The so-called 'Nuptial Number' is by far Plato's most intractable mathematical problem. It is so dense and so obscure that many commentators, both ancient and modern, have decided that it has no rational solution but is instead an item of satire, not to be taken seriously. Others - including the present author (an admitted obscurantist) - see it as one of the great keys of the Platonic canon, if only it is understood correctly. Like the impossibly obscure formulae of the alchemists, it is the key to the Mysteries.
THE PROBLEM: WHAT CAUSES STATES TO DECLINE?
“Come, then,” I said, “and let us try to discern the way a timocracy will develop from of an aristocracy. Or is it the simple and unvarying law that in every form of government disturbance begins among the ruling class itself, when sedition arises among it, but as long as it is at one with itself, disturbance will not occur?”
“Yes, that is so.”
“How, then, Glaucon,” I said, “will disturbance arise in our ideal city, and how will our auxiliaries and rulers fall out and be at odds with one another and with themselves?
INVOCATION TO THE MUSES
Shall we invoke the Muses as Homer does in order to tell us “‘how faction first fell upon them,’” and shall we, like a tragic poet, picture the Muses speaking in an elevated style, as if they are speaking seriously when in fact they are playing with us and teasing us like one child teases another?
“How?”
“In some such fashion as follows:
THE WORDS OF THE MUSES
[Having been invoked in the Homeric manner - albeit ambiguously - the Muse now speaks, and it is through this quasi-Homeric Muse that we are given the Nuptial Number. It is an open question as to how seriously we should take this entire construction, although let us note that Plato uses this "Muse" in other contexts in other dialogues. Many of the 'Platonic myths' are introduced in this way.]
ALL THINGS PASS
It is indeed difficult for a state ideally constituted as we have described to be shaken and disturbed; but since for everything that is born there is also a time of destruction, even a constitution such as ours will not last forever, but it shall surely be dissolved eventually.
This is the manner of its dissolution:
THINGS MOVE IN CYCLES
Not only for plants that grow from the earth but also for animals that live upon it there is a cycle of fertility and barrenness of soul and body as often as the revolutions of their orbs come full circle, in short cycles for the short-lived things and long cycles for long-lived things.
BEGETTING CHILDREN OUT OF SEASON
Concerning human beings, there are also laws of fecund birth and of infertility, and there will come a time when these laws will escape the men you have bred to be the rulers of your city. For all their wisdom, and combining calculation with observation, they will beget children out of season.
A PERIOD COMPREHENDED BY A NUMBER
Now for divine creatures there is a period comprehended by a number that is final and perfect, but for a mortal the number is the first in which multiplications of root by square - when they have attained three distances, with four limits, of that which makes like and unlike and waxes and wanes - have rendered all things commensurable with one another.
THE BASE PRODUCES TWO HARMONIES
The base of this, containing the ratio of four to three, yoked with five, produces two harmonies when increased three times.
1. One of them is equal an equal number of times, so many times a hundred.
2. The other is equal length one way but oblong
- one dimension of a hundred numbers determined by the rational diameters of five diminished by one in each case, or of the irrational by two; the other dimension of a hundred cubes of three.
LORD OF BETTER AND WORSE BIRTHS
The sum of these - this entire geometrical number, a number measuring the earth - is lord of better and worse births.
THE NEGLECTFUL GUARDIANS
When the guardians of your ideal state neglect this and marry brides to bridegrooms out of season children of ill-nature and ill-fortune will be born.
NEGLECT OF THE MUSES
The best of their predecessors will indeed make rulers but these offspring, being unworthy, when they have succeeded to their fathers' offices of power, will begin to neglect us muses, though they are our guardians, and will pay too little heed to music, and then to gymnastics, so that the children will deteriorate and grow up without us.
THE METALLIC RACES
And the rulers who come after them will have little of the guardian in them for testing Hesiod's races and your own - races of gold and silver and copper and iron. And iron will be mixed with silver, and copper will be mixed with gold, and this will engender unlikeness within them, and an unevenness which is disharmonious, which things always create war and enmity wherever they are found. This surely is the pedigree of sedition, wherever it arises. As Homer says, "Of this lineage, look you!" (Iliad 6:211)
CONCLUSION
It is not the purpose of this post to offer any sort of detailed solution to the 'problem' posed by this most difficult of passages from Plato's Republic; rather its limited purpose is simply to present the passage in question in a clear, accurate translation usefully divided into its component parts. The Greek text is impossibly convoluted and many of the words used are both rare and esoteric. No translation can save the passage from what seems to be deliberate obfuscation on the part of the author. But much of it can be disentangled and made clearer by some explanatory additions and a careful break down of its parts.
What the infamous 'number' might be and what its significance might be is another matter, but the drift of the passage in itself - leaving aside the formula of the number - can be straightened out into a largely lucid account. That is what is attempted below. The rendering is based upon those of Benjamin Jowett and James Adam (with copious references to the lexicon of Liddel & Scott. We begin at Republic 545D:
What the infamous 'number' might be and what its significance might be is another matter, but the drift of the passage in itself - leaving aside the formula of the number - can be straightened out into a largely lucid account. That is what is attempted below. The rendering is based upon those of Benjamin Jowett and James Adam (with copious references to the lexicon of Liddel & Scott. We begin at Republic 545D:
THE PROBLEM: WHAT CAUSES STATES TO DECLINE?
“Come, then,” I said, “and let us try to discern the way a timocracy will develop from of an aristocracy. Or is it the simple and unvarying law that in every form of government disturbance begins among the ruling class itself, when sedition arises among it, but as long as it is at one with itself, disturbance will not occur?”
“Yes, that is so.”
“How, then, Glaucon,” I said, “will disturbance arise in our ideal city, and how will our auxiliaries and rulers fall out and be at odds with one another and with themselves?
INVOCATION TO THE MUSES
Shall we invoke the Muses as Homer does in order to tell us “‘how faction first fell upon them,’” and shall we, like a tragic poet, picture the Muses speaking in an elevated style, as if they are speaking seriously when in fact they are playing with us and teasing us like one child teases another?
“How?”
“In some such fashion as follows:
THE WORDS OF THE MUSES
[Having been invoked in the Homeric manner - albeit ambiguously - the Muse now speaks, and it is through this quasi-Homeric Muse that we are given the Nuptial Number. It is an open question as to how seriously we should take this entire construction, although let us note that Plato uses this "Muse" in other contexts in other dialogues. Many of the 'Platonic myths' are introduced in this way.]
ALL THINGS PASS
It is indeed difficult for a state ideally constituted as we have described to be shaken and disturbed; but since for everything that is born there is also a time of destruction, even a constitution such as ours will not last forever, but it shall surely be dissolved eventually.
This is the manner of its dissolution:
THINGS MOVE IN CYCLES
Not only for plants that grow from the earth but also for animals that live upon it there is a cycle of fertility and barrenness of soul and body as often as the revolutions of their orbs come full circle, in short cycles for the short-lived things and long cycles for long-lived things.
BEGETTING CHILDREN OUT OF SEASON
Concerning human beings, there are also laws of fecund birth and of infertility, and there will come a time when these laws will escape the men you have bred to be the rulers of your city. For all their wisdom, and combining calculation with observation, they will beget children out of season.
A PERIOD COMPREHENDED BY A NUMBER
Now for divine creatures there is a period comprehended by a number that is final and perfect, but for a mortal the number is the first in which multiplications of root by square - when they have attained three distances, with four limits, of that which makes like and unlike and waxes and wanes - have rendered all things commensurable with one another.
THE BASE PRODUCES TWO HARMONIES
The base of this, containing the ratio of four to three, yoked with five, produces two harmonies when increased three times.
1. One of them is equal an equal number of times, so many times a hundred.
2. The other is equal length one way but oblong
- one dimension of a hundred numbers determined by the rational diameters of five diminished by one in each case, or of the irrational by two; the other dimension of a hundred cubes of three.
LORD OF BETTER AND WORSE BIRTHS
The sum of these - this entire geometrical number, a number measuring the earth - is lord of better and worse births.
THE NEGLECTFUL GUARDIANS
When the guardians of your ideal state neglect this and marry brides to bridegrooms out of season children of ill-nature and ill-fortune will be born.
NEGLECT OF THE MUSES
The best of their predecessors will indeed make rulers but these offspring, being unworthy, when they have succeeded to their fathers' offices of power, will begin to neglect us muses, though they are our guardians, and will pay too little heed to music, and then to gymnastics, so that the children will deteriorate and grow up without us.
THE METALLIC RACES
And the rulers who come after them will have little of the guardian in them for testing Hesiod's races and your own - races of gold and silver and copper and iron. And iron will be mixed with silver, and copper will be mixed with gold, and this will engender unlikeness within them, and an unevenness which is disharmonious, which things always create war and enmity wherever they are found. This surely is the pedigree of sedition, wherever it arises. As Homer says, "Of this lineage, look you!" (Iliad 6:211)
CONCLUSION
[Here the Muses finish and we return to the conversation between the interlocutors of the dialogue, namely Socrates and Glaucon.]
"And quite right too," said Glaucon. "We affirm what the Muses say as correct."
"Indeed," I said, "because they are Muses."
Yours,
Harper McAlpine Black
"And quite right too," said Glaucon. "We affirm what the Muses say as correct."
"Indeed," I said, "because they are Muses."
* * *
Yours,
Harper McAlpine Black
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