242 



UN POLYGONAL NUMBERS. 



answered. 



essay, Tliut the propositions following the third are derived 

 from those which precede them is it fact, that is proved by 

 the references ; consequently, if my paper contain a gene- 

 ral demonstration of the theorem proposed, the necessity of 

 the first three propositions is proved. 

 '2d objection Mr. Barlow's second objection charges me with false lo- 

 gic; and this gentleman states a sopliism, which he consi- 

 ders to be similar to the argument used in cor. 2, prop. 4, 

 of the essay on polygonal numbers. lie observes, " that 

 *' the author of this essay might, with as much propriety, 

 *• have said, that every natural number is either even or 

 ** odd, and every aggregate of polygonals being also either 

 *' even or odd, therefore every natural number is the ag- 

 •* gregate of polygonals." Mr. B. rests his refutation of the 

 argument used in qor. 2, prop. 4, on the supposed similarity 

 of it and the preceding sophism ; if then I can show these 

 two to be dissimilar, his second objection mvist be pro- 

 nounced futile. To do this, I may observe, that numbers* 

 like most other things, are aggregates of qualities, not sin- 

 gle qualities, otherwise tliere could be no more numbers 

 than qualities; that is, a number, beside being odd or even, 

 is prime or composi*^e, rational or irrational. This consider- 

 ation shows the nature of the intended fallacy contained in 

 the preceding sophism ; for it maintains two aggregates of 

 qualities to be the same ; because they have one of these 

 qualities in common. This I presume is an objection, to 

 which the demonstration in question is not liable : for 

 equality constitutes identity in numbers ; that is, if anv one 

 of two or more equal numbers possess any three of the qua- 

 lities pointed out above, or any of the properties contained 

 in the definitions to the 7th Book of the Elements, all the 

 rest of them possess just the same, neither more nor less 

 (by axiom 1st of the same book). Now it is shown in the 

 first corollary to the 4th proposition, that every aggreo-ate 

 of polygons of the denomination 7« is of the form p -\- 



m-^2 . s; where p is limited by and m — 3 ; and .v is in- 

 definite: hence it follows, that each aggregate of such po- 

 lygons is equal to an assignable value of p -\- m' — 2 . s. 

 Moreover it appears fioui the second coiollary to the same 



proposition, 



