and containing many multiple points.' The locus of HPL will be found almost entirely to coincide with this.

Great results were expected from the assumption of (E + R) as a function of *v*: but the opponents of this theorem having actually succeeded in demonstrating that the *v*-element did not even enter into the function, it appeared
hopeless to obtain any real value of π by this method.

III. Penrhyn's Method.

This was an exhaustive process for extracting the value of π, in a series of terms, by repeated divisions. The series so obtained appeared to be convergent, but the residual quantity was always negative, which of course made the process of extraction impossible.

This theorem was originally derived from a radical series in Arithmetical Progression: let us denote the series itself by A.P., and its sum by (A.P.)S. It was found that the function