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
than by the finite traversing the infinite either by locomotion over
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.