294 
PLATO 
directions are at the same distance from the middle (centre) ’.* 
The ‘ sphere ’ is similarly defined as ‘ that which has the 
distances from its centre to its terminations or ends in every 
direction equal ’, or simply as that which is £ equal (equidistant) 
from the centre in all directions’. 2 
The Parmenides contains certain phrases corresponding to 
what we find in Euclid’s preliminary matter. Thus Plato 
speaks of something which is ‘ a part ’ but not 1 parts ’ of the 
One, 8 reminding us of Euclid’s distinction between a fraction 
which is ‘ a part ’, i. e. an aliquot part or submultiple, and one 
which is 1 parts i. e. some number more than one of such 
parts, e. g. f. If equals be added to unequals, the sums differ 
by the same amount as the original unequals did: 4 an axiom 
in a rather more complete form than that subsequently inter 
polated in Euclid. 
Summary of the mathematics in Plato. 
The actual arithmetical and geometrical propositions referred 
to or presupposed in Plato’s writings are not such as to suggest 
that he was in advance of his time in mathematics; his 
knowledge does not appear to have been more than up to 
date. In the following paragraphs I have attempted to give 
a summary, as complete as possible, of the mathematics con 
tained in the dialogues. 
A proposition in proportion is quoted in the Parmenides 
namely that, if a > b, then (a + c): (b + c) < a :b. 
In the Laius a certain number, 5,040, is selected as a most 
convenient number of citizens to form a state; its advantages 
are that it is the product of 12, 21 and 20, that a twelfth 
part of it is again divisible by 12, and that it has as many as 
59 different divisors in all, including all the natural numbers 
from 1 to 12 with the exception of 11, while it is nearly 
divisible by 11 (5038 being a multiple of ll). 6 
(a) Regular and semi-regular solids. 
The ‘so-called Platonic figures’, by which are meant the 
five regular solids, are of course not Plato’s discovery, for they 
had been partly investigated by the Pythagoreans, and very 
1 Parmenides, 137 E. 2 Timaeus, 33 B, 34 B, 
• 3 Parmenides, 153 n. 4 lb. 154 b. 
5 lb. 154 d. 6 Laws, 537 e-538 a.