Commentary Aristotle Physics
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. 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 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. 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.