Dialogues

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 (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 (Epistulae ad Atticum 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 (Politics 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. (Compare “Cratylus,” passim; “Protagoras” 342 and following)

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 ὧν ἐπίτριτος πυθμὴν, κ.τ.λ.) describes a solid figure.139 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 (3² + 4² = 5², or 9 + 16 = 25).

Plato begins by speaking of a perfect or cyclical number (compare “Timaeus” 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 (compare Aristotle). 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 8,000. This explanation derives a certain plausibility from the circumstance that 8,000 is the ancient number of the Spartan citizens (Herodotus 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 (Republic 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 explained as follows. A perfect number (τέλειος ἀριθμός), 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 ὄροι, “terms” or “notes,” and ἀποστάσεις, “intervals,” are applicable to music as well as to number and figure. Πρώτῳ is the “base” on which the whole calculation depends, or the “lowest term” from which it can be worked out. The words δυνάμεναί τε καὶ δυναστευόμενοι have been variously translated⁠—“squared and cubed” (Donaldson), “equalling and equalled in power” (Weber), “by involution and evolution,” i.e. by