BOOK VI

Lecture 1

No continuum is composed of indivisibles

750. After the Philosopher has finished dividing niotion into its species and discussing the unity and contrariety of motions and of states of rest, he proposes in this Sixth Book to discuss the things that pertain to the division of motion precisely as it is divisible into quantitative parts.

The whole book is divided into two parts.

In the first he shows that motion, as every continuum, is divisible;

In the second he shows how motion is divided, at L. 5.

The first part is subdivided into two sections:

In the first he shows that no continuum is composed solely of indivisibles; at L. 4.

In the second that no continuum is indivisible, near the end

The first is further subdivided into two parts:

In the first he shows that no continuum is composed of indivisibles only;

In the second (because the proofs for the first seem to be applicable mainly to magnitudes) he shows that the same proofs apply to magnitudes, to motion and to time, at L, 2.

In regard to the first part he does two things:

First he recalls some definitions previously given, with a view to using them in demonstrating his proposition;

Secondly, he proves the proposition, at 752.

751. He says therefore first (562) that if the previously given definitions of continuum, of that which is touched, of that which is consecutive to are correct (namely, that continua are things whose extremities are one; contigua are things whose extremities are together; consecutive things are those between which nothing of the same type intervenes), then it would follow that it is impossible for any continuum to be composed solely of indivisibles; i.e., it is impossible, for example, for a line to be composed of points only, provided, of course, that a line is conceded to be a continuum and that a point is an indivisible. This proviso is added to prevent other meanings being attached to point and line.

752. Then at (563) he proves the proposition:

First he gives two proofs of the proposition;

Secondly, he explains things that might be misunderstood in his proofs, at 756.

In regard to the first proof he does two things:

First he shows that no continuum is composed solely of indivisibles, either after the manner of continuity or of contact;

Secondly, or after the manner of things that are consecutive, at 754.

In regard to the first he gives two reasons, of which the first is: Whatever things a unit is composed of, either after the manner of continuity or of contact, the extremities must either be one or they must be together. But the extremities of points cannot be one, because an extremity is spoken of in relation to a part, whereas in an indivisible it is impossible to distinguish that which is an extremity and something else that is a part. Similarly, it cannot be said that the extremities are together, because nothing can be the extremity of a thing that cannot be divided into parts, whereas an extremity must always be distinct from that of which it is the extremity. But in a thing that cannot be divided into parts, there is no way of distinguishing one thing and another. It follows therefore that a line cannot be composed of points either after the manner of continuity or after the manner of contact.

753. The second reason is given at (564). If a continuum is composed solely of points, they must be either continuous with one another or touch (and the same is true of all other indivisibles, i.e., that no continuum is composed solely of them).

To prove that they are not continuous with one another, the first argument suffices.

But to prove that they cannot touch one another, another argument is adduced, which is the following: Everything that touches something else does so either by the whole touching the other wholly, or by a part of one touching a part of the other or the whole of the other. But since an indivisible does not have parts, it cannot be said that part of one touches either a part or the whole of the other. Hence if two points touch, the whole point touches another whole point. But when a whole touches a whole, no continuum can be formed, because every continuum has distinct parts so that one part is here and another there, and is divisible into parts that are different and distinct in regard to place, i.e., position (in things that have-position)—whereas things that touch one another totally are not distinguished as to place or position, It therefore follows that a line cannot be composed of points that are in contact.

754. Then at (565) he shows that no continuum is composed of indivisibles after the manner of things that are consecutive. For no point will be consecutive to another so as to form a line; and no “now” is consecutive to another “now” so as to form a period of time, because consecutive things are by definition such that nothing of the same kind intervenes between any two. But between any two points there is always a line, and so, if a line is composed of points only, it would follow that between any two points there is always another, mediate, point. The same is true for the “now’s’”. if a period of time is nothing but a series of “now’s”, then between any two “now’s” there would be another “now”. Therefore, no line is composed solely of points, and no time is composed solely of “now’s”, after the manner of things that are consecutive.

755. The second reason is given at (566) and is based on a different definition of continuum—the one given at the beginning of Book III—that a continuum is “that which is divisible ad infinitum”. Here is the proof: A line or time can be divided into whatsoever they are composed of. If, therefore, each of them is composed of indivisibles, it follows that each is divided into indivisibles. But this is false, since neither of them is divisible into indivisibles, for that would mean they would not be divisible ad infinitum. No continuum, therefore, is composed of indivisibles.

756. Then at (567) he explains two statements he made in the course of his proofs. The first of these was that between two points there is always a line and that between two “now’s” there is always time. He explains it thus:

If two points exist, they must differ in position; otherwise, they would not be two, but one. But they cannot touch one another, as was shown above; hence they are distant, and something is between them. But no other intermediate is possible, except a line between two points, and time between “now’s”: for if the intermediate between two points were other than a line, that intermediate must be either divisible or indivisible. If indivisible, it must be distinct from the two points—at least in position—and, since it touches neither, there must be another intermediate between that indivisible and the original extremities and so on ad infinitum, until a divisible intermediate is found. However, if the intermediate is divisible, it will be divisible into indivisibles or into what are further divisible. But it cannot be divided into indivisibles only, because then the same difficulty returns—how a divisible can be composed solely of indivisibles. It must be granted, then, that the intermediate is divisible into what are further divisible. But that is what a continuum is. Therefore, that intermediate will be a continuum. But the only continuous intermediate between two points is a line. Therefore, between any two points there is an intermediate line. Likewise, between two “now’s” there is time; and the same for other types of continua.

757. Then at (568) he explains the second statement referred to at the beginning of 756, that every continuum is divisible into divisibles. For on the supposition that a continuum is divisible solely into indivisibles, it would follow that two indivisibles would have to be in contact in order to form the continuum. For continua have an extremity that is one, as appears from the definition thereof; moreover, the parts of a continuum must touch, because if the extremities are one, they are together, as was stated in Book V. Therefore, since it is impossible for two indivisibles to touch, it is impossible for a continuum to be divided into indivisibles.

Lecture 2

Motion composed of indivisibles follows a continuum composed of indivisibles—impossibility of the former

758. Because the arguments presented in the previous lecture clearly apply to lines and other continua having position, in which continua contact is properly found, the Philosopher now wishes to show that the same reasoning applies to magnitudes and time and motion. And it is divided into two parts:

First he proposes his intention;

Secondly, he proves his proposition at 759.

He says therefore first (569) that any argument which shows that a magnitude is composed or not composed of indivisibles, and divided or not divided into indivisibles, applies also to time and motion; for whatever is granted in regard to any of them would necessarily be true of the others.

759. Then at (570) he proves this proposition:

First in regard to magnitude and motion;

Secondly in regard to time and magnitude, in L. 3.

About the first he does three things:

First he presents his proposition;

Secondly, he gives an example, at 760;

Thirdly, he proves his proposition, at 761.

The proposition is this: If a magnitude is composed of indivisibles, likewise the motion that traverses it will be composed of indivisible motions, equal in number to the indivisibles of which the magnitude is composed.

760. Of this he gives the following example at (571): Let the line ABC be composed of the 3 indivisibles A, B and C, and let 0 be an object in motion over the distance of the line ABC, so that DEZ is its motion. Now if the parts of the distance or of the line are indivisibles, then the parts of the motion are indivisibles.

Then at (572) he proves his proposition. About which he does three things:

First he lays down some premisses necessary for his proof;

Secondly, he proves that if a magnitude is composed of points, then the motion is composed not of motions but of moments, at 762;

Thirdly, he shows that it is impossible for motion to be composed of moments, at 763.

761. Therefore first he lays down two presuppositions. The first at (572) is that according to each part of the motion under consideration something must be in motion, and, conversely, if something is in motion, a motion must be in it. Now if this is true, then the mobile 0 is being moved through A which is part of the entire magnitude by means of that part of the motion that is D, and through B, another part of the magnitude) by that part of the motion that is E, and through C (the third part of the magnitude) by that part of the motion that is Z. In other words, single parts of motion correspond to single parts of the magnitude.

The second presupposition at (573) is that what is being moved from one terminus to another is not at the same time being moved and finished moving, any more than a man going to Thebes is, at the time while he is going, already there.

He presupposes these two statements as per se evident. For, as to the statement that when motion is present, something must be in the state of being moved, a like situation is apparent in all accidents and forms; for in order that something be white it must have whiteness, and, conversely, if whiteness exists, something is white. As to the statement that “being moved” and “having been moved” are not simultaneous, we appeal to the very successive nature of motion; for it is impossible that any two elements of time co-exist, as we explained in Book IV. Hence it is impossible that “having been moved”, which is the terminus of motion, be simultaneous with “being in motion”.

762. Then at (574) he uses these presuppositions to prove his proposition: For if it is true that whenever a part of motion is present, something has to be in motion, and if it is in motion, there must be motion present, then if the mobile 0 is in motion with respect to an indivisible part of the magnitude, namely, A, there is in 0 that part of the motion we called D. Accordingly, 0 is being moved through A and has completed its motion, either at the same time or not at the same time. If not at the same time but later, it follows that A is divisible; because while 0 was in motion, it was neither resting at A (with the rest preceding motion) nor had passed through the entire distance A—for then it would not still be in motion through A, since nothing is in motion through a distance it has already traversed. Consequently, it must be midway. Therefore, when it is in motion through A, it has already passed through part of A and is now in another part of A. Consequently, A is divisible —contrary to our supposition.

But if it is in motion through A and in the state of completed motion at the same time, it follows that it arrived while it was coming, and it will have completed its motion while it was being moved, which is against the second presupposition.

From this it is clear that no motion is possible when the magnitude is indivisible; for there are only two choices: either things can be in motion at the same time that their motion is over, or, the magnitude must be divisible.

Therefore, assuming that nothing can be in motion through the indivisible A, if someone should say that a mobile is in motion through the entire magnitude ABC and that the whole motion by which it is in motion is DEZ, and moreover, that nothing can be in motion but only in the state of completed motion through the indivisible A, it follows that the motion consists not of motions but of moments. Now the reason why we say that “it follows that the motion is not composed of motions” is that, since the part of the motion that is D corresponds to the part of the magnitude that is A, then if D were a motion, the mobile should be in motion through A, because when motion is present, the mobile is being moved. But it was proved that the mobile is not in motion through A as indivisible, but in the state of having completed its motion when it had traversed this indivisible. Consequently, what remains is that D is not a motion but a moment. (The state of completed motion is called “moment”, just as being moved is called “motion”; moreover, moment is related to motion as point is related to line). And the same holds for the other parts of the motion and of the magnitude. Consequently, it follows necessarily that if a magnitude is composed of indivisibles, then a motion is composed of indivisibles, i.e., of moments; and this is what he intended to show.

763. But since it is not possible for a motion to be composed of moments any more than a line be composed of points, then at (575) he exposes this impossibility by concluding to three impossibilities. The first of these is that if motion is composed of moments, and a magnitude of indivisibles, in such a way that through an indivisible part of a magnitude things are not in motion but in the state of completed motion, it will follow that something has completed a motion without having been in motion. For it was assumed that in regard to the indivisible, something arrived without going, because it was not able to be in motion at that indivisible. Hence it follows that something has finished a motion without previously being in motion. But this is no more possible than for an event to be past without having been present.

764. But because a person who claimed that motion is composed of moments might grant this strange state of affairs, Aristotle concludes to another impossibility, in the following argument: Anything capable of being in motion and at rest must be either in motion or at rest. But in our original example, while the mobile is in A, it is not being moved; likewise, when it is at B, and when it is at C; therefore, it must be at rest while at A and while at B and while at C. Therefore, it follows that a thing is at the same time continually at rest and continually in motion.

That this follows, he now proves; We have agreed that it is in motion throughout the entire length ABC and again that it is at rest in relation to each part. But what is at rest in relation to each and every part is at rest throughout the whole. Consequently, it is at rest throughout the entire length. Thus, it follows, that throughout the entire length it is continually in motion and continually at rest—which is wholly impossible.

765. He gives the third impossibility at (577): It has been shown that if a magnitude is composed of indivisibles, so also the motion. Now those indivisibles of motion, namely, D and E and Z, are such that each of them is either a motion or not. If each is a motion, then, since each of them corresponds to an indivisible part of the magnitude (in which something is not in motion but in the state of completed motion), it will follow that a mobile is not in motion but at rest, even though a motion exists—which is against the first presupposition. If each is not a motion, it follows that motion is composed of non-motions, which is no more possible than that a line be composed of non-lines.

Lecture 3

Time follows magnitude in divisibility and conversely

766. After showing that it is for a same reason that a magnitude and a motion traversing it would be composed of indivisibles, the Philosopher shows the same for time and magnitude. And the treatment falls into two parts:

In the first he shows that division of time follows upon division of magnitude, and vice versa;

In the second that the infinity of one follows upon the infinity of the other, in L. 4.

About the first he does two things:

First he states his proposition;

Secondly, he demonstrates it, at 767.

He says therefore first (578) that time, too, is divisible and indivisible, and composed of indivisibles, just as length and motion are.

767. Then he proves his proposition, giving three reasons:

The first of which is based on things equally fast;

The second is based on the faster and the slower, at 769;

The third uses one and the same mobile, at 776.

He says therefore first (579) that a mobile which is as fast as another traverses a smaller magnitude in less time. Therefore, let us take a divisible magnitude which a mobile traverses in a given time. It follows that an equally fast mobile traverses part of that magnitude in less time. Consequently, the given time must be divisible. Conversely, if the time is given as divisible and a given mobile is in motion over a given magnitude, it follows that a mobile equally fast traverses a smaller magnitude in less time, urhich is part of the whole time. Consequently, the magnitude A is divisible.

768. Then at (580) he proves the same thing with two mobiles, one of which is faster and the other slower.

But first he lays down some presuppositions to be used in proving his proposition.

Secondly, he proves the proposition at 774.

About the first he does two things:

First he explains how the faster and the slower compare with regard to being moved over a larger magnitude;

Secondly, how they compare with regard to being moved over an equal magnitude, at 772.

About the first he does two things:

First he states his proposition, repeating something mentioned previously but needed for the demonstrations that follow;

Secondly, he demonstrates his proposition, at 770.

769. He repeats therefore (580) that every magnitude is divisible into magnitudes. And this is evident from a previous conclusion that it is impossible for a continuum to be composed of atoms, i.e., indivisibles; and every magnitude is a kind of continuum. From these it follows that a faster body is moved through a greater magnitude in equal time and even in less time. Indeed, that is the way in which some have defined the faster, that it is moved more in. equal and even in less time.

770. Then at (581) he proves his two presuppositions:

First, that a faster thing is moved a greater distance in equal time;

Secondly, that it is moved a greater distance in less time, at 771.

He says therefore first (581): Let A and B be two mobiles, of which A is faster than B, and let CD be the magnitude traversed by A in time ZI. Now let B, which is slower, and A, which is faster, pass over the same magnitude, and let them start together.

Therefore, under these conditions, the following argument is given: The faster is the one moved more in equal time; but A is faster than B. Therefore, when A shall have arrived at D, B will not have arrived at D (which is the terminus of the magnitude) but will be some distance from it; yet it will have covered part of tho magnitude. Now, since every part is less than the whole, what remains is that A is moved through a greater distance in time ZI than B, which in the same time has traversed part of the magnitude. Consequently, the faster traverses more distance in equal time.

77l. Then at (582) he shows that the faster traverses more space in less time. For it was said that at the time when A arrived at D, B, which is slower, was still distant from D. Let us grant, therefore, that B arrived at E when A arrived at D. Now, since every magnitude is divisible, let us divide the remaining magnitude ED (which is how much the faster exceeds the slower) at T. It is eviJent that the magnitude CT is less than CD. But one and the same mobile traverses a smaller magnitude in less time. Therefore, since A arrived at D in the total time ZI, it arrived at T in less time.

Let that less time be ZK. Then the argument continues: the magnitude CT which A traversed is greater than the magnitude CE which B traversed. But the time ZK in which A traversed CT is less than the whole time ZI, in which the slower B traversed CE. Therefore, it follows that the faster traverses a larger space in less time.

772. Then at (593) he shows how the faster compares with the slower in regard to being moved through an equal magnitude.

First he states his intention;

Secondly, he proves his proposition here at 772.

He says therefore first (583) that from the foregoing it could be clear that a faster thing traverses an equal space in less time. Then he proves this with two arguments, to the first of which he prefaces two facts: one of which has already been proved, namely, that a faster thing traverses a greater magnitude in less time than a slower. The second is per se evident, namely, that one and the same mobile traverses a greater magnitude in a given time than in a shorter time. For let the mobile A, which is faster, traverse the magnitude LM in time PR and the part LX of the magnitude in less time PS, which is less than PR in which it traverses LM just as LX is less than LM.

From the first supposition he takes it that the whole time PR in which A traverses the entire magnitude LM is less than time H in which B (which is slower) traverses the smaller magnitude LX. For it was said that a faster object traverses a greater magnitude in less time.

With this background he proceeds to his argument: The time PR is less than time H (in which B, which is slower, traverses magnitude LX); moreover, time PS is less than time PR. Therefore, it follows that time PS is less than time H, for what is less than the lesser is less than the greater. Therefore, since it was granted that in the time PS the faster traverses magnitude LX and the slower traverses the same LX in time H, it follows that the faster traverses an equal magnitude in less time.

773. Then, after these preliminaries, he gives his second argument, which is this: A thing that traverses an equal magnitude along with another mobile is moved through that magnitude either in equal time or less or more. If it is moved through that equal magnitude in greater time, it is slower, as was proved above; if it is moved in equal time through the equal magnitude, it is equally fast, as is per se evident. Therefore, since what is faster is neither equally fast nor slower, it follows that it is moved through an equal magnitude neither in more time nor in equal time. Therefore, in less time.

Thus, we have proved that necessarily the faster traverses an equal magnitude in less time.

774. Then at (586) he proves the proposition that one and the same reason proves that both time and magnitude are always divided into divisibles, or are composed of indivisibles. About this he does three things:

First he lays down premisses to be used in the proof;

Secondly, he states his proposition at 775;

Thirdly, he proves it at 775,

Therefore (586) he lays down the premisses that every motion exists in time—this was proved in Book IV—and that motion is possible in any time—this is evident from the definition of time given in Book IV. Secondly, that whatever is being moved can be moved faster and slower, i.e., among mobiles some are moved faster and some slower. But this statement seems false, because the speeds of motions are fixed in nature; for there is one motion so fast that none could be faster, namely, the motion of the first mobile.

In reply it must be said that we can speak of the nature of anything in two ways: either according to its general notion or insofar as it is applied to its proper matter. Now, there is nothing to forbid something which is possible in the light of a thing’s general definition to be prevented from happening when application is made to some definite matter; for example, it is not the general definition of the sun that precludes many suns, but the fact that the total matter of this nature is contained under one sun, Likewise, it is not the general nature of motion that prevents the existence of a speed greater than any given speed; rather it is the particular powers of the mobiles and movers.

Now, Aristotle is here discussing motion from the viewpoint of its general nature without application to particular movers and mobiles. Indeed, he frequently uses such propositions in this Sixth Book and they are true, if you limit yourself to a general consideration of motion, but not necessarily true, if you get down to particular mobiles.

Likewise, it is not against the nature of magnitude that. every magnitude be divisible into smaller ones. Therefore, in this Book he goes on the assumption that it is possible to take a magnitude smaller than any given magnitude, even though in every particular nature there is always a minimum magnitude, since each nature has limits of largeness and smallness, as was mentioned even in Book I.

From these two premisses he concludes to a third one, namely, that in any given time, faster and slower motions than a given motion are possible.

775. Then at (587) from the foregoing he concludes to his proposition. And he says that since the foregoing are true, time must be a continuum, i.e., divisible into parts that are further divisible. For if that is the definition of a continuum, then if a magnitude is a continuum, time must be continuous, because the division of time follows upon division of magnitude, end vice versa.

Then at (588) he proves the proposition, namely, that time and magnitude are divided in a similar way. For since we have shown that a faster thing traverses an equal space in less time, let A be the faster and B the slower, and let B be moved more slowly through magnitude CB in time ZI.

It is plain that A, which is faster, traverses the same magnitude in less time ZT.

But again, since A, which is faster, has in time ZT traversed the entire magnitude CD, B, the slower, traversed a smaller magnitude CK in the same time. And because B, the slower, traversed the magnitude CK in time ZT, A, the faster, traversed the same magnitude in even less time. Thus the time ZT will be further divided. And when it is, the magnitude CK will also be divided, because the slower traverses less space in part of that time. And if the magnitude is divided, the time also is divided, because the faster will cover that part of the magnitude in less time. So we continue in this manner, taking a slower mobile after the motion of the faster, and after the slower taking the faster, and making use of the statement already proved that the faster traverses an equal space in less time and that the slower traverses a smaller magnitude in equal time. For by thus taking what is faster, we will divide the time, and by taking what is slower, we will divide the magnitude.

Therefore, it is true that such a conversion can be made by going from the faster to the slower and from the slower to the faster. And if such switching causes the magnitude and then the time to be divided, then it will be clear that time is continuous, i.e., divisible into times that are further divisible, and the same for magnitude; for both time and magnitude will receive the same and equal divisions, as we have already shown.

776. Then at (589) he gives a third reason to show that magnitude and time are correspondingly divided. But this time we shall consider one and the same mobile. And he says that it is clear from the ordinary reasons that if time is continuous, i.e,, divisible into parts that are further divisible, then a magnitude is likewise divisible: because one and the same mobile in uniform motion, since it traverses the whole magnitude in a given time, will traverse half in half the time, and a smaller part in less than half the time. And the reason why this happens is that time is divided as magnitude is.

Lecture 4

Proof that no continuum is indivisible

777. After showing that magnitude and time are subject to similar divisions, the Philosopher now shows that if either is finite or infinite, so is the other. About this he does three things:

First he states the proposition;

Secondly, from this he settles a doubt at 779;

Thirdly, he proves the proposition at 780.

778. He says therefore first (590) that if either of these two, namely, time and magnitude, is infinite, so is the other; likewise, both will be infinite in the same manner.

He explains this by distinguishing two ways of being infinite, saying that if time is infinite in respect of its extremities, the magnitude, too, is infinite in that way. Now time and magnitude are said to be infinite in their extremities, because they lack extremities. It is as though we imagined that a line is not terminated at any points, or that time is not terminated at a first or final instant. Moreover, if time is infinite through division, so also is a length. And this is the second way in which something is infinite. But something is said to be infinite through division, because it can be divided ad infinitum; which, of course, pertains to the definition of a continuum, as was said. Consequently, if time is infinite both ways, so, too, is length.

It is fitting that these two ways of being infinite be set in contrast: for the first way is taken from the viewpoint of indivisible extremities that are absent; the second is taken from the viewpoint of the indivisibles which are intermediate, for a line is divided according to points within the line.

779. Then at (591) he uses these facts to refute Zeno, who tried to prove that nothing is woved from one place to another, for example, from A to B.

For it is clear that between A and B there is an infinitude of intermediate points, since a continuum is divisible ad infinitum. Therefore, if something were to be moved from A to B, it would have to bridge the infinite and touch each of the infinites, and this cannot be done in finite time. Therefore, nothing can be moved through even the smallest distance during a period of finite time, however great.

The Philosopher, therefore, says that this argument is based on a false opinion, for length and time and any magnitude are said to be infinite in two ways, as we have said; namely, according to division and according to their extremities. Accordingly, if there were things (namely, a mobile and a distance) infinite in regard to quantity, which is to be infinite at the extremities, they could not touch one another in finite time. But if they are infinite in respect of division, they will touch, because time also, which is finite in respect of quantity, is infinite in respect of division.

Hence two things follow: that the infinite can be traversed not in finite but in infinite time, and that the infinite points of a magnitude are traversed in the infinite “now’s” of time but not in the finite “now’s”.

But it should be noted that this solution is ad hominem and not ad veritatem, as Aristotle will explain in Book VIII, L. 17.

78C. Then at (592) he proves what he stated above as a proposition.

First he restates the proposition;

Secondly, he proves it at 781.

He says therefore first (592) that no mobile can traverse an infinite distance in finite time nor a finite distance in infinite time; rather, if the time is infinite, then the magnitude must be infinite, and vice versa.

Then at (593) he proves the proposition:

First that the time cannot be infinite, if the magnitude is finite;

Secondly, that if the length is infinite, the time cannot be finite at 784.

781. He proves the first part of the proposition with two reasons, the first of which (593) is this: Let AB be a finite magnitude and let G be an infinite time. Take GD as a finite part of this infinite time. Now, since the mobile traverses the entire magnitude AB in the entire time G, then in part of this time, which is GD, it will traverse the part BE of the magnitude. But since the magnitude AB is finite and greater than BE, which is finite and less, then BE is either an exact measure of AB or it will be less or greater. (These are the only relationships that a lesser finite quantity can bear to a greater finite quantity, as is evident in numbers. For 3, which is less than 6, measures it twice, but 3 taken twice does not measure 5, which is greater than 3, but exceeds it, nor does it measure 7, but is less than 7. But if 3 were taken thrice, that product would exceed even 7). Now it makes no difference in which of these three ways BE is related to AB, for the same mobile will always traverse a magnitude equal to BE in a time equal to GD. But BE is either an exact measure of AB or will exceed it, if taken a sufficient number of times. Therefore, also GD should exactly measure the entire time G or exceed it, if GD is repeated frequently enough. Consequently, the whole time G (in which the entire finite magnitude was traversed) must be finite; because for every segment of magnitude there was a corresponding segment of time.

782. The second reason is given at (782). It is this: Although it be granted that a mobile traverse the finite magnitude AB in infinite time, it cannot be granted that it will traverse any magn1tude at random in infinite time, because we see finite magnitudes being traversed in finite times.

So let BE be the finite magnitude which is traversed In finite time. But BE, since it is finite, will measure AB, which is also finite. Now, the same mobile will traverse a magnitude equal to BE in a finite time equal to that in which it traversed BE. Thus the number of magnitudes equal to BE that will form AB corresponds to the number of equal times required to form the entire time consumed. Hence the entire time was finite.

?83. This second reason is different from the first, because in the first, BE was taken to be part of the magnitude AB, but here it is taken as a separate magnitude.

Then at (595) he shows the necessity of this second reason. For someone could cavil by saying that just as the whole magnitude AB is traversed in infinite time, so would every part of it, and thus the part BE would not be traversed in finite time. But because it cannot be granted that any magnitude at random is traversed in infinite time, it was necessary to present the second reason in which BE is a different magnitude which is traversed in finite time. For if the time in which BE is traversed is finite and less than the infinite time in which AB is traversed, then necessarily, BE is less than AB, and must be finite, since AB is finite.

784. Then at (596) he posits that the same proof leads to an impossibility if the length is said to be infinite and the time finite, because a part of the infinite length will be taker, as finite, just as a finite part of infinite time was taken,

785. Then at (597) he proves that no continuum is indivisible.

First he says that an inconsistency would otherwise follow;

Secondly, he gives the demonstrations that lead to that inconsistency, at 786.

He says therefore first (597) that it is clear from what has been said that no line or plane or any continuum is indivisible: first of all on account of the foregoing, namely, that it is impossible for any continuum to be composed of indivisibles, although a continuum can be composed of continua; secondly, because it would follow that an indivisible can be divided.

786. Then at (598) he gives the proof which leads to this inconsistency. In this proof he makes use of certain facts already established. One of these is that in any finite time the faster and the slower can be in motion. The second is that the faster will traverse more distance in equal time. The third is that there can be excess of speed over speed and of length traversed over length traversed according to varying proportions; for example, according to the proportion of 2 to 1, or 3 to 2, or any other proportion.

With these presuppositions he proceeds thus: Let this be the ratio of the faster to the fast, that the one is faster in the ratio of 3 to 2; and let the faster traverse one magnitude ABCD composed of 3 indivisible magnitudes AB, BC and CD. During the same time according to the given ratio, tae slower will traverse a magnitude of two indivisible magnitudes, which form the magnitude EZI. And because the time is divided as the magnitude, the time in which the faster traverses the 3 indivisible magnitudes must be divided into 3 indivisibles, because the equal magnitude must be traversed in equal time. So let the time be KLMN divided into 3 indivisibles. But because the slower, during that time, traverses EZI, which are 2 indivisible magnitudes, the time can be divided into 2 halves. Consequently, it follows that an indivisible has been divided. For the slower had to traverse one indivisible magnitude in 1 and a half indivisibles of time, since it cannot be said that it traverses one indivisible magnitude in one indivisible time, for then the faster would not have been moved ahead of the slower. Therefore what remains is that the slower traverses an indivisible magnitude in more than one indivisible and less than two indivisibles of time. Thus the indivisible time will have had to be divided.

In like manner, it follows that an indivisible magnitude is divided, if the slower manages to move through three indivisible magnitudes in three indivisible times. For the faster will in one indivisible time be moved through more than one indivisible of magnitude and less than two.

Therefore, it is clear that no continuum can be indivisible.

Lecture 5

The “now” as the indivisible of time. Everything that moves is divisible. Difficulties solved

787. After showing that no continuum is composed of indivisibles and that no continuum is indivisible, thus making it seem that motion is divisible, the Philosopher now determines about the division of motion.

First he states certain facts necessary for the division of motion;

Secondly, he treats of the division of motion, L. 6.

About the first he does two things:

First he shows that in an indivisible of time, there is neither motion nor rest;

Secondly, that an indivisible cannot be moved, at 796.

About the first he does two things:

First he shows that the indivisible of time is the “now”;

Secondly that in the “now” nothing is being moved or is at rest, at 794.

About the first he does three things:

First he states his intention;

Secondly, he states facts from which his proposition can be reached, at 789.

Thirdly, he shows what follows from his proposition, ?90.

788. About the first (599) we must take into account that something is called “now” in relation to something else and not in relation to itself; for example, we say that what is being done in the course of a whole day is being done “now”, yet the whole day is not said to be present according to its entirety but according to some part of itself. For it is evident that part of a whole day has passed and part is still to come, and neither of them is “now”. Thus it is evident that the entire present day is not a “now” primarily and per se but only according to something of itself—and what is true of the day is true of an hour or any period of time.

He says therefore that what is “now” primarily and per se and not according to something else is necessarily indivisible and present in every time.

789. Then at (600) he proves his proposition, For it is evident that it is possible in regard to any finite continuum to take an extremity outside of which there is existing nothing of that of which it is the extremity, just as nothing of a line is outside the point which terminates the line. But past time is a continuum which is terminated at the present. Therefore it is possible to take something as the extremity of the past, so that beyond it there is nothing of the past, and previous to it nothing of the future. In like manner, it is possible to take an extremity of the future, beyond which there is nothing of the past. Now that extremity will be the limit of both, i.e., of the past and of the future; for since the totality of time is a continuum, the past and the future must be joined at one term. And if the “now” fits the description just given, it is clear that it is indivisible.

790. Then he shows what follows from these premisses. About this he does two things:

First he shows that on the supposition that the “now” is indivisible, the limit of the past and the limit of the future must be one and the same “now”.

Secondly, that on the other hand, if each is the “now1l, then the 11now” must be indivisible, at 79-3.

About the first he does two things:

First at (601) he concludes from the foregoing that it must be the same “now” which is the limit of the past and of the future.

791. Secondly, at (602) he proves this statement with the following argument: If the “now” which is the beginning of the future is other than the “now” which is the end of the past, then either these two “now’s” are consecutive and immediately follow one upon the other or one is apart from and distinct from the other. But it cannot be that they are immediately consecutive, because then it will follow that time is composed of an aggregate of “now’s”—which cannot be, because no continuum is composed of indivisible parts, as was said above. Neither can it be that one “now” is apart from the other and distant from it, because then there would have to be a time between those two “now’s”. For it is the very nature of a continuum that there is something continuous between any two given indivisibles, just as there is line between any two given points of a line.

But that this is impossible, he proves in two ways. First of all, because if there were a period of time between the two “now’s” in question, it would follow that something of the same kind would exist between the two, which is impossible, for it is not possible that between the extremities of two lines that touch or are consecutive, there be a line between. For that is against the nature of consecutive things, which were defined as things between which nothing like them occurs. And so, since future time is consecutive to past time, it is impossible that between the end of the past and the beginning of the future there be an intervening time.

He proves the same point in another way: Whatever is intermediate between the past and the future is called “now”. If, therefore, there is any time between the limits of the past and future, it will follow that that will also be called a “now.” But all time is divisible, as has been proved. Consequently, it would follow, that the “now” is divisible.

792. Although in the immediately foregoing he had laid down the principles from which it could be proved that the Now is indivisible, yet because he had not derived the conclusion from these principles, he now shows that the Now is indivisible at (603). And he does this with three arguments.

The first of these is that if Now be divisible, it will follow that something of the past is in the future and something of the future in the past. For since the Now is the extremity of the past and the extremity of the future, and every extremity is in that of which it is the extremity, as a point in a line, then necessarily the entire Now is both in the past as its end and in the future as its beginning. But if the Now be divided, that division must determine the past and the future. For any division made in time distinguishes past and future, since among any parts of time taken at random, one is related to the other as past to future. It will follow, therefore, that part of the Now is past and part future. And so, since the Now is in the past and in the future, it will follow that something of the future is in the past and something of the past in the future.

The second argument he gives at (604): If the Now be divisible, it will be such, not according to itself, but according to something else. For no divisible is the very division by which it is divided. But the division of time is the Now. For that by which a continuum is divided is nothing but a term common to two parts. But that is what we understand by the Now, that it is a term common to the past and future. Thus, therefore, it is clear that what is divisible cannot be the Now according to itself.

The third argument is given at (605): Whenever time is divided, one part is always past and the other future. If, therefore, the Now is divided, necessarily part of it will be past and part future. But past and future are not the same. It will follow, therefore, that the Now is not the same as itself, i.e., something existing as a whole all at once (which is against the definition of the Now: for when we speak of the Now, we consider it as existing completely in the present); rather there will be much diversity and even succession in the Now, just as there is in time, which can be divided any number of times.

793. Therefore, having thus shown that the Now is divisible (as a consequence of supposing that the Now which is the extremity of the past and of the future is not identical), and having rejected this consequent, he concludes to the rejection of the antecedent. And that is what he says: If it is impossible for the Now to be divisible, then it must be admitted that the Now which is the extremity of the past and of the future is one and the same.

Then at (606) he shows that conversely, if the Now of the past and of the future is the same, then it must be indivisible; because if it were divisible, all the aforementioned inconsistencies would follow, And so, from the fact that the Now cannot be admitted to be divisible (as though the Now of the past were something distinct from the Now of the future) and is indeed not divisible, if the Now of the present is the same as the Now of the future, he concludes from the foregoing that it is clear that in time there must be something indivisible which is called the Now.

794. Then at (607) he shows that in the Now there can be neither motion nor rest.

First he shows it for motion;

Secondly, for rest, at 795.

He says therefore first (607) that it is clear from what follows that in the Now nothing can be in motion, for if anything were in motion in the Now, two things could be in motion then, one of which is faster than the other. So let N be the Now, and let there be a faster body being moved in N through the magnitude AB. In an equal time, a slower body is moved a smaller distance. Therefore, in this instant, it traverses the smaller magnitude AG. But the faster will. cover the same distance in less time than the slower. Therefore, because the slower body traversed the magnitude AG in the very Now, the faster traversed the same magnitude in less than the Now. Hence the Now is divided. But it was already proved that the Now cannot be divided. Therefore, nothing can be moved in a Now.

795. Then at (608) he proves the same thing for rest, giving three arguments. The first of which is this: It was said in Book V that an object at rest is one that is naturally capable of being in motion, but is not in motion when it is capable of being in motion and in respect to the part by which it is capable of being in motion and in the manner in which it is apt to be in motion. For if a thing lacks what it is not naturally capable of having (as a stone lacks sight) or lacks it when it is not naturally due to have it (as a dog lacks sight before the ninth day) or in the part in which it is not naturally capable of having it (as sight in the foot or in the hand) or in the way in which it is not apt to have it (as for a man to have sight as keen as an eaglets), none of these reasons is sufficient for saying that a thing is deprived of sight. Now rest is privation of motion. Hence nothing is at rest except what is apt to be moved and when and as it is apt. But it has been shown that nothing is naturally capable of being moved in the very Now. Therefore, it is clear that nothing is at rest in the Now.

The second argument is given at (609): That which is being moved in an entire period of time is being moved in each part of that time, in which it is apt to be moved; likewise, what is at rest in a given period of time is at rest in each period of that time in which it is apt to be at rest. But the same Now is in two periods of time, in one of which the mobile is totally at rest and in the other of which it is totally in motion (as appears in that which is in motion after rest or at rest after motion). Therefore, if in the Now something is apt to rest and be in motion, it will follow that something is at once in motion and at rest which is impossible.

The third argument is given at (610): Rest is said of things which maintain themselves now just as they were previously, but in their entirety and in respect of all their parts. For it is on this account that a thing is said to be in motion, that now it is different from what it was previously, either in respect to place or quantity or quality. But in the Now itself, there is nothing previous; otherwise, the Now would be divisible, because the word “previous” refers to the past. Therefore, it is impossible to rest in the Now.

From this he further concludes that necessarily anything that is being moved and anything that is at rest, is being moved and is at rest in time.

796. Then at (611) he shows that whatever is in motion is divisible: For every change is from this to that. But when something is at the goal, it is no longer being changed but has been changed, for nothing can be at the same time in the state of being changed and having been changed, as was said above. But when something is at the starting-point of change both in its entirety and in regard to all its parts, then it is not being changed; for it was said above that whatever maintains itself constant in its entirety and in regard to all its parts is not being changed but is at rest. He adds “In regard to all its parts”, because when a thing is beginning to be changed, it does not emerge in its entirety from the place it previously occupied, but part emerges after part.

Moreover, it cannot be said that it is in both terms in its entirety and in regard to its parts, while it is being moved; for then something would be in two places at one time.

Nor, again, can it be said that it is in neither of the terms: for we are now speaking of the nearest goal into which a thing is being changed and not of the remotest; for example, if something is being changed from white to black, black is the remote goal, but grey is the nearer one. In like manner, if a line ABCD is divided into three equal parts, it is clear that a mobile, which in the beginning of the motion was in AB as in a place equal to itself, can during the motion be neither in AB nor in CD; for at some time it is in its entirety in BC.

Therefore, when it is said that what is being moved cannot happen to be in neither extremity while it is being moved, must be understood as referring not to the remotest extremity but to a nearer one.

What is left, therefore, is that whatever is being changed is, while it is being changed, partly in one and partly in the other; for example, when something is being changed from AB to BC, then during the motion, the part leaving the place AB is entering the place BC; likewise, when something is being moved from white to black, the part which ceases to be white becomes grey or light grey.

Consequently, it is clear that anything that is being moved, since it is partly in one and party in the other, is divisible.

797. But it should be mentioned that the Commentator here raises the problem that if Aristotle does not intend in this place to demonstrate that every mobile is divisible but only what is mobile in regard to motion (which he said is present in only three genera; namely, quantity, quality and where), then his demonstration will not be universal but particular; because even the subject of substantial change is found to be divisible. Hence, he seems to be speaking of what is subject to any and every type of change, including even generation and ceasing-to-be in the genus of substance. And this is evident from his very words: for he does not say “what is being moved”, but “what is being changed”.

But in that case his demonstration has no value, because some changes are indivisible, such as generation and ceasing-to-be of substance, which do not consume time. In such changes it is not true that what is being changed is partly in one extremity and partly in the other, for when fire is generated, it is not partly fire and partly non-fire.

798. In the face of this problem he proposes a number of solutions, one of which is Alexander’s, who says that no change is indivisible or in non-time, But this must be rejected, because it conflicts with an opinion that is held as probable and famous with Aristotle and all Peripatetics, namely, that certain changes are in non-time, such as illumination and the like.

He mentions also the solution of Themistius, who says that even if there be changes in non-time, they are hidden, whereas Aristotle appeals to what is evident, namely, that change occurs in time. But this he also rejects, because change and the changeable are divided in the same way, and the divisibility of a mobile is more hidden than the divisibility of change. Hence Aristotle’s demonstration would not be valid, because someone could say that although things which changed by changes evidently divisible are themselves divisible, yet there are some hidden changeable beings which are indivisible.

He mentions, too, the solution of Avempace [Ibn-Bajja], who says that the problem here is not about the quantitative division of the things capable of change but of that division whereby the subject is divided by contrary accidents, one of which is changed into the other.

799. Then the Commentator adds his own solution: namely, that those changes which are said to occur in non-time are the extremities of certain divisible motions, It happens, therefore, that something should be changed in non-time, insofar as every motion is terminated in an instant. And because what is accidental is ignored when it comes to demonstrating, for that reason Aristotle proceeds in this demonstration as though every change were divisible and in time.

800. But if you consider the matter correctly, you will see that this objection is not to the point. For in his demonstration Aristotle does not use as his principle the statement that every change is divisible (since he proceeds rather from the divisibility of the mobile to the division of change, as will be clear later, for as he says later, divisibility is first in the mobile, before it is in motion or change). Rather he uses principles that are evident and which must be conceded in any and every case of change; namely, (1) that what is being changed in regard to a certain matter is not being changed in regard to that matter as long as it is totally and according to all its parts still in the starting point, and (2) when it is in the goal, it is not being changed but has been changed, and (3) that it cannot be entirely in both terms or entirely in either of them, as was explained. Hence, it necessarily follows that in any change whatsoever, what is being changed is, during the change, partly in one extremity and partly in the other.

But this occurs in various ways in various changes. For in changes between whose extremities there is something intermediate, it can happen that the mobile is, during the change, partly in one extreme and partly in the other extreme, precisely as extremes. But in those between whose extremes there is nothing intermediate, that which is being changed does not have different parts in different extremities precisely as extremities, but by reason of something connected with the extremities. For example, when matter is being changed from privation of fire to the form of fire, then while it is in the state of being changed, it is indeed under privation as to itself, but yet it is partly under the form of fire, not inasmuch as it is fire, but according to something connected with it, i.e., according to the particular disposition for fire, which disposition it partly receives before it has the form of fire. That is why Aristotle will later prove that even generation and ceasing-to-be are divisible, because what is generated was previously being generated, and what ceases-to-be was previously ceasing-to-be.

Perhaps this was the sense in which Alexander understood the statement that every change is divisible; namely, either according to itself or according to a motion connected with it. So also Themistius understood by the statement that Aristotle took what was evident and abstracted from what was hidden, that the proper place for treating of the divisibility or indivisibility of changes would not be reached until later.

Nevertheless, in all divisibles and indivisibles, what Aristotle says here is true: because even changes that are called indivisible are in a sense divisible, not by reason of their extremities but by reason of something connected to them. And this is what Averroes wanted to say when he said that it is per accidens that some changes occur in non-time.

801. However, there is here another difficulty. For when it comes to alteration, it does not seem to be true that what is being altered is partly in one term and partly in the other, during the alteration. For the motion of alteration does not take place in such a way that first one part and then another is altered; rather the entire thing that was less hot becomes hotter. For which reason Aristotle even says in the book On Sense and the Thing Sensed that alterations are not like local motions. For in the latter, the subject reaches the intermediate before the goal, but such is not the case with things that are altered; for some things are altered all at once and not part by part, for it is the entire water that all at once freezes.

802. But to this it must be replied that in this Sixth Book Aristotle is treating of motion as continuous. And continuity is primarily and per se and strictly found only in local motion, which alone can be c;_ntinuous and regular, as will be shown in Book VIII. Therefore, the demonstrations given in Book VI pertain perfectly to local motion but imperfectly to other motions, i.e., only to the extent that they are continuous and regular.

Consequently, it must be said that what is mobile in respect of place always enters a new place part by part before it is there in its entirety; but in alteration, that is only partially true. For it is clear that every alteration depends on the power of the agent that causes the alteration—as its power is stronger it is able to alter a greater body. Therefore, since the cause of the alteration has finite power, a body capable of being altered is subject to its power up to a certain limit of quantity, which receives the impression of the agent all at once; hence the whole is altered all at once, and not part after part. Yet that which is altered can in turn alter something else conjoined to it, although its power in acting will be less forceful, and so on, until the power involved in the series of alterations is depleted. An example of this is fire which all at once heats one section of air, which in turn heats another, and thus part after part is altered. Hence in the book On Sense and the Thing Sensed, after the above-quoted passage, Aristotle goes on to say: “Yet if the object heated or frozen is large, part after part will be affected. But the first part had to be altered all at once and suddenly by the agent”.

Yet even in things that are altered all at once, it is possible to discover some kind of succession, because since alteration depends on contact with the cause which alters, the parts closer to the body that causes the alteration will more perfectly receive at the very beginning an impression from the agent: and thus the state of perfect alteration is reached successively according to an order of parts. This is especially true when the body to be altered has something which resists the power of the altering cause.

Consequently, the conclusion (that what is being changed, is, while it is being changed, partly in the terminus a quo and partly in the terminus ad quem, in the sense that one part reaches the terminus ad quem before another does) is unqualifiedly and absolutely true in local motion. But in alteration it is qualifiedly true, as we have said.

803. Some on the other hand have held that the present doctrine is truer when applied to alteration than when applied to local motion, For they hold that the statement “what is being changed is partly in the terminus a quo and partly in the terminus ad quem is not to be interpreted as meaning that one part of the thing in motion is in one term and another in the other, but that reference is being made to the parts of the termini, i.e., that what is being moved has part of the terminus a quo and part of the terminus ad quem, as something in motion from white to black, is at the very beginning neither perfectly white nor perfectly black, but imperfectly partakes of both; whereas in local motion this does not seem to be true, except in the sense that the thing in motion, while it is between the two extremities, somehow partakes of both extremities. For example, if earth were to be moved to the place normal to fire, then while it was in the region proper to air, it would have a part of each extremity, (i.e., earth and fire), in the sense that the place of air is above that of earth, and below that of fire,

804. But this is a forced explanation and against Aristotle’s opinion. For in the first place we need only look at the very words of Aristotle. For he says as a conclusion: “it follows therefore that part of that which is being changed must be at the starting-point and part at the goal”. He is speaking therefore about the parts of the mobile and not about the parts of the termini.

In the second place it is against Aristotle’s intention, For Aristotle brings to light facts that will prove that what is being changed is divisible—a statement that could not be proved, if you held to the interpretation given. Hence Avempace said that Aristotle does not intend here to prove that a mobile can be divided into quantitative parts but according to forms, in the sense that what is being moved from contrary to contrary has, while it in being changed, something from each contrary. But the intention of Aristotle is expressly to show that a mobile can be divided into its quantitative parts, just as any continuum, for he makes use of that fact in the demonstrations that will follow.

Nor can we heed the opinion that such an interpretation will help to prove that a mobile can be divided on the basis of continuity. Because the very fact that a mobile, while it is being moved, partakes of each terminus and does not perfectly possess the terminus ad quem all at once, reveals that change is divisible on the basis of continuity. And thus, since a divisible cannot exist in an indivisible, it follows that the mobile also can be divided as a continuum. For in the matters to follow, Aristotle will clearly prove that motion is divisible, because the mobile is divisible. Hence, if he intended to conclude that a mobile is divisible because motion is divisible, he would be arguing in a circle.

Thirdly, such an interpretation appears to conflict with Aristotle’s own interpretation at (611 bis) where he says “here by ‘goal’ of change I mean that into which it is first changed during the process of change”. This shows that he does not intend to say that it is partly in the terminus a quo and partly in the terminus add quem just because it is midway and, as it were, sharing in both extremities, but because in regard to one part of itself it is in one extreme, and according to another part in what is midway.

805. But with respect to this explanation of Aristotle, one might wonder why he says “that into which it is first changed” for it seems impossible to discover that into which it is first changed, since a magnitude can be divided ad infinitum.

Therefore, it must be said that “That into which it is first changed” in local motion is the place next to but not part of the place from which the local motion starts. For if we took it to mean a place that included part of the original place, we would not be assigning the first place into which it is being moved. The following example will illustrate this: Let AB be the place whence a mobile is being moved, and let BC be the adjacent place equal to AB. Now, since AB can be divided, let it be divided at D and take a point G near C so that the place GC is equal to BD. It is clear that the mobile will arrive at DG before it reaches BC. Moreover, since AD can be divided, a place prior to DG can be take n, and so on ad infinitum.

Similarly, in regard to alteration, “the first into which something is changed” must be considered to be an intermediate; for example, when something is changed from white to black, the first into which the subject is changed is into grey, not into less white.

Lecture 6

Two manners of dividing motion. What things are co-divided with motion

806. Having established the facts needed for dividing motion, he now begins to treat of the division of motion. And the treatment is divided into two parts.

In the first he treats of the division of motion;

In the second he uses his conclusions to refute errors about motion, at L. 11.

The first part is divided into two sections:

In the first he discusses division of motion;

In the second, division of rest, at L. 10.

The first section is divided into two parts:

In the first he deals with division of motion;

In the second he discusses finite and infinite with respect to motion (for both, namely, “divisible” and “infinite” seem to belong to the continuum), at L. 9.

The first is divided into two parts:

In the first he shows how motion is divided;

In the second he treats of the order of the parts of motion, at L. 7.

In regard to the first he does two things:

First he lists two ways by which motion is divided;

Secondly, he mentions what else is divided when motion is divi-ded, at 812.

In regard to the first he does two things:

First he mentions the ways in which motion is divided;

Secondly, he explains them, at 808.

807. He says therefore first (612) that motion is divided in two ways. In one way it is divided according to time, because it has been shown that motion occurs not in the “now” but in time. In a second way, it is divided according to the motions of the parts of the mobile. For let the mobile AC be divided, for any mobile can be divided, as we have shown. If therefore the entire mobile AC is being moved, then each of its parts AB and BC is in motion,

But notice that the dividing of motion according to the parts of the mobile can be understood in two ways, First of all, that part is being moved after part—which is not possible in that which is in motion per se in its entirety, for in the case of such a mobile all the parts are moved together, not in isolation from the whole, but in the whole. In the second sense, the dividing of motion according to parts of the mobile can be taken in the same sense that the division of an accident whose subject is divisible depends on the division of that subject; for example, if a whole body is white, then as the body is divided, the whiteness will be divided per accidens. And it is in this sense that we are taking division of motion according to the parts of the mobile, i.e., just as both parts of the mobile are in motion at the same time as the whole is, so the motions of both parts occur at the same time. This shows that division of motion according to the parts of the mobile is different from, that which is according to time, in which division two given parts of a motion do not occur at the same time. But if we were to compare the motion of one part to that of another part not absolutely but according to a fixed stage to be reached, then the motion of one part, will precede in time the motion of another part. For if the mobile ABC is moved in the magnitude EFG, so that. EY is equal to length ABC of the mobile, it is clear that BC will reach F before AB does. According to this the division of motion according to the parts of time and according to the parts of the mobile will be concurrent.

808. Then at (613) he explains these ways of dividing motion:

First he shows that motion is divided according to the parts of the mob-ile;

Secondly, that it is divided according to the parts of time, 8-1.71.

The first he shows by three arguments, of which the first is this: Since the parts are in motion by the fact of the wholes being in motion, let DE be the motion of the part AB and EZ the motion of the part BC. Therefore, just as the whole mobile is composed of AB and BC, so the whole motion DZ is composed of DE and EZ. Since, therefore, both of the parts of the mobile are being moved in accordance with both of the parts of the motion in such a way that neither part of the mobile is being moved in accordance with the motion of the other part (because then the entire motion would be the motion of one part, which would be moved by its own motion and by the motion of the other part), then it must be admitted that the whole motion DZ is the motion of the whole mobile AC; and thus the motion of the whole is divided by means of the motion of the parts.

809. At (614) he gives the second argument, which is this: Every motion belongs to some mobile. But the entire motion DZ does not belong to either of the parts, because neither is being moved according to the entire motion, but both are being moved according to the parts of the motion, as we have said. Nor can it be said that the whole motion DZ is the motion of some other mobile separated from AC, because, if the whole of this motion were the motion of some other whole mobile, it would follow that the parts of this motion would belong to the parts of that mobile; whereas we have already agreed that the parts of the motion DZ belong to the parts of the original mobile, which are AB and BC, and to no other parts (for if they belonged to these and to others as well, it would follow that one motion would belong to several things, which is impossible), What remains, therefore, is that the entire motion belongs to the entire magnitude just as the parts of it belong to the parts of the magnitude. And thus the motion of the whole mobile is divided according to the parts off the mobile.

810. At (615) he gives the third argument, which is this: Everything that is being moved has a position. Therefore, if the whole motion DZ does not belong to the whole mobile AC, then some of the motion does, and let it be TI. Now, from this motion TI take away by division the motions of both parts, which must be equal to the motions that form DEZ, for the following reason: One mobile does not have but one motion, and, consequently, the parts’ motions which are taken away from the motion TI (which is the motion of a whole) cannot be said to be greater or less than DE and EZ, which we agreed are the motions of those same parts. Now the motions of the parts consume the whole motion TI or they are less or greater. If they consume the entire TI and are neither greater nor less, it follows that the motion TI is equal to the motion DZ (which is the motion of the parts) and does not differ from it. But if the motions of the parts are less than TI so that TI exceeds DZ by the amount KI, then the part KI of the motion does not belong to any mobile. For it is neither the motion of AC nor of any of its parts, because one thing has only one motion, and we have already assigned a different motion both to the whole AC and to its parts. Nor can we say that KI belongs to some other mobile, because the entire motion TI is one continuous motion and a continuous motion must belong to a thing that is continuous, as we have shown in Book V. Hence it cannot be that the part KI of this continuous motion belongs to a mobile not continuous with ABC.

A like difficulty follows, if it is said that the motion of the parts exceeds the divided motion TI, because it will follow that the parts exceed the whole—which is impossible. Consequently, if it is impossible that the parts either exceed or are less than to the whole, then necessarily the motion of the parts is equal to and is the same as the motion of the whole.

And so this division is based on the motions of the parts and such a partition must be found in motion, because everything that is being moved is capable of being divided into parts.

811. Then at (616) he shows in the following argument that motion is divided according to the division of time; Every motion occurs in time and every time is divisible, as we have proved. Therefore, since there is less motion in less time, every motion must be capable of being divided according to time.

812. Then at (617) he shows what other things are divided when motion is divided. About this he does three things:

First he mentions five things that are co-divided;

Secondly, he shows that if the finite or infinite is found in any of them, it is found in all the others, at 816;

Thirdly, he shows in which of them is first found division and infinite, at 817.

About the first he does two things:

First he states his proposition;

Secondly, he explains the proposition, at 813.

He says therefore first (617) that since everything that is being moved is being moved in respect to some genus or species as well as in time and, moreover, since every mobile is capable of some motion, then necessarily the following five things must be divided at the same time that any one of them is divided: time and motion and the very “act of being moved” and the mobile which is being moved and “the sphere of motion”, i.e., the genus or species in regard to which there is motion, i.e., place or quality or quantity.

Nevertheless, the divisions of the “spheres of motion” do not all occur in the same way but in some the division is per se and in others per accidens. The division is per se, if it is in the sphere of quantity, as it is in local motion and also in growth and decrease; but it is per accidens in the sphere of quality, as in the motion called “alteration”.

813. Then at (618) he explains what he has said:

First the statement that time and motion are co-divided;

Secondly, that motion and the t1act of being moved” are, at 814.

Thirdly, that motion and the sphere of motion are, at 815,

About the first he does two things:

First he shows that with division of time, motion is divided;

Secondly, vice versa, at 814.

He says therefore first (618): Let A be the time in which something Is being moved, and let B be the motion occurring in this time. Now it is evident that if something is being moved through an entire magnitude in the whole time A, then in half the time, it will be moved through a smaller magnitude. But to be moved through the entire motion is the same as being moved through the entire magnitude, just as to be moved through part of the motion is the same as being moved through part of the magnitude. Therefore, it is clear that if in the entire time it is moved through the whole motion, then in part of the time it will be moved through a smaller motion. And if the time be again divided, a smaller motion will be found, and so on indefinitely. And so it is evident that according to the division of time, motion is divided.

Then at (619) he shows that on the other hand, if the motion is divided, the time is divided. Because if it is being moved through the entire time, then through half the motion it will be moved through half the time and so on, as the motion is smaller, the corresponding time is also, provided of course that we are dealing with the same mobile or one equally fast.

814. Then at (620) he shows that motion and the “act of being moved” are co-divided. Regarding this he does two things:

First he shows that “being moved” is divided according to the division of motion;

Secondly, that motion is divided in accordance with the division of “being moved”, at 814.

He says therefore first (620) that in the same way, it is proved that “being moved” is divided in accordance with the division of time and motion. For let “being moved” be C. Now it is evident that a thing is not moved as much according to part of the motion as according to the whole of the motion. Therefore, according to half of the motion, part of the factor called “being moved” will be less than the whole factor and still less according to half of the half, and so on. Therefore, as time and motion are continually subdivided, so also the factor called “being moved”.

Then at (621) he proves that conversely motion is divided according to the division of “being moved”. For let DC and CE be two parts of a motion, according to both of which something is being moved. Then if the parts of the motion correspond to the parts of “being moved”, then the whole corresponds to the whole, because if there were more in one than in the other, then the same argument would apply here that applied when we proved that the motion of a whole can be divided into motions of the parts in such a way that there is neither excess nor defect. In like manner, the parts of “being moved” can neither be less nor greater than the parts of tho motion; for since we must admit a “being moved” for each part of the. motion, then necessarily the entire factor called “being moved” is continuous and corresponds to the entire motion. And thus, the parts of “being moved” correspond to the parts of the motion and the whole to the whole. Consequently, one is divided in accordance with the other.

815. Then at (622) he shows the same for the sphere of motion, i.e., for the genus or species in which the motion takes place. And he says that in the same way it can be demonstrated that the length in which something is moved locally can be divided according to the division of time and of motion and of “being moved”. And what we say of the length in local motion is to be understood of every sphere in which there is motion, except that in some spheres the division is per accidens, as in the case of qualities in the motion, of alteration, as was said. And hence it is that all those things are divided, because the subject of change can be divided, as was explained above. Consequently, if one is divided, all the others must.

816. Then at (623) he says that just as the above-mentioned things follow upon one another in divisibility, so also in being finite or infinite, so that if one of them is finite, all the others are, and if one is infinite, so are all the others.

817. Then at (624) he shows in which of the five above-mentioned things divisibility and finite and infinite are first found. And he says that the subject of change is the first root from which the divisibility and finiteness and infinity of the others flow, because what is naturally first in motion is the mobile, which of its very nature has the properties called “divisibility”, “finiteness” and “infinity”. Hence from it divisibility and finiteness flow to the others.

But how the mobile is divisible and how the others are divided through it, we have already shown. How the mobile is infinite will be explained later in this Book VI.

Lecture 7

The time in which something is first changed is indivisible. How a first may, and may not, be taken in motion

818, After explaining how motion is divided, the Philosopher now discusses the order of the parts of motion.

First he asks whether there is a first in motion;

Secondly, he shows how the factors involved in motion precede one another, in L. 8.

About the first he does two things:

First he shows that that into which something is first changed is indivisible;

Secondly, how in motion a first can and cannot be found, 822.

About the first he does two things:

First he mentions facts to be used in explaining the proposition;

Secondly, he proves the proposition, at 821.

About the first he does two things:

First he mentions his proposition;

Secondly, he proves it, at 819.

819. He says therefore first (625) that because whatever is being changed is being changed from one term to the other, then when the subject of change has now been changed, it has to be in the terminus ad quem.

Then at (626) he proves this proposition with two arguments, the first of which is particular and the second universal.

The first argument is this: Everything being changed must either (1) be distant from the term at which the change starts, as is evident in local motion, in which the place from which the motion starts remains and the mobile gets to be distant from it; or (2) the terminus a quo must cease to be, as in the motion called alteration: for when something white becomes black, the whiteness ceases to be.

In order to explain this proposition he adds that either the process of being changed is the same as departing, or the latter is a consequence of change and, therefore, “to have departed” (from the terminus a quo) is a consequence of having been changed. But it is evident that they are the same in reality but different in conception. For “departing” is spoken of in relation to the terminus a quo, whereas “change” gets its name from the terminus ad quem. And in explanation of this, Aristotle adds that “both are related to both in a similar way”, i.e., as “departing” is related to “being changed”, so “having departed” is related to “having been changed”.

From these premisses he argues to the conclusion, using as his example the species of change that involves terms contradictorily opposed, where the transition is between being and non-being, as in generation and ceasing-to-be. For it is evident from the foregoing that whatever is being changed departs from the terminus a quo and that whatever has been changed has already departed. When, therefore, something has been changed from non-being to being, it has already departed from non-being. But of anything at all it is true to say that it either is or is not. Therefore, what has been changed from non-being to being is in being, when the change is over. Likewise, what has been changed from being to non-being must be in non-being. Therefore, it is evident that in the change which involves contradictories, the thing which has been changed exists in that into which it has been changed. And if it is true in that type of change, then for an equal reason it is true in other changes. From this the first proposition is clear.

820. Then at (627) the second argument, a general one, is given, And he says that the same conclusion can be proved by considering any change at all. And he picks local motion; Whatever has been changed must be somewhere, i.e., either in the terminus a quo or in some other. But since what has been changed has already departed from that from which it has been changed, it must be elsewhere. Therefore, it must be either in that in which we are trying to prove it is, i.e., in the terminus ad quem or elsewhere. If it is in the former, our point is proved; if not, then let us suppose that something is being moved into B and when the change is finished the thing is not in B but in C. Then we must say that from C it is also changed into B, because B and C are not consecutive. For a change of the type under discussion is continuous, and in continua one part is not consecutive to another, because between two parts there occurs a part that is similar to those two, as was proved above. Hence, it will follow, if that which has been changed is in C when it has been changed and from C it is being changed to B (which is the terminus ad quem), that when it has been changed, it is also being changed into what it has already become—which is impossible, For “being changed” and “having been changed” are never simultaneous, as we have shown above. Now it makes no difference whether the termini C and B are applied to local motion or to any other change. Consequently, it is universally true that what has been changed is (when it has been changed) in that into which it has been changed, i.e., in the terminus ad quem.

From this he further concludes that what has been changed is, as soon as it has been changed, in that into which it has been changed. He added “as soon as”, because after it has been changed into something, it could depart from it and not be there; but as soon as it has been changed, it must be there.

821. Then at (628) he shows that “to have been changed” is first and per se in an indivisible; and he says that that time in which what has been changed was first changed must be indivisible. ‘Why he adds “first” he explains by saying that A is said to have been first changed as soon as it is not said to have been changed merely by reason of any of its parts. For example, if we say that a mobile has been changed in a day, because it was changed in some part of the day. in that case it was not first changed in the day.

But that the time in which something has been first changed is indivisible he now proves: If the said time were divisible, let it be AC and let it be divided at B. Now three things are possible: either (1) the change is over in each part or (2) it is going on in each part or (3) in one part it is going on and in the other it is over. Now, if in each part it is over, then it was first completely changed not in the whole but in the part; but if it is being changed in each part, then it is also being changed in the whole (for the reason why something is said to be changing in a whole period of time is that the change was going on during each part of the whole time). But this is against our assumption that in the whole of AC it had been changed.

On the other hand, if it be supposed that in one part of the time it is being changed and in the other part it has been changed, the same difficulty ensues; namely, that it was not first changed in the whole time, because since the part is prior to the whole and something is in motion in a part of time before it is moved in the entire time, it follows that there was something prior to the first, which is impossible. Consequently, it must be admitted that the time in which the thing was first completely changed is indivisible,

From this he further concludes that everything that has ceased to be and everything that has been completely made, was made and ceased to be in an indivisible of time, because generation and ceasing to be are the termini of alteration. Consequently, if a motion is terminated in an instant (for these two things are the same, i.e., the termination of a motion and to have been first changed), it follows that generation and ceasing-to-be occur in an instant.

822. Then at (629) he shows how to discern in a motion, that which is first. About this he does two things:

First he proposes the truth;

Secondly, he proves it at 823.

He says therefore first (629) that the expression “in which something has been first changed” has two interpretations: first, it can mean that in which the change is first complete or terminated —in which case it is true to say that something has been changed, when the change is now over. Secondly, it can mean that in which it first began to be changed, and not that in which it was first true to say that it has been changed.

Taken in the first sense, namely, according to the termination of the change, it is applied to instances of motion in which there exists a first in which something has been changed. For a change can be first terminated some time, because every change has a termination. It was in this sense that we understood that “that in which something was first changed” is an indivisible—which was proved on the ground that it is the end, i.e., the terminus, of the motion—and we know that every terminus of a continuum is an indivisible.

But if it is taken in the second sense, namely, according to the beginning of the change, i.e., according to the first part of the motion, then there is no first in which something has been changed. For no beginning of a change can be definitely pointed out, i.e., no part that is not preceded by some other part. In like manner, it is not possible to isolate a first time in which something is first being moved.

823. Then at (630) he proves that if one looks at the beginning of a motion, it is not possible to assign “a first in which something has been changed”.

First with an argument from time;

Secondly, with an argument from the mobile, at 824;

Thirdly, with an argument from the sphere in which the motion occurs, at 825.

As to the first he gives this reason: If there is any element of time in which something has been first changed, let it be AD. Now AD must be either divisible or indivisible. If the latter, two difficulties ensue. The first is that the “now’s” in time are consecutive. This difficulty follows from the fact that time is divided just like motion, as was shown above. But if any part of the motion was present in AD, then AD must have been a part of time and, consequently, time will be composed of indivisibles. However, the indivisibles of time are the “now’s”. It will follow, therefore, that the “now’s” are consecutive in time.

And there is a second difficulty. Let us suppose that in the time CA, which preceded AD, the same mobile that was being moved in time AD was entirely at rest. If, therefore, it was at rest in the entire time CA, it was at rest in A, which is an element of the time CA. If, therefore, (as we supposed) AD is indivisible, it’ will follow that a thing is at rest and in motion at the same time; for we have already concluded that it was at rest in A and assumed that it was in motion in AD. But if AD is indivisible, then A is the same as AD. It will follow, therefore, that a thing is at rest and in motion in the same time.

It should be noted, however, that if a thing was at rest throughout an entire time, it does not follow that it was at rest in the last indivisible of that time; for we have already shown that in the “now” things are neither at rest nor in motion. But Aristotle concludes this here by arguing from what his adversary has proposed, namely, that the element of time in which the object was first being moved is an indivisible. And if it can be in motion in an indivisible of time, there is no reason why it could not also be at rest.

Therefore, having rejected the indivisibility of time AD, we are left with the fact that it is divisible. And since it is in AD that the object is said to be first moved, then it is being moved in any part of AD. This he now proves:

Let AD be divided into two parts, Then the object is being moved either in neither part or in both parts or in one part only. If in neither part, then not in the whole time. If in both parts, then it could be granted that it is being moved in the whole time. But if in one part only, it will follow that it is being moved in the whole time but not first, but by reason of the part. Therefore, since it is agreed to be moving in the whole time, it has been in motion in each part of the whole time. But time is divided infinitely just like any continuum; consequently, it is possible always to consider a part smaller than a previous one; for example, a day before a month and an hour before the day. Therefore, it is evident that it is impossible to find a time in which it is first being moved so that a previous could not be found. For if you were to assume that it is in a day that the object is first moved, that assumption would not be true, because it would have been first moved in the first part of the day, before it was moved in the whole day.

824, Then at (631) he establishes the same point by considering the mobile, and he concludes from the foregoing that neither in that which is being changed is it possible to take something that is first changed. Now this is to be understood in the sense that some definite point is to be crossed ‘through ‘the motion of the whole or of the part: for it is evident that the first part of the mobile will first pass a given point, and a second part will pass it after that, and so on. Otherwise, if it were understood in the sense of the absolute nature of motion, what we have to say would not be ad rem: for it is clear that the whole is being moved at the same time as all the parts, but the whole does not pass a certain point all at once but part before part continuously. Hence, just as it is impossible to find a first part of the mobile than which there is not a previous smaller part, so also is it impossible to isolate a part of the mobile that would be first moved. And because time and mobile are correspondingly divided, as we have shown above, then what was concluded about time, he now concludes about the mobile. Here is his proof:

Let DE be a mobile and (because every mobile can be divided, as was proved above) let DZ be the part that is first being moved. And let DZ be moved so that it passes a definite point in the time TI. if, therefore, DZ has been changed in this whole time, it follows that what has been changed in half the time is both less than DZ and moved prior to DZ. And for the same reason there will be something prior to that and so on forever, because time can be divided infinitely, it is evident, therefore, that in the mobile one cannot find something that has been first changed.

Hence it is clear that a first cannot be found in motion, whether we consider the time or the mobile.

825. Then at (632) he proves the same thing by considering the sphere in which the motion occurs. But first he mentions that the situation with respect to the sphere in which the motion occurs is not exactly the same as it was with respect to time and the mobile. For since there are three things to be considered in change; namely, the mobile which is being changed (for example, a man) and that in which it is being changed, i.e., the time, and that into which it is being changed (for example, into white), two of these, namely, the time and the mobile are always divisible. But with white it is another story, because a white thing is not divisible per se, but it, and things like it, are divisible per accidens, inasmuch as the subject of whiteness or of any other quality is divisible.

Now the per accidens division of white can take place in two ways. In one way according to the quantitative parts, as when a white surface is split into two parts, the white will be divided per accidens. In another way, according to greater or less intensity, for the fact that one and the same part is whiter or less white is not due to the nature of whiteness (because if it existed in isolation, whiteness would be constant and never subject to more and less, any more than a substance is susceptible of more and less) but to the varying degrees in which a divisible subject participates whiteness. Therefore, neglecting what is divided per accidens in the sphere of motion and considering only what is divided per se in those spheres, it is impossible to find a first.

And he proves this first of all in magnitudes in which there is local motion. Let the magnitude AC be divided at B, and suppose that C is that into which something is first moved from B. Now BC is either divisible or indivisible. If the latter, it follows that an indivisible will be touching an indivisible, for there is no reason why the second part of the motion will not be into an indivisible, since we can divide a magnitude just as the motion was divided, and as time was.

But if BC is divisible, it is possible to take a stage nearer to B than to C, and so the thing will be changed from B into that stage before it is changed into C and into a stage prior to that one, and so on, because there is no limit to the division of a magnitude. It is therefore evident that it is impossible to find a first stage into which a thing has been changed in local motion.

The same is true in change of quantity, i.e., growing and decreasing. For even these changes are in terms of a continuum, i.e., in terms of added quantity or subtracted quantity, in which no first is to be found, since there can be division ad infinitum.

And so it is clear that it is only in qualitative change that something is per se indivisible. But inasmuch as in this per accidens divisibility is found, likewise no first is discernible in such change. This is true whether the succession consists in part being altered after part (for it is evident that no first part of white can be found any more than a first part of magnitude can) or whether the succession is based on one and the same thing becoming more and more white or less and less white, for a subject can be modified in an infinite number of ways with regard to degrees of whiteness, Thus the motion involved in alteration can be continuous and not possess a first.

Lecture 8

Before every “being moved” is a “having been moved,” and conversely

826. After explaining how a first is to be taken in motion and how not, the Philosopher now explains the order of precedence among the things present in motion.

First he premises facts needed for explaining the proposition;

Secondly, he explains the proposition, at 828.

827. He says therefore first (633) that whatever is being changed is being changed in time, as we have explained. But something is being changed in a time in two ways: in one way, first and per se; in another way, by reason of something else, i.e., by reason of a part, as when something is said to be changed in a year, because it is being changed in a day.

With this distinction in mind, he states what he intends to prove: namely, that if something is being first moved in a time, it is necessarily being moved in some part of that time. This he proves in two ways:

First, from the definition of “first”, for here something is said to be in a thing “first”, if it belongs to it by reason of each and every part, as was said in the beginning of Book V.

Secondly, he proves the same thing with an argument: Let XR be the time in which something is being first moved and, since time is divisible, let XR be divided at K. Then of necessity in the part XK of the time, the object is either being moved or not, and likewise for the part KR. Now if it be said that it is being moved in neither of those parts, it follows that it is not being moved in the whole time but is at rest throughout that time, for it is impossible for a thing to be in motion in a time without being in motion in some part of it. But if it be supposed that it is being moved in just one part of the time, it will follow that it is not being first moved in the time called XR; because that would require motion in respect to both parts and not in respect to just one. Therefore, of necessity, it must be in motion in each part of the time XR. And that is what we want to demonstrate: namely, that if something is being first moved in a time, it is being moved in every part of it.

828. Then at (634) he sets about proving the main proposition. And about this he does two things:

First he introduces the proofs of the proposition;

Secondly, he concludes to the truth, at 838.

About the first he does two things:

First he shows that before each state of being moved there was a state of completed motion;

Secondly, that, conversely, before each state of completed motion there was a state of being moved, at 832.

829. He proves the first with three arguments, of which the first is: Let KL be the magnitude through which a mobile has been moved in the first time XR. It is clear that an equally fast mobile, which began its motion with the first one, will have covered half the magnitude in half the time. Since the first mobile (which we have said covers the entire magnitude) is as fast as the second, it follows that even it has in half the time already been moved through, half the magnitude KL. It will follow, therefore, that what is being moved has been previously moved.

To get a better understanding of what we mean, it must be considered that just as “point” is a name for the terminus of a line, so “completed motion” is a name for the terminus of a motion. Now, no matter what line or what part of a line you take, it is always true that before the consummation of the whole line, you can take a point according to which the line can be divided. Likewise, before any motion or part of a motion, you can take a “state of completed motion”; because while the mobile is being moved to its terminus, it has already passed a certain stage in respect to which the mobile is said to have been already changed. But just as a point within a line is in potency before the line is actually divided (for a point is the very division of a line), so also the thing called “completed motion” (within a motion) is in potency as long as the motion does not stop there; but if it does stop there, it will be actual. And since what is in act is better known than what is in potency, therefore Aristotle proves his proposition (that what is being continually moved has already been moved) by referring to an equally fast mobile whose motion has already been completed. This is like proving that in a certain line there is a point in potency by showing that a like line has been actually divided.

830. The second argument, which he gives at (635), is this: In the whole time XR or in any other, something is said to have been changed by the very fact that a final “now” of the time is taken, not that something is being moved in that “now”, but that the motion is terminated then. Hence “having been moved” is taken here not for that which is at some time being moved but for the fact that the motion is ended. Now the reason why the motion must be terminated in the final “now” of the time that measures the motion is that that “now” terminates the time, just as a point terminates a line. And all time is midway between two “now’s”, just as a line is between two points. Therefore, since “being moved” occurs in time, it follows that “having been moved” occurs in the “now” which is the terminus of time. And if that is the case with a motion in a whole period of time, the same must be true of the parts of motion that occur in the parts of time. Now, we have already shown that if something is being first moved in the whole time, it is being moved in each part of the time. But whichever part of time you take, it is terminated at some “now”. For the terminus of half of the time is the “now” which divided the time into two parts. Therefore, it follows that what is being moved through the whole is previously moved at the middle of time, on account of the 11nowlt which determines the middle. And the same reasoning applies to any part of time. For no matter how the time is divided, it will always be found that each part of the time is determined by two “now’s”, and after the first “now” of the time measuring the motion, no matter which other “now” is taken, the object has already been moved in that part of the time, for that “now”—whichever it is—is the terminus of the time measuring the motion.

Now, because every period of time is divisible into times and each period exists between two “now’s”, and because in any “now” that happens to be the ending of a time measuring the motion, something has been moved, it follows that whatever is being changed has been changed an infinite number of times, because “having been changed” is the terminus of a motion, just as a point is of a line and a “now” is of a time.

Therefore, just as it is possible in any line to pick out point ahead of point ad infinitum and in any period of time “now” before “now” (because both line and time are divisible ad infinitum), so in any “being moved” it is possible to pick out infinitely many “having been moved’s”, because motion, too, is divisible ad infinitum, just as the line and time, as was previously proved.

831. The third argument is in (636): In the case of anything that is being changed (if it is not ceasing-to-be and does not cease to be moved, but is being continually changed), it is necessary that i