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:
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