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.