Hence Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to traverse or severally come in contact with illimitable things in a limited time. For there are two senses in which a distance or a period of time (or indeed any continuum) may be regarded as illimitable, viz., in respect to its divisibility or in respect to its extension.” Now it is not possible to come in contact with quantitively illimitable things in a limited time, but it is possible to traverse what is illimitable in its divisibility; for in this respect time itself is also illimitable. Accordingly, a distance which is (in this sense) illimitable is traversed in a time which is (in this sense) not limited but illimitable; and the contacts with the illimitable (points) are made at “nows’ which are not limited but illimitable in number.® Thus it is impossible for an illimitable distance to be covered in a limited time, or for an illimitable time to be occupied in covering a limited distance; if the distance a Pe ὙΠ Ύ ΠΡΕ: time wap Ge, din a Hy Oo 5 time is illimitable so must the distance be, and if the distance, the time. To prove this, let the line AD divisible in exactly the same way: the distance at points into shorter distances, the time at nows info shorter periods. A point marks off a stage in the journey, and the correspond- ing now marks off the time taken to accomplish that stage. Neither the points nor the nows are limitable.
ARISTOTLE 238 86 ἔστω yap πεπερασμένον μέγεθος ἐφ᾽ οὗ AB, χρόνος δὲ ἄπειρος ἐφ᾽ ᾧ Τ᾽ εἰλήφθω δέ τι τοῦ 8880 χρόνου πεπερασμένον, ἐφ’ ᾧ TA. ἐν τούτῳ οὖν δίεισί Te τοῦ μεγέθους, Kal ἔστω διεληλυθὸς ἐφ᾽ ᾧ BE. (τοῦτο δὲ ἢ καταμετρήσει τὸ ἐφ’ ᾧ ΑΒ ἢ 2 a A ς 7 a ae \ oP ae J ἐλλείψει ἢ ὑπερβαλεῖ: διαφέρει yap οὐθέν). εἰ yap 5 ἀεὶ TO ἰσον TH BE μέγεθος ἐν ἴσῳ χρόνῳ δίεισι (τοῦτο δὲ καταμετρεῖ τὸ ὅλον), πεπερασμένος ἔσται ὁ πᾶς χρόνος ἐν ᾧ διῆλθεν: εἰς ἴσα γὰρ διαιρε- θήσεται Kal’ τὸ μέγεθος. ἔτι δὲ εἰ μὴ πᾶν μέγεθος ἐν ἀπείρῳ χρόνῳ δίεισιν, ἀλλ᾽ ἐνδέχεταί τι καὶ ἐν πεπερασμένῳ διελθεῖν, οἷον τὸ BE (τοῦτο δὲ κατα- 10 μετρήσει τὸ πᾶν), καὶ τὸ ἴσον ἐν ἴσῳ δίεισιν, ὥστε πεπερασμένος ἔσται καὶ ὁ χρόνος. ὅτι δὲ οὐκ ἐν ἀπείρῳ δίεισι τὸ BE, φανερόν, εἰ ληφθείη ἐπὶ 1 [καὶ E and Alexander, Aspasius, and Themistius (Simplic. 950. 3): ὡς καὶ cett.—C.]
ὦ For in any case a limited multiple of AC will equal or exceed AD; and the same limited multiple of the time OS must equal or exceed the time occupied in traversing AD. This time itself, therefore, must be limited.
> The argument that a time must be finite if it is less than infinite time is unsound, and inconsistent with the definition of infinity in Bk. ΠῚ. chs. v., vi. and vii. (see especially 207 a 7 sqg.). For being less than infinite does not preclude it from exceeding any definite time whatever, and therefore imposes no limit on its duration. One infinite can have any ratio (finite or infinite) to another, as is recognized in the proof that the universe is limited in De caelo, i. 8, 271 Ὁ 27 sqq. But the argument is not only unsound but unnecessary, for to deny that there is any distance which can be traversed in a limited time is to deny that any movement at all can ever be completed, and this is to deny the reality of motion, which is axiomatic to Aristotle (see Bk. VIII. chap. i. p. 269 and iii.
PHYSICS, VI. i.
PHYSICS, VI. i.
represent the limited distance, and the line from O an illimitable time. Take 5 anywhere on the line from O. Then in the limited time OS the mobile will traverse a certain part of the (hmited) distance; let that part be represented by Ac. (AC will either be an exact measure of AD or be less or greater than some exact measure of it, and it doesn’t matter which it is.*) For since a distance equal to AC will always be traversed in an equal time, then (taking the case in which AC is an exact measure of AD) the whole time occupied in covering AD will be limited; for it will be divided into the same (limited) number of parts-that the distance AD has been divided into. Again, if it be granted that it does not require an illimitable time to cover any distance whatever, but that there are distances for which a limited time suffices, such as AC (which we will suppose to be an exact measure of AD), then, if equal distances are traversed in equal times, it follows that the time taken in traversing AD must, like the distance AD itself, be finite. That a limited time does suffice for AC is evident,® for if we take as one limit of the time the instant at which the mobile p.293). The reader who has only an elementary acquaintance with mathematics is recommended to consult De Morgan’s Algebra, Chapter vi, where he discusses the meaning of the term ‘ infinite ’ and its application to limits and series.
At 238 a 8 sq. the same argument is applied to motion whose velocity is not uniform. It is assumed, however, that the velocity varies within limits or rather that it could not fall below any assignable limit whatever, for otherwise the length of time that would suffice for the final stage of the journey might rise above any assignable limit (notwith- standing that it would require less time than the whole dis- tance requires). And if there is any part of the distance for which no limited time suffices, 1t follows that no limited time will suffice for the whole.
ARISTOTLE 233 "θάτερα πεπερασμένος ὃ χρόνος" εἰ yap ἐν ἐλάττονι τὸ μέρος δίεισι, τοῦτο ἀνάγκη πεπεράνθαι, θατέρου γε πέρατος ὑπάρχοντος. ἡ αὐτὴ δ᾽ ἀπόδειξις καὶ τό εἰ τὸ μὲν μῆκος ἄπειρον ὁ δὲ χρόνος πεπερασμένος. ανερὸν οὖν ἐκ τῶν εἰρημένων ὡς οὔτε γραμμὴ οὔτε ἐπίπεδον οὔτε ὅλως τῶν συνεχῶν οὐθὲν ἐ ἔσται ἄτομον, οὐ μόνον διὰ τὸ νῦν “λεχθὲν ἀλλὰ καὶ ὅτι συμβήσεται διαιρεῖσθαι τὸ ἄτομον. ἐπεὶ γὰρ ἐν 20 ἅπαντι χρόνῳ τὸ θᾶττον καὶ βραδύτερόν ἐστι, τὸ δὲ θᾶττον πλεῖον διέρχεται ἐν τῷ ἴσῳ χρόνῳ, ἐνδέχεται. καὶ δυπλάσιον καὶ ἡμιόλιον διιέναι μῆκος" εἴη γὰρ ἂν οὗτος ὃ λόγος τοῦ τάχους. ἐνηνέχθω AB τ΄ δ E Z H K A M N a οὖν τὸ θᾶττον ἡμιόλιον ἐν τῷ αὐτῷ χρόνῳ, καὶ διῃρήσθω τὰ μεγέθη TO" μὲν τοῦ θάττονος ἐφ᾽ Ὁ ΑΒΓΔ εἰς τρία ἄτομα, τὸ δὲ τοῦ βραδυτέρου εἰς δύο ἐφ᾽ ὧν EZ, ZH. οὐκοῦν καὶ ὁ χρόνος διαιρεθή- σεται εἰς τρία ἄτομα" τὸ “γὰρ ἴσον ἐν τῷ ἴσῳ χρόνῳ δέεισιν. διῃρήσθω οὖν ὁ χρόνος εἰς τὰ KA, AM, MN. πάλιν δ᾽ ἐπεὶ τὸ βραδύτερον ἐνήνεκται τὴν 1 [τὸ Oxf. Trans. Cf. Simplic. 954. 1.—C.]
a ¢ (Literally, ‘ For since in any and every time (occupied by movement) there can be a distinction of faster and slower.’
ΩΣ eo PHYSICS, VI. τι.
is at the point A, then, since by hypothesis time without the other limit suffices for the transit of AD in its totality, something less than that time (1.6. a period which has its other limit) will suffice for the lesser AC. The same proof applies to the case of an unlimited distance and limited time.
Evidently, then, neither length nor surface nor any continuum whatever can be indivisible, since (in addition to what has been said already) the contrary supposition would involve a division of the indivisible. For since there may be a faster or slower transit over any distance whatever,® and the mobile at the greater velocity covers more distance in an equal time, then according to the ratio of the velocities, the one mobile may cover twice or once and a half the distance covered by the other in the same time.
tume O R 5 Τ Let it be once and a half, and let AD, which the faster covers in the given time, consist of the three indivisibles AB, BC, CD. Then the distance covered in the same time by the slower will be two indivisibles, EF and FG. Now to the three atomic distances AB, BC, CD, there would correspond three atoms of time OR, RS, ST, making up the whole time taken by the swifter to cover AD. But this same time would be what the slower takes to cover EF and FG. It follows that half that distance (say ARISTOTLE 288 EZ, ZH, καὶ ὁ χρόνος τμηθήσεται. δίχα. διαιρε- 80 θήσεται ἄρα τὸ ἄτομον, καὶ τὸ ἀμερὲς οὐκ ἐν ἀτόμῳ δίεισιν ἀλλ᾽ ἐν πλείονι. φανερὸν οὖν ὅτι οὐδέν ἐστι τῶν συνεχῶν ἀμερές.
CHAPTER III ARGUMENT The present or ‘now’ which divides the past from the Suture must be indwisible. For the final limit of the past and the initial limit of the future cannot be‘ nets,’ but must either be (i) at the same instant, or (ii) be separated by a (divisible) period of time. If ‘now’ were a period of time, at would not be the authentic limit between the past and the future, but a period of time embracing that limit; for it would itself be divisible into past and future, and no part of it could definitely be said to be either past or future, for 88 ᾿Ανάγκη δὲ καὶ τὸ νῦν τὸ μὴ καθ᾽ ἕτερον ἀλλὰ καθ᾽ αὑτὸ καὶ πρῶτον. λεγόμενον ἀδιαίρετον εἶναι, 86 καὶ ἐν ἅπαντι τὸ τοιοῦτο χρόνῳ ἐνυπάρχειν. ἔστι 284 α γὰρ ἔσχατόν τι τοῦ γεγονότος, οὗ ἐπὶ τάδε οὐθέν ἐστι τοῦ μέλλοντος, καὶ πάλιν τοῦ μέλλοντος, οὗ ἐπὶ τάδε οὐθέν ἐστι τοῦ γεγονότος" ὃ δὴ ἔφαμεν ἀμφοῖν εἶναι πέρας. τοῦτο δὲ ἂν δειχθῇ ὅτι τοιοῦτόν ἐστι καθ᾽ αὑτὸ καὶ ταὐτόν, ἅμα φανερὸν ὅ ἔσται καὶ ὅτι ἀδιαίρετον. ᾿Ανάγκη δὴ τὸ αὐτὸ εἶναι τὸ νῦν τὸ ἔσχατον ἀμφοτέρων τῶν χρόνων" εἰ γὰρ ἕτερον, ἐφεξῆς μὲν οὐκ ἂν εἴη θάτερον θατέρῳ διὰ τὸ μὴ εἶναι συνεχὲς α 2,6. not as we speak of ‘ this present year,’ in virtue of the actual ‘ now’ being included in that period.
PHYSICS, VI. τι.--τῶι.
EF) would take the slower half the time OT. But that will involve bisecting the atom of time RS, and the mobile will take not an atom of time but more (an atom and a half) to cover its atomic distance EF. There is then no continuum that cannot be divided into parts.
CHAPTER ITI ARGUMENT (continued) the authentic limit would be indeterminate as it might come at any point within the period (283 Ὁ 33-234 a 24).
Why nothing can be either in motion or at rest in a ‘now,’ but all movement and rest must occupy time (a 24-b 9).
FurtTHER, what we call ‘now’ or ‘the present’ (not in any derivative sense,” but primarily and on its own account) must be indivisible, and there must be such a ‘now’ embraced in any defined period of time whatever. For the ‘now’ may be regarded as the limit up to which the past has run, none of the future being this side of it, and also as the limit from which the future runs, none of the past being that side of it. And accordingly we have defined it ® as the limit alike of the past on one side and of the future on the other. If, then, we can make good that it is really one and the same thing, namely the authentic ‘now,’ that limits the past and future, it will be clear also that it is indivisible.
Granting that the ‘present now’ is what divides the past and future, the limits of the past and future as determined above must be the same as each other and as the present; for if they were different from each other, they could not be next-in-succession to each other, for it has been shown that two limits can- ARISTOTLE 5844 ἐξ ἀμερῶν, εἰ δὲ χωρὶς ἑκάτερον, μεταξὺ ἔσται 10 15 χρόνος" πᾶν γὰρ τὸ συνεχὲς τοιοῦτον ὥστ᾽ εἶναί τι συνώνυμον μεταξύ τῶν περάτων. ἀλλὰ μὴν εἰ χρόνος τὸ μεταξύ, διαιρετὸν ἔσται (πᾶς γὰρ χρόνος δέδεικται ὅτι διαιρετός)" ὥστε διαιρετὸν τὸ νῦν. εἰ δὲ διαιρετὸν τὸ νῦν, ἔσται τι τοῦ γεγονότος ἐν τῷ μέλλοντι Kat τοῦ μέλλοντος ἐν τῷ γεγονότι" καθ᾽ ὃ γὰρ ἂν διαιρεθῇ, τοῦτο διοριεῖ τὸν παρ- ἥκοντα καὶ τὸν μέλλοντα χρόνον. ἅμα δὲ καὶ οὐκ av καθ᾽ αὑτὸ εἴη τὸ νῦν, ἀλλὰ Kal” ἕτερον: ἡ yap διαίρεσις οὐ καθ᾽ αὑτό. πρὸς δὲ τούτοις τοῦ νῦν \ TO μέν TL γεγονὸς ἔσται τὸ δὲ μέλλον, Kal οὐκ ἀεὶ TO αὐτὸ γεγονὸς ἢ μέλλον. οὐδὲ δὴ τὸ νῦν TO 4 (Literally, ‘ because a continuum (such as time has been shown to be) cannot be composed of indivisibles’ (which might be ‘ successive’), as we saw in chap. i—C.]
> [καθ ὃ yap κτλ. ‘ For at whatever point (this extended ‘now ’) is actually divided, that point will be the boundary between past and future.’ Accordingly, Aristotle argues, that segment of the extended now which lies before the actual point of division is really a part of the past, but it will be ‘in the future,’ since it is part of the supposed extended present, which, as it is the present, must be after the past and so in the future.—C. ] ¢ (Literally, ‘for the division (which carves out this ex- tended now) is not a division in the proper sense’ (viz. an indivisible boundary).—C. ] @ Aristotle is not now speaking of the present as something which is always moving with the passage of time, so that what was future becomes past—but as something which is regarded as a period of time, divisible at any point into past and future. Let RT be the divisible present, then it could be divided at any point into past and future. If it were divided at §,, then RS, would be past and §,T would be future. But it might equally well be divided at 8,. in which case RS, would be past and 8,17 would be future; so that 8,8, might equally well be claimed as past or as future. Thus there PHYSICS, VI. m1.
not be ‘nexts’ to each other”; and if the two limits of time were separated, there must lie between them something of the denomination of the continuum they limit, which in this case is time. The ‘now’ that separates the past from the future would in that case itself be a period of time, and as such it would be divisible. Thus the ‘now’ that separates past and future would be divisible, and so would contain past and future. But all that was beyond the past was declared to be future with no past in it, whereas this supposed extended ‘ now’ lies beyond the limit of the past and is therefore future, but it has some past in it. And by analogous reasoning, being this side of the future, it is past, but has some of the future in Yt post 17 2244 present νὰν ιίμρεςς γε PUsb spp sss Lisp Ρεδκπὶ BASSALAS \Juture VS Upasty 2,222 td A SAS AAAS flere νὰ it.2 Again, such a ‘now’ would not be the proper but the derivative ‘now,’ for it would not be the limit but only the segment of a continuum within which the limit occurs.° Besides, a portion of this ‘now’ would be past and another portion future, and it would not in every case be the same portion that was past or future.“ In fact the ‘now’ itself would would be nothing to determine which part of the ‘ present’ was past and which future.
R Ss, τ “4 past future 5 2 ARISTOTLE e 2848 αὐτό" πολλαχῇ γὰρ διαιρετὸς 6 χρόνος. ὥστ᾽ εἶ 0 ταῦτα ἀδύνατον ὑπάρχειν τῷ νῦν, ἀνάγκη τὸ αὐτὸ εἶναι TO ἐν ἑκατέρῳ viv. ἀλλὰ μὴν εἰ ταὐτό, φανερὸν ὅτι Kal ἀδιαίρετον" εἰ yap διαιρετόν, πάλιν ταὐτὰ συμβήσεται ἃ καὶ ἐν τῷ πρότερον. ὅτι μὲν τοίνυν ἐστί τι ἐν TH χρόνῳ ἀδιαίρετον o φαμεν εἶναι τὸ νῦν, δῆλόν ἐστιν ἐκ τῶν εἰρημένων. Ore δ᾽ οὐδὲν ἐν τῷ νῦν κινεῖται, ἐκ τῶνδε os φανερόν ἐστιν. εἰ γάρ, ἐνδέχεται Kat θᾶττον κινεῖσθαι ἐν αὐτῷ καὶ βραδύτερον. ἔστω δὴ τὸ νῦν ἐφ᾽ ᾧ Ν, κεκινήσθω δ᾽ ἐν αὐτῷ τὸ θᾶττον τὴν A " ω a4 age ee ee Ὅ»- ¢ Ne ee ne ee ee ee ge Ζ ΑΒ. οὐκοῦν τὸ βραδύτερον ἐν τῷ αὐτῷ ἐλάττω τῆς ΑΒ κινηθήσεται, οἷον τὴν AT. ἐπεὶ δὲ τὸ βραδύτερον ἐν ὅλῳ τῷ viv κεκίνηται τὴν AT, τὸ 80 θᾶττον ἐν ἐλάττονι τούτου κινηθήσεται. ὥστε διαιρεθήσεται τὸ νῦν. GAA “Hv ἀδιαίρετον. οὐκ ἄρα ἔστι κινεῖσθαι ἐν τῷ νῦν.
ἃ μὴν οὐδ᾽ ἠρεμεῖν. ἠρεμεῖν γὰρ ἐλέγομεν τὸ πεφυκὸς κινεῖσθαι μὴ κινούμενον ὅτε πέφυκε PHYSICS, VI. 11.
not be identical with itself, because the time of which it consisted could be divided at many points. Thus, if all these contradictions follow from the hypo- thesis that the limit to which the past reaches is other than the limit from which the future starts, it follows that the two definitions of ‘now’ define one and the same ‘now.’ And if that be so, obviously the present ‘now’ is indivisible; for if it were divisible, we should be back again in the same impossibilities. Thus we have shown that there is a something pertaining to time which is indivisible, and this something is what we mean by the ‘present’ or ‘now.’
That nothing can move in the ‘present now,’ the following considerations will show. If it is possible for there to be motion in the ‘now,’ then there can be faster or slower motion in it. Let N be the Cc D distance ‘| tzme N ‘now,’ and let the faster movement cover AD in N. Then the slower will cover something less, say AC, in N. But since the slower covers AC in the whole of N, the faster will cover it in less than N. But this means dividing the ‘now,’ which we have found to be indivisible. Motion in the ‘now’ is therefore impossible.
And neither is rest or ‘station’ possible in the ‘now.’ For, as we were saying, a thing is ‘ at rest ’ only if it is naturally capable of such and such motion, ARISTOTLE 234a καὶ οὗ καὶ ὥς" ὥστ᾽, ἐπεὶ ἐν TH νῦν οὐδὲν πέφυκε κινεῖσθαι, δῆλον ὡς οὐδ᾽ ἠρεμεῖν. 3 "Eri δ᾽, εἰ τὸ αὐτὸ μὲν ἔστι τὸ νῦν ἐν ἀμφοῖν τοῖν 234b χρόνοιν, ἐνδέχεται δὲ τὸν μὲν κινεῖσθαι τὸν ὃ ἠρεμεῖν ὅλον, TO δ᾽ ὅλον κινούμενον TOV χρόνον ἐν ὁτῳοῦν κινηθήσεται τῶν τούτου καθ᾽ ὃ πέφυκε κινεῖσθαι, Kal TO ἠρεμοῦν ὡσαύτως ἠρεμήσει, συμβήσεται TO αὐτὸ ἅμα ἠρεμεῖν καὶ κινεῖσθαι" TO δ γὰρ αὐτὸ ἔσχατον τῶν χρόνων ἀμφοτέρων, τὸ νῦν. τι δ᾽ ἠρεμεῖν μὲν λέγομεν TO ὁμοίως ἔχον Kat αὐτὸ Kal τὰ μέρη νῦν καὶ πρότερον" ἐν δὲ TH νῦν οὐκ ἔστι TO πρότερον, ὥστε OVD ἠρεμεῖν. Ανάγκη ἄρα καὶ κινεῖσθαι τὸ κινούμενον ἐν χρόνῳ Kal ἠρεμεῖν TO ἠρεμοῦν. α * Beginning’ here means the initial limit, not (as at 236 a 15) the first part. Ὁ 2,6. they cannot be instantaneous.
PHYSICS, VI. x11.
but at this specified time and this specified place (though still retaining its natural capacity for such motion) it is not actually moving. So that if (as just proved) nothing has a natural capacity of moving in the ‘now,’ nothing can be said to be at rest ‘in it.’
Moreover, if a stretch of time be divided by a ‘now’ into two periods, we have seen that this ‘now ἡ qua end of the first period and qua beginning of the second is the same. Now suppose a mobile to be in motion in the whole of the first period and at rest in the whole of the second. Being in motion in the whole of the first period, it will be in motion in any of it which is naturally capable of being moved in; so that if its end were capable of being moved in, it would be moving in that. And, by parity of reasoning, if the beginning of the second period were capable of being rested in, it would be at rest in the beginning of that. But the end of one and the beginning of the other are the same ‘now’: there- fore if a ‘now’ were capable of being moved in and rested in, the same mobile would be at once moving in it qua end of one section and at rest in it qua be- ginning of the other.
Yet again, we say a thing is at rest when it has not changed its position, either in respect to its total- ity or in respect to its parts, between now and then: but there is no ‘then’ in ‘now’; so there is no being at rest.
Both motion and rest, then, must necessarily occupy time.® ARISTOTLE CHAPTER IV INTRODUCTORY NOTE The demonstration that anything that changes must be divisible begins with an examination of the condition of a thing ‘in motion’ which includes changing in quality, quantity or position.
There are four logical alternatives: ῳ to be both entirely in one contrary and entirely in the other, (2) to be either entirely in one contrary ov entirely in the other, (3) not to be in either at all, (4) to be partially in one and partially in the other.
Of these the first is not possible under any circum- stances; the second is not possible when the subject is actually ‘in motion’; the third is not possible when the motion is between these contraries only. There remains the fourth—the only one possible when the thing is in the act of changing from the one contrary to the other.
There is no relevant distinction between the various processes of change, but their incidental differences must be taken into account when applying the general argu- ment. To make this clear we may vary the illustration ARGUMENT [Everything that can be in a process af change must be divisible into parts, for it must, while changing, be partially in the one condition, partially in the other (234 b 10-20).
A movement is divisible both (i) in respect of the teme it occupies and (ii) according to the movements of the parts of the thing moved. (ii) Three proofs that the movement of the whole is equivalent to the sum of the movements of the paris (Ὁ 20-285 a 10). (4) The divisibility of movement in re- spect of time (a 10-18).
PHYSICS, VI. rv.
CHAPTER IV INTRODUCTORY NOTE (continued) given in the text, and suppose the contraries to be light and darkness The change from the one to the other of these may occur in either of two ways.
Ist way: A black shadow may gradually invade a bright surface and eventually eclipse it entirely; in this case the immediate subject of change is the area of the shadow. Now when any part of the surface comes under shadow it remains so, and for it the change is over. Similarly the change has not yet begun for any part that is still in the light; but during any period of the change the shadow progresses over some part of the surface, and any such progress is divisible because any part of the surface is divisible.
2nd way: the light may gradually fade from the whole surface simultaneously; in this case the immediate sub- ject of change is the brightness. As the shadow deepens the whole surface passes simultaneously from lighter to darker and the progress of the change is measured by the degree in which it does so. Any such progress is divisible because any degree is divisible.
Change of position is measured by linear distance, and is divisible because any distance is so. Similarly increase and decrease of magnitude is divisible because any magnitude is so.
ARGUMENT (continued) The movement, the time it occupies, the actual ἡ being in motion,’ and the field in which the mavement occurs are all susceptible of being divided correspondingly with the thing moved (a 138-18). This is established on detail. (i) The divisions of movement and time correspond (a 18~24). (ii) The ‘being in motion’ (i.e. the condition of the thing moved in so far as it is actually being moved) can be ARISTOTLE ARGUMENT (continued) divided into parts in the same way as the movement; as a whole it is, like the movement, continuous (a 25-384). (dit) Whatever forms a field of change (e.g. distance, quantity, quality) is correspondingly divisible, though a quale is 234h10 Lo δὲ μεταβάλλον ἅπαν ἀνάγκη διαιρετὸν εἶναι. ἐπεὶ γὰρ ἔκ τινος εἴς τι πᾶσα μεταβολή, καὶ ὅταν μὲν ἢ ἐν τούτῳ εἰς ὃ μετέβαλλεν, οὐκέτι μετα- βάλλει, ὅταν δὲ ἐξ οὗ μετέβαλλε, καὶ αὐτὸ καὶ τὰ 15 μέρη πάντα, οὐ μεταβάλλει (τὸ yap ὡσαύτως ἔχον καὶ αὖτὸ καὶ τὰ μέρη οὐ μεταβάλλει) ἀνάγκη οὖν τὸ μέν τι ἐν τούτῳ εἶναι τὸ δὲ ἐν θατέρῳ τοῦ μετα- βάλλοντος" οὔτε yap ἐν ἀμφοτέροις οὔτ᾽ ἐν μηδετέρῳ ὅλον δυνατον. (λέγω δὲ εἰς 6 μεταβάλλει τὸ πρῶτον κατὰ τὴν μεταβολήν, οἷον ἐκ τοῦ λευκοῦ TO φαιόν, οὐ τὸ μέλαν" οὐ yap ἀνάγκη τὸ μεταβάλλον 20 ἐν ὁποτερῳοῦν εἶναι τῶν ἄκρων.) φανερὸν οὖν ὅτι πᾶν τὸ μεταβάλλον ἔσται διαιρετόν. Κίνησις δ᾽ ἐστὶ διαιρετὴ διχῶς, ἕνα μὲν τρόπον τῷ χρόνῳ, ἄλλον δὲ κατὰ τὰς τῶν μερῶν τοῦ κινου- μένου κινήσεις" οἷον, εἰ τὸ AI’ κινεῖται ὅλον, καὶ τὸ ΑΒ κινηθήσεται καὶ τὸ BI’. ἔστω δὴ τοῦ μὲν 2 ΑΒ ἡ ΔΕ, τοῦ δὲ ΒΓ ἡ EZ, κίνησις τῶν μερῶν. ἀνάγκη δὴ τὴν ὅλην, ἐφ᾽ ἧς ΔΖ, τοῦ ΑΤ εἶναι ’ὔ κίνησιν: κινήσεται γὰρ κατὰ ταύτην, ἐπείπερ 4 See p. 81 note ὁ.
PHYSICS, VI. tv. ARGUMENT (continued) divisible only accidentally (a 34-37). All these entities also go together in respect of being limited or unlimited, the attributes ‘ divisible’ and ‘unlimited’ belonging primarily to the thing that changes (a 37-b 5).—C.]
ANYTHING that changes must be divisible (with respect to that which is primarily affected by the change). For every change moves along a definite line from this condition to that; and when the mobile has already reached the ‘ condition to which’ then the change is no longer in progress; and as long as it remains, as a whole and in all its parts, in the ‘ condition from which’ so long the change is not yet in progress (for such is the definition of the static as opposed to the changing state). It follows, then, that during the whole progress of the change it must be partly under one condition and partly under the other, for it can- not be entirely under both or under neither. (I am speaking, of course, not of the extreme, but of a proximate degree of change—from white to grey, for instance, not from white to black—for it would not follow that it must be under one or other of the extreme conditions.) It is clear therefore that any- thing that changes must be divisible into parts.* A movement is divisible in two ways: accord- ing to the time it occupies; according to the move- ments of the several parts of the moving thing. Thus, if AC moves as a whole there will be a move- ment of AB and also of BC.
Let LM represent the movement of the part AB, and MN the movement of the part BC. Then the whole LN must necessarily represent the movement of the whole AC. LN will constitute its movement, ARISTOTLE 234b ἑκάτερον τῶν μερῶν κινεῖται καθ᾽ ἑκατέραν, οὐδὲν Α Β Γ ee Στ ΝΡ ΘΝΕΝΕΝΗΝΝΒΒΟΒΒΒΒΘΒΒΒΘΉΟΝ nae ἘΠ semen | A E 7 δὲ κινεῖται κατὰ τὴν ἄλλου κίνησιν. ὥστε ἡ ὅλη κίνησις τοῦ ὅλου ἐστὶ μεγέθους κίνησις.
8.. Ἔστι δ᾽ εἰ πᾶσα μὲν κίνησις τινός, ἡ δ᾽ ὅλη κίνησις ἡ ἐφ᾽ ἧς AZ μήτε τῶν μερῶν ἐστι μηὃ- ετέρου (μέρους yap ἑκατέρα) μήτε ἄλλου μηδενός (οὗ yap ἡ ὅλη ὅλου, Kat τὰ μέρη τῶν μερῶν" τὰ δὲ μέρη τοῦ ΔΖ τῶν AB, BY καὶ οὐδένων ἄλλων" πλειόνων γὰρ οὐκ ἦν μία κίνησις), κἂν ἡ ὅλη κίνησις εἴη ἂν τοῦ ΑΒΤ' μεγέθους.
2858. Eire δ᾽ εἰ μὲν ἔστιν ἄλλη τοῦ ὅλου κίνησις, οἷον ep ἧς OI, ἀφαιρεθήσεται am’ αὐτῆς ἡ ἑκατέρων τῶν μερῶν κίνησις" αὗται δ᾽ ἴσαι ἔσονται ταῖς AE, ¢ The corresponding parts of the movement and of the mobile are here represented by the same length for the sake of simplicity; but all that is necessary is that the ratio of a part of the mobile to the whole of the mobile shall be the same as the ratio of the movement of that part to the movement of the whole.
> But to change along some other line—z.e. in some other category. [Literally, ‘for, as we have seen (at 228 Ὁ 1 ff.), a movement that is single, cannot be the movement of more than one thing.’-—C.]
PHYSICS, VI. τν.
since LM and MN respectively constitute the move- ments of each of its parts, and a thing’s movement cannot be constituted by the movement of anything else. Accordingly the whole movement is the move- ment of the whole magnitude.?
Besides, every movement is the movement of some | mobile. Now the whole movement represented by LN is not the movement of either of the parts taken by itself, for each part of LN is the movement of one part of AC. Neither is it the movement of anything else than the sum of the parts; for that same whole whose movement constitutes the whole movement is made up of those parts whose movements con- stitute the parts of the movement. That is to say, the parts of the movement which make up LN are the movements of the parts AB, BC, and of nothing else; for if there were any other movement, not thus accounted for, it would not pertain to the move- ment along the single path that we are considering.” So the movement represented by the sum of LM and MN is the movement of the mobile ABC.
Besides, if there is a movement of the whole AC other than that represented by LN, let it be repre- sented by GK. Now subtract from GK successively the movements of the several parts; these will be equal to LM and MN, for one thing must have one VOL. II K 129 ARISTOTLE 25a EZ: μία γὰρ ἑνὸς κίνησις. ὥστ᾽ εἰ μὲν ὅλη διαιρε- θήσεται ἡ ΘΙ εἰς τὰς τῶν μερῶν κινήσεις, ἴση ἔσται ἡ ΘΙ τῇ ΔΖ: εἰ δ᾽ ἀπολείπει τι, οἷον τὸ KI, δ αὕτη οὐδενὸς ἔσται κίνησις" οὔτε γὰρ τοῦ ὅλου οὔτε τῶν μερῶν (διὰ τὸ μίαν εἶναι τοῦ ἕνός), οὔτε ἄλλου οὐθενός: ἡ yap συνεχὴς κίνησίς ἐστι συνεχῶν τινῶν: ὡσαύτως δὲ καὶ εἰ ὑπερβάλλει κατὰ τὴν διαίρεσιν. ὥστ᾽ «εἰ τοῦτο ἀδύνατον, ἀνάγκη τὴν αὐτὴν εἶναι καὶ ἴσην. Αὕτη μὲν οὖν ἡ διαίρεσις κατὰ τὰς τῶν μερῶν 10 κινήσεις ἐστίν, καὶ ἀνάγκη παντὸς εἶναι τοῦ μερι- στοῦ αὐτήν. AdAn δὲ κατὰ τὸν χρόνον" ἐπεὶ γὰρ ἅπασα κίνησις ἐν χρόνῳ, χρόνος δὲ πᾶς διαιρετός, ἐν δὲ τῷ ἐλάττονι ἐλάττων ἡ κίνησις, ἀνάγκη πᾶσαν κίνησιν διαιρεῖσθαι κατὰ τὸν χρόνον. Ἐπεὶ δὲ πᾶν τὸ κινούμενον ἔν τινι κινεῦται καὶ Ἁ 18 χρόνον τινά, καὶ παντὸς ἔστι κίνησις, ἀνάγκη τὰς α i.e. any other subject, or of the given subject on any other part of the given line, or on any other line of change.
PHYSICS, VI. τν.
movement. Now if by this process GK has been exactly resolved into parts, then it is equal to LN mobile A isiB ς movement Ee ες ὅὕὙ τ ι-. Μ Ν which is also so resolved. But if, after the sub- traction, there were any remainder HK it would be a movement with no mobile, for it could not be the movement either of the whole AC or of the two parts that make it up, since one thing has one move- ment. But it cannot be the movement of anything else,” for by hypothesis we are considering a finite and continuous movement, and therefore the move- ment of a finite and continuous mobile, on a finite and continuous path. And by parity of reasoning the successive subtractions cannot cover more than GK. Since then GK is neither more nor less than LN, it must be equal to it.
Such then is the division of a movement accord- ing to the movements of the several parts of the mobile, and it applies of necessity to everything that is divisible.
Movement can be divided in another way: by time; for since every movement occurs in time, and time (however short) is always divisible, andin less time the movement is less, it follows that any movement can be divided in correspondence with divisions of time.
And again since anything that moves, moves along a definite line and during a certain time, and has a ARISTOTLE 285 αὐτὰς εἶναι διαιρέσεις τοῦ τε χρόνου Kal τῆς κινή- σεως καὶ τοῦ κινεῖσθαι καὶ τοῦ κινουμένου καὶ ἐν ᾧ ἡ κίνησις (πλὴν οὐ πάντων ὁμοίως, ἐν οἷς ἡ κίνησις, ἀλλὰ τοῦ μὲν ποσοῦ καθ᾽ αὗτό, τοῦ δὲ ποιοῦ κατὰ συμβεβηκός).
Εϊὐλήφθω γὰρ ὁ χρόνος ἐν ᾧ κινεῖται ἐφ᾽ ᾧ A, 20 καὶ ἡ κίνησις ἐφ᾽ ᾧ Β. εἰ οὖν τὴν ὅλην ἐν τῷ παντὶ χρόνῳ κεκίνηται, ἐν τῷ ἡμίσει ἐλάττω, καὶ πάλιν τούτου διαιρεθέντος ἐλάττω ταύτης, καὶ ἀεὶ οὕτως. ὁμοίως δὲ καὶ ἡ κίνησις διαιρετὴ καὶ ὁ χρόνος διαιρετός: εἰ γὰρ τὴν ὅλην ἐν τῷ παντί, τὴν ἡμίσειαν ἐν τῷ ἡμίσει, καὶ πάλιν τὴν ἐλάττω ἐν τῷ ἐλάττονι.
25 Τὸν αὐτὸν δὲ τρόπον καὶ τὸ κινεῖσθαι διαιρε- θήσεται. ἔστω γὰρ ἐφ᾽ ᾧ Τ' τὸ κινεῖσθαι: κατὰ δὴ τὴν ἡμίσειαν κίνησιν ἔλαττον ἔσται τοῦ ὅλου, καὶ πάλιν κατὰ τὴν τῆς ἡμισείας ἡμίσειαν, καὶ ἀεὶ οὕτως. ἔστι δὲ καὶ ἐκθέμενον τὸ καθ᾽ ἑκατέραν τῶν κινήσεων κινεῖσθαι, οἷον κατά τε τὴν ΔΙ᾽ καὶ 80. τὴν TE, λέγειν ὅτι τὸ ὅλον ἔσται κατὰ τὴν ὅλην (εἰ γὰρ ἄλλο, πλείω ἔσται κινεῖσθαι κατὰ τὴν αὐτὴν κίνησιν), ὥσπερ ἐδείξαμεν καὶ τὴν κίνησιν διαιρετὴν εἰς τὰς τῶν μερῶν κινήσεις οὖσαν" ληφθέντος γὰρ τοῦ κινεῖσθαι καθ᾽ ἑκατέραν, συνεχὲς ἔσται τὸ ὅλον.
Ὡσαύτως δὲ δειχθήσεται καὶ τὸ μῆκος δι- ¢ Aristotle is here thinking, not of the continuous modifi- cation of the quality itself (from white to black, for instance), but of the division, e.g. of a blue area, which does not divide the blueness except in the sense of making it the blueness of two areas. See Introd. Note.
PHYSICS, VI. τν.
certain motion, it must be possible to make corre- sponding divisions: of the time and of the movement, of the actual being-in-motion and of the mobile, and of that in respect to which it moves (the line along which the movement occurs), though not in the same way in all cases, for a quantum is divisible primarily and directly, but a quale by concomitance only.@ For let the time occupied be represented by A and the movement by B. Then if the whole movement has occupied the whole time, less than the whole movement occupied half the time, and less again a subdivision of that half, and so on without limit. And conversely the divisibility of time follows upon the divisibility of movement; for if the whole time is long enough for the whole movement, half the time will suffice for half of it, and a shorter time still for still less change.
And so too with divisions of the actual being- in-motion. Let C represent the whole of the being- in-motion. Then something short of the whole will correspond to half the movement, and so with the corresponding quarter of the being-in-motion and quarter of the movement and so on. And if we set out the being-in-motion corresponding to each of the two movements, DC and CE, we may argue that the whole of the being-in-motion would correspond with the whole movement (for otherwise there would be a being-in-motion without any corresponding movement) just as we showed a total movement to be divisible into the movements of its parts; for if the being-in-motion corresponds divisionally with the divisions of movement which is continuous, it is itself continuous, and there is no interval between its parts.
And in like manner it can be shown that the ARISTOTLE 285835 αἱρετόν, καὶ ὅλως πᾶν ἐν ᾧ ἐστιν ἡ μεταβολὴ 4 (πλὴν ἔνια κατὰ συμβεβηκός, ὅτι TO μεταβάλλον 2 ἐστὶ διαιρετόν) évds γὰρ διαιρουμένου πάντα δι- αἱρεθήσεται. 288» Kat ἐπὶ τοῦ πεπερασμένα εἶναι ἢ ἄπειρα ὁμοίως v4 ἕξει κατὰ πάντων. ἠκολούθηκε δὲ μάλιστα τὸ διαιρεῖσθαι πάντα καὶ ἄπειρα εἶναι ἀπὸ τοῦ μετα- LAA - ἡθὴ Ἁ > 7 “~ LAA ἄλλοντος“ εὐθὺς γὰρ ἐνυπάρχει τῷ μεταβάλλοντι τὸ διαιρετὸν καὶ τὸ ἄπειρον. τὸ μὲν οὖν διαιρετὸν 5 δέδεικται πρότερον, τὸ δ᾽ ἄπειρον ἐν τοῖς ἑπομένοις ἔσται δῆλον.
1 [Punctuation corrected, as in the Oxf. Trans.—C.]
CHAPTER V ARGUMENT In this chapter Aristotle begins by examining the end (i.e. the completion) of a change, and shows that this is a limit and is therefore indivisible, and occurs at an indivisible instant. He gives two independent proofs, one referring specially to contradictory changes, but also applicable to other kinds, and one referring specially to changes of place and He shows that if the completion were not the (indwisible) limit but a (divisible) part of the change, the change would arrive at completion while ἐξ was still going on, which is impossible (Ὁ 30-286 a 7).
He then turns to the ᾿ beginning,’ and shows that neither (a) the initial limit nor any subsequent (b) instant or (c) period can be the primary ‘when’ of the beginning of a ¢ Ἐπεὶ δὲ πᾶν τὸ μεταβάλλον ἔκ τινος εἴς τι μετα- βάλλει, ἀνάγκη τὸ μεταβεβληκός, ὅτε πρῶτον μετα- PHYSICS, VI. 1v.-v.
distance is divisible, and generally anything in respect to which a thing changes (with the reservation that the division is sometimes concomitant, depending on the fact that the subject that changes is itself divisible): for if any one of the connected group we are examining is divisible, so are all the others.
And they all go together as to being limited or without limit; and it is best to consider them all as following the changing subject itself in these respects, for divisibility and the absence of limitation primarily appertain to the subject that changes. As to divisi- bility, then, our examination is finished, but as to the absence of limitation it lies before us.
CHAPTER V ARGUMENT (continued) change. For (a) at the initial limit the subject is still unchanged; (b) no instant follows immediately on the initial limit (for two indivisibles cannot touch); and (c) any period is divisible, and the whole of a divisible period cannot be the primary ‘when’ of the beginning (a 7-27).
Nor is there any primary part of the subject affected, for everything that changes is divisible (a 27-26).
This argument applies without qualification to change of quantity, for anything which is directly divisible has no primary (or least possible) part. In change of quality, however, there may be a factor not directly divisible, but, as even this is indirectly divisible, there is no exception to the rule of divisibility in the factors of change (236 b 1-18).
Now since everything that changes, changes out of something into something else, it follows that the ARISTOTLE Ὄ 286 βέβληκεν, εἶναι ev ᾧ μεταβέβληκεν. τὸ yap μεταβάλλον ἐξ οὗ μεταβάλλει ἐξίσταται ἢ ἀπο- 10 λείπει αὐτό: καὶ ἤτοι ταὐτόν ἐστι τὸ μεταβάλλειν καὶ TO ἀπολείπειν, ἢ ἀκολουθεῖ τῷ μεταβάλλειν τὸ ἀπολείπειν: εἰ δὲ TH μεταβάλλειν τὸ ἀπολείπειν, τῷ μεταβεβληκέναι τὸ ἀπολελοιπέναι: ὁμοίως γὰρ ἑκάτερον ἔχει πρὸς ἑκάτερον. Ἐπεὶ οὖν pia τῶν μεταβολῶν ἡ κατ᾽ ἀντίφασιν, ὅτε μεταβέβληκεν ἐκ τοῦ μὴ ὄντος εἰς τὸ ὄν, 1s ἀπολέλοιπε τὸ μὴ ὄν. ἔσται ἄρα ἐν τῷ ὄντι" πᾶν γὰρ ἀνάγκη ἢ εἶναι ἢ μὴ εἶναι. φανερὸν οὖν ὅτι ἐν τῇ κατ᾽ ἀντίφασιν μεταβολῇ τὸ μεταβεβληκὸς - ἔσται ἐν ᾧ μεταβέβληκεν. εἰ δ᾽ ἐν ταύτῃ, καὶ Ὁ ΜΒ ness 2) ἐν ταῖς ἄλλαις" ὁμοίως yap ἐπὶ μιᾶς Kal τῶν ἄλλων. "Ere δὲ καθ᾽ ἑκάστην λαμβάνουσι φανερόν, εἴπερ 20 ἀνάγκη TO μεταβεβληκὸς εἶναί που ἢ ἔν τινι. ἐπεὶ γὰρ ἐξ οὗ μεταβέβληκεν ἀπολέλοιπεν, ἀνάγκη δ᾽ εἶναί που, ἢ ἐν τούτῳ ἢ ἐν ἄλλῳ ἔσται. εἰ μὲν οὖν ἐν ἄλλῳ, οἷον ev τῷ Τ᾽, τὸ εἰς TO Β μεταβεβληκός, πάλιν ἐκ τοῦ I’ μεταβάλλει εἰς τὸ Β (οὐ γὰρ ἦν 28 ἐχόμενον τῷ Β’ ἡ γὰρ μεταβολὴ συνεχής). ὥστε α΄ * Quitting ἡ here means ‘ getting free from,’ ‘ to have en- tirely quitted,’ ‘to be entirely free from’; thus ‘to have entirely quitted whiteness ᾿ would mean ‘ to be entirely free from any trace of whiteness.’