LECTURE 16 (261 b 27-262 b 9)

NO LOCAL MOTION OTHER THAN CIRCULAR MOTION CAN BE CONTINUOUS AND ETERNAL

1104. After the Philosopher has shown that no mutation can be continuous and eternal except local motion, he shows here that no local motion can be continuous and eternal except circular motion.

             Concerning this he makes two points. First he proves his position demonstratively, and secondly, he proves it logically, where he says, 'If we look . . .' (264 a 8).

             Concerning the first part he makes two points. First he proves his position. Secondly, where he says, 'So this is how . . .' (262 b 10), he answers certain objections by means of the truth he has demonstrated.

             Concerning the first part he makes three points. First he states his principal intention. He intends to show that it is possible for a certain motion, existing as one, to be infinitely continuous, and this motion is circular motion. He proves the latter point first.

             1105. Second, where he says, 'The motion of everything . . .' (261 b 27), he explains what procedure must be used.

             Everything which is moved locally is moved either by a circular motion, or by a straight motion, or by a motion composed of both, as when something is moved through a chord and an arc. It is clear that if one of the two simple motions, namely, either circular or straight motion, cannot be infinitely continuous, then much less will that which is composed of both be infinitely continuous. Hence it is permissible to omit composite motions and discuss simple ones.

             1106. Thirdly, where he says, 'Now it is plain . . .' (261 b 31), he shows that straight motion through a straight and finite magnitude cannot be infinitely continuous. And thus no straight, continuous motion can be infinite unless an actually infinite magnitude is granted. But this was rejected above in Book III.

             He proves this with two arguments, the first of which is as follows.

             If there is an infinite motion over a straight and finite magnitude, this must be a reflex motion. For it was shown in Book VI that a thing passes over a finite magnitude in a finite time. Therefore, when it arrives at the end of the finite magnitude, the motion will cease unless the mobile object is returned by a reflex motion to the beginning of the magnitude from which its motion began. But that which is reflected in a straight motion is moved by contrary motions. He proves this as follows.

             Contrary motions are those which have contrary termini, as was proven in Book V. But the contrarieties of place are up and down, before and behind, right and left. Everything which is reflected must be reflected in respect to one of these contrarieties. Hence whatever is reflected is moved by contrary motions.

             Moreover, it was shown above in Book V which motion is one and continuous, namely, that which is of one subject, and in one time, and in the same thing which does not differ in species. For three things are involved in every motion: the first is the time; the second is the subject which is moved, for example, a man or a god, according to those who call celestial bodies gods; and the third is that in which the motion occurs, which, of course, in local motion is place; in alteration, passion, that is, passive quality; in generation and corruption, species; in increase and decrease, magnitude.

             It is also clear that contraries differ according to species. Hence, contrary motions cannot be one and continuous. Moreover, as was said, there are six differentiae of place. Hence, they must be contraries, because the differentiae of any genus are contraries. Therefore it follows that it is impossible for that which is reflected to be moved by one, continuous motion.

             1107. But someone might question whether that which is reflected is moved by contrary motions. For contrariety in place does not appear to be as clear and determined as in the other genera in which there is motion, as was said above in Book V. Hence, over and above the argument just given which deals with the contrariety of the termini, he adds an example to prove the same thing.

             He says that an example of this is that motion from A to B is contrary to motion from B to A, as occurs in reflex motion. For if such motions occur at the same time, they arrest and stop each other, that is, one impedes the other and causes it to stop.

             This occurs in both reflex straight motion and reflex circular motion. Let A B C be three signs on a circle. It is clear that if a motion starting from A toward B later becomes a motion from A to C in the other direction, this will be a reflex motion. And these two motions impede each other, and the one arrests the other, that is, it causes it to stop. But there will not be reflex motion if a thing is moved continuously from A to B and through B to C.

             Therefore, both circular and straight reflex motions impede each other. For contraries naturally impede and destroy each other.

             But motions which are diverse and are not contraries do not impede each other. For example, upward motion and lateral motion, either to the right or to the left, do not impede each other. Rather a thing can be moved upward and to the right at the same time.

             1108. Next where he says, 'But what shows . . .' (262 a 12), he gives the second argument to show that reflex motion cannot be infinitely continuous. This argument deals with the state of rest which must intervene.

             He says that it is particularly clear that straight motion cannot be infinitely continuous because of the fact that whatever is reflected must be at rest between the two motions. And this is true not only if it is moved through a straight line, but also if it is moved through a circle.

             But someone might think that 'being moved through a circle' means the same thing as 'being moved circularly'. To reject this he adds that 'being moved circularly', that is, in respect to the property of a circle, is not the same as 'being moved through a circle', that is, going through a circle by one's own motion.

             For it sometimes happens that the motion of that which is moved is continuous while it goes through part after part according to the order of the parts of a circle. This is 'being moved circularly'.

             But sometimes, when that which goes through a circle comes to the starting-point of its motion, it does not continue on according to the order of the parts of a circle, but goes back in the other direction. This is reflex motion.

             And so, whether the reflex motion is in a straight line or in a circular line, an intermediate state of rest must intervene.

             1109. The truth of this point may be established not only from sensation, since it is apparent to the senses, but also from reason.

             A principle that must be granted for this argument is that there are three things in the magnitude which is crossed; namely, the beginning, the middle, and the end. Now the middle is both a beginning and an end. For with respect to the end, it is a beginning, and with respect to the beginning, it is an end. Thus, although it is one in subject, it is two in reason [ratio]. Another principle that must be granted is the distinction between that which is in potency and that which is in act.

             Granting these points, it should be realized from what has been said that any sign, that is, any designated point between the termini of a line over which something is moved, is potentially a middle. It is not actually a middle unless the motion divides the line such that that which is moved stops at that point and then begins to be moved from that point. Then this middle becomes actually a beginning and an end. It is the beginning of the subsequent motion insofar as the motion begins again; and it is the end of the first motion insofar as the first motion was terminated by rest. Take a line the beginning of which is A, the midpoint B, and the end C. Let a motion from A to B stop at B. And then let the motion begin again from B up to C. It is obvious that B is actually the end of the prior motion and the beginning of the subsequent motion.

             But if a thing is moved continuously from A to C without any intervening rest, it is not possible to say that this mobile object has either arrived at or left from the point which is A or the point which is B. It can only be said that it is at A or at B in some 'now'. (It cannot be said to be there in some time, unless perhaps a thing is said to be somewhere in time because it is there in a 'now' of time. Hence, that which is moved continuously in some time from A to C will be at B in a 'now', which is a division of that time. And thus it is said to be at B in that whole time according to that manner of speaking in which a thing is said to be moved in a day because it is moved in a part of that day.)

             But there seems to be difficulty in saying that that which is moved is not present in or absent from any point in a designated magnitude which is crossed by a continuous motion. Hence to explain this, he says that if one were to concede that a mobile object is present in and absent from some point in a designated magnitude, it follows that it is at rest there. For it is impossible for the mobile object to be present in and absent from B at the same instant. For to be present and to be absent somewhere are contraries, and these cannot exist at the same instant.

             Therefore, the mobile object must be present in and absent from a point in the magnitude in different 'nows' of time. But between any two 'nows' there is an intermediary time. It follows, therefore, that the mobile object A is at rest at B. For everything which is somewhere during some time is in the same place before and after. And the same thing may be said of all other signs or points, for the same argument applies to all.

             Hence it is clear that that which is moved continuously through a magnitude is not present and absent, that is, it does not arrive and leave, at any intermediate point of the magnitude. For when it is said that a mobile object is present at some point, or comes to be there, or arrives there, all such expressions signify that that point is the terminus of the motion.

             And to say that it is absent from or leaving is to signify the beginning of a motion. The middle point of a magnitude is not actually either the beginning or the end of a motion, since the motion does not begin or end there except potentially (for the motion can begin or end there). Hence, the mobile object is neither present nor absent at a middle point, but is simply said to be there at a 'now'. For the existence of a mobile object at some point of a magnitude is related to the total motion as the 'now' is related to time.

             1110. But when the mobile object A uses B as a mid-point and as actually a beginning and an end, it must stop there. For by moving and stopping, it makes the one point two, namely, beginning and end, as occurs also in understanding. For at the same time we can understand one point as one in a subject. But if we understand it separately as a beginning and separately as an end, this does not occur at the same time. In the same way when that which is moved uses a sign as one, it will not be there except in one 'now'. But if it uses it as two, that is, as an actual beginning and end, it will necessarily be there in two 'nows', and consequently in an intermediate time. Therefore, it is at rest.

             It is clear, then, that what is moved continuously from A to C was never present or absent, that is, it never arrived at or left, at the midpoint B. Rather it left or departed from the first point A, as if from an actual beginning. And at the last point C it arrived or is present, for the motion is completed there and the mobile object is at rest.

             It should be noted that in the foregoing 'A' is sometimes assumed to be the mobile object and sometimes the beginning of the magnitude.

             1111. From this it is clear that reflex motion, either in a circular or in a straight magnitude, cannot be continuous. Rather a state of rest intervenes. For the same point is actually the end of the first motion and the beginning of the reflex motion. But in circular motion the mobile object does not use any point as an actual beginning or end, but uses every point of the magnitude as a mid-point. Therefore, circular motion can be continuous, but reflex motion cannot.