1^4 



ON POLYGONAL NUMBERS. 



bc-\-bd-\-cd::ib^ + c^ + d^-\-2Xbc{-bd+cd-k-a*. 

 Theorem con- Thus it appears, that the present proposition generalizes a 

 fined to squares ^l^gQj.gjj^ formerly confined to squares, and extends it to 

 lied. polygons of all denominations. 



Prop. 4. Prop. 4th. Put e and v respectively equal to g -{- h -{- If 



&c., and bc + bd-^cd'm prop. 3 ; then v — -\ 



Cor.t, 



Cor. 2. 



Tdp 



Cor. 1. 



For e -f in— 2 X vzzk (by prop. 3) ~ a -\- m — 2 X 



(by prop. 2) ; hence v — -1 . Q E D. 



Cor. 1st. Since b, c, d, Sec, are integers, v is an inte- 

 ger ; but f_JZ^is an integer, therefore llZf is an integer ; 

 2 m — 2 



c . 

 hence if give a quotient 5 and a remainder p, a zz p 



m — 2 ^ '■ 



or gives a quotient r and a remainder p ; from which 



we have the following general expressions c — p + nt — 2X5; 



-' — ^ a* — a 

 «— p + m — 2 X r; v zz — \- r — s. 



Cor. 2d. Putp;=0, 1, 2, 3 ... 7n — 3 successively, then 

 e will be expressed as in the following table. 



0=m-~2 — 2wj— 4 — 3 W2— 6, &c. 



l—m—2^-l — 2m—A^-l—3 7n—6\-l, &c. 



2zzm — 2-|-2zi2m — 4+2=3 ?n— 6+2, &c. 



3z=;n— 2+3 = 2 m— 4 + 3=3 ffj— 6 1-3, &c. 



It appears from the table, that the values of e, taken 

 vertically, constitute the series of natural numbers ; there- 

 fore every integer is either a polygon of a given denomina- 

 tion 7n, or it may be resolved into polygons of that denomi- 

 nation. 



Prop. 5th. i^ + c^ + cZ«, &c. = a + 2 .? — 2 r." For 

 Z>* + c^ + d' + 2 w = a- by involution ; but 2 v — a^ — a 

 + 2r — 2 5 (cor. 1 , prop. 4) ; therefore b^ + c^ + d^ zza-h 

 2s— 2 r. Q E D. 



Cor, 1st. b" — 6 ]- c* — c-yd"^ — dzz2s — 2r, because 

 b ■{- c -{• d zz a. 



Cou 



v- 







ezz 



p- 



1 



ezzi 



p- 



2 



ezz: 



p~ 



3 



ezz. 



