34* ON POLYGONAL NUMBERS. 



GouV ^ number; tnis gentleman has divided ray paper into three 

 objections, which he has pronounced to be futile, and I 

 must add without proving them to be so. Mr. Gough will 

 recollect, that in my letter, so far as related to the first four 

 propositions and their corollaries, I did not object to the 

 conclusion, but to the manner of obtaining it; my object 

 was only to show, that an unnecessary number of proposi- 

 tions and corollaries were introduced into the essay, to de- 

 monstrate that which needed no demonstration. Mr. Gough 

 does not deny this, except that 1 have called that a postulate, 

 to which I have given the importance of; a theorem, and de- 

 monstrated it as such ; it is true, as it stands in my letter, 

 it has the appearance of a proposition, but it was unneces- 

 sary to give it this form : it might have stood thus ; 



Definition. Unity is a polygon of every denomination. 



Postulate. Let it be granted, that every integer is an ag- 

 gregate of units, or of polygonals. 



Here it is evident that no demonstration is necessary, and 

 had Mr. Gough begun his essay with this postulate, he 

 would at all events have saved himself and readers consider- 

 able trouble. 



My third, and only objection that effects the truth of the 

 demonstration, Mr. Gough has evaded, by charging me 

 with putting a false construction upon his words; and now 

 I am under the necessity of retorting the charge. I never 

 said, nor intended to say, that it was necessary to show, that 

 e ~y + t can be resolved into m -~-f polygon in all cases ; 

 the question which I proposed was this; " If e r: y -f- 1 

 cannot be resolved into m —>/ polygons, how does it appear 

 from the demonstration, that e + f can be resolved into m 

 polygons ?" This question I again repeat, and unless it 

 can be satisfactorily answered, the theorem will be still with- 

 out a demonstration, however unwilling Mr. Gough may be 

 to acknowledge it. This gentleman must also be aware, 

 that he cannot be allowed to introduce his examples, which 

 are the moment before derived from the demonstration, to 

 prove the truth of the supposition on which that demonstra* 

 tion is founded. 



Mr. Gough appears to have deceived himself by consi- 

 dering only small numbers, in which the polygons, being 

 1 also 



