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 (see infra, 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. vii. 13, 5). 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. v. 12, § 7), 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. (Cp. Cratylus _passim_; Protag. 342 ff.)

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 (from [Greek: ô(=n e)pi/tritos puthmê\n, k.t.l.]) describes a

solid figure. 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 {cxxxi} triangle, the squares of the two

lesser sides equal the square of the hypotenuse (3^2 + 4^2 = 5^2, or

9 + 16 = 25).

[Footnote 2: Pol. v. 12, § 8:--'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.']

Plato begins by speaking of a perfect or cyclical number (cp. Tim. 39 D),

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 (cp. 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. vii. 34), and would be what Plato might have called 'a number

which nearly concerns the population of a city' (588 A); 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 (Rep. iv. 443 D),

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 {cxxxii} explained

as follows. A perfect number ([Greek: te/leios a)rithmo/s]), 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:

o)/roi], 'terms' or 'notes,' and [Greek: a)posta/seis], 'intervals,' are

applicable to music as well as to number and figure. [Greek: Prô/tô|] is

the 'base' on which the whole calculation depends, or the 'lowest term'

from which it can be worked out. The words [Greek: duna/menai/ te kai\

dunasteuo/menoi] 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: o(moiou=nte/s

te kai\ a)nomoiou=ntes]) 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 = 2^3 and 3^3; and conversely. 'Waxing' ([Greek: au)/xontes]) numbers,

called also 'increasing' ([Greek: u(pertelei=s]) are those which are

exceeded by the sum of their divisors: e.g. 12 and 18 are less than 16 and

21. 'Waning' ([Greek: phthi/nontes]) numbers, called also 'decreasing'

([Greek: e)llipei=s]) 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: prosê/gora kai\ r(êta/]) 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: A(rmoni/a] 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: i)/sên i)sa/kis]); the second harmony is an

'oblong' number ([Greek: promê/kê]), i.e. a number representing a figure

of which the opposite sides only are equal. [Greek: A)rithmoi\ a)po\

diame/trôn] = 'numbers squared from' or 'upon diameters'; [Greek: r(êtô=n]

= 'rational,' i.e. omitting fractions, [Greek: a)r)r(ê/tôn], '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, vol. i. p. 81 ff.).

{cxxxiii} 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[3]; and if we take this as the basis of our computation, we

shall have two cube numbers ([Greek: au)xê/seis duna/menai/ te kai\

dunasteuo/menai]), 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 = 3^3 + 4^3 + 5^3, and 3^2 + 4^2

= 5^2, 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.' The two [Greek: a(rmoni/ai] he elsewhere explains as follows:

'The first [Greek: a(rmoni/a] is [Greek: i)/sên i)sa/kis e(kato\n

tosauta/kis], in other words (4/3 x 5)^2 = 100 x 2^2/3^2. The second

[Greek: a(rmoni/a], a cube of the same root, is described as 100

multiplied ([Greek: a]) by the rational diameter of 5 diminished by unity,

i.e., as shown above, 48: ([Greek: b]) by two incommensurable diameters,

i.e. the two first irrationals, or 2 and 3: and ([Greek: g]) by the cube

of 3, or 27. Thus we have (48 + 5 + 27) 100 = 1000 x 2^3. This second

harmony is to be the cube of the number of which the former harmony is the

square, and therefore must be divided by the cube of 3. In other words,

the whole expression will be: (1), for the first harmony, 400/9: (2), for

the second harmony, 8000/27.'

[Footnote 3: The Platonic Tetractys consisted of a series of seven terms,

1, 2, 3, 4, 9, 8, 27.]

The reasons which have inclined me to agree with Dr. Donaldson and also

with Schleiermacher in supposing that 216 is the Platonic number of births

are: (1) that it coincides with the description of the number given in the

first part of the passage ([Greek: e)n ô(=| prô/tô| ... {cxxxiv}

a)pe/phêsan]): (2) that the number 216 with its permutations would have

been familiar to a Greek mathematician, though unfamiliar to us: (3) that

216 is the cube of 6, and also the sum of 3^3, 4^3, 5^3, the numbers 3, 4,

5 representing the Pythagorean triangle, of which the sides when squared

equal the square of the hypotenuse (3^2 + 4^2 = 5^2): (4) that it is also

the period of the Pythagorean Metempsychosis: (5) the three ultimate terms

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.,

373 E), Proclus (super prima Eucl. iv. p. 111), and Quintilian (de Musica

iii. p. 152) 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: gamê/lion

dia/gramma]).

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: a)r)r(ê/tôn de\

duei=n] can mean 'two incommensurables,' which he arbitrarily assumes to

be 2 and 3, but rather, as the preceding clause implies, [Greek: duei=n

a)rithmoi=n a)po\ a)r)r(ê/tôn diame/trôn pempa/dos], 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: e)pi/tritos puthmê/n k.t.l.], '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 {cxxxv} 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 have been a pupil of Plato). Of the degree of importance

or of exactness to be attributed to the problem, the number of the tyrant

in Book ix. (729 = 365 x 2), and the slight correction of the error in the

number 5040/12 (Laws, 771 C), may furnish a criterion. There is nothing

surprising in the circumstance that those who were seeking for order in

nature and had found order in number, should have imagined one to give law

to the other. Plato believes in a power of number far beyond what he could

see realized in the world around him, and he knows the great influence

which 'the little matter of 1, 2, 3' (vii. 522 C) exercises upon

education. He may even be thought to have a prophetic anticipation of the

discoveries of Quetelet and others, that numbers depend upon numbers;

e.g.--in population, the numbers of births and the respective numbers of

children born of either sex, on the respective ages of parents, i.e. on

other numbers.

* * * * *

[Sidenote: _Republic IX._ Analysis.]