SECTION II

PHYSICAL AND MATHEMATICAL FRAGMENTS

7. As Eudemus reports, Archytas used to ask this question: If I was situated at the extreme and immovable limit of the world, could I, or not, extend a wand outside of it? To say I could not, is absurd; but if I can, there must be something, outside of the world, be it body or space; and in whatever manner we reason, by the same reasoning we will ever return to this limit. I will still place myself there, and ask, is there anything else on which I may place my wand. Therefore, the infinite exists; if it is a body, our proposition is demonstrated; if it its space, place is that in which a body could be; and if it exists potentially, we will have to place it among, classify it among the eternal things, and the infinite will then be a body and a place.

8. The essence of place is that all other things are in it, while itself is not in anything. For if it was in a place, there would be a place in a place, and that would continue to infinity. All other beings must therefore be in place, and place in nothing. Its relation to things is the same as limit to limited things; for the place of the entire world is the limit of all things.

9a. Some say that time is the sphere of the world; such was the sentiment of the Pythagoreans, according to those who had no doubt heard Archytas give this general definition of time: "Time is the interval of the nature of all."

9b. The divine Iamblichus, in the first book of his commentaries on the Categories, said that Archytas thus defined time: "It is the number of movement, or in general the interval of the nature of all."

9c. We must combine these two definitions, and recognise time as both continuous and discrete, though it is properly continuous. Iamblichus claims that Archytas taught the distinction of time physical, and time psychic; so at least Iamblichus interpreted Archytas; but we must recognise that there, and often elsewhere, he adds to his commentaries to explain matters.

10. The general proper essence of "When-ness" and time is to be indivisible and insubstantial. For, being indivisible, the present time has passed, while expressing it, and thinking of it; naught remains of it, becoming continuously the same, it never subsists numerically, but only specifically. In fact, the actually present time and the future are not identical with former time. For the one has past, and is no more; the other passes while being produced and thought. Thus the present is never but a bond; it perpetually becomes, changes, and perishes; but nevertheless it remains identical in its own kind.

In fact, every present is without parts, and indivisible; it is the term of past time, the beginning to come; just as in a broken line, the point where the break occurs becomes the beginning of a line, and the end of the other. Time is continuous, and not discrete as are number, speech and harmony. In speech, the syllables are parts, and distinct parts; in the harmony, they are the sounds; in number; the unities. The line, place and space are continuous; if they are divided, their parts form common sections. For the line divides into points, the surface into lines, the solid into surfaces. Therefore time is continuous. In fact, there was no nature, when time was not; and there was no movement, when the present was not. But the present has always been, it will always be, and will never fail; it changes perpetually, and becomes an other according to the number, but remains the same according to kind. The line differs from the other continua, in that if you divide the line, place, and space, its parts will subsist; but in time, the past has perished, and the future will. That is why either time does absolutely not exist, or it hardly exists, and has but an insensible existence. For of its parts one, the past, is no more; the future is not yet, how then could the present, without parts and indivisible, possess true reality?

11. Plato says that the movement is the great and the small, the non-being. the unequal and all that reduces to these; like Archytas we had better say that it is a cause.

12.Why do all natural bodies take the spherical form? Is it, as said Archytas, because the natural movement is the proportion of equality? For everything moves in proportion; and this proportion of equality is the only one which, when it occurs, produces circles and spheres because it returns on itself.

13. 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.

14. The Pythagorean opinion about sciences to me seems correct, and they seen to show an exact judgment about each of then. Having known how to form a just idea of the nature of all, they should have likewise seen the essential nature of the parts. They have left us certain evident theories about arithmetic, geometry, spherics; also about music; for all these sciences seem to be kindred, in fact, the first two kinds of being are indistinguishable.

15a. 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 an opposite direction and those that are moved without equal 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, same, -- 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. This 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 then 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 in 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 [give] a shrill noise. Let us yet adduce the reed: 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 this 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.

15b. Eudoxus and Archytas believed that the reasons of the agreement of the sounds was in the numbers; they agree in thinking that these reasons consist in the movements, the shrill movement being quick, because the agitation of the air is continuous, and the vibration more rapid; the low pitch movement being slow, because it is calmer. 16. 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. 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.