LECTURE 3 (232 a 19-233 a 16)

THE DIVISIBILITY OF TIME FOLLOWS FROM THE DIVISIBILITY OF MAGNITUDE, AND VICE VERSA

             766. After the Philosopher has shown that both magnitude and the motion which crosses it are, for the same reasons [ratio], not composed of indivisible parts, he shows here that the same is true of time and magnitude.

             This discussion is divided into two parts. First he shows that the division of time follows from the division of magnitude, and vice versa. Secondly, where he says, 'And if either is infinite . . .' (233 a 17), he shows that the infinity of one of these follows from the infinity of the other.

             Concerning the first part he makes two points. First he states his position, and secondly he demonstrates it, where he says, '. . . for if the whole . . .' (232 a 20).

             He says, therefore, first that time also must be divisible or indivisible and composed of indivisible parts, as was the case with magnitude and motion.

             767. Next where he says, '. . . for if the whole . . .' (232 a 20), he proves his position with three arguments. The first is taken from equal velocities. The second is taken from faster and slower velocities, where he says, 'And since every magnitude . . .' (232 a 23). The third is taken from the mobile object itself, where he says, 'Moreover, the current popular . . .' (233 a 13).

             He says, therefore, first that the nature [ratio] of an equal velocity is such that an object will cross a smaller magnitude in less time. Hence, let there be given a divisible magnitude which a mobile object crosses in some given time. It follows that the mobile object, having equal velocity, crosses part of the magnitude in less time. And thus it is necessary for the given time to be divisible. Conversely, let there be given a divisible time in which a given mobile object is moved through some given magnitude. It follows that the mobile object, having equal velocity, is moved through less of the magnitude in a smaller part of the total time. And thus it follows that the magnitude A is divisible.

             768. Next where he says, 'And since every magnitude . . .' (232 a 23), he shows the same thing by means of two mobile objects, one of which is faster than the other.

             First he sets forth certain things which are necessary to prove his point. Secondly, where he says, 'And since every motion . . .' (232 b 21),

he proves his point.

             Concerning the first part he makes two points. First he shows how the faster is related to the slower in regard to being moved through a greater magnitude. Secondly, where he says, 'And from this . . .' (232 b 5), he shows how the faster is related to the slower in regard to being moved through an equal magnitude.

             Concerning the first part he makes two points. First he states his position by repeating a certain thing said above which is necessary for the proofs which follow. Secondly, where he says, 'Suppose that A . . .' (232 a 28), he proves his position.

             769. Hence he states again that every magnitude is divisible into magnitudes. This is clear from what was shown above; namely, it is impossible for a continuum to be composed of indivisible parts. It is clear that every magnitude belongs to the genus of continuous things. From this it follows that in an equal time a faster body must be moved through a greater magnitude. It even happens that in a smaller time a faster body is moved through a greater magnitude. And this is how some people define 'faster'; namely, that which is moved farther in an equal time and even in a smaller time.

             770. Next where he says, 'Suppose that A . . .' (232 a 28), he proves the two points made above.

             First he proves that in an equal time a faster body is moved through a greater space. Secondly, where he says, 'More than this . . .' (232 a 32), he proves that even in a smaller time a faster body is moved through a greater space.

             Let there be two mobile objects, A and B, and let A be faster than B. And let there be a magnitude C D which A crosses in the time F G. Moreover both A, which is faster, and B, which is slower, are moved through the same magnitude, and begin to be moved simultaneously.

             Granting these suppositions, he argues as follows. In an equal time a faster body is moved farther. But A is faster than B. Hence when A arrives at D, B has not yet arrived at D, which is the terminus of the magnitude. Rather B will be at some distance from D. Nevertheless in this time B will have been moved through some part of the magnitude. Hence, since any part is less than the whole, it follows that, in the time F G, A is moved through a greater magnitude than B, which in the same time is moved through part of the magnitude. Thus it follows that in an equal time the faster body crosses through more space.

             771. Next where he says, 'More than this . . .' (232 a 32), he shows that in a smaller time the faster body crosses through more space.

             It was said that at the time when A has already arrived at D, B which is slower is still at some distance from D. Let it be granted then that at the same time B has arrived at E. Now since every magnitude is divisible, as was said above, the remainder of the magnitude--E D--which is the distance that the faster body exceeded the slower one, is divided into two parts at the point H. It is clear that the magnitude C H is less than the magnitude C D. But the same mobile body is moved through a smaller magnitude in less time. Hence, since A arrived at D in the total time F G, it arrives at the point H in less time. Let this time be F J.

             The argument continues as follows. The magnitude C H, which A crosses, is greater than the magnitude C E, which B crosses. But the time F J in which A crosses the magnitude C H is less than the total time F G, in which the slower B crosses the magnitude C E. Hence it follows that the faster body crosses through a greater space in less time.

             772. Next where he says, 'And from this . . .' (232 b 5), he shows how the faster is related to the slower in regard to being moved through an equal magnitude.

             First he states his position, and secondly he proves it, where he says, 'For since it passes . . .' (232 b 7).

             He says, therefore, first that from the foregoing it can be made clear that a faster body crosses an equal space in less time.

             Secondly, where he says, 'For since it passes . . .' (232 b 7), he proves this with two arguments.

             The first argument has two suppositions. The first of these has already been proven; namely, a faster body crosses a greater magnitude in less time than a slower body. The second supposition is per se evident; namely, a mobile body, considered in itself, in a greater time crosses a greater magnitude than in a smaller time.

             For the mobile body A, which is faster, crosses the magnitude K L in the time P Q. And it will cross part of the magnitude, K N, in a smaller time, P R. This latter time is less than the time P Q in which it crosses K L, just as K N is also smaller than K L.

             From the first supposition it follows that the total time P Q, in which A crosses the whole magnitude K L, is less than the time V in which B, which is slower, crosses the smaller magnitude K N. For it was said that a faster body crosses a greater magnitude in less time.

             From these suppositions he proceeds as follows. The time P Q is less than the time V in which B, which is slower, crosses the magnitude K N. And the time P R is less than the time P Q. Therefore it follows that the time P R is less than the time V. For if something is less than that which is smaller, it will also be less than that which is greater. Hence, since it was granted that in the time P R the faster body was moved through the magnitude K N, and since the slower body was moved through the same magnitude in the time V, it follows that the faster body was moved through an equal space in less time.

             773. He gives the second argument where he says, 'Again, since the motion . . .' (232 b 15). The argument is as follows.

             Anything which is moved through an equal magnitude with some other mobile body is moved through that magnitude in either an equal time or less time or more time. Moreover, that which is moved through an equal magnitude in the greater time is the slower body, as was proven above. And that which is moved through an equal magnitude in an equal time is equal in velocity, as is per se evident. Now since that which is faster has neither an equal nor a slower velocity, it follows that it is not moved through an equal magnitude in either a greater or an equal time. Hence it follows that it is moved in less time.

             Thus it has been proven that a faster body must cross an equal magnitude in less time.

             774. Next where he says, 'And since every motion . . .' (232 b 21), he proves his position; namely, for the same reasons [ratio] both time and magnitude are either divided into infinitely divisible parts or else they are composed of indivisible parts.

             Concerning this he makes three points. First he sets forth certain things which are necessary for the following proof. Secondly, where he says, '. . . and this being so . . .' (232 b 23), he states his position. Thirdly, where he says, 'For since it has been . . .' (232 b 27), he proves his position.

             Therefore, he states first that every motion is in time. This was proven in Book IV. Further he states that in any time there can be motion. This is clear from the definition of time given in Book IV.

             A second point is that whatever is moved can happen to move faster or slower. This means that any mobile object can be found to be moved sometimes faster and sometimes slower.

             But this statement seems to be false because the velocities of motions in nature are determined. For there is a motion which is so fast that nothing can be faster; namely, the motion of the first mobile object.

             To this it must be answered that we can speak of the nature [natura] of things in two ways, either according to their common nature [ratio] or according to their connection with proper matter. Nothing prevents that which is not impeded by its common nature [ratio] from being impeded in regard to its connection with some determinate matter. For example, the existence of many suns is not impeded by the nature [ratio] of the form of the sun. Rather this occurs because the whole matter of the species is contained under one sun. And likewise the common nature [natura] of motion does not prevent the possibility of finding a velocity greater than any given velocity. Rather this is prevented by the determinate powers of movers and mobile bodies. Aristotle is here treating motion according to its common nature [ratio], and is not yet applying motion to determinate movers and mobile bodies. Hence here in Book VI he frequently uses propositions which are true in respect to the common consideration of motion, but are not true in respect to the application of motion to determinate mobile bodies.

             And likewise it is not contrary to the nature [ratio] of magnitude that every magnitude is divided into smaller ones. Therefore in this book he uses the proposition that for any given magnitude there is another smaller magnitude. But in applying magnitude to a determinate nature [natura], there is some minimum magnitude. For every nature [natura] requires a determinate magnitude and smallness, as was said in Book I.

             From the two points above he concludes to a third. In any given time there is a faster and slower motion than the motion given in that time.

             775. Next where he says, '. . . and this being so . . .' (232 b 23), he draws his conclusion from the foregoing.

             He says that, since the foregoing is true, time must be a continuum; that is, it must be divisible into infinitely divisible parts.

             Granting that this is the definition of a continuum, then time must be continuous if magnitude is continuous. For the division of time follows from the division of magnitude, and vice versa.

             Next where he says, 'For since it has been . . .' (232 b 27), he proves this point; namely, that time and magnitude are divided in the same way.

             It was shown that a faster body crosses an equal space in less time. Let A be the faster body and B the slower body. The slower body B is moved through the magnitude C D in the time F G.

             It is clear that A, which is faster, is moved through the same magnitude in less time. Let this time be F H.

             Further, since A, which is faster, crosses the whole magnitude C D in the time F H, then B, which is slower, in the same time crosses a smaller magnitude C J. And since B, which is slower, crosses the magnitude C J in the time F H, then A, which is faster, will cross this same magnitude in a still shorter time. And thus the time F H will be divided again. And when the time is divided, the magnitude C J will be divided for the same reason [ratio]. For in part of that time the slower body is moved through a smaller magnitude. And if the magnitude is divided, the time will also be divided again. For the faster body will cross that part of the magnitude in a shorter time. This goes on to infinity by taking a slower mobile body after the motion of the faster body, and then a faster after the slower, etc., and by using what has been demonstrated; namely, a faster body crosses an equal magnitude in less time, and a slower body crosses a smaller magnitude in an equal time.

             Thus we divide the time in terms of the faster body and the magnitude in terms of the slower body.

             Hence, if it is true that this alternation from faster to slower and from slower to faster can occur to infinity, and if a division of time and of magnitude always results from such an alternation, then it is clear that every time, and likewise every magnitude, is a continuum; that is, is divisible into infinitely divisible parts. For time and magnitude are divided by the same equal divisions, as was shown.

             776. Next where he says, 'Moreover, the current . . .' (233 a 13), he gives the third argument to show that time and magnitude are divided in the same way. This argument considers only one mobile body.

             It is also clear from arguments that are customarily used that if time is a continuum, that is, is divisible into infinitely divisible parts, then magnitude is also a continuum in the same way. For when one and the same mobile body is moved regularly, then just as it crosses a whole magnitude in a whole time, it crosses half the magnitude in half the time. And universally it crosses a smaller magnitude in a smaller time. This occurs because both time and magnitude are divided in the same way.