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 p3 = 3V5 /2, P3 = 315, p4 = 2V2, and P4 = 4. It is geometrically obvious that the sequences (p N ) and (P N ) are respectively monotonic increasing and monotonic decreasing, with common limit rr. This is the basis of Archimedes’ method for approximating to or. (See, for example, Heath (2J.) Using elementary geometrical reasoning, Archimedes obtained the following recurrence relation, in which the two sequences remain entwined:
KeywordsDecimal Place Irrational Number Common Limit Geometrical Reasoning Chebyshev Series
Unable to display preview. Download preview PDF.
- 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.T. L. Heath, The Works of Archimedes, Cambridge University Press, 1897.Google Scholar
- 3.G. M. Phillips and P. J. Taylor, Theory and Applications of Numerical Analysis, Academic Press, 1973.Google Scholar