Lecture 12 — It is shown from speed and slowness that there is no separated void

Further, the truth of what we assert is plain from the following considerations. We see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing to excess of weight or of lightness.

Now, the medium causes a difference because it impedes the moving thing, most of all if it is moving in the opposite direction, but in a secondary degree even if it is at rest, and especially a medium that is not easily divided, that is, a medium that is somewhat dense. A, then, will move through B in time C, and through D, which is thinner, in time E (if the length of B is equal to D), in analogy to the density of the hindering body. For let B be water and D air; then, by so much as air is thinner and more incorporeal than water, A will move through D more quickly than through B. Let the speed have the same ratio to the speed, then, that air has to water. Then, if air is twice as thin, the body will traverse B in twice the time that it does D, and the time C will be twice the time E. And always, by so much as the medium is more incorporeal and less resistant and more easily divided, the faster will be the motion.

Now, there is no ratio in which the void is exceeded by body, as there is no ratio of zero to a number. For if four exceeds three by one and exceeds two by more than one, and exceeds one by still more than it exceeds two, still there is no ratio by which it exceeds zero; for that which exceeds must be divisible into the excess plus that which is exceeded, so that will be what it exceeds zero by plus-zero. For this reason, too, a line does not exceed a point unless it is composed of points. Similarly the void can bear no ratio to the full.

Therefore, neither can motion through the one [bear a ratio] to motion through the other.

But if a thing moves such and such a distance through the thickest medium in such and such a time, it moves through the void with a speed beyond any ratio.

For let Z be void, equal in magnitude to B and to D. Then, if A is to traverse and move through it in a certain time I—a time less than E, however—the void will bear this ratio to the full. But, in a time equal to I, A will traverse the part D itself of A. And it will surely also traverse in that time any substance Z that exceeds air in thinness in the ratio that the time E bears to the time I. For if the body Z be as much thinner than D as E exceeds I, then A, if it moves through Z, will traverse it in a time inverse to the speed of the motion, that is, in a time equal to I. If, then, there is no body in Z, A will traverse Z still more quickly. But we supposed that its traverse of Z when Z was void occupied the time I, such that it will traverse Z in an equal time whether Z be full or void. But this is impossible.

It is plain, then, that, if there is a time in which it will move through any part of the void, this impossible result will follow: it will be found to traverse a certain distance, whether this be full or void, in an equal time; for there will be some body that is in the same ratio to the other body as the time is to the time.

To sum the matter up, the cause of this result is obvious: between any two motions, there is a ratio (for they occupy time, and there is a ratio between any two times, so long as both are finite), but there is no ratio of void to full.

These are the consequences that result from a difference in the media.

The following depend upon an excess of one moving body over another. We see that bodies that have a greater impulse either of weight or of lightness, if they are alike in other respects, move faster over an equal space, and in the ratio that their magnitudes bear to each other. Therefore, they will also move through the void with this ratio of speed.

But that is impossible, for why should one move faster? (In moving through plena it must be so, for the greater divides them more quickly by its force. For a moving thing cleaves the medium either by its shape or by the impulse that the body that is carried along or is projected possesses.)

Therefore, all will possess equal velocity. But this is impossible.

It is evident from what has been said, then, that, if there is a void, a result follows that is the very opposite of the reason for which those who believe in a void set it up. They think that, if motion in respect of place is to exist, the void must exist, separated all by itself; but this is the same as to say that place is a separate cavity, which has already been stated to be impossible.

527. Here the Philosopher, arguing from the fast and slow in motion, shows that the void does not exist.

About this, he does two things:

first, he assigns the causes of fastness and slowness in motion;

second, he uses these causes to argue to his point, at now, the medium causes (215a29; [528]).

He therefore says first (215a24) that one and the same heavy body—or any other thing such as a stone or something of this sort—is carried more quickly for two reasons:

either on account of the medium in which it is being moved, such as air or earth or water;

or on account of differences in the object, such as being heavier or lighter, all other things being equal.

528. Then, at now, the medium causes (215a29), he argues to his point from the causes given.

First, from the differences of the medium;

second, from the differences in the mobile object, at the following depend upon (216a12; [539]).

As to the first, he does two things:

first, he gives an argument;

second, he recapitulates, at to sum the matter up (216a8; [533]).

Concerning the first, he does two things:

first, he gives his argument;

second, he shows that the conclusion follows from the premises, at for let Z be void (215b22; [532]).

529. Therefore, he first gives this argument. The ratio of motion to motion in regard to speed is equal to the ratio of medium to medium in respect of subtlety. But there is no ratio between empty space and full space. Therefore, motion in a void has no ratio to motion in the full.

First of all, he explains the first proposition of this argument. And he says that the medium through which a body is in motion is the cause of its fastness or slowness because it acts as an obstacle to the body in motion.

The greatest obstacle occurs when the medium is in a contrary motion, as is evident in the case of a ship whose motion is impeded by the wind.

The medium is an obstacle in a secondary way even if it is not in motion, for if it were in motion with the object, it would not be an obstacle but a help, as the water that carries a ship downstream.

But, among obstacles, a greater impediment is offered by things that are not easy to divide, such as the denser bodies.

He explains this by an example. Let the body in motion be A, and let the space through which it is being moved be B, and let the time in which A is being moved through B be C. Let us posit another space, D, of the same length as B, but let D be full of a subtler body than the one in B, such that a certain analogy, or ratio, exists between the bodies that impede the motion (for example, let B be full of water and D full of air). Now, to the extent that air is subtler than water and less condensed, to that extent will the mobile A be more quickly moved through D than through B. Therefore, the ratio of the velocities will equal the ratio of the subtlety of air to the subtlety of water. And, the greater the velocity, the less the time: because that motion is faster that covers the same interval in less time, as will be shown in book 6. (Lect. 3.) Hence, if air is twice as subtle as water, the time it takes A to be moved through B, which is full of water, will be twice the time for A to pass through D, which is full of air. Consequently, the time C in which it travels the distance B will be twice the time it takes E to pass through D. Therefore, we can take it as a general rule that, in whatever ratio the medium (in which something is in motion) is subtler and less resistant and more easily divisible, in that ratio will the motion be faster.

530. Then, at now, there is no ratio (215b12), he explains the second proposition, saying that the void is not exceeded by the full according to some certain ratio. And he proves this by the fact that a number does not exceed nothing by any ratio, for ratios can exist only between one number and another, or between a number and unity, as four exceeds three by one, and exceeds two by more yet, and exceeds one by still more.

Hence there is said to be a greater ratio existing between four and one than between four and two or between four and three. But four does not exceed nothing according to any ratio.

This is so because anything exceeding is necessarily divided into that which is exceeded and into the excess, namely, that in which it exceeds; for example, four can be divided into three and one, which latter is the amount by which four exceeds three.

But, if four exceeds nothing, it will follow that four can be divided into so much and nothing, which is unacceptable. For the same reason, one could not say that a line exceeds a point unless it were composed of points and divided into points. In like manner, it cannot be said that the void has any ratio to the full, because the void is not a part of the full.

531. Then, at therefore, neither can motion (215b20), he concludes that there can be no ratio between a motion in the void and a motion in the full, but that, if any body is in motion in the subtlest of mediums over such and such a distance for such and such a time, the motion in the void will exceed any given ratio.

532. Then, at for let Z be void (215b22), because he had directly deduced the above conclusion from the assumed principles, and lest any doubt arise about those principles, and to make the process more certain, he now proves the same conclusion by deducing to the impossible. For if it be claimed that the speed of a motion taking place in the void has a ratio to the speed of a motion taking place in the full, then let the empty space be Z, which shall be equal in magnitude to the space B, full of water, and to the space D, full of air.

Now, if it is supposed that a motion through Z has a certain ratio in respect of speed to the motions through B and D, then it must be admitted that the motion through Z, the void, takes place in some definite portion of time, because velocities are distinguished according to the quantities of the times consumed, as was said above. If, therefore, we say the object A passes through the empty space Z in a definite time, then let that time be I, which must be less than the time E required for A to pass through D, which is full of air. Then the ratio of the motion through the empty to the motion through the full will equal the ratio of time E to time I. But, during time I, the mobile A will pass through a definite space that is full of a subtler body than exists in D itself, that is, than D. And this will happen if one can find a body that differs in subtlety from air (of which D is full) in the ratio that the time E has to the time I. For example, say the space Z, which had been originally empty space, is now full of fire. If the body of which Z is full is subtler than the body of which D is full, it will follow that, in the amount that the time E exceeds the time I, the mobile A, if it is in motion through Z (which is the space now full of a most subtle body), and through D (which is the space full of air), will pass through Z conversely at a greater speed in a time I.

If, therefore, no body exists in Z, but it is again considered to be empty space as previously, it will have to move even faster. But this is against what was posited, namely, that the motion through Z, which is empty space, required time I. Consequently, since it passes over the same space in time I when it is full of the most subtle of bodies, it follows that, during the same time, the same mobile passes through one and the same space when that space is empty and when it is full. It is thus clear that, if it took a definite time for the mobile to pass through an empty space, the impossibility follows that, in equal time, it will pass through full and empty space, because there will be some body having the same ratio to some other body as one time has to another time.

533. Then, at to sum the matter up (216a8), he summarizes that in which the force of the previous reasoning consists. And he says that we can now say in recapitulation that the reason that the conflict mentioned in the above occurs is clear: it is because every motion has a ratio to every other motion in respect to speed. For every motion requires time, and any two periods of time, if they are finite, have a ratio one to the other. But there is no ratio between the empty and the full, as we proved. Hence, the supposition that motion occurs in the void leads to the conflict mentioned.

In a final summary (216a11), he concludes that the above mentioned conflicts occur if the different species of motion are taken according to differences of the media.

534. But several difficulties arise against this reasoning of Aristotle.

The first is that it does not seem to follow that, if motion takes place in the void, it has no ratio to motion in the full. For every motion has its definite velocity from the ratio of the motive energy to the mobile, even if no obstacle exists. And this is evident both from an example and from reason.

The example is that, indeed, the motion of the heavenly bodies encounters no obstacle and yet they have a definite velocity depending on the amount of time.

It is evident from reason also, since it is possible to point out a “before” and “after” in the magnitude through which the motion takes place, and so also one can take a “before” and “after” in the motion, from which it follows that motion is in a determined time. But it is true that this velocity can be diminished on account of an obstacle. Yet it is not therefore necessary to make the ratio of motion to motion in respect of velocity be as the ratio of obstacle to obstacle, such as to make the motion occur in no time if there be no obstacle; rather, the ratio of one slowing up to another slowing up must correspond to the ratio of obstacle to obstacle. Hence, on the assumption that motion takes place in the void, it follows that no slowing up happens to the natural speed, but it does not follow that a motion in the void will have no ratio to motion in the full.

535. Averroes attempts to counter this objection in his commentary.

First, he tries to show that this objection proceeds from false imagination. For he says that those who make the above objection imagine that an addition in slowness of motion occurs just like an addition in the magnitude of a line, where the added part is distinct from the part to which the addition is made.

For the above objection seems to proceed as though slowing up takes place by adding one motion to another motion in such a way that, if you were to subtract the motion that was added through the obstacle that slows, the quantity of natural motion would then be left.

But this is not the case, for when a motion is slowed up, each part of the motion becomes slower, whereas each part of a line does not become larger.

Then he attempts to show how Aristotle’s argument concludes with necessity. And he says that the speed or slowness of a motion does indeed arise from the proportion of the mover to the mobile; but the mobile must in some manner resist the mover, as the patient is in a certain way contrary to the agent.

This resistance can arise from three sources.

It can arise first from the position of the mobile; for from the very fact that the mover intends to transfer the mobile to some certain place, the mobile, existing in some other place, resists the intention of the mover.

It can arise second from the nature of the mobile, as is evident in forced motions, as when a heavy object is thrown upward.

It can arise third from the medium.

All three are taken together as one resistance to constitute one cause of slowing up in the motion. Therefore, when the mobile, considered in isolation as different from the mover, is a being in act, the resistance of the mobile to the mover can be traced either to the mobile only, as happens in the heavenly bodies, or to the mobile and medium together, as happens in the case of animate bodies on this earth.

But, in heavy and light objects, if you take away what the mobile receives from the mover—namely, the form that is the principle of motion given by the generator (the mover)—nothing remains but the matter that can offer no resistance to the mover. Hence, in light and heavy objects, the only source of resistance is the medium.

Consequently, in heavenly bodies, differences in velocity arise only on account of the ratio between mover and mobile; in animate bodies, they arise from the proportion of the mover to the mobile and to the resisting medium—both together.

And it is in these latter cases that the given objection would have effect—that, if you remove the slowing up caused by the impeding medium, there still remains a definite amount of time in the motion, according to the proportion of the mover to the mobile. But, in heavy and light bodies, there can be no slowing up of speed except what the resistance of the medium causes—and in such cases, Aristotle’s argument applies.

536. But all this seems quite frivolous.

First, although the quantity of slowing up is not parallel to the mode of continuous quantity (such that motion is added to motion), but rather parallel to the mode of intensive quantity, as when something is whiter than something else, still the quantity of time from which Aristotle argues is parallel to the manner of continuous quantity—and time becomes greater by the addition of time to time. Hence, if you subtract the time that was added on account of the obstacle, the time of the natural velocity remains.

Next, if you remove the form that the generator gives to light and heavy bodies, there still remains in the understanding “quantified body,” which, from the very fact that it is a quantified body existing in an opposite position, offers resistance to the mover. For we cannot suppose in heavenly bodies any other resistance to their movers. Hence, as Averroes presents the case, even in the case of heavy and light bodies, the reasoning of Aristotle would not follow.

Therefore, it is better and briefer to say that the argument brought forward by Aristotle is an argument aimed at contradicting his opponent’s position and not a demonstrative argument in the absolute sense. For those who posited a void did so in order that motion be not prevented.

Thus, according to them, the cause of motion was on the part of a medium that did not impede motion. And therefore Aristotle argued against them as though the total cause of fastness and slowness derived from the medium, as he clearly shows above when he says that, if nature is the cause of the motion of simple bodies, it is not necessary to posit the void as the cause of their motion. (Lect. 11, n. 521.)

In this way, he gives us to understand that they supposed the total cause of the motion to depend on the medium and not on the nature of the mobile.

537. The second difficulty against his argument is that, if the medium that is full impedes, as he says it does, then it follows that there will not be any pure unimpeded motion in this lower medium, and this seems unfitting.

To this, the Commentator replies that the impediment that arises from the medium is required by the natural motion of heavy and light bodies, such that there can be resistance of the mobile to the mover, at least on the side of the medium. But it is better to say that every natural motion begins from a place that is not natural and tends to a natural place. Hence, until it reaches its natural place, it is not unfitting if something unnatural be attached to it. For it gradually departs from what is against its nature and tends to what is in keeping with its nature. And, for this reason, a natural motion is intended by its end.

538. The third objection is that, since there is a fixed limit of rarity in natural bodies, it does not seem that one can keep supposing a rarer and rarer body according to any given proportion of time to time.

In reply, it should be said that a fixed rarity in natural things is not due to the nature of the mobile body insofar as it is mobile, but to the nature of specific forms that require specific rareness and density. But, in this book, we are dealing with mobile body in general, and therefore Aristotle frequently uses in his arguments things that are false if the specific natures of bodies are considered, but possible if the nature of body in general is considered.

Or it can be replied that he is here also proceeding according to the opinion of the earlier philosophers who posited the rare and the dense as the first formal principles. According to them, rarity and density could be increased to infinity, since these did not depend on other previous forms according to whose exigencies they would be determined.

539. Then, at the following depend upon (216a12), he shows that there is no separated void, arguing from the speed and slowness of motion, insofar as the cause is taken entirely from the viewpoint of the mobile. And he says that what he is about to say will follow logically if we attend to the difference of speed and slowness insofar as bodies in motion exceed one another. For we see that, over a given equal space, greater speed is shown by bodies having a greater inclination due either to heaviness or lightness, whether they are greater in quantity but equally heavy or light, or whether they are equal in quantity but unequal in heaviness or lightness.

And I say this if they are similar in shape. For a wide body, if it be deficient in heaviness or size, is moved more slowly than a body with a pointed shape. And the ratio of the velocity corresponds to the ratio that the moving magnitudes have to one another in respect to their weight or in respect to their magnitude. And this will have to be true also if the motion occurs in the void, namely, that a heavier body or a lighter body or a more pointed body will be moved faster through an empty medium.

But this cannot be, since it is impossible to explain why one body would be moved faster than another. For if the motion takes place within a space filled with some body, an explanation for the greater or lesser speed can be given—it will be due to any of the named causes. The explanation is that a greater body will, on account of its strength, divide the medium more quickly; or on account of its shape, because what is sharp has greater penetrating power; or on account of a greater inclination, due either to the heaviness or lightness of the body, or even to the force imparted by that which projects it. But the void cannot be cleaved more quickly or more slowly. Hence, it will be moved through a void with equal speed. But this clearly appears as impossible. And so, from a consideration of the velocity of motion, it is evident that the void does not exist.

It should be observed that, in this reasoning process, there exists the same difficulty as in the first one. For he seems to suppose that difference in velocity in motions is due only to the different ways in which the medium can be cleaved, whereas the fact is that there are differences of velocity among the heavenly bodies in which there is no full medium resisting that has to be cleaved by the motion of the heavenly body. But this difficulty should be solved as the above one was.

540. Finally (216a21), he summarizes, and concludes that, from the foregoing, it is clear that, in regard to the philosophers who posit a void, it is the contrary of what they supposed as a reason for proving it. For they proceeded on the assumption that motion could not take place unless there was a void. But the contrary has been proved: if there be a void, there is no motion. (Lect. 11, nn. 523–24, 526.) Thus, those philosophers believe that the void is some distinct and separate thing—a space having separate dimensions—and they believed it was such a space that had to exist if local motion were to be possible. However, to posit such a separated void is the same as saying that place is a kind of space distinct from bodies—which is impossible, as was shown in the treatise on place. (Lect. 6, nn. 461–62.)