LECTURE 9 (237 b 24-238 b 22)

THE FINITE AND THE INFINITE ARE FOUND TOGETHER IN MAGNITUDE, TIME, MOTION, AND THE MOBILE BODY

             841. After the Philosopher has treated the division of motion, he here treats the finite and the infinite in motion. For since division pertains to the nature [ratio] of a continuum, so do the finite and the infinite. Moreover, he has shown above that division is found simultaneously in motion, magnitude, time, and the mobile body. He now shows that the same is true of the infinite.

             Concerning this he makes three points. First he shows that the infinite is found in the same way in magnitude and in time. Secondly, where he says, 'This having been proved . . .' (238 a 32), he shows that the infinite is also found in the mobile body in the same way. Thirdly, where he says, 'Since, then, it is . . .' (238 b 18), he shows that the infinite is found in motion in the same way.

             Concerning the first part he makes two points. First he shows that if magnitude is finite, then time cannot be infinite. Secondly, where he says, 'The same reasoning . . .' (238 a 20), he shows conversely that if time is finite, then magnitude cannot be infinite.

             Concerning the first part he makes two points. First he states his intention. Secondly, where he says, 'In all cases where . . .' (237 b 27), he proves his position.

             842. He first repeats two things which are necessary to prove his position. One of these is that whatever is moved is moved in time. The second is that in a longer time a greater magnitude will be crossed by the same mobile body. From these two suppositions he intends to prove a third proposition; namely, it is impossible to cross a finite magnitude in an infinite time. This is not to be understood as meaning that the mobile body re-crosses the same magnitude or some part thereof many times. Rather he means that in the whole time the body is moved through the whole magnitude. He adds this in order to exclude circular motion, which always occurs in a finite magnitude but which can occur in an infinite time, as he will say in Book VIII.

             843. Next where he says, 'In all cases where . . .' (237 b 27), he proves his position. He does this first on the assumption that the mobile body is moved through the whole magnitude with equal velocity. Secondly, where he says, 'But it makes no . . .' (237 b 34), he proves his point on the assumption that the body is not moved uniformly and regularly.

             He says, therefore, first that if there be a mobile body which is moved through a whole magnitude with equal velocity, then, if it crosses a finite magnitude, this must occur in a finite time. For let there be one part of the magnitude which measures the whole; for example, a third or a fourth of the magnitude. Hence, if the mobile body is moved through the whole magnitude with equal velocity, and if 'equal velocity' means that an equal space is crossed in an equal time, it follows that the mobile body crosses the whole magnitude in as many equal times as there are parts of the magnitude. For example, if we take a fourth part of the magnitude, the body will cross it in a certain time. And it will cross another fourth in another equal time. And hence it will cross the whole magnitude in four equal times.

             Therefore, since the parts of the magnitude are finite in number, and since each part is finite in quantity, and since the body crosses all the parts in as many equal times, it follows that the whole time in which the body crosses the whole magnitude is finite. For it will be measured by a finite time, because there are as many times in which the body crosses a part of the magnitude as there are parts of the magnitude. And thus the total time will have been multiplied in respect to the multiplication of the parts. And every multiple is measured by that which is under the multiple. For example, a double is measured by a half, and a triple by a third, and so forth. Furthermore, the time in which the body crosses part of the magnitude is finite. For if it were infinite, it would follow that the body would cross both the whole and the part in an equal time. This is contrary to what was supposed. Hence the total time must be finite, because no infinity is measured by something which is finite.

             844. But someone might say that, although the parts of the magnitude are equal and measure the whole magnitude, nevertheless it can happen that the parts of the time are not equal; for example, when there is not an equal velocity in a whole motion. And thus the time in which the body is moved through part of the magnitude will not measure the time in which it is moved through the whole magnitude. Consequently, where he says, 'But it makes no . . .' (237 b 34), he shows that this does not affect his position.

             Let there be a finite space A B which is crossed in an infinite time C D. In this whole motion one part must be crossed prior to another. And it is also clear that different parts of the magnitude are crossed in different parts of the time. Thus it is necessary that two parts of the magnitude are not crossed in one and the same part of the time, and in two parts of the time one and the same part of the magnitude is not crossed. And thus, if some part of the magnitude is crossed in some time, then in a longer time not only that part but also other parts of the magnitude must be crossed. And this is true whether the mobile body is moved with equal velocity or not. The latter occurs either when the velocity is increased more and more, as in natural motions, or when it is decreased more and more, as in violent motions.

             On these suppositions let A E be a part of the space A B, and let it measure the whole of A B such that it is some part of A B, either a third or a fourth. Hence this part of the space is crossed in some finite time. It cannot be held that A E is crossed in an infinite time. For the whole space is crossed in an infinite time, and a part is crossed in less time than the whole.

             If we take another part of the space equal to A E, this part must be crossed in a finite time for the same reason, for the whole space is crossed in an infinite time.

             By continuing this process one will take as many finite times as there are parts of the space. The total time in which the body is moved through the whole space will be constituted by these times.

             But it is impossible for any part of an infinite to measure a whole, either in respect to magnitude or in respect to multitude. For it is impossible for an infinite to consist of parts which are finite in number, each one of which is finite in quantity, either equal or unequal. For whatever is measured by some one thing, in respect to either multitude or magnitude, must be finite.

             I say 'multitude and magnitude' because something is measured because it has a finite magnitude, whether the measuring parts are equal or unequal. When they are equal, the part measures the whole in both multitude and magnitude. But when the measuring parts are unequal, the part measures the whole in multitude but not in magnitude. Thus it is clear that any time which has parts which are finite in number and in quantity, either equal or unequal, is finite. But the finite space A B is measured by as many finite things as happen to compose A B. And the parts of the time and the parts of the magnitude must be equal in number. And each part must be finite in quantity. Therefore it follows that the body is moved through the whole space as in a finite time.

             845. Next where he says, 'The same reasoning . . .' (238 a 20), he shows conversely that if time is finite, then magnitude is also finite.

             He says that it can be shown with the same argument that an infinite space cannot be crossed in a finite time. Nor indeed can rest be infinite in a finite time. It makes no difference whether the body is moved regularly, that is, with equal velocity, or irregularly. Since the time is given as finite, let us take a part in this time which measures the whole time. In this part of the time the mobile body crosses part of the magnitude (but not the whole magnitude, for it crosses the whole magnitude in the whole time). Further in another equal time it crosses another part of the magnitude. And in the same way there is a part of the magnitude for each part of the time. It makes no difference whether the second part of the magnitude is equal to the first part (which happens when the body is moved with equal velocity) or is not equal to the first part (which happens when the body is not moved with equal velocity). This makes no difference as long as each part taken in the magnitude is finite. This must be said, for otherwise the body would be moved in part of the time as much as it is moved in the whole. Thus it is clear that by a division of the time the whole infinite space is exhausted by a finite subtraction. For since the time is divided into equal finite parts, and since there must be as many parts of the magnitude as there are of the time, it would follow that an infinite space would be consumed by a finite subtraction. For the magnitude must be divided in as many ways as the time. But this is impossible. Therefore it is clear that an infinite space is not crossed in a finite time. And it makes no difference whether the magnitude of the space is infinite in one direction or in both. For the same argument applies to both cases.

             846. Next where he says, 'This having been proved . . .' (238 a 32), he shows that the finite and the infinite are found in the mobile body in the same way that they are found in magnitude and in time.

             Concerning this he makes three points. First he shows that, if time and magnitude are finite, then the mobile body is not infinite. Secondly, where he says 'Nor again will . . .' (238 b 14), he shows that if magnitude is infinite and time finite, then the mobile body is not infinite. Thirdly, where he says, 'We can further prove . . .' (238 b 16), he shows that, if magnitude is finite and time infinite, then the mobile body cannot be infinite.

             He proves the first point with two arguments. In the first argument he says that since it has been demonstrated that a finite magnitude is not crossed in an infinite time, nor is an infinite magnitude crossed in a finite time, for the same reason it is clear that an infinite mobile body cannot cross a finite magnitude in a finite time. Let us take some part of the finite time. In that part of the time the finite space will be crossed by a part, but not the whole, of the mobile body. And in another part of the time the same thing will happen, and so forth. And thus there must be as many parts of the mobile body as there are parts of the time. But the infinite is not composed of finite parts, as was shown. Therefore it follows that a mobile body which is moved in a total finite time is finite.

             847. He gives the second argument where he says, 'And since a finite . . .' (238 a 37).

             This second argument differs from the first one because in the first argument he assumed as a principle the middle proposition which he used in a proof above. But here he uses as a principle the conclusion which he demonstrated above. It was shown above that a finite mobile body cannot cross an infinite space in a finite time. Hence for the same reason it is clear that an infinite mobile body cannot cross a finite space in a finite time.

             If an infinite mobile body crosses a finite space, it follows that a finite mobile body also crosses a finite space. For since both the mobile body and the space are quantified, when these two quantities are given, it makes no difference which one is moved and which one is at rest. In this case the space is at rest, and the mobile body is moved.

             It is clear that, whichever is given as being moved, the finite crosses the infinite. Let the infinite body A be in motion. And let C D be some finite part of A. When the whole is being moved, the finite part C D will be in the space signified by B. And when the motion is continued, another part of the infinite mobile body will become located in that space, and so forth to infinity. Hence, just as the mobile body crosses the space, the space also in a way crosses the mobile body insofar as the diverse parts of the mobile body are successively alternated with the corresponding space. Hence it is clear that an infinite mobile body is moved through a finite space, and simultaneously the finite crosses the infinite. For the infinite cannot be moved through a finite space unless the finite crosses the infinite, either in such a way that the finite is carried through the infinite (as when the mobile body is finite and the space is infinite) or at least in such a way that the finite measures the infinite (as when the space is finite and the mobile body infinite). For in the latter case, although the finite is not carried through the infinite, nevertheless the finite measures the infinite insofar as the finite space becomes joined to the individual parts of the infinite mobile body. Now since this is impossible, it follows that the infinite mobile body does not cross a finite space in a finite time.

             848. Next where he says, 'Nor again will . . .' (238 b 14), he shows that the mobile body cannot be infinite when the space is infinite and the time finite. He says that an infinite mobile body does not cross an infinite space in a finite time. For in every infinite there is a finite. If, therefore, an infinite mobile body crosses an infinite space in a finite time, it follows that it crosses a finite space in a finite time. This is contrary to what was shown above.

             849. Next where he says, 'We can further prove . . .' (238 b 16), he says that the same demonstration applies if the time is infinite and the space finite. For if an infinite mobile body crosses a finite space in an infinite time, it follows that in a finite part of that time it crosses a part of the space. And thus the infinite will cross the finite in a finite time, which is contrary to what was shown above.

             850. Next where he says, 'Since, then, it is . . .' (238 b 18), he shows that the finite and the infinite are found in motion in the same way as in the foregoing.

             He says that a finite mobile body does not cross an infinite space, and an infinite mobile body does not cross a finite space, and an infinite mobile body does not cross an infinite space in a finite time. From these points it follows that there cannot be an infinite motion in a finite time. For the quantity of motion is taken according to the quantity of space. Hence it makes no difference whether the motion or the magnitude is said to be infinite. If one of these is infinite, the other must be infinite, for there cannot be a part of local motion outside of place.