TWO GEOMETRICAL PASSAGES IN THE ME NO 303 
to the other square. If this were the correct interpretation, 
Plato is using much too general language about the applied 
rectangle and that by which it is deficient; it would be 
extraordinary that he should express the condition in this 
elaborate way when he need only have said that the radius 
of the circle must be equal to the side of the square and 
therefore 2 feet in length. The explanation seems to me 
incredible. The criterion sought by Socrates is evidently 
intended to be a real Siopio-pos, or determination of the 
conditions or limits of the possibility of a solution of the pro 
blem whether in its original form or in the form to which 
it is reduced; but it is no real SLopLa-pos to say what is 
equivalent to saying that the problem is possible of solution 
if the circle is of a particular size, but impossible if the circle 
is greater or less than that size. 
The passage incidentally shows that the idea of a formal 
Siopurpos defining the limits of possibility of solution was 
familiar even before Plato’s time, and therefore that Proclus 
must be in error when he says that Leon, the pupil of 
Neoclides, ‘ invented SLopurpoi (determining) when the problem 
which is the subject of investigation is possible and when 
impossible V although Leon may have been the first to intro 
duce the term or to recognize formally the essential part 
played by SiopLcr/xoi in geometry. 
(e) Plato and the doubling of the cube. 
The story of Plato’s relation to the problem of doubling 
the cube has already been told (pp. 245-6, 255). Although the 
solution attributed to him is not his, it may have been with 
this problem in view that he complained that the study of 
solid geometry had been unduly neglected up to his time. 2 
and he maintains that, if Plato had meant it in this sense, he should 
have added that the * defect ’, although ‘ similar ’, is not similarly situated. 
I see no force in this argument in view of the want of fixity in mathe 
matical terminology in Plato’s time, and of his own habit of varying his 
phrases for literary effect. Benecke makes the words mean ‘ of the same 
hind', e. g. a square with a square or a rectangle with a rectangle. But 
this would have no point unless the figures are squares, which begs the 
whole question. 
1 Proclus on Eucl. I, p. 66. 20-2. 
2 Republic, vii, 528 a-c.