Archytas

The mathematicians (τοὶ περὶ τὰ μαθήματα) seem to me to have arrived at correct conclusions, and it is not therefore surprising that they have a true conception of the nature of each individual thing: for, having reached such correct conclusions regarding the nature of the universe, they were bound to see in its true light the nature of particular things as well. Thus they have handed down to us clear knowledge about the speed of the stars, their risings and settings, and about geometry, arithmetic, and sphaeric, and last, not least, about music; for these μαθήματα seem to be sisters.

for the first two kinds of being are indistinguishable.

First they have seen that it was not possible that there should be any noise, unless there was a shock of one body against another; they said there is a shock, when moving bodies meet and strike each other. The bodies moved in the air in opposite direction and those that are moved with unequal swiftness, — in the same direction, — the first, when overtaken, make a noise, because struck. Many of these noises are not susceptible of being perceived by our organs; some because of the slightness of the shock, the others because of their too great distance from us, some even because of the very excess of their intensity; for noises too great do not enter into our ears, as one cannot introduce anything into jars with too narrow an opening when one pours in too much at a time.

Of the sounds that fall within the range of our senses, some, — those that come quickly from the bodies struck, — seem shrill; those that arrive slowly and feebly, seem of low pitch. In fact, when one agitates some object slowly and feebly, the shock produces a low pitch; if the waving is done quickly, and with energy, the sound is shrill. That is not the only proof of the fact; which we can prove when we speak or sing; when we wish to speak loud and high, we use a great force of breath. So also something thrown; if you throw them hard, they go far; if you throw them without energy, they fall near, for they air yields more to bodies moved with much force, than to those thrown with little. This phenomenon is also reproduced in the sound of the voice; for the sounds produced by an energetic breath are shrill, while those produced by a feeble breath are weak and low pitch.

This same observation can be seen in the force of a signal given from any place; if you pronounce it loud, it can be heard far; if you pronounce the same signal low, we do not hear it even from near. So also in flutes, the breath emitted by the mouth and which presents itself to the holes nearest the mouthpiece, produces a shriller sound, because the impulsive force is greater; further, they are of lower pitch. It is therefore evident that the swiftness of the movement produces shrillness, and slowness, lower pitch. The same thing is seen in the magic tops which are spun in the mysteries; those that move slowly produce a low pitch, while those that move quickly with force make a shrill noise. Let us yet adduce the reeds. If you close the lower opening, and blow into it, it will produce a certain sound; and if you stop it in the centre, or in the front, the sound will be shrill. For the same breath traversing a long space weakens, while traversing a shorter, it remains of the same power. After having developed his opinion that the movement of the voice is measured by the intervals, he resumes his discussion, saying, that the shrill sounds are the result of a swifter movement, the lower sounds, of a slower movement, this is a fact which numerous experiments demonstrate clearly.

Explaining himself about the means, Archytas writes: In music there are three means: the first is the arithmetical mean, the second is the geometrical, the third is the subcontrary mean, which is called harmonic. The mean is arithmetical, when the three terms are in a relation of analogical excess, that is to say, when the difference between the first and second is the same as between second and third; in this proportion, the relation of the greater terms is smaller, and the relation of the smaller is greater. The geometric mean exists when the first term is to the second as the second is to the third; here the relation of the greater is identical with the relation of the smaller terms. The subcontrary mean, which we call Harmonic, exists when the first term exceeds the second by a fraction of itself, identically with the fraction by which the second exceeds the third; in this proportion the relation of the greatest terms is greater, and that of the smaller, smaller.

He who knows must have learned from another, or have found his knowledge by himself. The science that you learn from another, is as you might say, exterior; what you find by yourself belongs to ourselves individually. To find without seeking is something difficult and rare; to find what one is seeking is commodious and easy; to ignore, and seek what you ignore, is impossible.

When the art of reflexion was discovered, diminished dissensions, and increased concord; those who possess it feel the pride of predominance yielding to the sentiment of equality. It is by reflexion that we succeed in adjusting our affairs in a friendly fashion; through it the poor receive riches, and the rich give to the poor, each possessing the confidence that he possesses the equality of rights.

Reflexion is like a rule which hinders and turns aside the people who know how to reflect from committing injustices, for it convinces them that they cannot remain hidden if they carry out their purposes, and the punishment which has overtaken those who do not know how to abstain makes them reflect and not become back-sliders.

Logic, compared with the other sciences is by far the most successful, and succeeds in demonstrating its objectives even better than geometry. Where geometric demonstration fails, the logic succeeds; and logic treats not only with general classes, but with their exceptions.

The decad determines every number, including the nature of everything, of the even and the odd, of the movable and immobile, of good and evil. It has been the subject of long discussions by Archytas, and of Philolaus, in his work on nature.

And likewise, many of the Pythagoreans practised the art of flute-playing, as Euphranor, and Archytas, and Philolaus, and many others. But Euphranor has also left behind an essay on Flutes, and so too has Archytas.

others again on machinery, such as Diades, Archytas, Archimedes, Ctesibius, Nymphodorus, Philo of Byzantium …

Democritus the physicist; Xenophon the disciple of Socrates; Aristotle and Theophrastus, the peripatetics; Archytas the pythagorean