ON POLYGONAL NUMBERS, 

 proposition, that every natural number is of the form p 4- 

 »« — 'i.5, limited as above; where s may be found, the 

 number being given with m and p; but this value of * sub* 

 stituted in the form p -{- m — 2 . s, gives an aggregate of 

 polygons of the denomination m, which is true in all cases; 

 it is therefore a universal truth not admitting of one excep- 

 tion. The preceding facts appear to give indisputable ac- 

 curacy to the following syllogism : every natural number is 

 equal to an assignable value of the form p + tn — 2 . s; 

 and there is an aggregate of polygons of the denomination 

 m equal to the same value of the same form ; therefore every 

 natural number is an aggregate of such polygons ; because 

 things, which are equal to the same thing, are equal to one 

 another; Euclid, Axiom 1, Book 1; and equal numbers 

 have been shown to have the same qualities neither more 

 nor less. The supposed similarity betwixt my critic's so- 

 phism and the preceding mode of argument appears to be 

 done away; for he proceeds on the supposition, that the 

 sameness of one quality constitutes identity in numbers; 

 but the first axiom of the 7th book of Euclid is the founda- 

 tion of my reasoning ; namely, that a perfect agreement in 

 qualities pi-oduces the same thing, namely, identity of num- 

 bers. My opponent, in fact, does not rely altogether on 

 the similarity of his intended, and my accidental sophistry; 

 for he produces a second sophism, and pronounces it to be 

 strictly analogous to mine, though it differs in every parti- 

 cular from his former parody of my supposed mistake. 

 Mr. B. observes, that *' every natural number is of the form 

 ** p -\- 7n — 2 X s; and, every square number being also of 

 ** the form p + m — 2.5, therefore every natural number is 

 '♦ a square number." It is true, that every square number 



is of the form p 4- m — 2.5; but then s is limited, being 



Q* 4* 2 Q w + t)*— p 

 of the form s — -, where q^ = p or the 



m — 2 



next greater square when p is not a square, and v is to be 

 taken so as to make 5 a whole number; but s is unlimited 

 in the case of natural numbers; therefore, by the rules of 

 logic, every square integer may be proved to be a natural 

 R 2 numbier 



US 



