Aristotle

by Various Works [lit]

  points can neither be one (since of an indivisible there can be no

  extremity as distinct from some other part) nor together (since that

  which has no parts can have no extremity, the extremity and the

  thing of which it is the extremity being distinct).

  Moreover, if that which is continuous is composed of points, these

  points must be either continuous or in contact with one another: and

  the same reasoning applies in the case of all indivisibles. Now for

  the reason given above they cannot be continuous: and one thing can be

  in contact with another only if whole is in contact with whole or part

  with part or part with whole. But since indivisibles have no parts,

  they must be in contact with one another as whole with whole. And if

  they are in contact with one another as whole with whole, they will

  not be continuous: for that which is continuous has distinct parts:

  and these parts into which it is divisible are different in this

  way, i.e. spatially separate.

  Nor, again, can a point be in succession to a point or a moment to a

  moment in such a way that length can be composed of points or time

  of moments: for things are in succession if there is nothing of

  their own kind intermediate between them, whereas that which is

  intermediate between points is always a line and that which is

  intermediate between moments is always a period of time.

  Again, if length and time could thus be composed of indivisibles,

  they could be divided into indivisibles, since each is divisible

  into the parts of which it is composed. But, as we saw, no

  continuous thing is divisible into things without parts. Nor can there

  be anything of any other kind intermediate between the parts or

  between the moments: for if there could be any such thing it is

  clear that it must be either indivisible or divisible, and if it is

  divisible, it must be divisible either into indivisibles or into

  divisibles that are infinitely divisible, in which case it is

  continuous.

  Moreover, it is plain that everything continuous is divisible into

  divisibles that are infinitely divisible: for if it were divisible

  into indivisibles, we should have an indivisible in contact with an

  indivisible, since the extremities of things that are continuous

  with one another are one and are in contact.

  The same reasoning applies equally to magnitude, to time, and to

  motion: either all of these are composed of indivisibles and are

  divisible into indivisibles, or none. This may be made clear as

  follows. If a magnitude is composed of indivisibles, the motion over

  that magnitude must be composed of corresponding indivisible

  motions: e.g. if the magnitude ABG is composed of the indivisibles

  A, B, G, each corresponding part of the motion DEZ of O over ABG is

  indivisible. Therefore, since where there is motion there must be

  something that is in motion, and where there is something in motion

  there must be motion, therefore the being-moved will also be

  composed of indivisibles. So O traversed A when its motion was D, B

  when its motion was E, and G similarly when its motion was Z. Now a

  thing that is in motion from one place to another cannot at the moment

  when it was in motion both be in motion and at the same time have

  completed its motion at the place to which it was in motion: e.g. if a

  man is walking to Thebes, he cannot be walking to Thebes and at the

  same time have completed his walk to Thebes: and, as we saw, O

  traverses a the partless section A in virtue of the presence of the

  motion D. Consequently, if O actually passed through A after being

  in process of passing through, the motion must be divisible: for at

  the time when O was passing through, it neither was at rest nor had

  completed its passage but was in an intermediate state: while if it is

  passing through and has completed its passage at the same moment, then

  that which is walking will at the moment when it is walking have

  completed its walk and will be in the place to which it is walking;

  that is to say, it will have completed its motion at the place to

  which it is in motion. And if a thing is in motion over the whole

  KBG and its motion is the three D, E, and Z, and if it is not in

  motion at all over the partless section A but has completed its motion

  over it, then the motion will consist not of motions but of starts,

  and will take place by a thing's having completed a motion without

  being in motion: for on this assumption it has completed its passage

  through A without passing through it. So it will be possible for a

  thing to have completed a walk without ever walking: for on this

  assumption it has completed a walk over a particular distance

  without walking over that distance. Since, then, everything must be

  either at rest or in motion, and O is therefore at rest in each of the

  sections A, B, and G, it follows that a thing can be continuously at

  rest and at the same time in motion: for, as we saw, O is in motion

  over the whole ABG and at rest in any part (and consequently in the

  whole) of it. Moreover, if the indivisibles composing DEZ are motions,

  it would be possible for a thing in spite of the presence in it of

  motion to be not in motion but at rest, while if they are not motions,

  it would be possible for motion to be composed of something other than

  motions.

  And if length and motion are thus indivisible, it is neither more

  nor less necessary that time also be similarly indivisible, that is to

  say be composed of indivisible moments: for if the whole distance is

  divisible and an equal velocity will cause a thing to pass through

  less of it in less time, the time must also be divisible, and

  conversely, if the time in which a thing is carried over the section A

  is divisible, this section A must also be divisible.

  2

  And since every magnitude is divisible into magnitudes-for we have

  shown that it is impossible for anything continuous to be composed

  of indivisible parts, and every magnitude is continuous-it necessarily

  follows that the quicker of two things traverses a greater magnitude

  in an equal time, an equal magnitude in less time, and a greater

  magnitude in less time, in conformity with the definition sometimes

  given of 'the quicker'. Suppose that A is quicker than B. Now since of

  two things that which changes sooner is quicker, in the time ZH, in

  which A has changed from G to D, B will not yet have arrived at D

  but will be short of it: so that in an equal time the quicker will

  pass over a greater magnitude. More than this, it will pass over a

  greater magnitude in less time: for in the time in which A has arrived

  at D, B being the slower has arrived, let us say, at E. Then since A

  has occupied the whole time ZH in arriving at D, will have arrived

  at O in less time than this, say ZK. Now the magnitude GO that A has

  passed over is greater than the magnitude GE, and the time ZK is

  less than the whole time ZH: so that the quicker will pass over a

  greater magnitude in less time. And from this it is also clear that

  the quicker will pass over an equ
al magnitude in less time than the

  slower. For since it passes over the greater magnitude in less time

  than the slower, and (regarded by itself) passes over LM the greater

  in more time than LX the lesser, the time PRh in which it passes

  over LM will be more than the time PS, which it passes over LX: so

  that, the time PRh being less than the time PCh in which the slower

  passes over LX, the time PS will also be less than the time PX: for it

  is less than the time PRh, and that which is less than something

  else that is less than a thing is also itself less than that thing.

  Hence it follows that the quicker will traverse an equal magnitude

  in less time than the slower. Again, since the motion of anything must

  always occupy either an equal time or less or more time in

  comparison with that of another thing, and since, whereas a thing is

  slower if its motion occupies more time and of equal velocity if its

  motion occupies an equal time, the quicker is neither of equal

  velocity nor slower, it follows that the motion of the quicker can

  occupy neither an equal time nor more time. It can only be, then, that

  it occupies less time, and thus we get the necessary consequence

  that the quicker will pass over an equal magnitude (as well as a

  greater) in less time than the slower.

  And since every motion is in time and a motion may occupy any

  time, and the motion of everything that is in motion may be either

  quicker or slower, both quicker motion and slower motion may occupy

  any time: and this being so, it necessarily follows that time also

  is continuous. By continuous I mean that which is divisible into

  divisibles that are infinitely divisible: and if we take this as the

  definition of continuous, it follows necessarily that time is

  continuous. For since it has been shown that the quicker will pass

  over an equal magnitude in less time than the slower, suppose that A

  is quicker and B slower, and that the slower has traversed the

  magnitude GD in the time ZH. Now it is clear that the quicker will

  traverse the same magnitude in less time than this: let us say in

  the time ZO. Again, since the quicker has passed over the whole D in

  the time ZO, the slower will in the same time pass over GK, say, which

  is less than GD. And since B, the slower, has passed over GK in the

  time ZO, the quicker will pass over it in less time: so that the

  time ZO will again be divided. And if this is divided the magnitude GK

  will also be divided just as GD was: and again, if the magnitude is

  divided, the time will also be divided. And we can carry on this

  process for ever, taking the slower after the quicker and the

  quicker after the slower alternately, and using what has been

  demonstrated at each stage as a new point of departure: for the

  quicker will divide the time and the slower will divide the length.

  If, then, this alternation always holds good, and at every turn

  involves a division, it is evident that all time must be continuous.

  And at the same time it is clear that all magnitude is also

  continuous; for the divisions of which time and magnitude respectively

  are susceptible are the same and equal.

  Moreover, the current popular arguments make it plain that, if

  time is continuous, magnitude is continuous also, inasmuch as a

  thing asses over half a given magnitude in half the time taken to

  cover the whole: in fact without qualification it passes over a less

  magnitude in less time; for the divisions of time and of magnitude

  will be the same. And if either is infinite, so is the other, and

  the one is so in the same way as the other; i.e. if time is infinite

  in respect of its extremities, length is also infinite in respect of

  its extremities: if time is infinite in respect of divisibility,

  length is also infinite in respect of divisibility: and if time is

  infinite in both respects, magnitude is also infinite in both

  respects.

  Hence Zeno's argument makes a false assumption in asserting that

  it is impossible for a thing to pass over or severally to come in

  contact with infinite things in a finite time. For there are two

  senses in which length and time and generally anything continuous

  are called 'infinite': they are called so either in respect of

  divisibility or in respect of their extremities. So while a thing in a

  finite time cannot come in contact with things quantitatively

  infinite, it can come in contact with things infinite in respect of

  divisibility: for in this sense the time itself is also infinite:

  and so we find that the time occupied by the passage over the infinite

  is not a finite but an infinite time, and the contact with the

  infinites is made by means of moments not finite but infinite in

  number.

  The passage over the infinite, then, cannot occupy a finite time,

  and the passage over the finite cannot occupy an infinite time: if the

  time is infinite the magnitude must be infinite also, and if the

  magnitude is infinite, so also is the time. This may be shown as

  follows. Let AB be a finite magnitude, and let us suppose that it is

  traversed in infinite time G, and let a finite period GD of the time

  be taken. Now in this period the thing in motion will pass over a

  certain segment of the magnitude: let BE be the segment that it has

  thus passed over. (This will be either an exact measure of AB or

  less or greater than an exact measure: it makes no difference which it

  is.) Then, since a magnitude equal to BE will always be passed over in

  an equal time, and BE measures the whole magnitude, the whole time

  occupied in passing over AB will be finite: for it will be divisible

  into periods equal in number to the segments into which the

  magnitude is divisible. Moreover, if it is the case that infinite time

  is not occupied in passing over every magnitude, but it is possible to

  ass over some magnitude, say BE, in a finite time, and if this BE

  measures the whole of which it is a part, and if an equal magnitude is

  passed over in an equal time, then it follows that the time like the

  magnitude is finite. That infinite time will not be occupied in

  passing over BE is evident if the time be taken as limited in one

  direction: for as the part will be passed over in less time than the

  whole, the time occupied in traversing this part must be finite, the

  limit in one direction being given. The same reasoning will also

  show the falsity of the assumption that infinite length can be

  traversed in a finite time. It is evident, then, from what has been

  said that neither a line nor a surface nor in fact anything continuous

  can be indivisible.

  This conclusion follows not only from the present argument but

  from the consideration that the opposite assumption implies the

  divisibility of the indivisible. For since the distinction of

  quicker and slower may apply to motions occupying any period of time

  and in an equal time the quicker passes over a greater length, it

  may happen that it will pass over a length twice, or one and a half

  times, as great as that passed over by the slower: for their

  resp
ective velocities may stand to one another in this proportion.

  Suppose, then, that the quicker has in the same time been carried over

  a length one and a half times as great as that traversed by the

  slower, and that the respective magnitudes are divided, that of the

  quicker, the magnitude ABGD, into three indivisibles, and that of

  the slower into the two indivisibles EZ, ZH. Then the time may also be

  divided into three indivisibles, for an equal magnitude will be passed

  over in an equal time. Suppose then that it is thus divided into KL,

  LM, MN. Again, since in the same time the slower has been carried over

  EZ, ZH, the time may also be similarly divided into two. Thus the

  indivisible will be divisible, and that which has no parts will be

  passed over not in an indivisible but in a greater time. It is

  evident, therefore, that nothing continuous is without parts.

  3

  The present also is necessarily indivisible-the present, that is,

  not in the sense in which the word is applied to one thing in virtue

  of another, but in its proper and primary sense; in which sense it

  is inherent in all time. For the present is something that is an

  extremity of the past (no part of the future being on this side of it)

  and also of the future (no part of the past being on the other side of

  it): it is, as we have said, a limit of both. And if it is once

  shown that it is essentially of this character and one and the same,

  it will at once be evident also that it is indivisible.

  Now the present that is the extremity of both times must be one

  and the same: for if each extremity were different, the one could

  not be in succession to the other, because nothing continuous can be

  composed of things having no parts: and if the one is apart from the

  other, there will be time intermediate between them, because

  everything continuous is such that there is something intermediate

  between its limits and described by the same name as itself. But if

  the intermediate thing is time, it will be divisible: for all time has

  been shown to be divisible. Thus on this assumption the present is