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Ancilla to the Pre-Socratic Philosophers, by Kathleen Freeman, [1948], at


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,

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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 on 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 exhalation as we utter . . .

This happens also with missiles. Those which are vigorously thrown are carried far, those weakly thrown (fall) 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 further away, a lower note results. Clearly swift motion produces a high-pitched sound, slow motion a low-pitched sound.

Moreover, the 'whirlers' 1 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.

2. There are three 'means' in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary,

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which they call 'harmonic'. The arithmetic mean 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. 1 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. 2 The subcontrary, which we call 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. 3

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

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is able to treat more clearly any problem it will . . . and—a thing in which geometry fails—arithmetic adds proofs, and at the same time, if the problem concerns 'forms' (i.e. numerical first principles), arithmetic treats of the forms also.

(Doubtful titles attributed to Archytas)

5-8. 'On the Decad.' 'On Nature.' 'On Flutes.' 'On Mechanics.' 'On Agriculture.'

9. (List of spurious titles).


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).

Next: 48. Occelus (or Ocellus) of Lucania