it. Again, our use of the phrase 'being at rest' also implies that the
previous state of a thing is still unaltered, not one point only but
two at least being thus needed to determine its presence: consequently
that in which a thing is at rest cannot be without parts. Since,
then it is divisible, it must be a period of time, and the thing
must be at rest in every one of its parts, as may be shown by the same
method as that used above in similar demonstrations.
So there can be no primary part of the time: and the reason is
that rest and motion are always in a period of time, and a period of
time has no primary part any more than a magnitude or in fact anything
continuous: for everything continuous is divisible into an infinite
number of parts.
And since everything that is in motion is in motion in a period of
time and changes from something to something, when its motion is
comprised within a particular period of time essentially-that is to
say when it fills the whole and not merely a part of the time in
question-it is impossible that in that time that which is in motion
should be over against some particular thing primarily. For if a
thing-itself and each of its parts-occupies the same space for a
definite period of time, it is at rest: for it is in just these
circumstances that we use the term 'being at rest'-when at one
moment after another it can be said with truth that a thing, itself
and its parts, occupies the same space. So if this is being at rest it
is impossible for that which is changing to be as a whole, at the time
when it is primarily changing, over against any particular thing
(for the whole period of time is divisible), so that in one part of it
after another it will be true to say that the thing, itself and its
parts, occupies the same space. If this is not so and the aforesaid
proposition is true only at a single moment, then the thing will be
over against a particular thing not for any period of time but only at
a moment that limits the time. It is true that at any moment it is
always over against something stationary: but it is not at rest: for
at a moment it is not possible for anything to be either in motion
or at rest. So while it is true to say that that which is in motion is
at a moment not in motion and is opposite some particular thing, it
cannot in a period of time be over against that which is at rest:
for that would involve the conclusion that that which is in locomotion
is at rest.
9
Zeno's reasoning, however, is fallacious, when he says that if
everything when it occupies an equal space is at rest, and if that
which is in locomotion is always occupying such a space at any moment,
the flying arrow is therefore motionless. This is false, for time is
not composed of indivisible moments any more than any other
magnitude is composed of indivisibles.
Zeno's arguments about motion, which cause so much disquietude to
those who try to solve the problems that they present, are four in
number. The first asserts the non-existence of motion on the ground
that that which is in locomotion must arrive at the half-way stage
before it arrives at the goal. This we have discussed above.
The second is the so-called 'Achilles', and it amounts to this, that
in a race the quickest runner can never overtake the slowest, since
the pursuer must first reach the point whence the pursued started,
so that the slower must always hold a lead. This argument is the
same in principle as that which depends on bisection, though it
differs from it in that the spaces with which we successively have
to deal are not divided into halves. The result of the argument is
that the slower is not overtaken: but it proceeds along the same lines
as the bisection-argument (for in both a division of the space in a
certain way leads to the result that the goal is not reached, though
the 'Achilles' goes further in that it affirms that even the
quickest runner in legendary tradition must fail in his pursuit of the
slowest), so that the solution must be the same. And the axiom that
that which holds a lead is never overtaken is false: it is not
overtaken, it is true, while it holds a lead: but it is overtaken
nevertheless if it is granted that it traverses the finite distance
prescribed. These then are two of his arguments.
The third is that already given above, to the effect that the flying
arrow is at rest, which result follows from the assumption that time
is composed of moments: if this assumption is not granted, the
conclusion will not follow.
The fourth argument is that concerning the two rows of bodies,
each row being composed of an equal number of bodies of equal size,
passing each other on a race-course as they proceed with equal
velocity in opposite directions, the one row originally occupying
the space between the goal and the middle point of the course and
the other that between the middle point and the starting-post. This,
he thinks, involves the conclusion that half a given time is equal
to double that time. The fallacy of the reasoning lies in the
assumption that a body occupies an equal time in passing with equal
velocity a body that is in motion and a body of equal size that is
at rest; which is false. For instance (so runs the argument), let A,
A...be the stationary bodies of equal size, B, B...the bodies, equal
in number and in size to A, A...,originally occupying the half of
the course from the starting-post to the middle of the A's, and G,
G...those originally occupying the other half from the goal to the
middle of the A's, equal in number, size, and velocity to B, B....Then
three consequences follow:
First, as the B's and the G's pass one another, the first B
reaches the last G at the same moment as the first G reaches the
last B. Secondly at this moment the first G has passed all the A's,
whereas the first B has passed only half the A's, and has consequently
occupied only half the time occupied by the first G, since each of the
two occupies an equal time in passing each A. Thirdly, at the same
moment all the B's have passed all the G's: for the first G and the
first B will simultaneously reach the opposite ends of the course,
since (so says Zeno) the time occupied by the first G in passing
each of the B's is equal to that occupied by it in passing each of the
A's, because an equal time is occupied by both the first B and the
first G in passing all the A's. This is the argument, but it
presupposed the aforesaid fallacious assumption.
Nor in reference to contradictory change shall we find anything
unanswerable in the argument that if a thing is changing from
not-white, say, to white, and is in neither condition, then it will be
neither white nor not-white: for the fact that it is not wholly in
either condition will not preclude us from calling it white or
not-white. We call a thing white or not-white not necessarily
because it is be one or the other, but cause most of its parts or
the most essential part
s of it are so: not being in a certain
condition is different from not being wholly in that condition. So,
too, in the case of being and not-being and all other conditions which
stand in a contradictory relation: while the changing thing must of
necessity be in one of the two opposites, it is never wholly in
either.
Again, in the case of circles and spheres and everything whose
motion is confined within the space that it occupies, it is not true
to say the motion can be nothing but rest, on the ground that such
things in motion, themselves and their parts, will occupy the same
position for a period of time, and that therefore they will be at once
at rest and in motion. For in the first place the parts do not
occupy the same position for any period of time: and in the second
place the whole also is always changing to a different position: for
if we take the orbit as described from a point A on a circumference,
it will not be the same as the orbit as described from B or G or any
other point on the same circumference except in an accidental sense,
the sense that is to say in which a musical man is the same as a
man. Thus one orbit is always changing into another, and the thing
will never be at rest. And it is the same with the sphere and
everything else whose motion is confined within the space that it
occupies.
10
Our next point is that that which is without parts cannot be in
motion except accidentally: i.e. it can be in motion only in so far as
the body or the magnitude is in motion and the partless is in motion
by inclusion therein, just as that which is in a boat may be in motion
in consequence of the locomotion of the boat, or a part may be in
motion in virtue of the motion of the whole. (It must be remembered,
however, that by 'that which is without parts' I mean that which is
quantitatively indivisible (and that the case of the motion of a
part is not exactly parallel): for parts have motions belonging
essentially and severally to themselves distinct from the motion of
the whole. The distinction may be seen most clearly in the case of a
revolving sphere, in which the velocities of the parts near the centre
and of those on the surface are different from one another and from
that of the whole; this implies that there is not one motion but
many). As we have said, then, that which is without parts can be in
motion in the sense in which a man sitting in a boat is in motion when
the boat is travelling, but it cannot be in motion of itself. For
suppose that it is changing from AB to BG-either from one magnitude to
another, or from one form to another, or from some state to its
contradictory-and let D be the primary time in which it undergoes
the change. Then in the time in which it is changing it must be either
in AB or in BG or partly in one and partly in the other: for this,
as we saw, is true of everything that is changing. Now it cannot be
partly in each of the two: for then it would be divisible into
parts. Nor again can it be in BG: for then it will have completed
the change, whereas the assumption is that the change is in process.
It remains, then, that in the time in which it is changing, it is in
AB. That being so, it will be at rest: for, as we saw, to be in the
same condition for a period of time is to be at rest. So it is not
possible for that which has no parts to be in motion or to change in
any way: for only one condition could have made it possible for it
to have motion, viz. that time should be composed of moments, in which
case at any moment it would have completed a motion or a change, so
that it would never be in motion, but would always have been in
motion. But this we have already shown above to be impossible: time is
not composed of moments, just as a line is not composed of points, and
motion is not composed of starts: for this theory simply makes
motion consist of indivisibles in exactly the same way as time is made
to consist of moments or a length of points.
Again, it may be shown in the following way that there can be no
motion of a point or of any other indivisible. That which is in motion
can never traverse a space greater than itself without first
traversing a space equal to or less than itself. That being so, it
is evident that the point also must first traverse a space equal to or
less than itself. But since it is indivisible, there can be no space
less than itself for it to traverse first: so it will have to traverse
a distance equal to itself. Thus the line will be composed of
points, for the point, as it continually traverses a distance equal to
itself, will be a measure of the whole line. But since this is
impossible, it is likewise impossible for the indivisible to be in
motion.
Again, since motion is always in a period of time and never in a
moment, and all time is divisible, for everything that is in motion
there must be a time less than that in which it traverses a distance
as great as itself. For that in which it is in motion will be a
time, because all motion is in a period of time; and all time has been
shown above to be divisible. Therefore, if a point is in motion, there
must be a time less than that in which it has itself traversed any
distance. But this is impossible, for in less time it must traverse
less distance, and thus the indivisible will be divisible into
something less than itself, just as the time is so divisible: the fact
being that the only condition under which that which is without
parts and indivisible could be in motion would have been the
possibility of the infinitely small being in motion in a moment: for
in the two questions-that of motion in a moment and that of motion
of something indivisible-the same principle is involved.
Our next point is that no process of change is infinite: for every
change, whether between contradictories or between contraries, is a
change from something to something. Thus in contradictory changes
the positive or the negative, as the case may be, is the limit, e.g.
being is the limit of coming to be and not-being is the limit of
ceasing to be: and in contrary changes the particular contraries are
the limits, since these are the extreme points of any such process
of change, and consequently of every process of alteration: for
alteration is always dependent upon some contraries. Similarly
contraries are the extreme points of processes of increase and
decrease: the limit of increase is to be found in the complete
magnitude proper to the peculiar nature of the thing that is
increasing, while the limit of decrease is the complete loss of such
magnitude. Locomotion, it is true, we cannot show to be finite in this
way, since it is not always between contraries. But since that which
cannot be cut (in the sense that it is inconceivable that it should be
cut, the term 'cannot' being used in several senses)-since it is
inconceivable that that which in this sense cannot be cut should be in
process of being cut, and generally that that which cannot come to
be should be in process of coming to be, it follows that it is
inconceivable that that which cannot complete a change should be in
process of changing to that to which it cannot complete a change.
If, then, it is to be assumed that that which is in locomotion is in
process of changing, it must be capable of completing the change.
Consequently its motion is not infinite, and it will not be in
locomotion over an infinite distance, for it cannot traverse such a
distance.
It is evident, then, that a process of change cannot be infinite
in the sense that it is not defined by limits. But it remains to be
considered whether it is possible in the sense that one and the same
process of change may be infinite in respect of the time which it
occupies. If it is not one process, it would seem that there is
nothing to prevent its being infinite in this sense; e.g. if a process
of locomotion be succeeded by a process of alteration and that by a
process of increase and that again by a process of coming to be: in
this way there may be motion for ever so far as the time is concerned,
but it will not be one motion, because all these motions do not
compose one. If it is to be one process, no motion can be infinite
in respect of the time that it occupies, with the single exception
of rotatory locomotion.
Book VII
1
EVERYTHING that is in motion must be moved by something. For if it
has not the source of its motion in itself it is evident that it is
moved by something other than itself, for there must be something else
that moves it. If on the other hand it has the source of its motion in
itself, let AB be taken to represent that which is in motion
essentially of itself and not in virtue of the fact that something
belonging to it is in motion. Now in the first place to assume that
AB, because it is in motion as a whole and is not moved by anything
external to itself, is therefore moved by itself-this is just as if,
supposing that KL is moving LM and is also itself in motion, we were
Aristotle Page 22