J ^2 ON POLYGONAL NUMBERS. 



2cl, The difference of two adjacent terms in the generat- 

 ing series increased by two is called the denomination of 

 the polygonal number ;. and the number is said to be di- 

 gonal, trigonal, tetragonal, &c., accordingly as the denomi- 

 nation w zr 2, 3, 4, &c. 



3d, The riumber of terms «, which are added together 

 to form any polygonal number, is called its index ; and the 

 polygon is said to be of the first, second, or third order, &c., 

 accordingly as a — 1, 2, 3, &c. 

 Prop. t. Proposition IsL Lemma. Let i be the greater of two 



adjacent terms, a &, in a series of digonal or natural num- 



bers; then we have — -\-b — 1. For &:=«+ 1, 



2 2 



and 6~l=a; but ^lz±-b:::rT^L-ay^t±2- 



2 2 2 



:!!+.".. again :il=.'' + i _i ^ ^« + „= £!+f = 



Corollary. Corollary. Hence if we put a=: 1, 2, 3, &c. successively, 



each succeeding value of ? ^~- may be found, by adding 



the next value of a in succession to that of 'LSH^ last found, 



2 



and subtracting unity from the sum ; where it is evident, 



that when a zr 1 , rr 0. In this manner the annexed 



2 



table is constructed, the use of which will appear in the 

 sequel. 



Table. a~\. 2 .3.4. 5. 6 . 7 . 8 . 9 . 10 . 11 



a* —a = . 1 . 3 . 6 . 10 . 15 . 21 . 28 . 36 . 45 . 55 



Prop.'2. Proposition 'id. If A; be a polygonal number, of which 



the denomination —m, and index "= a, we have k zza + 

 ifi — 2 X «* — «. For the first term of the generating se- 

 ries — I (by Def. 1st); and the common difference of the 

 terms r: m — 2 (by Def. 2d) ; but the term of which the 



number 



