Aristotle · a new plain-English translation from the original language
1| Book Z. If continuous, touching, and successive are as they have been defined earlier — continuous things being those whose extremities are one, touching things being those whose extremities are together, and successive things being those between which there is nothing of the same kind in between — then it is impossible for anything continuous to be composed of indivisibles, for example a line to be composed of points, if indeed the line is continuous and the point is indivisible. For the extremities of points are not one (since there is no extremity of an indivisible distinct from some other part of it), nor are their extremities together (since there is no extremity at all of what has no parts, for the extremity and that of which it is the extremity are different things). Further, it is necessary that the points of which the continuous thing is composed be either continuous with one another or touching one another, and the same account holds for all indivisibles alike. Now they cannot be continuous, for the reason just stated; and everything touches either whole to whole, part to part, or whole to part. But since the indivisible has no parts, it must touch whole to whole. And what touches whole to whole will not be continuous: for the continuous has one part here and another there, and is divided into parts that are different in this way and separated in place. But neither, again, will point be successive to point, or the now to the now, so that length or time could be composed of these —
2| for successive things are those between which nothing of the same kind lies in between, and between points there is always a line, and between nows there is always time. Further, length or time would be divisible into indivisibles, if indeed each thing is divided into the things of which it is composed; but none of the continuous things was found to be divisible into partless things. And it is not possible for anything of another kind to lie between. For it will be either indivisible or divisible, and if divisible, either into indivisibles or into things always further divisible; and this last is what is continuous. And it is also clear that everything continuous is divisible into parts that are always further divisible; for if it were divisible into indivisibles, there would be an indivisible touching an indivisible, since the extremities of continuous things are one and it is by these that they touch.
3| By the same reasoning, magnitude, time, and motion are all alike composed of indivisibles and divided into indivisibles, or else none of them is. This is clear from the following. If a magnitude is composed of indivisibles, then the motion over it will also be composed of an equal number of indivisible motions — for example, if the magnitude ΑΒΓ is composed of the indivisibles Α, Β, Γ, then the motion ΔΕΖ, by which Ω was moved over ΑΒΓ, has each of its parts indivisible. If, then, wherever motion is present something must be in motion, and wherever something is in motion, motion must be present, then being-in-motion too will be composed of indivisibles. Ω, then, was moved over Α by the motion Δ, over Β by the motion Ε, and likewise over Γ by the motion Ζ. Now if it is necessary that what is moved from one place to another cannot at the same time be moving and have moved to the place it was moving toward, at the time it was moving there (for example, if something is walking to Thebes, it is impossible for it to be walking to Thebes and to have walked to Thebes at the same time), and Ω was moving over the partless Α by the motion Δ, which was present to it — then, if it had finished passing through Α at a time later than when it was passing through it, the motion would be divisible (for while it was passing through, it was neither at rest nor had it finished passing through, but was in between); whereas if it passes through and has passed through at the same time, then the thing walking, at the moment it is walking, will already have walked to that point, and will have completed its motion over the very thing it is still moving over.
4| But if something is moving over the whole of ΑΒΓ, and the motion by which it moves is ΔΕΖ, while over the partless Α it is not moving at all but has already moved, then the result would be that motion is composed not of motions but of completed movements, and that something has moved without ever being in motion — for it has completed its passage through Α without traversing it. So there will be something that has walked without ever walking: for it has walked this stretch without walking it. If, then, everything must be either at rest or in motion, it is at rest with respect to each of Α, Β, Γ, so that there will be something continuously at rest and in motion at the same time — for it was moving over the whole of ΑΒΓ, and at rest with respect to any given part of it, and therefore with respect to the whole. And if the indivisibles of ΔΕΖ are motions, then it would be possible, while motion is present, not to be in motion but to be at rest; but if they are not motions, then motion will not be composed of motions. And in the same way as with length and motion, time too must be indivisible, and composed of indivisible nows. For if the motion is wholly divisible, and a thing moving at equal speed traverses a lesser distance in a lesser time, then time too will be divisible. But if the time in which something is carried over Α is divisible, then Α too will be divisible.
5| Since every magnitude is divisible into magnitudes (for it has been shown that nothing continuous can be made of indivisibles, and every magnitude is continuous), it is necessary that the faster thing moves a greater distance in an equal time, an equal distance in a lesser time, and a greater distance in a lesser time, just as some people define the faster. For let A be faster than B. Since, then, the faster is what changes earlier, in the time in which A has changed from C to D, say ZH, B will not yet have reached D in this time but will fall short of it, so that in an equal time the faster covers more. But indeed it also covers more in a lesser time: for in the time in which A has arrived at D, let B, being the slower, be at E. Then, since A has arrived at D in the whole time ZH, it will be at the point Θ in a time less than this; let this time be ZK. Now CΘ, which A has traversed, is greater than CE, while the time ZK is less than the whole time ZH.
6| So it covers more in less time. And it is clear from this too that the faster traverses an equal distance in less time. For since it traverses the greater distance in less time than the slower does, while it itself, taken on its own, traverses a greater distance in more time than a lesser distance — say LM in more time than LΞ — the time PR, in which it traverses LM, would be more than PS, in which it traverses LΞ. So that if the time PR is less than X, the time in which the slower traverses LΞ, then PS too will be less than X: for it is less than PR, and what is less than the lesser is itself also less, so that the faster will move over the equal distance in less time. Further, since everything must move either in an equal time, a lesser time, or a greater time, and what moves in more time is slower, what moves in an equal time is of equal speed, and the faster is neither of equal speed nor slower, the faster could move in neither an equal time nor a greater time. It remains, then, that it moves in a lesser time, so that the faster must also traverse an equal magnitude in less time. Since every motion occurs in time, and in
7| any time whatever it is possible to move, and everything that moves admits of moving both faster and slower, so in any time whatever there will be motion both faster and slower. Given this, time too must be continuous. By continuous I mean what is divisible into parts that are always further divisible; for once this is laid down as what continuous means, time must be continuous. For since it has been shown that the faster traverses an equal distance in less time, let A be the faster and B the slower, and let the slower have moved over the magnitude CD in the time ZH. It is clear, then, that the faster will move over the same magnitude in less time than this; let it have moved over it in ZΘ. Again, since the faster has traversed the whole of CD in ZΘ, the slower in the same time traverses less; let this be CK. And since the slower, B, has traversed CK in the time ZΘ, the faster traverses it in less time, so that the time ZΘ will again be divided. And as this time is divided, the magnitude CK will also be divided in the same ratio. And if the magnitude is divided, so is the time.
8| And this will always happen if we keep alternating from the faster to the slower and from the slower to the faster, using what has been demonstrated: for the faster will divide the time, and the slower the length. If, then, this alternation always holds, and a division always results from the alternation, it is clear that every time will be continuous. And at the same time it is also clear that every magnitude is continuous: for time is divided into the same divisions, and equal ones, as the magnitude. Further, it is also clear from the arguments customarily given that if time is continuous, so is magnitude, since a thing traverses half the distance in half the time, and in general a lesser distance in a lesser time: for the divisions of time and of magnitude will be the same. And if either one is infinite, so is the other, and in whatever way the one is infinite, so is the other: for instance, if time is infinite at its extremities, the length is also infinite at its extremities; and if it is infinite by division, the length is also infinite by division; and if in both ways, the magnitude is also infinite in both ways.
9| For this reason too Zeno's argument assumes something false, namely that it is not possible to traverse infinitely many things, or to touch infinitely many things one by one, in a finite time. For both length and time, and in general everything continuous, are called infinite in two senses: either by division or at their extremities. Now it is not possible to touch things that are infinite in quantity in a finite time, but it is possible to touch things infinite by division: for time itself is infinite in this way too. So it turns out that a thing traverses the infinite in an infinite time, not a finite one, and touches infinitely many things by infinitely many contacts, not by finitely many. So it is not possible to traverse the infinite in a finite time, nor the finite in an infinite time; but if the time is infinite, the magnitude will also be infinite, and if the magnitude is infinite, so is the time. For let AB be a finite magnitude, and C an infinite time; and let some finite part of the time be taken, CD. In this, then, something traverses some part of the magnitude; let what it has traversed be BE. This will either measure AB exactly, or fall short of it, or exceed it: it makes no difference which.
10| For if it always traverses a magnitude equal to BE in an equal time, and BE measures the whole exactly, then the whole time in which it traversed the magnitude will be finite: for the magnitude too will be divided into equal parts corresponding to the time. Further, if not every magnitude is traversed in infinite time, but it is possible for some part to be traversed even in a finite time — for instance BE — and this measures the whole exactly, and it traverses an equal part in an equal time, then the time too will be finite. And that BE is not traversed in an infinite time is clear, if a finite time is taken on the other side: for if it traverses the part in less time, this time must be finite, since the other limit exists. And the same demonstration applies if the length is infinite but the time is finite.
11| It is clear, then, from what has been said, that neither a line nor a plane nor, in general, anything continuous will be indivisible — not only for the reason just given, but also because it will follow that the indivisible is divided. For since in every time there is a faster and a slower, and the faster traverses more in an equal time, and it is possible for it to traverse a length twice as great, or one and a half times as great as the slower — for this could be the ratio of the speed — let the faster, then, have covered one and a half times the distance in the same time, and let the magnitudes be divided: that of the faster into three indivisibles, AB, BC, CD, and that of the slower into two, EZ, ZH. Then the time too will be divided into three indivisibles: for it covers an equal part in an equal time. Let the time, then, be divided into KL, LM, MN. Again, since the slower has traversed EZH, the time too will be cut in two. The indivisible, then, will be divided, and the partless thing will traverse its distance not in an indivisible time but in more than one. It is clear, then, that nothing continuous is without parts.
12| The now — the one spoken of not in virtue of something else but in its own right, and spoken of as primary — must also be indivisible, and something of this kind must be present in every time. For there is a last point of what has come to be, beyond which nothing of what is going to be exists, and again, of what is going to be, beyond which nothing of what has come to be exists; and this we say is a limit of both. And if it is shown that this limit is one and the same thing in both cases, it will at once be clear that it is also indivisible. Now the now that is the last point of both times must be the same one; for if it were different, the one could not be next in succession to the other, since a continuum cannot be composed of things without parts, and if the two are separate, there will be time between them; for every continuum is such that there is something of the same kind between its limits. But then, if what lies between is time, it will be divisible; for every time has been shown to be divisible. So the now would be divisible. But if the now is divisible, part of what has come to be will lie in the future and part of what is going to be will lie in the past; for whatever point it is divided at will mark off the time that has elapsed from the time still to come. And at the same time the now would no longer be a now in its own right, but in virtue of something else —
13| for the division would not be in its own right. Besides this, part of the now will be past and part future, and it will not always be the same part that is past or future. Nor, then, will the now be the same now, since time is divisible in many different ways. So since these consequences are impossible, the now in each of the two times must be one and the same. But if it is the same, it is clear that it is also indivisible; for if it were divisible, the same consequences as before would follow all over again. That there is, then, something indivisible in time — what we call the now — is clear from what has been said; and that nothing moves in the now is clear from the following. For suppose something were moving in it: then nothing would stop that mover from going faster at one moment and slower at another, within that very now. Let the now, then, be N, and let the faster thing have moved through AB in it. Then the slower thing will move through less than AB in that same now — say through AC. Since the slower thing has moved through AC in the whole of the now, the faster thing will move through this distance AC in less time than that, so that the now will be divided. But it was indivisible. Therefore it is not possible for anything to move in the now. But
14| it is not at rest in it either. For we call a thing at rest when it is naturally such as to move but is not moving, when, where, and in the way it is naturally such as to move; so since nothing is naturally such as to move in the now, clearly nothing is at rest in it either. Further, if the now is the same in both times, and it is possible for the whole of one time to be a time of motion and the whole of the other a time of rest, then the thing moving through the whole time will be moving in any part of it whatsoever, in accordance with that in respect of which it is naturally such as to move, and the thing at rest will likewise be at rest in any part of its time; and it will follow that the same thing is at once at rest and in motion, since the same now is the last point of both times. Further, we call a thing at rest when it, and its parts, are in the same condition now as they were before; but in the now there is no before, so that it cannot be at rest in the now either. It is necessary, then, both that what is moving be moving in time and that what is at rest be at rest in time.
15| Everything that changes must be divisible. For since every change is from something to something, then when a thing is already in that to which it was changing, it is no longer changing, and when it — itself and all its parts — is still in that from which it was changing, it is not yet changing (for what is in the same condition, itself and all its parts, is not changing); so it is necessary that part of the changing thing be in the one state and part in the other, since it cannot be in both, nor in neither. By that into which it changes I mean the first thing reached in the course of the change — for example, from white it changes into gray, not into black; for the changing thing need not be at either of the two extremes. It is clear, then, that everything that changes will be divisible.
16| Motion is divisible in two ways: in one way by time, and in another according to the motions of the parts of the thing moved — for example, if the whole AC is moving, then AB will be moving and BC will be moving too. Let DE, then, be the motion of AB, and EF the motion of BC. Then the whole motion, DF, must be the motion of AC; for AC will move in accordance with it, since each of the two parts moves in accordance with each of the two motions, and nothing moves in accordance with the motion of something else; so the whole motion is the motion of the whole magnitude. Further, if every motion is the motion of something, and the whole motion, DF, is neither the motion of either of the two parts (for each of the two motions belongs to a part) nor the motion of anything else (for that of which the whole is the motion, of that too the parts are the motions of the parts; and the parts DE and EF are the motions of AB and BC and of nothing else, since there could not be one motion belonging to more than one thing), then the whole motion must indeed be the motion of the magnitude ABC. Further, if there is some other motion of the whole — say HI — the motion belonging to each of the two parts will be subtracted from it; and these subtracted motions will be equal to DE and EF —
17| for one motion belongs to one thing. So if the whole of HI is divided out into the motions of the parts, HI will be equal to DF; but if something is left over — say KI — this will be the motion of nothing (for it will be the motion neither of the whole nor of the parts, since one motion belongs to one thing, nor of anything else, since a continuous motion is the motion of some one continuous thing), and the same holds if the parts exceed it in the division; so since this is impossible, the two must be one and the same, and equal. This, then, is the division according to the motions of the parts, and it must hold of everything divisible; there is also another division, according to time. For since a motion of any given amount takes place in time, and every time is divisible, and in the lesser time there is less motion, every motion must be divisible according to time. And since everything that
18| is moved is moved in something and for some period of time, and there is a motion belonging to everything moved, the divisions of the time, of the motion, of the being-moved, of the thing moved, and of that in which the motion takes place must all be the same — except that they do not hold alike of all the things in which the motion takes place, but the division of place belongs to it in its own right, while the division of the quality belongs to it only accidentally. For let the time in which the thing is moved be taken as A, and the motion as B. If, then, it has moved through the whole distance in the whole time, it will move through less than the whole in half the time, and again, when this half is divided, through less than this in the smaller part, and so on always. Likewise too, if the motion is divisible, the time is also divisible; for if it moves through the whole distance in the whole time, it moves through half the distance in half the time, and again through a smaller distance in a smaller time. In the same way the being-moved will also be divided. For let the being-moved be taken as C. Then in accordance with the half of the motion it will be less than the whole, and again in accordance with the half of that half, and so on always. And it is also possible, having set out the being-moved corresponding to each of the two motions — say corresponding to DC and to CE — to say that the being-moved corresponding to the whole will correspond to the whole motion (for if it were something else, there would be more than one instance of being-moved corresponding to the same motion), just as we showed
19| that motion too is divisible into the motions of the parts. For once the being-moved corresponding to each of the two motions is taken, the whole will be continuous. In the same way it will also be shown that length is divisible, and in general everything in which change occurs (except that some things are divisible only accidentally, because the thing changing is divisible); for when the one thing is divided, all the rest will be divided along with it. And the same will hold alike of all of them as regards being finite or infinite. And it is above all from the changing thing that the divisibility and infinity of all these has followed; for the divisible and the infinite belong to the changing thing immediately. The divisible, then, has already been shown; the infinite will be made clear in what follows.
20| Since everything that changes changes from something into something, what has changed must, when it has first changed, be in that into which it has changed. For what changes departs from, or leaves behind, that from which it changes, and either changing and leaving behind are the same thing, or leaving behind follows upon changing. And if leaving behind follows upon changing, then having left behind follows upon having changed; for each stands to the other in the same way. Since, then, change according to contradiction is one kind of change, when a thing has changed from not-being into being, it has left behind not-being. It will therefore be in being; for everything must either be or not be. It is clear, then, that in change according to contradiction, what has changed will be in that into which it has changed. And if this holds in this case, it holds in the others too; for it is the same with one as with the rest. Further, this is also clear if we take each kind of change on its own, given that what has changed must be somewhere or in something. For since it has left behind that from which it has changed, and it must be somewhere, it will be either in this or in something else. Now if what has changed into B is in something else, say C, then it changes again from C into B; for B was not adjacent to C, and change is continuous. So what has changed, at the moment it has changed, is changing into that into which it has changed. But this is impossible; therefore what has changed must be in this very thing into which it has changed. It is clear, then, that what has come to be, at the moment it has come to be, will exist, and what has perished will not exist; for this has been stated universally, about every kind of change, and it is most evident in change according to contradiction.
21| That, then, what has changed, at the moment it has first changed, is in that into which it has changed, is clear. But that in which what has changed has first changed must be indivisible. By "first" I mean that which is not so in virtue of some part of it being so. For let AC be divisible, and let it be divided at B. If, then, it has changed in AB, or again in BC, it will not have changed in AC as a first interval. But if it was changing in each of the two — for it must either have changed or be changing in each of them — then it would be changing in the whole too; but it was assumed to have already changed. The same account holds if it is changing in the one and has changed in the other; for then there will be something prior to the first, so that that in which it has changed cannot be divisible. It is clear, then, that both what has perished and what has come to be do so in something indivisible — the one having perished, the other having come to be, in an indivisible interval. Now "that in which it has first changed" is spoken of in two ways:
22| on the one hand, that in which the change was first completed (for then it is true to say that it has changed), and on the other, that in which it first began to change. Now the "first" spoken of with reference to the end of the change does hold good and does exist (for a change can be completed, and there is an end of change, which has in fact been shown to be indivisible, because it is a limit); but the "first" spoken of with reference to the beginning does not exist at all. For there is no beginning of change, nor did it change in a first period of time. For let AD be supposed to be first. Now this cannot be indivisible; for it would follow that the nows are adjacent to one another. Further, if throughout the whole time CA it is at rest — let it be assumed to be at rest — then it is also at rest at A, so that if AD is without parts, it will at once be at rest and will have changed; for at A it is at rest, and at D it has changed. But since it is not without parts, it must be divisible, and it must have changed in some part of it, whatever that part may be. For if AD is divided, and it has changed in neither part, then it has not changed in the whole either; but if it is changing in both, then it is changing in the whole too; or if it has changed in one of the two, then it has not changed in the whole as a first interval. So it must have changed in some part or other. It is clear, then, that there is no first interval in which it has changed; for the divisions are infinite. Nor, again, is there any first part of what has changed that has changed. For let DZ be the first part of DE that has changed; for it has been shown that whatever changes is throughout divisible. And let the time in which DZ has changed be HI. If, then, DZ has changed in the whole of this time, then in half the time something less will have changed, and prior to DZ, and again something else prior to that, and something else prior to that, and so on forever. So there will be nothing first belonging to what is changing that has changed.
23| That, then, there is nothing first either of what is changing, or of the time in which it changes, is clear from what has been said. But the thing itself that changes, or that in respect of which it changes, will not be in the same case. For there are three things spoken of in connection with change: the thing that changes, that in which it changes, and that into which it changes — for example, the man, the time, and the white. Now the man and the time are divisible, but the case of the white calls for a different account. Except that everything is divisible in an accidental sense; for that to which the white or the quality is an accidental attribute is itself divisible. Since, in fact, whatever is called divisible in its own right, and not accidentally, there will be no first thing among these either — for example, among magnitudes. For let AB be a magnitude, and let it have moved first from B to C. Then if BC is indivisible, a partless thing will be adjacent to a partless thing; but if it is divisible, there will be something prior to C into which it has changed, and again something else prior to that, and so on forever, because the division never gives out. So there will be no first thing into which it has changed. The same holds for change in quantity too; for this too takes place in something continuous. It is clear, then, that only in motion in respect of quality is it possible for something to be indivisible in its own right.
24| Since everything that changes changes in time, and "to change in time" is said both in the sense of "in a primary time" and in the sense of "in virtue of another" — as, for instance, we say a thing changes in the year because it changes in the day — what changes, changing in a primary time, must be changing in any part whatever of that time. This is clear from the definition itself (for this is how we defined "primary"); but it is also evident from the following. For let XP be the primary time in which the moving thing is moving, and let it be divided at K; for every time is divisible. Now in the time XK it either is moving or is not moving, and likewise again in KP. If, then, it is moving in neither, it would be at rest in the whole (for it is impossible for a thing to be moving while moving in none of its parts); and if it is moving in only one of the two, then it would not be moving in XP as a primary time; for then the motion would be "in virtue of another." It is necessary, then, that it be moving in any part whatever of XP.
25| Now that this has been shown, it is clear that everything that is moving must have moved previously. For if in the primary time XP the magnitude KL has been moved through, then in half the time a thing moving at the same speed and starting at the same moment will have moved through half of it. And if a thing moving at the same speed has moved through something in the same time, then the other thing too must have moved through the same magnitude, so that what is moving will have already moved. Further, if we say that it has moved in the whole time XP, or in general in any time whatever, we say so by taking the extreme now of that time (for this is what marks it off, and what lies between the nows is time), and the same account would apply to saying that it has moved in the other parts too. And the extreme of the half is the point of division. So it will have moved in the half too, and in general in any part whatever; for at every point of division there is always a time marked off by the nows. If, then, all time is divisible, and what lies between the nows is time, then everything that changes will have changed an infinite number of times. Further, if it is necessary that whatever changes continuously, and has neither perished nor ceased from its change, either be changing or have changed in any part whatever, and it is not possible to be changing in a now, then it must have changed at each of the nows; so that, if the nows are infinite in number, everything that changes will have changed an infinite number of times.
26| And not only must what is changing have changed,
27| But also, what has changed must have been changing earlier; for everything that has changed from one thing into another has changed in time. For let it have changed from A to B in the now. Then, in the very now in which it is in A, it has not changed (for it would then be in A and in B at the same time); for it has been shown earlier that what has changed, when it has changed, is not in that from which it changed. But if it is in another now, there will be a time in between; for the nows were not next to one another. Since, then, it has changed in time, and every time is divisible, in the half of that time something else will have changed, and again in the half of that half something else, and so on forever; so that it would have been changing earlier. Further, what has been said is clearer in the case of magnitude, because the magnitude in which what changes changes is continuous. For let something have changed from Γ to Δ. Now if ΓΔ is indivisible, a partless thing will be next to a partless thing; but since this is impossible, the interval between them must be a magnitude, and divisible into infinitely many parts; so that it changes into those parts earlier. Everything that has changed, then, must have been changing earlier. For the same demonstration holds also for things that are not continuous, for instance among contraries and in contradiction; for we will take the time in which it has changed, and again say the same things. So it is necessary that what has changed be changing, and that what is changing have changed, and the having-changed will be prior to the changing, and the changing prior to the having-changed, and the first point will never be grasped. The cause of this is that a partless thing is not next to a partless thing; for the division is infinite, just as with lines that are being increased or decreased.
28| It is clear, then, that what has come to be must have been coming to be earlier, and what is coming to be must have come to be, in the case of everything divisible and continuous — not, however, always the very thing that is coming to be, but sometimes something else, some part of it, as the foundation comes to be before the house. Likewise too with what is perishing and what has perished; for something infinite is present at once in what is coming to be and in what is perishing, given that it is continuous, and it is not possible for anything to be coming to be without something having come to be, nor to have come to be without something coming to be; and likewise too with perishing and having perished, for the having-perished will always be prior to the perishing, and the perishing prior to the having-perished. It is clear, then, that what has come to be must have been coming to be earlier, and what is coming to be must have come to be; for every magnitude and every time is always divisible. So that, whatever it is in, it will not be in that as in a first thing.
29| Since everything that is moved is moved in time, and moves through a greater magnitude in more time, it is impossible for a finite magnitude to be moved through an infinite time — not in the sense that the same part is always in motion while only some part of the magnitude moves, but in the sense that the whole is in motion during every part of the time. That, if something moves at a uniform speed, the finite magnitude must be moved in a finite time is clear: for if a part is taken that will measure the whole, the whole will have been moved through in as many equal times as there are parts, so that since these are finite both in how great each is and in how many they all are, the time too would be finite; for it will be that many times as great as the time of the part, multiplied by the number of the parts. But even if it does not move at a uniform speed, it makes no difference. For let AB be a finite interval, which has been moved through in the infinite time, and let ΓΔ be the infinite time. If, then, one part of it must have been moved before another (and this is clear, because in the earlier and later part of the time a different part has been moved; for always in the greater time a different part will have been moved, whether the change is at a uniform speed or not, and whether the motion intensifies or slackens or stays the same, none the less), let some part of the interval AB be taken, AE, which will measure AB.
30| This, then, occurred in some part of the infinite time; for it cannot occur in an infinite time, since it is the whole that is moved through in the infinite time. And again, if I take another part as great as AE, it too must occur in a finite time; for again it is the whole that is moved through in the infinite time. And continuing to take parts in this way — since no part of the infinite has a measure that will measure it out (for it is impossible for the infinite to be composed of finite parts, whether equal or unequal, because the finite parts, being determinate both in number and in magnitude, would then be measured by some one thing, whether they are equal or unequal, provided each is bounded in magnitude, none the less) — while the finite interval is measured by so many parts equal to AE, AB would be moved through in a finite time (and likewise too with rest); so that it is not possible for anything one and the same to be always coming to be or perishing.
31| The same argument shows also that an infinite magnitude cannot be moved, nor brought to rest, in a finite time, whether it moves uniformly or not. For if some part is taken that will measure out the whole time, in that part the moving thing will traverse a certain amount of the magnitude, and not the whole (for in the whole time it traverses the whole, and again in an equal part it traverses another such amount, and likewise in each part, whether that amount is equal or unequal to the first; for it makes no difference, so long as each is finite); for it is clear that as the time is used up the infinite magnitude will not be used up, since the amount subtracted is finite both in size and in the number of times it is subtracted; so that the infinite is not traversed in a finite time. And it makes no difference whether the magnitude is infinite in one direction or in both, for the argument will be the same. Now that these points have been demonstrated, it is clear that a finite magnitude cannot traverse an infinite one in a finite time either, for the same reason; for in a part of the time it traverses a finite amount, and likewise in each part, so that in the whole time it traverses a finite amount. And since the finite does not traverse the infinite in a finite time, it is clear that the infinite does not traverse the finite either; for if the infinite traversed the finite, the finite would also have to traverse the infinite.
32| For it makes no difference which of the two is the thing moved; either way the finite traverses the infinite. For whenever the infinite, on which let A stand, is moved, there will be some finite part of it at B, for instance ΓΔ, and again another and another, and so on forever. So that it will follow at once both that the infinite has been moved past the finite and that the finite has traversed the infinite; for presumably it is not possible for the infinite to be moved past the finite in any other way than by the finite traversing the infinite, either by being carried along it or by measuring it out. So that, since this is impossible, the infinite cannot traverse the finite. But neither, again, does the infinite traverse the infinite in a finite time; for if the infinite magnitude does, so does the finite; for the finite is contained within the infinite. Further, if the time is also taken as infinite, the same demonstration will apply. Since, then, neither the finite traverses the infinite, nor the infinite the finite, nor the infinite the infinite, in a finite time, it is clear that motion too cannot be infinite in a finite time; for what difference does it make whether one makes the motion infinite or the magnitude? For necessarily, if either one is infinite, so is the other;
33| for every locomotion occurs in place.
34| Since everything naturally disposed to be in motion or at rest is in motion or at rest when, where, and as it is naturally disposed to be, what is coming to rest must be in motion while it is coming to rest; for if it is not in motion, it will be at rest, but what is already at rest cannot be in the process of coming to rest. Now that this has been demonstrated, it is clear that a thing must also come to rest in time (for what is in motion is in motion in time, and what is coming to rest has been shown to be in motion, so it must come to rest in time). Further, since we speak of faster and slower with reference to time, and coming to rest admits of faster and slower, then whatever time is the primary time in which a thing comes to rest, it must be coming to rest in any part whatever of that time. For if the time is divided and the thing comes to rest in neither part, then it does not come to rest in the whole either, so that what is coming to rest would not be coming to rest at all; but if it comes to rest in one of the two parts, then the whole would not be the primary time in which it comes to rest, since in that case it comes to rest in virtue of something else, namely the part, as was said earlier in the case of what is in motion. And just as there is no primary time in which what is in motion is in motion, so too there is none in which what is coming to rest is coming to rest; for there is nothing primary either of being in motion or of coming to rest. For let AB be the primary time in which it comes to rest. Now this cannot be without parts (for there is no motion in what is without parts, since something of the thing would then have to have moved, and what is coming to rest has been shown to be in motion); but then, if it is divisible, the thing comes to rest in any part whatever of it, for this was shown earlier — that whatever the primary time is in which a thing comes to rest, it comes to rest in any part whatever of that time. Since, then, the time in which it primarily comes to rest is a time, and not something indivisible, and every time is divisible into infinitely many parts, there will be no time that is the primary time in which it comes to rest.
35| Nor, again, is there a first moment at which what is at rest came to rest. For it did not come to rest in something without parts, since there is no motion in an indivisible; and wherever there is being-at-rest there is also being-in-motion (for we say a thing is at rest when, in the time in which it is naturally disposed to be in motion, the thing so disposed is not in motion; and further, we also say a thing is at rest when it is in the same condition now as before, judging this not by reference to one single now but to two nows at the least — so that the time in which a thing is at rest cannot be without parts). But if it is divisible, it will be a time, and the thing will be at rest in any part whatever of it; for this will be shown in the same way as in the previous cases, so that there will be nothing primary. The cause of this is that everything is at rest and in motion in time, and there is no primary time, nor any primary magnitude, nor indeed anything continuous at all that is primary; for everything is divisible into infinitely many parts.
36| Since everything that is in motion is in motion in time and changes from something into something, then in the time in which it is in motion in its own right, and not by virtue of being in some part of that time, it is impossible for what is in motion then to be, as a primary state, at some determinate place. For being at rest is being in the same place for some time, both the thing itself and each of its parts. For this is what we mean by being at rest: that at one now after another it is true to say that both the thing itself and its parts are in the same place. But if this is what being at rest is, then it is not possible for what is changing to be, as a whole, at some determinate place during the primary time; for every time is divisible, so that it will be true to say, at one part of it after another, that both the thing itself and its parts are in the same place. For if this is not so, but the thing is at some determinate place only at a single now, then it will be at that place for no time at all, but only at the limit of the time. Now at a now the thing always is, to be sure, at some determinate place, but it is not thereby at rest; for it is not possible either to be in motion or to be at rest at a now — rather, it is true that at a now the thing is not in motion and is at some determinate place, but it is not possible for a thing to be at rest, at some determinate place, throughout a stretch of time; for that would mean that what is being carried along is at rest.
38| Zeno's reasoning is fallacious. For, he says, if everything is always at rest whenever it occupies a space equal to itself, and what is being carried is always occupying such a space in the now, then the flying arrow is motionless. But this is false; for time is not composed of indivisible nows, any more than any other magnitude is. There are four arguments of Zeno's about motion that create difficulties for those who try to solve them. The first is the one denying motion on the ground that what is moving must arrive at the half-way point before it arrives at the end; we have dealt with this in our earlier discussions. The second is the one called the Achilles. It holds that the slowest runner will never be caught, as it runs, by the fastest; for the pursuer must first reach the point from which the pursued started, so that the slower must always be some distance ahead. This is the same argument as the dichotomy, but it differs in that the magnitude successively taken is not divided in half. So the conclusion that the slower is not caught does follow from the argument, and it arises from the same cause as in the dichotomy (for in both cases the result is a failure to reach the limit, because the magnitude is divided in some way; but this argument has the added feature that not even the fastest thing — for all the drama of the case — catches the slowest as it pursues it), so the solution must be the same as well. The claim that what is ahead is not caught is false: it is true that it is not caught while it is still ahead, but nevertheless it is caught, provided one grants that it gets through the finite distance. These, then, are the first two arguments. The third is the one just stated, that the flying arrow stands still. It results from taking time to be composed of nows; for if this is not granted, the deduction will not go through.
39| The fourth is the argument about the bodies moving in the stadium, in opposite directions, past equal bodies, at equal speed — one group starting from the end of the stadium, the other from the middle — in which he thinks it results that half a time is equal to double that time. The fallacy lies in requiring that a body of equal magnitude, moving at equal speed, takes an equal time to pass a moving body as to pass a body at rest; and this is false. For example: let the bodies at rest be equal, the A's; let the B's, equal to these in number and in magnitude, start from the middle; and let the Γ's, equal to these in number and in magnitude, start from the end, moving at the same speed as the B's. Then it results that the first B is, at the same moment, at the last A as the first Γ is, since they are moving past one another. And it results that the Γ has passed all of the B's, while the B has passed only half of the A's, so that on this reckoning the time is supposedly half as long; for each of the two takes an equal time to pass each single one. But at the same time it also results that the first B has passed all of the Γ's; for the first Γ and the first B will be at the opposite ends at the same moment, because both take an equal time to pass the A's.
40| This, then, is the argument, and it results from the falsehood already stated. Nor, again, will anything be impossible for us in the case of change between contradictories — for example, if a thing changes from not-white to white and is in neither state, it does not follow that it will then be neither white nor not-white; for it is not the case that, because it is not wholly in either state, it will therefore not be called white or not-white. We call a thing white or not-white not because it is wholly such, but because most of its parts, or the most decisive parts, are such; and not being in this state is not the same as not being wholly in this state. The same holds likewise for being and not-being, and for the other cases governed by contradiction: the thing will necessarily be in one or the other of the two opposed states, but it will not always be wholly in either. Again, take the case of the circle and the sphere, and in general of things that move within their own place: it might seem that it will result that they are at rest, since both the thing itself and its parts will be in the same place for some time, so that it would be at rest and in motion at once. But in the first place, the parts are not in the same place for any time at all,
41| Moreover, the whole is also always changing into something different. For the arc taken from A is not the same as the arc taken from B, or from C, or from each of the other points, except in the way that "the musical man" and "man" are the same — that is, accidentally. So one arc is always changing into another, and it will never be at rest. And the same holds, in the same way, for the sphere too, and for the other things that move within them.
42| Now that these points have been demonstrated, we say that what has no parts cannot move except accidentally — for example, when the body or magnitude in which it is present is moving, just as something in a ship might move because of the ship's motion, or a part might move because of the motion of the whole. (By "what has no parts" I mean what is indivisible in quantity.) For the motions of the parts are different, both from one another and from the motion of the whole. One can see this difference most clearly in the case of the sphere: the speed of the points near the center is not the same as that of the points on the outside, nor the same as that of the whole — which shows that the motion is not one. So then, as we said, what has no parts can move in this way, the way a person sitting in a ship moves while the ship runs its course, but it cannot move in its own right. For let it be changing from AB to BC — whether from one magnitude to another, or from one form to another, or between contradictories — and let the time in which it first changes be D. Then necessarily, during the time in which it is changing, it must be either in AB or in BC, or partly in the one and partly in the other —
43| for everything that changes is in this condition. Now no part of it can be in each of the two, for then it would be divisible. But it cannot be in BC either, since then it would have already changed, whereas it is assumed to be changing. It remains, then, that it is in AB during the time in which it is changing. It will therefore be at rest, since being in the same place for some time was what being at rest meant. So it is not possible for what has no parts to move, or to change at all. For its motion could occur in only one way: if time were made up of nows. For then it would always have been moving and have already changed in the now, so that it would never be moving, but would always have moved. That this is impossible has already been shown earlier: neither is time made up of nows, nor a line made up of points, nor motion made up of movements. For anyone who says this is doing nothing other than making motion out of things without parts, just as if one made time out of nows, or length out of points. It is clear from the following considerations too that —
44| — neither a point nor anything else indivisible can move. For nothing that moves can traverse a greater distance before it has traversed one equal to or less than itself. If this is so, it is clear that the point too will first traverse a distance equal to or less than itself. But since it is indivisible, it cannot first traverse a lesser distance; so it will traverse one equal to itself. This means the line will be made up of points, since the point, always traversing a distance equal to itself, will measure out the whole line. But if this is impossible, then it is also impossible for the indivisible to move. Further, if everything moves in time, and nothing moves in the now, and every time is divisible, then for anything in motion there would be some time shorter than the time in which it traverses its own extent. For this will be a time in which it is moving, since everything moves in time, and it has been shown earlier that every time is divisible. If, then, a point is moving, there will be some time shorter than the time in which it traversed itself. But that is impossible, for in a lesser time it must traverse a lesser distance. So the indivisible will turn out to be divisible into something smaller, just as the time is divisible into a smaller time. For what has no parts and is indivisible could move in only one way: if it were possible for the indivisible thing to move in the now. For to move in the now, and for something indivisible to move, come to the same thing.
45| No change is infinite. For however many changes there were, each was from something to something — both the change between contradictories and the change between contraries. So for changes by way of contradiction, the affirmation and the negation are the limits (for instance, being is the limit of coming-to-be, and not-being the limit of perishing), while for changes between contraries, the contraries themselves are the limits, since these are the extremes of the change. This holds too for every alteration, since alteration proceeds from certain contraries, and likewise for increase and decrease: the limit of increase is the complete magnitude proper to the thing's own nature, and the limit of decrease is departure from this. Locomotion, however, will not be bounded in this way, since not every locomotion is between contraries. But since what cannot be traversed in this way is impossible because it is not possible to traverse it (for "impossible" is said in several ways), what is impossible in this sense cannot be traversed; and just as, in general, what cannot come to be does not come to be, so what cannot change cannot change into that which it is impossible to change into. If, then, what is being carried along is changing into something, it must also be possible for it to change into it. So motion is not infinite, and it will not traverse the infinite, since it is impossible to get through it. That there is no change so infinite, in this sense, that it is not bounded by limits, is clear. But we must consider whether change can be infinite in this other sense: infinite in time while remaining one and the same. For if it is not one, presumably nothing prevents this — for instance, if after locomotion there were alteration, and after the alteration increase, and then coming-to-be again; for in this way motion will always exist through time, but it will not be a single motion, since no single motion is made up of all of these. But for it to become one motion, it is not possible for it to be infinite in time, except in one case — and that is circular locomotion.