LECTURE 11 (239 b 5-240 b 7)

THE ARGUMENTS OF ZENO, WHO TRIED TO DENY ALL MOTION, ARE ANSWERED

             860. After the Philosopher has treated the division of motion and rest, he here refutes certain things which have caused some to fall into error concerning motion.

             Concerning this he makes three points. First he answers the arguments of Zeno, who denied all motion. Secondly, where he says, 'Our next point is . . .' (240 b 8), he shows that the indivisible is not moved. This is contrary to Democritus, who held that indivisibles are always in motion. Thirdly, where he says, 'Our next point is . . .' (241 a 27), he shows that every mutation is finite. This is contrary to Heraclitus, who held that everything is always in motion.

             Concerning the first part he makes two points. First he gives and answers a certain argument of Zeno which pertains to what he has just set forth in regard to motion. Secondly, where he says, 'Zeno's arguments . . .' (239 b 10), he explains all of Zeno's arguments in order.

             861. He says, therefore, first that Zeno reasons fallaciously and uses an apparent syllogism to show that nothing is moved, not even that which seems to be moved with the greatest speed, like a flying arrow. His argument is as follows. Whatever is in a place equal to itself is either in motion or at rest. But in each 'now' whatever is being moved is in a place equal to itself. Therefore in each 'now' it is either moved or at rest. But it is not moved. Therefore it is at rest. Moreover, if it is not moved in any 'now', but rather seems to be at rest, then it follows that in the total time it is not moved, but rather is at rest.

             This argument could be answered by means of that which was shown above; namely, in the 'now' there is neither motion nor rest. But this answer does not refute Zeno's intention. For it is sufficient for Zeno's purpose if it can be shown that there is no motion in a whole time. And this seems to follow from the fact that there is no motion in the 'now' of time. Therefore Aristotle answers in another way, saying that the conclusion of the argument is false and does not follow from the premises.

             In order that a thing be moved in some time, it must be moved in each part of that time. But the 'now' is not a part of time. For time is not composed of indivisible 'nows', just as magnitude is not composed of indivisibles, as was proven above. Hence from the fact that there is no motion in the 'now', it does not follow that there is no motion in time.

             862. Next where he says, 'Zeno's arguments . . .' (239 b 10), he gives in order all the arguments which Zeno used to deny motion.

             Concerning this he makes three points. First he shows how Zeno denied local motion with his arguments. Secondly, where he says, 'Nor in reference to . . .' (240 a 19), he shows how Zeno denied the other species of mutation. Thirdly, where he says, 'Again, in the case . . .' (240 a 28), he shows how Zeno denied circular motion in a special way.

             863. Concerning the first part he gives four arguments. He says that Zeno used four arguments against motion which have caused difficulty for many who wished to answer them.

             The first argument is as follows. If a thing is being moved through a whole space, it must first cross half that space before it comes to the end. But since that half is divisible, it must first cross half of that half, and so forth to infinity, since magnitude is infinitely divisible. But the infinite cannot be crossed in a finite time. Therefore nothing can be moved.

             Aristotle says that he has answered this argument above at the beginning of Book VI where he said that time is divided to infinity in the same way that magnitude is divided to infinity. But this answer applies to one who asks if the infinite can be crossed in a finite time rather than to the question itself, as he will say in Book VIII. In this latter place he answers this argument by saying that a mobile body does not use the infinities in magnitude as if they existed in act, but rather in potency. If the mobile body uses a point in space as a beginning and as an end, then it would use that point as existing in act. It would then be necessary for the body to be at rest there, as he will show. And thus, if the body must cross an infinity existing in act, then it would never arrive at the end.

             864. He gives the second argument where he says, 'The second is . . .' (239 b 14). He says that this is called the 'Achilles' argument, as though it is invincible and insoluble.

             The argument is as follows. If a thing is being moved, it follows that that which runs slower, if it began to be moved first, is never joined or touched by that which is very fast.

             He proves this as follows. If a slow body begins to be moved in some time before a very fast body, then in that time the slow body has crossed some space. And before a very fast body which pursues can touch the very slow body which flees, the very fast body must go from the place from which the fleeing body began up to the place which the fleeing body reached in the time that the pursuing body was not being moved. But the very fast body must cross that space in some time. In this time the slower body has crossed another space, and so forth to infinity. Therefore the slower body will always have some space by which it precedes the very fast body which pursues it. And thus the faster body will never touch the slower body. But this is impossible. Therefore, it should rather be said that nothing is moved.

             865. In answering this argument Aristotle says that this argument is the same as the first, which proceeded by a division of space into two halves insofar as there is a middle. But this second argument differs from the first as follows. The magnitude of space in the second argument is not divided into two halves but is divided according to the proportion of excess of the faster body to the slower in motion. For in the first time in which only the slower body was moved there is a greater magnitude. Since the second time in which the faster body crosses this space is shorter, the magnitude crossed by the slower body is smaller, and so forth to infinity. Hence, since the time and the magnitude are infinitely divided, it seems from this argument that the slower body is never joined by the faster body.

             But this comes to the same thing as was said in the first argument about the division of magnitude into two halves. For in each argument it seems that the mobile body cannot arrive at a terminus because of the division of magnitude to infinity, in whatever way it is divided. That is, the magnitude is either divided into two halves, as in the first argument, or it is divided according to the excess of the faster to the slower, as in the second argument. But the second argument adds that the fastest body cannot catch up with a slower body while it is pursuing it. This is said with a certain grandeur of words in order to cause wonderment, but does not contribute to the force of the argument.

             Hence it is clear that the same answer must be given to both the first and the second arguments. In the first argument it was falsely concluded that the mobile body would never arrive at a terminus of the magnitude because of the infinite division of the magnitude. Likewise, the conclusion of the second argument to the effect that a preceding slower body is not joined by a pursuing faster body is false. For this is nothing else than that the mobile body does not arrive at some terminus.

             It is true that, while the slower body precedes, the faster body is not joined with it. Nevertheless, at some time it will be joined with it, if it is granted that a mobile body can cross a finite magnitude in a finite time. For the faster pursuing body will cross the whole magnitude by which the slower fleeing body precedes it. And it will cross a greater magnitude in less time than the time in which the slower body is moved through some determinate quantity. Thus the faster body will not only catch up with the slower body, but will also pass it. Therefore these two arguments of Zeno are answered.

             866. He gives the third argument where he says, 'The third is . . .' (239 b 30). He says that Zeno's third argument; namely, that the flying arrow is at rest, was given above before he began to enumerate the arguments. As he said above, this seems to follow because Zeno supposed that time is composed of 'nows'. If this is not granted, one cannot argue to this conclusion.

             867. He gives the fourth argument where he says, 'The fourth argument . . .' (239 b 32).

             Concerning this he makes three points. First he gives the argument. Secondly, where he says, 'The fallacy of . . .' (240 a 1), he answers the argument. Thirdly, where he says, 'For instance . . .' (240 a 4), he clarifies the argument with an example.

             He says, therefore, first that Zeno's fourth argument deals with things which are moved on a race-course. Two equal magnitudes are moved through a part of the space of the race-course which is equal to each magnitude in quantity. The motions are contrary, that is, one of the equal magnitudes is moved through this space of the race-course in one direction, and the other in the other direction. Also one of these mobile magnitudes begins to be moved from the end of the race-course equal to it. The other begins to be moved from the middle of the race-course, or from the middle of the space in the given race-course. Further, each body is moved with equal velocity. Having constructed this situation, Zeno thought that half of the time would be equal to double the time. And since this is impossible, he wished to infer further from this that motion is impossible.

             868. Next where he says, 'The fallacy of . . .' (240 a 1), he answers the argument.

             He says that Zeno was deceived because he has on the one hand a mobile body being moved next to a moved magnitude, and on the other hand he has a body being moved next to a magnitude at rest which is equal to the moved magnitude. And since he supposed that the velocities of the mobile bodies are equal, he held that in an equal time there is motion of equal velocity for the equal magnitudes, of which one is in motion and the other is at rest. This is clearly false. For when a thing is moved next to a magnitude which is at rest, there is only one motion. But when a thing is moved next to a moved magnitude, there are two motions. And if these motions are in the same direction, the time should be added. But if these motions are in opposite directions the time should be decreased according to the quantity of the other motion. For if a magnitude is moved with equal or greater velocity in the same direction as an adjacent mobile body, the mobile body will never be able to cross it. But if the magnitude is moved with a smaller velocity, the mobile body will cross it at some time. However, the time would be shorter if the magnitude were at rest. However, the contrary is the case if the magnitude is moved in the opposite direction than the mobile body. For the faster the magnitude is moved, the shorter the time in which the mobile body crosses it. For the motions operate in such a way that they cross each other.

             869. Next when he says, 'For instance . . .' (240 a 4), he clarifies what he has said with an example. Let A refer to three magnitudes equal to each other. These magnitudes are at rest. This is to be understood as though A refers to a space of three cubits. And let B refer to three other magnitudes equal to each other, for example, a mobile body of three cubits. These magnitudes begin to be moved from the middle of the space. And also let C refer to three other magnitudes which are equal in number, magnitude, and velocity to B. These magnitudes begin to be moved from the end of the space, that is, from the last A.

             On these assumptions the first B by its motion will come together with the last A, and the first C by its motion will come together with the first A at the opposite extremity. And the first C will also be together with the last B, having crossed all the B's which are moved in the opposite direction from C. When this has been done, the first C has crossed all the A's, but B has crossed only half the A's. Hence, since B and C have equal velocity, and since an equal velocity crosses a smaller magnitude in a shorter time, it follows that the time in which B has arrived at the last A is half the time in which C has arrived at the first A at the opposite end. For both B and C cross each A in an equal time.

             Therefore, supposing that the time in which B arrives at the last A is half the time in which C arrives at the first A at the opposite end, we must also consider how Zeno wished to conclude that this half time is equal to this double time. Granting that the time of the motion of C is twice the time of the motion of B, then in the first half of the time B was at rest and C was in motion. Thus in that half of the time C arrived at the middle of the space where B is. And then B began to be moved in one direction and C in the other direction. Moreover when B arrives at the last A it must have crossed all of the C's. For the first B and the first C are simultaneously in the opposite extremes; that is, C is in the first A, and B is in the last A. And as Zeno said, C comes next to each B in the same time that it crosses each of the A's. This is true because both the B and the C cross one A in an equal time. And thus, if B and C cross the same space in an equal time, it seems that C crosses both a B and an A in an equal time. Therefore the time in which C crosses all the B's is equal to the time in which it crosses all the A's. But the time in which C crosses all the B's is equal to the time in which C or B crosses half of the A's, as was said. Moreover it was proven that the time in which B crosses half of the A's is half the time in which C crosses all of the A's. Therefore it follows that a half is equal to a double, which is impossible.

             This, therefore, is Zeno's argument. But it falls into the error mentioned above; namely, that C crosses B, which is moved in the opposite direction, and A, which is at rest, in the same time. This is false, as was said above.

             870. Next where he says, 'Nor in reference to . . .' (240 a 19), he gives the argument by which Zeno denied mutation between contradictories.

             The argument is as follows. Whatever is changed, while it is being changed, is in neither of the termini. For while it is in the terminus from which, it is not yet being changed. And while it is in the terminus to which, it already has been changed. Therefore, if a thing is being changed from one contradictory to another, for example, from nonwhite to white, it follows that while it is being changed, it is neither white nor non-white. This is impossible.

             Now although this impossibility follows for those who hold that the indivisible is moved, nevertheless for us who hold that whatever is moved is divisible no impossible results here. For if the moving body is not wholly in one of the extremes, it is not necessary that it cannot be called either white or non-white. For one part of it is white, and another part is not white. Moreover, a body is not called white because the whole is white, but because many of the main parts are white. These are the parts which are more properly and naturally white. For not being white or non-white is not the same as not being totally white or nonwhite.

             What has been said of white or non-white must also be understood of unqualified being or non-being and of all things which are opposed as contradictories, like hot and non-hot, and such things. For that which is being changed must always be in one of the contrary opposites, and it will be named by that in which it is principally present. But it does not follow that the whole is always in neither of the extremities, as Zeno thought.

             However it must be realized that this answer is sufficient to refute the argument which Zeno primarily intends here. But the truth in regard to this matter will be made clearer in Book VIII. For it is not true that in every case a part is altered or generated before another part. Rather sometimes the whole is changed all at once, as was said above. The answer given here does not apply to this case. Rather the answer which is given in Book VIII applies.

             871. Next where he says, 'Again, in the case . . .' (240 a 28), he answers the argument by which Zeno denied spherical motion.

             Zeno said that it is impossible for a thing to be moved in a circle or in a sphere or in any other way in which a body does not progress from the place in which it is, but rather is moved in the same place. He proved this with the following argument. Whatever is through some time in one and the same place in respect to its whole and its parts is at rest. But things which are given as being moved in a circle are through some time in the same place in respect to the whole and its parts. Therefore it follows that they are simultaneously in motion and at rest, which is impossible.

             The Philosopher objects to this argument in two ways.

             First, Zeno said that the parts of a moved sphere are in the same place through some time. Against this Aristotle says that the parts of a moved sphere are at no time in the same place. For Zeno was dealing with the place of the whole. And it is true that, while the sphere is being moved, no part of it is outside the place of the whole sphere. But Aristotle is speaking of the proper place of a part insofar as a part can have a place. It was said in Book IV that the parts of a continuum are in place potentially. Moreover, it is clear that in spherical motion a part changes its proper place, but not the place of the whole. For where one part was, another part follows.

             Secondly, he objects to the above argument insofar as Zeno said that the whole remains in the same place through some time. Against this Aristotle says that even the whole is always being changed to another place. This is clear as follows. In order for there to be two diverse places, it is not necessary that one of these places be totally outside of the other. Rather sometimes the second place is partially joined to the first place and partially divided from the first place. This can be seen in things which are moved in a straight line. Let a body of one cubit be moved from the place A B, which is one cubit, to the place B C, which is one cubit. While this body is being moved from the one place to the other, it must partly leave the one place and enter the other. Thus, if it leaves the place A B as far as A D, it will enter the place B C as far as B E. Hence it is clear that the place D E is different from the place A B. But it is not totally separated from A B, but only partially.

             If it be granted that the part of the mobile body which enters the second place is returned to the part of the place which the body left, then there would be two places which are separated from each other in no way. These places differ only according to reason [ratio] insofar as the beginning of the place is taken at diverse signs where the body is; that is, some sign is taken in the mobile body as a beginning. Thus there are two places according to reason [ratio], but one place in respect to the subject.

             Thus what he says here must be understood to mean that there are different circles here insofar as the circle begins at A, or B, or C, or at any other sign. On the other hand, it might be said that the circles are the same in subject, as a man and a musical man are the same in subject, for the one is accidental to the other. Hence it is clear that the body is always moved from one circular place to another, and it is not at rest, as Zeno tried to prove. And the same thing applies to the sphere and to all other things like wheels, pillars, and such things, which are moved within a proper place.