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 {cxxx} 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 (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).