Archimedes the Numerical Analyst

  • George M. Phillips


Let p N and P N denote half the lengths of the perimeters of the inscribed and circumscribed regular N-gons of the unit circle. Thus , and P 4 = 4. It is geometrically obvious that the sequences {p N } and {P N } are respectively monotonic increasing and monotonic decreasing, with common limit π. This is the basis of Archimedes’ method for approximating to π. (See, for example, Heath [2].) Using elementary geometrical reasoning, Archimedes obtained the following recurrence relation, in which the two sequences remain entwined:


Recurrence Relation Decimal Place Common Limit National Physical Laboratory Numerical Skill 


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  1. 1.
    C. W. Clenshaw, Chebyshev Series for Mathematical Functions, Mathematical Tables, vol. 5, National Physical Laboratory, H.M.S.O., London, 1962.Google Scholar
  2. 2.
    T. L. Heath, The Works of Archimedes, Cambridge University Press, 1897.Google Scholar
  3. 3.
    G. M. Phillips and P. J. Taylor, Theory and Applications of Numerical Analysis, Academic Press, 1973.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • George M. Phillips
    • 1
  1. 1.The Mathematical InstituteUniversity of St. AndrewsSt. AndrewsScotland

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