Archytas of Tarentum, son of Mnesagoras, or of Hestius, according to Aristoxenus, also was a Pythagorean. It was he who, by a letter, saved Plato from the death threatened by Dionysus. He possessed all the virtues, so that, being the admiration of the crowd, he was seven times named general, in spite of the law which prohibited reelection after one year. Plato wrote him two letters, in response to this one of Archytas:
It is fortunate for you that you have recovered from your illness, for I have heard of it not only from you, but also from Lamiscus. I have busied myself about those notes, and took a trip into Lucania, where I met descendants of Ocellus. I have in my possession the treatises On Law and On Kingship, On Sanctity, and On the Origin of All Things, and I am sending them to you. The others could not be discovered. Should they be found, they will be sent to you.
I am delighted to have received the works which you have sent me, and I acknowledge a great admiration for him who wrote them. He seems to be worthy of his ancient and glorious ancestors, who are said to be from Myra, and among the number of those Trojans who emigrated under the leadership of Laomedon, all worthy people, as the legend proves. Those works of mine about which you wrote me are not in a sufficient state of perfection, but I send them such as they are. Both of us are in perfect agreement on the subject of protecting them. No use to renew the request. May your health improve.
Such are these two letters.
There were four Archytases; the first, of whom we have just spoken; the second, from Mytilene, was a musician; the third wrote On Agriculture ; the fourth is an author of epigrams. Some mention a fifth, an architect, who left a treatise On Mechanics , beginning as follows: "This book contains what I have been taught by Teucer of Carthage." The musician is said to have made this joke: on being reproached for not advertising himself more, he said "It is my instrument which speaks for me."
Aristoxenus claims that the philosopher Archytas was never defeated during his command. Once, overcome by envy, he had been obliged to resign his command, and his fellow-citizens were immediately conquered. he was the first who methodically applied the principles of mathematics to mechanics; who imparted an organic motion to a geometric figure, by the section of the semi-cylinder seeking two means that would be proportional, in order to double the cube. He also first, by geometry, discovered the properties of the cube, as Plato records in the Republic [528B].
Mathematicians seem to me to have excellent discernment, and it is in no way strange that they should think correctly concerning the nature of particular existences. For, since they have passed an excellent judgment on the nature of the Whole, they were bound to have an excellent view of separate things. Indeed they have handed on to us a clear judgment on the speed of the constellations and their rising and setting, as well as one geometry and numbers and solid geometry, and not least on music; for these mathematical studies appear to be related. For they are concerned with things that are related, namely the two primary forms of being.
First of all, therefore, mathematicians have judged that sound is impossible unless there occurs a striking of objects against one another. This striking, they said, occurs when moving objects meet one another and collide. Now things moving in opposite directions, when they meet, produce a sound by simultaneously relaxing. But things moving in the same direction, though at unequal speeds, create a sound by being struck when overtaken by what is following behind. Now many of these sounds cannot be recognized by our nature, some because of the faintness of the sound, others because of their great distance from us, and some even because of their excessive loudness, for the big sounds cannot make their way into our hearing—just as, in vessels with narrow necks, when one pours in too much, nothing enters. Thus when things impinge upon the perception, those which reach us quickly and powerfully from the source of sound seem high-pitched, while those which reach us slowly and feebly seem low-pitched. For if one takes a rod and strikes an object slowly and feebly, he will produce a low note with the blow, but if he strikes quickly and powerfully, a high note. Moreover, we can know this not only by this means, but also because when in speaking or singing we wish to produce a loud high sound, we employ strong exhalations as we utter. . . .
This also happens with missiles. Those which are vigorously thrown are carried far, those weakly thrown near; for the air yields more readily before those which are vigorously thrown, whereas it yields less readily to those which are weakly thrown. This is bound to happen also with the notes of the voice: if a note is expelled by a forcible breath, it will be loud and high, if by a feeble breath, soft and low. But we can also see it from this, the strongest piece of evidence, namely, that when a man shouts loudly, we can hear him from a distance, whereas if the same man speaks softly, we cannot hear him even near at hand. Further, in flutes, when the breath expelled from the mouth falls on the holes nearest the mouth, a higher note is given out because of the greater force, but when it falls on the holes farther away, a lower note results. Clearly swift motion produces a high-pitched sound, slow motion a low-pitched sound.
Moreover, the 'whirlers' which are swung round at the Mysteries, if they are whirled gently, they give out a low note, if vigorously, a high note. So too with the reed: if one stops its lower end and blows, it gives out a low kind of note; but if one blows into the middle or some part of it, it will sound high; for the same breath passes weakly through the long distance, powerfully through the lesser.
That high notes are in swift motion, low notes in slow motion, has become clear to us from many examples.
There are three 'means' in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call 'harmonic'. The arithmetic means is when there are three terms showing successively the same excess: the second exceeds the third by the same amount as the first exceeds the second. In this proportion, the ratio of the larger numbers is less, that of the smaller numbers greater. The geometric mean is when the second is to the third as the first is to the second; in this, the greater numbers have the same ratio as the smaller numbers. The subcontrary, which we call the harmonic, is as follows: by whatever part of itself the first term exceeds the second, the middle term exceeds the third by the same part of the third. In this proportion, the ratio of the larger numbers is larger, and of the lower numbers less.
In subjects of which one has no knowledge, one must obtain knowledge either by learning from someone else, or by discovering it for oneself. That which is learnt, therefore, comes from another and by outside help; that which is discovered comes by one's own efforts and independently. To discover without seeking is difficult and rare, but if one seeks, it is frequent and easy; if, however, one does not know how to seek, discovery is impossible.
Right Reckoning, when discovered, checks civil strife and increases concord; for where it has been achieved, there can be no excess of gain and equality reigns. It is this that brings us to terms over business contracts, and through it the poor receive from the men of means, and the rich give to the needy, both trusting that through it, they will be treated fairly. Being the standard and the deterrent of wrongdoers, it checks those who are able to reckon before they do wrong, convincing them that they will not be able to avoid detection when they come against it; but when they are not able, it shows them that in this lies their wrongdoing, and so it prevents them from committing the wrong deed.
Arithmetic, it seems, in regard to wisdom is far superior to all the other sciences, especially geometry, because arithmetic is able to treat more clearly any problem it wishes . . . and —a thing in which geometry fails —arithmetic adds proofs, and at the same time, if the problem concerns 'forms' arithmetic treats of the forms also.
The laws of the wicked and atheists are opposed by the unwritten laws of the gods, who inflict evils and terrible punishments on the disobedient. it is these divine laws which have developed and directed the laws and written maxims given to men.
The relation of law to the soul and human life is identical to that of harmony to the sense of hearing, and the voice: for the law instructs the soul, and thereby, the life, as harmony regulates the voice through education of the ear. In my opinion every society is composed of the commander, the commanded, and the laws. Among the latter, one is living, namely the king, and the other is inanimate, this being the written letter. The law is therefore the most essential; only through it is the king legitimate, the magistrate regularly instituted, the commanded free, and the whole community happy. When it is violated, the king is no more than a tyrant, the magistrate is illegitimate, the commanded becomes a slave, and the whole community becomes unhappy. Human acts are like a mingled tissue, formed of command, duty, obedience, and force sufficient to overcome resistance. Essentially, the command belongs to the better, being commanded to the inferior, and force belongs to both. For the reasonable part of the soul commands, and the irrational part is commanded. Both have the force to conquer the passions. Virtue is born from the harmonious cooperation of both and leads the soul to rest and indifference by turning it away from pleasures and sorrows.
Law must conform to nature, exercise an efficient power over things, and be useful to the social community; for if it lacks one, two, or all of these characteristics, it is no longer a law, or at least it s no longer a perfect law. It conforms to nature if it is the image of natural right, which fits itself, and distributes to each according to his deserts. It prevails if it harmonizes with the men who are to be subject thereto; for there are many people who are not apt to receive what by nature is the first of goods, and who are fitted to practice only the good which is in relation with them, and possible for them, for that is how the sick and the suffering have to be nursed. Law is useful to the political society if it is not monarchical, if it does not constitute privileged classes, if it is made in the interest of all, and is equally imposed on all. Law must also regard the country and the lands, for not all soils can yield the same returns, neither all human souls the same virtues. That is why some establish the aristocratic constitution, while others prefer the democratic or oligarchic. The aristocratic constitution is founded on the subcontrary proportion, and is the most just, for this proportion attributes the greatest results to the greatest terms, and the smallest to the smallest. The democratic constitution is founded on the geometrical proportion, in which the results of the great and small are equal.
John Paul Adams, CSUN