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
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