The Republic
and who in his utter helplessness and suspiciousness is leading an
almost impossible existence, without that remnant of good which, in
Plato’s opinion, was required to give power to evil (Book I). This
ideal of wickedness living in helpless misery, is the reverse of that
other portrait of perfect injustice ruling in happiness and splendour,
which first of all Thrasymachus, and afterwards the sons of Ariston had
drawn, and is also the reverse of the king whose rule of life is the
good of his subjects.
Each of these governments and individuals has a corresponding ethical
gradation: the ideal State is under the rule of reason, not
extinguishing but harmonizing the passions, and training them in
virtue; in the timocracy and the timocratic man the constitution,
whether of the State or of the individual, is based, first, upon
courage, and secondly, upon the love of honour; this latter virtue,
which is hardly to be esteemed a virtue, has superseded all the rest.
In the second stage of decline the virtues have altogether disappeared,
and the love of gain has succeeded to them; in the third stage, or
democracy, the various passions are allowed to have free play, and the
virtues and vices are impartially cultivated. But this freedom, which
leads to many curious extravagances of character, is in reality only a
state of weakness and dissipation. At last, one monster passion takes
possession of the whole nature of man—this is tyranny. In all of them
excess—the excess first of wealth and then of freedom, is the element
of decay.
The eighth book of the Republic abounds in pictures of life and
fanciful allusions; the use of metaphorical language is carried to a
greater extent than anywhere else in Plato. We may remark,
(1), the description of the two nations in one, which become more and
more divided in the Greek Republics, as in feudal times, and perhaps
also in our own;
(2), the notion of democracy expressed in a sort of Pythagorean formula
as equality among unequals;
(3), the free and easy ways of men and animals, which are
characteristic of liberty, as foreign mercenaries and universal
mistrust are of the tyrant;
(4), the proposal that mere debts should not be recoverable by law is a
speculation which has often been entertained by reformers of the law in
modern times, and is in harmony with the tendencies of modern
legislation. Debt and land were the two great difficulties of the
ancient lawgiver: in modern times we may be said to have almost, if not
quite, solved the first of these difficulties, but hardly the second.
Still more remarkable are the corresponding portraits of individuals:
there is the family picture of the father and mother and the old
servant of the timocratical man, and the outward respectability and
inherent meanness of the oligarchical; the uncontrolled licence and
freedom of the democrat, in which the young Alcibiades seems to be
depicted, doing right or wrong as he pleases, and who at last, like the
prodigal, goes into a far country (note here the play of language by
which the democratic man is himself represented under the image of a
State having a citadel and receiving embassies); and there is the
wild-beast nature, which breaks loose in his successor. The hit about
the tyrant being a parricide; the representation of the tyrant’s life
as an obscene dream; the rhetorical surprise of a more miserable 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 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 ‘lowest term’ from which it can be worked out. The
words (Greek) have been variously translated—‘squared and cubed’
(Donaldson), ‘equalling and equalled in power’ (Weber), ‘by involution
and evolution,’ i.e. by raising the power and extracting the root (as
in the translation). Numbers are called ‘like and unlike’ (Greek) when
the factors or the sides of the planes and cubes which they represent
are or are not in the same ratio: e.g. 8 and 27 = 2 cubed and 3 cubed;
and conversely. ‘Waxing’ (Greek) numbers, called also ‘increasing’
(Greek), are those which are exceeded by the sum of their divisors:
e.g. 12 and 18 are less than 16 and 21. ‘Waning’ (Greek) numbers,
called also ‘decreasing’ (Greek) are those which succeed the sum of
their divisors: e.g. 8 and 27 exceed 7 and 13. The words translated
‘commensurable and agreeable to one another’ (Greek) seem to be
different ways of describing the same relation, with more or less
precision. They are equivalent to ‘expressible in terms having the same
relation to one another,’ like the series 8, 12, 18, 27, each of which
numbers is in the relation of (1 and 1/2) to the preceding. The ‘base,’
or ‘fundamental number, which has 1/3 added to it’ (1 and 1/3) = 4/3 or
a musical fourth. (Greek) is a ‘proportion’ of numbers as of musical
notes, applied either to the parts or factors of a single number or to
the relation of one number to another. The first harmony is a ‘square’
number (Greek); the second harmony is an ‘oblong’ number (Greek), i.e.
a number representing a figure of which the opposite sides only are
equal. (Greek) = ‘numbers squared from’ or ‘upon diameters’; (Greek) =
‘rational,’ i.e. omitting fractions, (Greek), ‘irrational,’ i.e.
including fractions; e.g. 49 is a square of the rational diameter of a
figure the side of which = 5: 50, of an irrational diameter of the
same. For several of the explanations here given and for a good deal
besides I am indebted to an excellent article on the Platonic Number by
Dr. Donaldson (Proc. of the Philol. Society).
The conclusions which he draws from these data are summed up by him as
follows. Having assumed that the number of the perfect or divine cycle
is the number of the world, and the number of the imperfect cycle the
number of the state, he proceeds: ‘The period of the world is defined
by the perfect number 6, that of the state by the cube of that number
or 216, which is the product of the last pair of terms in the Platonic
Tetractys (a series of seven terms, 1, 2, 3, 4, 9, 8, 27); and if we
take this as the basis of our computation, we shall have two cube
numbers (Greek), viz. 8 and 27; and the mean proportionals between
these, viz. 12 and 18, will furnish three intervals and four terms, and
these terms and intervals stand related to one another in the
sesqui-altera ratio, i.e. each term is to the preceding as 3/2. Now if
we remember that the number 216 = 8 x 27 = 3 cubed + 4 cubed + 5 cubed,
and 3 squared + 4 squared = 5 squared, we must admit that this number
implies the numbers 3, 4, 5, to which musicians attach so much
importance. And if we combine the ratio 4/3 with the number 5, or
multiply the ratios of the sides by the hypotenuse, we shall by first
squaring and then cubing obtain two expressions, which denote the ratio
of the two last pairs of terms in the Platonic Tetractys, the former
multiplied by the square, the latter by the cube of the number 10, the
sum of the first four digits which constitute the Platonic Tetractys.’