of which the particular moment is a dividing-point. To maintain that
it has come to be and ceased to be there will involve the
consequence that A in the course of its locomotion will always be
coming to a stand: for it is impossible that A should simultaneously
have come to be at B and ceased to be there, so that the two things
must have happened at different points of time, and therefore there
will be the intervening period of time: consequently A will be in a
state of rest at B, and similarly at all other points, since the
same reasoning holds good in every case. When to A, that which is in
process of locomotion, B, the middle-point, serves both as a
finishing-point and as a starting-point for its motion, A must come to
a stand at B, because it makes it two just as one might do in thought.
However, the point A is the real starting-point at which the moving
body has ceased to be, and it is at G that it has really come to be
when its course is finished and it comes to a stand. So this is how we
must meet the difficulty that then arises, which is as follows.
Suppose the line E is equal to the line Z, that A proceeds in
continuous locomotion from the extreme point of E to G, and that, at
the moment when A is at the point B, D is proceeding in uniform
locomotion and with the same velocity as A from the extremity of Z
to H: then, says the argument, D will have reached H before A has
reached G for that which makes an earlier start and departure must
make an earlier arrival: the reason, then, for the late arrival of A
is that it has not simultaneously come to be and ceased to be at B:
otherwise it will not arrive later: for this to happen it will be
necessary that it should come to a stand there. Therefore we must
not hold that there was a moment when A came to be at B and that at
the same moment D was in motion from the extremity of Z: for the
fact of A's having come to be at B will involve the fact of its also
ceasing to be there, and the two events will not be simultaneous,
whereas the truth is that A is at B at a sectional point of time and
does not occupy time there. In this case, therefore, where the
motion of a thing is continuous, it is impossible to use this form
of expression. On the other hand in the case of a thing that turns
back in its course we must do so. For suppose H in the course of its
locomotion proceeds to D and then turns back and proceeds downwards
again: then the extreme point D has served as finishing-point and as
starting-point for it, one point thus serving as two: therefore H must
have come to a stand there: it cannot have come to be at D and
departed from D simultaneously, for in that case it would
simultaneously be there and not be there at the same moment. And
here we cannot apply the argument used to solve the difficulty
stated above: we cannot argue that H is at D at a sectional point of
time and has not come to be or ceased to be there. For here the goal
that is reached is necessarily one that is actually, not
potentially, existent. Now the point in the middle is potential: but
this one is actual, and regarded from below it is a finishing-point,
while regarded from above it is a starting-point, so that it stands in
these same two respective relations to the two motions. Therefore that
which turns back in traversing a rectilinear course must in so doing
come to a stand. Consequently there cannot be a continuous rectilinear
motion that is eternal.
The same method should also be adopted in replying to those who ask,
in the terms of Zeno's argument, whether we admit that before any
distance can be traversed half the distance must be traversed, that
these half-distances are infinite in number, and that it is impossible
to traverse distances infinite in number-or some on the lines of
this same argument put the questions in another form, and would have
us grant that in the time during which a motion is in progress it
should be possible to reckon a half-motion before the whole for
every half-distance that we get, so that we have the result that
when the whole distance is traversed we have reckoned an infinite
the question of motion we put forward a solution of this difficulty
turning on the fact that the period of time occupied in traversing the
distance contains within itself an infinite number of units: there
is no absurdity, we said, in supposing the traversing of infinite
distances in infinite time, and the element of infinity is present
in the time no less than in the distance. But, although this
solution is adequate as a reply to the questioner (the question
asked being whether it is possible in a finite time to traverse or
reckon an infinite number of units), nevertheless as an account of the
fact and explanation of its true nature it is inadequate. For
suppose the distance to be left out of account and the question
asked to be no longer whether it is possible in a finite time to
traverse an infinite number of distances, and suppose that the inquiry
is made to refer to the time taken by itself (for the time contains an
infinite number of divisions): then this solution will no longer be
adequate, and we must apply the truth that we enunciated in our recent
discussion, stating it in the following way. In the act of dividing
the continuous distance into two halves one point is treated as two,
since we make it a starting-point and a finishing-point: and this same
result is also produced by the act of reckoning halves as well as by
the act of dividing into halves. But if divisions are made in this
way, neither the distance nor the motion will be continuous: for
motion if it is to be continuous must relate to what is continuous:
and though what is continuous contains an infinite number of halves,
they are not actual but potential halves. If the halves are made
actual, we shall get not a continuous but an intermittent motion. In
the case of reckoning the halves, it is clear that this result
follows: for then one point must be reckoned as two: it will be the
finishing-point of the one half and the starting-point of the other,
if we reckon not the one continuous whole but the two halves.
Therefore to the question whether it is possible to pass through an
infinite number of units either of time or of distance we must reply
that in a sense it is and in a sense it is not. If the units are
actual, it is not possible: if they are potential, it is possible. For
in the course of a continuous motion the traveller has traversed an
infinite number of units in an accidental sense but not in an
unqualified sense: for though it is an accidental characteristic of
the distance to be an infinite number of half-distances, this is not
its real and essential character. It is also plain that unless we hold
that the point of time that divides earlier from later always
belongs only to the later so far as the thing is concerned, we shall
be involved in the consequence that the same thing is at the same
moment existent and not existent, and that a thing is not existent
at the moment when it has become. It is true that the point is
common to both times, the earlier as well as the later, and that,
while numerically one and the same, it is theoretically not so,
being the finishing-point of the one and the starting-point of the
other: but so far as the thing is concerned it belongs to the later
stage of what happens to it. Let us suppose a time ABG and a thing
D, D being white in the time A and not-white in the time B. Then D
is at the moment G white and not-white: for if we were right in saying
that it is white during the whole time A, it is true to call it
white at any moment of A, and not-white in B, and G is in both A and
B. We must not allow, therefore, that it is white in the whole of A,
but must say that it is so in all of it except the last moment G. G
belongs already to the later period, and if in the whole of A
not-white was in process of becoming and white of perishing, at G
the process is complete. And so G is the first moment at which it is
true to call the thing white or not white respectively. Otherwise a
thing may be non-existent at the moment when it has become and
existent at the moment when it has perished: or else it must be
possible for a thing at the same time to be white and not white and in
fact to be existent and non-existent. Further, if anything that exists
after having been previously non-existent must become existent and
does not exist when it is becoming, time cannot be divisible into
time-atoms. For suppose that D was becoming white in the time A and
that at another time B, a time-atom consecutive with the last atom
of A, D has already become white and so is white at that moment: then,
inasmuch as in the time A it was becoming white and so was not white
and at the moment B it is white, there must have been a becoming
between A and B and therefore also a time in which the becoming took
place. On the other hand, those who deny atoms of time (as we do)
are not affected by this argument: according to them D has become
and so is white at the last point of the actual time in which it was
becoming white: and this point has no other point consecutive with
or in succession to it, whereas time-atoms are conceived as
successive. Moreover it is clear that if D was becoming white in the
whole time A, the time occupied by it in having become white in
addition to having been in process of becoming white is no more than
all that it occupied in the mere process of becoming white.
These and such-like, then, are the arguments for our conclusion that
derive cogency from the fact that they have a special bearing on the
point at issue. If we look at the question from the point of view of
general theory, the same result would also appear to be indicated by
the following arguments. Everything whose motion is continuous must,
on arriving at any point in the course of its locomotion, have been
previously also in process of locomotion to that point, if it is not
forced out of its path by anything: e.g. on arriving at B a thing must
also have been in process of locomotion to B, and that not merely when
it was near to B, but from the moment of its starting on its course,
since there can be, no reason for its being so at any particular stage
rather than at an earlier one. So, too, in the case of the other kinds
of motion. Now we are to suppose that a thing proceeds in locomotion
from A to G and that at the moment of its arrival at G the
continuity of its motion is unbroken and will remain so until it has
arrived back at A. Then when it is undergoing locomotion from A to G
it is at the same time undergoing also its locomotion to A from G:
consequently it is simultaneously undergoing two contrary motions,
since the two motions that follow the same straight line are
contrary to each other. With this consequence there also follows
another: we have a thing that is in process of change from a
position in which it has not yet been: so, inasmuch as this is
impossible, the thing must come to a stand at G. Therefore the
motion is not a single motion, since motion that is interrupted by
stationariness is not single.
Further, the following argument will serve better to make this point
clear universally in respect of every kind of motion. If the motion
undergone by that which is in motion is always one of those already
enumerated, and the state of rest that it undergoes is one of those
that are the opposites of the motions (for we found that there could
be no other besides these), and moreover that which is undergoing
but does not always undergo a particular motion (by this I mean one of
the various specifically distinct motions, not some particular part of
the whole motion) must have been previously undergoing the state of
rest that is the opposite of the motion, the state of rest being
privation of motion; then, inasmuch as the two motions that follow the
same straight line are contrary motions, and it is impossible for a
thing to undergo simultaneously two contrary motions, that which is
undergoing locomotion from A to G cannot also simultaneously be
undergoing locomotion from G to A: and since the latter locomotion
is not simultaneous with the former but is still to be undergone,
before it is undergone there must occur a state of rest at G: for
this, as we found, is the state of rest that is the opposite of the
motion from G. The foregoing argument, then, makes it plain that the
motion in question is not continuous.
Our next argument has a more special bearing than the foregoing on
the point at issue. We will suppose that there has occurred in
something simultaneously a perishing of not-white and a becoming of
white. Then if the alteration to white and from white is a
continuous process and the white does not remain any time, there
must have occurred simultaneously a perishing of not-white, a becoming
of white, and a becoming of not-white: for the time of the three
will be the same.
Again, from the continuity of the time in which the motion takes
place we cannot infer continuity in the motion, but only
successiveness: in fact, how could contraries, e.g. whiteness and
blackness, meet in the same extreme point?
On the other hand, in motion on a circular line we shall find
singleness and continuity: for here we are met by no impossible
consequence: that which is in motion from A will in virtue of the same
direction of energy be simultaneously in motion to A (since it is in
motion to the point at which it will finally arrive), and yet will not
be undergoing two contrary or opposite motions: for a motion to a
point and a motion from that point are not always contraries or
opposites: they are contraries only if they are on the same straight
line (for then they are contrary to one another in respect of place,
as e.g. the two motions along the diameter of the circle, since the
ends of this are at the greatest possible distance from one
another), and they are o
pposites only if they are along the same line.
Therefore in the case we are now considering there is nothing to
prevent the motion being continuous and free from all intermission:
for rotatory motion is motion of a thing from its place to its
place, whereas rectilinear motion is motion from its place to
another place.
Moreover the progress of rotatory motion is never localized within
certain fixed limits, whereas that of rectilinear motion repeatedly is
so. Now a motion that is always shifting its ground from moment to
moment can be continuous: but a motion that is repeatedly localized
within certain fixed limits cannot be so, since then the same thing
would have to undergo simultaneously two opposite motions. So, too,
there cannot be continuous motion in a semicircle or in any other
arc of a circle, since here also the same ground must be traversed
repeatedly and two contrary processes of change must occur. The reason
is that in these motions the starting-point and the termination do not
coincide, whereas in motion over a circle they do coincide, and so
this is the only perfect motion.
This differentiation also provides another means of showing that the
other kinds of motion cannot be continuous either: for in all of
them we find that there is the same ground to be traversed repeatedly;
thus in alteration there are the intermediate stages of the process,
and in quantitative change there are the intervening degrees of
magnitude: and in becoming and perishing the same thing is true. It
makes no difference whether we take the intermediate stages of the
process to be few or many, or whether we add or subtract one: for in
either case we find that there is still the same ground to be
traversed repeatedly. Moreover it is plain from what has been said
that those physicists who assert that all sensible things are always
in motion are wrong: for their motion must be one or other of the
motions just mentioned: in fact they mostly conceive it as
alteration (things are always in flux and decay, they say), and they
go so far as to speak even of becoming and perishing as a process of
alteration. On the other hand, our argument has enabled us to assert
the fact, applying universally to all motions, that no motion admits