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 (Greek) describes a solid figure. (Pol. – '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.') 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 (9 + 16 = 25).

Plato begins by speaking of a perfect or cyclical number (Tim.), 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 (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.), and would be what Plato might have called 'a number which nearly concerns the population of a city'; 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, 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 (Greek), 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), 'terms' or 'notes,' and (Greek), 'intervals,' are applicable to music as well as to number and figure. (Greek) is the 'base' on which the whole calculation depends, or the 'lowest term' from which it can be worked out. The words (Greek) 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) 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 cubed and 3 cubed; and conversely. 'Waxing' (Greek) numbers, called also 'increasing' (Greek), are those which are exceeded by the sum of their divisors: e.g. 12 and 18 are less than 16 and 21. 'Waning' (Greek) numbers, called also 'decreasing' (Greek) 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) 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) 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); the second harmony is an 'oblong' number (Greek), i.e. a number representing a figure of which the opposite sides only are equal. (Greek) = 'numbers squared from' or 'upon diameters'; (Greek) = 'rational,' i.e. omitting fractions, (Greek), '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).

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 (a series of seven terms, 1, 2, 3, 4, 9, 8, 27); and if we take this as the basis of our computation, we shall have two cube numbers (Greek), 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 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 new-fangled 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, cut-purses, 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 we not ask the same question about the men themselves, requesting some one to look into them who is able to penetrate the inner nature of man, and will not be panic-struck by the vain pomp of tyranny? I will suppose that he is one who has lived with him, and has seen him in family life, or perhaps in the hour of trouble and danger.

Assuming that we ourselves are the impartial judge for whom we seek, let us begin by comparing the individual and State, and ask first of all, whether the State is likely to be free or enslaved – Will there not be a little freedom and a great deal of slavery? And the freedom is of the bad, and the slavery of the good; and this applies to the man as well as to the State; for his soul is full of meanness and slavery, and the better part is enslaved to the worse. He cannot do what he would, and his mind is full of confusion; he is the very reverse of a freeman. The State will be poor and full of misery and sorrow; and the man's soul will also be poor and full of sorrows, and he will be the most miserable of men. No, not the most miserable, for there is yet a more miserable. 'Who is that?' The tyrannical man who has the misfortune also to become a public tyrant. 'There I suspect that you are right.' Say rather, 'I am sure;' conjecture is out of place in an enquiry of this nature. He is like a wealthy owner of slaves, only he has more of them than any private individual. You will say, 'The owners of slaves are not generally in any fear of them.' But why? Because the whole city is in a league which protects the individual. Suppose however that one of these owners and his household is carried off by a god into a wilderness, where there are no freemen to help him – will he not be in an agony of terror? – will he not be compelled to flatter his slaves and to promise them many things sore against his will? And suppose the same god who carried him off were to surround him with neighbours who declare that no man ought to have slaves, and that the owners of them should be punished with death. 'Still worse and worse! He will be in the midst of his enemies.' And is not our tyrant such a captive soul, who is tormented by a swarm of passions which he cannot indulge; living indoors always like a woman, and jealous of those who can go out and see the world?

Having so many evils, will not the most miserable of men be still more miserable in a public station? Master of others when he is not master of himself; like a sick man who is compelled to be an athlete; the meanest of slaves and the most abject of flatterers; wanting all things, and never able to satisfy his desires; always in fear and distraction, like the State of which he is the representative. His jealous, hateful, faithless temper grows worse with command; he is more and more faithless, envious, unrighteous, – the most wretched of men, a misery to himself and to others. And so let us have a final trial and proclamation; need we hire a herald, or shall I proclaim the result? 'Made the proclamation yourself.' The son of Ariston (the best) is of opinion that the best and justest of men is also the happiest, and that this is he who is the most royal master of himself; and that the unjust man is he who is the greatest tyrant of himself and of his State. And I add further – 'seen or unseen by gods or men.'

This is our first proof. The second is derived from the three kinds of pleasure, which answer to the three elements of the soul – reason, passion, desire; under which last is comprehended avarice as well as sensual appetite, while passion includes ambition, party-feeling, love of reputation. Reason, again, is solely directed to the attainment of truth, and careless of money and reputation. In accordance with the difference of men's natures, one of these three principles is in the ascendant, and they have their several pleasures corresponding to them. Interrogate now the three natures, and each one will be found praising his own pleasures and depreciating those of others. The money-maker will contrast the vanity of knowledge with the solid advantages of wealth. The ambitious man will despise knowledge which brings no honour; whereas the philosopher will regard only the fruition of truth, and will call other pleasures necessary rather than good. Now, how shall we decide between them? Is there any better criterion than experience and knowledge? And which of the three has the truest knowledge and the widest experience? The experience of youth makes the philosopher acquainted with the two kinds of desire, but the avaricious and the ambitious man never taste the pleasures of truth and wisdom. Honour he has equally with them; they are 'judged of him,' but he is 'not judged of them,' for they never attain to the knowledge of true being. And his instrument is reason, whereas their standard is only wealth and honour; and if by reason we are to judge, his good will be the truest. And so we arrive at the result that the pleasure of the rational part of the soul, and a life passed in such pleasure is the pleasantest. He who has a right to judge judges thus. Next comes the life of ambition, and, in the third place, that of money-making.

Twice has the just man overthrown the unjust – once more, as in an Olympian contest, first offering up a prayer to the saviour Zeus, let him try a fall. A wise man whispers to me that the pleasures of the wise are true and pure; all others are a shadow only. Let us examine this: Is not pleasure opposed to pain, and is there not a mean state which is neither? When a man is sick, nothing is more pleasant to him than health. But this he never found out while he was well. In pain he desires only to cease from pain; on the other hand, when he is in an ecstasy of pleasure, rest is painful to him. Thus rest or cessation is both pleasure and pain. But can that which is neither become both? Again, pleasure and pain are motions, and the absence of them is rest; but if so, how can the absence of either of them be the other? Thus we are led to infer that the contradiction is an appearance only, and witchery of the senses. And these are not the only pleasures, for there are others which have no preceding pains. Pure pleasure then is not the absence of pain, nor pure pain the absence of pleasure; although most of the pleasures which reach the mind through the body are reliefs of pain, and have not only their reactions when they depart, but their anticipations before they come. They can be best described in a simile. There is in nature an upper, lower, and middle region, and he who passes from the lower to the middle imagines that he is going up and is already in the upper world; and if he were taken back again would think, and truly think, that he was descending. All this arises out of his ignorance of the true upper, middle, and lower regions. And a like confusion happens with pleasure and pain, and with many other things. The man who compares grey with black, calls grey white; and the man who compares absence of pain with pain, calls the absence of pain pleasure. Again, hunger and thirst are inanitions of the body, ignorance and folly of the soul; and food is the satisfaction of the one, knowledge of the other. Now which is the purer satisfaction – that of eating and drinking, or that of knowledge? Consider the matter thus: The satisfaction of that which has more existence is truer than of that which has less. The invariable and immortal has a more real existence than the variable and mortal, and has a corresponding measure of knowledge and truth. The soul, again, has more existence and truth and knowledge than the body, and is therefore more really satisfied and has a more natural pleasure. Those who feast only on earthly food, are always going at random up to the middle and down again; but they never pass into the true upper world, or have a taste of true pleasure. They are like fatted beasts, full of gluttony and sensuality, and ready to kill one another by reason of their insatiable lust; for they are not filled with true being, and their vessel is leaky (Gorgias). Their pleasures are mere shadows of pleasure, mixed with pain, coloured and intensified by contrast, and therefore intensely desired; and men go fighting about them, as Stesichorus says that the Greeks fought about the shadow of Helen at Troy, because they know not the truth.

The same may be said of the passionate element: – the desires of the ambitious soul, as well as of the covetous, have an inferior satisfaction. Only when under the guidance of reason do either of the other principles do their own business or attain the pleasure which is natural to them. When not attaining, they compel the other parts of the soul to pursue a shadow of pleasure which is not theirs. And the more distant they are from philosophy and reason, the more distant they will be from law and order, and the more illusive will be their pleasures. The desires of love and tyranny are the farthest from law, and those of the king are nearest to it. There is one genuine pleasure, and two spurious ones: the tyrant goes beyond even the latter; he has run away altogether from law and reason. Nor can the measure of his inferiority be told, except in a figure. The tyrant is the third removed from the oligarch, and has therefore, not a shadow of his pleasure, but the shadow of a shadow only. The oligarch, again, is thrice removed from the king, and thus we get the formula 3 x 3, which is the number of a surface, representing the shadow which is the tyrant's pleasure, and if you like to cube this 'number of the beast,' you will find that the measure of the difference amounts to 729; the king is 729 times more happy than the tyrant. And this extraordinary number is NEARLY equal to the number of days and nights in a year (365 x 2 = 730); and is therefore concerned with human life. This is the interval between a good and bad man in happiness only: what must be the difference between them in comeliness of life and virtue!

Perhaps you may remember some one saying at the beginning of our discussion that the unjust man was profited if he had the reputation of justice. Now that we know the nature of justice and injustice, let us make an image of the soul, which will personify his words. First of all, fashion a multitudinous beast, having a ring of heads of all manner of animals, tame and wild, and able to produce and change them at pleasure. Suppose now another form of a lion, and another of a man; the second smaller than the first, the third than the second; join them together and cover them with a human skin, in which they are completely concealed. When this has been done, let us tell the supporter of injustice that he is feeding up the beasts and starving the man. The maintainer of justice, on the other hand, is trying to strengthen the man; he is nourishing the gentle principle within him, and making an alliance with the lion heart, in order that he may be able to keep down the many-headed hydra, and bring all into unity with each other and with themselves. Thus in every point of view, whether in relation to pleasure, honour, or advantage, the just man is right, and the unjust wrong.

But now, let us reason with the unjust, who is not intentionally in error. Is not the noble that which subjects the beast to the man, or rather to the God in man; the ignoble, that which subjects the man to the beast? And if so, who would receive gold on condition that he was to degrade the noblest part of himself under the worst? – who would sell his son or daughter into the hands of brutal and evil men, for any amount of money? And will he sell his own fairer and diviner part without any compunction to the most godless and foul? Would he not be worse than Eriphyle, who sold her husband's life for a necklace? And intemperance is the letting loose of the multiform monster, and pride and sullenness are the growth and increase of the lion and serpent element, while luxury and effeminacy are caused by a too great relaxation of spirit. Flattery and meanness again arise when the spirited element is subjected to avarice, and the lion is habituated to become a monkey. The real disgrace of handicraft arts is, that those who are engaged in them have to flatter, instead of mastering their desires; therefore we say that they should be placed under the control of the better principle in another because they have none in themselves; not, as Thrasymachus imagined, to the injury of the subjects, but for their good. And our intention in educating the young, is to give them self-control; the law desires to nurse up in them a higher principle, and when they have acquired this, they may go their ways.

'What, then, shall a man profit, if he gain the whole world' and become more and more wicked? Or what shall he profit by escaping discovery, if the concealment of evil prevents the cure? If he had been punished, the brute within him would have been silenced, and the gentler element liberated; and he would have united temperance, justice, and wisdom in his soul – a union better far than any combination of bodily gifts. The man of understanding will honour knowledge above all; in the next place he will keep under his body, not only for the sake of health and strength, but in order to attain the most perfect harmony of body and soul. In the acquisition of riches, too, he will aim at order and harmony; he will not desire to heap up wealth without measure, but he will fear that the increase of wealth will disturb the constitution of his own soul. For the same reason he will only accept such honours as will make him a better man; any others he will decline. 'In that case,' said he, 'he will never be a politician.' Yes, but he will, in his own city; though probably not in his native country, unless by some divine accident. 'You mean that he will be a citizen of the ideal city, which has no place upon earth.' But in heaven, I replied, there is a pattern of such a city, and he who wishes may order his life after that image. Whether such a state is or ever will be matters not; he will act according to that pattern and no other…