In Chapter vi. he defines the * proper time ’ of a change as the time during any part of which change is occurring. From this definition and the illimitable divisibility of time it follows that any change occupying time must be divis- ible. (Note that the indivisible unit of numerical change occurs all at once.) The proposition cannot be demon- strated directly (because, until the whole change is ac- complished, the mutabile never reaches a stable condition, and is therefore never found at, but always approaching or receding from any intermediate point on its course.) He therefore introduces a second mutabile, which changes at the same rate as the first, but for only part of the time, at the end of which it reaches a stable condition and is found (according to the definition of proper time) to have accomplished a corresponding part of the change; the original mutabile changing at equal speed must have 87 ARISTOTLE accomplished an equal part of the change in an equal time. This reasoning applies to any part of the change and of the time it occupies; so that anything which is in process of changing must have been changing previously; anything which has accomplished a change must have accomplished some change previously; anything which is in process of changing must have accomplished some change previously; anything which has accomplished a change must have been changing previously.
He gives change of magnitude as a particularly obvious example, and notes that the above demonstration also applies to genesis and perishing of continuous and divisible things.
ie Chapter vii. he gives a demonstration that (i) a limited movement, whether uniform or not, cannot occupy unlimited time, and (ii) there could not be an unlimited movement in a limited time. His proof follows the same line of argument as the demonstration in Chapter ii. (see p- 112 note ὃ). The same reasoning is applied to genesis and extinction.
There is no‘ primary,’ i.e. smallest-possible or irreducible first component of time, or of dimension, or of anything that is continuous.
Chapter viii. Both coming to rest and being at rest (a) occupy divisible periods of time; (Ὁ) are going on during any part of that proper time; and (c) have no irreducible earliest stage within that proper time. A moving thing never exactly ‘ covers’ any definite station- ary object during any part of the proper time of its move- ment, but at any indivisible instant it must necessarily do In Chapter ix. he discusses Zeno’s four dilemmas and two others, all designed to prove that belief in the reality of motion leads to impossible consequences. He refutes Zeno’s 1st and 2nd dilemmas (provisionally) by pointing out the complete parallelism between time and distance 88 PHYSICS, VI. INTRODUCTION with respect to extension and divisibility. In his final refutation in Bk. VIII. he recognizes that this does not dispose of the deeper underlying problem of the reality, not only of motion, but of time and space themselves and of any continuum whatever. He refutes Zeno’s 3rd and 4th dilemmas by denying the assumptions on which they rest. The contention that if change is possible a thing could be e.g. both white and not white simultaneously, is refuted by pointing out that the supposed proof depends on the equivocal use of terms. The same criticism applies to the contention that a rotating sphere is both in motion and at rest simultaneously.
Chapter x. Proof that an indivisible cannot move (or change) on its own account, but only concomitantly; and that no change can be unlimited except rotary locomotion.
89 Z, CHAPTER I ARGUMENT The terms ‘ continuity, ‘ contiguity’ (or touching), and * neat - in - succession’ defined and distinguished. These definitions suffice to show that a continuum (such as length, time, movement) cannot be constituted by indivisibles (points, “nows, stations) or be resolved into them. Nor can two points (nows, stations) be continuous or contiguous one with Further demonstration that points cannot by contiguity form a continuum; for indivisibles must either be in the same proper place (i.e. be positionally identical) or else be entirely isolated (they cannot occupy different places without having intervals between them), whereas the successive parts of a continuum occupy different places but have nothing between them (a 29-b 6).
A line cannot be constituted by a succession of points which are ‘next without contact’; for between ‘ nexts’ there is nothing of their own category, and what lies between any two points is linear extension which is divisible at intermediate points (see above). So that points lie between any two points, and no point is next to any other (Ὁ 6-18).
A formal proof that the argument holds equally for spatial magnitude, time, and motion, and that all three hang together. It rests on two axioms: The first: That when motion is taking place, something is moving from here to there and vice versa.
The second: That the mobile or subject which experi- ences the motion cannot simultaneously be in the act of moving towards a given position and in the state of being already at it.
90 BOOK VI CHAPTER I ARGUMENT (continued) The steps of the argument are: (1) If L, a component of motion, is itself a motion, then (by axiom ii) after L has started and before it has finished, P (the mobile) is (by axiom i) past the start and short of the finish of A (the distance). Therefore A is divisible in correspondence with L; and so likewise are B and C with Mand N. (2) If it were still maintained that A, etc., need not be (divisible) dis- tances but might be (indivisible) ‘ terms’ in the distance, it would involve one or other of the following impossibilities: (a) If L, etc., were motions, P would be wn motion (while L was in progress) without moving from A; and so with M and N,and Band C. (Ὁ) If L, etc., were not motions, P would never be in motion but would accomplish the motion without moving. Therefore both distance and motion must be divisible (Ὁ 18-232 a 18).
Time is divisible if distance and motion are, and vice versa, for if the whole of the length A is traversed in time T, a part of it would be traversed (at equal speed) in less than T. Or if the whole time T were occupied in traversing the distance A, then in part of the time less than A would be traversed (a 18-22).
Note.—The absence of method in the system of lettering in the Greek teat makes the discussion in this and the following chapter unnecessarily difficult to follow. Therefore an en- tirely independent system of lettering has been adopted in the translation. But for purposes of comparison duplicate diagrams are given showing the two systems of lettering side by side. Where there is no diagram the Greek letters are given in brackets after the English.
91 ARISTOTLE @3ia21 Hi δ᾽ ἐστὶ συνεχὲς καὶ ἁπτόμενον καὶ ἐφεξῆς ὡς διώρισται πρότερον---συνεχῇ μὲν ὧν τὰ ἔσχατα ἕν, ἁπτόμενον δὲ ὧν ἅμα, ἐφεξῆς δὲ ὧν μηδὲν μεταξὺ συγγενές---ἀδύνατον ἐξ ἀδιαιρέτων εἶναί τι συνεχές, 96 οἷον γραμμὴ ἐκ στιγμῶν, εἴπερ ἡ γραμμὴ μὲν συνεχὲς ἡ στιγμὴ δ᾽ ἀδιαίρετον. οὔτε γὰρ ἕν τὰ ἔσχατα τῶν στιγμῶν (οὐ γάρ ἐστι τὸ μὲν ἔσχατον τὸ δ᾽ ἄλλο τι μόριον τοῦ ἀδιαιρέτου), οὔθ᾽ ἅμα τὰ ἔσχατα (οὐ γὰρ ἔστιν ἔσχατον. τοῦ ἀμεροῦς οὐδέν, ἕτερον γὰρ τὸ ἔσχατον καὶ οὗ ἔσχατον).
80 Ἔτι δὲ ἀνάγκη ἤτοι συνεχεῖς. εἶναι τὰς στιγμὰς ἢ ἁπτομένας ἀλλήλων, ἐξ ὧν ἐστι τὸ συνεχές" ὁ 231 δ᾽ αὐτὸς λόγος καὶ ἐπὶ πάντων τῶν ἀδιαιρέτων. συνεχεῖς, “μὲν δὴ οὐκ ἂν εἶεν διὰ τὸν εἰρημένον λόγον" ἅπτεται δ᾽ ἅπαν ἢ ὅλον ὅλου ἢ μέρος μέρους ἢ ὅλου μέρος. ἐπεὶ δ᾽ ἀμερὲς τὸ ἀδιαίρετον, ἀνάγκη ὅλον ὅλου ἅπτεσθαι. ὅλον δ᾽ ὅλου ἁπτό- ὄὅμενον οὐκ ἔσται συνεχές" τὸ γὰρ συνεχὲς ἔχει τὸ μὲν ἄλλο τὸ δ᾽ ἄλλο μέρος, καὶ διαιρεῖται εἰς οὕτως ἕτερα καὶ τόπῳ κεχωρισμένα.
α That is to say if their limits are not only ‘ together ’ but united, or in other words bound to be together.
> These definitions are given in Bk. V., 226 Ὁ 18 sqq., and constitute the only portion of that Book which is relevant to thecontext. Thatsome such exposition of the terms did occur elsewhere in Aristotle’s work and probably occupied a place between Bks. [V.and VI. may be taken for granted. But this by no means implies that it was the whole of this Book, or even this relevant portion of it in the form in which we have it, that originally furnished the previous definition here referred to.
¢ The Greek does not imply that a limit is itself a part of that which it limits. See Vol. I. Introd. pp. lxxxiii ff. and Vol. II., List of Corrigenda.
92 PHYSICS, VI. 1.
Tue terms ‘ continuous,’ ‘ contiguous,’ and ‘ next-in- succession’ have been defined above as follows: things are ‘continuous’ if (while they are them- selves distinct in the sense of occupying different places) their limits are one,* ‘contiguous’ if their limits are together, “next-in-succession’ if they have nothing of the same nature as themselves between them.® If these definitions are accepted, it follows that no continuum can be made up of indivisibles, as for instance a line out of points, granting that the line is continuous and the point indivisible. For two points cannot have identical limits, since in an indi- visible there can be no distinction of a limit from some part other than the limit ¢; and (for the same reason) neither can the limits be together, for a thing that has no parts has no limit, since a limit must be dis- tinct from what it limits.
Yet the points would have to be either continuous or contiguous if they were to make a continuum. And the same is true of any indivisible. As to the impossi- bility of their being continuous, the proof just given will suffice; but we will consider the alternative of con- tiguity further. If A is contiguous with B, either A in its entirety must touch B in its entirety, or a part of one must touch a part of the other, or a part of one the other in its entirety. But since the indivisible has no parts, if two indivisibles touched each other at all it must be in their entirety. But if they were touching in their entirety, they could not make a continuum, for a continuum is divisible into parts which are distinguishable from each other in the sense of being in different places.@ 93 ARISTOTLE 281} ἀλλὰ μὴν οὐδ᾽ ἐφεξῆς ἔσται στιγμὴ στιγμῆς, ἢ τὸ νῦν τοῦ νῦν, ὥστ᾽ ἐκ τούτων εἶναι τὸ μῆκος ἢ τὸν χρόνον" ἐφεξῆς μὲν γάρ ἐστιν ὧν μηθέν ἐστι μεταξὺ συγγενές, στιγμῶν δ᾽ ἀεὶ τὸ μεταξὺ γραμμή, 10 καὶ τῶν νῦν χρόνος. ἔτι διαιροῦτ᾽ ἂν εἰς ἀδιαίρετα, εἴπερ ἐξ ὧν ἐστιν ἑκάτερον, εἰς ταῦτα διαιρεῖται. ἀλλ᾽ οὐθὲν ἦν τῶν συνεχῶν εἰς ἀμερῆ διαιρετόν. ἄλλο δὲ γένος οὐχ οἷόν 7 εἶναι μεταξὺ τῶν στιγμῶν καὶ τῶν νῦν οὐθέν. εἰ γὰρ ἔσται, δῆλον ὡς ror ἀδιαίρετον ἔσται 7 διαιρετόν: καὶ εἰ 4 {I have partly re-written this paragraph on the assump- tion that it contains a series of arguments against the thesis that a continuous linear magnitude or stretch of time could be made up of a row of successive points or moments.—C.]
¢ [Successive points (nows) which cannot touch must have something between them. It has been asserted above that what does lie between them is something of the same kind (cvyyevés)—a line(time). Aristotle now forestalls the objection that what lies between might be something of a different kind, ¢.g. a ‘void’ such as some Pythagoreans (213 b 24) supposed to separate the distinct points composing a line. Such a ‘void’ was not a line, and did not itself contain points. On that view our continuous line is to be composed of points that will be ‘ successive’ (with nothing of the same kind between them), each point being separated from the next by a stretch of ‘void.’ Against this Aristotle argues as follows. This void stretch must be either (a) indivisible, or (6) divisible either (a) into indivisible parts or (8) infinitely. If (8) it is infinitely divisible, ‘then it isa continuous magni- tude,’ τοῦτο δὲ συνεχές (it is, in fact, a linear magnitude, such as we said did lie between points, and the alleged ‘ successive ’ points are not successive, for they have something of the same kind between them). (And it must be infinitely divisible. Alternatives (a) and (a) can be dismissed; for this ‘void’ is a constituent part of our line, which is ex hypothest to be a continuum), ‘and it is manifest that any continuum (like our line) is divisible infinitely, for if there 94 PHYSICS, VI. 1.
Again,* one point, so far from being continuous or contiguous with another point, cannot even be the next-in-succession to it, or one ‘now’ to another ἡ now,’ in such a way as to make up a length or a space of time; for things are ‘next’ to each other when there is nothing of their own sort between them, and two points have always a line (divisible at intermediate points) between them, and two ‘nows 8. space of time (divisible at intermediate “nows ’). Moreover, if a succession of indivisibles could make up a continuum either of magnitude or time, that continuum could be resolved into its indivisible constituents. But, as we have seen,? no continuum can be resolved into elements which have no parts. Further, there cannot be anything of 8, different kind between the points or the ‘ nows.’¢ For if there could be such a thing, clearly it must be either (a) something indivisible or (6) something were indivisible parts they would have to touch one another’ (in order to make up a continuum; and that we have seen to be impossible, 231 b 2 ff.); ‘for the extremities of con- tinuous things meet and become one.’ The last statement is explicable if it is remembered that ἅπτεσθαι was, from Euclid onwards, the technical term in geometry for the ‘ meeting’ of two lines, ἐφάπτεσθαι being used for the ‘ touching’ of two circles (Heath, Thirteen Books of Euclid, ii. 2). Two lines laid end to end ‘ meet ’ in such a way that their end-points coalesce, and the two lines unite into one continuous line. It was stated at 227 a 24 that ‘the extremities must meet (ἅψασθαι), if they are to coalesce.’ Similarly when a line is divided, the ends of the two sections are not two distinct points in contact with, or successive to, one another. If, however, two lines meet at an angle, the end point of the one and the beginning point of the other are the same point, but the lines do not unite into a continuous line. This is important in view of the argument (Bk. VIII. chapters vi.—viii.) on continuity of movement. The conclusion here established is used at 234 a 6 ff.—C.]
95 ARISTOTLE 231b 15 διαιρετόν, ἢ εἰς ἀδιαίρετα ἢ εἰς ἀεὶ διαιρετά. τοῦτο δὲ συνεχές. φανερὸν δὲ καὶ ὅτι πᾶν συνεχὲς διαιρετὸν εἰς ἀεὶ διαιρετά (εἰ γὰρ εἰς ἀδιαίρετα, ἐσται ἀδιαίρετον ἀδιαιρέτου ἁπτόμενον): ἕν γὰρ TO ἔσχατον καὶ ἅπτεται τῶν συνεχῶν. Tod δὲ αὐτοῦ λόγου καὶ μέγεθος Kal χρόνον Kal 20 Κίνησιν ἐξ ἀδιαιρέτων συγκεῖσθαι καὶ διαιρεῖσθαι εἰς ἀδιαίρετα, ἢ μηθέν. δῆλον δὲ ἐκ τῶνδε. εἰ γὰρ τὸ μέγεθος ἐξ ἀδιαιρέτων σύγκειται, καὶ ἡ κίνησις ἡ τούτου ἐξ ἴσων κινήσεων ἔσται ἀδιαι- ρέτων: οἷον εἰ τὸ ABI’ ἐκ τῶν A, Β, Γ ἐστὶν Ὄ, A ἀδιαιρέτων, ἡ κίνησις ἐφ᾽ ἧς ΔΕΖ, ἣν ἐκινήθη τὸ 45 ἐπὶ τῆς ABT’, ἕκαστον τὸ μέρος ἔχει ἀδιαίρετον.
εἰ δὴ παρούσης κινήσεως ἀνάγκη κινεῖσθαί τι καὶ εἰ κινεῖταί τι, παρεῖναι κίνησιν, καὶ τὸ κινεῖσθαι ἔσται ἐξ ἀδιαιρέτων. τὸ μὲν δὴ A ἐκινήθη τὸ ἢ τὴν τὸ Δ κινούμενον κίνησιν, τὸ δὲ Β τὴν τὸ E, καὶ τὸ Τ' ὡσαύτως τὴν τὸ ΖΦ. εἰ δὴ ἀνάγκη TO κινούμενόν ποθέν ποι μὴ ἅμα κινεῖσθαι καὶ 96 PHYSICS, VI. 1.
divisible; and if divisible, divisible either (a) into indivisibles or (β) into divisibles that are divisible without limit. But in the latter case it is a continuum. And it is manifest that any continuum is divisible into parts that are divisible without limit—for if the parts were indivisible, we should have one indivisible touching another—since the extremities of things that are continuous meet and become one.
Now the same argument applies to spatial mag- nitude and time and motion, so that if it can be shown that any one of them cannot be so built up and broken down, it follows that none of them can. The following proof will make this clear. If a distance is composed of indivisibles, the motion through it must be composed of an equal number of indivisible motions. Let ABC be the distance traversed by P A B ς distance LM N μμαικακατααρεόμσεας.. εν ναῖρσετα αν memmncanemenmmemed movement in course of the movement LMN. Then if ABC is composed of the indivisibles A, B, and C, the motion of P will be, throughout, composed of indi- visible motions. And if you grant that while the movement is taking place, there must be something in motion, and that if there is something in motion there must be a movement in progress, then (on the hypothesis) the whole movement of the moving thing must be composed of indivisibles. So P was in passage over A when experiencing the motion L, and so for B and M, and for C and N. Now, when anything moves from here to there it cannot have VOL. II H 07 ARISTOTLE 231b30 κεκινῆσθαι οὗ ἐκινεῖτο ὅτε ἐκινεῖτο (οἷον εἰ Θήβαζέ τις βαδίζει, ἀδύνατον ἅμα βαδίζειν Θήβαζε 232 καὶ βεβαδικέναι Θήβαζε), τὴν δὲ τὸ A τὴν ἀμερῆ 1 ἐκινεῖτο τὸ QL, 4 ἡ τὸ A κίνησις παρῆν, ὥστ᾽ εἰ μὲν ὕστερον διῆλθεν ἢ Sines, διαιρετὴ ἂν εἴη (ὅτε γὰρ διήει, οὔτε ἠρέμει οὔτε διεληλύθει ἀλλὰ μεταξὺ ἦν)" εἰ δ᾽ ἅμα διέρχεται καὶ διελήλυθε, τὸ δ βαδίζον ὅτε βαδίζει βεβαδικὸς ἐκεῖ ἔσται, καὶ oS κεκινημένον οὗ κινεῦται.
Εἰ δὲ τὴν μὲν ὅλην τὴν ΑΒΓ κινεῖταΐ τι, καὶ ἡ κίνησις ἣν κινεῖται τὰ Δ, E, Z ἐστί, τὴν δ᾽ ἀμερῆ τὴν A οὐθὲν κινεῖται ἀλλὰ κεκίνηται, εἴη ἂν ἡ κίνησις οὐκ ἐκ κινήσεων ἀλλ᾽ ἐκ κινημάτων, καὶ τὸ" κεκινῆσθαί τι μὴ κινούμενον (τὴν yap A δι- ελήλυθεν οὐ διεξιόν)" ὥστε ἔσται τι βεβᾳδικέναι μηδέποτε βαδίζον: ταύτην γὰρ βεβάδικεν οὐ βαδίζον ταύτην. εἰ οὖν ἀνάγκη ἢ ἠρεμεῖν ἢ κινεῖσθαι πᾶν, ἠρεμεῖ δὲ καθ᾽ ἕκαστον τῶν A, Β, Γ΄, ὥστ᾽ ἔσται τι συνεχῶς ἠρεμοῦν ἅμα καὶ κινούμενον: τὴν γὰρ 1 [ὥστε in apodosi, as below, 1. 13.—C.]
2 [ro om. FHI: τῴ Oxf. Trans. Either of the ms. readings is possible with εὖ) ἂν meaning (with τὸ) ‘we should have’ or (without τὸ) ‘it would be possible.’—C. ] * {A distinct argument, refuting the hypothesis that a movement can consist of a series of indivisible components occupying atoms of time and complete as soon as begun. Seley oe upheld this view. (Bailey, Greek Atomists, ra [Literally, “and if P is not in motion at all over the section A (which has no parts) but has moved.’—C.]
98 PHYSICS, VI. 1.
already got there while still moving thither (for instance, the man who is walking to Thebes cannot have already got to Thebes and be there at the same time as he is still walking to Thebes). But P was moving over the indivisible A exactly while its motion L was in progress. Accordingly if (i) P had not completed the movement while experiencing it, but only when it had ceased, then A must be divisible; because while P was moving, it was neither at rest (since it had already started to cover A) nor had it covered A (for it was still in the act of covering it). But if Gi) when P were moving, at one and the same time it both had accomplished and was accom- plishing the movement, it would both have arrived at the term of the motion and be moving towards it: the walker, while still walking, would have finished his walk and be at his destination.
α Again, if anything moved over the whole distance ABC and the motions were L, M, and N, and if the indivisible A did not mark a distance moved over, but a (potential) term of movement,’ (and so with B and C,) then the whole corresponding motion LMN would consist not in experiencing motions, but in having experienced them, and that which never was in motion would have accomplished the movement; for without ever passing through A, it would have passed through it. It would then be possible for our walker to have finished his walk without ever taking it: he would have walked this distance without walking over it. So if we grant that every mobile must be either actually moving or at rest, and if P is at rest in each of the com- ponents of ABC, we shall have P both resting and moving at one and the same time continuously; 99 ARISTOTLE e32a15 ABL' ὅλην ἐκινεῖτο καὶ ἠρέμει ὁτιοῦν μέρος, ὥστε καὶ πᾶσαν. καὶ εἰ μὲν τὰ ἀδιαίρετα τῆς ΔΕΖ, κινήσεις, κινήσεως παρούσης ἐνδέχοιτ᾽ ἂν μὴ κινεῖσθαι ἀλλ᾽ ἠρεμεῖν: εἰ δὲ μὴ κινήσεις, τὴν κίνησιν μὴ ἐκ κινήσεων εἶναι. Opoiws δ᾽ ἀνάγκη τῷ μήκει καὶ TH κινήσει ἀδιαίρετον εἶναι τὸν χρόνον καὶ συγκεῖσθαι ἐκ τῶν νῦν ὄντων ἀδιαιρέτων. εἰ γὰρ πᾶσα διαιρετός, ἐν τῷ ἐλάττονι δὲ τὸ ἰσοταχὲς δίεισιν ἔλαττον, διαιρετὸς ἐσται καὶ ὁ χρόνος" εἰ δὲ 6 γρόνος διαιρετὸς ἐν ᾧ φέρεταί τι τὴν Α, καὶ ἡ τὸ Α ἔσται διαιρετή.
1 [πᾶσα: Themistius (184. νι apparently read ἅπας (se. χρόνος), with copies known to Alexander (Simplic. 936. 22): πᾶς ἀδιαίρετος is recorded by Aspasius. Simplicius doubted whether γραμμή should be supplied with πᾶσα, or (with less ‘violence’) κίνησις, in which case the argument is from motion to time, and from time to distance. Did Aristotle write πᾶσ᾽ ἡ A— If A is divisible throughout’? Cf. Alex. ap. Simpl. (936. 25) δείξας ὅτι ἂν τὸ μέγεθος πάντῃ διαιρετὸν ἧ, καὶ ὁ χρόνος ἔσται διαιρετός.---(,} CHAPTER II ARGUMENT Development of the implications of the continuous nature of linear magnitudes, of tume, and of motion, established in the last chapter. If P is moving at a higher velocity than Q, then not only does P (i) cover a greater distance in a given time, and (ii) cover a given distance in a lesser time, but also (iii) covers a greater distance wn a lesser tume (232 a 23-b 20).
Every motion occupies time, and any period of tyme can be occupied by motion; and the motion occupying any period (however short) may be quicker or slower. It follows that time must be continuous. This is proved by PHYSICS, VI. 1-11.
since it was at once moving over the whole ABC and at rest in each several component, and therefore in the whole, of ABC. Further, if the indivisible components of the motion LMN were to be con- sidered as motions, we should have to say that a thing, while in motion, might be not moving but at rest; whereas if they were not motions, we should have to say that a movement might be made of components that were not movements but stations.
Again, from indivisible components of distance and motion would follow indivisible components of time, which would, on that hypothesis, be made up of indivisible ‘ nows.’ But if on the other hand we admit that every distance or motion is divisible, so must the corresponding periods of time be, since a thing moving at a uniform velocity will cover a part of any distance in less time than the whole. And conversely if the time in which the distance A is covered is divisible, so must A be.
CHAPTER II ARGUMENT (continued) taking two bodies moving at different speeds. However short a time the slower takes to cover a given distance, the quicker will always take still less time, and in that lesser time the slower will always cover ὦ stil shorter distance; in this way both the time and the distance may be reduced without limit; therefore the continuity of time follows from the continuity of magnitude and vice versa. This conclusion agrees with popular belief (b 20-233 a 17).
Further, of either time or linear magnitude is illimitable in extension or divisible without limit, so must the other ARISTOTLE ARGUMENT (continued) The distinction between wlimitable extension and unlimited divisibility gives the clue to the fallacy of Zeno’s argument against the possibility of motion ἡ because it would involve that a moving body would have to pass an illimitable number of points and thus establish illimitable contacts in a limited time. For both the distance and the time are limited in extent but unlimited in divisibility, and the contacts do not occupy time any more than they occupy space, but are estab- lished by divisibility (a 21-31).
Two proofs that an illimitable time cannot be required to 232203 “Eres δὲ πᾶν μέγεθος εἰς μεγέθη διαιρετόν (δέδεικται γὰρ ὅτι ἀδύνατον ἐξ ἀτόμων εἶναί τι 25 συνεχές, μέγεθος δ᾽ ἐστὶν ἅπαν συνεχές), ἀνάγκη τὸ θᾶττον ἐν τῷ ἴσῳ χρόνῳ μεῖζον καὶ ἐν τῷ ἐλάττονι ἴσον καὶ ἐν τῷ ἐλάττονι πλεῖον κινεῖσθαι, καθάπερ ὁρίζονταί τινες τὸ θᾶττον.
ἐπεὶ τοίνυν θᾶττόν ἐστι τὸ πρότερον μεταβάλλον, ν ᾧ χρόνῳ τὸ A μεταβέβληκεν ἀ ἀπὸ τοῦ Γ εἰς τὸ 80 Δ (οἷον τῷ ΖΗ), ἐν τούτῳ τὸ Β οὕπω ἔσται πρὸς τῷ Δ ἀλλ᾽ ἀπολείψει" ὥστε ἐν τῷ ἴσῳ χρόνῳ πλεῖον δίεισι τὸ θᾶττον. ᾿Αλλὰ μὴν καὶ ἐν τῷ ἐλάττονι πλεῖον. ἐν ᾧ yap τὸ A γεγένηται πρὸς τῷ A, τὸ B ἔστω πρὸς * Though not so much greater as in case (i). Vide infra.
PHYSICS, VI. τι.
ARGUMENT (continued) traverse any limited distance. Both depend on there being some distance for which a limited time suffices. In the first this is taken for granted (233 a 32 ff.); in the second mentioned as a necessary condition (223 Ὁ 8). Then follows a proof that such distances do east (see p. 112 note b). The same line of argument shows that illimitable distance cannot be traversed in a limited time (a 31-b 15).
Further proof that there is no continuum whieh cannot be divided (Ὁ 15-32).
SINCE any magnitude can be divided into magnitudes (for it has been shown that nothing continuous can be composed of atomic constituents, and all magnitude is continuous), it follows that if P is quicker than Q it will (i) cover a greater distance in the same time; (ii) cover the same distance in a lesser time; (iii) cover a greater distance in a lesser time.* ‘ Quicker’ has been defined in this way.
For (i) since P is quicker than Q and therefore is in advance of it, in the time OT which it has taken P to change from A to D, Q will not have reached D but will still be short of it. So in the same time the quicker will cover a greater distance than the slower.
But (iii) we may go further and say that the quicker will cover a greater distance in less time. For in the time it has taken P to reach D, let Q have ARISTOTLE - gan τῷ Εἰ τὸ βραδύτερον ὄν. οὐκοῦν ἐπεὶ τὸ Α πρὸς τῷ Δ γεγένηται ἐν ἅπαντι τῷ ΖΗ χρόνῳ, “πρὸς τῷ Θ ἔσται ἐν ἐλάττονι τούτου" καὶ ἔστω ἐν τῷ ZK. τὸ μὲν οὖν ΓΘ, ὃ διελήλυθε τὸ A, μεῖζόν ἐστι τοῦ TE, 6 δὲ χρόνος ὁ ZK ἐλάττων τοῦ smavros τοῦ ZH: wore ἐν ἐλάττονι μεῖζον δίεισιν. Φανερὸν δὲ ἐκ τούτων καὶ ὅτι τὸ θᾶττον ἐν ἐλάττονι χρόνῳ δίεισι τὸ ἴσον. ἐπεὶ γὰρ τὴν μείζω ἐν ἐλάττονι διέρχεται τοῦ βραδυτέρου, αὐτὸ δὲ καθ᾽ αὑτὸ λαμβανόμενον ἐν πλείονι χρόνῳ THY μείζω τῆς ἐλάττονος (οἷον τὴν ΛΜ τῆς ΛΕ), 10 πλείων ἂν εἴη 6 χρόνος 6 ΠΡ ἐν ᾧ τὴν AM διέρ- χεται ἢ ὁ ΠΣ ἐν ᾧ τὴν Ag. ὥστε εἰ ὁ ITP χρόνος ἐλάττων ἐστὶ τοῦ ΠΧ ἐν ᾧ τὸ βραδύτερον δι- ἔρχεται τὴν ΛΕ, καὶ ὁ ΠΣ ἐλάττων ὁ ἔσται τοῦ ἐφ᾽ ᾧ ITX- τοῦ γὰρ ΠΡ ἐλάττων, τὸ δὲ τοῦ ἐλάττονος ἔλαττον καὶ αὐτὸ ἔλαττον. ὥστε ἐν ἐλάττονι 16 κινήσεται τὸ ἴσον.
"Ere δ᾽ εἰ πᾶν ἀνάγκη ἢ ἐν ἴσῳ χρόνῳ ἢ ἐν ἐλάττονι ἢ ἐν πλείονι κινεῖσθαι, καὶ τὸ μὲν ἐν πλείονι βραδύτερον τὸ δ᾽ ἐν ἴσῳ ἰσοταχὲς τὸ δὲ θᾶττον οὔτε ἰσοταχὲς οὔτε βραδύτερον, οὔτ᾽ ἂν ἐν ἴσῳ οὔτ᾽ ἐν πλείονι κινοῖτο τὸ θᾶττον. λείπεται ARISTOTLE 232 b οὖν ἐν ἐλάττονι. ὥστ᾽ ἀνάγκη καὶ τὸ ἴσον μέγεθος 20 ἐν ἐλάττονι χρόνῳ διιέναι. τὸ θᾶττον. πεὶ δὲ πᾶσα μὲν κίνησις ἐν “χρόνῳ καὶ ἐν ἅπαντι χρόνῳ δυνατὸν κινηθῆναι, πᾶν δὲ τὸ κινού- μενον ἐνδέχεται καὶ θᾶττον κινεῖσθαι καὶ βραδύ- τερον, ἐν ἅπαντι χρόνῳ ἔσται τὸ θᾶττον κινεῖσθαι καὶ βραδύτερον. τούτων δ᾽ ὄντων ἀνάγκη καὶ τὸν 95 χρόνον συνεχῆ εἶναι. λέγω δὲ συνεχὲς τὸ διαιρετὸν εἰς ἀεὶ διαιρετά: τούτου γὰρ ὑποκειμένου τοῦ συνεχοῦς, ἀνάγκη συνεχῆ εἶναι τὸν χρόνον. ἐπεὶ γὰρ δέδεικται ὅτι τὸ θᾶττον ἐν ἐλάττονι χρόνῳ δίεισι τὸ ἴσον, ἔστω τὸ μὲν ἐφ᾽ ᾧ A θᾶττον τὸ δ᾽ ἐφ᾽ ᾧ Β βραδύτερον, καὶ κεκινήσθω τὸ βραδύτερον 80) τὸ ἐφ᾽ ᾧ TA μέγεθος ἐν τῷ ZH χρόνῳ. δῆλον τοίνυν ὅτι τὸ θᾶττον ἐν ἐλάττονι τούτου κινήσεται τὸ αὐτὸ μέγεθος" καὶ κεκινήσθω ev τῷ ZO. πάλιν δ᾽ ἐπεὶ τὸ θᾶττον ἐν τῷ ΖθΘ διελήλυθε τὴν ὅλην τὴν TA, τὸ βραδύτερον ἐ ἐν τῷ αὐτῷ χρόνῳ τὴν ἐλάττω 238 a δίεισιν" ἔστω οὖν ἐφ᾽ ἧς ΓΚ. ἐπεὶ δὲ τὸ βρα- δύτερον τὸ Β ἐν τῷ ZO χρόνῳ τὴν ΓΚ διελήλυθε, τὸ θᾶττον ἐν ἐλάττονι δίεισιν, wore πάλιν διαιρεθήσεται 6 ZO χρόνος. τούτου δὲ διαιρουμένου * Τῃ the figure the unit of distance and the unit of time are PHYSICS, VI. τι.
take a shorter time. Consequently the quicker must cover the same distance in a shorter time.
Again, since every movement takes place in time, and in any period of time movement can take place, and everything that moves can move at a greater or lesser velocity, quicker or slower movements may take place within any period of time however small; whence it follows that time must be continuous. I mean by continuous ‘capable of being divided into parts that can in their turn be divided again, and so on without limit’; and on this definition I say that time is of necessity continuous.
It has been shown that the quicker will cover the same distance in a lesser time. Let P be the quicker, Q the slower, and let Q have moved over the distance AD in the time OV. It is clear then that the quicker P will cover that same distance in a shorter time; let this time be OT. Again, since the quicker P has covered the whole distance AD in the time OT, the slower Q in that same time covers a shorter distance, say AC. And since the slower Q has covered AC in the time OT, P will cover that same distance in less than OT, so that the time will be divided again (at 8). And the time being so supposed to be represented by equal lines, and the velocity of P is plotted as twice that of Q. The proof would hold on any other convention or hypothesis.
ARISTOTLE 838 καὶ TO ΓΚ μέγεθος διαιρεθήσεται κατὰ τὸν αὐτὸν ὅ λόγον. εἰ δὲ τὸ μέγεθος, καὶ ὁ χρόνος. καὶ ἀεὶ τοῦτ᾽ ἔσται μεταλαμβάνουσιν ἀπὸ τοῦ θάττονος τὸ βραδύτερον καὶ ἀπὸ τοῦ βραδυτέρου τὸ θᾶττον, καὶ τῷ ἀποδεδειγμένῳ. χρωμένοις" διαιρήσει γὰρ τὸ μὲν θᾶττον τὸν χρόνον, τὸ δὲ βραδύτερον τὸ μῆκος. εἰ οὖν ἀεὶ μὲν ἀντιστρέ ew ἀληθές, ἀντιτὸ στρεφομένου δὲ ἀεὶ γίγνεται διαίρεσις, φανερὸν ὅτι πᾶς χρόνος ἔσται συνεχής.
Ἅμα δὲ δῆλον καὶ ὅτι μέγεθος ἅπαν ἐστὶ συνεχές" τὰς αὐτὰς γὰρ καὶ τὰς ἴσας διαιρέσεις 6 χρόνος διαιρεῖται καὶ τὸ μέγεθος. ἔτι δὲ καὶ ἐκ τῶν εἰωθότων λόγων λέγεσθαι Φανερὸν ὡς εἴπερ ὁ χρόνος ἐστὶ συνεχής, ὅτι καὶ τὸ μέγεθος, εἴπερ ἐν τῷ ἡμίσει χρόνῳ ἥμισυ διέρχεται καὶ ἁπλῶς ἐν τῷ ἐλάττονι ἔλαττον' αἱ γὰρ αὐταὶ διαιρέσεις ἔσονται τοῦ χρόνου καὶ τοῦ μεγέθους.
Καὶ εἰ ὁποτερονοῦν ἄπειρον, καὶ θάτερον, καὶ ὡς θάτερον, καὶ θάτερον" οἷον εἰ μὲν τοῖς ἐσχάτοις ἄπειρος ὃ χρόνος, καὶ τὸ μῆκος τοῖς ἐσχάτοις, εἰ 0 δὲ τῇ διαιρέσει, τῇ διαιρέσει καὶ τὸ μῆκος, εἰ δὲ ἀμφοῖν 6 χρόνος, ἀμφοῖν καὶ τὸ μέγεθος.
pt ao * Length and distance are used indifferently in this section according to convenience to represent μῆκος.
ὃ If Aristotle here uses ‘ magnitude ° as a variant for ‘len th,’ he means by ‘any’ magnitude ‘any however small.’ If he means to distinguish between ἡ length ’ (or * distance ’) and ‘ area,’ ‘ volume’ or "weight ’ or any other _ measurable ἢ (in which sense, ‘time,’ ‘ temperature,’ * pressure,’ etc., are all ‘ magnitudes ’), then he assumes that his reader understands that a ratio between two lengths, for instance, can be identical with, greater, or less than, a ratio between two areas etc., so that what is proved of time and distance is proved of time and any other magnitude, viz. that PHYSICS, VI. τι.
divided, the distance AC will also be divided in the same proportion. And if the distance is divided, the time is correspondingly divided. And this pro- cess may be carried on without limit, if you determine the lesser time P takes to cover a given distance as compared with Q, and then determine the lesser distance that Q covers in that lesser time as com- pared with P. For in comparison with Q, P will always curtail the tame, and in comparison with P, Q will always curtail the distance. But if this conversion always holds, however many divisions have been made, and every conversion leads to a further division, it is evident that (1 length is continuous) so is time.
Thus the continuity of time follows on that of magnitude and also the continuity of magnitude ὃ on that of time, for divisions and subdivisions of the given time and the given magnitude can always be made to keep pace in number and in ratio without limit. Moreover our ordinary way of talking assumes that the continuity of time carries with it that of magnitude, for we do not hesitate to say that half the time suffices to cover half the distance, or generally the lesser time the lesser distance; for the divisions of the distance can always be made in the same ratio as the divisions of the time.
Likewise, if either time or magnitude is unlimited in any respect, so is the other in the same respect. For instance, if the time extends in both directions without limit, so will the distance; and if time is divisible without limit, so will distance be; and if time is both extended without limit and divisible without limit, so will distance be.
the one is divisible, without limit, in the same ratio as the other.
25 80 ARISTOTLE Διὸ καὶ ὃ Ζήνωνος λόγος ψεῦδος λαμβάνει τὸ μὴ ἐνδέχεσθαι τὰ ἄπειρα διελθεῖν ἢ ἅψασθαι τῶν ἀπείρων καθ᾽ ἕκαστον ἐν πεπερασμένῳ χρόνῳ. διχῶς γὰρ λέγεται καὶ τὸ μῆκος καὶ ὁ χρόνος ἄπειρον, καὶ ὅλως πᾶν τὸ συνεχές---ἦτοι κατὰ διαίρεσιν ἢ τοῖς ἐσχάτοις. τῶν μὲν οὖν κατὰ ποσὸν ἀπείρων οὐκ ἐνδέχεται ἅψασθαι ἐν πεπερα- σμένῳ χρόνῳ, τῶν δὲ κατὰ διαίρεσιν ἐνδέχεται" καὶ γὰρ αὐτὸς 6 χρόνος οὕτως ἄπειρος. ὥστε ἐν τῷ ἀπείρῳ καὶ οὐκ ἐν τῷ πεπερασμένῳ συμβαίνει δωέναι τὸ ἄπειρον, καὶ ἅπτεσθαι τῶν ἀπείρων τοῖς ἀπείροις, οὐ τοῖς πεπερασμένοις.
Οὔτε δὴ τὸ ἄπειρον οἷόν τε ἐν πεπερασμένῳ χρόνῳ διελθεῖν, οὔτ᾽ ἐν ἀπείρῳ τὸ πεπερασμένον" Β A E rf ee eS ee ee ἀλλ᾽ ἐάν τε ὁ χρόνος ἄπειρος 7, καὶ τὸ μέγεθος A bid 3 A ἔσται ἄπειρον, ἐάν τε TO μέγεθος, Kal ὃ χρόνος.
4 Zeno makes it appear as though the number of the con- tacts to be established accord in the case of the distance with its divisibility (which is unlimited), but in the case of time with its extension (which is limited). So that the set of con- tingents would be illimitable on one side and limited on the other. Whereas, in reality it accords with divisibility (which is unlimited) in both cases. So that the contingents are illimitable on both sides.
* A definite distance and a definite period of time are PHYSICS, VI. τι.