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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Pi: A Source Book - Part A
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
_____________________________________________________________
Lennart Berggren, Jonathan Borwein, and Peter Borwein (eds),
Pi: A Source Book.
Springer-Verlag, New York-Berlin-Heidelberg 1997
______________________________________________________________
1. THE AUTHORS (read: EDITORS)
______________________________
Lennart Berggren
Professor and Chairman of
Department of Mathematics and Statistics.
Simon Fraser University
Burnaby, BC V5A 1S6
Canada
Web Page:
http://www.math.sfu.ca/mast/people/faculty/berggren/berggren.html
e-mail address:
berggren@sfu.ca
Jonathan Borwein
Department of Mathematics and Statistics.
Simon Fraser University
Burnaby, BC V5A 1S6
Canada
Web Page:
http://www.cecm.sfu.ca/~jborwein/
e-mail address:
jborwein@cecm.sfu.ca
Peter Borwein
Department of Mathematics and Statistics.
Simon Fraser University
Burnaby, BC V5A 1S6
Canada
Web Page:
http://www.cecm.sfu.ca/~pborwein/
e-mail address:
pborwein@cecm.sfu.ca
2. THE BOOK
____________
Lennart Berggren, Jonathan Borwein, and Peter Borwein, Pi: A Source Book.
Springer-Verlag, New York-Berlin-Heidelberg 1997
ISBN 0-387-94924-0
Hardcover; 25X17.5; 736(= xix+[1]+716)pp; 82 figures, 67 tables
Price: $59.95
"The aim of this book is to provide a complete history of pi from the dawn
of mathematical time to the present."
Book's Web Page:
http://www.springer-ny.com/catalog/np/apr97np/DATA/0-387-94924-0.html
Contents:
Preface
Acknoledgenents
Introduction
#1 - #70 [Papers/Experts Listing]
Appendix I - On the Early History of Pi.
Appendix II - A Computational Chronology of Pi.
Appendix III - Selected Formulae for Pi.
Bibliography
Credits
Index
(Continued)
Antreas
http://users.hol.gr/~xpolakis/
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Pi: A Source Book - Part B (long)
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
_____________________________________________________________
Lennart Berggren, Jonathan Borwein, and Peter Borwein (eds),
Pi: A Source Book.
Springer-Verlag, New York-Berlin-Heidelberg 1997
______________________________________________________________
3. PAPERS / EXTRACTS LISTING:
____________________________
Notes:
a. This listing is a compiling of the Book's Contents #1-#70 and Bibliography
listings.
b. The listing is of mine, so maybe some misspellings etc. are of mine too.
-----------------------------------------------------------------------------
1. The Rhind Mathematical Papyrus-Problem 50
A.B. Chase, The Rhind Mathematical Papyrous. The NCTM, Boston 1979, pp. 92-93
2. H. Engels, Quadrature of the Circle in Ancient Egypt.
Historia Mathematica 4(1977) 137 - 140
3. Archimedes, Measurement of a Circle.
T. L. Heath, The Works of Archimedes with the Method of Archimedes.
Dover Publications Inc., New York 1953, pp. 91-98
4. G. M. Phillips, Archimedes the Numerical Analyst.
The American Mathematical Monthly 88(1981) 165-169
5. Lam Lay-Yong and Ang Tian-Se, Circle Measurements in Ancient China.
Historia Mathematica 13(1986) 325-340
6. The Banu Musa: The Measurement of Plane and Solid Figures.
M. Clagett, Archimedes in the Middle Ages, Volume I: The Arabo-Latin
Tradition. University of Wisconsin Press, Madison 1964, pp. 261-269
7.[Madhava, The Power Series for Arctan and Pi.]
K. Mukunda Matar and C. T. Rajagonal, On the Hindu Quadrature of the Circle.
Appendix by K. Balagangadharan.
Journal of the Bombay Branch of the Royal Asiatic Society 20(1944) 77-82
8. W. Hope-Jones, Ludolph (or Ludolf or Lucius) van Ceulen.
[Letter] To the Editor of the Mathematical Gazette.
Mathematical Gazette 22(1938) 281-282
9. F. Viete, Variorum de Rebus Mathematicis Reponsorum Liber VII.
F. Viete, Opera Mathematica (repr.). Georg Olms Verlag, Hildesheim,
New York 1970, pp. 398-401, 436-446
10. Analysis Before Newton and Leibniz.
A Source Book in Mathematics 1200-1800, Princeton University Press 1986,
pp. 244 - 253
11. J. Wallis, Arithmetica Infinitorum, sive Nova Methodus Inquirendi
in Curvilineorum Quadratum, aliaque difficiliora Matheseos Problemata.
Oxford 1655/6.
Reprinted in vol. 1 (pp. 357-478) of Opera Mathematica. Oxford 1693
12. C. Huygens, De Circuli Magnitudine Inventa.
Christiani Hugenii Opera Varia I. Leiden 1724, pp. 384-388
13. [J. Gregory, Correspondence with John Collins.]
H.W. Turnbull (ed.), James Gregory Tercentenary Memorial Volume.
Bell, London 1939, pp. 168-172
14. Ranjan Roy, The Discovery of the Series Formula for p by Leibniz,
Gregory, and Niakantha.
Mathematics Magazine 63(1990) 291-306
15. William Jones, The First Use of pi for the Circle Ratio.
D. E. Smith (ed), A Source Book in Mathematics II.
Dover Publications Inc, New York 1959, pp. 346 - 347
16. I. Newton, Of the Method of Fluxions and Infinite Series.
The Mathematical Works of Isaac Newton I. Johnson Reprint Corp.
New York 1964, pp. 128 - 131
17. L. Euler, On the Use of the Discovered Fractions to Sum Infinite Series.
L. Euler, Introduction to Analysis of the Infinite. Book I.
Translated from the Latin by J. D. Blanton. Springer-Verlag, New York 1964,
pp. 137 - 153
18. A.M. Lambert, Memoire Sur Quelques Proprietes Remarquables Des Quantites
Transcedentes Circulaires et Logarithmiques.
Histoire de l' Academie Royale des Sciences et des Belles-Lettres der
Berlin 17(1761) 265-276
19. A.M. Lambert, Irrationality of pi.
A Source Book in Mathematics 1200-1800, Princeton University Press 1986,
pp. 369 - 374
20. W. Shanks, Contributions to Mathematics Comprising Chiefly of the
Rectification of the Circle to 607 Places of Decimals. London 1853,
pp. i-xvi, 10
21. C. Hermite, Sur La Fonction Exponentielle.
C. Hermite, Oeuvres, vol. 3. Paris 1912, pp. 150-181
22. F. Lindemann, Ueber die Zahl pi.
Mathematische Annalen 20(1882) 213-225
23. K. Weirstrass, Zu Lindemann's Abhandlung "Ueber die Ludolph'sche Zahl"
Berliner Akademie 5(1885 1067 - 1085
24. D. Hilbert, Ueber die Transcendenz der Zahlen e und pi.
Mathematische Annalen 43(1893) 216-219
25. E. J. Goodwin, Quadrature of the Circle.
The American Mathematical Monthly 1(1894) 246
26. W.E. Edington, House Bill No. 246, Indiana State Legislature, 1897.
Proceedings of the Indiana Academy of Science 45(1935) 206-210
27. D. Sigmaster, The Legal Values of Pi.
The Mathematical Intelligencer 7:2(1985) 69-72
28. S. Ramanujan, Squaring the Circle.
Journal of the Indian Mathematical Society, V, 1913, 132
G.H. Hardy, P.V. Sehuigar, and B. M. Wilson (eds), S. Ramanujan: Collected
Papers. Chelsea, New York 1962, p. 22
29. S. Ramanujan, Modular Equations and Approximations to pi.
Journal of the Indian Mathematical Society, XLV, 1914, 350-372
G.H. Hardy, P.V. Sehuigar, and B. M. Wilson (eds), S. Ramanujan:
Collected Papers. Chelsea, New York 1962, pp. 23-29
30. G.N. Watson, The Marquis and the Land Agent; A Tale of the
Eighteenth Century.
The Mathematical Gazette 17(1933) 5-17
31. J.P. Ballantine, The Best (?) Formula for Computing pi
to a Thousand Places.
The American Mathematical Monthly 46(1939) 499-501
32. R.H. Birch, An Algorithm for Construction of Arctangent Relations.
Journal of the London Mathematical Society 21(1946) 173-174
33. I. Niven, A Simple Proof that pi Is Irrational.
Bulletin of the American Mathematical Society, vol. 53, no. 7, July (1947),
p. 507
34. G.W. Reitwiesner, An ENIAC Determination of pi and e to
2000 Decimal Places.
Mathematical Tables and Other Aids to Computation 4(1950) 11-15
35. H.C. Schepler, The Chronology of Pi.
Mathematics Magazine 23(1950) 165-170, 216-228, and 279-283
36. K. Mahler, On the Approximation of pi.
Indagationes Mathematicae 15(1953) 30-42
37. J. W. Wrench, Jr, The Evolution of Extended Decimal Approximations to pi.
The Mathematics Teacher 53(1960) 644-650
38. D. Shanks and J. W. Wrench, Jr, Calculation of pi to 100,000 Decimals.
Mathematics of Computation 16(1962) 76-99
39. D.W. Sweeney, On the Computation of Euler's Constant.
Mathematics of Computation 17(1963) 170-178
40. A. Baker, Approximations to the Logarithms of Certain Rational Numbers.
Acta Arithmetica 10(1964) 315-323
41. W. W. Adams, Asymptotic Diophantine Appoximations to E.
Proceedings of the National Academy of Sciences 55(1966) 28-31
42. K. Mahler, Applications of Some Formulae by Hermite to the Approximations
of Exponentials of Logarithms.
Mathematische Annalen 168(1967) 200-277
43. H. W. Eves, In Mathematical Circles: A Selection of Mathematical Stories
and Anecdotes.
Prindle, Weber & Schmidt Publishing Co., Inc. Boston 1969, pp. 81-82
44. H. W. Eves, Mathematical Circles Revisited: A Second Selection of
Mathematical Stories and Anecdotes.
Prindle, Weber & Schmidt Publishing Co., Inc. Boston 1971, pp. 25-34
45. J. Todd, The Lemniscate Constants.
Communications of the ACM 18(1975) 14-19
46. E. Salamin, Computation of pi Using Arithmetic-Geometric Mean.
Mathematics of Computation 30(1976) 565-570
47. R. P. Brent, Fast Multiple-Precision Evaluation of Elementary Functions.
Journal of the ACM 23(1976) 242-251
48. F. Beukers, A Note on the Irrationality of zeta(2) and zeta(3).
Bulletin of the London Mathematical Society 11(1979) 268-272
49. A. J. van der Poorten, A Proof that Euler Missed ... Apery's Proof
of Irrationality of zeta(3).
The Mathematical Intelligencer 1(1979) 195-203
50. R.P. Brent and E.M. McMillan, Some New Algorithms for High-Precision
Computation of Euler's Constant.
Mathematics of Computation 34(1980) 305-312
51. T.M. Apostol, A Proof that Euler Missed: Evaluating zeta(2) the Easy Way.
The Mathematical Intelligencer 5:3(1983) 59-60
52. L. O'Shaghnessy, Putting God Back in Math.
The Mathematical Intellingencer 5:4(1983) 76-77
53. M.D. Stern, A Remarkable Approximation to pi.
The Mathematical Gazette 69(1985) 218-219
54. M. Newman and D. Shanks, On a Sequence Arising in Series for pi.
Mathematics of Computation 42(1984) 199-217
55. D.A. Cox, The Arithmetic-Geometric Mean of Gauss.
L' Ensignement Mathematique 30(1984) 275-330
56. J.M. Borwein and P. B. Borwein, The Arithmetic-Geometric Mean and Fast
Computation of Elementary Functions.
SIAM Review 26(1984) 351-365
57. D.J. Newman, A Simplified Version of the Fast Algorithms of
Brent and Salamin.
Mathematics of Computation 44(1985) 207-210
58. S. Wagon, Is Pi Normal?
The Mathematical Intelligencer 7:3(1985) 65-67
59. M. Keith, Circle Digits: A Self-Referential Story.
The Mathematical Intelligencer 8:3(1986) 56-57
60. D.H. Bailey, The Computation of pi to 29,360,000 Decimal Digits Using
Borwein's Quatrically Convergent Algorithm.
Mathematics of Computation 50(1988) 283-296
61. Y. Kanada, Vectorization of Multiple-Precision Arithmetic Program and
201,326,000 Decimal Digits of pi Calculation.
Supercomputing 88: Volume II, Sciences and Applications.
IEEE 1988, pp. 117-128
62. J.M. Borwein and P.B. Borwein, Ramanujan and pi.
Scientific American, no. 256, 1988, 112-117
63. D.V. Chudnovsky and G.V. Chudnovsky, Approximations and Complex
Multiplication According to Ramanujan.
G.E. Andrews et al (eds), Ramanujan Revisited. Academic Press, Boston 1988,
pp. 375-396, 468-472
64. J.M. Borwein, P.B. Borwein and D.H. Bailey, Ramanujan, Modular Equations,
and Approximations to Pi or How to Compute One Billion Digits of Pi.
The American Mathematical Monthly 96(1989) 201-219
65. J.M. Borwein, P.B. Borwein and K. Dilcher, Pi Euler Numbers, and
Asymptotic Expansions.
The American Mathematical Monthly 96(1989) 681-687
66. F. Beukers, J.P. Bezivin, and P. Robba, An Alternative Proof of the
Lindemann-Weirstrass Theorem.
The American Mathematical Monthly 97(1990) 193-197
67. R. Webster, The Tail of Pi.
Fourteenth Annual IMO Lecture at the Royal Society) (17th September 1991)
68. U. Eco, Foucault's Pendulum.
Ballantine, 1988, p. 3
69.M. Keith, Pi Mnemonics and the Art of Constrained Writing (1996).
Unpublished
70. D.H. Bailey, P.B. Borwein and S. Plouffe, On the Rapid Computation of
Various Polylogarithmic Constants.
Mathematics of Computation 66(1997) 903-913
(Continued)
Antreas
http://users.hol.gr/~xpolakis/
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Pi: A Source Book - Part C
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
_____________________________________________________________
Lennart Berggren, Jonathan Borwein, and Peter Borwein (eds),
Pi: A Source Book.
Springer-Verlag, New York-Berlin-Heidelberg 1997
______________________________________________________________
4. REMARKS - COMMENTS
_____________________
Generally speaking, The Book is a wonderful one! I think that every
mathematician and everyone interested in History of Mathematics will
find it useful.
Now, some comments/remarks from a first reading.
1. On Contents:
In general, no historically/mathematically "first degree"-contribution
to Pi is missed. However, many other contributions are not included.
Not only N. Americans, Britishmen, Frenchmen, Germans, Japanese have in modern
times contributed to Pi. There are also contributions by Russians and other
E. Europeans, N. Europeans etc.
Two papers come in my mind: one from former Chechoslovak and one from
Norway (there are, of course, many-many more):
Jaroslav Hancl, Two Proofs of Transcendency of pi and e.
Chechoslovak Mathematical Journal 35(110) 1985, pp. 543-549
(Btw, many thanks to Jaroslav who has sent me his paper)
[He also has written one more pi article:
Jaroslav Hancl, A Simple Proof of the irrationality of Pi^4.
The American Mathematical Monthly 93(1986) 374-375]
Viggo Brun, On the problem of partitioning the circle so as to visualize
Leibniz's formula for Pi. Nordisk Matematisk Tidskrift 1955
(Also, by the same author: The formula of Leibniz for Pi deduced by a
"mapping" of the circular disk. op. cit. 1970)
Brun gave a geometrical proof of Leibniz's formula for Pi/4.
Btw, he had given a lecture on 26 February 1970 at Trondheim, Norway,
(in Norwegian) under the title: "The Circle through the Ages".
A very interesting lecture with rich math./hist. material on the circle and
Pi.
(I know it from its Greek translation, published in a Greek periodical).
On the Contents #69:
"69. M. Keith, Pi Mnemonics and the Art of Constrained Writing.
Unpublished."
Actuall, it is published in the web at:
http://users.aol.com/s6sj7gt/mikerav.htm
Mike has written one larger story ("Cadaeic Cadenza") encoding the first 3835
digits of Pi. As he says:
"The 3835 digits of Cadaeic Cadenza exceeds my previous world record of 740
digits, which was held by the version of The Raven contained in Section One
of this very tale (originally composed in July 1995)."
It can be found at: http://users.aol.com/s6sj7gt/cadenza.htm
Mike has informed me that he is writing a book on the Pi literature.
2. On Bibliography:
The same comment as above: There are important papers/books on Pi, besides
the ones written by N. Americans, Britishmen, Germans, Frenchmen, which are
not included. Just two (by two Latinoamericans):
F. J. Duarte, Monografia sobre los numeros Pi y e.
Boletin de la Academia de Ciencias Fisicas, Matematicas y Naturales,
Ano XIV, v. XI, June-December 1948 (Caracas)
(Btw, many thanks to Edgardo L. Fernandez Stacco, Argentina, who has sent
me a photocopy)
Dario Castellanos, Ubiquitous Pi.
Mathematics Magazine 61(1988) 67-98, and 148-163
Also, a good reference to pi paramathematics ("Indiana's Pi", Circle squarers)
is: Underwood Dudley, Mathematical Cranks.
The Math. Ass. of America, Washington, D.C., 1992
(Legislating Pi: pp. 191-197
Quadrature of the Circle: pp. 279-321)
3. Typographical Errors etc.
In p. 699, #70: Read (1997) instead of (1977).
In p. 694, Read K.M. Marar instead of K.M. Marur.
In the Bibliography: The references:
K. M. Marur [read Marar] etc. (p. 694) and H. W. Turnbull (p. 695) should be
asterisk-ized, that is, they should read: * K. M. Marur etc.; * H. W. Turnbull.
[they are Contents #7 and #13 respectively]
(Btw, in my listing [Part B] there are two misspellings (of mine) in #7:
Read: Marar instead of Matar and Rajagopal instead of Rajagonal)
There is a misreferencing to The Mathematical Intellingencer:
The references to, should read n:m(year) instead n(year)
(where n = the volume number, and m = the issue number, since every issue
has its own pages numeration). I corrected it in my listing (previous Part B).
(Continued)
Antreas
http://users.hol.gr/~xpolakis/
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: New World Record of Pi : 51.5 billion Decimal Digits.
Comments: cc: olletg@ts.umu.se
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
Since I am writing on a book on Pi, edited by Jonathan Borwein (et al),
it isn't, I think, a bad idea to post one of Jon's web pages, that one at:
http://www.cecm.sfu.ca/personal/jborwein/Kanada_50b.html
Enjoy !
==============================================================================
From: kanada@pi.cc.u-tokyo.ac.jp (Yasumasa KANADA)
Subject: New world record of pi : 51.5 billion decimal digits
Dear pi people;
Now is the time for the announcement of new world record of pi. It took
longer time than our expectation. Nearly two years has passed since we got
new world record of 6.4 billion. Now, we got eight times more record than
6.4 billion as the following texts which you can get with anonymous ftp to
'www.cc.u-tokyo.ac.jp'
Yasumasa KANADA, Computer Centre, University of Tokyo
Our latest record was established as follows:
Declared record: 51,539,600,000
decimal digits
Two independent calculations based on two different algorithms generated
51,539,607,552 (=3*2^34) decimal digits of pi and comparison of two
generated sequences matched 51,539,607,510 decimal digits, e.g., a 42
decimal digits difference. Then we are declaring 51,539,600,000 decimal
digits as the new world record. (See related lecture on Pi.)
Main program run:
Job start : 6th June 1997 22:29:06
Job end : 8th June 1997 03:32:17
Elapsed time : 29:03:11
Main memory : 212 GB
Algorithm : Borwein's 4-th order convergent algorithm
(Run the algorithm.)
Verification program run:
Job start : 4th July 1997 22:11:42
Job end : 6th July 1997 11:19:58
Elapsed time : 37:08:16
Main memory : 188 GB
Algorithm : Gauss-Legendre algorithm (Brent-Salamin)
Machine used: HITACHI SR2201 at the Computer Centre, University of
Tokyo, with 1024 Processors.
50,000,000,000-th digits of pi and 1/pi:
pi : 85133 98712 75109 30042
1/pi: 1191 08624 25640 78042
(First digit '3' for pi or '0' for 1/pi is not included in the above count.)
Frequency distribution for pi-3 up to 50,000,000,000 decimal
places:
'0' : 5000012647; '1' : 4999986263; '2' : 5000020237; '3' : 4999914405
'4' : 5000023598; '5' : 4999991499; '6' : 4999928368; '7' : 5000014860
'8' : 5000117637; '9' : 4999990486;
Chi square = 5.60
Frequency distribution for 1/pi up to 50,000,000,000 decimal
places:
'0' : 4999969955; '1' : 5000113699; '2' : 4999987893; '3' : 5000040906
'4' : 4999985863; '5' : 4999977583; '6' : 4999990916; '7' : 4999985552
'8' : 4999881183; '9' : 5000066450;
Chi square = 7.04
51,539,600,000-th digits of pi and 1/pi
pi : 70532 46569 86142 12904
1/pi: 0081 50624 62192 72973
(First digit '3' for pi or '0' for 1/pi is not included in the above count.)
Some interesting digit sequences
0123456789 : from 17,387,594,880-th of pi
0123456789 : from 26,852,899,245-th of pi
0123456789 : from 30,243,957,439-th of pi
0123456789 : from 34,549,153,953-th of pi
0123456789 : from 41,952,536,161-th of pi
0123456789 : from 43,289,964,000-th of pi
9876543210 : from 21,981,157,633-th of pi
9876543210 : from 29,832,636,867-th of pi
9876543210 : from 39,232,573,648-th of pi
9876543210 : from 42,140,457,481-th of pi
9876543210 : from 43,065,796,214-th of pi
09876543210 : from 42,321,758,803-th of pi
27182818284 : from 45,111,908,393-th of pi
0123456789 : from 6,214,876,462-th of 1/pi
01234567890 : from 50,494,465,695-th of 1/pi
9876543210 : from 15,603,388,145-th of 1/pi
9876543210 : from 51,507,034,812-th of 1/pi
999999999999 : from 12,479,021,132-th of 1/pi
(First digit '3' for pi or '0' for 1/pi is not included in the above count.)
Programs were written by Mr. Daisuke TAKAHASHI, a Research Associate at
our Computer Centre.
Message passing routines were written by myself.
CPU used was the HITACHI SR2201 at the Computer Centre, University of
Tokyo.
1024 PE's were definitely used through single job parallel processing for
total of two programs run.
Yasumasa KANADA
Computer Centre, University of Tokyo
Bunkyo-ku Yayoi 2-11-16
Tokyo 113 Japan
Fax : +81-3-3814-7231 (office)
E-mail: kanada@pi.cc.u-tokyo.ac.jp
July 26:
QUESTION(JMB): Do you have an estimate for the same method
(i) in serial
ANSWER(YK): $\inf$ because we can't access machines with 256GB main
memory and minimum of 72 GB disk storage. (If we can access the
machine with more memory, elapsed time is even shorter. Both
calculations will be less than half a day with 300GFlops peak performance
machine.)
(ii) without fast multiplication?
ANSWER: $\inf$ because man-made machines can easily be collapsed or
give incorrect answers with the duration of long long calculations.
=============================================================================
PS. to my Swedish friend Ole, I am Cc-ing this to:
Time to change your page (http://www.ts.umu.se/~olletg/pi/pirec.html), friend !
Don't think so? :-))))
Antreas
http://users.hol.gr/~xpolakis/
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Pi: A Source Book - Part D
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
_____________________________________________________________
Lennart Berggren, Jonathan Borwein, and Peter Borwein (eds),
Pi: A Source Book.
Springer-Verlag, New York-Berlin-Heidelberg 1997
______________________________________________________________
5. Pi Literature/Art
____________________
When I once asked one of the Editors whether the Book will contain sources
of the rich Pi-related literature/art etc, he replied that they leave it
to me ... (that is, for my collection PiPhilology). However, the Book contains
some very nice pieces of (Mike Keith's excellent poems etc).
Besides those included ones there are many more, and among them the famous poem
Liczba Pi (= Number Pi) by the Polish Nobelist (Nobel Prize for the Literature
1996) poet, namely Wislawa Szymborska.
Here is the poem:
Number Pi
The admirable number pi:
three point one four one.
All the following digits are also initial,
five nine two because it never ends.
It can't be comprehended six five three five at a glance,
eight nine by calculation,
seven nine or imagination,
not even three two three eight by wit, that is, by comparison
four six to anything else
two six four three in the world.
The longest snake on earth calls it quits at about forty feet.
Likewise, snakes of myth and legend, though they may hold out a bit
longer. The pageant of digits comprising the number pi doesn't stop at
the page's edge. It goes on across the table, through the air, over a
wall, a leaf, a bird's nest, clouds, straight into the sky, through
all the bottomless, bloated heavens. Oh how brief - a mouse tail, a
pigtail - is the tail of a comet! How feeble the star's ray, bent by
bumping up against space! While here we have two three fifteen three
hundred nineteen my phone number your shirt size the year nineteen
hundred and seventy-three the sixth floor the number of inhabitants
sixty-ripe cents hip measurement two fingers a charade, a code, in
which we find hail to thee, blithe spirit, bird thou never wert
alongside ladies and gentlemen, no cause for alarm, as well as heaven
and earth shall pass away, but not the number pi, oh no, nothing
doing, it keeps right on with its rather remarkable five, its
uncommonly fine eight, its far from final seven, nudging, always
nudging a sluggish eternity to continue.
(Translation: Stanislaw Baranczak and Clare Cavanagh)
I would like to add one more pi-related poem. That one by Rebecca Carlson.
Enjoy!
Poesy
My love is like a source
code
that compiles on the first
shot.
My love is like a PDE [= partial differential equation]
when all nonlinear terms
drop out.
And so you can
extrapolate,
so locked in phase am I,
That I will love you till
they find
the last digit of Pi.
Note: Those interested in the Pi literature may download from the Internet
my collection PiPhilology (as a free macintosh or plain text file).
More information at: http://users.hol.gr/~xpolakis/piphil.html
PS. I just received Fred's message ("Our purpose").
Thank you, Fred !.
(Continued)
Antreas
http://users.hol.gr/~xpolakis/
From owner-math-history-list@ENTERPRISE.MAA.ORG Sat Aug 2 20:29:54 1997
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Date: Sat, 2 Aug 1997 21:21:00 -0300
Reply-To: "Antreas P. Hatzipolakis"
From: xpolakis@HOL.GR
Subject: Pi: A Source Book - Part E
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
_____________________________________________________________
Lennart Berggren, Jonathan Borwein, and Peter Borwein (eds),
Pi: A Source Book.
Springer-Verlag, New York-Berlin-Heidelberg 1997
______________________________________________________________
6. Pi-Paramathematics
_____________________
There are two kinds of pi-paramathematicians:
1. Those who "invented" new pi's (!), and
2. Those who "discovered" pi (approximations) in Ancient Civilizations's
monuments/literature.
>From the first category: The Book contains three sources on the Goodwin's
different Pi's (#25,#26, and #27)
>From the second category the Book hasn't any sources (unless the "rabbinical"
Pi = 3X(111/106) is but a pi-paramathematician's discovery; #53)
However, there are many such sources: for the Atlantean Pi
[ Don't you believe me? Well .... have a look at:
http://www.lynx.bc.ca/~jirimruzek/atma.htm ]
for the Egyptian Pi (of the pyramidoPARAlogists) etc.
Let's see now a Greek Pi:
Everyone knows the famous Platonic saying:
A)ei\ o( Qeo\s gewmetrei=
(Ever God geometrizes)
Someone extended this saying as follows (I shall omit the gramm. signs:
accents etc): "Aei o Qeos o Megas gewmetrei" [Ever Great God geometrizes],
in order to be a Pi mnemonic for the first 6 digits of the Pi
(that is, Aei = 3 letters; o = 1 letter; megas = 4 letters; o = 1 letter;
megas = 5 letters; gewmetrei = 9 letters)
When? I don't know exactly. Probably in the last century. The earliest
source that I know is of the year 1919)
The modern Greek pi-paramahematicians now say that the Ancient Greeks knew
the Pi up to 6 decimal places, since they (= the Anc. Greeks) had the
saying above, which encodes the first 6 digits of the Pi !!!
A variation of that I read recently:
The Greek language is a mathematical one (for the same as above reason) !
7. FINAL WORDS
______________
The Book, as I already have said, is a wonderful one. It will be a handy
tool for everyone interested in mathematics and/or mathematics history.
Antreas
http://users.hol.gr/~xpolakis/
From owner-math-history-list@ENTERPRISE.MAA.ORG Sun Aug 3 15:53:56 1997
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Re: Pi: A Source Book - Part E
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
From: xpolakis@prometheus.hol.gr (Antreas P. Hatzipolakis)
Subject: Re: Pi: A Source Book - Part E
On Sun, 3 Aug 1997, I wrote:
>Someone extended this saying as follows (I shall omit the gramm. signs:
>accents etc): "Aei o Qeos o Megas gewmetrei" [Ever Great God geometrizes],
>in order to be a Pi mnemonic for the first 6 digits of the Pi
>(that is, Aei = 3 letters; o = 1 letter; megas = 4 letters; o = 1 letter;
^^^^^
Qeos
>megas = 5 letters; gewmetrei = 9 letters)
Additionally, in 1924 the Professor of Mathematics at Athens University
Nikolaos Hatzidakis extended it up to 23 places:
Non-accented Greek Text * (My bad) English Translation
___________________________________* __________________________________________
Aei o Qeos o Megas gewmetrei * Great God ever geometrizes
to kuklou mhkos ina orish diametrw * To define the circle length by its diameter
parhgagen ariqmon aperanton * Produced an endless number
kai on feu oudepote olon * Which whole, alas, mortals
qnhtoi qa eurwsi. * Will never find.
BTW, a fine on-line "Pi Trivia Game" by Eve Andersson (with math/historical
information on Pi), is available at:
http://eveander.com/~eveander/trivia/
Really interesting !
Antreas
http://users.hol.gr/~xpolakis/
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Date: Sun, 3 Aug 1997 22:59:29 +0200
Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Re: Pi: A Source Book - Part E
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
On Sun, 3 Aug 1997, I wrote:
>Someone extended this saying as follows (I shall omit the gramm. signs:
>accents etc): "Aei o Qeos o Megas gewmetrei" [Ever Great God geometrizes],
>in order to be a Pi mnemonic for the first 6 digits of the Pi
>(that is, Aei = 3 letters; o = 1 letter; megas = 4 letters; o = 1 letter;
^^^^^
Qeos
>megas = 5 letters; gewmetrei = 9 letters)
Additionally, in 1924 the Professor of Mathematics at Athens University
Nikolaos Hatzidakis extended it up to 23 places:
Non-accented Greek Text * (My bad) English Translation
___________________________________* __________________________________________
Aei o Qeos o Megas gewmetrei * Great God ever geometrizes
to kuklou mhkos ina orish diametrw * To define the circle length by its diameter
parhgagen ariqmon aperanton * Produced an endless number
kai on feu oudepote olon * Which whole, alas, mortals
qnhtoi qa eurwsi. * Will never find.
BTW, a fine on-line "Pi Trivia Game" by Eve Andersson (with math/historical
information on Pi), is available at:
http://eveander.com/~eveander/trivia/
Really interesting !
Antreas
http://users.hol.gr/~xpolakis/
From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 6 18:34:52 1997
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Zimaths: An on-line math. magazin
Comments: cc: dinoj@usa.net
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: RO
Zimaths Magazine (of Math. Department, Univ. of Zimbabwe) is available at:
http://www.geocities.com/CapeCanaveral/Lab/3550/zimaths.htm
The first three issues contain interesting math. articles.
Some of them which are math-history related:
Biography of I. Newton.
Herbert Fleischner's Experiences with P. Erdos.
Herbert Fleischner's Solving a problem posed by Paul Erdos.
A History of Pi
Bravo friends !
Antreas
http://users.hol.gr/~xpolakis
From owner-math-history-list@ENTERPRISE.MAA.ORG Sun Aug 10 12:25:52 1997
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Reply-To: Discussion List on the History of Mathematics
From: Boaz Tsaban
Subject: A new paper on the History of Pi
Comments: To: math-history-list@maa.org
Comments: cc: David Garber
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: OR
Shalom all,
There is a new paper on the history of Pi in my homepage
(http://www.cs.biu.ac.il:8080/~tsaban/).
Comments are welcome.
Boaz Tsaban and David Garber
-------------------------------------------------------------------------------
Boaz Tsaban
-------------------------------------------------------------------------------
Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, ISRAEL
Tel: (972-3) 5318407 Fax: (972-3) 535-3325 E-mail: tsaban@macs.biu.ac.il
Home page: http://www.cs.biu.ac.il:8080/~tsaban/
-------------------------------------------------------------------------------
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Reply-To: Discussion List on the History of Mathematics
From: Avinoam Mann
Subject: Re: A new paper on the History of Pi
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Content-Type: text
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Status: RO
>Shalom all,
>There is a new paper on the history of Pi in my homepage
>(http://www.cs.biu.ac.il:8080/~tsaban/).
>
>Comments are welcome.
>Boaz Tsaban and David Garber
-------------------------------------------------------------------------------
>Boaz Tsaban
-------------------------------------------------------------------------------
>Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, ISRAEL
>Tel: (972-3) 5318407 Fax: (972-3) 535-3325 E-mail: tsaban@macs.biu.ac.il
>Home page: http://www.cs.biu.ac.il:8080/~tsaban/
-------------------------------------------------------------------------------
The homepage in question includes two papers on pi. One by Tsaban-Garber,
and one by S.E.G.Belaga. The latter discusses a Biblical approximation to
pi, and this approximation is also mentioned in the Tsaban-Garber paper,
which discusses also Talmudic approximations.
The discussion is based on the verse in Kings I, 7, 23 (repeated in
Chronicles II, 4, 2). Describing a huge vessel ("the molten sea") the verse
says: "...ten cubits from its end to its end circular around...and a string
of thirty cubits will go around it" (my translation). This seems to suggest
a value of 3 for pi.
For me, an interesting fact is that the author of this verse detailed both
the diameter and the circumference, so he did not expect his readers to be
able to work out for themselves the circumference, given the diameter. A
similar phenomenon occurs in Numbers 31, 28-47. There a long list of numbers
is given, and it is stated that they all have to be divided by 500, and the
results of the division are also stated explicitely. However, the results
of division by 50, which also has to be done, are not stated.
Return to the Belaga and Tsaban-Garber papers. They note that in the verse
in Kings the word "string" (literally "line") is written in a slightly
non-standard way, but is supposed to be read in the standard way. Specifically,
the word should be written "kv", but is actually written "kvh". In Chronicles
it is written kv. Now the Hebrew letters have numerical values, like the
Greek ones. Adding these values, "kv" has a value of 106, while "kvh" has
the value 111. Multiplying 3 by 111/106 gives 333/106, a very good
approximation to pi, indeed the 3rd convergent in the continued fraction
for pi, following 3 and 22/7, and preceding 355/113. The authors assume that
this is no coincidence, but that this difference between the written and
oral versions (or between Kings and Chronicles) is meant to give a hint of
this much better value of pi.
This speculation is traced by the authors to papers of M.H.Munk (1962). It
is by now reasonably well known to Israeli mathematicians, I first heard
it from E.Rips years ago. I have two difficulties with it (besides its
speculative nature). First, one has to assume that the system of assigning
numerical values to letters was already in use at the time of writing of
Kings. Belaga certainly assumes that, and Tsaban-Garber point to a similar
system used Sargon II in the 8th century BC. On the other hand, I do not
recall any hint in the Bible for such a system. Numbers in the Bible are
spelled out by words. The other difficulty is that the same difference
between written and oral versions occurs elsewhere. The word "kv" occurs
19 times in the Bible, in 15 verses, and of these, 3 times it is written
"kvh". So one has to explain this spelling in the two other places (Jer.
31, 38 and Zech. 1, 16), they certainly have nothing to do with pi.
Avinoam Mann
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Reply-To: Discussion List on the History of Mathematics
From: Julio Gonzalez Cabillon
Subject: Re: A new paper on the History of Pi
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For those interested in pi I'd like to remark that
Michael Deakin and Hans Lausch have written a fine
article entitled "The Bible and pi", published in
_Function_ (Monash University), vol. 21, part 2,
pp. 59-64, April 1997.
Julio Gonzalez Cabillon
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Reply-To: Discussion List on the History of Mathematics
From: Michael_Deakin
Organization: RMIT Mathematics
Subject: Re: A new paper on the History of Pi
Comments: cc: Hans.Lausch@sci.monash.edu.au
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Just to follow this up. The _Function_ article is a popular version
of a more technical discussion to appear in _The Mathematical
Gazette_.
Off list, Hans & I have been in discussion with Avi Mann and it
appears that there is more (much more) to be said.
One of the sad aspects of Monash's decision to get rid of Hans &
myself is that we had lined up a student to do his honours project on
all this. He had the necessary background and abilities (e.g.
familiarity with Herbrew) to do a superb job. It now seems he will
have to look elsewhere for supervision!
Mike D
> For those interested in pi I'd like to remark that
> Michael Deakin and Hans Lausch have written a fine
> article entitled "The Bible and pi", published in
> _Function_ (Monash University), vol. 21, part 2,
> pp. 59-64, April 1997.
>
> Julio Gonzalez Cabillon
>
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Reply-To: Discussion List on the History of Mathematics
From: "Antreas P. Hatzipolakis"
Subject: Re: Medhava-Gregory-Leibnitz Formula in Kerala
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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David Fowler wrote:
>There's a detailed account of these trigonometric expansions, which gives
>the Sanskrit verses with (fairly modernised and adapted? -- I'd like to
>know) English translations, in
>CT Rajagopal & MS Rangachchari, On an untapped source of medieval Keralese
>mathematics, Archive for History of Exact Sciences 18 (1978) 89-102.
>Also see the article by the same authors:
>On medieval Kerala mathematics, AHES 35 (1986) 91-9.
>
>David Fowler
There is also a whole chapter (CHAPTER VII) on the circle (THE CIRCLE) in:
Dr. T.A. Sarasvati Amma, _Geometry in Ancient and Medieval India_
Motilal Banarsidass, Delhi - Varanasi - Patna 1979.
The Sanskrit verses translated into English (pp. 182-183):
The product of the given sine-chord and the radius, divided by the cosine
chord, is the first result. Then a series of results are to be obtained
from this first result and the succeedings ones by making the square
of the sine-chord the multiplier and the square of the cosine-chord the
divisor. When these are divided in order by the odd numbers 1,3, etc., the
sum of the terms in the even places is to substracted from the sum of the
terms in the odd places to get the arc. The smaller of the sine and
cosine-chords is to be used for this calculation.
BTW, some Indian Pi approximations (also in verses) from this book:
p. 155: 62,832/20,000
p. 157: 2,827,433,388,233/900,000,000,000
p. 158: 104,348/33,215
Greetings from Athens.
Antreas
http://users.hol.gr/~xpolakis/
The first four thousand Decimals of Pi
Pi=
3.14159265358979323846264338327950288419716939937510582097494459230781
64062
8620899862803482534211706798214808651328230664709384460955058223172535
94081284
8111745028410270193852110555964462294895493038196442881097566593344612
84756482
3378678316527120190914564856692346034861045432664821339360726024914127
37245870
0660631558817488152092096282925409171536436789259036001133053054882046
65213841
4695194151160943305727036575959195309218611738193261179310511854807446
23799627
4956735188575272489122793818301194912983367336244065664308602139494639
52247371
9070217986094370277053921717629317675238467481846766940513200056812714
52635608
2778577134275778960917363717872146844090122495343014654958537105079227
96892589
2354201995611212902196086403441815981362977477130996051870721134999999
83729780
4995105973173281609631859502445945534690830264252230825334468503526193
11881710
1000313783875288658753320838142061717766914730359825349042875546873115
95628638
8235378759375195778185778053217122680661300192787661119590921642019893
80952572
0106548586327886593615338182796823030195203530185296899577362259941389
12497217
7528347913151557485724245415069595082953311686172785588907509838175463
74649393
1925506040092770167113900984882401285836160356370766010471018194295559
61989467
6783744944825537977472684710404753464620804668425906949129331367702898
91521047
5216205696602405803815019351125338243003558764024749647326391419927260
42699227
9678235478163600934172164121992458631503028618297455570674983850549458
85869269
9569092721079750930295532116534498720275596023648066549911988183479775
35663698
0742654252786255181841757467289097777279380008164706001614524919217321
72147723
5014144197356854816136115735255213347574184946843852332390739414333454
77624168
6251898356948556209921922218427255025425688767179049460165346680498862
72327917
8608578438382796797668145410095388378636095068006422512520511739298489
60841284
8862694560424196528502221066118630674427862203919494504712371378696095
63643719
1728746776465757396241389086583264599581339047802759009946576407895126
94683983
5259570982582262052248940772671947826848260147699090264013639443745530
50682034
9625245174939965143142980919065925093722169646151570985838741059788595
97729754
9893016175392846813826868386894277415599185592524595395943104997252468
08459872
7364469584865383673622262609912460805124388439045124413654976278079771
56914359
9770012961608944169486855584840635342207222582848864815845602850601684
27394522
6746767889525213852254995466672782398645659611635488623057745649803559
36345681
7432411251507606947945109659609402522887971089314566913686722874894056
01015033
0861792868092087476091782493858900971490967598526136554978189312978482
16829989
4872265880485756401427047755513237964145152374623436454285844479526586
78210511
4135473573952311342716610213596953623144295248493718711014576540359027
99344037
4200731057853906219838744780847848968332144571386875194350643021845319
10484810
0537061468067491927819119793995206141966342875444064374512371819217999
83910159
1956181467514269123974894090718649423196156794520809514655022523160388
19301420
9376213785595663893778708303906979207734672218256259966150142150306803
84477345
4920260541466592520149744285073251866600213243408819071048633173464965
14539057
9626856100550810665879699816357473638405257145910289706414011097120628
04390397
5951567715770042033786993600723055876317635942187312514712053292819182
61861258
6732157919841484882916447060957527069572209175671167229109816909152801
73506712
7485832228718352093539657251210835791513698820914442100675103346711031
41267111
3699086585163983150197016515116851714376576183515565088490998985998238
73455283
3163550764791853589322618548963213293308985706420467525907091548141654
98594616
3718027098199430992448895757128289059232332609729971208443357326548938
23911932
5974636673058360414281388303203824903758985243744170291327656180937734
44030707
4692112019130203303801976211011004492932151608424448596376698389522868
47831235
5265821314495768572624334418930396864262434107732269780280731891544110
10446823 252716201052652272111660396
Exiting, isn't it? War auch auf der Lichtzeile zu bewundern.
Smeg.
_________________________________________________________________
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