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.]