106 
PYTHAGOREAN ARITHMETIC 
The gnomons of triangles are therefore the successive natural 
numbers. Squares (c. 9) are obtained by adding any number 
of successive terms of the series of odd numbers, beginning 
with 1, or 
1, 3, 5, ... 2n— 1,.... 
The gnomons of squares are the successive odd numbers. 
Similarly the gnovions of pentagonal numbers (c. 10) are the 
numbers forming an arithmetical progression with 3 as com 
mon difference, or 
1, 4, 7,... 1 +(w—l)3,...; 
and generally (c. 11) the gnomons of polygonal numbers of a 
sides are 
1, 1 + (a — 2), 1+2 (a—2),... 1 + (r — 1) (a-2),... 
and the a-gonal number with side n is 
1 + 1 + (ci — 2) + 1 + 2 (ct — 2) + ... + 1 + (n — l) {cl — 2) 
= n + ^n (n — 1) {a —2) 
The general formula is not given by Nicomachus, who con 
tents himself with writing down a certain number of poly 
gonal numbers of each species up to heptagons. 
After mentioning (c. 12) that any square is the sum of two 
successive triangular numbers, i.e. 
n 2 = \ {n - 1) ti + -| n {n+ 1), 
and that an a-gonal number of side n is the sum of an 
(a—l)-gonal number of side n plus a triangular number of 
side 7i—l, i.e. 
n + \n{n—\) [a — 2) = n + ^n (n— 1) {a - 3) + ^n {n — 1), 
he passes (c. 13) to the first solid number, the pyramid. The 
base of the pyramid may be a triangular, a square, or any 
polygonal number. If the base has the side n, the pyramid is 
formed by similar and similarly situated polygons placed 
successively upon it, each of which has 1 less in its side than 
that which precedes it; it ends of course in a unit at the top, 
the unit being ‘potentially’ any polygonal number. Nico 
machus mentions the first triangular pyramids as being 1, 4, 
10, 20, 35, 56, 84, and (c. 14) explains the formation of the 
series of pyramids with square bases, but he gives no general