The Republic, Plato [simple e reader txt] 📗

3/2. Now if we remember that the number 216 = 8 x 27 = 3 cubed + 4 cubed + 5 cubed, and 3 squared + 4 squared = 5 squared, 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) he elsewhere explains as follows: ‘The first (Greek) is (Greek), in other words (4/3 x 5) all squared = 100 x 2 squared over 3 squared. The second (Greek), a cube of the same root, is described as 100 multiplied (alpha) by the rational diameter of 5 diminished by unity, i.e., as shown above, 48: (beta) by two incommensurable diameters, i.e. the two first irrationals, or 2 and 3: and (gamma) by the cube of 3, or 27. Thus we have (48 + 5 + 27) 100 = 1000 x 2 cubed. 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.’

 

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…): (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 cubed, 4 cubed, 5 cubed, the numbers 3, 4, 5 representing the Pythagorean triangle, of which the sides when squared equal the square of the hypotenuse (9 + 16 = 25): (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.), Proclus (super prima Eucl.), and Quintilian (de Musica) 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).

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) can mean ‘two incommensurables,’ which he arbitrarily assumes to be 2 and 3, but rather, as the preceding clause implies, (Greek), 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), ‘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 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), 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’ 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.

BOOK IX. Last of all comes the tyrannical man, about whom 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 before going to rest, and has satisfied his desires just enough to prevent their perturbing his reason, which remains clear and luminous, and when he is free from quarrel and heat,—the visions which he has on his bed are least irregular and abnormal. Even in good men there is such an irregular wild-beast nature, which peers out in sleep.

To return:—You remember what was said of the democrat; that he was the son of a miserly father, who encouraged the saving desires and repressed the ornamental and expensive ones; presently the youth got into fine company, and began to entertain a dislike to his father’s narrow ways; and being a better man than the corrupters of his youth, he came to a mean, and led a life, not of lawless or slavish passion, but of regular and successive indulgence. Now imagine that the youth has become a father, and has a son who is exposed to the same temptations, and has companions who lead him into every sort of iniquity, and parents and friends who try to keep him right. The counsellors of evil find that their only chance of retaining him is to implant in his soul a monster drone, or love; while other desires buzz around him and mystify him with sweet sounds and scents, this monster love takes possession of him, and puts an end to every true or modest thought or wish. Love, like drunkenness and madness, is a tyranny; and the tyrannical man, whether made by nature or habit, is just a drinking, lusting, furious sort of animal.

And how does such an one live? ‘Nay, that you must tell me.’ Well then, I fancy that he will live amid revelries and harlotries, and love will be the lord and master of the house. Many desires require much money, and so he spends all that he has and borrows more; and when he has nothing the young ravens are still in the nest in which they were hatched, crying for food. Love urges them on; and they must be gratified by force or fraud, or if not, they become painful and troublesome; and as the new pleasures succeed the old ones, so will the son take possession of the goods of his parents; if they show signs of refusing, he will defraud and deceive them; and if they openly resist, what then? ‘I can only say, that I should not much like to be in their place.’ But, O heavens, Adeimantus, to think that for some newfangled and unnecessary love he will give up his old father and mother, best and dearest of friends, or enslave them to the fancies of the hour! Truly a tyrannical son is a blessing to his father and mother! When there is no more to be got out of them, he turns burglar or pickpocket, or robs a temple. Love overmasters the thoughts of his youth, and he becomes in sober reality the monster that he was sometimes in sleep. He waxes strong in all violence and lawlessness; and is ready for any deed of daring that will supply the wants of his rabble-rout. In a well-ordered State there are only a few such, and these in time of war go out and become the mercenaries of a tyrant. But in time of peace they stay at home and do mischief; they are the thieves, footpads, cutpurses, man-stealers of the community; or if they are able to speak, they turn false-witnesses and informers. ‘No small catalogue of crimes truly, even if the perpetrators are few.’ Yes, I said; but small and great are relative terms, and no crimes which are committed by them approach those of the tyrant, whom this class, growing strong and numerous, create out of themselves. If the people yield, well and good, but, if they resist, then, as before he beat his father and mother, so now he beats his fatherland and motherland, and places his mercenaries over them. Such men in their early days live with flatterers, and they themselves flatter others, in order to gain their ends; but they soon discard their followers when they have no longer any need of them; they are always either masters or servants,—the joys of friendship are unknown to them. And they are utterly treacherous and unjust, if the nature of justice be at all understood by us. They realize our dream; and he who is the most of a tyrant by nature, and leads the life of a tyrant for the longest time, will be the worst of them, and being the worst of them, will also be the most miserable.

Like man, like State,—the tyrannical man will answer to tyranny, which is the extreme opposite of the royal State; for one is the best and the other the worst. But which is the happier? Great and terrible as the tyrant may appear enthroned amid his satellites, let us not be afraid to go in and ask; and the answer is, that the monarchical is the happiest, and the tyrannical the most miserable of States. And may