82 
PYTHAGOREAN ARITHMETIC 
in order that h (and c also) may be a whole number. ‘ Similar 
plane ’ numbers are of course numbers which are the product 
of two factors proportional in pairs, as mp. np and mq. nq, or 
mnp 2 and mivf. Provided, then, that these numbers are both 
even or both odd, 
is the solution, which includes both the Pythagorean and the 
Platonic formulae. 
(£) Oblong numbers. 
Pythagoras, or the earliest Pythagoreans, having discovered 
that, by adding any number of su&lessive terms (beginning 
from 1) of the series 1 + 2 + 3 + ... + n = (n + 1), we obtain 
triangular numbers, and that by adding the successive odd 
numbers 1 + 3 + 5 + ... + (2 n— 1) = n 2 we obtain squares, it 
cannot be doubted that in like manner they summed the 
series of even numbers 2 + 4 + 6 + ... +2n = n(n+ 1) and 
discovered accordingly that the sum of any number of succes 
sive terms of the series beginning with 2 was an ‘ oblong ’ 
number {erepo [ig Kgs), with ‘sides’ or factors differing by 1. 
They would also see that the oblong number is double of 
a triangular number. These facts would be brought out by 
taking two dots representing 2 and then placing round them, 
gnomon-wise and successively, the even numbers 4, 6, &c., 
thus: 
The successive oblong numbers are 
2.3=6, 3.4 = 12, 4.5 = 20..., n(n+1)..., 
and it is clear that no two of these numbers are similar, for 
the ratio n:{n + l) is different for all different values of n. 
We may have here an explanation of the Pythagorean identi 
fication of ‘ odd ’ with ‘ limit ’ or ‘ limited ’ and of ‘ even ’ with