ON POLYGONAL NUMBERS. 1]^ 



a so afford Mr. G. an opportunity of explaining the ambi- On Fermat's 

 guous parts; and will at the same time much oblige proposition on 



Yours, &c. numbers. 



Royal Mil. Academy, P. BARLOW. 



Sept. 13, 1808. 



The curious theorem, which Mr. Gough has undertaken 

 to demonstrate, was first announced by Fermat, in one of 

 his notes at page 180 of his edition of Diophantus; and the 

 demonstration for the particular case of squares was given 

 first by Lagrange, in the (Mem. de Berlin, IJJOj, and af- 

 terwards in a simpler form by Euler, in the (Acta Petrop, 

 Ann, Mil), as we are informed by le Gendre, in his Essai 

 stir la Theorie des Nomhres, at page 202; where there is 

 also given a demonstration for the same particular case. Le 

 Gendre has likewise in another part extended it to triangu- 

 lar numbers, and this is the most that has ever been done 

 by any mathematician. If therefore the ingenious author 

 of the abovementioned essay has failed in his demonstration, 

 he has the satisfaction of having failed in an attempt, in 

 which many of t4ie ablest mathematicians in Europe have 

 succeeded no better than himself; and if, on the contrary, 

 he can clear up those parts, which appear at present to be 

 defective, the greater degree of merit will be due to his in- 

 genuity and ability, of which I have always entertained the 

 highest opinion : and I feel confident, that he will not mis- 

 take my intentions in the following criticism, but rather at- 

 tribute it to my love for mathematical truth, than to any 

 invidious desire of criticising his paper. 



The first thx-ee propositions and their corollaries are in 

 themselves correct, although I am at a loss to see in what 

 manner they are intended to be applied to the general de- 

 monstration. The first part that I shall examine is the con- 

 clusion drawn at cor. 2, prop, 4. Tn cor. 1 of the same 

 prop, it is proved, that e, which is taken to represent any 

 aggregate of polygonals of the denomination m, is of the 

 form ezzp + m — 2.s; and then in cor. 2, having shown 

 that every natural number is also of the same form, p +■ 



m «-=« 2. s, the^converse of the prop, is inferred to be true 



likewise 3 



