19
Views
6
CrossRef citations to date
0
Altmetric
Original Articles

Of Calculations Past and Present: The Archimedean Algorithm

Pages 17-35 | Published online: 05 Feb 2018

References

  • A. Aho, J. Hopcroft, and J. Ullman, the Design and Analysis of Algorithms, Addison-Wesley, Reading, Massachusetts, 1974.
  • D. V. Anderson, A polynomial for π, Math. Gaz., 55 (1971) 67–68.
  • L. Baxter, Are π,e, and √2 equally difficult to compute?, this MONTHLY, 88 (1981) 165–169.
  • P. Beckmann, A History of π, The Golem Press, Boulder, Colorado, 1977.
  • C. W. Borchardt, Sur deux algorithmes analogues à celui de la moyenne arithmético-géométrique de deux éléments, In Memoriam Dominici Chelini, Collect. Math., L. Cremona, Editor, U. Hoepli, Milan, 1881, pp. 206–212.
  • B. C. Carlson, On computing elliptic integrals and functions, J. Math, and Phys., 44 (1965) 36–51.
  • B. C. Carlson. Hidden symmetries of special functions, SIAM Rev., 12 (1970) 332–345.
  • B. C. Carlson. Algorithms involving arithmetic and geometric means, this Monthly, 78 (1971) 496–505.
  • B. C. Carlson. An algorithm for computing logarithms and arctangents, Math. Comp., 26 (1972) 543–549.
  • B. C. Carlson. Special Functions of Applied Mathematics, Academic Press, 1977.
  • B. C. Carlson. Computing elliptic integrals by duplication, Numer. Math., 33 (1979) 1–16.
  • K. Y. Choong, D. E. Daykin, and C. R. Rathbone, Rational approximations to π Math. Comp., 25 (1971) 387–392.
  • K. Y. Choong, D. E. Daykin, and C. R. Rathbone, Regular continued fractions for π and γ, UMT 23, Math. Comp., 25 (1971) 403.
  • Collected Algorithms from ACM, ACM Algorithms Distribution Service, IMSL, Inc., 7500 Bellaire Blvd., Houston, TX 77036.
  • P. J. Davis, The Lore of Large Numbers, New Mathematical Library No. 6, Mathematical Association of America, Washington, D.C., 1961.
  • M. Dehn and E. D. Hellinger, Certain mathematical achievements of James Gregory, this Monthly, 50 (1943) 149–163.
  • J. Ercolano, A diagram for a square root algorithm, Math. Gaz., 59 (1975) 189–190.
  • H. Eves, A Survey of Geometry, Allyn and Bacon, Boston, 1972.
  • L. Fox and L. Hayes, A further helping of π Math. Gaz., 59 (1975) 38–40.
  • J. S. Frame, Continued fractions and matrices, this Monthly, 56 (1949) 98–103.
  • M. Gardner, New Mathematical Diversions from Scientific American, Simon and Schuster, 1966, pp. 91–102.
  • M. Gardner. Mathematical Games, Scientific American (Sept. 1979) 22ff.
  • M. Gardner. Mathematical Games, Scientific American (Nov. 1979) 20ff.
  • C. F. Gauss, Werke, vol. 3, Teubner, Leipzig, 1876.
  • M. Gardner. Werke, vol. 10, part 1, Teubner, Leipzig, 1917.
  • J. R. Goggins, Formula for π/4, Math. Gaz., 57 (1973) 134.
  • C. Goldsmith, Calculation of in 2 and π, Math. Gaz., 55 (1971) 434–436.
  • R. W. Gosper, Jr., Table of the Simple Continued Fraction for π and the Derived Decimal Approximation, Artificial Intelligence Laboratory, Stanford University, Oct. 1975, deposited in UMT file.
  • J. Guilloud and M. Bouyer, Un million de decimales de π. Commissariat à l'Energie Atomique, Paris, 1974.
  • T. L. Heath, The Works of Archimedes, Cambridge University Press, 1897.
  • T. L. Heath. A History of Greek Mathematics, Oxford University Press, 1921.
  • P. Henrici, Elements of Numerical Analysis, Wiley, New York, 1964.
  • E. W. Hobson, “Squaring the Circle,” a History of the Problem, Cambridge, 1913; reprinted by Chelsea, New York, 1953.
  • J. -L. Lagrange, Oeuvres, vol. 2, Gauthier-Villars, Paris, 1868.
  • E. Maor, The history of π on the pocket calculator, J. College Science Teaching, Nov. 1976, 97–99.
  • A. I. Markushevich, The Remarkable Sine Functions, American Elsevier, New York, 1966.
  • K. O. May, Index of the American Mathematical Monthly, vols. 1–80 (1894–1973), The Mathematical Association of America, 1977.
  • G. Miel, An algorithm for the calculation of π, this Monthly, 86 (1979) 694–697.
  • G. Miel. Evaluation of complex logarithms and related functions with interval arithmetic, in Interval Mathematics 1980, K. Nickel, Editor, Academic Press, New York, 1980, pp. 407–415.
  • G. Miel. Evaluation of complex logarithms and related functions, SIAM J. Numer. Anal., 18 (1981) 744–750.
  • L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, 1960.
  • A. J. Moakes, The calculation of π, Math. Gaz., 54 (1970) 261–264.
  • A. J. Moakes. A further note on machine computation for π, Math. Gaz., 55 (1971) 306–310.
  • R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1966.
  • R. E. Moore. Methods and Applications of Interval Analysis, Society for Industrial and Applied Mathematics, Philadelphia, 1979.
  • K. Nickel, Editor, Interval Mathematics 1980, Academic Press, New York, 1980.
  • T. S. Nanjundiah, On Huygens' approximation to π, Math. Magazine, 44 (1971) 221–223.
  • C. D. Olds, Continued Fractions, New Mathematical Library No. 9. Mathematical Association of America, Washington, D.C., 1963.
  • G. M. Phillips, Archimedes the numerical analyst, this Monthly, 88 (1981) 165–169.
  • C. W. Puritz, An elementary method of calculating π, Math. Gaz., 58 (1974) 102–108.
  • E. Salamin, Computation of π using arithmetic-geometric mean, Math. Comp., 30 (1976) 565–570.
  • H. C. Schepler, The chronology of pi, Math. Magazine, 23 (1950) 165–170, 216–228,279–283.
  • L. Schlesinger, Uber Gauss' Jugendarbeiten zum arithmetisch-geometrischen Mittel, Jber. Deutsch. Math.- Verein., 20(1911) 396–403.
  • D. Shanks. Review 15, Math. Comp., 31 (1977) 1044.
  • D. Shanks. Quartic approximations for π, Abstract No. 80T-A183, Abstracts Amer. Math. Soc., 1 (1980) 558.
  • D. Shanks and J. W. Wrench, Jr., Calculation of π to 100,000 decimals, Math. Comp., 16 (1962) 76–99.
  • C. L. Siegel, Topics in Complex Function Theory, vol. 1, Wiley-Interscience, New York, 1969.
  • H. C. Thacher, Jr., Iterated square root expansions for the inverse cosine and inverse hyperbolic cosine, Math. Comp., 15 (1961) 399–403.
  • J. Todd, A problem on arc tangent relations, this Monthly, 56 (1949) 517–528.
  • J. Todd. The lemniscate constants, Commun. ACM, 18 (1975) 14–19. 462.
  • J. Todd. Basic Numerical Mathematics, Academic Press, New York. 1981.
  • H. W. Turnbull, Editor, James Gregory Tercentenary Memorial Volume, published for the Royal Society of Edinburgh, London, 1939.
  • G. N. Watson, The marquis and the land-agent; a tale of the eighteenth century, Math. Gaz., 17 (1933) 5–17.
  • D. T. Whiteside, Patterns of mathematical thought in the latter seventeenth century. Archive for History of Exact Sciences, vol.1, 1960–1962.
  • J. W. Wrench, Jr., The evolution of extended decimal approximations to π. Math. Teacher, 53 (1960) 644–650.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.