Lecture 9 — The finite and the infinite found together in magnitude, time, motion, and the mobile body

Now, since the motion of everything is in motion in time, and a greater magnitude is traversed in a longer time, it is impossible that a thing should undergo a finite motion in an infinite time, if this does not mean that the same motion or a part of it is continually repeated, but that the whole infinite time is occupied by the whole finite motion.

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, (Lect. 6, n. 812ff.) 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. (Cf. lect. 8, n. 827ff.)

The second is that, in more time, a greater magnitude is traversed by the same mobile. (Cf. lect. 3, n. 772.)

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. (Lect. 19ff.)

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.

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.