Archŷtas of Tarentum: first half of fourth century B.C.
Wrote in literary Doric a work on Mathematical Science, and on Harmony; possibly also one on Mechanics.
1. 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 judgement 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 judgement on the speed of the constellations and their rising and setting, as well as on (plane) geometry and Numbers (arithmetic) 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 (i.e. checking each other's speed). 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 recognised 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,
That high notes are in swift motion, low notes in slow motion, has become clear to us from many examples.
2. There are three 'means' in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary,
3. 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 (Right Reckoning) 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 (Right Reckoning) they will be treated fairly. Being the standard and the deterrent of wrongdoers, it checks those who are able to reckon (consequences) 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 (to reckon) it shows them that in this 4 lies their wrongdoing, and so it prevents them from committing the wrong deed.
(Attributed to a work entitled 'Conversations')
4. Arithmetic, it seems, in regard to wisdom is far superior to all the other sciences, especially geometry, because arithmetic
79:1 ῥόμβος, an instrument whirled round on a string at the Mysteries. 80:1 e.g., 6, 4, 2; 6 - 4 = 4 - 2, and 6/4 < 4/2. 80:2 e.g., 8, 4, 2; 2:4 = 4:2, and 4/2 = 8/4. 80:3 e.g., 6, 4, 3; 6 - 4 =2 ,4 - 3 = 1, and 2:6 = 1:3; 6/4 > 4/3. 80:4 i.e., in their inability to reckon consequences (Kranz).