76 
PYTHAGOREAN ARITHMETIC 
3 ; 2 (the fifth), and 2 : 1 (the octave). Speusippus observes 
further that 10 contains in it the ‘ linear ‘ plane ’ and ‘ solid ’ 
varieties of number; for 1 is a point, 2 is a line, 1 3 a triangle, 
and 4 a pyramid. 2 
Figured numbers. 
This brings us once more to the theory of figured numbers, 
which seems to go back to Pythagoras himself. A point or 
dot is used to represent 1; two dots placed apart represent 
2, and at the same time define the straight line joining the 
two dots; three dots, representing 3, mark out the first 
rectilinear plane figure, a triangle; four dots, gne of which is 
outside the plane containing the other three, represent 4 and 
also define the first rectilineal solid figure. It seems clear 
that the oldest Pythagoreans were acquainted with the forma 
tion of triangular and square numbers by means of pebbles or 
dots 3 ; and we judge from the account of Speusippus’s book, 
On the Pythagorean Numbers, which was based on works of 
Philolaus, that the latter dealt with linear numbers, polygonal 
numbers, and plane and solid numbers of all sorts, as well as 
with the five regular solid figures. 4 The varieties of plane 
numbers (triangular, square, oblong, pentagonal, hexagonal, 
and so on), solid numbers (cube, pyramidal, &c.) are all dis 
cussed, with the methods of their formation, by Nicomachus 5 
and Theon of Smyrna. 6 
(a) Triangular numbers. 
To begin with triangular numbers. It was probably 
Pythagoras who discovered that the sum of any number of 
successive terms of the series of natural numbers 1, 2, 3 . . . 
beginning from 1 makes a triangular number. This is obvious 
enough from the following arrangements of rows of points; 
Thus 1 + 2 + 3 + .. ,+n = ^n{n+l) is a triangular number 
1 Cf. Arist. Metaph. Z. 10, 1036 b 12. 2 Theol. Ar. (Ast), p. 62. 17-22. 
3 Cf. Arist. Metaph. N. 5, 1092 b 12. 4 Theol. Ar. (Ast), p. 61. 
5 Nicom. i. 7-11, 13-16, 17, e Theon of Smyrna, pp. 26-42.