Commentary Aristotle Physics
In all cases where a thing is in motion with uniform velocity, it is clear that the finite magnitude is traversed in a finite time.
For if we take a part of the motion that shall be a measure of the whole, the whole motion is completed in as many equal periods of the time as there are parts of the motion.
Consequently, since these parts are finite, both in size individually and in number collectively, the whole time must also be finite, for it will be a multiple of the portion, equal to the time occupied in completing the stated part multiplied by the number of the parts.
But it makes no difference even if the velocity is not uniform.
For let us suppose that the line AB represents a finite stretch over which a thing has been moved in the given time, and let CD be the infinite time.
Now, if one part of the stretch must have been traversed before another part (it is clear that, in the earlier and in the later part of the time, a different part of the stretch has been traversed, for as the time lengthens, a different part of the motion will always be completed in it, whether the thing in motion changes with uniform velocity or not, and whether the rate of motion increases or diminishes or remains stationary, this is none the less so), let us then take AE, a part of the whole stretch of motion AB, which shall be a measure of AB.
Now, this part of the motion occupies a certain period of the infinite time; it cannot itself occupy an infinite time, for we are assuming that that is occupied by the whole AB.
And if, again, I take another part equal to AE, that also must occupy a finite time in consequence of the same assumption.
And, if I go on taking parts in this way, on the one hand, there is no part that will be a measure of the infinite time (for the infinite cannot be composed of finite parts, whether equal or unequal, because there must be some unity that will be a measure of things finite in multitude or in magnitude, which, whether they are equal or unequal, are none the less limited in magnitude), while, on the other hand, the finite stretch of motion AB is a certain multiple of AE.
Consequently, the motion AB must be accomplished in a finite time. Moreover, it is the same with coming to rest as it is with motion. And so it is impossible for one and the same thing to be infinitely in process of becoming or of corruption.
The reasoning will prove that, in a finite time, there cannot be an infinite extent of motion or of coming to rest, whether the motion is regular or irregular.
For if we take a part that shall be a measure of the whole time, in this part, a certain fraction (not the whole) of the magnitude will be traversed, because we assume that the traversing of the whole occupies all the time.
Again, in another equal part of the time, another part of the magnitude will be traversed, and similarly in each part of the time that we take, whether equal or unequal to the part originally taken.
It makes no difference whether the parts are equal or not, if only each is finite, for it is clear that, while the time is exhausted by the subtraction of its parts, the infinite magnitude will not be thus exhausted, since the process of subtraction is finite both in respect of the quantity subtracted and of the number of times a subtraction is made.
Consequently, the infinite magnitude will not be traversed in finite time; and it makes no difference whether the magnitude is infinite in only one direction or in both, for the same reasoning will hold good.
This having been proved, it is also evident that a finite magnitude cannot traverse an infinite magnitude in a finite time, the reason being the same as that given above: in part of the time, it will traverse a finite magnitude, and in each part likewise, such that, in the whole time, it will traverse a finite magnitude.
And, since a finite magnitude will not traverse an infinite in a finite time, it is clear that neither will an infinite traverse a finite in a finite time.
For if the infinite could traverse the finite, the finite could traverse the infinite; for it makes no difference which of the two is the thing in motion: either case involves the traversing of the infinite by the finite. For when the infinite magnitude A is in motion, a part of it, say CD, will occupy the finite, and then another, and then another, and so on to infinity.
Thus, the two results will coincide: the infinite will have completed a motion over the finite and the finite will have traversed the infinite.
For it would seem to be impossible for the motion of the infinite over the finite to occur in any way other than by the finite traversing the infinite either by locomotion over it or by measuring it. Therefore, since this is impossible, the infinite cannot traverse the finite.
Nor will the infinite traverse the infinite in a finite time. Otherwise, it would also traverse the finite, for the infinite includes the finite.
We can further prove this in the same way by taking the time as our starting point.
Since, then, it is established that, in a finite time, neither will the finite traverse the infinite nor will the infinite the finite, nor the infinite the infinite, it is evident also that, in a finite time, there cannot be infinite motion. For what difference does it make whether we take the motion or the magnitude to be infinite? If either of the two is infinite, the other must be so likewise, for all locomotion is in space.
841. After determining the division of motion, the Philosopher now determines about the infinite and finite in motion, for just as division pertains to the account of continuum, so also do finite and infinite. But, just as he said above that division is found simultaneously in motion, magnitude, time, and mobile, so now he shows that the same is true of the infinite.
Thus, about this, he does three things:
first, he shows that the infinite is found in the same way in magnitude and in time;
second, he shows that it is found in the same way in the mobile, at “this having been proved” (238a32; [846]);
and third, in motion, at, since, then, it is (238b17; [850]).
About the first, he does two things:
first, he shows that, if a magnitude is finite, the time cannot be infinite;
second, he shows that, if the time is finite, the magnitude cannot be infinite, at “the reasoning will prove” (238a20; [845]).
In regard to the first, he does two things:
first, he proposes what he intends;
second, he proves his proposition, at “in all cases where” (237b26; [843]).
842. First, therefore (237b23), he repeats two things that are needed for proving the proposition.
One of these is that whatever is being moved is being moved in time.
The second is that, in more time, a greater magnitude is traversed by the same mobile.
From these two suppositions, he intends to prove a third: it is impossible to traverse a finite magnitude in infinite time. This is to be understood in the sense that the thing in motion is not to re-traverse the same magnitude repeatedly or any part of it, but must be moved through the entire magnitude in the entire time. And he added this to save himself from circular motion over a finite magnitude, which can occur in infinite time, as will be explained in book 8.
843. Then, at “in all cases where” (237b26), he proves his proposition:
first, by assuming a mobile of equal speed being moved over the whole magnitude;
second, if it is not being moved with a regular and uniform motion, at “but it makes no” (237b34; [844]).
He says first (237b26) that, if a mobile of equal speed is traversing a whole, then, if the whole is a finite magnitude, it must be traversed in finite time. For we can take one part of the magnitude and make it measure the whole, such as a part that is one third or one fourth of the magnitude. If, therefore, a mobile is moved with equal speed over the whole and if the equally fast is what traverses an equal space in equal time, it follows that, in a number of equal times that are determined by the number of parts into which the magnitude was divided, it will traverse the whole magnitude; for example, if one fourth of the magnitude is taken, it will traverse it in a certain time and another fourth in an equal time, and so it will traverse the entire magnitude in four equal times.
Because, therefore, the parts of the magnitude are finite in number and each is finite in quantity, and in a given number of equal times, the whole magnitude is traversed, it follows that the whole time in which the entire magnitude is traversed is finite. For it will be measured by a finite time, since it will be as many times as much as the time required to traverse one part, the whole magnitude being as many times as the quantity of each part. And, thus, the whole time will be the multiplication product of the length multiplied by the number of parts. But every product of multiplication is measured by a denominator, as double is measured by half and triple by third, and so on. But the time required to traverse a part is finite, for if it were infinite, it would follow that the whole and the part were traversed in equal time, which is against the original assumption. Therefore, the whole time has to be finite, because nothing infinite can be measured by the finite.
844. But someone could say that, although the parts of the magnitude are equal and measure the whole magnitude, it could happen that the parts of time are not equal, as when an equal speed is not maintained through the entire motion, and so the time required to traverse a part of the magnitude will not be a measure of the time required to traverse the whole. Therefore, at “but it makes no” (237b34), he shows that this makes no difference to the proposition.