36 Air. Nicholson^s Answer 



an unknown quantity in a transformed equation, will be the proper 

 divisor for ascertaining the next figure of the root ; but certainly 

 nothing can be more absurd or destitute of principle, and this one 

 instance is sufficient to show that he never had any clear ideas of 

 treating the subject. 



The manner in which I first formed my ideas on the extraction 

 of the roots of equations, will be found in the Introduction to the 

 Essay on Involution and Evolution, I shall therefore not take up 

 the reader's time by repeating it here, but proceed to give an ex- 

 ample of my method of demonstrating the rule, as I have now just 

 done by the method of Mr. Holdred, which the candid reader will 

 no doubt feel to be the most eligible and satisfactory method of 

 settling the dispute. 



I take a, b, c, &c. for each part of the root found by the corre- 

 sponding step. In the first equation I substitute a-\-u for x, and 

 obtain an equation in u ; but in the next equation, instead of u, 

 I substitute b + v, and obtain a new equation in v and so on. So 

 that in the different steps the equations which I produce are the 

 same as Sir Isaac Newton's, except that the orders of figurate 

 numbers are clearly exhibited ; and thus to prove every portion of 

 the root, only one equation is required. But by the method of 

 Mr. Holdred, every portion of the root after the first would re- 

 quire as many equations, and one more, as the exponent of the 

 highest power has units. As my method is founded on the prin- 

 ciple of Sir Isaac, I have, from its simplicity, been able to com- 

 prise the demonstration of the figurate method within the limits 

 of two octavo pages. 



With regard to the non-figurate method, the fact is, that after 

 the rule has been demonstrated by my method, or any other, no 

 other demonstration is necessary to perform the operation, either 

 by Mr. Horner's method, by Mr. Holdred's, or my own, as all the 

 different forms arise from the manner of summing up the quan- 

 tities ; viz. by writing the numbers and their sums under them, 

 or writing down the sums as each number is added, with the num- 

 ber to be added, or writing the first number and the successive 

 sums onlv. As, fo; example, let 771 and w be any two consecutive 

 coefficients of the original c(|uation, and let /;, q, r, &c. Ixe any 

 numbers to be added to m, and let N = m + p. These different 

 forms of addition are explained by the following operations : — 



No. 1. 



Here every number on 



the right-hand of the 



line, after the second, is 



found by multiplying the 



5Na-f 2ya-|-7a + «=3Q opposite number on the 



left 



N + ^ + r 



