to Mr. Holdied. 35 



to me, and is to be foiiiKl in chap. x. sec. 2. where the same 

 rule is applied to a (ju fid rati c equation*. 

 This idea only wanted generalizing. 



He was acquainted with the numeral exegesis to wnich his 

 method bears a striking affinity, but he was an entire stranger 

 to what had been done by either Newton or Rr.lphson, before I 

 pointed out their methods to him. 



If he had known Sir Isaac's method of transforming equations, 

 which is that from which I derived my ideas, and not from his 

 crude notions, he would Iiave been ashamed to say that " Mr. 

 Nicholson extracted all from me, even his demonstrations are 

 mine." (Vide The Philosophical Magazine, Nov. 1820.) He cer- 

 tainly must entertain a very mean opinion of such as might be 

 interested at all in the discovering and maturing of the subject, to 

 suppose that thiy would take either his ipse dixit or mine, with- 

 out convincing themselves by a diligent perusal and investigation 

 of the method, and an attentive perusal of the Essay which I 

 published on Involution and Evolution, with the Postscript. 



With respect to what he calls his demonstration, the first step is 

 the same with what has been done both by Newton and Ralphson. 

 Suppose, for instance, that the equation is in x, he takes r for 

 the first figure of the root and y for the remaining part of it, and 

 substitutes r-\-y for x, and thus the original equation is trans- 

 formed to another in y ; he then takes a for the second figure of 

 the root, and 2/ for the remaining part of it; so that x=r + a-[-z/, 

 and to reduce this to the binomial form, he puts ^r = r-\-a and 

 substitutes c^r + u for x. By this means he gets an equation in 

 u where the coefficients are expressed by the coefficients of the 

 original equation and „r, and therefore reinstatesr + a for or in all 

 the coefficients of the powers of u, and thus forms as many sub- 

 .sidiary equations as there are units in the exponent of the highest 

 power : therefi)re if n be the exponent of the power, the number 

 of ccjuations that each digitical figure of the root will require after 

 the first will be n+\. 



In these equations the coefficients do not exhibit the figurates 

 which it is his object to elicit ; consequently, he has recourse to a 

 mosttedious and circumlocutory explanation in words. This demon- 

 stration, as Mr. Iloldred calls it, is extended to eight quarto pages. 



In page 5 of this work he asserts, that '■^n-irC, + D + &c. is 

 an imjjerfcct divisor by which the next figure of the root may be 

 discovered, which when found I call a." Similar assertions will 

 be found in pages 12, 15, 17, 18, 20, 25 and 20. This is equi- 

 valent to saying, that the sum of the coefficients of the powers of 



I liave seen since the same rule njjplicd to u (luudiiitic equation in 

 l'^UKi.4on'» Algebra. 



E 2 an 



