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#### ii) Calculus Based Methods

Calculus provides the basis for the second technique. The underlying method relies on Gregory's series of 1671

coupled with a formula which allows small x to be used, like

This particular formula is due to Machin and was employed by him to compute 100 digits of in 1706. Variations on this second theme are the basis of all the calculations done until the 1970's including William Shanks' monumental hand-calculation of 527 digits. In the introduction to his book [32], which presents this calculation, Shanks writes:

Towards the close of the year 1850 the Author first formed the design of rectifying the circle upwards of 300 places of decimals. He was fully aware at that time, that the accomplishment of his purpose would add little or nothing to his fame as a Mathematician though it might as a Computer: nor would it be productive of anything in the shape of pecuniary recompense.

Shanks actually attempted to hand-calculate 707 digits but a mistake crept in at the 527th digit. This went unnoticed until 1945, when D. Ferguson, in one of the last ``nondigital'' calculations, computed 530 digits. Even with machine calculations mistakes occur, so most record-setting calculations are done twice --- by sufficiently different methods.

The advent of computers has greatly increased the scope and decreased the toil of such calculations. Metropolis, Reitwieser, and von Neumann computed and analyzed 2037 digits using Machin's formula on ENIAC in 1949. In 1961, Dan Shanks and Wrench calculated 100,000 digits on an IBM 7090 [31]. By 1973, still using Machin-like arctan expansions, the million digit mark was passed by Guilloud and Bouyer on a CDC 7600.

Contents Next: iii) Transformation Methods Up: Matters Computational Previous: i) Archimedes Method