the
divine or perfect number in which all lesser cycles or revolutions are
complete. He also speaks of a human or imperfect number, having four
terms and three intervals of numbers which are related to one another in
certain proportions; these he converts into figures, and finds in
them when they have been raised to the third power certain elements of
number, which give two 'harmonies,' the one square, the other oblong;
but he does not say that the square number answers to the divine, or the
oblong number to the human cycle; nor is any intimation given that the
first or divine number represents the period of the world, the second
the period of the state, or of the human race as Zeller supposes; nor
is the divine number afterwards mentioned (Arist.). The second is the
number of generations or births, and presides over them in the same
mysterious manner in which the stars preside over them, or in which,
according to the Pythagoreans, opportunity, justice, marriage, are
represented by some number or figure. This is probably the number 216.

The explanation given in the text supposes the two harmonies to make up
the number 8000. This explanation derives a certain plausibility from
the circumstance that 8000 is the ancient number of the Spartan citizens
(Herod.), and would be what Plato might have called 'a number which
nearly concerns the population of a city'; the mysterious disappearance
of the Spartan population may possibly have suggested to him the first
cause of his decline of States. The lesser or square 'harmony,' of 400,
might be a symbol of the guardians,--the larger or oblong 'harmony,'
of the people, and the numbers 3, 4, 5 might refer respectively to the
three orders in the State or parts of the soul, the four virtues, the
five forms of government. The harmony of the musical scale, which
is elsewhere used as a symbol of the harmony of the state, is also
indicated. For the numbers 3, 4, 5, which represent the sides of the
Pythagorean triangle, also denote the intervals of the scale.

The terms used in the statement of the problem may be explained as
follows. A perfect number (Greek), as already stated, is one which is
equal to the sum of its divisors. Thus 6, which is the first perfect or
cyclical number, = 1 + 2 + 3. The words (Greek), 'terms' or 'notes,' and
(Greek), 'intervals,' are applicable to music as well as to number and
figure. (Greek) is the 'base' on which the whole calculation depends,
or the 'low