LECTURE 6 (234 b 21-235 b 5)

TWO WAYS IN WHICH MOTION IS DIVIDED

             806. Having set forth certain things which are necessary for the division of motion, he begins here to treat the division of motion.

             This discussion is divided into two parts. First he treats the division of motion. Secondly, where he says, 'Zeno's reasoning . . .' (239 b 5), he refutes from what has been determined certain errors about motion.

             The first part is divided into two parts. First he treats the division of motion, and secondly the division of rest, where he says, 'Since everything to which . . .' (238 b 23).

             The first part is divided into two parts. First he treats the division of motion. Secondly, where he says, 'Now since the motion . . .' (237 b 24), he treats the finite and the infinite in motion. For both the divisible and the infinite seem to pertain to the nature [ratio] of a continuum.

             The first part is divided into two parts. First he explains how motion is divided. Secondly, where he says, 'Since everything that changes . . .' (235 b 6), he treats the order of the parts of motion.

             Concerning the first part he makes two points. First he states two ways in which motion is divided. Secondly, where he says, 'And since everything . . .' (235 a 13), he shows what things are divided together with motion.

             Concerning the first part he makes two points. First he states the ways in which motion is divided. Secondly, where he says, 'That being so . . .' (234 b 24), he explains these ways.

             807. He says, therefore, first that motion is divided in two ways. First, motion is divided in respect to time. For it was shown that motion is not in the 'now', but in time. Secondly, motion is divided in respect to the motions of the parts of the mobile body. Let A C be a mobile body which is divided. It was shown that whatever is moved is divisible. Hence, if the whole of A C is moved, then both of its parts, A B and B C, must also be moved.

             However, it must be realized that the division of motion in respect to the parts of the mobile body can be understood in two ways. First this can mean that the body is moved part after part. This cannot occur when the whole in itself is moved. For in this case all the parts are moved together, not separately from the whole, but within the whole itself. The second way in which the division of motion in respect to the parts of the mobile body can be understood is as follows. The division of an accident, whose subject is divisible, occurs by a division of its subject. For example, if this whole body is white, then whiteness is accidentally divided by a division of the body. The division of motion in respect to the parts of the mobile body is here taken in this second way. Thus, each part of the mobile body is moved together with the whole so that the motions of all the parts are simultaneous. In this way the division of motion in respect to the parts of the mobile body differs from the division of motion in respect to time. For in the latter the two parts of the motion are not simultaneous. However, if the motion of one part is related to the motion of another part, not simply, but in respect to some determinate sign, then the motion of the one part precedes the motion of the other part in time. For if the mobile body A B C is moved through the magnitude E F G such that E F is equal to the whole of A C, then it is clear that B C will cross through this sign F before A B will. In this way the divisions of motion in respect to the parts of time and in respect to the parts of the mobile body run together.

             808. Next where he says, 'That being so . . .' (234 b 24), he explains the two ways in which motion is divided. First he shows that motion is divided in respect to the parts of the mobile body. Secondly, where he says, 'Motion is also susceptible . . .' (235 a 10), he shows that motion is divided in respect to the parts of time.

             He establishes the first point with three arguments. The first is as follows.

             Of the parts which are moved by the whole motion, let D E be the motion of the part A B. And let E F be the motion of the other part B C. Just as the whole mobile body A C is composed of A B and B C, in the same way the whole motion D F is composed of D E and E F. Hence, each part of the mobile body is moved by the corresponding part of the motion, such that no part of the mobile body is moved by the motion of another part (for otherwise the whole motion would pertain to one part which would be moved by its own motion and by the motion of the other part). Therefore it must be said that the whole motion D F pertains to the whole mobile body A C. And thus the motion of the whole is divided by the motion of the parts.

             809. He gives the second argument where he says, 'Again, since every motion . . .' (234 b 28). The argument is as follows.

             Every motion is a motion of some mobile body. But the whole motion D F is not the motion of either of the parts. For neither is moved by the whole motion, rather each is moved by a part of the motion, as was said. Nor can it be said that D F is the motion of some other mobile body separated from A C. For if this whole motion were the motion of some other whole body, then it would follow that the parts of this motion would be the motions of the parts of that other mobile body. But the parts of this motion D F are the motions of the parts of the mobile body, A B and B C, and of nothing else. For if they were the motions of both these and other parts, it would follow that one motion would be many, which is impossible. Hence it follows that the whole motion is the motion of the whole magnitude, just as the parts of the motion are the motions of the parts. And thus the motion of the whole is divided by the parts of the mobile body.

             810. He gives the third argument where he says, 'Again, if there is . . .' (234 b 34). The argument is as follows.

             Whatever is moved has some motion. Hence, if the whole motion D F is not the motion of the whole mobile body A C, then there must be some other motion of A C. Let this motion be H I. The motions of each of the parts is subtracted from the motion H I. The motions of the parts must be equal to D E F because for one mobile body there is only one motion. Hence, it cannot be said that the motions of the parts, which were subtracted from H I (given as the motion of the whole), are greater or less than D E and E F (given as the motions of these very parts). Hence, the motions of the parts either exhaust the whole of H I when subtracted, or they fall short of it, or else they exceed it. If they exhaust the whole of H I and neither fall short nor exceed it, then it follows that the motion H I is equal to the motion D F, which is the motion of the parts, and does not differ from D F. If the motions of the parts falls short of H I, such that H I exceeds D F by J I, then this motion J I is the motion of no mobile body. For it is not the motion of the whole of A C nor of its parts. For of one mobile body there is only one motion, and another motion has already been assigned to both the whole and the parts. Nor can it be said that J I is the motion of some other mobile body. For the whole motion H I is a continuous motion, and a continuous motion must be a motion of a continuous body, as was shown in Book V. Hence, the part J I of this continuous motion cannot be the motion of a mobile body which is not continuous with A B C.

             An inconsistency also follows if it be said that the motions of the parts exceed H I by division. For it would then follow that the parts exceed the whole, which is impossible. Hence, if it is impossible for the motion of the parts to either fall short of or to exceed the motion of the whole, then the motion of the parts must be equal to and the same as the motion of the whole.

             This is a division in respect to the motions of the parts. Such a division must also be found in the motion, because whatever is moved is divisible.

             811. Next where he says, 'Motion is also susceptible . . .' (235 a 10), he shows with the following argument that motion is divided according to the division of time.

             Every motion is in time. And every time is divisible, as was proven. Therefore, since there is less motion in a smaller time, every motion must be divided in respect to time.

             812. Next where he says, 'And since everything . . .' (235 a 13), he shows what things are divided together with motion.

             Concerning this he makes three points. First he points out five things which are divided together. Secondly, where he says, 'So, too, in the matter . . .' (235 a 37), he shows that the finite and the infinite are found together in all five of these. Thirdly, where he says, 'And we now see . . .' (235 b 1), he shows in which one of these five is primarily found division and the infinite.

             Concerning the first part he makes two points. First he states his intention. Secondly, where he says, 'For suppose that A . . .' (235 a 19), he explains his position.

             He says, therefore, first that whatever is moved is moved in 'something', that is, in respect to some genus or species. And it is moved in some time. And there is some motion for every mobile body. Therefore, the following five things must be divided together: the time, the motion, the 'being moved', the mobile body which is moved, and that in which there is motion, either place or quality or quantity.

             Nevertheless, there is not the same kind of division for all the things in which there is motion. Rather in some of these cases there is division per se and in some per accidens. There is division per se of all things which pertain to the genus of quantity, that is, local motion and increase and decrease. But there is division per accidens in things which pertain to quality, that is, alteration.

             813. Next where he says, 'For suppose that A . . .' (235 a 19), he explains what he has said.

             First he shows that time and motion are divided together. Secondly, where he says, 'In the same way . . .' (235 a 25), he shows that motion and 'being moved' are divided together. Thirdly, where he says, 'The same reasoning . . .' (235 a 34), he shows that the same is true of motion and of that in which there is motion.

             Concerning the first part he makes two points. First he shows that motion is divided by the division of time. Secondly, where he says, 'Again, the time will be . . .' (235 a 23), he shows conversely that time is divided by the division of motion.

             He proceeds as follows. Let A be the time in which something is moved. And let B be the motion which occurs in this time. Now if something is moved through a whole magnitude in a whole time, it is clear that in half the time it will be moved through less of the magnitude. Moreover to be moved by the whole motion is the same as being moved through the whole magnitude. And to be moved by part of the motion is the same as being moved through part of the magnitude. Hence it is clear that if a body is moved in the whole time by the whole motion, then in part of the time it will be moved by a smaller motion. And if the time is divided again, there will be found a still smaller motion, and so on to infinity. From this it is clear that motion is divided by the division of time.

             Next where he says, 'Again, the time will be . . .' (235 a 23), he shows conversely that if motion is divided, time is also divided. For if a body is moved through a whole motion in a whole time, then it will be moved through half the motion in half the time. And there will always be a lesser motion in a smaller time, if the mobile body has the same or equal velocity.

             814. Next where he says, 'In the same way . . .' (235 a 25), he shows that motion and 'being moved' are divided together.

             Concerning this he makes two points. First he shows that 'being moved' is divided by the division of motion. Secondly, where he says, 'Moreover by setting out . . .' (235 a 28), he shows that motion is divided by the division of 'being moved'.

             He says, therefore, first that in the same way it is proven that 'being moved' is divided by the division of time and motion. Let C be the 'being moved'. It is clear that the body is moved not only in respect to part of the motion but also in respect to the whole motion. Hence it is clear that, in respect to half of the motion, part of the 'being moved' is less than the whole 'being moved'. And it will be still less in respect to half of this half, and so on to infinity. Hence just as the time and the motion are infinitely divided, so is the 'being moved'.

             Next where he says, 'Moreover by setting out . . .' (235 a 28), he proves conversely that motion is divided by the division of 'being moved'.

             Let there be two parts of a motion, D C and C E, in respect to both of which a body is moved. And thus, if the parts of the motion correspond to the parts of 'being moved', then the whole motion must correspond to the whole 'being moved'. For if there were more of one than the other, we would argue here about motion and 'being moved' as we argued above when we showed that the motion of the whole is divisible into the motion of parts which neither fall short of nor exceed the whole. In the same way the parts of the 'being moved' cannot fall short of nor exceed the parts of the motion. For since there must be a 'being moved' which corresponds to each part of the motion, it must be that the whole 'being moved' is a continuum corresponding to the whole motion. And thus the parts of the 'being moved' will always correspond to the parts of the motion, and the whole will correspond to the whole. And hence the one is divided by the other.

             815. Next where he says, 'The same reasoning . . .' (235 a 34), he shows that the same is true of that in which there is motion.

             He says that it can be demonstrated in the same way that the length in which a body is moved in place is divisible in respect to the division of the time, and of the motion, and of the 'being moved'.

             What we say about length in local motion must be understood to apply also to everything in which there is motion, except that certain things are divisible per accidens, for example, qualities in alteration, as was said.

             Further, all these things are so divided because that which is changed is divisible, as was shown above. Hence, when one of these is divided, all of them must be divided.

             816. Next where he says, 'So, too, in the matter . . .' (235 a 37), he shows that, just as the foregoing are so related in regard to divisibility, they are also so related in regard to being either finite or infinite. Thus, if one of these is finite, they are all finite. And if one is infinite, so are the others.

             817. Next where he says, 'And we now see . . .' (235 b 1), he shows in which of the foregoing are primarily found divisibility and the finite or the infinite.

             He says that the fact that all the others are divided and are finite or infinite follows primarily from that which is changed. For that which is naturally first in motion is the mobile body itself; and divisibility and being finite or infinite are immediately present in it from its very nature. And thus from the mobile body is derived the divisibility or the finitude of the others.

             It was shown above how the mobile body is divisible and how the others are divided by it. But it will be shown below in this Book how the mobile body is related to the infinite.