124 SCIENCE IN GRECO-ROMAN ANTIQUITY 



According to G. Milhaud it is possible to explain 

 by arithmetic the table of metaphysical categories 

 framed by the Pythagoreans. 1 This table sets forth, 

 on the one hand, the ideas of " finite," " odd," " unity," 



A 



/A 



/ / \ 



/ T 



i 



Fig. 13. 



" square," etc., with, on the other hand, the opposite 

 ideas of " infinite," " even," " plurality," " hetero- 

 geneous factors," etc. In order to understand these 

 oppositions we must remember this : if we build up 

 the odd numbers with the gnomon, we obtain a square, 

 i.e., a finite and complete figure, whose sides have a 



ratio - always identical and equal to unity. On the 

 n 



contrary, the construction of the even numbers by the 



gnomon gives a rectangle, a figure indefinite in this 



sense that its sides n and n + 1 have a ratio changing 



with the value of n, namely : ,-,... 



3 4 » + 1 

 We know also that, in their arithmetic, the Pytha- 

 goreans w r ent so far as to consider that even moral 

 realities were formed of numbers. 2 Justice, for 



1 20 Milhaud, Phi. geo., p. 116 et seq. 



2 22a Robin, La pensee grecque, p. 73. 



