Timaeus — Plato

a-Kk^-nvaiy) : (2) that the number 216 with its permutations would have been famihar to a Greek mathematician, though unfamiliar to us: (3) that 216 is the cube of 6, and also the sum of 3^, 4^^ ^3^ the numbers 3, 4, 5 representing the Pythagorean triangle, of which the sides when squared equal the square of the hypotenuse (3^ + 4^ = 5^) • (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. iii), 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 {yanifALov 6idypafj.fj.a).

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 apprjroyv 6e Svelv can mean 'two incommensurables,' which he arbitrarily assumes to be 2 and 3, but rather, as the preceding clause implies, dvelv apid/uolv a-ird dpprjruv diajuerpuv ireju- 7rd(hg, 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 kniTpiTog nvd/uT^v K.T.I. , * 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 harmpny 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

The Number of the State.

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 + 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 i, 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.

Republic VIII. Introditc- TION.

Steph. BOOK IX. Last of all comes the tyrannical man, about whom p^^, 571 we have to enquire. Whence is he, and how does he live — in happiness or in misery ? There is, however, a previous question of the nature and number of the appetites, which I should like to consider first. Some of them are unlawful, and yet admit of being chastened and weakened in various degrees by the power of reason and law. ' What appetites do you mean ? ' I mean those which are awake when the reasoning powers are asleep, which get up and walk about naked without any self-respect or shame ; and there is no conceivable folly or crime, however cruel or unnatural, of which, in imagination, they may not be guilty. 'True,' he said; * very true.' But when a man's pulse beats temperately, and he has supped on a feast of reason and come to a knowledge of himself

ALYSIS.

CXXXVl

Analysis 572-574.