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#### i) Archimedes Method

The first technique due to Archimedes of Syracuse (287--212 B.C.) is, recursively, to calculate the length of circumscribed and inscribed regular -gons about a circle of diameter 1. Call these quantities and . respectively. Then , and, as Gauss's teacher Pfaff discovered in 1800,

Archimedes, with n=4, obtained

While hardly better than estimates one could get with a ruler, this is the first method that can be used to generate an arbitrary number of digits, and to a nonnumerical mathematician perhaps the problem ends here. Variations on this theme provided the basis for virtually all calculations of for the next 1800 years, culminating with a 34 digit calculation due to Ludolph van Ceulen (1540--1610). This demands polygons with about sides and so is extraordinarily time consuming.

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