NICOMACHUS 
105 
are the original terms, a being the smallest, we take three 
terms of the form 
a, h~ a, {c — a— 2(6 — <x)} = c + <x — 2b, 
then apply the same rule to these three, and so on. 
In cc. 3-4 it is pointed out that, if 
1, r, r 2 ..., r n ... 
be a geometrical progression, and if 
then 
and similarly, if 
p n = 1 + 
Pn r+1 
r 
, an èm/xopcos ratio, 
P n Pn-1 "b Pn ’ 
P n 
Pn 
r + 1 
r 
and so on. 
If we set out in rows numbers formed in this way, 
* 
r, r 2 , r 3 ... r n 
r+1, 7’ 2 + r, T d + r 2 .,. T n - 
r 2 + 2r+l, r 3 + 2r 2 + r... i>»|2r n ' 1 + r n_2 
r 3 + 3 r 2 + 3 r + 1... r n + 3 r n-1 + 3 r n ~ 2 + r n ~ z 
r n r>ij r n~2 
+ ... 
the vertical rows are successive numbers in the ratio r/(r + 1), 
while diagonally we have the geometrical series 1, r+1, 
(r+1) 2 , (r+1) 3 .... 
Next follows the theory of polygonal numbers. It is pre 
faced by an explanation of the quasi-geometrical way of 
representing numbers by means of dots or a’s. Any number 
from 2 onwards can be represented as a line ; the plane num 
bers begin with 3, which is the first number that can be 
represented in the form of a triangle ; after triangles follow 
squares, pentagons, hexagons, &c. (c. 7). Triangles (c. 8) arise 
by adding any number of successive terms, beginning with 1, 
of the series of natural numbers