An Algorithm for the Construction of Arctangent Relations

Abstract

An important source of solutions for the quadrature of the circle has been the use of Gregory’s series

$$ {\text{arctan}}{\chi ^{{\text{ - 1}}}}{\text{ = }}{\chi ^{{\text{ - 1}}}}{\text{ - }}\frac{{\text{1}}}{{\text{3}}}{\chi ^{{\text{ - 3}}}}{\text{ + }}\frac{{\text{1}}}{{\text{3}}}{\chi ^{{\text{ - 5}}}}{\text{ - }}... $$

in certain identities, for instance in those expressing 1/4 π as a sum of nrctangents, of which the following, due to Euler, is typical:

$$\frac{1}{4}\pi = 4\arctan \frac{1}{5} - \arctan \frac{1}{{70}} + \arctan \frac{1}{{99}}. $$

Received 16 May. 1946; read 20 June. 1946.