Introduction

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

2 Engels. Quadrature of the Circle in Ancient Egypt (1977)

3 Archimedes. Measurement of a Circle (~250 BC)

4 Phillips. Archimedes the Numerical Analyst (1981)

5 Lam and Ang. Circle Measurements in Ancient China (1986)

6 The Banu Musa: The Measurement of Plane and Solid Figures (~850)

7 Madhava. The Power Series for Arctan and Pi (~1400)

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

9 Viète. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)

10 Wallis. Computation of ? by Successive Interpolations (1655)

11 Wallis. Arithmetica Infinitorum (1655)

12 Huygens. De Circuli Magnitudine Inventa (1724)

13 Gregory. Correspondence with John Collins (1671)

14 Roy. The Discovery of the Series Formula for; by Leibniz, Gregory, and Nilakantha (1990)

15 Jones. The First Use of ? for the Circle Ratio (1706)

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

17 Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748)

18 Lambert. Mèmoire Sur Quelques Propriètès Remarquables Des Quantitès Transcendentes Circulaires et Legarithmiques (1761)

19 Lambert. Irrationality of ? (1969)

20 Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853)

21 Hermite. Sur La Foncion Exponentielle (1873)

22 Lindemann. Ueber die Zahl ? (1882)

23 Weierstrass. Zu Lindemann''s Abhandlung "Über die Ludolphsche Zahl" (1885)

24 Hilbert. Ueber dieTrancendenz der Zahlen e und ? (1893)

25 Goodwin. Quadrature of the Circle (1894)

26 Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935)

27 Singmaster. The Legal Values of Pi (1985)

28 Ramanujan. Squaring the Circle (1913)

29 Ramanujan. Modular Equations and Approximations to ? (1914)

30 Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933)

31 Ballantine. The Best (?) Formula for Computing ? to a Thousand Places (1939)

32 Birch. An Algorithm for Construction of Arctangent Relations (1946)

33 Niven. A Simple Proof that ? Is Irrational (1947)

34 Reitwiesner. An ENIAC Determination of ? and e to 2000 Decimal Places (1950)

35 Schepler. The Chronology of Pi (1950)

36 Mahler. On the Approximation of ? (1953)

37 Wrench, Jr. The Evolution of Extended Decimal Approximations to ? (1960)

38 Shanks and Wrench, Jr. Calculation of ? to 100,000 Decimals (1962)

39 Sweeny. On the Computation of Euler''s Constant (1963)

40 Baker. Approximations to the Logarithms of Certain Rational Numbers (1964)

41 Adams. Asymptotic Diophantine Approximations to E (1966)

42 Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967)

43 Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)(

44 Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971)

45 Todd. The Lemniscate Constants (1975)

46 Salamin. Computation of ? Using Arithmetic-;Geometric Mean (1976)

47 Brent. Fast Multiple-;Precision Evaluation of Elementary Functions (1976)

48 Beukers. A Note on the Irrationality of &zgr;(2) and &zgr;(3) (1979)

49 van der Poorten. A Proof that Euler Missed...Apèry''s Proof of the Irrationality of &zgr;(3) 1979)

50 Brent and McMillan. Some New Algorithms for High-;Precision Computation of Euler''s Constant (1980)

51 Apostol. A Proof that Euler Missed: Evaluating &zgr;(2) the Easy Way (1983)

52 O''Shaughnessy. Putting God Back in Math (1983)

53 Stern. A Remarkable Approximation to ? (1985)

54 Newman and Shanks. On a Sequence Arising in Series for ? (1984)

55 Cox. The Arithmetic-;Geometric Mean of Gauss (1984)

56 Borwein and Borwein. The Arithmetic-;Geometric Mean and Fast Computation of Elementary Functions (1984)

57 Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984)

58 Wagon. Is Pi Normal? (1985)

59 Keith. Circle Digits: A Self-;Referential Story (1986)

60 Bailey. The Computation of ? to 29,360,000 Decimal Digits Using Borweins'' Quartically Convergent Algorithm (1988)

61 Kanada. Vectorization of Multiple-;Precision Arithmetic Program and 201,326,000 Decimal Digits of ? Calculation (1988)

62 Borwein and Borwein. Ramanujan and Pi (1988)

63 Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988)

64 Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989)

65 Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989)

66 Beukers, Bèzivin, and Robba. An Alternative Proof of the Lindemann-;Weierstrass Theorem (1990)

67 Webster. The Tail of Pi (1991)

68 Eco. An excerpt from Foucault''s Pendulum (1993)

69 Keith. Pi Mnemonics and the Art of Constrained Writing (1996)

70 Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996)

Appendix I-On the Early History of Pi

Appendix II-A Computational Chronology of Pi

Appendix III-Selected Formulae for Pi

Bibliography

Credits

Index

## Reading Group Guide

reface

Acknowledgments

Introduction

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

2 Engels. Quadrature of the Circle in Ancient Egypt (1977)

3 Archimedes. Measurement of a Circle (~250 BC)

4 Phillips. Archimedes the Numerical Analyst (1981)

5 Lam and Ang. Circle Measurements in Ancient China (1986)

6 The Banu Musa: The Measurement of Plane and Solid Figures (~850)

7 Madhava. The Power Series for Arctan and Pi (~1400)

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

9 Viète. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)

10 Wallis. Computation of ? by Successive Interpolations (1655)

11 Wallis. Arithmetica Infinitorum (1655)

12 Huygens. De Circuli Magnitudine Inventa (1724)

13 Gregory. Correspondence with John Collins (1671)

14 Roy. The Discovery of the Series Formula for; by Leibniz, Gregory, and Nilakantha (1990)

15 Jones. The First Use of ? for the Circle Ratio (1706)

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

17 Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748)

18 Lambert. Mèmoire Sur Quelques Propriètès Remarquables Des Quantitès Transcendentes Circulaires et Legarithmiques (1761)

19 Lambert. Irrationality of ? (1969)

20 Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853)

21 Hermite. Sur La Foncion Exponentielle (1873)

22 Lindemann. Ueber die Zahl ? (1882)

23 Weierstrass. Zu Lindemann''s Abhandlung "Über die Ludolphsche Zahl" (1885)

24 Hilbert. Ueber dieTrancendenz der Zahlen e und ? (1893)

25 Goodwin. Quadrature of the Circle (1894)

26 Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935)

27 Singmaster. The Legal Values of Pi (1985)

28 Ramanujan. Squaring the Circle (1913)

29 Ramanujan. Modular Equations and Approximations to ? (1914)

30 Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933)

31 Ballantine. The Best (?) Formula for Computing ? to a Thousand Places (1939)

32 Birch. An Algorithm for Construction of Arctangent Relations (1946)

33 Niven. A Simple Proof that ? Is Irrational (1947)

34 Reitwiesner. An ENIAC Determination of ? and e to 2000 Decimal Places (1950)

35 Schepler. The Chronology of Pi (1950)

36 Mahler. On the Approximation of ? (1953)

37 Wrench, Jr. The Evolution of Extended Decimal Approximations to ? (1960)

38 Shanks and Wrench, Jr. Calculation of ? to 100,000 Decimals (1962)

39 Sweeny. On the Computation of Euler''s Constant (1963)

40 Baker. Approximations to the Logarithms of Certain Rational Numbers (1964)

41 Adams. Asymptotic Diophantine Approximations to E (1966)

42 Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967)

43 Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)(

44 Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971)

45 Todd. The Lemniscate Constants (1975)

46 Salamin. Computation of ? Using Arithmetic-;Geometric Mean (1976)

47 Brent. Fast Multiple-;Precision Evaluation of Elementary Functions (1976)

48 Beukers. A Note on the Irrationality of &zgr;(2) and &zgr;(3) (1979)

49 van der Poorten. A Proof that Euler Missed...Apèry''s Proof of the Irrationality of &zgr;(3) 1979)

50 Brent and McMillan. Some New Algorithms for High-;Precision Computation of Euler''s Constant (1980)

51 Apostol. A Proof that Euler Missed: Evaluating &zgr;(2) the Easy Way (1983)

52 O''Shaughnessy. Putting God Back in Math (1983)

53 Stern. A Remarkable Approximation to ? (1985)

54 Newman and Shanks. On a Sequence Arising in Series for ? (1984)

55 Cox. The Arithmetic-;Geometric Mean of Gauss (1984)

56 Borwein and Borwein. The Arithmetic-;Geometric Mean and Fast Computation of Elementary Functions (1984)

57 Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984)

58 Wagon. Is Pi Normal? (1985)

59 Keith. Circle Digits: A Self-;Referential Story (1986)

60 Bailey. The Computation of ? to 29,360,000 Decimal Digits Using Borweins'' Quartically Convergent Algorithm (1988)

61 Kanada. Vectorization of Multiple-;Precision Arithmetic Program and 201,326,000 Decimal Digits of ? Calculation (1988)

62 Borwein and Borwein. Ramanujan and Pi (1988)

63 Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988)

64 Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989)

65 Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989)

66 Beukers, Bèzivin, and Robba. An Alternative Proof of the Lindemann-;Weierstrass Theorem (1990)

67 Webster. The Tail of Pi (1991)

68 Eco. An excerpt from Foucault''s Pendulum (1993)

69 Keith. Pi Mnemonics and the Art of Constrained Writing (1996)

70 Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996)

Appendix I-On the Early History of Pi

Appendix II-A Computational Chronology of Pi

Appendix III-Selected Formulae for Pi

Bibliography

Credits

Index

## Interviews

reface

Acknowledgments

Introduction

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

2 Engels. Quadrature of the Circle in Ancient Egypt (1977)

3 Archimedes. Measurement of a Circle (~250 BC)

4 Phillips. Archimedes the Numerical Analyst (1981)

5 Lam and Ang. Circle Measurements in Ancient China (1986)

6 The Banu Musa: The Measurement of Plane and Solid Figures (~850)

7 Madhava. The Power Series for Arctan and Pi (~1400)

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

9 Viète. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)

10 Wallis. Computation of ? by Successive Interpolations (1655)

11 Wallis. Arithmetica Infinitorum (1655)

12 Huygens. De Circuli Magnitudine Inventa (1724)

13 Gregory. Correspondence with John Collins (1671)

14 Roy. The Discovery of the Series Formula for; by Leibniz, Gregory, and Nilakantha (1990)

15 Jones. The First Use of ? for the Circle Ratio (1706)

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

17 Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748)

18 Lambert. Mèmoire Sur Quelques Propriètès Remarquables Des Quantitès Transcendentes Circulaires et Legarithmiques (1761)

19 Lambert. Irrationality of ? (1969)

20 Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853)

21 Hermite. Sur La Foncion Exponentielle (1873)

22 Lindemann. Ueber die Zahl ? (1882)

23 Weierstrass. Zu Lindemann''s Abhandlung "Über die Ludolphsche Zahl" (1885)

24 Hilbert. Ueber dieTrancendenz der Zahlen e und ? (1893)

25 Goodwin. Quadrature of the Circle (1894)

26 Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935)

27 Singmaster. The Legal Values of Pi (1985)

28 Ramanujan. Squaring the Circle (1913)

29 Ramanujan. Modular Equations and Approximations to ? (1914)

30 Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933)

31 Ballantine. The Best (?) Formula for Computing ? to a Thousand Places (1939)

32 Birch. An Algorithm for Construction of Arctangent Relations (1946)

33 Niven. A Simple Proof that ? Is Irrational (1947)

34 Reitwiesner. An ENIAC Determination of ? and e to 2000 Decimal Places (1950)

35 Schepler. The Chronology of Pi (1950)

36 Mahler. On the Approximation of ? (1953)

37 Wrench, Jr. The Evolution of Extended Decimal Approximations to ? (1960)

38 Shanks and Wrench, Jr. Calculation of ? to 100,000 Decimals (1962)

39 Sweeny. On the Computation of Euler''s Constant (1963)

40 Baker. Approximations to the Logarithms of Certain Rational Numbers (1964)

41 Adams. Asymptotic Diophantine Approximations to E (1966)

42 Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967)

43 Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)(

44 Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971)

45 Todd. The Lemniscate Constants (1975)

46 Salamin. Computation of ? Using Arithmetic-;Geometric Mean (1976)

47 Brent. Fast Multiple-;Precision Evaluation of Elementary Functions (1976)

48 Beukers. A Note on the Irrationality of &zgr;(2) and &zgr;(3) (1979)

49 van der Poorten. A Proof that Euler Missed...Apèry''s Proof of the Irrationality of &zgr;(3) 1979)

50 Brent and McMillan. Some New Algorithms for High-;Precision Computation of Euler''s Constant (1980)

51 Apostol. A Proof that Euler Missed: Evaluating &zgr;(2) the Easy Way (1983)

52 O''Shaughnessy. Putting God Back in Math (1983)

53 Stern. A Remarkable Approximation to ? (1985)

54 Newman and Shanks. On a Sequence Arising in Series for ? (1984)

55 Cox. The Arithmetic-;Geometric Mean of Gauss (1984)

56 Borwein and Borwein. The Arithmetic-;Geometric Mean and Fast Computation of Elementary Functions (1984)

57 Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984)

58 Wagon. Is Pi Normal? (1985)

59 Keith. Circle Digits: A Self-;Referential Story (1986)

60 Bailey. The Computation of ? to 29,360,000 Decimal Digits Using Borweins'' Quartically Convergent Algorithm (1988)

61 Kanada. Vectorization of Multiple-;Precision Arithmetic Program and 201,326,000 Decimal Digits of ? Calculation (1988)

62 Borwein and Borwein. Ramanujan and Pi (1988)

63 Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988)

64 Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989)

65 Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989)

66 Beukers, Bèzivin, and Robba. An Alternative Proof of the Lindemann-;Weierstrass Theorem (1990)

67 Webster. The Tail of Pi (1991)

68 Eco. An excerpt from Foucault''s Pendulum (1993)

69 Keith. Pi Mnemonics and the Art of Constrained Writing (1996)

70 Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996)

Appendix I-On the Early History of Pi

Appendix II-A Computational Chronology of Pi

Appendix III-Selected Formulae for Pi

Bibliography

Credits

Index

## Introduction

In the following introduction to the papers from the three periods we have resisted the temptation to turn our Source Book into a "History of Pi and the Methods for Computing it." Accordingly, we have made no attempt to give detailed accounts of any of the papers selected, even when the language or style might seem to render such accounts desirable. Instead, we urge the reader seeking an account of 'what's going on' to either consult a reliable general history of mathematics, such as that of C. Boyer (in its most recent up-date by U. Merzbach) or V. Katz, or P. Beckmann's more specialized and personalized history of pi.

**The Pre-Newtonian Period (Papers [1] to [15])**

The primary sources for this period are, not surprisingly, more problematic than those of later periods, and for this reason we have included an additional appendix on this material. Our selections visit Egyptian, Greek, Chinese, and Medieval Arabo-European traditions. We commence with an excerpt from the Rhind Mathematical Papyrus from theMiddle Kingdom in Egypt, circa 1650 B.C., representing some of what the ancient Egyptians knew about mathematics around 1800 B.C. By far the most significant ancient work-that of Archimedes of Syracuse (277-212 B.C.), which survives under the title On the Measurement of the Circle follows. It is hard to overemphasize how this work dominated the subject prior to the advent of the calculus.

We continue with a study of Liu Hui's third century A.D. commentary on the Chinese classic Nine Chapters in the Mathematical Art and of the lost work of the fifth century astronomer Zu Chongzhi. Marshall Clagett's translation of Verba Filiorum, the Latin version of the 9th century Arabic Book of Knowledge of the Measurement of Plane and Spherical Figures completes our first millenium extracts.

The next selection jumps forward 500 years and discusses the tombstone of Ludolph van Ceulen which recorded the culminating computation of pi by purely Archimedian techniques to 35 places as performed by Ludolph, using 2^{30}gons, before 1615. We complete this period with excerpts from three great transitional thinkers: Frangois Viete (1540-1603) whose work greatly influenced that of Fermat; John Wallis (1616-1703), to whom Newton indicated great indebtedness; and the Dutch polymath Christian Huygens (1629-1695), who correctly formalized Willebrord Snell's acceleration of Archimedes' method and was thus able to recapture Van Ceulen's computation with only 2^{30} gons. In a part of this work, not reproduced here, Huygens vigorously attacks the validity of Gregory's argument for the transcendence of pi.

**From Newton to Hilbert (Papers [16] to [24])**

These comprise many of the most significant papers on pi. After visiting Newton's contribution we record a discussion of the arctangent series for pi variously credited to the Scot James Gregory, the German Leibniz, and to the earlier Indian Madhava. In this period we move from the initial investigations of irrationality, by Euler and Lambert, to one of the landmarks of nineteenth century mathematics, the proof of the transcendence of pi.

The first paper is a selection from Euler and it demonstrates Euler's almost unparalleled - save for Ramanujan ability to formally manipulate series, particularly series for pi. It is followed by an excerpt from Lambert and a discussion by Struik of Lambert's proof of the irrationality of pi, which is generally credited as the first proof of its irrationality. Euler had previously proved the irrationality of e. Lambert's proof of the irrationality of pi is based on a complicated continued fraction expansion. Much simpler proofs are to be found in [33], [48].

There is a selection from Shank's self-financed publication that records his hand calculation of 607 digits of pi. (It is in fact correct only to 527 places, but this went unnoticed for almost a century.) The selection is included to illustrate the excesses that this side of the story has evoked. With a modern understanding of accelerating calculations this computation, even done by hand, could be considerably simplified. Neither Shanks's obsession with the computation of digits nor his error are in any way unique. Some of this is further discussed in [64].

The next paper is Hermite's 1873 proof of the transcendence of e. It is followed by Lindemann's 1882 proof of the transcendence of pi. These are, arguably, the most important papers in the collection. The proof of the transcendence of pi laid to rest the possibility of "squaring the circle," a problem that had been explicit since the late 5th c. B.C. Hermite's seminal paper on e in many ways anticipates Lindemann, and it is perhaps surprising that Hermite did not himself prove the transcendence of pi. The themes of Hermite's paper are explored and expanded in a number of later papers in this volume. See in particular Mahler [42]. The last two papers offer simplified proofs of the transcendence. One is due to Weierstrass in 1885 and the other to Hilbert in 1893. Hilbert's elegant proof is still probably the simplest proof we have.

** The Twentieth Century (Papers [26] to [70])**

The remaining forty-five papers are equally split between analytic and computational selections, with an interweaving of more diversionary selections.

On the analytic side we commence with the work of Ramanujan. His 1914 paper, [29], presents an extraordinary set of approximations to pi via "singular values" of elliptic integrals. The first half of this paper was well studied by Watson and others in the 1920s and 1930s, while the second half, which presents marvelous series for pi, was decoded and applied only more than 50 years later. (See [61], [62], [63].) Other highlights include: Watson's engaging and readable account of the early development of elliptic functions, [30]; several very influential papers by Kurt Mahler; Fields Medalist Alan Baker's 1964 paper on "algebraic independence of logarithms," [40]; and two papers on the irrationality of ζ(3) ([48], [49]) which was established only in 1976.

The computational selections include a report on the early computer calculation of pi - 2037 places on ENIAC in 1949 by Reitwiesner, Metropolis and Von Neumann [34] and the 1961 computation of pi to 100,000 places by Shanks and Wrench [38], both by arctangent methods. Another highlight is the independent 1976 discovery of arithmeticgeometric mean methods for the computation of pi by Salamin and by Brent ([46], [47], see also [57]). Recent supercomputational applications of these and related methods by Kanada, by Bailey, and by the Chudnovsky brothers are included (see [60] to [64]). As of going to press, these scientists have now pushed the record for computation of pi beyond 17 billion digits. (See Appendix 11.) One of the final papers in the volume, [701, describes a method of computing individual binary digits of pi and similar polylogarithmic constants and records the 1995 computation of the ten billionth hexadecimal digit of pi.

## Recipe

reface

Acknowledgments

Introduction

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

2 Engels. Quadrature of the Circle in Ancient Egypt (1977)

3 Archimedes. Measurement of a Circle (~250 BC)

4 Phillips. Archimedes the Numerical Analyst (1981)

5 Lam and Ang. Circle Measurements in Ancient China (1986)

6 The Banu Musa: The Measurement of Plane and Solid Figures (~850)

7 Madhava. The Power Series for Arctan and Pi (~1400)

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

9 Viète. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593)

10 Wallis. Computation of ? by Successive Interpolations (1655)

11 Wallis. Arithmetica Infinitorum (1655)

12 Huygens. De Circuli Magnitudine Inventa (1724)

13 Gregory. Correspondence with John Collins (1671)

14 Roy. The Discovery of the Series Formula for; by Leibniz, Gregory, and Nilakantha (1990)

15 Jones. The First Use of ? for the Circle Ratio (1706)

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

17 Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use of the Discovered Fractions to Sum Infinite Series) (1748)

18 Lambert. Mèmoire Sur Quelques Propriètès Remarquables Des Quantitès Transcendentes Circulaires et Legarithmiques (1761)

19 Lambert. Irrationality of ? (1969)

20 Shanks. Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals (1853)

21 Hermite. Sur La Foncion Exponentielle (1873)

22 Lindemann. Ueber die Zahl ? (1882)

23 Weierstrass. Zu Lindemann''s Abhandlung "Über die Ludolphsche Zahl" (1885)

24 Hilbert. Ueber dieTrancendenz der Zahlen e und ? (1893)

25 Goodwin. Quadrature of the Circle (1894)

26 Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935)

27 Singmaster. The Legal Values of Pi (1985)

28 Ramanujan. Squaring the Circle (1913)

29 Ramanujan. Modular Equations and Approximations to ? (1914)

30 Watson. The Marquis and the Land Agent: A Tale of the Eighteenth Century (1933)

31 Ballantine. The Best (?) Formula for Computing ? to a Thousand Places (1939)

32 Birch. An Algorithm for Construction of Arctangent Relations (1946)

33 Niven. A Simple Proof that ? Is Irrational (1947)

34 Reitwiesner. An ENIAC Determination of ? and e to 2000 Decimal Places (1950)

35 Schepler. The Chronology of Pi (1950)

36 Mahler. On the Approximation of ? (1953)

37 Wrench, Jr. The Evolution of Extended Decimal Approximations to ? (1960)

38 Shanks and Wrench, Jr. Calculation of ? to 100,000 Decimals (1962)

39 Sweeny. On the Computation of Euler''s Constant (1963)

40 Baker. Approximations to the Logarithms of Certain Rational Numbers (1964)

41 Adams. Asymptotic Diophantine Approximations to E (1966)

42 Mahler. Applications of Some Formulae by Hermite to the Approximations of Exponentials of Logarithms (1967)

43 Eves. In Mathematical Circles; A Selection of Mathematical Stories and Anecdotes (excerpt) (1969)(

44 Eves. Mathematical Circles Revisited; A Second Collection of Mathematical Stories and Anecdotes (excerpt) (1971)

45 Todd. The Lemniscate Constants (1975)

46 Salamin. Computation of ? Using Arithmetic-;Geometric Mean (1976)

47 Brent. Fast Multiple-;Precision Evaluation of Elementary Functions (1976)

48 Beukers. A Note on the Irrationality of &zgr;(2) and &zgr;(3) (1979)

49 van der Poorten. A Proof that Euler Missed...Apèry''s Proof of the Irrationality of &zgr;(3) 1979)

50 Brent and McMillan. Some New Algorithms for High-;Precision Computation of Euler''s Constant (1980)

51 Apostol. A Proof that Euler Missed: Evaluating &zgr;(2) the Easy Way (1983)

52 O''Shaughnessy. Putting God Back in Math (1983)

53 Stern. A Remarkable Approximation to ? (1985)

54 Newman and Shanks. On a Sequence Arising in Series for ? (1984)

55 Cox. The Arithmetic-;Geometric Mean of Gauss (1984)

56 Borwein and Borwein. The Arithmetic-;Geometric Mean and Fast Computation of Elementary Functions (1984)

57 Newman. A Simplified Version of the Fast Algorithms of Brent and Salamin (1984)

58 Wagon. Is Pi Normal? (1985)

59 Keith. Circle Digits: A Self-;Referential Story (1986)

60 Bailey. The Computation of ? to 29,360,000 Decimal Digits Using Borweins'' Quartically Convergent Algorithm (1988)

61 Kanada. Vectorization of Multiple-;Precision Arithmetic Program and 201,326,000 Decimal Digits of ? Calculation (1988)

62 Borwein and Borwein. Ramanujan and Pi (1988)

63 Chudnovsky and Chudnovsky. Approximations and Complex Multiplication According to Ramanujan (1988)

64 Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi (1989)

65 Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic Expansions (1989)

66 Beukers, Bèzivin, and Robba. An Alternative Proof of the Lindemann-;Weierstrass Theorem (1990)

67 Webster. The Tail of Pi (1991)

68 Eco. An excerpt from Foucault''s Pendulum (1993)

69 Keith. Pi Mnemonics and the Art of Constrained Writing (1996)

70 Bailey, Borwein, and Plouffe. On the Rapid Computation of Various Polylogarithmic Constants (1996)

Appendix I-On the Early History of Pi

Appendix II-A Computational Chronology of Pi

Appendix III-Selected Formulae for Pi

Bibliography

Credits

Index