A History of Pi
Page 18
71. Lodge.
72. Andrade.
73. Newton, Treatise on the Method of Fluctions and Infinite Series, London, 1737, pp. 129-131. See Whiteside, vol. I, pp. 100-101.
74. Smith (1929), p. 346.
75. Leonard Euler, Perepiska (Leonhard Euler, Correspondence), Nauka, Leningrad, 1967.
76. Freyman.
77. Tremble, enemies of France,
Kings drunk with blood and conceit!
The sovereign people advances,
Tyrants, descend into the grave!
78. Hammersley and Hanscomb.
79. Beckmann, P., Elements of Applied Probability Theory, Harcourt, Brace & World, New York, 1968; pp. 37-38.
80. On the Librascope of the Electr. Engrg. Dept., Univ. of Colorado.
81. This line of reasoning works with sufficient, not necessary conditions. Actually, the final equation will be quadratic, since it can be shown that each new step results in a quadratic equation with coefficients that are either rational or square roots. For details, see Hobson, pp. 47-51.
82. From A Concise History of Mathematics by Dirk J. Struik, Dover Publications, Inc., New York. Reproduced by permission of the Publisher.
83. Coxeter, p. 53.
84. Tropfke, p. 217
85. Lindemann, F: “Über die Zahl π,” Mathematische Annalen, vol. 20, pp. 212-225 (1882); also “Über die Ludolphsche Zahl,” Berichte der Berliner Akademie, vol. 2, pp. 679-682 (1882).
86. Weierstrass, K.W.: “Zu Lindemanns Abhandlung ‘Über die Ludolphsche Zahl’,” Berichte der Berliner Akademie, vol. 5, pp.1067-1085 (1885).
87. Hobson, pp. 53-57.
88. Schubert, p. 143.
89. The jacket of the handwritten copy is thus entitled, and with the same capitalization.
90. W.E. Eddington, “House Bill No. 246, Indiana State Legislature, 1897,” Proc. of the Indiana Academy of Sciences, vol. 45, pp. 206-210 (1935).
91. See bibliography under Parker.
92. Before the advent of the computer, that is. The natural logarithm and square root of 2 are now known to 3,683 and 1,000,082 decimals, respectively. See note 95.
93. Wrench (1960).
94. Shanks and Wrench (1962).
95. This paragraph (and also note 92) is based on a private communication by Dr. J.W. Wrench, Jr., which is gratefully acknowledged.
96. Funk and Wagnall’s Standard College Dictionary, Harcourt, Brace & World, New York, 1963.
97. J.R. Slagle, Artificial Intelligence: The Heuristic Programming Approach, McGraw-Hill, New York, 1971.
BIBLIOGRAPHY
ANDRADE, E.N. da C., Sir Isaac Newton, Doubleday, New York, 1958.
BALL, W.W.R., A Short Account of the History of Mathematics, Macmillan, London, 1888. Reprinted by Dover Publications, 1960.
BARNES, H.E., An Intellectual Cultural History of the World. In 3 vols., 3rd ed., Dover Publications, New York, 1965.
BELL, E.T., Men of Mathematics, Simon and Schuster, New York, 1937.
— — —, The Development of Mathematics, McGraw-Hill, New York, 1940.
BOYER, C.B., The History of the Calculus and Its Conceptual Development, 2nd ed., Hafner, New York, 1949; reprinted by Dover Publications, 1959.
— — —, A History of Mathematics, Wiley, New York, 1968.
BRANDON, W., The American Heritage Book of Indians, American Heritage Publ. Co., New York, 1961; reprinted by Dell Publ. Co., New York, 1964.
BUTKEVICH, A.V., GANSHIN, V.N., KHRENOV, L.S., Vremya i kalendar’ (Time and the Calendar), Vysshaya shkola, Moscow, 1961.
CANTOR, M., Vorlesungen über Geschichte der Mathematik, Teubner, Leipzig; vol. 1, 1894, vol.2, 1900; vol. 3, 1901; vol. 4, 1908.
COLLIER, J., Indians of the Americas, New American Library (Mentor Books), New York, 1948.
COOLIDGE, J.L., The Mathematics of Great Amateurs, Clarendon Press, Oxford, 1949; reprinted by Dover Publications, 1963.
— — —, A History of the Conic Sections and Quadric Surfaces, University Press, Oxford, 1945; reprinted by Dover Publications, 1968.
COXETER, H.S.M., Introduction to Geometry, Wiley, New York, 1969.
DANTZIG, T., Number: The Language of Science, London, 1940; Macmillan, New York, 1943.
DE CAMP, L.S., The Ancient Engineers, Doubleday, Garden City, N.J., 1963.
DEPMAN, I.Y., Rasskazy o matematike (Tales about Mathematics), Gosdetizdat, Leningrad, 1954.
EDINGTON, E., House Bill No. 246, Indiana State Legislature, 1897. Proceedings of the Indiana Academy of Sciences, vol. 45, pp. 206-210 (1935).
FREYMAN, L.S., Tvortsy vysshey matematiki (Creators of Higher Mathematics), Nauka, Moscow, 1968.
GEDDIE, W., and GEDDIE, L. (Ed.), Chambers Biographical Dictionary, 2nd ed., Chambers, London, 1938.
GREENBLATT, M.H., The “legal” value of π and some related mathematical anomalies, American Scientist, vol. 53, pp. 427A-432A, Dec. 1965.
HAMMERSLEY, J.M., and HANDSCOMB, D.C., Monte Carlo Methods, Wiley, New York, 1964.
HEATH, T.L., The Works of Archimedes, Cambridge University Press, Cambridge, 1897 and 1912. Reprinted by Dover Publications, 1953.
— — —, The Thirteen Books of Euclid’s Elements, 3 vols., Cambridge University Press, Cambridge, 1908. Reprinted by Dover Publications, 1956.
— — —, A Manual of Greek Mathematics, Oxford University Press, Oxford, 1931. Reprinted by Dover Publications, New York, 1963.
HEISEL, C.T., FABER, C.T., Behold! The Grand Problem, The Circle Squared Beyond Refutation, No Longer Unsolved, (Printed by) S.J. Monck, Cleveland, Ohio, 1931.
HOBSON, E.W., Squaring the Circle. Oxford University Press, 1913. Reprinted by Chelsea Publ. Co., New York (no date given).
HOGBEN, L., Mathematics for the Million, W.W. Norton & Co., New York, 1937; reprinted by Pocket Books, New York, 1965.
— — —, Mathematics in the making, Rathbone Books, London, 1960.
KOLMAN, E., Istoriya matematiki v drevnosti (History of Mathematics in Antiquity), Fizmatgizdat, Moscow, 1961.
LIETZMANN, W., Methodik des mathematischen Unterrichtes, vol. 2, Quelle & Meyer, Leipzig, 1923.
— — —, Altes und Neues vom Kreis, Teubner, Leipzig, 1935.
LINDEMANN, F., Über die Ludolphsche Zahl, Berichte der Berliner Akademie, vol. 2, pp. 679-682 (1882).
— — —, Über die Zahl π, Mathematische Annalen, vol. 20, pp. 221-225 (1882).
LODGE, Sir Oliver, Pioneers of Science, Macmillan & Co., London, 1893. Reprinted by Dover Publications, New York, 1960.
MAISTROV, L.E., Teoriya veroyatnostey — istoricheski ocherk (Probability Theory — An Historical Outline), Nauka, Moscow, 1967.
MIDONICK, H.O., (Ed.), A Treasury of Mathematics, Philosophical Library, New York, 1965.
MORLEY, S.G., The Ancient Maya, Stanford University Press, 1947.
NEEDHAM, J., Science and Civilization in China, vol. III, Cambridge University Press, Cambridge, 1959.
NEUGEBAUER, O., The Exact Sciences in Antiquity, 2nd ed., Brown University Press, 1957. Reprinted by Dover Publ., New York, 1969.
NEWMAN, J.R., The World of Mathematics, 4 vols., Simon & Schuster, N.Y., 1969.
ORE, O., Cardano the Gambling Scholar, Princeton University Press, Princeton, N.J., 1953. Reprinted by Dover Publications, New York, 1965.
PARKER, J.A., Quadrature of the Circle, John Wiley & Son, New York, 1874.
RAPPORT, S., WRIGHT, H., Mathematics, New York University Press, New York, 1963. Reprinted by Washington Square Press, New York, 1964.
RUDIO, F., Archimedes, Huygens, Lambert, Legendre. Vier Abhandlungen über die Kreismessung, Leipzig, 1892.
SCHUBERT, H., The Squaring of the Circle. Contained in Mathematical Essays and Recreations, Translation from German, Open Court Publishing Company, Chicago, 1899.
SIMON, M., Uber die Entwicklung der Elementargeometrie im XIX. Jahrhundert, Leipzig, 1906.
SHANKS, D., WRENCH, J.W., Jr., Calculation of π to 100,000 decimal
s, Mathematics of Computation, vol. 16, pp. 76-99, January 1962.
SMITH, D.E., A Source Book in Mathematics, 1929. Reprinted by Dover Publ., New York, 1959.
SMITH, D.E., and MIKAMI, Y., A History of Japanese Mathematics, Open Court Publishing Co., Chicago, 1914.
STRUIK, D.J., A Concise History of Mathematics, 3rd rev. ed., Dover Publications, New York, 1967.
— — —, A Source Book in Mathematics, 1200-1800. Harvard University Press, Cambridge, Mass., 1969
TROPFKE, J., Geschichte der Elementarmathematik. Vierter Band: Ebene Geometrie. Vereinigung wissenschaftlicher Verleger, Berlin-Leipzig, 1923.
TURNBULL, H.W., The Great Mathematicians, Simon & Schuster, New York, 1962.
WEIERSTRASS, K.W., Ζu Lindemanns Abhandlung “Uber die Ludolphsche Zahl,” Berichte der Berliner Akademie, vol. 5, pp. 1067-1085 (1885).
WHITESIDE, D.T. (Ed.), The Mathematical Works of Isaac Newton, Johnson Reprint Corporation, New York, vol.1, 1964; vol. 2, 1967.
WILDER, R.L., evolution of Mathematical Concepts, Wiley, New York, 1968.
WILLIAMS, T.I., (Ed.), A Biographical Dictionary of Scientists, Wiley-Interscience, New York, 1969.
WRENCH, J.W., Jr., The Evolution of Extended Decimal Approximations to π, The Mathematics Teacher, vol. 53, pp. 644-650, December 1960.
ZVORYKIN, A.A. (Ed.), Biograficheski slovar’ deyateley estestvoznaniya i tekhniki (Biographical Dictionary of Scientists and Engineers), Izd. Bolshoy Sovetskoy Entsiklopedii, Moscow, vol 1, 1958; vol. 2, 1959.
CHRONOLOGICAL TABLE
ca. 2000 B.C.
Babylonians use π = 3 1/8
ca. 2000 B.C.
Egyptians use π = (16/9)2 = 3.1605
12th century B.C.
Chinese use π = 3
ca. 550 B.C.
I Kings vii, 23 implies π = 3
ca. 440 B.C.
Hippocrates of Chios squares the lune
ca. 434 B.C.
Anaxagoras attempts to square the circle
ca. 430 B.C.
Antiphon enunciates the principle of exhaustion
ca. 420 B.C.
Hippias discovers the quadratrix
ca. 335 B.C.
Dinostratos uses the quadratrix to square the circle
3rd century B.C.
Archimedes establishes 3 10/71 < π < 3 1/7 and π ≈ 211875 : 67441 = 3.14163
Archimedes uses the Archimedean Spiral to rectify the circle
ca. 225 B.C.
Appolonius improves the Archimedean value, unknown to what extent
2nd century A.D.
Ptolemy uses π = 377/120 = 3.14166 …
3rd century A.D.
Chung Hing uses π = √10 = 3.16 …
Wang Fau uses π = 142/45 = 3.1555 …
263 A.D.
Liu Hui uses π = 157/50 = 3.14
5th century
Tsu Chung-Chi establishes 3.1415926 < π < 3.1415927
ca. 500
Aryabhatta uses π = 62832/2000 = 3.1416
6th century
Brahmagupta uses π = √10 = 3.16 …
1220
Leonardo of Pisa (Fibonacci) finds π ≈ 3.141818
before 1436
Al-Kashi of Samarkand calculates π to 14 places
1450
Cusanus finds approximation for length of arc
1573
Valentinus Otho finds π ≈ 355/113 = 3.1415929
1583
Simon Duchesne finds π = (39/22)2 = 3.14256 …
1593
François Viète finds π as an infinite irrational product
1593
Adriaen van Roomen finds π to 15 decimal places
1596
Ludolph van Ceulen calculates π to 32 places, later to 35 decimal places
1621
Snellius refines the Archimedean method
1654
Huygens proves the validity of Snellius’ refinement
1655
Wallis finds an infinite rational product for π; Brouncker converts it to a continued fraction
1665–1666
Newton discovers the calculus and calculates π to at least 16 decimal places; not published until 1737 (posthumously)
1671
Gregory discovers the arctangent series
1674
Leibniz discovers the arctangent series for π
1705
Sharp calculates π to 72 decimal places
1706
Machin calculates π to 100 places
1706
Jones uses the symbol π for the circle ratio
1719
De Lagny calculates π to 127 places
1748
Euler publishes the Introductio in analysin infinitorum, containing Euler’s Theorem and many series for π and π2
1755
Euler derives a very rapidly converging arctangent series
1766
Lambert proves the irrationality of π
1775
Euler suggests that π is transcendental
1794
Legendre proves the irrationality of π and π2
1794
Vega calculates π to 140 decimal places
1840
Liouville proves the existence of transcendental numbers
1844
Strassnitzky and Dase calculate π to 200 places
1855
Richter calculates π to 500 decimal places
1873
Hermite proves the transcendence of e
1873-74
Shanks calculates π to 707 decimal places
1882
Lindemann proves the transcendence of π
1945
Ferguson finds Shanks’ calculation erroneous from the 527th place onward
1946
Ferguson publishes 620 decimal places
1947
Ferguson calculates 808 places using a desk calculator
1949
ENIAC is programmed to compute 2,037 decimals
1954-1955
NORC is programmed to compute 3,089 decimals
1957
Pegasus computer (London) computes 7,480 places
1959
IBM 704 (Paris) computes 16,167 decimal places
1961
Shanks and Wrench improve computer program for π, use IBM 7090 (New York) to compute 100,000 decimal places
1966
IBM 7030 (Paris) computes 250,000 decimal places
1967
CDC 6600 (Paris) computes 500,000 decimal places
Index
The index that appeared in the print version of this title does not match the pages in your eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.
Ahmes papyrus
Albert von Sachsen
Alexander the Great
Alexandria, University of
Algebraic numbers
Al-Ghazzali
Al-Khowarizmi
American Indians
Amr ibn-al-As
Anaxagoras of Cazomenae
Antiphon
Anthoniszoon, Adriaan
Apollonius of Perga
Arab mathematics
Arabic numerals
Aristotle
Archimedean polygons
Archimedean Spiral
Archimedes of Syracuse
Arsinoe of Egypt
Aryabhata
Aurelianus
Axioms of geometry
Babylonian value of π
Bar-Kokba
Barrow, Isaac
BASIC
Baskhara
Bernoulli, Daniel I
— —, Jacques I
— —, Jean I
— —, Jean II
— —, Nicolaus I
— —, Nicolaus III
Biblical value of π
Bidder, George Parker
Boethiu
s
Bolyai, János
Brachistochrone
Brahe, Tycho
Brahmagupta
Brouncker, William
Bruno, Giordano
Buffon, Comte de
Bürgi, Jobst
Buteo, Johannes
Buxton, Jedediah
Caesar, Julius
Calculating prodigies
Calendar
Cantor, Georg
Cardano, Gerolamo
Carnot, Lazare
Cartesius, see Descartes
Catherine II of Russia
Cato, Marcus Porcius
Cavalieri, Bonaventura
Cicero
Cleopatra
Cochleoid
Columbus, Christopher
Computer evaluation of π
Conon of Alexandria
Copernicus, Nicolas
Continued fractions
Coss, cossike
Cusanus, Cardinal