than the most miserable of
men in Book IX; the hint to the poets that if they are the friends of
tyrants there is no place for them in a constitutional State, and that
they are too clever not to see the propriety of their own expulsion; the
continuous image of the drones who are of two kinds, swelling at last
into the monster drone having wings (Book IX),--are among Plato's
happiest touches.

There remains to be considered the great difficulty of this book of the
Republic, the so-called number of the State. This is a puzzle almost as
great as the Number of the Beast in the Book of Revelation, and though
apparently known to Aristotle, is referred to by Cicero as a proverb of
obscurity (Ep. ad Att.). And some have imagined that there is no answer
to the puzzle, and that Plato has been practising upon his readers.
But such a deception as this is inconsistent with the manner in which
Aristotle speaks of the number (Pol.), and would have been ridiculous to
any reader of the Republic who was acquainted with Greek mathematics.
As little reason is there for supposing that Plato intentionally used
obscure expressions; the obscurity arises from our want of familiarity
with the subject. On the other hand, Plato himself indicates that he is
not altogether serious, and in describing his number as a solemn jest of
the Muses, he appears to imply some degree of satire on the symbolical
use of number. (Compare Cratylus; Protag.)

Our hope of understanding the passage depends principally on an accurate
study of the words themselves; on which a faint light is thrown by the
parallel passage in the ninth book. Another help is the allusion in
Aristotle, who makes the important remark that the latter part of the
passage (Greek) describes a solid figure. (Pol.--'He only says that
nothing is abiding, but that all things change in a certain cycle; and
that the origin of the change is a base of numbers which are in the
ratio of 4:3; and this when combined with a figure of five gives two
harmonies; he means when the number of this figure becomes solid.')
Some further clue may be gathered from the appearance of the Pythagorean
triangle, which is denoted by the numbers 3, 4, 5, and in which, as in
every right-angled triangle, the squares of the two lesser sides equal
the square of the hypotenuse (9 + 16 = 25).

Plato begins by speaking of a perfect or cyclical number (Tim.), i.e.
a number in which the sum of the divisors equals the whole; this is