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 in each “now” of the time in which it is being moved, it is being changed or has been changed. But in the “now” nothing is being changed, as we have shown. Therefore, in each “now” of the time which measures continuous motion, the object has been changed, But in any portion of time there are an infinitude of “now’s”, because the “now” divides time, and time is infinitely divisible. Therefore, everything that is being changed has been changed an infinite number of times. And so it follows that before every state called “being moved” is a state called “having been moved”, which, however, does not exist outside the state of “being changed” but is in it and terminates a part of it.
832. Then at (637) he proves that on the other hand a state of “being changed” precedes each state of “having been changed”.
First he proves it from the viewpoint of the time;
Secondly, from the viewpoint of the sphere in which the motion occurs, at 836.
About the first he does three things:
First he states the proposition;
Secondly, he proves certain things needed for proving the proposition, at 833.
Thirdly, he gives the proof of the main proposition, at 835.
He says therefore first that not only is it true that whatever is being changed had already been changed, but that every state of “having been changed” must be preceded by a state of “being changed”, because the former is the terminus of the latter. Therefore, every “having been changed” must be preceded by a “being changed”.
633. Then at (638) he states something needed for his proof of the proposition, i.e., that whatever is being changed from something to something was changed in time. But note carefully that here the words, “was changed”, do not refer to the termination of motion, for it was explained above that the time in which a thing “was changed” is an indivisible. But here “was changed” signifies that something was previously being moved, as though he said: “Whatever was being moved was being moved in time”.
This he now proves: If our proposition is not true, then let there be something that was changed from A to B, i.e., from one term to another, in a “now”. From this it follows that when it is in A, i.e., in the terminus a quo in the same “now” it was not yet changed, because it has already been proved that what was changed, when it was being changed is not in the terminus a quo but more in he terminus ad quem. Otherwise, it would follow that it was at once in A and in B. Therefore, it is necessary to say that in one “now” it is in A, and in another it was being changed. But between two “now’s” there is a time, because two “now’s” cannot be immediately connected, as we have shown. What remains, therefore, is that whatever is being changed is being changed in time.
334. But it seems that this conclusion has no application in generation and ceasing-to-be, between whose two termini there is nothing intermediate. For if between the “now” in which something is at the terminus a quo and the “now” in which it is at the terminus ad quem a period of time occurs, it will follow that there is something between being and non-being, because in that intermediate time the subject of change would be neither being nor non-being.
Nevertheless, because the argument which Aristotle gives here is demonstrative, it must be said that it applies somehow even to generation and ceasing-to-be but in the sense that such changes are also instantaneous, since there can be no medium between the termini.
So it must be said that whatever is being changed from non-being to being or vice versa is not in being and non-being at the same time. But, as will be said in Book VIII, there is no final instant in which what is generated is a non-being, but there is a first instant in which it is a being, so that in the entire time preceding that instant, it is non-being. However, between that “now” and the time preceding, there is nothing intermediate, so that between being and non-being there is no medium. Now, since the time which precedes the instant in which something is generated first, is the measure of some motion, it follows that just as that instant in which something is first generated is the terminus of the preceding time that measures the motion, so the first instant of the being of the thing generated is the terminus of a preceding change. If, therefore, generation is said to be the very beginning of being, it must be the terminus of a motion, and thus it takes place in an instant, because a motion’s being terminated—which is the same as having been changed—occurs in an indivisible of time, as we have shown.
However, if generation is taken as the very beginning of being plus the entire preceding motion of which it is the terminus, then it occurs not in an instant but in time, so that what is being generated is a non-being during the entire preceding time and a being in the final instant. And the same applies to ceasing-to-be.
835. Then at (639) he proves the main proposition with the following reason: Whatever has been changed was being changed in time, as we have proved; but time is divisible and whatever is being changed in time is being changed in part of time. Therefore, it is necessary to say that what has been changed in some entire period of time was previously being changed during half of the time and again during half of that half and so on, because time is divisible infinitely. Therefore, it follows that what has been changed was previously being changed. Consequently, before every state of “having been changed” there is a previous state of “being changed”.
836. Then at (640) he proves the same point with an argument based on the sphere of motion.
First as to motions in quantity;
Secondly, as to other changes, at 837.
He says therefore first (640) that what was said, from the viewpoint of time, to be common to every change, becomes clearer from the viewpoint of magnitude, for magnitude is better known than time, and magnitude is continuous, as a line, and in it something is changed, namely, that which is according to place, or according to increase and decrease. Therefore, consider something changed from C to D. Now, it cannot be said that the whole of CD is indivisible, because CD has to be part of a magnitude, just as the motion from C to D is part of a whole motion, for there is a correspondence between division of magnitude and division of motion, as we have shown. But if an indivisible is a part of a magnitude, it follows that two indivisibles are immediate neighbors—which is impossible, as we have shown. Therefore, the whole CD cannot be an indivisible, Consequently, that which is between C and D is a magnitude and can be infinitely divided. And something is always first changed in part of a magnitude before it has been changed throughout the entire magnitude. Therefore, anything that has been changed was previously being changed, just as before any whole magnitude there are its parts.
837. Then at (641) he shows that the same point is true in those changes which do not take place in terms of a continuum; for example, alteration, which is between contrary qualities, and generation and ceasing-to-be, which are between contradictories. And although in those changes the demonstration is not derived from things in which the motion is, yet it is possible to take the time in which the changes occur, and then the demonstration will proceed the same way.
Thus in the three changes, which are alteration, generation and ceasing-to-be, only the first argument holds, while in the other three, namely, growth, decrease and local motion, both arguments hold.
838. Then at (642) he concludes to the main proposition:
First, in general;
Secondly, with special application to generation and ceasing-to-be, at 839.
He concludes therefore first (642) from the foregoing that everything which has been changed was previously being changed, and that everything which is being changed has previously been changed. Consequently, it is true that a state of “having been changed” preceded a state of “being changed”, and vice versa. And so it is clear that a first something cannot be definitely pointed to.
The reason for this is that in motion an indivisible is not joined to an indivisible so as to make a motion be composed of indivisibles, because, if that were the case, we could discover a first. But it is not true, for motion is infinitely divisible just as a line is, which can be infinitely decreased by division and increased by addition opposite to the decrease, in the sense that what is taken from one is being added to another, as was shown in Book III. For it is evident that in a line, before each part of a line, one can take a point in its midst, and before that midpoint is a part of the line, and so on ad infinitum. However, the line is not infinite, because no part of the line is in front of the first point of the line.
Well, the same thing is true of motion. For since each part of motion is divisible, before each part of the motion there is in the midst of that part an indivisible, which is called “having been changed”, and before that indivisible there is a part of the motion, and so on ad infinitum. Yet it does not follow that the motion is infinite, cause in front of the first indivisible of motion there was no part of motion. But note that the first indivisible is not one called “having been changed”, any more than the first point of a line is a dividing point.
839. ‘Then at (643) he comes to the same conclusion with reference to generation and ceasing-to-be. And he makes a special point of these changes, because the relation of “having been changed” to “being changed” in generation and ceasing-to-be is not the same as it is in other changes.
For in the others, the state of “having been changed” and the state of “being changed” occur in respect to the same thing; for example, to whiteness, in the case of alteration. For “to be being altered” is to be being changed in respect to whiteness, and “to have been altered” is to have been changed in regard to whiteness; and the same is true in local motion, in growth and in decrease. But in generation “having been changed” refers to one thing and “being changed” to another. For the former is based on the form, but the latter, though not based on negation of a form (which is not of itself susceptible of more and less) is based on something joined to such a negation, something, that is, which is susceptible of more and less, namely, a quality. Therefore, “to have been generated” is the terminus of “being altered” and the same is true of “having been corrupted”. And because motions get their name from the terminus ad quem, as we have said in the beginning of Book V, “to be altered” (since it has two termini, namely, substantial form and quality) has two names: for it can be called “to be altered”, and “to come to be and cease to be”.
And this is the sense in which coming-to-be and ceasing-to-be are substituted for “being altered”, i.e., because the alteration terminates at being or non-being. And consequently, Aristotle says that what has been made was previously being made, and what is being made must necessarily have been made, provided that divisible and continuous things are involved. And Aristotle makes that addition (as the Commentator says) in order to exclude things that indivisibly come to be without continuous motion; for example, understanding and sensing, which are motions only in an analogous sense, as will be shown in Book III of On the Soul. But it could be that Aristotle made this addition in order to show that generation should include the entire continuous motion that precedes it.
840. But the statement “what is being made has been previously made” applies in different ways to different things. For some things, such as air and water, are simple and have simple generation—in these cases, part is not generated after part, but the whole and the parts are altered and generated at once. And it is in such that what has been made was previously being made and what is being made has been previously made, on account of the preceding alteration being continuous.
But other things are composites of unlike parts. In these cases, part is generated after part, as in an animal the heart is first generated, and in a house the foundation. In such things what is being made was not itself previously made, but a part was. And this is what he adds, namely, that it is not always so that what is being made has been itself previously made but something pertaining to it has been made, as the foundation of a house. But since we must come to a part that is entirely being made at once, then in some part, that which is being made has been made in relation to a terminus taken in the preceding alteration; for example, in the generation of an animal, the heart has already been made and while the heart is being generated, something has already been made—not indeed that there has been made some part of the heart, but some alteration ordained to the generation of the heart.
And what has been said of generation is to be understood with regard to ceasing-to-be. For immediately there is in something that is produced in being and is corrupted, something infinite, since it is continuous. For the very coming-to-be and the ceasing-to-be are continuous. Therefore, there is no “being produced in being”, unless something has been previously made, and nothing has been made unless it was previously being produced in being. And the same is true of ceasing-to-be and having ceased-to-be. For a “having-ceased-to-be” is always prior to a “ceasing-to-be” and a “ceasing-to-be” prior to a “having ceased-to-be”.
From this it is evident that whatever has been made was previously being made, and that all that is being made has in some way previously been made. And the reason is that every magnitude and every period of time are infinitely divisible. Consequently, in whatever period of time something comes to be, it is not coming to be in that time as in a first time, because it always possible to find a period previous. And what we have said of generation and ceasing-to-be is true also of illumination, which is the termination of the local motion of the illuminating body, just as generation and ceasing-to-be is the terminus of an alteration.

Lecture 9
Finite and infinite are found simultaneously in magnitude, time, mobile, and motion
841. After determining the division of motion, the Philosopher now determines about the infinite and finite in motion; for just as division pertains to the notion of continuum, so also do finite and infinite. But just as above he said that division is found simultaneously in motion, magnitude, time and mobile, so now he shows that the same is true of the infinite. Hence about this he does three things:
First he shows that the infinite is found in the same way in magnitude and in time;
Secondly, that it is found in the same way in the mobile, 846;
Thirdly, and in motion, at 652.
About the first he does two things:
First he shows that if a magnitude is finite, the time cannot be infinite;
Secondly, that if the time is finite, the magnitude cannot be infinite, at 845.
in regard to the first he does two things:
First he proposes what he intends;
Secondly, he proves his proposition, at 843.
842. First, therefore, (644) he repeats two things that are needed for proving the proposition. One of which is that whatever is being moved is being moved in time. The second is that in more time a greater magnitude is traversed by the same mobile. From these two suppositions he intends to prove a third, namely, that it is impossible to traverse a finite magnitude in infinite time. This is to be understood in the sense that the thing in motion is not to retraverse the same magnitude repeatedly or any part of it, but must be moved through the entire magnitude in the entire time. And he added this to save himself from circular motion over a finite magnitude, which can occur in infinite time, as will be explained in Book VIII.
843. Then at (645) he proves his proposition:
First by assuming a mobile of equal speed being moved over the whole magnitude;
Secondly, if it is not being moved with a regular and uniform motion, at 844.
He says therefore first (645) that if a mobile of equal speed is traversing a whole, then if the whole is a finite magnitude, it must be traversed in finite time. For we can take one part of the magnitude and make it measure the whole; for example, a part that is one-third or one-fourth of the magnitude, If, therefore, a mobile is moved with equal speed over the whole and if the equally fast is what traverses an equal space in equal time, it follows that in a number of equal times that are determined by the number of parts into which the magnitude was divided, it will traverse the whole magnitude; for example, if one-fourth of the magnitude is taken, it will traverse it in a certain time and another fourth in an equal time, and so it will traverse the entire magnitude in four equal times.
Because, therefore, the parts of the magnitude are finite in number and each is finite in quantity, and in a given number of equal times the whole magnitude is traversed, it follows that the whole time in which the entire magnitude is traversed is finite. For it will be measured by a finite time, since it will be as many times as much as the time required to traverse one part, the whole magnitude being as many times as the quantity of each part. And thus the whole time will be the multiplication product of the length multiplied by the number of parts. But every multiplication product is measured by a denominator, as double is measured by half and triple by third, and so on. The time, however, required to traverse a part is finite, because if it were infinite, it would follow that the whole and the part were traversed in equal time, which is against the original assumption. Therefore, the whole time has to be finite, because nothing infinite can be measured by the finite.
844. But someone could say that although the parts of the magnitude are equal and measure the whole magnitude, it could happen that the parts of time are not equal, as when an equal speed is not maintained through the entire motion, and so the time required to traverse a part of the magnitude will not be a measure of the time required to traverse the whole.
Therefore at (646) He shows that this makes no difference to the proposition. For let AB be a finite space that has been traversed in infinite time CD. Now in every motion, one part must be traversed ahead of another and also one part of the magnitude is traversed in the prior part of time and another part in a subsequent part of time. And so, no two parts of the magnitude are ever traversed in one and the same part of time, and no two parts of time correspond to one and the same part of the magnitude. Consequently, if a certain part of the magnitude is traversed in a certain time, then in more time is traversed not only that part of the magnitude but that part and another. And this will happen whether the mobile maintains constant speed or not, for in natural motions the speed is continually increased, while in compulsory motions it is diminished.
With these suppositions in mind, let AE be a part of the space AB and let it be an exact measure, say, one third or one fourth of AB. Therefore, this part of space has been traversed in a finite time. For it cannot be assumed that it was traversed in infinite time, because the whole space was traversed in infinite time, whereas less time is required to traverse a part than to traverse the whole. Likewise, let us take another part of the space and let it equal the part AE. This part, too, must be traversed in finite time, for it is the whole space that is being traversed in infinite time. Proceeding in this manner, let us take, in accordance with the parts of the entire space, a corresponding number of such times. From these will be constituted the whole time in which the entire space is traversed.
Now it is impossible that a part of an infinite measure the whole, either in the case of a magnitude or in that of a multitude, because it is impossible for the infinite to be composed of a finite number of parts, each of which is finite in quantity, whether those parts are equal or unequal—for whatever things are measured by some one thing, either according to magnitude or multitude, must be finite.
Now, I say “magnitude and multitude”, because a thing of finite magnitude can still be measured, whether the measuring parts are of equal or unequal size. For when they are equal, then any part is a measure of the whole, whether the whole be a magnitude or a multitude; but when they are unequal parts, any part will measure a multitude but not a magnitude. So, therefore, it is evident that any time which has parts finite in number and quantity, whether they be equal or not, is finite. But a finite space is measured by as many finite parts as are necessary to form AB. Moreover, the parts of the time will be equal in number to the parts of the magnitude, and the parts will be finite in quantity. What remains, therefore, is that the entire space is traversed in finite time.
845. Then at (647) he shows that on the other hand, if the time is finite, so too the nagnitude, And he says that by the same reasoning it can be shown that infinite space cannot be traversed in finite time, and that rest cannot be infinite in finite time, no matter whether the motion is regular or not. For since the time posited is finite, it is possible to take as a measure of the whole time a part in which the mobile traverses a part of the magnitude but not the whole magnitude, which is traversed in the whole time. Then in an equal time it will traverse another part of the magnitude. And, in like manner, for each part of the time take a corresponding part of the magnitude, and let this be done whether the second part of the magnitude be equal to the first part (which happens when the speed is constant) or not equal to it (which happens when the speed varies). For whether they are equal or not makes no difference, as long as each part you take of the magnitude is finite, which it must be; otherwise as much will be traversed in a part of time as in the whole time. According to this procedure, it is clear that by dividing time the entire infinite space will be exhausted as the finite parts are used up. For since the time is divided into finite equal parts and the number of magnitudinal parts must equal the number of parts of time, it follows that the infinite space will be consumed by making finite subtractions, since the magnitude has to be divided according to the way the time is divided. But this is impossible. Therefore, it is clear that an infinite space cannot be traversed in finite time, whether the magnitude of space be infinite in one direction or more, because in either case the same reason would hold.
846. Then at (648) he shows that infinite and finite are found in the mobile in the same way as they are found in magnitude and time. About this he does three things:
First he shows that the mobile is not infinite, if the magnitude is finite and the time finite;
Secondly, that the mobile is not infinite, if the magnitude is infinite and the time finite, at 848;
Thirdly, that the mobile cannot be infinite, if the magnitude is finite and the time infinitel at 849.
He proves the first point with two arguments. In regard to the first of these he says that, since it has been demonstrated that a finite magnitude is not traversed in infinite time nor an infinite magnitude in finite time, it is clear from the same causes that an infinite mobile cannot traverse a finite magnitude in finite time. For if you take any part of finite time, then during that part of time the finite space will be traversed not by the whole mobile but by a part, and during another part, it will be traversed by another part of the mobile, and so on. And so, it will be necessary to take as many parts of the mobile as parts of time. But the infinite is not composed of finite parts. Therefore, the mobile that is moved in a whole finite time is finite.
847. The second argument is given at (649) and it differs from the first, because in the first he took as his principle the same medium that he used in the previous demonstrations, but here he takes as his principle the conclusion reached above. For it has been shown above that a finite mobile cannot traverse an infinite space in finite time, Hence it is clear that for the same reason neither can an infinite mobile traverse a finite space in finite time. For if an infinite mobile traverses a finite space, it follows that a finite mobile can traverse an infinite space, because both the mobile and the space have dimensions. Now when two things having dimensions are involved, it makes no difference which is in motion and which is at rest. For it is clear that whichever is assumed as being in motion, it follows that the finite traverses the infinite. For let A be the infinite that is in motion and let CD be a finite part of it. When the whole is being moved, this finite part will be at the part B of the space, and as the motion continues, another part of the infinite mobile will be at B and so on. Hence, just as the mobile traverses space, so space in a sense traverses the mobile, inasmuch as the various parts of the mobile are successively other and other in regard to the space. Hence it is evident that at the same time that an infinite mobile is being moved through a finite space, something finite is traversing something infinite. For there is no other possible way for an infinite to be moved through finite space than for the finite to traverse infinite space, either by having the finite moved over the infinite, as when the mobile is finite and the space infinite, or by making something finite measure the infinite, as when the space is finite and the mobile infinite. For then, even though the finite is not being moved over the infinite, yet the finite is measuring the infinite, inasmuch as a finite space is placed opposite each of the parts of the infinite mobile. Therefore, because this is impossible, it follows that an infinite mobile does not traverse a finite space in finite time.
848. Then at (650) he shows that there cannot be an infinite mobile, if the space is infinite and time finite. And this is what he says: that an infinite mobile cannot traverse an infinite space in finite time. For in every infinite there is something finite. Therefore, if an infinite mobile should traverse an infinite space in finite time, it follows that it traverses a finite space in finite time, which is against a previous conclusion.
849. Then at (651) ae says that the same demonstration holds if the time be infinite and the space finite. Because if an infinite mobile traverses a finite space in infinite time, it follows that in a part of that time it will traverse a part of the space. Consequently, the infinite will be traversing the finite in finite time, which is also against a previous conclusion.
850. Then at (652) ha shows that finite and.infinite are found in motion in the way that they are found in mobile, space and time. And he says that a finite mobile does not traverse an infinite space, nor an infinite mobile finite space, nor an infinite mobile infinite space, in finite time, From these facts, it follows that there cannot be an infinite motion in finite time. For the quantity of motion depends on the quantity of space. Hence there is no difference between saying that the motion is infinite and that the magnitude is. For it is necessary that if either is infinite, so is the other, because no part of a local motion can exist outside of a place.

Lecture 10
Things pertaining to the division of “coming to a stand” and “rest”
851. After finishing the things that pertain to the division of motion, the Philosopher now determines about things that pertain to the division of rest. And because coming to rest is generation of rest, as we have said in Book V.
First he determines the things that pertain to coming to rest;
Secondly, the things that pertain to rest, at 856.
About the first he does three things:
First he shows that whatever is coming to rest is being moved!
Secondly, whatever is coming to rest does so in time, at 853;
Thirdly, how a first is spoken of in coming to rest, at 854.
852. He shows the first at (653): Everything apt to be moved must be either in motion or at rest at the time when it is apt to be moved and in the place in which it is apt to be moved and in the way in which it is apt to be moved. But what is coming to rest is not yet at rest—otherwise, it would happen that a thing would be at the same time tending to rest and actually resting. Therefore, whatever is coining to rest is in motion, when it is coming to rest.
853. Then at (654) he proves by two arguments that whatever is coming to rest is doing so in time. For whatever is being moved is being inoved in time, as has been proved. But whatever is coming to rest is being moved, as we have just proved. Therefore, whatever is coming to rest is coming to rest in time.
The second argument is that swiftness and slowness are determined according to time. But it can happen that something comes to rest either more swiftly or more slowly. Therefore, whatever is coming to rest does so in time.
854. Then at (655) he shows how “first” is spoken of in coming to rest. About this he does two things:
First he shows how something is said to be “first” coming to rest in a given time, where “first” is opposed to what is spoken of in reference to a part;
Secondly, he shows that in coming to rest, it is not possible to discern a first part, at 855.
Fe says therefore first (655) that if at a certain time something is said to be coming to rest first and per se and not by reason of a part, then it must be coming to rest in each part of that time. For time can be divided into two parts, and if it is said that it is coming to rest in neither, it will follow that it is not coming to rest in the whole time, in which it was assumed to be coming to rest. Therefore, something coming to rest is not coming to rest. Nor can it be said that it is coming to rest in only one of the parts, because then it would not be coming to rest first, but only by reason of a part. Hence it will remain that it is coming to rest in both. For it is said to be coming to rest in the whole time only because it is coming to rest in each part, as was said above about things in motion.
855. Then at (656) he shows that there is no first part in coming to rest. And he says that just as it is not possible to find in time a first part in which a mobile is being moved, so also in regard to coming to rest, because in neither case can there be a first part.
If this is denied, then let AB be the first part of time in which something is coming to rest. This part cannot be indivisible, because it has been shown above that motion does not occur in an indivisible of time (for it is always true that whatever is being moved has already been moved, as we have shown above) and, moreover, whatever is coming to rest is being moved, as we have just now proved. Hence AB must be divisible. Therefore, there is a coming to rest in each part of it, for we have just shown that when in a given time something is coming to rest first and per se and not by reason of a part, it is coming to rest in each part of that given time. Therefore, since the part is prior to the whole, AB was not, the first in which there was a coming to rest. And because that in which something is coming to rest is a time and all time is divisible ad infinitum, it follows that it is impossible to find a first in which something is coming to rest.
856. Then at (657) he shows the same thing is true for rest. About this he does two things:
First he shows that there is no first in rest;
Secondly, he gives a method to distinguish motion from rest,
And because it is for the same reason that no first is found in notion and in coming-to-rest and in rest, therefore, he concludes the same thing for rest as he concluded for motion and coming-to rest. And he says that there is no first in which a thing at rest has been at rest. To prove this he repeats something previously proved, namely, that nothing is at rest in an indivisible of time. Likewise, he repeats the two reasons he used when he proved this. The first of which is that there is no motion in an indivisible of time. But to rest and to be in motion are in the same: because we do not say that something is resting, unless what is capable of being moved is not being moved when it is apt to be moved and in the sphere in which it is apt to be moved; for example, quality or place or something of thls sort. Hence it remains that nothing is at rest in an indivisible of time.
The second reason is that it is then that we say something is at rest when it maintains itself as it was previously: as if to say that we do not judge rest by reason of one factor only but by comparing two things to one another and seeing that there is a aimilar situation in both. But it is impossible to find in something indivisible a “now” and something previous, or any two things. Therefore, that element of time in which something is at rest is not indivisible.
Having established this, he proceeds further to prove the main. proposition. For if that in which something is at rest is divisible into parts that possess a prior and a subsequent, it follows that it is a time; for this is the very nature of time. And if it is time, then it must be resting in each part of it. And this will be demonstrated in the same way that it was demonstrated in motion and in coming to rest; namely, that if it is not at rest in each part, it will be at rest in no part or in one only. If in no part, then not in the whole; if in one only, then in that part first and not in the whole first. But if it is at rest in each part of the time, it will not be possible to discover a first in rest any more than in motion.
The reason for this is that things are at rest and in motion in time. But in time there is no first any more than in a magnitude or in any continuum, for every continuum is divisible ad infinitum and, consequently, it is always possible to find a part smaller than another. And that is why there is no first in motion or in coming-to-rest or in rest.
857. Then at (658) he gives a way through which what is in motion is distinguished from what is at rest.
First he mentions it;
Secondly, he proves it, at 858.
In regard to the first he premises two suppositions, the first of which is that whatever is being moved is being moved in time. The second is that whatever is being changed is being changed from one terminus to another. From these two facts he intends to conclude a third; namely, that if you take a mobile, which is being moved first and per se and not by reason of its part only, it cannot remain one and the same with respect to that in which the motion is—for example, it cannot remain in one and the same place, or retain one and the same degree of whiteness—during a given period of time, provided that you take it as being in time according to itself and not according to something which is in time.
The reason why you must take a mobile which is being moved first and per se is that there is nothing to prevent a thing from being moved according to a part even though it remains in one and the same place throughout the entire time, as when a man sitting down moves his foot.
And the reason why he speaks of a time throughout which something is being moved per se and not by reason of some element of time is that while a thing is being moved it can be said that on such and such a day it is in one and the same place; but this would be said, because it was in that place not throughout the day but in some “now” of that day.
858. Then at (659) he proves the proposition. And he says that if what is being changed is throughout a definite period of time in one and the same state—for example, in one place—it follows that it is at rest, due to the fact that in that time there is present in one and the same place the entire mobile and each part of it; for we have already said that to be at rest means to be able to say of something that it and its parts are in one and the same state in different “now’s”, If, therefore, this is the definition of being at rest and if nothing can be at rest and in motion at the same time, it follows that the whole which is being moved cannot be totally in one state, e.g., in one and the same place, during the whole time and not only in something of it.
Why this follows he now explains. Every period of time is divisible into diverse parts, of which one is prior to another. Hence if something is in one state throughout the entire period, it will be true to say that in one and in another part of the time the whole mobile and its parts are in one and the same state, e.g., place—and this is to be at rest. For if it is said to be in one and the same state not in different parts of time but throughout one “now”, it does not follow that there is a time in which it is in one and the same state, but that there is a “now” in which it is in one and the same state.
For although from the fact that if something remains in one and the same state during a period of time, the conclusion can be drawnthat it is at rest, that conclusion cannot be drawn if it remains in one and the same state in just one “now”. For whatever is being moved is always stable, i.e., existing, vis-a-vis something of that in which it is being moved in each “now” of the time in which it is being moved; for example, place or quality or quantity, Yet it is not for that reason at rest, because it has already been proved that neither rest nor motion can occur in a “now”. But it is true to say that in the very “now” something is not being moved and that in the “now” even what is being moved is somewhere or according to something. But what is being moved in time cannot be under any aspect at rest, for then it would happen that something is at rest while it is in motion—which is impossible. What remains, therefore, is that whatever is being moved is never, as long as it is being moved, in one and the same state for two “now’s” but for only one.
859. And this point is clear in local motion. For let AC be a magnitude divided in half at B and let 0 be a body equal to each half, i.e., to AB and to BC, and let that body be moved from AB to BC. If no part of one of these two places can be a part of the other, there will be only two places for that body on AC. But it is evident that 0 does not relinquish its first place and enter the second all at once but successively. Hence, because place is divisible ad infinitum, the places also are multiplied ad infinitum. For if the half part AB is again halved at D and the other half part BC at E, it is evident that DE will be a place distinct from both AB and BC. By continuing such divisions other and other places will be found.
The same point is clear in alteration. For what passes from white to black passes through an infinitude of shades of whiteness and blackness and intermediate colors.
However, it does not follow that since there are an infinitude of intermediates, the ultimate cannot be reached, because these intermediate places are infinite not in act but only in potency, just as a magnitude is not actually divided infinitely but is potentially divisible.

Lecture 11
Zeno's arguments excluding all motion are resolved
860. After finishing with the division of motion and of rest, the Philosopher now refutes certain opinions that have been the source of error in regard to motion. About this he does three things:
First he answers the arguments of Zeno who absolutely denies that motion exists;
Secondly, he shows that an indivisible is not moved, against Democritus, who said that they are always in motion, at L. 12;
Thirdly, he shows that all change is finite, against Heraclitus, who said that all things are eternally moved, at L. 13.
About the first he does two things:
First he gives and rejects one of Zeno’s arguments, which pertains to what Zeno had accepted about motion;
Secondly, he explains all his arguments in order, at 863.
861. He says therefore first (660) that Zeno reasoned badly and used what had only the appearance of a syllogism to show that nothing is being moved, even what seems to be in rapid motion, as an arrow in flight. And this was his argument: Anything that is in a place equal to itself is either being moved or is at rest. But whatever is being moved is at each instant in a place equal to itself. Therefore, even at each instant it is either in motion or at rest. But it is not in motion, Therefore, it is at rest. But if it is not in motion at any instant but at rest, as it seems, then throughout the entire time it is at rest and not in motion.
Now this argument could be answered by appealing to something already proved; namely, that in an instant there is neither motion nor rest. But such a solution would not cripple Zeno’s intention, for he is satisfied to show that through the entire time there is no motiona fact that seems to follow, if there is no motion at any instant of the time. Therefore Aristotle answers in a different manner and says that the conclusion is both false and does not follow from the premisses.
For in order that something be moved in a given period of time, it has to be moved in each part of the time. But instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time.
862. Then at (661) he lists in order all the arguments that Zeno used for destroying motion. About this he does three things:
First he shows how he destroyed local motion with his arguments;
Secondly, how he destroyed the other types of change, at 870;
Thirdly, how in particular he destroyed circular motiong at 871.
863- In regard to the first he lists four reasons, and this is what he says: Zeno used against motion four arguments which have caused difficulty for many of those1who tried to answer them. The first of which is this: If anything is being moved through a certain space, it must reach the middle before it reaches the end. But since the first half is divisible, half of it must be first traversed and so on indefinitely, since a magnitude can be divided ad infinitum. Infinites, however, cannot be traversed in finite time. Therefore, nothing can be moved.
Therefore Aristotle says that he has already answered this argument (in the beginning of this Book VI), when he proved that time is divided ad infinitum in the same way as a magnitude is. This answer is directed more to one who asks whether infinites can be traversed in finite time than to the question, as he will say in Book VIII, (L. 17) where he answers this argument by showing that a mobile does not use the infinites which exist in a magnitude as though they were actually existing, but only as existing potentially. For a thing in motion uses a point in space as actually existing, when it uses it as a beginning and as an end, and it is then that the mobile must be at rest, as will be explained in Book VIII. But if it had to traverse infinites that were actually existing, then it would never reach the end.
864. The second argument is given at (662) and he says that they called this one the “Achilles”, as though it were invincible and unanswerable. The argument was this: If anything is being moved, it follows that a slower thing, if it started earlier, will never be caught by anything moving most rapidly. And it was proved in the following way: If a slower object began to be moved for some time before a very swift one, then in that time it has traversed some distance. Therefore, before the very swift one in pursuit could reach the slower, which is still running, it must leave the place first left by the pursued and reach the place which the pursued reached during the time the pursuer was not in motion. But the very fast pursuer must traverse this space in some time, during which the slower has meanwhile traversed a certain space, and so on forever. Therefore, the slower always has “something ahead”, i.e., is always some distance ahead of the most swift pursuer, and so the swifter will never catch the slower. But this is unacceptable. Therefore, it is better to say that nothing is moved.
865. In solving this argument he says that it is the same as the first, which proceeded by dividing the distance into two halves and then continually halving one part of the remainder. But the difference between them is that in the second the given magnitude of space is not divided into halves but according to the difference between the swift and the slower motion. For in the first period of time in which only the slower was in motion, there is a greater magnitude involved; in the second period (in which the faster traversed the distance covered by the slower between its start and the start of the faster), which is a shorter time period, a smaller magnitude was traversed by the slower, and so on forever. Hence, the.time and the magnitude are always being divided and that seems to be the reason why the slower is never caught by the swifter,
But this tends to the same thing as what was said of the division of the magnitude into halves; because in both arguments it seems that the mobile cannot reach a certain goal on account of the magnitude’s being infinitely divided, no matter how it happens to be divided, i.e., whether according to halves, as happens in the first argument, or according to the excess of the faster over the slower, as in the second argument. However, in this second argument it is further added that the very swift cannot reach the slower, which it is pursuing. This “tragic” phraseology employs inflated language in order to excite wonder, but it does not do anything to the force of the argument.
Hence it is clear that the solution of the two arguments is the same. For just as a false conclusion was reached in the first argument, namely, that the mobile would never reach the end of the magnitude on account of the infinite division of the magnitude, so also what the second argument tries to conclude is false, namely, that the slower will never be,caught by the swifter, which is just another way of saying that a mobile will never reach its goal.
Now, it is true that as long as the slower is ahead, it is not yet reached by the swifter. But yet it will at some time be reached, if you concede that a finite magnitude can be traversed in finite time. For the swifter pursuing mobile will traverse the whole distance by which the slower is ahead and even more, in less time than the slower was meantime moving farther ahead. Proceeding in this way the swifter will not only catch but pass the slower. These therefore are the solutions to two of Zeno’s arguments.
866. The third argument is given at (663) and he says that the third argument of Zeno was the one cited above (before he began to give the arguments): namely, that an arrow in flight is always at rest. And as was said above, this seems to happen, because Zeno supposed that time is made up of instants. For unless that be granted, the syllogism fails.
867. He sets out the fourth argument at (664). Concerning which he doea three things:
First he sets out the argument;
Secondly, the solution, at 868;
Thirdly, he explains it by an example, at 869.
First therefore he says that the fourth argument of Zeno proceeded from some bodies which move in a stadium so that there are two equal magnitudeE which are moved in an equal manner, that is, through a space in the stadium equal to both in quantity, and this motion is a contrary one, i.e., one of the equal magnitudes is moved through that space of the stadium toward one part, and the other toward the other part, in such a way, however, that one of the mobile magnitudes begins to move from the terminus of the stadium which is equal to it, and the other begins to move from the middle of the stadium or of a space in the given stadium; both move with equal velocity. This being given, Zeno held that it would result in a half time equalling a double time. Since this is impossible, he intended to infer further that it is impossible for anything to be moved.
868: Then at (665) he gives the solution. He says that Zeno was deceived in this, that he held that on the one hand the mobile is moved according to the moved magnitude, and on the other he held that it was moved according to a quiescent magnitude equal to the moved magnitude. Because an equal velocity of the moved bodies is supposed, he wanted to conclude that the motion of equally swift bodies in regard to equal magnitudes, one of which is in motion and the other standing still, is done in equal times This is seen to be false for the following reason:
When something is moved in relation to a quiescent magnitude, there is only one motion; but when something is moved in relation to a moving magnitude, there are two motions. If they are moving in the same direction it takes more, if they are moving in opposite directions it takes less time according to the amount of either motion. If the magnitude in relation to which something.mobile is moved, is moved in the same direction with an equal velocity or even a greater velocity, the other moving body can never pass it. If the magnitude moves with less speed, it will pass by it at a certain time, but it will take more time than if it were quiescent. It is quite the contrary if the magnitude is moved opposite the direction of the other body. The more swiftly the magnitude moves, the less time the other body takes to pass it, because both motions work together to pass each other.
869. Then at (666) he makes clear what he said in the latter part. Suppose that there are three magnitudes equal to each other, each designated as A, and these magnitudes are standing still; thus there might be a space of three cubits, each one of which is marked by an A. There are another three magnitudes all equal and designated as B, as there might be one moving unit of three cubits. These magnitudes begin to move from the middle of the space. There are also three other magnitudes, equal in number, size and velocity to B, and designated as 0. These begin to move from the last space, that is from the last A.
This being given, it occurs that the first R by its motion arrives at the last A and likewise the first C by its motion arrives at the first A, opposite the last. When this has been done, it is evident that this first C has passed all the A’s, but B has passed by only half. Since, therefore, R and C are equal in velocity, and an equal velocity passes by a smaller magnitude in less time, it follows that the time in which B travels to the last A is half the time in which C arrives at the first A opposite; in equal times C and B pass each section of A.
This being supposed, namely, that the time in which B arrives at the last A is half the time in which 0 arrives at the first A opposite, it must be further considered how Zeno wished to conclude that this half time is equal to its double, For from the supposition that the time of the motion of C is double the time of the motion of B, it is supposed that, in the first half of the time, B was still and C moved, and thus C in that half of the time arrived at the middle of the space, where B was; and then B began to move to one part and C to another. When Rarrived at the last A it had to pass all the C’s, because at the same time the first B and the first C are at contrary ultimates; namely, one at the first A and the other at the last, and as he said, C is next to each B, in the same amount of time as it takes to pass each one of the A’s. This is so, because both, namely Rand C, pass one A in the same interval of time. Thus it seems that if B covers a time equal to that in which it passes C, that C, in an equal interval of time, passes B and A. Therefore, the interval in which C passes all B’s is equal to the time in which it passed all the A’s. The time in which C passed all the B’s is equal to the time in which C or B passed the middle of the A’s, as was said. But it was proved that the time in which B passed the middle of the A’s is half the time in which C reached all the A’s. Therefore, it follows that the half is equal to the double, which is impossible.
This is the argument of Zeno. But he falls into the aforesaid error; namely, he assumes the fact that C in the same interval of time crosses B moving in a counter direction and A quiescent, which is false, as was said above.
[In this description of St. Thomas, since 0 is supposed to pass all the A’s, he is careful to have Cc first begin from the last A and come to the middle, while meanwhile B is motionless. This is not in the text of Aristotle.]
870. Then at (667) he gives the argument by which Zeno rejected change between contradictories. For he said: Whatever is being changed is in neither of the extremities while it is being changed, because while it is in the terminus a quo it is not yet being changed, and while it is in the terminus ad quem it has already been changed. Therefore, if something is being changed from one contradictory to another, as from non-white to white, it follows that while it is being changed, it is neither white nor black—which is impossible.
Now although this strange conclusion would follow for those who posit that an indivisible can be moved, yet for us who posit that whatever is being moved is divisible, nothing impossible follows. For even though it is not totally in one or other of the extremes, it is not for that reason neither white nor non-white, For one part could be white and the other non-white. For a thing is called white not only when all of it is white but also when very many or its main parts are white, i,e., the parts that are expected to receive whiteness, because it is one thing not to be something at all and another not to be entirely something, for example, white or non-white.
And what has been said of white and non-white is to be understood of unqualified being or non-being and of all things that are contradictorily opposed, as hot and non-hot and so on. For what is being changed must always be in one of the opposites, because it is described in terms of whichever opposite predominates in it, But it does not follow that it is always as a whole in neither of the extremities, as Zeno supposed.
Now it should be known that this answer is sufficient to refute Zeno’s argument and that is what Aristotle’s main intention is. But the truth of this matter will be more fully given in Book VIII. For it is not true in all cases that part is altered or generated after part, but sometimes the whole comes all at once, as was said above. In that case it is not this answer but the one in Book VIII that would apply.
871. Then at (668) he refutes the argument by which Zeno rejected spherical motion. For he said that it is not possible for anything to be moved circularly or spherically or in such a way that the motion is confined within the space occupied by the mobile. And he proved this with the following argument: Anything that is in its entirety and in respect of its parts in one and the same place for a period of time is not in motion but at rest. But all the above-mentioned fulfill these conditions, even when they are apparently in motion. Therefore, they are at once in motion and at rest—which is impossible.
The Philosopher attacks this argument on two points. First, as to the statement that the parts of the moving sphere are in the same place for some time. For Zeno was speaking of the place of the whole, and it is true that while the sphere is in motion, no part passes out of the place of the sphere, but Aristotle speaks of the particular place of each part, according as a part has a place. For it was said in Book IV that the parts of a continuum are in place potentially. But it is evident in spherical motion that a part does change its particular place, although it does not lose the place of the whole, because where one part was, another part succeeds.
Secondly, he attacks the statement that the whole remains in the same place for some time. Against this Aristotle says that even the whole is changing its place. For in order that two places be not the same, it is not required that one of them be entirely outside the other, but sometimes the second place is partly joined to part of the first and partly divided from it, as is clear in things moved in a straight line. For let a body of one cubit be moved from place AB to place BC—both places being one cubit each. While the mobile is being moved from one place to the other, it must partly desert one place and enter the other; for example, it could leave the portion AD of AB and enter the portion BE of BC. Therefore, it is clear that the place DE is distinct from AB, although not entirely, but only partly separated from it.
But if it were assumed that that part of the mobile which entered the second place re-entered part of the place deserted, there would be two places, yet in no way separated—they would differ not really but only in conception, i.e., in the sense that the beginning of the place might be successively called by different letters each time the mobile re-entered it, namely, where the beginning of the mobile is, i.e., some spot in the mobile which is taken as a beginning, Thus there would be two places conceptually but one and the same in reality.
This is how we must understand whatAristotle says here, namely, that it is not the same revolution, when it is taken as beginning at A and as beginning at B and as beginning at C or any other mark, unless you insist that it is the same revolution as to subject, as in the case of “musical man” and “man”, since one happens to the other. Hence it is clear that the mobile is always being moved from one circular place to another and is not at rest as Zeno tried to prove. And it is the same with the sphere and everything else whose motion is confined within the space it occupies, as in the case of a potter’s wheel and a (rotating) pillar or anything of that sort.

Lecture 12
What is indivisible according to quantity is moved only per accidens
872. After answering the arguments of Zeno who tried to disprove motion, the Philosopher now intends to show that a thing incapable of being divided into parts cannot be moved. This will answer the opinion of Democritus, who posited atoms that are per se mobile. About this he does two things:
First he proposes his intention;
Secondly, he proves his proposition, at 876.
He says therefore first (669) that assuming what we have proved above, it must be said that a thing incapable of being divided into parts cannot be moved, except perchance per accidens, as a point is moved in a whole body in which there is a point, for example, in a line or a surface.
873. To be in motion as a result of something else being in motion can occur in two ways. In one way, when what is moved as the result of something else being moved is not part of the latter, as what is on a ship is being moved when the ship is being moved, and as whiteness is moved with the motion of body, since it is not of the body. In a second way, as a part is moved when the whole is moved.
And because “what is incapable of being divided into parts” has many senses, just as what is capable of being divided into parts” has, he shows how he uses the phrase here and says that here it means what is indivisible in respect of quantity. For some things are indivisible according to species, as when we say that fire or air are indivisible, because they cannot be further resolved into several bodies that differ in species. But in regard to such an indivisible there is nothing to prevent it from being moved. Consequently, Aristotle intends to exclude motion from what is indivisible according to quantity.
874. Because he had said that the part is being moved when the whole is, and someone might say that the part is not moved at all, he adds that there are some motions of parts precisely as parts, that are diverse from the motion of the whole, as a motion of the whole.
This difference is particularly clear in the motion of a sphere, because the speed of the parts near the center is not the same as that of those outside, i.e,, on the exterior surface of the sphere, the speed of whose parts is considered to be the speed of the whole. It is as if there is not just one motion but the motions of many parts involved. For it is evident that whatever traverses a larger magnitude in an equal time is faster. Now, while the sphere is rotating, it is clear that an external part describes a larger circle than an interior part; hence the velocity of the external part is greater than that of an interior part. Yet the velocity of the whole sphere is the same as the velocity of the interior and exterior part.
But this diversity of motions is to be understood in the sense in which motion is ascribed to parts of a continuum, i.e., in a potential sense, Hence, actually there is one motion of the whole and of the parts, but potentially there are diverse motions: those of the parts being different from one another and from the motion of the whole. And so, when it is said that a part is being moved per accidens with the motion of the whole, it is a per accidens which is in potency per se —which is something not true of motion per accidens, when it is taken in the sense that accidents or forms are said to be moved per accidens.
875. Having made a distinction among things that are moved, he explains his intention. And he says that what is indivisible in respect of quantity can indeed be moved per accidens when something else is moved, but it is not moved as a part, for no magnitude is made up of indivisibles, as we have proved. Now, something not a part of another is moved along with the other in the same way that one sitting in a ship is moved along with the motion of a ship. But per se the indivisible cannot be moved.
He had proved this point previously, not as a main proposition but incidentally. Hence, in addition to the reason cited earlier, he now gives a further explanation of the truth and adds reasons that are strong enough to prove the proposition.
876. Then at (670) he proves his point with three arguments. The first of which is this: If it is insisted that an indivisible can be moved, let it be moved from AB into BC. (In this argument it makes no difference whether AB and BC are two magnitudes or two places, as in local motion and growth and decrease, or whether they are two qualities, as in the motion of alteration, or two things that are contradictorily opposed, as in generation and ceasing-to-be.) Let ED be the time in which something is changed from one terminus to the other first, i.e., not by reason of a part. In this time, then, it is necessary that what is being moved be either in AB, i.e., in the terminus a quo,or in BC, i.e., in the terminus ad quem; or else a part is in one terminus and a part in the other. For anything being moved must be in one of these three ways, as was said above.
Now the third situation is impossible; namely, that it be in each term according to its various parts, because then, the mobile would be divided into parts, and we have assumed that it is an indivisible mobile. Likewise, it cannot be the second alternative, i.e., that it be in BC, i.e., in the terminus ad quem, for when it is in the terminus ad quem, it has been already changed (as is clear from what we have said above), whereas we are assuming that it is being changed. What remains, therefore, is that in the entire time that the indivisible is being changed it remains at AB, i.e., in the terminus a quo, From which it follows that it is at rest, for resting is nothing more than to be in one and the same state throughout a definite period of time. For since there is a prior and a subsequent in time, if time is divisible, whatever for a period of time is in one and the same state keeps itself the same; namely, as it was previously—which is to rest. But it is impossible that a thing is at rest while it is being changed. Therefore, it cannot be that an indivisible is moved or changed in any way whatsoever.
The only way in which there could be motion of an indivisible thing is to have the time composed of “now’s”, because in the “now” there is always a condition called “having been moved” or “having been changed”.
And because what has been moved, precisely as such is not now being moved, it follows that in the “now” nothing is being moved, but has been moved. But if time were made up of “now’s”, there would be a way in which motion could be nosited in an indivisible, because it could be granted that in each of those “now’s” of which time is composed, it would be in one, and in the whole time, i.e., in all the “now’s”, it would be in many. And thus it would be in motion throughout the entire time, but not in one “now”.
But it has been proved above that it is impossible for time to be made up of “now’s”. Indeed, we have proved that neither is time composed of now’s nor a line of points, nor a motion of moments (where “moments” refers to states called “having been changed”). For anyone who says that an indivisible is being moved or that motion is composed of indivisibles is making time be composed of “now’s” or a magnitude of points—which is impossible. Therefore, it is also impossible that a thing incapable of being divided into parts be moved.
877. The second argument is given at (671). He says that if we look at the consequences, it is clear that neither a point nor any indivisible can be moved. And this special argument applies to local motion. For whatever is being moved according to place cannot traverse a distance greater than the mobile itself before traversing one that is equal to or less than it; rather, a mobile always traverses a magnitude equal to itself or less than itself before one greater than itself. If this is so, then it is clear that a point, if it is being moved will first traverse a length less than or equal to itself, before it traverses one greater than itself. But it is impossible for it to traverse something less than itself, since it is indivisible. So it has to traverse a length equal to itself. Consequently, it must number all the points in the line; for the point, since it is being moved through a motion equal to a line, is by that very fact being moved through the whole line, and, consequently, is always measuring the whole line—and this it does by counting all the points. Therefore, it follows that a line arises from points. Therefore, if this is impossible, it is impossible for an indivisible to be moved.
878. The third argument is at (672) and is this: Since motion is always in a period of time and never in a “now”, and since all time is divisible, as was shown above, then in every time in which something is. moved, there must be a lesser time in which a lesser mobile is moved. For, supposing the same speed, it is plain that in a lesser time the lesser mobile crosses a given mark than does a greater mobile, as in a lesser time the part than the whole, as is evident from what is above. If, therefore, a point is in motion, there must be a time less than that in which it is moved. But this is impossible, for it would follow that in that lesser time something less than a point would be moved, and thus the indivisible would be divisible into something less, just as time is divisible. This would be the only condition under which the indivisible could be in motion, namely, if it were possible for something to be moved in an indivisible “now”, for just as there is nothing smaller than the “now” in time, so one cannot take a smaller mobile.
And so it is evident that in the two questions—that of motion in a “now” and that of an indivisible being moved—the same principle is involved. But it is impossible for motion to occur in a “now”. Therefore, it is impossible for an indivisible to be moved.

Lecture 13
By nature, no change is infinite. How motion may be infinite in time
878. After showing that things which cannot be divided into parts are not moved, the Philosopher now intends to show that no change is infinite, This is against Heraclitus, who supposed that things are always in motion. About this he does two things:
First he shows that no change is infinite according to its own species;
Secondly, how there can be infinites in time, at 883,
About the first he does two things:
First he shows for all changes except local motion that no change is infinite according to its species;
Secondly, he shows the same thing for local motion, at 881.
880. The first reason is this: Every change is from something to something. Indeed, in some changes, namely, those which occur between contradictories, as do generation and ceasing-to-be, or between contraries, as do alteration and growing and decreasing, it is evident that they have pre-defined termini. Hence in changes that occur between contradictory termini, the terminus is either affirmation or negation, as the terminus of generation is a being, and that of ceasing-to-be, non-being.
Likewise, in regard to changes that are between contraries, the contraries are termini at which, as at ultimate goals, changes of this kind are terminated. Hence it follows, since every alteration is from contrary to contrary, that every alteration has some terminus.
The same must be said for growth and decrease, for the terminus of growth is perfect magnitude (and I say “perfect” in respect of the nature, for a different perfection of magnitude befits man from the one that befits a horse), and the terminus of decrease is the one that happens to a definite nature to be most removed from perfect magnitude.
Consequently, it is evident that each of the above-mentioned changes has a goal at which it is terminated. But such a situation precludes the infinite. Therefore, none of these changes can be infinite.
881. Then at (674) he proceeds to local motion. And first he shows that the argument in regard to local motion is not the same as for the other changes. For it cannot be proved that local motion is finite (as we have proved other motions are finite), because it is terminated at something contrary or contradictory, for not every local motion is between strict contraries, where contraries refer to things most distant.
There is a maximum distance in the strict sense in the natural motions of heavy and light bodies, for the place of fire is at a maximum distance from the center of the earth, in accordance with the distance that nature determines for such bodies. Hence, such changes are between strict contraries, Hence, it can be proved of such changes that they are not infinite any more than the others were.
But maximum distance in compulsory or voluntary motions does not depend strictly on certain definite termini but on the intention or energy of the one causing the motion, who either does not desire to or cannot physically move something any farther. Hence, it is only in a qualified sense., ere is maximum distance and a consequent contrariety. Hence, if you stick with the termini, it cannot be proved that no local motion is infinite.
882. Consequently, this must be proved by another argument, which is this: What is impossible to exist divided cannot be divided, And because things are said to be impossible in many senses, name, what never can occur or what cannot occur except with great difficulty, he therefore explains his meaning of “impossible” here. And he means it in the sense of that which cannot happen at all. For the same reason, what is impossible to have been made, is impossible to make; for example, if it is impossible that contradictories be together, it is impossible that this be brought about. For the same reason, what is impossible to have been changed into something cannot be changed into it, because nothing tends toward the impossible. But everything that is being changed according to place is being changed into something. Therefore, it is possible through that motion to arrive at something. But the infinite cannot be gone through. Therefore, nothing is moved through the infinite. Thus, therefore, no local motion is infinite.
And so it is universally evident that no change can be infinite in such a way that it be not terminated by definite termini from which it derives its species.
883. Then at (675) he shows how motion can be infinite in time. And he says that we must consider whether motion can be infinite in time in such a way that it remains numerically one and the same motion. For there is nothing to prevent motion from enduring through infinite time as long as it is not one and the same motion. But he leaves that matter in doubt when he adds “perhaps”, because he will settle the matter later. And he gives an example: Let us say that after a local motion there is an alteration, and after that a growing, and after that generation, and so on ad infinitum. In this way motion could always endure throughout infinite time. And it would not be one and the same numerical motion, because a series of such motions as are given in the example do not form one numerical motion, as we have proved in Book V. But that motion endure throughout infinite time in such a way that it remain one numerical motion can occur in only one species of motion, for a circular motion can endure as one and continuous throughout infinite time, as will be proved in Book VIII.

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