38 Mr. Nicholson's Answer 



some may appear trifles, and which consist in mere modes of ad- 

 dition, had he not noticed them himself. My object was, to di- 

 vest the operation of all numbers which were not expressed by my 

 formula, and I have succeeded so far as to do it. I cannot help 

 observing, that nothing but the desire of being thought the au- 

 thor of a useful discovery, ignorance of what has been before 

 published in the writings of men of superior intellect, and an over- 

 weening opinion of what he himself has understood but very im- 

 perfectly, could make him so roundly assert that I learned all from 

 him ; while, on the contrary, he availed himself of no hint of 

 mine. At the time I published the work now mentioned, I was 

 divided in my own opinion which of the methods I should adopt ; 

 but as I had not then wrought any example, it occurred to me that 

 the form which I had adopted was that which produced the fewest 

 numbers ; but I did not perceive, at that time, that it vvould oc- 

 casion more lines : however, I saw afterwards the advantage of 

 adopting the method used in the postscript, not oidy in saving 

 lines, but that it agreed exactly with the simplicity of the formula 

 which I had already investigated and adopted, and that it was 

 much better adapted to the transformations of equations, parti- 

 cularly in diminishing the root by unity at each step. These were 

 the motives that induced me to make the change in adding the 

 numbers together ; and that this was really the case, I have cre- 

 dible witnesses to confirm what I here assert. Had I adopted 

 Mr. Holdred's clumsy method of adding every two lines together, 

 I should not have been able to bring it within a moderate com- 

 pass, and should, besides, have obscured the principle. In 

 my arithmetical process, according to the figurate method, the 

 numbers stand exactly in the same manner as indicated in the 

 coefficients of my transformed equations ; and in my process 

 according to the non-figurate method, tlie different steps of 

 the operation are performed by a general formula, which is the 

 result of a property of figurate numbers, and which gives the very 

 same orders as are expressed in the coefficients of every general 

 transformed equation. On account of the course of conduct pur- 

 sued by Mr. Holdred, the non-figurate method would have been 

 published immediately after the rudiments of algebra, in my last 

 work entitled Analytical and Arithmetical Essays, published near 

 the end of Nov. 1 820, though dated 1821, whether Mr. Horner's 

 method of continuous approximation had appeared or not, as I 

 can prove if it were necessary ; l)ut understanding that Mr. Hol- 

 dred's book was in the press, I laid the work above alluded to 

 aside, and published my demonstrations and methods by them- 

 selves, in the Essay on Involution and Evolution, before Mr. Hol- 

 dred's work made its appearance. Let any one compare the 

 simple but general result in page 37 of my Essay, with his unsa- 



tisfactorv 



