292 
PLATO 
mentioned by Proclus is the method of successive bipartitions 
of genera into species such as we find in the Sophist and 
the Politicus, and has little to say to geometry; but the 
mention of it side by side with analysis itself suggests that 
Proclus confused the latter with the philosophical method 
referred to. 
(y) Definitions. 
Among the fundamentals of mathematics Plato paid a good 
deal of attention to definitions. In some cases his definitions 
connect themselves with Pythagorean tradition; in others he 
seems to have struck out a new line for himself. The division 
of numbers into odd and even is one of the most common of 
his illustrations; number, he says, is divided equally, i. e. 
there are as many odd numbers as even, and this is the true 
division of number; to divide number (e. g.) into myriads and 
what are not myriads is not a proper division. 1 An even 
number is defined as a number divisible into two equal parts 2 ; 
in another place it is explained as that which is not scalene 
but isosceles 3 : a curious and apparently unique application 
of these terms to number, and in any case a defective state 
ment unless the term ‘ scalene ’ is restricted to the case in which 
one part of the number is odd and the other even; for of 
course an even number can be divided into two unequal odd 
numbers or two unequaheven numbers (except 2 in the first 
case and 2 and 4 in the second). The further distinction 
between even-times-even, odd-times-even, even-times-odd and 
odd-times-odd occurs in Plato 4 : but, as thrice two is called 
odd-times-even and twice three is even-times-odd, the number 
in both cases being the same, it is clear that, like Euclid, 
Plato regarded even-times-odd and odd-times-even as con 
vertible terms, and did not restrict their meaning in the way 
that Nicomachus and the neo-Pythagoreans did. 
Coming to geometry we find an interesting view of the 
term ‘ figure ’. What is it, asks Socrates, that is true of the 
round, the straight, and the other things that you call figures, 
and is the same for all ? As a suggestion for a definition 
of ‘ figure ’, Socrates says, ‘ let us regard as figure that which 
alone of existing things is associated with colour Meno 
1 Politicus, 262 n, e, 2 Laws, 895 E. 
8 Euthyphro, 12 n. 4 Parmenides, 148 e-144 a.