11 - Series Transformation by Finite Differences

Summary

Preliminary Remarks

Around 1670, James Gregory found a large number of new infinite series, but his methods remain somewhat unclear. From circumstantial evidence and from the form of some of his series, it appears that he was the first mathematician to systematically make use of finite difference interpolation formulas in finding new infinite series. The work of Newton, Gregory, and Leibniz made the method of finite differences almost as important as calculus in the discovery of new infinite series. We observe that interpolation formulas usually deal with finite expressions because in practice the number of interpolating points is finite. By theoretically extending the number of points to infinity, Gregory found the binomial theorem, the Taylor series, and numerous interesting series involving trigonometric functions. Gregory most likely derived these theorems from the Gregory–Newton (or Harriot–Briggs) interpolation formula. Gregory's letter of November 23, 1670, to Collins explicitly mentions these results, and also contains some other series, not direct consequences of the Harriot–Briggs result. Instead, these other series seem to require the Newton–Gauss interpolation formula; one is compelled to conclude that Gregory must have obtained this interpolation formula, though it is not given anywhere in his surviving notes and letters. In a separate enclosure with his letter to Collins, Gregory wrote several formulas, including:

Given an arc whose sine is d, and sine of the double arc is 2d – e, it is required to find another arc which bears to the arc whose sine is d the ratio a to c. […]