rms or bases (3, 4, 5) of which 216 is composed answer to the third,
fourth, fifth in the musical scale: (6) that the number 216 is the
product of the cubes of 2 and 3, which are the two last terms in
the Platonic Tetractys: (7) that the Pythagorean triangle is said by
Plutarch (de Is. et Osir.), Proclus (super prima Eucl.), and Quintilian
(de Musica) to be contained in this passage, so that the tradition
of the school seems to point in the same direction: (8) that the
Pythagorean triangle is called also the figure of marriage (Greek).

But though agreeing with Dr. Donaldson thus far, I see no reason for
supposing, as he does, that the first or perfect number is the world,
the human or imperfect number the state; nor has he given any proof that
the second harmony is a cube. Nor do I think that (Greek) can mean
'two incommensurables,' which he arbitrarily assumes to be 2 and 3,
but rather, as the preceding clause implies, (Greek), i.e. two square
numbers based upon irrational diameters of a figure the side of which is
5 = 50 x 2.

The greatest objection to the translation is the sense given to the
words (Greek), 'a base of three with a third added to it, multiplied
by 5.' In this somewhat forced manner Plato introduces once more the
numbers of the Pythagorean triangle. But the coincidences in the numbers
which follow are in favour of the explanation. The first harmony of 400,
as has been already remarked, probably represents the rulers; the second
and oblong harmony of 7600, the people.

And here we take leave of the difficulty. The discovery of the riddle
would be useless, and would throw no light on ancient mathematics. The
point of interest is that Plato should have used such a symbol, and
that so much of the Pythagorean spirit should have prevailed in him. His
general meaning is that divine creation is perfect, and is represented
or presided over by a perfect or cyclical number; human generation is
imperfect, and represented or presided over by an imperfect number or
series of numbers. The number 5040, which is the number of the citizens
in the Laws, is expressly based by him on utilitarian grounds, namely,
the convenience of the number for division; it is also made up of
the first seven digits multiplied by one another. The contrast of the
perfect and imperfect number may have been easily suggested by the
corrections of the cycle, which were made first by Meton and secondly
by Callippus; (the latter is said to hav