Various Works

  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