If then there is that which is continued, that which touches, and that which is successive, as was before defined, (viz. that those things are continued whose extremes are one, that those touch whose extremes are together, and that those are successive, between which there is nothing of the same kind,) if this be the case it is impossible that any thing continuous should be composed from indivisibles; as, for instance, a line from points, since a line is a continued quantity, but a point is indivisible: for neither are the extremities of points one; since of that which is indivisible, one thing is not the extreme, and another some other part. Nor do the extremes suosist together; for there is no extremity of that which is without parts: for the extremity is different from that of which it is the extremity. Farther still it is necessary either that the points from which the continued quantity consists should be continued, or touch each other. The same reasoning also applies to all indivisibles. They will not, therefore, be continued, for the reason already assigned. But every thing touches, either the whole the whole, or a part a part, or a part the whole. Since, however, an indivisible is without parts, it is necessary that the whole should touch the whole. But the whole which touches the whole will not be continued; since the continued has different parts, and is divided into parts that are thus different and separated by places. Neither will a point be successive to a point, or the now to the now, so as that a length or time will consist from these: for things are successive, between which there is nothing of the same kind; but between points there is always a line; and between nows, time. Again, if this were the case, continued quantity might be divided into divisibles, since every thing may be divided into those things from which it consists; but nothing continued can be divided into impartibles. Nor can any other genus subsist between points and nows: for if this were possible, it is evident that it would be either divisible or indivisible. And if divisible, it will either be divisible into indivisibles, or into things always divisible. But this is something continued. It is also evident that every thing which is continued is divisible into things always divisible: for if into indivisibles, the indivisible would be touched by the indivisible; since the extreme of things continued is one, and touches. But there is the same reasoning with respect to magnitude, time, and motion; for either each, or no one of these consists from indivisibles, and is divided into indivisibles. And this is evident from the following considerations: for if magnitude were composed from indivisibles, the motion also through the space of this magnitude would be composed from equal indivisible motions. Thus, if the magnitude A B C were composed from the indivisibles A B C the motion also D E F, with which O is moved through the interval A B C, will have each of its parts an indivisible. But if motion being present, it is necessary something should be moved, and if it is necessary that motion should be present if something is moved; if this be the case, to be moved also will consist of indivisibles. Hence O is moved through A, when it is moved with the motion D; and through B with the motion E; and in like manner through G with the motion F. If, therefore, it is necessary that what is moved, should be moved from one place to another, and not at the same time be moved and have been moved, whither it was moved when it was moved; (just as if any one walks to Thebes, it is impossible that he should at the same time walk to Thebes and have walked to Thebes,)–if this be the case, O was moved through A, which is without parts, so far as the motion D was present to it. So that if it should pass through after it has passed through, it will be divisible: for when it was passing through, it was neither at rest, nor had passed through, but was between both. But if, at the same time, it both passes through and had passed through, that which proceeds when it proceeds, will be there, and will be moved to whither it is moved. And if any thing should be moved through the whole length A B C, and the motion with which it is moved is D E F; but nothing moves but has been moved through that which is without parts A;–if this be admitted, motion will consist not from motions, but from the boundaries of motions ( inēmata) and something will have been moved which was not in motion; for it will have passed through A by not passing through it. Hence there will be something which has proceeded, though it has never at any time proceeded; for it has proceeded through this, not proceeding through it. If, therefore, it is necessary that every thing should either be at rest, or be in motion, but it is at rest in each of the parts A B C, there will be something which is at the same time continually at rest, and continually moved: for it was moved through the whole magnitude A B C, and was at rest in every part of it, and therefore in the whole. And if the indivisible parts of D E F, are motions, it is possible that when motion is present, a thing may not be moved, but be at rest. But if they are not motions, it will come to pass that motion is not composed from motions. In like manner it is necessary, that as length, so time, should be indivisible and be composed from nows, which are indivisible: for if all motion is divisible, but that which is equally swift passes through a less space in a less time, time also will be divisible. And if the time is divisible in which any thing is borne along through A, A also will be divisible. But since every magnitude is divisible into magnitudes (for it has been shown that it is impossible for any thing continued to be composed from indivisibles; and every magnitude is continued) it is necessary that what is swifter should pass through a greater space in an equal time, an equal space in a less time, and a greater in a less time, just as some define that which is swifter: for let A be swifter than B. Since, therefore, that which is first changed is more swift; hence in the time in which A is changed from C to D, as for instance, in the time F G, in this time B will not have yet arrived at D, but will fall short of it. So that what is swifter will pass through more space in an equal time. In a less time also it will pass through more space than that which is slower: for in the time in which A arrives at D, D will arrive at E, since it is slower. Hence, because A arrived at D, in the whole time F G; it will arrive at H in a less time than this, and it will arrive at it in the time F G. The space, therefore, C H, which A will pass through, is greater than C E. But the time F K is less than the whole time F G; so that it passes through a greater space in a less time. Hence also, it is evident that what is swifter passes through an equal space in a less time: for since it passes through a greater length in a less time than that which is slower, but when considered itself by itself, it passes through a greater length in a less time; as, for instance, the length L M, which is greater than the length L X; hence the time P R, in which it passes through the length L M, is greater than the time P S, in which it passes through the length L X. So. that if the time P R, is less than the time P T, in which the slower moveable quantity passes through the length L X, the time also P S, is less than the time P T; for it is less than P R. But that which is less than the less, is itself less; so that in a less time, it will have moved through an equal space: farther still, if it is necessary that every thing in motion should be moved either in an equal or in a less, or in a greater time; and that which is moved in a greater time is slower; in an equal time, equally swift; and if that which is swifter is neither equally swift, nor slower; if this be the case, that which is swifter is neither moved in an equal nor in a greater time. It remains, therefore, that it is moved in a less time. Hence it is necessary that the swifter should pass through an equal magnitude in a less time. But since every motion is in time, and in every time it is possible for something to he moved, but every thing which is moved, may be moved swifter and slower; hence in every time it is possible for a thing to be moved swifter and slower. But this being the case, it is necessary that time should be continued. And I call that continued which is divisible into parts that are always divisible: for the continued being admitted to be this, it is necessary that time should be continued: for since it has been shown that what is more swift, passes through an equal space in a less time, let A be that which moves with a greater velocity, but B that which moves with a less. And let.that which moves with a less velocity, or the slower, pass through the magnitude C D in the time F G. It is evident, therefore, that the swifter will pass through the same magnitude in a less time than this; and let this time be F H. Again, because the swifter passes through the whole length C D, in the time F H; the slower in the same time will pass through a lesser length. Let this, therefore, be C K. But since the slower B passes through the length C K in the time F H, the swifter will pass through it in a less time. So that again, the time F H will be divided. But this being divided, the magnitude also C K will be divided in the same ratio. And if the magnitude, also the time; and this will always be the case if we assume a progression from the swifter to the slower, and from the slower to the swifter, and make use of that which has been demonstrated: for the swifter divides the time, and the slower the length. If, therefore, conversion is always true, and division is always produced by conversion; it is evident that all time is continued. But at the same time it is also manifest, that every magnitude is continued; for time and magnitude are divided according to the same and equal divisions. Again, from what is usually asserted, it is evident, that if time is continued, magnitude is likewise; since in the half of a certain time the half of a certain space is passed through; and in short, a less space in, a less time: for there will be the same divisions of time and magnitude. And if the one is infinite the other is also. As likewise is the one, so is the other: for instance, if time is infinite in the extremes, length also will be infinite in the extremes. And if time is infinite in division, this also will be the case with length; but if time is infinite in both these, length likewise will be infinite in both. Hence the reasoning of Zeno assumes that which is false, viz. that it is not possible to pass through infinites, or touch infinites one by one, in a finite time: for length, and time, and in short every thing continued, are said to be infinite in a twofold respect, viz. either according to division, or from the extremes. Things, therefore, that arc infinite according to quantity, cannot be touched in a finite time. But things according to division may; for time is thus infinite. So that it happens that the infinite may be passed through in an infinite, and not in a finite time, and that infinites may touch infinites in infinite, and not in finite, times. Neither, therefore, can the infinite be passed through in a finite time, nor the finite in an infinite time; but if time is infinite, magnitude also will be infinite; and if magnitude, likewise time: for let there be a finite magnitude A B, and an infinite time C. Let also some finite part of this thne be assume, as C D. In this time, therefore, it will pass through some part of the magnitude; and let it have passed through the part B E. But this part will either measure the magnitude A B, or will he deficient or exceed in measuring it. This, however, is of no consequence: for if it always passes through a magnitude equal to the part B E, and this measures the whole magnitude, the whole time in which it passes through will be finite; for it will be divided into equal parts, as also the magnitudes. Farther still, if it does not pass through the whole magnitude in an infinite time, but is able to pass through some portion of it in a finite time, as for instance, the part B E, and this measures the whole; — if this be the case, it will pass through an equal part in an equal time; so that the time will be finite. But that it does not pass through the part B E in an infinite time, will he evident if a finite time is assumed from the other part: for if it passes through a part in a less time, it is necessary that this part should be finite, since the other boundary is present. There is the same demonstration if the length is supposed to be infinite, but the time finite. It is evident, therefore, from what has been said, that neither a line, nor a superficies, nor, in short, any continued quantity, is indivisible, not only in consequence of what has been now said, but because it will happen that an indivisible will be divided: for since in all time there is the swifter and the slower, but the swifter passes through a greater space in an equal time, it is possible for it to pass through a double or sesquialter length; for this may be the ratio of the celerity. Let that which moves swifter then pass through a sesquialter length in the same time: and let the magnitudes be divided; that which moves swifter, into A B, B C, C D, three indivisibles; but that which moves slower into two, E F, F G. The time, therefore, will also be divided into three indivisibles; for it passes through an equal space in an equal time. Let the time then he divided into K L, L M, M N. But again, since that which moves more slow passes through E F, F G; hence the time will be cut into two parts. An indivisible, therefore, will be divided; and that which is without parts will not pass through in an indivisible, but in a longer time. Hence it is evident that nothing continued is without parts.
