Commentary Aristotle Physics
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.