Various Works Read online

  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

  number, which is admittedly impossible. Now when we first discussed

  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
<
br />   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