120 ON POLYGONAL NUMBERS. 



On Fermat's likewise ; namely, that every number is the aggregate of 

 pd^eonul" °" polygonals. Now it is easily seen, that this is false reason- 

 numbers, ing. Our author 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, there- 

 fore every natural number is the aggregate of polygonals : 

 or, to put it in a stronger light, every natural number is of 



the form jp -f m. — ■2. s; and, every square number being 



also of the form p -\- m — 2, s, therefore every natural num- 

 ber is a square number. No person can for a moment fail 

 - of detecting in those two last cases the fallacy of this rear 

 soning, nor of perceiving the strict analogy it bears with 

 that made use of in the cor. abovementioned. It is to be 

 observed, that I do not object to the conclusion, but to the 

 manner of obtaining it; for all that is drawn from the first 

 four propositions and their corollaries might have been 

 granted at first as a postulate, if any use could have been 

 made of it in the general demonstration. 



For unity is a polygon of every denomination, and every 

 natural number is composed of a number of units, there- 

 fore every natural number is composed of a number of po- 

 lygons of any denomination m, consequently every natural 

 number is either a polygon of a given denomination m, or 

 may be resolved into polygons of that denomination ; the 

 number of those polygons being unlimited, as in the corollary 

 alluded to. 



The next place, where any conclusion is drawn, is in the 

 cor. to prop. !6, where it is said, that If e zz y + t, can be 

 resolved into polygons, the number of which zz m — f, 

 € +/ may be resolved into m polygons of the same deno- 

 mination. Now either this supposition is necessary to com- 

 plete the demonstration, or it is not: if it is not necessary, 

 it ought to have been omitted; if it is necessary, it ought 

 to have been shown (but it no where is in the demon- 

 stration) that e zzy -\- t may be resolved into m — f po- 

 lygons, because the conclusion depends upon this suppo- 

 sition, and if the supposition is true, the conclusion is true; 

 on the coptrary, if the supposition is false, the conclusion 

 must necessarily be so likewise. This language is at all 

 events too vague for mathematical reasoning. I am willing 



to 



