XVI. Dr. Halley's quadrature of the circle improved: being a transformation of his series for that purpose to others which converge by the powers of 80

1. Dr. Halley's method of computing the ratio of the diameter of the circle to its circumference was considered by himself, and other learned mathematicians, as the easiest the problem admits of. And although, in the course of a century, much easier methods have been discovered, still a celebrated mathematician of our own times has expressed an opinion, that no other aliquot part of the circumference of a circle can be so easily computed by means of its tangent as that which was chosen by Dr. Halley, viz. the arch of 30 degrees. This opinion, whether it be just or not, I shall not now inquire; my present design being to show, how the series by which Dr. Halley computed the ratio of the diameter to the circumference of the circle, may be transformed into others of swifter convergency, and which, on account of the successive powers of 1/10 which occur in them, admit of an easy summation. 2. This transformation is obtained by means of different forms in which the fluents of some fluxions may be expressed. To proceed with the greater clearness, I will here set down the fluxion in a general form, and its fluent, in the two series which are used in the following particular instance, and maybe applied with advantage in similar cases.