Various Works Page 21

  which the magnitude has changed, and something else again prior to

  that, and so on to infinity, because the process of division may be

  continued without end. Thus there can be no primary 'where' to which a

  thing has changed. And if we take the case of quantitative change,

  we shall get a like result, for here too the change is in something

  continuous. It is evident, then, that only in qualitative motion can

  there be anything essentially indivisible.

  6

  Now everything that changes changes time, and that in two senses:

  for the time in which a thing is said to change may be the primary

  time, or on the other hand it may have an extended reference, as

  e.g. when we say that a thing changes in a particular year because

  it changes in a particular day. That being so, that which changes must

  be changing in any part of the primary time in which it changes.

  This is clear from our definition of 'primary', in which the word is

  said to express just this: it may also, however, be made evident by

  the following argument. Let ChRh be the primary time in which that

  which is in motion is in motion: and (as all time is divisible) let it

  be divided at K. Now in the time ChK it either is in motion or is

  not in motion, and the same is likewise true of the time KRh. Then

  if it is in motion in neither of the two parts, it will be at rest

  in the whole: for it is impossible that it should be in motion in a

  time in no part of which it is in motion. If on the other hand it is

  in motion in only one of the two parts of the time, ChRh cannot be the

  primary time in which it is in motion: for its motion will have

  reference to a time other than ChRh. It must, then, have been in

  motion in any part of ChRh.

  And now that this has been proved, it is evident that everything

  that is in motion must have been in motion before. For if that which

  is in motion has traversed the distance KL in the primary time ChRh,

  in half the time a thing that is in motion with equal velocity and

  began its motion at the same time will have traversed half the

  distance. But if this second thing whose velocity is equal has

  traversed a certain distance in a certain time, the original thing

  that is in motion must have traversed the same distance in the same

  time. Hence that which is in motion must have been in motion before.

  Again, if by taking the extreme moment of the time-for it is the

  moment that defines the time, and time is that which is intermediate

  between moments-we are enabled to say that motion has taken place in

  the whole time ChRh or in fact in any period of it, motion may

  likewise be said to have taken place in every other such period. But

  half the time finds an extreme in the point of division. Therefore

  motion will have taken place in half the time and in fact in any

  part of it: for as soon as any division is made there is always a time

  defined by moments. If, then, all time is divisible, and that which is

  intermediate between moments is time, everything that is changing must

  have completed an infinite number of changes.

  Again, since a thing that changes continuously and has not

  perished or ceased from its change must either be changing or have

  changed in any part of the time of its change, and since it cannot

  be changing in a moment, it follows that it must have changed at every

  moment in the time: consequently, since the moments are infinite in

  number, everything that is changing must have completed an infinite

  number of changes.

  And not only must that which is changing have changed, but that

  which has changed must also previously have been changing, since

  everything that has changed from something to something has changed in

  a period of time. For suppose that a thing has changed from A to B

  in a moment. Now the moment in which it has changed cannot be the same

  as that in which it is at A (since in that case it would be in A and B

  at once): for we have shown above that that that which has changed,

  when it has changed, is not in that from which it has changed. If,

  on the other hand, it is a different moment, there will be a period of

  time intermediate between the two: for, as we saw, moments are not

  consecutive. Since, then, it has changed in a period of time, and

  all time is divisible, in half the time it will have completed another

  change, in a quarter another, and so on to infinity: consequently when

  it has changed, it must have previously been changing.

  Moreover, the truth of what has been said is more evident in the

  case of magnitude, because the magnitude over which what is changing

  changes is continuous. For suppose that a thing has changed from G

  to D. Then if GD is indivisible, two things without parts will be

  consecutive. But since this is impossible, that which is

  intermediate between them must be a magnitude and divisible into an

  infinite number of segments: consequently, before the change is

  completed, the thing changes to those segments. Everything that has

  changed, therefore, must previously have been changing: for the same

  proof also holds good of change with respect to what is not

  continuous, changes, that is to say, between contraries and between

  contradictories. In such cases we have only to take the time in

  which a thing has changed and again apply the same reasoning. So

  that which has changed must have been changing and that which is

  changing must have changed, and a process of change is preceded by a

  completion of change and a completion by a process: and we can never

  take any stage and say that it is absolutely the first. The reason

  of this is that no two things without parts can be contiguous, and

  therefore in change the process of division is infinite, just as lines

  may be infinitely divided so that one part is continually increasing

  and the other continually decreasing.

  So it is evident also that that that which has become must

  previously have been in process of becoming, and that which is in

  process of becoming must previously have become, everything (that

  is) that is divisible and continuous: though it is not always the

  actual thing that is in process of becoming of which this is true:

  sometimes it is something else, that is to say, some part of the thing

  in question, e.g. the foundation-stone of a house. So, too, in the

  case of that which is perishing and that which has perished: for

  that which becomes and that which perishes must contain an element

  of infiniteness as an immediate consequence of the fact that they

  are continuous things: and so a thing cannot be in process of becoming

  without having become or have become without having been in process of

  becoming. So, too, in the case of perishing and having perished:

  perishing must be preceded by having perished, and having perished

  must be preceded by perishing. It is evident, then, that that which

  has become must previously have been in process of becoming, and

  that which is in process of becoming must previously have become:

  for all magnitudes and all periods of time are infinitely divisible.

&
nbsp; Consequently no absolutely first stage of change can be

  represented by any particular part of space or time which the changing

  thing may occupy.

  7

  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.

  8

  Since everything to which motion or rest is natural is in motion

  or at rest in the natural time, place, and manner, that which is

  coming to a stand, when it is coming to a stand, must be in motion:

  for if it is not in motion it must be at rest: but that which is at

  rest cannot be coming to rest. From this it evidently follows that

  coming to a stand must occupy a period of time: for the motion of that

  which is in motion occupies a period of time, and that which is coming

  to a stand has been shown to be in motion: consequently coming to a

  stand must occupy a period of time.

  Again, since the terms 'quicker' and 'slower' are used only of

  that which occupies a period of time, and the process of coming to a

  stand may be quicker or slower, the same conclusion follows.

  And that which is coming to a stand must be coming to a stand in any

  part of the primary time in which it is coming to a stand. For if it

  is coming to a stand in neither of two parts into whic
h the time may

  be divided, it cannot be coming to a stand in the whole time, with the

  result that that that which is coming to a stand will not be coming to

  a stand. If on the other hand it is coming to a stand in only one of

  the two parts of the time, the whole cannot be the primary time in

  which it is coming to a stand: for it is coming to a stand in the

  whole time not primarily but in virtue of something distinct from

  itself, the argument being the same as that which we used above

  about things in motion.

  And just as there is no primary time in which that which is in

  motion is in motion, so too there is no primary time in which that

  which is coming to a stand is coming to a stand, there being no

  primary stage either of being in motion or of coming to a stand. For

  let AB be the primary time in which a thing is coming to a stand.

  Now AB cannot be without parts: for there cannot be motion in that

  which is without parts, because the moving thing would necessarily

  have been already moved for part of the time of its movement: and that

  which is coming to a stand has been shown to be in motion. But since

  AB is therefore divisible, the thing is coming to a stand in every one

  of the parts of AB: for we have shown above that it is coming to a

  stand in every one of the parts in which it is primarily coming to a

  stand. Since then, that in which primarily a thing is coming to a

  stand must be a period of time and not something indivisible, and

  since all time is infinitely divisible, there cannot be anything in

  which primarily it is coming to a stand.

  Nor again can there be a primary time at which the being at rest

  of that which is at rest occurred: for it cannot have occurred in that

  which has no parts, because there cannot be motion in that which is

  indivisible, and that in which rest takes place is the same as that in

  which motion takes place: for we defined a state of rest to be the

  state of a thing to which motion is natural but which is not in motion

  when (that is to say in that in which) motion would be natural to