Mr. P. Nicholson on derivative Analysis. 4-35 



Upon my first examination of the figurate method, I per- 

 ceived that the same process which served to transform equa- 

 tions by diminishing their root would also extract their roots, 

 if each root were continually duninished by a figure at a 

 time. It was therefore evident to me, that whether Sir Isaac 

 Newton's method or my own were applied, the object would 

 be accomplished by either ; but as the theorem which I had 

 already discovered was also a theorem for the summation oi 

 figurate numbers, I immediately reduced Newton's method 

 to my own formula, and from this source alone I derived the 

 rule which I have applied to the non-figurate method in my 

 own publications. 



This general method of transforming fimctions first oc- 

 curred to me when writing my Introduction to the Method ot 

 Increments, pubhshed in 1817 by Davies and Dickson. In 

 pao-e 126 of this Introduction, my general method of trans- 

 fonnation is hinted at ; and in the same page an example of 

 its use is given exactly in the same form as it was published 

 in my Combinatorial Essays in 1818, the year following, and 

 as it is now published in the Philosophical Magazine of No- 

 vember 1823. 



When I first published the non-figurate method m my 

 treatise on Involution and Evolution, as also in the extraction 

 of roots in my analytical essays, I was induced, in order to 

 abrid^^e die operation, to omit the first column, which consisted 

 of repetitions of the figures of the root; and also the columns 

 of the multiples of the successive sums ; but the nunierous ob- 

 jections that were made to multiplying and adding in ore Ime, 

 which was rendered necessary by this method, have deter- 

 mined me to give the operation in full, as I had originally 

 published it in my Method of Increments, and more particu- 

 larly in my Essay on Binomial Factors, published along with 

 the Combinatorial Analysis. 



The application of my method of transformation to die ex- 

 traction of roots I have considered only as an improvement 

 on Mr. Holdred's figurate mode ; and although the principle 

 may be seen in my Introduction to the Method of Increments 

 and in the Combinatorial Analysis, which were both published 

 before Mr. Holdred communicated his general method of ex- 

 tracting the roots of equations ; yet it is doubtful whether my 

 method of transformation would ever have been applied to 

 the extraction of die roots of ecjuations, had I not previously 

 seen Mr. Holdred's figurate mode*. 



# 1 make this admission, because a similar iinprovenicnt occurred to 

 Mr. Holdred himseli' shortly alter he had communicated tiic figurate me- 

 thod to me. 



3 I 2 lo 



