Commentary Aristotle Physics
For let AB be a finite space that has been traversed in infinite time CD. Now, in every motion, one part must be traversed ahead of another, and also one part of the magnitude is traversed in the prior part of time and another part in a subsequent part of time. And so, no two parts of the magnitude are ever traversed in one and the same part of time, and no two parts of time correspond to one and the same part of the magnitude. Consequently, if a certain part of the magnitude is traversed in a certain time, then in more time is traversed not only that part of the magnitude, but that part and another. And this will happen whether the mobile maintains constant speed or not, for in natural motions, the speed is continually increased, while in forced motions, it is diminished.
With these suppositions in mind, let AE be a part of the space AB and let it be an exact measure, say, one third or one fourth of AB. Therefore, this part of space has been traversed in a finite time. For it cannot be assumed that it was traversed in infinite time, because the whole space was traversed in infinite time, but less time is required to traverse a part than to traverse the whole. Likewise, let us take another part of the space, and let it equal the part AE. This part, too, must be traversed in finite time, for it is the whole space that is being traversed in infinite time. Proceeding in this manner, let us take, in accordance with the parts of the entire space, a corresponding number of such times. From these will be constituted the whole time in which the entire space is traversed.
Now, it is impossible that a part of an infinite measure the whole, either in the case of a magnitude or in that of a multitude, because it is impossible for the infinite to be composed of a finite number of parts, each of which is finite in quantity, whether those parts are equal or unequal—for whatever things are measured by some one thing, either according to magnitude or multitude, must be finite.
Now, I say “magnitude” and “multitude” because a thing of finite magnitude can still be measured, whether the measuring parts are of equal or unequal size. For when they are equal, then any part is a measure of the whole, whether the whole be a magnitude or a multitude; but when they are unequal parts, any part will measure a multitude, but not a magnitude. So, therefore, it is evident that any time that has parts finite in number and quantity, whether they be equal or not, is finite. But a finite space is measured by as many finite parts as are necessary to form AB. Moreover, the parts of the time will be equal in number to the parts of the magnitude, and the parts will be finite in quantity. What remains, therefore, is that the entire space is traversed in finite time.
845. Then, at “the reasoning will prove” (238a20), he shows that, on the other hand, if the time is finite, so too the magnitude. And he says that, by the same reasoning, it can be shown that infinite space cannot be traversed in finite time, and that rest cannot be infinite in finite time, no matter whether the motion is regular or not. For since the time posited is finite, it is possible to take as a measure of the whole time a part in which the mobile traverses a part of the magnitude but not the whole magnitude, which is traversed in the whole time. Then in an equal time it will traverse another part of the magnitude. And, in like manner, for each part of the time, take a corresponding part of the magnitude, and let this be done whether the second part of the magnitude be equal to the first part (which happens when the speed is constant) or not equal to it (which happens when the speed varies). For whether they are equal or not makes no difference, as long as each part you take of the magnitude is finite—which it must be; otherwise, as much will be traversed in a part of time as in the whole time. According to this procedure, it is clear that, by dividing time, the entire infinite space will be exhausted as the finite parts are used up. For since the time is divided into finite equal parts and the number of magnitudinal parts must equal the number of parts of time, it follows that the infinite space will be consumed by making finite subtractions, since the magnitude has to be divided according to the way the time is divided. But this is impossible. Therefore, it is clear that an infinite space cannot be traversed in finite time, whether the magnitude of space be infinite in one direction or more, for in either case, the same reason would hold.
846. Then, at “this having been proved” (238a32), he shows that infinite and finite are found in the mobile in the same way as they are found in magnitude and time.
About this, he does three things:
first, he shows that the mobile is not infinite if the magnitude is finite and the time finite;
second, he shows that the mobile is not infinite if the magnitude is infinite and the time finite, at “nor will the infinite” (238b13; [848]);
third, he shows that the mobile cannot be infinite if the magnitude is finite and the time infinite, at “we can further prove” (238b16; [849]).
He proves the first point with two arguments.
In regard to the first of these, he says that, since it has been demonstrated that a finite magnitude is not traversed in infinite time, nor an infinite magnitude in finite time, it is clear from the same causes that an infinite mobile cannot traverse a finite magnitude in finite time. For if you take any part of finite time, then during that part of time, the finite space will be traversed not by the whole mobile, but by a part, and during another part, it will be traversed by another part of the mobile, and so on. And so, it will be necessary to take as many parts of the mobile as parts of time. But the infinite is not composed of finite parts. Therefore, the mobile that is moved in a whole finite time is finite.
847. The second argument is given at “and, since a finite” (238a36), and it differs from the first, because, in the first, he took as his principle the same middle term that he used in the previous demonstrations, but here he takes as his principle the conclusion reached above. For it has been shown above that a finite mobile cannot traverse an infinite space in finite time. Hence it is clear that, for the same reason, neither can an infinite mobile traverse a finite space in finite time. For if an infinite mobile traverses a finite space, it follows that a finite mobile can traverse an infinite space, because both the mobile and the space have dimensions. Now when two things having dimensions are involved, it makes no difference which is in motion and which is at rest. For it is clear that, whichever is assumed as being in motion, it follows that the finite traverses the infinite. For let A be the infinite that is in motion and let CD be a finite part of it. When the whole is being moved, this finite part will be at the part B of the space, and as the motion continues, another part of the infinite mobile will be at B, and so on. Hence, just as the mobile traverses space, so space in a sense traverses the mobile, inasmuch as the various parts of the mobile are successively different in regard to the space. Hence, it is evident that, at the same time that an infinite mobile is being moved through a finite space, something finite is traversing something infinite. For there is no other possible way for an infinite to be moved through finite space than for the finite to traverse infinite space, either by having the finite moved over the infinite, as when the mobile is finite and the space infinite, or by making something finite measure the infinite, as when the space is finite and the mobile infinite. For then, even though the finite is not being moved over the infinite, yet the finite is measuring the infinite, inasmuch as a finite space is placed opposite each of the parts of the infinite mobile. Therefore, because this is impossible, it follows that an infinite mobile does not traverse a finite space in finite time.
848. Then, at “nor will the infinite” (238b14), he shows that there cannot be an infinite mobile if the space is infinite and time finite: an infinite mobile cannot traverse an infinite space in finite time. For in every infinite, there is something finite. Therefore, if an infinite mobile should traverse an infinite space in finite time, it follows that it traverses a finite space in finite time, which is against a previous conclusion.
849. Then, at “we can further prove” (238b16), he says that the same demonstration holds if the time be infinite and the space finite. For if an infinite mobile traverses a finite space in infinite time, it follows that, in a part of that time, it will traverse a part of the space. Consequently, the infinite will be traversing the finite in finite time, which is also against a previous conclusion.
850. Then, at “since, then, it is” (238b17), he shows that finite and infinite are found in motion in the way that they are found in mobile, space, and time. And he says that, in finite time, a finite mobile does not traverse an infinite space, nor an infinite mobile finite space, nor an infinite mobile infinite space. From these facts, it follows that there cannot be an infinite motion in finite time. For the quantity of motion depends on the quantity of space. Hence, there is no difference between saying that the motion is infinite and saying that the magnitude is. For it is necessary that, if either is infinite, so is the other, because no part of a local motion can exist outside of a place.