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.