Republic Book VIII Arithmetic
Excerpt from Plato Republic Book VIII
Discussing the establishment of a perfect state
composed of a perfect government
Using Geometry and The Real relationships
of music intervals and structure
to Explain perfection
"Then, Glaucon," I said, "how will our city be moved and in what way will the auxiliaries and the rulers divide into factions against each other and among themselves? Or do you want us, as does Homer, to pray to the Muses to tell us how 'faction first attacked,'^ and shall we
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say that they speak to" us with high tragic talk, as though they were speaking seriously, playing and jesting with us like children?"
"How would they address us?"
"Something like this. After this mannner: A city so composed can hardly be shaken.
But, seeing that everything which has has a beginning has also an end, not even a constitution such as this will not remain for all time; it will in time be
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dissolved.
And this will be its dissolution:
b
In the plants that grow in the earth, as well as the animals that move on the earths surface, fertility and sterility of soul and body occur when the circumferences of the circles of each are completed, which in short lived existences pass over a short space, and in long-lived ones over a long space.
But to the knowledge of human fecundity and sterility all the wisdom and education of your rulers will not attain; the laws which regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and they will bring "children" into the world when they ought not.
3^2 + 4^2 = 5^2
9 + 16 = 25
3^3 + 4^3 + 5^3 = 6^3
27 + 64 + 125 = 216
3^3 X 4^3 = 12^3
27 X 64 = 1728
3^3 X 6 X 2^3 = 1296
27 is the root of all A notes 27 : 54 : 108 : 216 : 432 : 864 : 1728 : 3456
containing
the radius and dimeter of the sun
64 is the root of all C notes 64 128 256 512 1024 centered at 2^8
1728 is A6 in perfect music notation using 432 as the center of all eight octaves
‘Something like this. “It is no easy matter for a city founded in this way to be altered. But since destruction awaits everything that has come to be, even a foundation of this kind will not survive for the whole of time. It will fall apart, and this will be the manner of its falling. Both for plants in the ground and for animals above the ground it is a fact that souls and bodies are produced or not produced when the cycles of begetting for each species complete their revolutions — short revolutions for short-lived species, and the opposite for long-lived species. In the case of your species, wise though the people you have educated as leaders of the city are, still they will not quite hit the mark when they apply calculation - together with observation - to their programme of breeding and "birth" of legislation control.
Success will elude them, and they will sometimes produce "children" they should not produce. For the birth of a divine being there is a period embraced by a perfect number, while for a human being it is the first number in which: increase by the power of roots combined with squares - taking on three dimensions and four defining limits - of the numbers which create likeness and unlikeness, and which wax and wane, makes all things converse-able and rational with one another.
Of these numbers the ones that form the basis of the musical fourth, when coupled with five and three times increased, produce two harmonies. The first harmony is a square, the product of equal times equal, so many times 100. The second harmony is of equal length one way and forms a rectangle. One side is the square of the rational diagonal of a five-by-five square, minus one, times ioo, or the square of the irrational diagonal of a five-by-five square, minus two, times ioo. The other side is three cubed times ioo. Taken as a whole, this geometrical number is master of this domain - of better and worse "births"
Now for divine "begettings" there is a period (three complete octaves)
comprehended by a perfect number (1296 = 36 X 36) and
(1,296,000 = 360 degrees in a circle X 60 Minutes X 60 Seconds) and
180 + 144 + 216 + 324 + 432 = 1296
and for mortals by the first in which augmentations, dominating and dominated when they have attained to three distances and four degrees of the assimilating (increasing) and the dissimilating (decreasing), the waxing and the waning, render "all things" conversable and commensurable with one another,
[546c]
because the trisected depth of the five-part conjugation of two harmonies is provided with three increases,
4/3 X 1296 = 1728 / 3 = 576
the one with equal equal times, one hundred times the same 2.4 X 2.4 X 100 = 576
and the other with equal length but oblong,
one dimension one hundred times the numbers from the diameters of the five-part conjugations,
Lacking one for the rational,
2.3 X 2.5 X 100 = 575or of the irrational lacking two
and
the other dimension one hundred cubes of the triad. (3 X 3 X 3 = 27) (2X3 X 2X2X2 = 48)
And this entire geometrical number is determinative of this thing, of better and inferior births.
And when your guardians from ignorance of them cause "grooms" to live with "brides" out of season, the children will have neither good natures nor good luck.
Their predecessors will choose the best of these children; but, nevertheless, since they are unworthy, when they, in turn, come to the powers of their fathers, they will as guardians first begin to neglect us by having less consideration than is required, first, for music, and, second, for gymnastic; and from there your young will become more unmusical.
And rulers chosen from them won't be guardians very apt at testing Hesiod's races and yours—gold and silver and bronze and iron. And the chaotic mixing of iron with silver and of bronze with gold engenders unlikeness and inharmonious irregularity, which, once they arise, always breed war and hatred in the place where they happen to arise. Faction must always be said to be "of this ancestry" wherever it happens to rise."
"And we'll say," he said, "that what the Muses answer is right."
"Necessarily," I said.
"For they are Muses."
"What," he said, "do the Muses say next?"
"Once faction had arisen," I said, "each of these two races, the iron and bronze, pulled the regime toward money-making and the possession of land, houses, gold, and silver; while the other two, the gold and the silver—not being poor but rich by nature—led the souls toward virtue and the ancient establishment. Struggling and straining against one another, they came to an agreement on a middle way: they distributed land and houses to be held privately, while those
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who previously were guarded by them as free friends and supporters they then enslaved and held as serfs and domestics; and they occupied themselves with war and with guarding against these men."
"In my opinion," he said, "this is the source of this transformation."
"Wouldn't this regime," I said, "be a certain middle between aristocracy and oligarchy?"
"Most certainly." "This will be the way of the transformation. But once transformed, how will it be governed? Or is it evident that in some things it will imitate
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the preceding regime; in others oligarchy, because it is a middle; and that it will also have something peculiar to itself?"
"That's the way it is," he said.
"In honoring the rulers, and in the abstention of its war-making part from farming and the manual arts and the rest of money-making; in its provision for common meals and caring for gymnastic and the exercise of war—in all such ways won't it imitate the preceding regime?"
"Yes."
"But in being afraid to bring the wise to the ruling offices—
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be-cause the men of that kind it possesses are no longer simple and earnest, but mixed—and in leaning toward spirited and simpler men, men naturally more directed to war than to peace; in holding the wiles and stratagems of war in honor; and in spending all its time making
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war; won't most such aspects be peculiar to this regime?"
"Yes."
"And such men," I said, "will desire money just as those in oligarchies do, and under cover of darkness pay fierce honor to gold and silver, because they possess storehouses and domestic treasuries where they can deposit and hide them; and they will have walls around their houses, exactly like private nests, where they can make lavish expenditures on women and whomever else they might wish."
b
"Very true," he said. "Then they will also be stingy with money because they honor it and don't acquire it openly; but, pushed on by desire, they will love to spend other people's money; and they will harvest pleasures stealthily, running away from the law like boys from a father. This is because they weren't educated by persuasion but by force—the result of neglect of the true Muse accompanied by arguments and philosphy while giving more distinguished honor to gymnastic than music."
c
"You certainly speak of a reigme," he said, "which is a mixture of bad and good.
"Yes, it is mixed," I said, "but due to the dominance of spiritedness one thing alone is most distinctive in it: love of victories and of honors."
"Very much so," he said.
"Then " I said, "this is the way this regime would come into being and what it would be like—
given the fact that we are only outlining a
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regime's figure in speech and not working out its details precisely, since even the outline is sufficient for seeing the justest man and the un-justest one, and it is an impractically long job to go through all regimes and all dispositions and leave nothing out."
"Right," he said.
"Who, then, is the man corresponding to this regime? How did he come into being and what sort of man is he?"
"I suppose," said Adeimantus, "that as far as love of victory goes, he'd be somewhere near to Glaucon here."
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"Perhaps in that," I said, "but in these other respects his nature does not, in my opinion, correspond to Glaucon's."
"Which respects?"
"He must be more stubborn," 1 said, "and somewhat less apt at music although he loves it, and must be a lover of hearing although he's
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by no means skilled in rhetoric. With slaves such a man would be brutal, not merely despising slaves as the adequately educated man does. But with freemen he would be tame and to rulers most obedient. He is a lover of ruling and of honor, not basing his claim to rule on speaking or anything of the sort, but on warlike deeds and everything connected with war; he is a lover of gymnastic and the hunt."
"Yes," he said, "that is the disposition belonging to this regime."
"Wouldn't such a man," I said, "when he is young also despise
b
money, but as he grows older take ever more delight in participating in the money-lover's nature and not be pure in his attachment to virtue, having been abandoned by the best guardian?"
"What's that?" Adeimantus said.
"Argument mixed with music," I said. "It alone, when it is present, dwells within the one possessing it as a savior of virtue throughout life."
"What you say is fine," he said. "Such, then," I said, "is the timocratic youth, like the timocratic city."
c
"Most certainly." "And this is how he comes into being," I said. "Sometimes he is the young son of a good father who lives in a city that is not under a good regime, a father who flees the honors, the ruling offices, the law-suits, and everything of the sort that's to the busybody's taste, and who
549 c
is willing to be gotten the better of so as not to be bothered."
"In what way, then, does he come into being?" he said.
"When," I said, "in the first place, he listens to his mother complaining. Her husband is not one of the rulers and as a result she is at a disadvantage among the other women. Moreover, she sees that he isn't
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very serious about money and doesn't fight and insult people for its sake in private actions in courts and in public but takes everything of the sort in an easygoing way; and she becomes aware that he always turns his mind to himself and neither honors nor dishonors her very much. She complains about all this and says that his father is lacking in courage and too slack, and, of course, chants all the other refrains such as women are likely to do in cases of this sort."
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"Yes, indeed," said Adeimantus, "it's just like them to have many complaints."
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