On Newton's Rule for Imaginary Roots, 365 



Magazine*. As Professor Young was informed by the publishers 

 of this Journal that I directed an invitation to be forwarded to 

 him through their agency, he should, I think, have confined his 

 statement (if he conceived that anything turned upon a point 

 apparently so immaterial) to a denial of his having received it. 

 I was not informed that it had not been transmitted. As to the 

 other charge, Professor Young's memory must certainly be in 

 default, as my observations on his " supposed demonstration " 

 formed no part of my lecture, but were called forth by his inter- 

 pellations addressed to me at the close of it, and his express 

 " invitation " to state wherein the same was inaccurate. Surely 

 then u I was invited by Professor Young to express an opinion 

 on his supposed demonstration." 



Whilst guarding myself against being understood to acquiesce 

 in the correctness of the general impression which Professor 

 Young's last statement (however honestly meant) tends to con- 

 vey on the matter late in controversy, I think it right, under 

 the circumstances, to refrain from making any comment upon it. 

 A syllabus of my lecture, filling many pages of print, was 

 directed to be put into the hands of all who did me the honour 

 of attending it ; and if Professor Young received one he will find, 

 if he can spare time to consult it, that it does contain a proof in 

 eoctenso of Newton's theorem — not merely that which commonly 

 goes by the name, but of the complete theorem in the form given 

 to it by Newton himself, which had been lost sight of, without 

 exception, so far as I know, by all who have since treated of the 

 subject, in which form it is a refinement upon Descartes' s 

 rule of signs, and admits of being generalized still further into 

 a theorem which bears to itself the same kind of relation that 

 Fourier's theorem bears to Descartes' s, — this generalized theorem, 

 which is of a geminate character, being in its turn included 

 in one still more general, containing an arbitrary parameter 

 {limited). The late deeply lamented Mr. Purkiss has drawn 

 up a very full account of these investigations for the Number of 

 the Cambridge, Oxford, and Dublin Mathematical Messenger 

 which has just made its appearance. I do not like to conclude 

 without expressing my sincere regret if I have been unguardedly 



* Imagine the case that A. B., having boxed C. D.'s ears, immediately 

 afterwards offers a handsome apology for having just before unintentionally 

 trodden on his toes. This will afford a clue to the conflict of feeling occa- 

 sioned in my mind by Professor Young's concession as regards the right of 

 discovery, following an unredressed imputation upon my accuracy. In a 

 world and in a country where the experience of life has taught me that the 

 love of truth so loudly professed is so rarely acted upon as a rule of con- 

 duct, I shall perhaps hardly be believed when I assert that I prize a cha- 

 racter for truthfulness far above any merit that may be supposed to attach 

 to originality. 



