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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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** Next:** ii) Calculus Based
**Up:** Matters Computational
** Previous:** Matters Computational