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: