which the magnitude has changed, and something else again prior to
that, and so on to infinity, because the process of division may be
continued without end. Thus there can be no primary 'where' to which a
thing has changed. And if we take the case of quantitative change,
we shall get a like result, for here too the change is in something
continuous. It is evident, then, that only in qualitative motion can
there be anything essentially indivisible.
6
Now everything that changes changes time, and that in two senses:
for the time in which a thing is said to change may be the primary
time, or on the other hand it may have an extended reference, as
e.g. when we say that a thing changes in a particular year because
it changes in a particular day. That being so, that which changes must
be changing in any part of the primary time in which it changes.
This is clear from our definition of 'primary', in which the word is
said to express just this: it may also, however, be made evident by
the following argument. Let ChRh be the primary time in which that
which is in motion is in motion: and (as all time is divisible) let it
be divided at K. Now in the time ChK it either is in motion or is
not in motion, and the same is likewise true of the time KRh. Then
if it is in motion in neither of the two parts, it will be at rest
in the whole: for it is impossible that it should be in motion in a
time in no part of which it is in motion. If on the other hand it is
in motion in only one of the two parts of the time, ChRh cannot be the
primary time in which it is in motion: for its motion will have
reference to a time other than ChRh. It must, then, have been in
motion in any part of ChRh.
And now that this has been proved, it is evident that everything
that is in motion must have been in motion before. For if that which
is in motion has traversed the distance KL in the primary time ChRh,
in half the time a thing that is in motion with equal velocity and
began its motion at the same time will have traversed half the
distance. But if this second thing whose velocity is equal has
traversed a certain distance in a certain time, the original thing
that is in motion must have traversed the same distance in the same
time. Hence that which is in motion must have been in motion before.
Again, if by taking the extreme moment of the time-for it is the
moment that defines the time, and time is that which is intermediate
between moments-we are enabled to say that motion has taken place in
the whole time ChRh or in fact in any period of it, motion may
likewise be said to have taken place in every other such period. But
half the time finds an extreme in the point of division. Therefore
motion will have taken place in half the time and in fact in any
part of it: for as soon as any division is made there is always a time
defined by moments. If, then, all time is divisible, and that which is
intermediate between moments is time, everything that is changing must
have completed an infinite number of changes.
Again, since a thing that changes continuously and has not
perished or ceased from its change must either be changing or have
changed in any part of the time of its change, and since it cannot
be changing in a moment, it follows that it must have changed at every
moment in the time: consequently, since the moments are infinite in
number, everything that is changing must have completed an infinite
number of changes.
And not only must that which is changing have changed, but that
which has changed must also previously have been changing, since
everything that has changed from something to something has changed in
a period of time. For suppose that a thing has changed from A to B
in a moment. Now the moment in which it has changed cannot be the same
as that in which it is at A (since in that case it would be in A and B
at once): for we have shown above that that that which has changed,
when it has changed, is not in that from which it has changed. If,
on the other hand, it is a different moment, there will be a period of
time intermediate between the two: for, as we saw, moments are not
consecutive. Since, then, it has changed in a period of time, and
all time is divisible, in half the time it will have completed another
change, in a quarter another, and so on to infinity: consequently when
it has changed, it must have previously been changing.
Moreover, the truth of what has been said is more evident in the
case of magnitude, because the magnitude over which what is changing
changes is continuous. For suppose that a thing has changed from G
to D. Then if GD is indivisible, two things without parts will be
consecutive. But since this is impossible, that which is
intermediate between them must be a magnitude and divisible into an
infinite number of segments: consequently, before the change is
completed, the thing changes to those segments. Everything that has
changed, therefore, must previously have been changing: for the same
proof also holds good of change with respect to what is not
continuous, changes, that is to say, between contraries and between
contradictories. In such cases we have only to take the time in
which a thing has changed and again apply the same reasoning. So
that which has changed must have been changing and that which is
changing must have changed, and a process of change is preceded by a
completion of change and a completion by a process: and we can never
take any stage and say that it is absolutely the first. The reason
of this is that no two things without parts can be contiguous, and
therefore in change the process of division is infinite, just as lines
may be infinitely divided so that one part is continually increasing
and the other continually decreasing.
So it is evident also that that that which has become must
previously have been in process of becoming, and that which is in
process of becoming must previously have become, everything (that
is) that is divisible and continuous: though it is not always the
actual thing that is in process of becoming of which this is true:
sometimes it is something else, that is to say, some part of the thing
in question, e.g. the foundation-stone of a house. So, too, in the
case of that which is perishing and that which has perished: for
that which becomes and that which perishes must contain an element
of infiniteness as an immediate consequence of the fact that they
are continuous things: and so a thing cannot be in process of becoming
without having become or have become without having been in process of
becoming. So, too, in the case of perishing and having perished:
perishing must be preceded by having perished, and having perished
must be preceded by perishing. It is evident, then, that that which
has become must previously have been in process of becoming, and
that which is in process of becoming must previously have become:
for all magnitudes and all periods of time are infinitely divisible.
&
nbsp; Consequently no absolutely first stage of change can be
represented by any particular part of space or time which the changing
thing may occupy.
7
Now since the motion of everything that is in motion occupies a
period of time, and a greater magnitude is traversed in a longer time,
it is impossible that a thing should undergo a finite motion in an
infinite time, if this is understood to mean not that the same
motion or a part of it is continually repeated, but that the whole
infinite time is occupied by the whole finite motion. In all cases
where a thing is in motion with uniform velocity it is clear that
the finite magnitude is traversed in a finite time. For if we take a
part of the motion which shall be a measure of the whole, the whole
motion is completed in as many equal periods of the time as there
are parts of the motion. Consequently, since these parts are finite,
both in size individually and in number collectively, the whole time
must also be finite: for it will be a multiple of the portion, equal
to the time occupied in completing the aforesaid part multiplied by
the number of the parts.
But it makes no difference even if the velocity is not uniform.
For let us suppose that the line AB represents a finite stretch over
which a thing has been moved in the given time, and let GD be the
infinite time. Now if one part of the stretch must have been traversed
before another part (this is clear, that in the earlier and in the
later part of the time a different part of the stretch has been
traversed: for as the time lengthens a different part of the motion
will always be completed in it, whether the thing in motion changes
with uniform velocity or not: and whether the rate of motion increases
or diminishes or remains stationary this is none the less so), let
us then take AE a part of the whole stretch of motion AB which shall
be a measure of AB. Now this part of the motion occupies a certain
period of the infinite time: it cannot itself occupy an infinite time,
for we are assuming that that is occupied by the whole AB. And if
again I take another part equal to AE, that also must occupy a
finite time in consequence of the same assumption. And if I go on
taking parts in this way, on the one hand there is no part which
will be a measure of the infinite time (for the infinite cannot be
composed of finite parts whether equal or unequal, because there
must be some unity which will be a measure of things finite in
multitude or in magnitude, which, whether they are equal or unequal,
are none the less limited in magnitude); while on the other hand the
finite stretch of motion AB is a certain multiple of AE:
consequently the motion AB must be accomplished in a finite time.
Moreover it is the same with coming to rest as with motion. And so
it is impossible for one and the same thing to be infinitely in
process of becoming or of perishing. The reasoning he will prove
that in a finite time there cannot be an infinite extent of motion
or of coming to rest, whether the motion is regular or irregular.
For if we take a part which shall be a measure of the whole time, in
this part a certain fraction, not the whole, of the magnitude will
be traversed, because we assume that the traversing of the whole
occupies all the time. Again, in another equal part of the time
another part of the magnitude will be traversed: and similarly in each
part of the time that we take, whether equal or unequal to the part
originally taken. It makes no difference whether the parts are equal
or not, if only each is finite: for it is clear that while the time is
exhausted by the subtraction of its parts, the infinite magnitude will
not be thus exhausted, since the process of subtraction is finite both
in respect of the quantity subtracted and of the number of times a
subtraction is made. Consequently the infinite magnitude will not be
traversed in finite time: and it makes no difference whether the
magnitude is infinite in only one direction or in both: for the same
reasoning will hold good.
This having been proved, it is evident that neither can a finite
magnitude traverse an infinite magnitude in a finite time, the
reason being the same as that given above: in part of the time it will
traverse a finite magnitude and in each several part likewise, so that
in the whole time it will traverse a finite magnitude.
And since a finite magnitude will not traverse an infinite in a
finite time, it is clear that neither will an infinite traverse a
finite in a finite time. For if the infinite could traverse the
finite, the finite could traverse the infinite; for it makes no
difference which of the two is the thing in motion; either case
involves the traversing of the infinite by the finite. For when the
infinite magnitude A is in motion a part of it, say GD, will occupy
the finite and then another, and then another, and so on to
infinity. Thus the two results will coincide: the infinite will have
completed a motion over the finite and the finite will have
traversed the infinite: for it would seem to be impossible for the
motion of the infinite over the finite to occur in any way other
it or by measuring it. Therefore, since this is impossible, the
infinite cannot traverse the finite.
Nor again will the infinite traverse the infinite in a finite
time. Otherwise it would also traverse the finite, for the infinite
includes the finite. We can further prove this in the same way by
taking the time as our starting-point.
Since, then, it is established that in a finite time neither will
the finite traverse the infinite, nor the infinite the finite, nor the
infinite the infinite, it is evident also that in a finite time
there cannot be infinite motion: for what difference does it make
whether we take the motion or the magnitude to be infinite? If
either of the two is infinite, the other must be so likewise: for
all locomotion is in space.
8
Since everything to which motion or rest is natural is in motion
or at rest in the natural time, place, and manner, that which is
coming to a stand, when it is coming to a stand, must be in motion:
for if it is not in motion it must be at rest: but that which is at
rest cannot be coming to rest. From this it evidently follows that
coming to a stand must occupy a period of time: for the motion of that
which is in motion occupies a period of time, and that which is coming
to a stand has been shown to be in motion: consequently coming to a
stand must occupy a period of time.
Again, since the terms 'quicker' and 'slower' are used only of
that which occupies a period of time, and the process of coming to a
stand may be quicker or slower, the same conclusion follows.
And that which is coming to a stand must be coming to a stand in any
part of the primary time in which it is coming to a stand. For if it
is coming to a stand in neither of two parts into whic
h the time may
be divided, it cannot be coming to a stand in the whole time, with the
result that that that which is coming to a stand will not be coming to
a stand. If on the other hand it is coming to a stand in only one of
the two parts of the time, the whole cannot be the primary time in
which it is coming to a stand: for it is coming to a stand in the
whole time not primarily but in virtue of something distinct from
itself, the argument being the same as that which we used above
about things in motion.
And just as there is no primary time in which that which is in
motion is in motion, so too there is no primary time in which that
which is coming to a stand is coming to a stand, there being no
primary stage either of being in motion or of coming to a stand. For
let AB be the primary time in which a thing is coming to a stand.
Now AB cannot be without parts: for there cannot be motion in that
which is without parts, because the moving thing would necessarily
have been already moved for part of the time of its movement: and that
which is coming to a stand has been shown to be in motion. But since
AB is therefore divisible, the thing is coming to a stand in every one
of the parts of AB: for we have shown above that it is coming to a
stand in every one of the parts in which it is primarily coming to a
stand. Since then, that in which primarily a thing is coming to a
stand must be a period of time and not something indivisible, and
since all time is infinitely divisible, there cannot be anything in
which primarily it is coming to a stand.
Nor again can there be a primary time at which the being at rest
of that which is at rest occurred: for it cannot have occurred in that
which has no parts, because there cannot be motion in that which is
indivisible, and that in which rest takes place is the same as that in
which motion takes place: for we defined a state of rest to be the
state of a thing to which motion is natural but which is not in motion
when (that is to say in that in which) motion would be natural to