A History of Pi

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A History of Pi Page 18

by Petr Beckmann


  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

 

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