Lecture 6 — Two ways in which motion is divided

Now, motion is divisible in two senses.

In the first place, it is divisible in virtue of the time that it occupies.

In the second place, it is divisible according to the motions of the several parts of that which is in motion:

for instance, if the whole AC is in motion, there will be a motion of AB and a motion of BC.

That being so, let DE be the motion of the part AB, and EZ the motion of the part BC. Then, the whole DZ must be the motion of AC, for DZ must constitute the motion of AC inasmuch as DE and EZ severally constitute the motions of each of its parts. But the motion of a thing can never be constituted by the motion of something else, and consequently, the whole motion is the motion of the whole magnitude.

Again, since every motion is a motion of something and the whole motion DZ is not the motion of either of the parts (for each of the parts, DE or EZ is the motion of one of the parts AB or BC) or of anything else

(for the whole motion is the motion of a whole, and the parts of the motion are the motions of the parts; and the parts of DZ are the motions of AB and BC and of nothing else, for a motion that is one cannot be the motion of more than one thing).

Since this is so, the whole motion will be the motion of the magnitude ABC.

Again, if there is a motion of the whole other than DZ, parts may be subtracted from it, and these motions will be equal to DE and EZ, respectively, since the motion of that which is one must be one.

So, if the whole motion OI may be divided into the motions of the parts, OI will be equal to DZ; if, on the other hand, there is any remainder, say KI, this will be a motion of nothing.

For it can be the motion neither of the whole nor of the parts (as the motion of that which is one must be one) nor of anything else, for a motion that is continuous must be the motion of things that are continuous.

And the same result follows if the division of OI reveals a surplus on the side of the motions of the parts.

Consequently, if this is impossible, the whole motion must be the same as and equal to DZ.

This, then, is what is meant by the division of motion according to the motions of the parts, and it must be applicable to everything that is divisible into parts.

Motion is also susceptible to another kind of division, that according to time. For since all motion is in time and all time is divisible, and in less time, the motion is less, it follows that every motion must be divisible according to time.

And, since everything that is in motion is in motion to something and for a certain time and has a motion belonging to it, it follows that the time, the motion, the act of being in motion, the thing that is in motion, and the sphere of the motion must all be susceptible to the same divisions (though spheres of motion are not all divisible in a like manner, and thus quantity is essentially divisible, but quality accidentally).

For suppose that A is the time occupied by the motion B. Then, if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, and less again to occupy a further subdivision of the time, and so on to infinity.

Again, the time will be divisible similarly to the motion; for if the whole motion takes the whole time, then half the motion takes half the time; and, again, the lesser motion takes the lesser time.

In the same way, the being-in-motion will also be divisible. For let C be the whole being-in-motion.

Then the being-in-motion that corresponds to half the motion will be less than the whole being-in-motion, and that which corresponds to a quarter of the motion will be less again, and so on to infinity.

Moreover, by setting out successively the being-in-motion corresponding to each of the two motions DC (say) and CE, we may argue that the whole being-in-motion will correspond to the whole motion

(for if it were some other being-in-motion that corresponded to the whole motion, there would be more than one being-in motion corresponding to the same motion),

the argument being the same as that whereby we showed that the motion of a thing is divisible into the motions of the parts of the thing.

For if we take separately the being-in-motion corresponding to each of the two motions, we shall see that the whole being-in-motion is continuous.

The same reasoning will show the divisibility of the length, and in fact of everything that forms a sphere of change (though some of these are only accidentally divisible because that which changes is so), for the division of one term will involve the division of all.

So, too, in the matter of their being finite or infinite, they will all alike be either the one or the other.

And we now see that, in most cases, the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite.

For the attributes “divisible” and “infinite” belong in the first instance to the thing that changes. That divisibility does so we have already shown; that infinity does so will be made clear in what follows.

806. Having established the facts needed for dividing motion, he now begins to treat of the division of motion.

And the treatment is divided into two parts.

In the first, he treats of the division of motion;

in the second, he uses his conclusions to refute errors about motion, at Zeno’s reasoning (239b5; [860]).

The first part is divided into two sections:

in the first, he discusses division of motion;

in the second, he discusses division of rest, at since everything to which (238b23; [851]).

The first section is divided into two parts:

in the first, he deals with division of motion;

in the second, he discusses finite and infinite with respect to motion (for both the divisible and the infinite seem to belong to the continuum), at now, since the motion (237b23; [841]).

The first is divided into two parts:

in the first, he shows how motion is divided;

in the second, he treats of the order of the parts of motion, at since everything that changes (235b6; [818]).

In regard to the first, he does two things:

first, he lists two ways by which motion is divided;

second, he mentions what else is divided when motion is divided, at and, since everything (235a13; [812]).

In regard to the first, he does two things:

first, he mentions the ways in which motion is divided;

second, he explains them, at that being so (234b24; [808]).

807. He says first (234b21) that motion is divided in two ways.

In one way, it is divided according to time, because it has been shown that motion occurs not in the now, but in time. (Lect. 5, n. 794.)

In a second way, it is divided according to the motions of the parts of the mobile. Let the mobile AC be divided, for any mobile can be divided, as we have shown. (Lect. 5, n. 796.) If, therefore, the entire mobile AC is being moved, then each of its parts, AB and BC, is in motion.

But notice that the dividing of motion according to the parts of the mobile can be understood in two ways.

In the first way, part would be being moved after part, which is not possible in that which is in motion per se in its entirety, for in the case of such a mobile, all the parts are moved together—not in isolation from the whole, but in the whole.

In the second way, the dividing of motion according to parts of the mobile can be taken in the same sense as the division of an accident whose subject is divisible depends on the division of that subject; for example, if a whole body is white, then, as the body is divided, the whiteness will be divided per accidens.

And it is in this sense that we are taking division of motion according to the parts of the mobile: just as both parts of the mobile are in motion at the same time as the whole is, so the motions of both parts occur at the same time. This shows that division of motion according to the parts of the mobile is different from the division according to time, in which two given parts of a motion do not occur at the same time.

But, if we were to compare the motion of one part to that of another part not absolutely but according to a fixed stage to be reached, then the motion of one part will precede in time the motion of another part. For if the mobile ABC is moved in the magnitude EFG such that EF is equal to length ABC of the mobile, it is clear that BC will reach F before AB does. According to this, the division of motion according to the parts of time and according to the parts of the mobile will be the same.

808. Then, at that being so (234b24), he explains these ways of dividing motion:

first, he shows that motion is divided according to the parts of the mobile;

second, he shows that it is divided according to the parts of time, at motion is also susceptible (235a10; [811]).

The first he shows by three arguments.

The first is this. Since the parts are in motion because the wholes are in motion, let DE be the motion of the part AB, and EZ the motion of the part BC. Therefore, just as the whole mobile is composed of AB and BC, so the whole motion DZ is composed of DE and EZ.

Since, therefore, both of the parts of the mobile are being moved in accordance with both of the parts of the motion in such a way that neither part of the mobile is being moved in accordance with the motion of the other part

(because, then, the entire motion would be the motion of one part, which would be moved by its own motion and by the motion of the other part),

then it must be admitted that the whole motion DZ is the motion of the whole mobile AC. And, thus, the motion of the whole is divided by means of the motion of the parts.

809. He gives the second argument at again, since every motion (234b29). Every motion belongs to some mobile. But the entire motion DZ does not belong to either of the parts, because neither is being moved according to the entire motion, but both are being moved according to the parts of the motion, as we have said.

Nor can it be said that the whole motion DZ is the motion of some other mobile separated from AC, because, if the whole of this motion were the motion of some other whole mobile, it would follow that the parts of this motion would belong to the parts of that mobile;

but we have already agreed that the parts of the motion DZ belong to the parts of the original mobile, which are AB and BC, and to no other parts

(for if they belonged to these and to others as well, it would follow that one motion would belong to several things, which is impossible).

What remains, therefore, is that the entire motion belongs to the entire magnitude, just as the parts of it belong to the parts of the magnitude. And, thus, the motion of the whole mobile is divided according to the parts of the mobile.

810. He gives the third argument at again, if there is (234b34). Everything that is being moved has a motion. Therefore, if the whole motion DZ does not belong to the whole mobile AC, then some of the motion does, and let it be TI. Now, from this motion, TI, take away by division the motions of both parts, which must be equal to the motions that form DEZ for the following reason: one mobile does not have but one motion, and consequently, the parts’ motions that are taken away from the motion TI (which is the motion of a whole) cannot be said to be greater or less than DE and EZ, which we agreed are the motions of those same parts.

Now, the motions of the parts take up the whole motion TI, or they are less or greater. If they take up the entire TI and are neither greater nor less, it follows that the motion TI is equal to the motion DZ (which is the motion of the parts) and does not differ from it.

But, if the motions of the parts are less than TI such that TI exceeds DZ by the amount KI, then the part KI of the motion does not belong to any mobile. For it is the motion neither of AC nor of any of its parts, because one thing has only one motion, and we have already assigned a different motion both to the whole AC and to its parts.

Nor can we say that KI belongs to some other mobile, because the entire motion TI is one continuous motion and a continuous motion must belong to a thing that is continuous, as we have shown in book 5. (Lect. 7, n. 704.) Hence, it cannot be that the part KI of this continuous motion belongs to a mobile not continuous with ABC.

A like difficulty follows if it is said that the motion of the parts exceeds the divided motion TI, because it will follow that the parts exceed the whole—which is impossible.

Consequently, if it is impossible that the parts either exceed or are less than to the whole, then necessarily the motion of the parts is equal to and is the same as the motion of the whole.

And, so, this division is based on the motions of the parts, and such a partition must be found in motion, because everything that is being moved is capable of being divided into parts.

811. Then, at motion is also susceptible (235a10), he shows in the following argument that motion is divided according to the division of time. Every motion occurs in time, and every time is divisible, as we have proved. (Lect. 3.) Therefore, since there is less motion in less time, every motion must be capable of being divided according to time.

812. Then, at and, since everything (235a13), he shows what other things are divided when motion is divided.

About this, he does three things:

first, he mentions five things that are co-divided;

second, he shows that, if the finite or infinite is found in any of them, it is found in all the others, at so, too, in the matter (235a37; [816]);

third, he shows in which of them is first found division and the infinite, at and we now see (235b1; [817]).

About the first, he does two things:

first, he states his proposition;

second, he explains the proposition, at for suppose that A (235a18; [813]).

He says first (235a13) that, since everything that is being moved is being moved in respect to something—that is, to some genus or species as well as in time—and, moreover, since every mobile is capable of some motion, then necessarily the following five things must be divided at the same time that any one of them is divided:

time, motion, the very act of being moved, the mobile that is being moved, and that in which there is motion (that is, place or quality or quantity).

Nevertheless, the divisions of the spheres of motion do not all occur in the same way, but in some, the division is per se, and in others per accidens. The division is per se if it is in the genus of quantity, as it is in local motion and also in growth and decrease; but it is per accidens in the genus of quality, as in the motion called alteration.

813. Then, at for suppose that A (235a18), he explains what he has said:

first, the statement that time and motion are co-divided;

second, that motion and the act of being moved are divided, at in the same way (235a25; [814]);

third, that motion and that in which there is motion are divided, at the same reasoning (235a34; [815]).

About the first, he does two things:

first, he shows that, with division of time, motion is divided;

second, vice versa, at again, the time will be (235a22).

He first says, therefore (235a18), to let A be the time in which something is being moved, and let B be the motion occurring in this time.

Now, it is evident that, if something is being moved through an entire magnitude in the whole time A, then it will be moved through a smaller magnitude in half the time. But to be moved through the entire motion is the same as being moved through the entire magnitude, just as to be moved through part of the motion is the same as being moved through part of the magnitude.

Therefore, it is clear that, if it is moved through the whole motion in the entire time, then it will be moved through a smaller motion in part of the time. And, if the time be again divided, a smaller motion will be found, and so on indefinitely. And so, it is evident that motion is divided according to the division of time.

Then, at again, the time will be (235a22), he shows that, on the other hand, if the motion is divided, the time is divided. For if it is being moved through the entire time, then, through half the motion, it will be moved through half the time and so on; as the motion is smaller, the corresponding time is also, provided, of course, that we are dealing with the same mobile or one equally fast.

814. Then, at in the same way (235a25), he shows that motion and the act of being moved are co-divided.

Regarding this, he does two things:

first, he shows that being moved is divided according to the division of motion;

second, he shows that motion is divided in accordance with the division of being moved, at moreover, by setting out (235a28).

He therefore says (235a25) that, in the same way, it is proved that being moved is divided in accordance with the division of time and motion. For let “being moved” be C. Now, it is evident that a thing is not moved as much according to part of the motion as according to the whole of the motion. Therefore, according to half of the motion, part of the factor called “being moved” will be less than the whole factor, and still less according to half of the half, and so on. Therefore, as time and motion are continually subdivided, so also the factor called “being moved.”

Then, at moreover, by setting out (235a28), he proves that, conversely, motion is divided according to the division of being moved. For let DC and CE be two parts of a motion, according to both of which something is being moved. Then, if the parts of the motion correspond to the parts of being moved, then the whole corresponds to the whole, for if there were more in one than in the other, then the same argument would apply here that applied when we proved that the motion of a whole can be divided into motions of the parts in such a way that there is neither excess nor defect. In like manner, the parts of being moved can be neither less nor greater than the parts of the motion; for since we must admit a being moved for each part of the motion, then necessarily the entire factor called “being moved” is continuous and corresponds to the entire motion. And thus, the parts of being moved correspond to the parts of the motion, and the whole to the whole. Consequently, one is divided in accordance with the other.

815. Then, at the same reasoning (235a34), he shows the same for that in which the motion takes place. And he says that, in the same way, it can be demonstrated that the length in which something is moved locally can be divided according to the division of time and of motion and of being moved. And what we say of the length in local motion is to be understood of everything in which there is motion, except that, in some, the division is per accidens, as in the case of qualities in the motion of alteration, as was said. And hence it is that all those things are divided, because the subject of change can be divided, as was explained above. (Lect. 5, n. 796.) Consequently, if one is divided, all the others must be.

816. Then, at so, too, in the matter (235a37), he says that, just as the above-mentioned things follow upon one another in divisibility, so also in being finite or infinite, such that, if one of them is finite, all the others are, and if one is infinite, so are all the others. (Cf. lect. 9.)

817. Then, at and we now see (235b1), he shows in which of the five above-mentioned things divisibility and finite and infinite are first found. He says that the subject of change is the first root from which the divisibility and finiteness and infinity of the others flow, because what is naturally first in motion is the mobile, which of its very nature has the properties called divisibility, finiteness, and infinity. Hence, from it, divisibility and finiteness flow to the others.

But how the mobile is divisible and how the others are divided through it we have already shown. (N. 807ff.; cf. lect. 5, n. 796.) How the mobile is infinite will be explained later in this book 6. (Lect. 9.)