CLASSIFICATION OF NUMBERS 
71 
both odd or both even); while an odd number is that which, 
however divided, must in any case fall into two unequal parts, 
and those part*« always belonging to the two different kinds 
respectively (i.e. one being odd and one even). 51 
In the latter definition we have a trace of the original 
conception of 2 (the dyad) as being, not a number at all, but 
the principle or beginning of the even, just as one was not a 
number but the principle or beginning of number ; the defini 
tion implies that 2 was not originally regarded as an even 
number, the qualification made by Nicomachus with reference 
to the dyad being evidently a later addition to the original 
definition (Plato already speaks of two as even). 2 
With regard to the term ‘ odd-even it is to be noted that ? 
according to Aristotle, the Pythagoreans held that ‘ the One 
arises from both kinds (the odd and the even), for it is both 
even and odd ’. 3 The explanation of this strange view might 
apparently be that the unit, being the principle of all number, 
even as well as odd, cannot itself be odd and must therefore 
be called even-odd. There is, however, another explanation, 
attributed by Theon of Smyrna to Aristotle, to the effect that the 
unit when added to an even number makes an odd number, but 
when added to an odd number makes an even number : which 
could not be the case if it did not partake of both species ; 
Theon also mentions Archytas as being in agreement with this 
view. 4 But, inasmuch as the fragment of Philolaus speaks of 
‘ many forms ’ of the species odd and even, and ‘ a third ’ 
(even-odd) obtained from a combination of them, it seems 
more natural to take ‘ even-odd ’ as there meaning, not the 
unit, but the product of an odd and an even number, while, if 
‘ even ’ in the same passage excludes such a number, ‘ even ’ 
would appear to be confined to powers of 2, or 2 n . 
We do not know how far the Pythagoreans advanced 
towards the later elaborate classification of the varieties of 
odd and even numbers. But they presumably had not got 
beyond the point of view of Plato and Euclid. In Plato we 
have the terms ‘ even-times even ’ {cipria dpnaKis), ‘ odd- 
times odd’ (nepirrà nepirraKis), ‘odd-times even’ (cipria 
1 Nicom. i. 7. 4. 2 Plato, Parmenides, 143 D. 
3 Arist. Metaph. A. 5, 986 a 19. 
4 Theon of Smyrna, p. 22. 5-10.