THE REGULAR SOLIDS 
297 
latter into six scalene triangles. If we draw lines in a pen 
tagon as shown in the accompanying figure, we obtain such 
a set of triangles in a way which also shows the Pythagorean 
pentagram (cf. p, 161, above). 
(y) Geometric means between two square numbers 
or two cubes. 
In the Timaeus Plato, speaking of numbers ‘ whether solid 
or square ’ with a (geometric) mean or means between them, 
observes that between planes one mean suffices, but to connect 
two solids two means are necessary, 1 By planes and solids 
Plato probably meant square and cube numbers respectively, 
so that the theorems quoted are probably those of Eucl. VIII. 
11, 12, to the effect that between two square numbers there is 
one mean proportional number, and between two cube numbers 
two mean proportional numbers. Nicornachus quotes these 
very propositions as constituting ‘ a certain Platonic theorem ’. 2 
Here, too, it may be that the theorem is called ‘ Platonic ’ for 
the sole reason that it is quoted by Plato in the Timaeus; 
it may well be older, for the idea of two mean proportionals 
between two straight lines had already appeared in Hippo 
crates’s reduction of the problem of doubling the cube. Plato’s 
allusion does not appear to be to the duplication of the cube 
in this passage any more than in the expression kv(3ow av^rj, 
‘ cubic increase ’, in the Republic, 3 which appears to be nothing 
but the addition of the third dimension to a square, making 
a cube (cf. rpirq av£rj, ‘ third increase ’, 4 meaning a cube 
number as compared with Svvapus, a square number, pterins 
which are applied, e. g, to the numbers 729 and 81 respec 
tively). 
(S) The two geometrical passages in the Meno. 
We come now to the two geometrical passages in the Meno. 
In the first 5 Socrates is trying to show that teaching is only 
reawaking in the mind of the learner the memory of some 
thing. He illustrates by putting to the slave a carefully 
prepared series of questions, each requiring little more than 
1 Timaeus, 31 c-B2 B. 
3 Republic, 528 b. 
2 Nicom. ii. 24. 6. 
4 lb. 587 D. 
6 Meno, 82 B-85 b.