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Abstract

Practically all computations of the value of ? before 1600 were done using Archimedes' method. As is well-known, this method consists of approximation of the circle with diameter 1 by inscribed and circumscribed regular polygons. Denote the circumference of the inscribed and circumscribed regular N-gon by PN and QN respectively. Then PN < ?< QN and lim N !1 PN = lim N !1 QN = ? A little trigonometry shows us that QN = N tan ? N ; PN = N sin ? N (1) from which the duplication formulae Q2N = 2PNQN PN + QN ; P2N = p PNQ2N (2) follow readily.As is well-known, Archimedes, started with the values Q6 = 2p3 and P6 = 3 and calculated Q12; P12;Q24; : : : ; Q96; P96 consecutively using the duplication formulae (2). See [A]. We also know that Ludolph van Ceulen, around 1600, continued this procedure until he obtained 35 decimal places of ?. To get an idea of the accuracy of the approximation QN to ? we use the Taylor series expansion of tan x.