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

  • Ranjan Roy
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Keywords

Infinite Series Indian Work Fifteenth Century Successive Derivative English Mathematician 
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References

  1. 1.
    For further information about Leibniz’s mathematical development, the reader may consult: J. E. Hofmann, Leibniz in Paris 1672–1676 (Cambridge: The Cambridge University Press. 1974) and its review by A. Weil. Collected Papers Vol. 3 (New York: Springer-Verlag. 1979). An English translation of Leibniz’s own account, Historia et origo calculi differentialis, can be found in J. M. Child, The Early Mathematical Manuscripts of Leibniz (Chicago: Open Court, 1920). An easily available synopsis of Leibniz’s work in calculus is given in C. H. Edwards. Jr., The Historical Development of the Calculus (New York: Springer-Verlag. 1979).Google Scholar
  2. 2.
    The Early Mathematical Manuscripts, p. 215. Bonaventura Cavalieri (1598–1947) published his Geometria Indivisibilibus in 1635. This book was very influential in the development of calculus. Cavalieri’s work indicated that when n is a positive integer. Blaise Pascal (1623–1662) made important and fundamental contributions to projective geometry, probability theory and the development of calculus. The work to which Leibniz refers was published in 1658 and contains the first statement and proof of. This proof is presented in D. J. Struik’s A Source Book in Mathematics 1200–1800 (Cambridge: Harvard University Press, 1969), p. 239. Paul Guldin (1577–1643), a Swiss mathematician of considerable note, contributed to the development of calculus, and his methods were generally more rigorous than those of Cavalieri.Google Scholar
  3. 3.
    See H. W. Turnbull (ed.). The Correspondence of Isaac Newton (Cambridge: The University Press, 1960), Vol. 2, p. 130.Google Scholar
  4. 4.
    Peter Beckmann has persuasively argued that Gregory must have known the series for π/4 as well. See Beckmann’s A History of Pi (Boulder, Colorado: The Golem Press, 1977), p. 133.Google Scholar
  5. 5.
    The reader might find it of interest to consult: H. W. Turnbull (ed.), James Gregory Tercentenary Memorial Volume (London: C. Bell, 1939). This volume contains Gregory’s scientific correspondence with John Collins and a discussion of the former’s life and work.Google Scholar
  6. 7.
    James Gregory, p. 89.Google Scholar
  7. 8.
    See The Correspondence of Isaac Newton, Vol. 1, p. 52, note 1.Google Scholar
  8. 9.
    See D. T. Whiteside. “Henry Briggs: The Binomial theorem Anticipated,” The Mathematical Gazette, Vol. 15, (1962), p. 9. Whiteside shows how the expansion of (1 + x)1/2 arose out of Brigg’s work on logarithms.Google Scholar
  9. 10.
    Jantes Gregory, p. 92.Google Scholar
  10. 11.
    In their review of the Gregory Memorial Volume, M. Dehn and E. Hellinger explain how the binomial expansion comes out of the interpolation formula. See The American Mathematical Monthly, Vol. 50, (1943), p. 149.Google Scholar
  11. 12.
    James Gregory, p. 148.Google Scholar
  12. l3 Ibid., p. 170.Google Scholar
  13. 14.
    It should be mentioned that Newton himself discovered the Taylor series around 1691. See D. T. Whiteside (ed.). The Mathematical Papers of Isaac Newton, Vol. VII (Cambridge: The Cambridge University Press, 1976), p. 19. In fact, Taylor was anticipated by at least five mathematicians. However, the Taylor series is not unjustly named after Brook Taylor who published it in 1715. He saw the importance of the result and derived several interesting consequences. For a discussion of these matters see: L. Feigenbaum, “Brook Taylor and the Method of Increments,” Archive for History of Exact Sciences, Vol. 34, (1985), pp. 1–140.Google Scholar
  14. 15.
    James Gregory, p. 352.Google Scholar
  15. 16.
    Rajagopal’s work may be found in the following papers: (with M. S. Rangachari) “On an Untapped Source of Medieval Keralese Mathematics.” Archive for History of Exact Sciences, Vol. 18, (1977), pp. 89–102; “On Medieval Kerala Mathematics.” Archive for History of Exact Sciences, Vol. 35, (1986), pp. 91–99. These papers give the Sanskrit verses of the Tantrasangrahavakhya which describe the series for the aretan, sine and cosine. An English translation and commentary is also provided. A commentary on the proof of aretan series given in the Yuktibhasa is available in the two papers: “A Neglected Chapter of Hindu Mathematics,” Scripta Mathematics, Vol. 15, (1949), pp. 201–209; “On the Hindu Proof of Gregory’s Series,” Ibid. Vol. 17, (1951), pp. 65–74. A commentary on the Yuktibhasa’s proof of the sine and cosine series is contained in C. Rajagopal and A. Venkataraman, “The sine and cosine power series in Hindu mathematics.” Journal of the Royal Asiatic Society of Bengal, Science, Vol. 15, (1949), pp. 1–13.Google Scholar
  16. 17.
    See J. E. Hofmann, “Über eine alt indische Berechnung von π und ihre allgemeine Bedeutung,” Mathematische-Physikalische Semester Berichte, Bd. 3, II. 3/4, Hamburg (1953). See also D. T. Whiteside, “Patterns of Mathematical Thought in the later Seventeenth Century,” Archive for History of Exact Sciences, Vol. I, 1960–1962). pp. 179–388. For a discussion of medieval Indian mathematicians and the Tantrasangraha in particular, one might consult: A. P. Jushkevich, Geschichte der Mathematik in Mittelalter (German translation Lepzig, 1964, of the Russian original, Moscow, 1961).Google Scholar
  17. 18.
    See The Historical Development of the Calculus (mentioned in footnote 1), p. 84. Alhazen is the latinized form of the name Ibn Al-Haytham (c. 965–1039).Google Scholar
  18. 19.
    See Geschichte der Mathematik, p. 169.Google Scholar
  19. 20.
    These observations concerning the continued fraction expansion of f(n) and its relation to the Indian work and that of Brouncker, and concerning the decimal places in f(20), are due to D. T. Whiteside. See “On Medieval Kerala Mathematics” of footnote 13.Google Scholar
  20. 21.
    See “Patterns of Mathematical Thought in the later Seventeenth Century” of footnote 17. See also A. Weil, “History of Mathematics: Why and How” in Collected Papers, Vol. 3 (New York: Springer-Verlag, 1979), p. 435.Google Scholar
  21. 22.
    See D. E. Smith and Y. Mikami, A History of Japanese Mathematics (Chicago: Open Court, 1914). This series was also obtained by the French missionary Pierre Jartoux (1670–1720) in 1720. He worked in China and was in correspondence with Leibniz, but the present opinion is that Takebe’s discovery was independent. Leonhard Euler (1707–1783) rediscovered the same series in 1737. A simple evaluation of it can be given using Clausen’s formula for the square of a hypergeometric series.Google Scholar

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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Ranjan Roy
    • 1
  1. 1.Beloit CollegeBeloitUSA

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