Abstract
Archimedes used the perimeter of inscribed and circumscribed regular polygons to obtain lower and upper bounds of π. Starting with two regular hexagons he doubled their sides from 6 to 12, 24, 48, and 96. Using the perimeters of 96 side regular polygons, Archimedes showed that 3+10/71<π<3+1/7 and his method can be realized as a recurrence formula called the Borchardt-Pfaff-Schwab algorithm. Heinrich Dörrie modified this algorithm to produce better approximations to π than these based on Archimedes’ scheme. Lower bounds generated by his modified algorithm are the same as from the method discovered earlier by cardinal Nicolaus Cusanus (XV century), and again re-discovered two hundred years later by Willebrord Snell (XVII century). Knowledge of Taylor series of the functions used in these methods allows to develop new algorithms. Realizing Richardson’s extrapolation, it is possible to increase the accuracy of the constructed methods by eliminating some terms in their series. Two new methods are presented. An approximation of squaring the circle with high accuracy is proposed.
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