192 The Rev. F. H. Hummel on 



of subtrahend? is, in fact. 6 x the series of ' triangular ' num- 

 bers, 1. 3, 6, 10, &c. Here is the cube root of 729 found by 



this process. 



n 3 = 729 

 _6 



723 



16 



705 

 36 



669 

 60 



609 

 90 



519 

 126 



393 



168 



225 



216 



n= __9 



<: It is clear that similar rules may be framed for finding the 

 roots of higher powers:" 



Considering this method of evolution to be of great import- 

 ance, and believing it (under correction) to be wholly novel, 

 I must at once assign the credit of its invention to the real 

 author. To the foregoing portion of this paper I have contri- 

 buted nothing but the wording : the matter of it was commu- 

 nicated to me by my esteemed friend and neighbour the Rev. 

 TV. B. Cole, of Shanklin. In the belief that the method is 

 altogether new to the world (though its author seems to have 

 retained it in petto for some years). I have ventured to give it 

 the title of " Cole's Method of Evolution : " and with his per- 

 mission I now present it to the public, with a few remarks of 

 my own on its more extended application. 



The first point to which I shall refer is the principle on 

 which the method is based, so that we may obtain a general 

 formula whence to frame rules of evolution for the fourth, 

 fifth, or any higher roots. TVe shall then be in a position to 

 decide on the extent of its application, and to frame tables for 

 its employment. 



Having given a known power of an unknown number (call 

 it //"'). we subtract successively the terms of a series, and find 

 for a last remainder the required root n. Obviouslv this series 

 is the one whose sum to n terms =n m — n ; and obviouslv, 



