The Politics of Aristotle

by Aristotle

  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 [5] 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 put the same argument in another form, and would have us grant that in the time during which a motion is in progress we should first count the half-motion for every half-distance that we get, so that we have the result that when the whole distance is traversed we have counted an infinite number, which is admittedly [10] impossible. Now in our first discussions of motion we put forward a solution of this difficulty turning on the fact that the period of time 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 [15] questioner (the question asked being whether it is impossible in a finite time to traverse or count an infinite number of units), nevertheless as an account of the fact and the truth 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 [20] time 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. In the act of dividing the continuous distance into two halves one point is treated as two, since we make it a beginning and an end; and this same result is produced by the act of counting halves as well as by the act of dividing into halves. [25] 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 he makes the halves actual, he will get not a continuous but an intermittent motion. In the case of counting the halves, it is clear that this result [263b1] follows; for then one point must be reckoned as two: it will be the end of the one half and the beginning of the other, if he counts 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 [5] 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, it is different in essence and being.

  It is also plain that unless we hold that the point of time that divides earlier [10] 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 at the same moment is and is not, and that a thing is not 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 not so in definition, being the end of the one and the beginning of the other; but so far as the thing is concerned it always belongs to [15] the later affection. Let us suppose a time ACB and a thing D, D being white in the time A and not white in the time B. Then D is at C 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 C is in both A and B. We must not [20] 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 now C. C already belongs to the later period, and if in the whole of A not white was becoming and white perishing, at C it had become or perished. And so either that is the first moment at which it is true to call the thing not white;53 or a thing may not be at the moment when it has become and may be at the moment [25] when it has perished; or else things must at the same time be white and not white and in general be and not be. Further, if anything that is after having previously not been must become being and is not when it is becoming, time cannot be divisible into indivisible times. For suppose that D was becoming white at A and that at another indivisible time B, consecutive with A, D has already become white and so is white at that moment: then, inasmuch as at A it was becoming white and so was [30] not white and at 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, [264a1] those who deny indivisibles are not affected by this argument: according to them it has become and 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 indivisible times are successive. Moreover it is clear that if it was becoming [5] white in the whole time A, there was no more time in which it had become and was becoming than the total of the time in which it was merely becoming.

  These and such-like, then, are the arguments on which one might rely as being appropriate to the subject matter. If we look at the question generally, 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 [10] 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 and that when it arrives at C it comes again, moving [15] continuously, to A. Then when it is undergoing locomotion from A to C it is at the same time undergoing also its locomotion to A from C: consequently, it is simultaneously undergoing two contrary motions, since the two motions that follow the same straight line are contrary to each other. At the same time it changes from a state in which it is not: so, inasmuch as this is impossible, the thing must come to a stand at C. Therefore the motion is not a single motion, since motion that is [20] 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 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 [25] 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 C cannot also simultaneously be undergoing locomotion from C to A; and since the latter [30] 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 C; for this, as we found, is the state of rest that is the opposite of the motion from C. The foregoing argument, then, makes it plain that the motion is not continuous. [264b1]

  Again, there is the following argument, more appropriate than its predecessors. At the same time something has ceased to be not white and has become white. Then if the alteration to white and from white is continuous and does not persist for any time, at the same time it has ceased to be not white and [5] has become white and has become not white; for the time of the three will be the same.

  Again, from the continuity of the time in which the motion take
s 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, motion on a circular line will be one and continuous; for here we are met by no impossible consequence: that which is in motion from A will [10] 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 [15] the same straight line (for this has points contrary in place, e.g. the points on a diameter—for they are furthest from one another), and they are opposites only if they are along the same line. Therefore 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.

  [20] Moreover rotatory motion is never at the same points, whereas rectilinear motion repeatedly is so. Now a motion that is always shifting its ground can be continuous; but a motion that is repeatedly at the same points cannot be so, since then the same thing would have to undergo simultaneously two opposite motions. [25] 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. For the beginning and the termination do not coincide, whereas in motion over a circle they do coincide, and so this is the only perfect motion.

  This analysis shows that the other kinds of motion cannot be continuous either; [30] 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, 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 [265a1] of the change 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. Thus 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 [5] becoming and perishing as a process of alteration. On the other hand, our argument has shown universally of all motions, that no motion admits of continuity except rotatory motion: consequently neither alteration nor increase admits of continuity. [10] So much for the view that there is no change that admits of infinity or continuity except rotatory locomotion.

  9 · It can now be shown plainly that rotation is the primary locomotion. Every locomotion, as we said before, is either rotatory or rectilinear or a compound [15] of the two; and the two former must be prior to the last, since they are the elements of which the latter consists. Moreover rotatory locomotion is prior to rectilinear locomotion, because it is more simple and complete. For the line traversed in rectilinear motion cannot be infinite; for there is no such thing as an infinite straight line; and even if there were, it would not be traversed by anything in motion; for the [20] impossible does not happen and it is impossible to traverse an infinite distance. On the other hand rectilinear motion on a finite line is composite if it turns back, i.e. two motions, while if it does not turn back it is incomplete and perishable; and in the order of nature, of definition, and of time alike the complete is prior to the incomplete and the imperishable to the perishable. Again, a motion that admits of being eternal is prior to one that does not. Now rotatory motion can be eternal; but [25] no other motion, whether locomotion or motion of any other kind, can be so, since in all of them rest must occur, and with the occurrence of rest the motion has perished.

  The result at which we have arrived, that rotatory motion is single and continuous, and rectilinear motion is not, is a reasonable one. In rectilinear motion we have a definite beginning, end and middle, which all have their place in it in such [30] a way that there is a point from which that which is in motion will begin and a point at which it will end (for when anything is at the limits of its course, whether at the whence or at the whither, it is in a state of rest). On the other hand in circular motion there are no such definite points; for why should any one point on the line be a limit rather than any other? Any one point as much as any other is alike beginning, middle, and end, so that they are both always and never at a beginning [265b1] and at an end (so that a sphere is in a way both in motion and at rest; for it continues to occupy the same place). The reason of this is that in this case all these characteristics belong to the centre: that is to say, the centre is alike beginning, middle, and end of the space traversed; consequently since this point is not a point on the circular line, there is no point at which that which is in process of locomotion [5] can be in a state of rest as having traversed its course, because in its locomotion it is proceeding always about a central point and not to an extreme point; and because this remains still, the whole is in a sense always at rest as well as continuously in motion. Our next point gives a convertible result: on the one hand, because rotation is the measure of motions it must be the primary motion (for all things are measured [10] by what is primary); on the other hand, because rotation is the primary motion it is the measure of all other motions. Again, rotatory motion is also the only motion that admits of being regular. In rectilinear locomotion the motion of things in leaving the beginning is not uniform with their motion in approaching the end, since the velocity of a thing always increases proportionately as it removes itself farther from its position of rest; on the other hand rotatory motion alone has by nature no [15] beginning or end in itself but only outside.

  As to locomotion being the primary motion, this is a truth that is attested by all who have ever made mention of motion: they all assign their principles of motion to things that impart motion of this kind. Thus separation and combination are motions in respect of place, and the motion imparted by Love and Strife takes these [20] forms, the latter separating and the former combining. Anaxagoras, too, says that Mind, his first mover, separates. Similarly those who assert no cause of this kind but say that void accounts for motion—they also hold that the motion of natural substance is motion in respect of place; for their motion that is accounted for by void [25] is locomotion, and its sphere of operation may be said to be place. Moreover they are of opinion that the primary substances are not subject to any of the other motions, though the things that are compounds of these substances are so subject: the processes of increase and decrease and alteration, they say, are effects of the combination and separation of atoms. It is the same, too, with those who make out [30] that the becoming or perishing of a thing is accounted for by density or rarity; for it is by combination and separation that the place of these things in their systems is determined. Moreover to these we may add those who make soul the cause of motion; for they say that things that undergo motion have as their first principle [266a1] that which moves itself; and when animals and all living things move themselves, the motion is motion in respect of place. Finally, we say that a thing is in motion in the strict sense of the term only when its motion is motion in respect of place: if a thing is in process of increase or decrease or is undergoing some alteration while remaining at rest in the same place, we say that it is in motion in some particular [5] respect: we do not say that it is in motion without qualification.

  We have argued that there always was motion and always will be motion throughout all time, and we have explained what is the first principle of this eternal motion; we have explained further which is the primary motion and which is the only motion that can be eternal; and we have pronounced the first mover to be unmoved.

  [10] 10 · We have now to assert that the first mover must be without parts and without magnit
ude, beginning with the establishment of the premisses on which this conclusion depends.

  One of these premisses is that nothing finite can cause motion during an infinite time. We have three things, the mover, the moved, and thirdly that in which the motion takes place, namely the time; and these are either all infinite or all finite [15] or some—that is to say two of them or one of them—finite and some infinite. Let A be the mover, B the moved, and C infinite time. Now let us suppose that D moves E, a part of B. Then the time occupied by this motion cannot be equal to C; for the greater the amount moved, the longer the time occupied. It follows that the time F is not infinite. Now we see that by continuing to add to D I shall use up A and by [20] continuing to add to E I shall use up B; but I shall not use up the time by continually subtracting a corresponding amount from it, because it is infinite. Consequently the part of C which is occupied by all A in moving the whole of B, will be finite. Therefore a finite thing cannot impart to anything an infinite motion. It is clear, then, that it is impossible for the finite to cause motion during an infinite time.

  [25] That in no case is it possible for an infinite force to reside in a finite magnitude, can be shown as follows: we take it for granted that the greater force is always that which in less time does an equal amount of work—heating, for example, or sweetening or throwing, or in general causing motion. Then that on which the forces act must be affected to some extent by the finite magnitude possessing an infinite [30] force—in fact to a greater extent than by anything else, since the infinite force is greater than any other. But then there cannot be any time in which its action could take place. Suppose that A is the time occupied by the infinite power in the performance of an act of heating or pushing, and that AB is the time occupied by a [266b1] finite power in the performance of the same act: then by adding to the latter another finite power and continually increasing the magnitude of the power so added I shall at some time or other reach a point at which the finite power has completed the motive act in the time A; for by continual addition to a finite magnitude I must arrive at a magnitude that exceeds any assigned limit, and in the same way by continual subtraction I must arrive at one that falls short of any assigned limit. So we get the result that the finite force will occupy the same amount of time in performing the motive act as the infinite force. But this is impossible. Therefore [5] nothing finite can possess an infinite force. So it is also impossible for a finite force to reside in an infinite magnitude. It is true that a greater force can reside in a lesser magnitude; but then a still greater force will reside in a greater. Now let AB be an infinite magnitude. Then BC possesses a certain force that occupies a certain time, let us say the time EF, in moving D. Now if I take a magnitude twice as great as BC, [10] the time occupied by this magnitude in moving D will be half of EF (assuming this to be the proportion): so we may call this time FG. That being so, by continually taking a greater magnitude in this way I shall never arrive at AB, whereas I shall always be getting a lesser fraction of the time originally given. Therefore the force must be infinite; for it exceeds any finite force if the time occupied by the action of [15] any finite force must also be finite (for if a given force moves something in a certain time, a greater force will do so in a lesser time, but still a definite time, in inverse proportion). But a force must always be infinite—just as a number or a magnitude is—if it exceeds all definite limits. This point may also be proved in another [20] way—by taking a finite magnitude in which there resides a force the same in kind as that which resides in the infinite magnitude, so that this force will be a measure of the finite force residing in the infinite magnitude.