ON POLYGOKAI. NUMBERS. iSS 



Cor. 2(1. Since a is any term of an arithmetical pro- Cor. 2, 

 gression, bounded by p and e, and having w — 2 for its 

 common difference, prop. 4. cor. 1 ; it will be easily under- 

 stood, that r is also a corresponding term of an anthmeti- 

 cal progression, of which the extremes are o and S, and 

 common difference 1; hence it follows, that 2*-— 2/" in- 

 creases, while r and a diminish; therefore b^ — b -{- c^ — c 

 + d^—d increases, while the sum of the roots b 4- c + d 

 diminishes ; consequently, the number of the parts b, c, d, 

 into which a is divided, decreases at the same time. 



Cor. 3d. If a can be so taken that a 4- '2 s — 2r ~ a^, Cor. S. 

 b — a; and e is a polygon to the index a and denomina- 

 tion m. 



Prop. 6th. If e be an aggregate of polygons of any de- pf^p. s, 

 nomination »i, and y that polygon, which is less than e, but. 

 greater than any polygon of an inferior order and the same 

 denomination ; then the polygons, into which e can be re- 

 solved, are equal to y or less than y. For the next superior 

 polygon is greater than y (by prop. 2) ; it is therefore 

 greater than e by hypothesis, and cannot constitute a part 

 of it. QED. 



Cor. Ifeiry-4-m — l,it may be resolved into m po- Cor. 

 lygons of the denomination m ; namely, into y and m — \ 

 units; again, \^ e zz y -\- m it may be resolved into poly- 

 gons, the number of which is less than m -f- 1 ; this is evi- 

 dent from cor. 2, prop. 4 : lastly, \i e, zz y -^ t^ can be re-, 

 solved into polygons, the number of which zz m —f, e \f 

 may be resolved into m polygons of the same denomination. 

 It is evident, from the properties of polygonal numbers 

 contained in this corollary, that every whole number piay 

 be resolved into polygons of the denorhination w, the rium- 

 ber of which does not exceed m. ,. 



Prop. 7. Problem. To resolve a given number e into Pron, 7, 

 polygons of a given denomination m, by help of the table 

 in prop. 1. 



Method. 1st, Write down all the successive Aalues of ar, 

 beginning with e and diminishing them progressively by 

 m — 2, until the series terminates with 0, or with p^less 

 than m — 2; 2d, under each value of a place the coi:r<5- 



sponding' 



