Various Works

  the past in the future and part of the future in the past: for past

  time will be marked off from future time at the actual point of

  division. Also the present will be a present not in the proper sense

  but in virtue of something else: for the division which yields it will

  not be a division proper. Furthermore, there will be a part of the

  present that is past and a part that is future, and it will not always

  be the same part that is past or future: in fact one and the same

  present will not be simultaneous: for the time may be divided at

  many points. If, therefore, the present cannot possibly have these

  characteristics, it follows that it must be the same present that

  belongs to each of the two times. But if this is so it is evident that

  the present is also indivisible: for if it is divisible it will be

  involved in the same implications as before. It is clear, then, from

  what has been said that time contains something indivisible, and

  this is what we call a present.

  We will now show that nothing can be in motion in a present. For

  if this is possible, there can be both quicker and slower motion in

  the present. Suppose then that in the present N the quicker has

  traversed the distance AB. That being so, the slower will in the

  same present traverse a distance less than AB, say AG. But since the

  slower will have occupied the whole present in traversing AG, the

  quicker will occupy less than this in traversing it. Thus we shall

  have a division of the present, whereas we found it to be indivisible.

  It is impossible, therefore, for anything to be in motion in a

  present.

  Nor can anything be at rest in a present: for, as we were saying,

  only can be at rest which is naturally designed to be in motion but is

  not in motion when, where, or as it would naturally be so: since,

  therefore, nothing is naturally designed to be in motion in a present,

  it is clear that nothing can be at rest in a present either.

  Moreover, inasmuch as it is the same present that belongs to both

  the times, and it is possible for a thing to be in motion throughout

  one time and to be at rest throughout the other, and that which is

  in motion or at rest for the whole of a time will be in motion or at

  rest as the case may be in any part of it in which it is naturally

  designed to be in motion or at rest: this being so, the assumption

  that there can be motion or rest in a present will carry with it the

  implication that the same thing can at the same time be at rest and in

  motion: for both the times have the same extremity, viz. the present.

  Again, when we say that a thing is at rest, we imply that its

  condition in whole and in part is at the time of speaking uniform with

  what it was previously: but the present contains no 'previously':

  consequently, there can be no rest in it.

  It follows then that the motion of that which is in motion and the

  rest of that which is at rest must occupy time.

  4

  Further, everything that changes must be divisible. For since

  every change is from something to something, and when a thing is at

  the goal of its change it is no longer changing, and when both it

  itself and all its parts are at the starting-point of its change it is

  not changing (for that which is in whole and in part in an unvarying

  condition is not in a state of change); it follows, therefore, that

  part of that which is changing must be at the starting-point and

  part at the goal: for as a whole it cannot be in both or in neither.

  (Here by 'goal of change' I mean that which comes first in the process

  of change: e.g. in a process of change from white the goal in question

  will be grey, not black: for it is not necessary that that that

  which is changing should be at either of the extremes.) It is evident,

  therefore, that everything that changes must be divisible.

  Now motion is divisible in two senses. In the first place it is

  divisible in virtue of the time that it occupies. In the second

  place it is divisible according to the motions of the several parts of

  that which is in motion: e.g. if the whole AG is in motion, there will

  be a motion of AB and a motion of BG. That being so, let DE be the

  motion of the part AB and EZ the motion of the part BG. Then the whole

  DZ must be the motion of AG: for DZ must constitute the motion of AG

  inasmuch as DE and EZ severally constitute the motions of each of

  its parts. But the motion of a thing can never be constituted by the

  motion of something else: consequently the whole motion is the

  motion of the whole magnitude.

  Again, since every motion is a motion of something, and the whole

  motion DZ is not the motion of either of the parts (for each of the

  parts DE, EZ is the motion of one of the parts AB, BG) or of

  anything else (for, the whole motion being the motion of a whole,

  the parts of the motion are the motions of the parts of that whole:

  and the parts of DZ are the motions of AB, BG and of nothing else:

  for, as we saw, a motion that is one cannot be the motion of more

  things than one): since this is so, the whole motion will be the

  motion of the magnitude ABG.

  Again, if there is a motion of the whole other than DZ, say the

  the of each of the arts may be subtracted from it: and these motions

  will be equal to DE, EZ respectively: for the motion of that which

  is one must be one. So if the whole motion OI may be divided into

  the motions of the parts, OI will be equal to DZ: if on the other hand

  there is any remainder, say KI, this will be a motion of nothing:

  for it can be the motion neither of the whole nor of the parts (as the

  motion of that which is one must be one) nor of anything else: for a

  motion that is continuous must be the motion of things that are

  continuous. And the same result follows if the division of OI

  reveals a surplus on the side of the motions of the parts.

  Consequently, if this is impossible, the whole motion must be the same

  as and equal to DZ.

  This then is what is meant by the division of motion according to

  the motions of the parts: and it must be applicable to everything that

  is divisible into parts.

  Motion is also susceptible of another kind of division, that

  according to time. For since all motion is in time and all time is

  divisible, and in less time the motion is less, it follows that

  every motion must be divisible according to time. And since everything

  that is in motion is in motion in a certain sphere and for a certain

  time and has a motion belonging to it, it follows that the time, the

  motion, the being-in-motion, the thing that is in motion, and the

  sphere of the motion must all be susceptible of the same divisions

  (though spheres of motion are not all divisible in a like manner: thus

  quantity is essentially, quality accidentally divisible). For

  suppose that A is the time occupied by the motion B. Then if all the

  time has been occupied by the whole motion, it will take less of the

  motion to occupy half the time, less again to occupy a further

  subdivision of the time, and so on to infinity. Ag
ain, the time will

  be divisible similarly to the motion: for if the whole motion occupies

  all the time half the motion will occupy half the time, and less of

  the motion again will occupy less of the time.

  In the same way the being-in-motion will also be divisible. For

  let G be the whole being-in-motion. Then the being-in-motion that

  corresponds to half the motion will be less than the whole

  being-in-motion, that which corresponds to a quarter of the motion

  will be less again, and so on to infinity. Moreover by setting out

  successively the being-in-motion corresponding to each of the two

  motions DG (say) and GE, we may argue that the whole being-in-motion

  will correspond to the whole motion (for if it were some other

  being-in-motion that corresponded to the whole motion, there would

  be more than one being-in motion corresponding to the same motion),

  the argument being the same as that whereby we showed that the

  motion of a thing is divisible into the motions of the parts of the

  thing: for if we take separately the being-in motion corresponding

  to each of the two motions, we shall see that the whole being-in

  motion is continuous.

  The same reasoning will show the divisibility of the length, and

  in fact of everything that forms a sphere of change (though some of

  these are only accidentally divisible because that which changes is

  so): for the division of one term will involve the division of all.

  So, too, in the matter of their being finite or infinite, they will

  all alike be either the one or the other. And we now see that in

  most cases the fact that all the terms are divisible or infinite is

  a direct consequence of the fact that the thing that changes is

  divisible or infinite: for the attributes 'divisible' and 'infinite'

  belong in the first instance to the thing that changes. That

  divisibility does so we have already shown: that infinity does so will

  be made clear in what follows?

  5

  Since everything that changes changes from something to something,

  that which has changed must at the moment when it has first changed be

  in that to which it has changed. For that which changes retires from

  or leaves that from which it changes: and leaving, if not identical

  with changing, is at any rate a consequence of it. And if leaving is a

  consequence of changing, having left is a consequence of having

  changed: for there is a like relation between the two in each case.

  One kind of change, then, being change in a relation of

  contradiction, where a thing has changed from not-being to being it

  has left not-being. Therefore it will be in being: for everything must

  either be or not be. It is evident, then, that in contradictory change

  that which has changed must be in that to which it has changed. And if

  this is true in this kind of change, it will be true in all other

  kinds as well: for in this matter what holds good in the case of one

  will hold good likewise in the case of the rest.

  Moreover, if we take each kind of change separately, the truth of

  our conclusion will be equally evident, on the ground that that that

  which has changed must be somewhere or in something. For, since it has

  left that from which it has changed and must be somewhere, it must

  be either in that to which it has changed or in something else. If,

  then, that which has changed to B is in something other than B, say G,

  it must again be changing from G to B: for it cannot be assumed that

  there is no interval between G and B, since change is continuous. Thus

  we have the result that the thing that has changed, at the moment when

  it has changed, is changing to that to which it has changed, which

  is impossible: that which has changed, therefore, must be in that to

  which it has changed. So it is evident likewise that that that which

  has come to be, at the moment when it has come to be, will be, and

  that which has ceased to be will not-be: for what we have said applies

  universally to every kind of change, and its truth is most obvious

  in the case of contradictory change. It is clear, then, that that

  which has changed, at the moment when it has first changed, is in that

  to which it has changed.

  We will now show that the 'primary when' in which that which has

  changed effected the completion of its change must be indivisible,

  where by 'primary' I mean possessing the characteristics in question

  of itself and not in virtue of the possession of them by something

  else belonging to it. For let AG be divisible, and let it be divided

  at B. If then the completion of change has been effected in AB or

  again in BG, AG cannot be the primary thing in which the completion of

  change has been effected. If, on the other hand, it has been

  changing in both AB and BG (for it must either have changed or be

  changing in each of them), it must have been changing in the whole AG:

  but our assumption was that AG contains only the completion of the

  change. It is equally impossible to suppose that one part of AG

  contains the process and the other the completion of the change: for

  then we shall have something prior to what is primary. So that in

  which the completion of change has been effected must be

  indivisible. It is also evident, therefore, that that that in which

  that which has ceased to be has ceased to be and that in which that

  which has come to be has come to be are indivisible.

  But there are two senses of the expression 'the primary when in

  which something has changed'. On the one hand it may mean the

  primary when containing the completion of the process of change- the

  moment when it is correct to say 'it has changed': on the other hand

  it may mean the primary when containing the beginning of the process

  of change. Now the primary when that has reference to the end of the

  change is something really existent: for a change may really be

  completed, and there is such a thing as an end of change, which we

  have in fact shown to be indivisible because it is a limit. But that

  which has reference to the beginning is not existent at all: for there

  is no such thing as a beginning of a process of change, and the time

  occupied by the change does not contain any primary when in which

  the change began. For suppose that AD is such a primary when. Then

  it cannot be indivisible: for, if it were, the moment immediately

  preceding the change and the moment in which the change begins would

  be consecutive (and moments cannot be consecutive). Again, if the

  changing thing is at rest in the whole preceding time GA (for we may

  suppose that it is at rest), it is at rest in A also: so if AD is

  without parts, it will simultaneously be at rest and have changed: for

  it is at rest in A and has changed in D. Since then AD is not

  without parts, it must be divisible, and the changing thing must

  have changed in every part of it (for if it has changed in neither

  of the two parts into which AD is divided, it has not changed in the

  whole either: if, on the other hand, it is in process of change in

  both parts, it is likewise in process of change in the whole: a
nd

  if, again, it has changed in one of the two parts, the whole is not

  the primary when in which it has changed: it must therefore have

  changed in every part). It is evident, then, that with reference to

  the beginning of change there is no primary when in which change has

  been effected: for the divisions are infinite.

  So, too, of that which has changed there is no primary part that has

  changed. For suppose that of AE the primary part that has changed is

  AZ (everything that changes having been shown to be divisible): and

  let OI be the time in which DZ has changed. If, then, in the whole

  time DZ has changed, in half the time there will be a part that has

  changed, less than and therefore prior to DZ: and again there will

  be another part prior to this, and yet another, and so on to infinity.

  Thus of that which changes there cannot be any primary part that has

  changed. It is evident, then, from what has been said, that neither of

  that which changes nor of the time in which it changes is there any

  primary part.

  With regard, however, to the actual subject of change-that is to say

  that in respect of which a thing changes-there is a difference to be

  observed. For in a process of change we may distinguish three

  terms-that which changes, that in which it changes, and the actual

  subject of change, e.g. the man, the time, and the fair complexion. Of

  these the man and the time are divisible: but with the fair complexion

  it is otherwise (though they are all divisible accidentally, for

  that in which the fair complexion or any other quality is an

  accident is divisible). For of actual subjects of change it will be

  seen that those which are classed as essentially, not accidentally,

  divisible have no primary part. Take the case of magnitudes: let AB be

  a magnitude, and suppose that it has moved from B to a primary 'where'

  G. Then if BG is taken to be indivisible, two things without parts

  will have to be contiguous (which is impossible): if on the other hand

  it is taken to be divisible, there will be something prior to G to