ὃ [Since the whole passage from A to Ὁ is continuous, any intermediate point C must not be regarded as ‘ next-without- an-interval ’ to D (cf. 231 Ὁ 15 sq., and the definition of ἐχόμενον 227 a 6), so that the change from Ὁ to D might occur all at PHYSICS, VI. v.
instant it has finished changing, it already is that which it has been changing into. For that which is in the process of changing is coming out of or quitting that which it is changing from, so that if ‘quitting’ is not the same thing as ‘changing from,’ at any rate it follows from it; and if ‘to be quitting’ is involved in ‘to be changing from,’ then ‘ to have completely quitted’ is involved in ‘to have completed the change from,’ for the relation is identical in either case.4 And since one sort of change is the change from one state to the flatly contradictory state, it follows that if a thing has passed out of non-existence into existence it has quitted non-existence. So it will have passed into existence, for everything must either exist or not exist. It is clear then that in the case of contradictory states the thing that has finished changing must already be in the state it was changing into. And if this is true in this particular case of change, it must be true in the other cases too, for there is no pertinent difference between them.
It is easy, moreover, to demonstrate the proposi- tion directly for changes of place or quality, if we admit that everything which has finished changing must be somewhere or in some state. For since it has passed out of where it was and must be some- where, that somewhere must be either the goal of the change or some other ‘where.’ Ifit be another, let the thing which has finished changing to D(B) be now at c(I'). In that case, once more, it is changing from C to Ὁ (for the point Ο is not ‘next’ to Ὁ, since the change is continuous).’ So the thing that had finished once; but the thing must be in process of changing all the way till Ὁ is reached.—C.]
ARISTOTLE 285» τὸ μεταβεβληκός, ὅτε μεταβέβληκε, μεταβάλλει εἰς 6 μεταβέβληκεν. τοῦτο δ᾽ ἀδύνατον: ἀνάγκη ἄρα τὸ μεταβεβληκὸς εἶναι ἐν τούτῳ εἰς ὃ μεταβέβληκεν. φανερὸν οὖν ὅτι καὶ τὸ γεγονός, ὅτε γέγονεν, ἔσται, καὶ τὸ ἐφθαρμένον οὐκ ἔσται: καθόλου τε γὰρ 80 εἴρηται περὶ πάσης μεταβολῆς, καὶ μάλιστα δῆλον ἐν τῇ κατὰ ἀντίφασιν.
Ὅτι μὲν τοίυν τὸ μεταβεβληκός, ὅτε μετα- βέβληκε πρῶτον, ἐν ἐκείνῳ ἐστί, δῆλον: ἐν ᾧ δὲ πρώτῳ μεταβέβληκε τὸ μεταβεβληκός, ἀνάγκη ἄτομον εἶναι. (λέγω δὲ πρῶτον ὃ μὴ τῷ ἕτερόν τι αὐτοῦ εἶναι πρῶτον τοιοῦτόν ἐστιν.) ἔστω γὰρ 86 διαιρετὸν τὸ AT, καὶ διῃρήσθω κατὰ τὸ Β. εἰ μὲν οὖν ἐν τῷ ΑΒ μεταβέβληκεν ἢ πάλιν ἐν τῷ ΒΓ, οὐκ ἂν ἐν πρώτῳ τῷ AT μεταβεβληκὸς εἴη. εἰ 2864 δ᾽ ἐν ἑκατέρῳ μετέβαλλεν (ἀνάγκη γὰρ ἢ μετα- βεβληκέναι ἢ μεταβάλλειν ἐν ἑκατέρῳ), κἂν ἐν τῷ ὅλῳ μεταβάλλοι" ἀλλ᾽ ἣν μεταβεβληκός. ὁ αὐτὸς δὲ λόγος καὶ εἰ ἐν τῷ μὲν μεταβάλλει, ἐν δὲ τῷ μεταβέβληκεν: ἔσται γάρ τι τοῦ πρώτου πρότερον" α (Cf. Ross, Aristotle (1923), Ὁ. 92: ‘An event is in a nest of times as a body is in a nest of places; the death of Caesar took place in March 8.6. 44, and also in B.c. 44, and also in the first century 8.c. The “first ’’ time of an event is the time it precisely occupies, its exact or commensurate time. There is in this respect a close analogy between Aristotle’s treatment of time and his treatment of place.’—C.]
PHYSICS, VI. v.
changing would at the instant at which it finished changing be in process of approaching the ‘where’ at which it had already arrived. But this is impossible. So the thing that has finished changing must already be in the state into which it has changed. So also it is clear that that which has come into existence at the instant when it has come into existence, exists, and that which has gone out of existence does not exist. The principle which has been laid down as applying to every kind of change is especially obvious in the case of the contradiction between existence and non-existence.
It is evident, then, that the thing that has changed, at the very instant ‘at which’ it has completed the change, is in the state ‘into which.’ But we must further note that the time ‘at’ which, in the primary sense, the change is completed is atomic, 1.6. indivis- ible. (I mean by ‘in the primary sense’ that it roust be directly and primarily the time ‘ at’ which, not some period that includes that precise instant.) @ For if we take a divisible OT(AT) to represent the time, let it be divided at S(B). Then, if the change came to its completion either in OS(AB) or in ST(BI’), in neither case can it be primarily located at OT(AT). If, on the other hand, the subject were in process of change in both parts (and it must have been either in process of change or in the state of having com- pleted the change in each of them), it would be in process of change in the whole OT(AT), which is contrary to the hypothesis that the change reached its completion ‘at’ the time OT(AT). And the argument may be repeated if it be said that the change was in progress during OS(AB), but was com- pleted ‘at’ sT(BI); for then OT(AT), the supposed ARISTOTLE 286a5 ὥστε οὐκ ἂν εἴ διαιρετὸν ἐν ᾧ μεταβέβληκεν.
φανερὸν οὖν ὅτι καὶ τὸ ἐφθαρμένον καὶ τὸ γεγονὸς ἐν ἀτόμῳ τὸ μὲν ἔφθαρται τὸ δὲ γέγονεν.
“Λέγεται δὲ τὸ ἐν ᾧ πρώτῳ μεταβέβληκε διχῶς--- τὸ μὲν ἐν ᾧ πρώτῳ ἐπετελέσθη ἡ μεταβολή (τότε γὰρ ἀληθὲς εἰπεῖν ὅτι μεταβέβληκε), τὸ δὲ ἐν ᾧ 10 πρώτῳ ἤρξατο μεταβάλλειν. τὸ μὲν οὖν κατὰ τὸ 1 σι τέλος τῆς μεταβολῆς πρῶτον λεγόμενον ὑπάρχει τε καὶ ἔστιν' ἐνδέχεται γὰρ ἐπιτελεσθῆναι μετα- βολήν, καὶ ἔστι μεταβολῆς τέλος" ὃ δὴ καὶ δέδεικται ἀδιαίρετον ¢ ὃν διὰ τὸ πέρας εἶναι. τὸ δὲ κατὰ τὴν ἀρχὴν ὅλως οὐκ ἔστιν: οὐ γὰρ ἔστιν ἀρχὴ μετα- βολῆς, οὐδ᾽ ἐν ᾧ πρώτῳ τοῦ χρόνου μετέβαλλεν. ἔστω γὰρ πρῶτον ἐφ᾽ ᾧ τὸ ΑΔ. τοῦτο δὴ ἀ- Γ᾽ A ἃ διαίρετον μὲν οὐκ ἔστιν: συμβήσεται yap ἐχόμενα εἶναι τὰ νῦν. ἔτι δ᾽ εἰ ἐν τῷ TA χρόνῳ παντὶ * [Aristotle does not mean that a process of change has not an anterior (as well as a posterior) limit. ‘ Beginning’ here means ‘ initial part,’ falling within the limits (πρώτη κίνησις rather than ἀρχὴ (= πέρας) κινήσεως, Them. 195. 20). Such a part would necessarily be infinitely divisible into earlier and earlier parts and occupy a divisible time.—C. ] ὃ [Literally, " Then OT cannot be indivisible; for, if it were, (the?) “ nows ” would be consecutive’ (which is impossible). What ‘nows’? τὰ νῦν isambiguous, (1) It may mean ‘the moments,’ as in the interpretation adopted by the Oxf. Trans.: ‘the moment immediately preceding the change and the moment in which the change begins would be con- secutive (and moments cannot be consecutive). But how does this consequence follow from the indivisibility of OT? (2) τὰ νῦν may mean moments generally (as at 237 a25 ov PHYSICS, VI. v.
PHYSICS, VI. v.
‘at’ in the primary sense, would only be such in virtue of containing the more primary ST(BI), and so on indefinitely. Thus the ‘time at which’ a change is completed cannot be divisible. It is also clear that that which passes out of existence or comes into existence must do so ‘at’ an indivisible moment. But there is still an ambiguity to clear up. Wecan speak of the primary ‘when’ ‘at’ which the change has been completed (for at that instant it is true to say that the change has been accomplished); but can we speak of the primary ‘when’ in which it began changing? As to what we call the primary ‘ when’ in reference to the end or completed change, we may safely assert that there really is such a thing; for a change can actually be completed and there is such a thing as its end, and we have shown that end to be indivisible, because it is a limit. But with reference to the beginning the phrase has no meaning, for there is no beginning of a process of change,? and no primary ‘ when’ in which the change was (first) in progress. For if there be, let OT re- O T time present that ‘when.’ Then OT cannot be indivisible, because it must be continuous with the initial limit (and indivisibles cannot be continuous).? Or again, γὰρ ἣν ἐχόμενα τὰ νῦν). The argument may then be: OT is supposed to be an earliest time occupied by an initial part of the process of change. We cannot regard OT as indivis- ible without falling back on the false view that the time occupied by a motion can be an atom of time corresponding to an atomic component of motion, and that all stretches of time are made up of such consecutive atoms.—C.]
ARISTOTLE 236 a ἠρεμεῖ (κείσθω γὰρ ἠρεμοῦν), καὶ ἐν τῷ A ἠρεμεῖ, ὥστε εἰ ἀμερές ἐστι τὸ ΑΔ, ἅμα ἠρεμήσει καὶ 20 μεταβεβληκὸς ἔσται: ἐν μὲν γὰρ τῷ Α ἠρεμεῖ, ἐν δὲ τῷ Δ μεταβέβληκεν. ἐπεὶ δ᾽ οὐκ ἔστιν ἀμερές, ἀνάγκη διαιρετὸν εἶναι καὶ ἐν ὁτῳοῦν τῶν τούτου μεταβεβληκέναι (διαιρεθέντος γὰρ τοῦ ΑΔ, εἰ μὲν ἐν μηδετέρῳ μεταβέβληκεν, οὐδ᾽ ἐν τῷ ὅλῳ: εἰ δ᾽ ἐν ἀμφοῖν μεταβάλλει, καὶ ἐν τῷ παντί: εἰ δ᾽ ἐν 25 θατέρῳ μεταβέβληκεν, οὐκ ἐν τῷ ὅλῳ πρώτῳ. ὥστε ἀνάγκη ἐν ὁτῳοῦν μεταβεβληκέναι). φα- νερὸν τοίνυν ὅτι οὐκ ἔστιν ἐν ᾧ πρώτῳ μετα- βέβληκεν’ ἄπειροι γὰρ αἱ διαιρέσεις. Οὐδὲ δὴ τοῦ μεταβεβληκότος ἔστι τι πρῶτον ὃ μεταβέβληκεν. ἔστω γὰρ τὸ ΔΖ πρῶτον μετα- 80 βεβληκὸς τοῦ ΔῈ (πᾶν γὰρ δέδεικται διαιρετὸν τὸ μεταβάλλον): 6 δὲ χρόνος ἐν ᾧ τὸ ΔΖ μεταβέβληκεν ἔστω ἐφ᾽ ᾧ ΘΙ. εἰ οὖν ἐν τῷ παντὶ τὸ ΔΖ μετα- βέβληκεν, ἐν τῷ ἡμίσει ἔλαττον ἔσται τι μετα- ) 1 [rc Oxf. Trans. Cf. Simplic. 987. 24: om. EH: τὸ cett, ] ----ὄἡὉὉ αἼΡΟΤ is indivisible there is nothing between O and T, thatis to say they are ‘in the same place,’ Bk. V., chap. iii. 226 Ὁ 20, 227 a 10, 1.6. they are the same instant. If no part of the change had occurred before O, the thing would still be unchanged at O. And if some part of it were already accomplished at T, the thing would simultaneously not have begun to change and have changed at O=T.
This argument is easily followed with respect to changes in which the affection itself does not change but pradually ‘invades’ the subject; itis also applicable, mutatis mutandis, to changes in which the affection is simultaneously in- tensified throughout the entire subject. See Introduction to chapter iv.
ARISTOTLE 236a βεβληκὸς καὶ πρότερον τοῦ AZ, καὶ πάλιν τούτου ἄλλο, κἀκείνου ἕτερον, καὶ ἀεὶ οὕτως. ὥστε A zZ E (easareronieneneleseatassnagrpe asters ΡΟΜΒΒΙΒΗΒΒΟΝΗΝΟΒΟΒΝΝΟΒΒΕΝΟΒΝΘΝΗΒΝΗΡῚ Leann eee elena | ovlev ἔσται πρῶτον τοῦ μεταβάλλοντος ὃ μετα- 35 βέβληκεν. TL μὲν οὖν οὔτε τοῦ μεταβάλλοντος οὔτε ἐν ᾧ μεταβάλλει χρόνῳ πρῶτον οὐθέν ἐστι, φανερὸν 3 σι ἐκ τῶν εἰρημένων. 236b Αὐτὸ δὲ 6 μεταβάλλει ἢ καθ᾽ ὃ μεταβάλλει οὐκέθ᾽ ὁμοίως ἕξει. τρία γάρ ἐστιν ἃ λέγεται κατὰ τὴν μεταβολήν---τό τε μεταβάλλον καὶ ἐν ᾧ καὶ ὃ μεταβάλλει: οἷον 6 ἄνθρωπος Kal ὁ χρόνος Kal τὸ 5 λευκόν. ὁ μὲν οὖν ἄνθρωπος καὶ ὁ χρόνος διαιρε- Tol, περὶ δὲ τοῦ λευκοῦ ἄλλος λόγος (πλὴν κατὰ συμβεβηκός γε πάντα διαιρετά" ᾧ γὰρ συμβέβηκε τὸ λευκὸν ἢ τὸ ποιόν, ἐκεῖνο διαιρετόν ἐστιν) ὦ ἐπεὶ ὅσα ye καθ᾽ αὑτὰ λέγεται διαιρετὰ καὶ μὴ κατὰ συμβεβηκός, οὐδ᾽ ἐν τούτοις ἔσται τὸ πρῶτον" 1 [Punctuation corrected as in Oxf. Trans. The next sentence, ἐπεὶ xrh., explains why ‘it is otherwise with that in respect of which the change takes place,’ viz. place or quality or quantity, in that a distinction is needed between changes of quality and other changes.—C. ] @ [As Simplic. 988. 9 explains, ὃ μεταβάλλει here has the PHYSICS, VI. v.
the period OS is completed, and again still less in less, and less again in less without limit. So that A ς D Cn nS SE TN Y O S there will be no part of the subject of the change which has in its entirety been the very first part that has changed.
We have now shown that in the subject of change there is no (irreducible) part that can be the first to have completed the change, and in the time occupied by the change no (irreducible) earliest part in which change has been effected.
But the case is different with that in * which, or with respect to which, the actual change takes place. For there are three things which are concerned whenever change is spoken οἷ: (1) the subject of the change; (2) the duration of it; and (3) that in respect of which it changes: for instance, (1) the man, (2) the time, and (3) the pallor. Now the man (as a bulk) and the time are divisible, but it is another thing with the pallor (though they are all, it is true, incidentally divisible, for the subject in which the pallor or other quality inheres is divisible). For when that with respect to which the thing changes is not incidentally but primarily divisible, it, too, has same meaning’ as τὸ καθ᾽ ὃ μεταβάλλει---ἰῃς place, quality (e.g. pallor), or quantity, in respect of which the subject of change x ARISTOTLE οἷον ἐν τοῖς μεγέθεσιν. ἔστω yap τὸ ἐφ᾽ ᾧ AB ͵ μέγεθος, κεκινήσθω δ᾽ ἐκ τοῦ B εἰς τὸ Τ' πρῶτον.
A Bek Inevesenntnemnstsinninindntinsnnitopmnaesmemmamrenrweramrsiartetarenibeanlinwmnivend οὐκοῦν ef μὲν ἀδιαίρετον ἔσται τὸ BI, ἀμερὲς ἀμεροῦς ἔσται ἐχόμενον" ef δὲ διαιρετόν, ἔσται τι τοῦ I’ πρότερον εἰς ὃ μεταβέβληκε, κἀκείνου πάλιν ἄλλο, καὶ ἀεὶ οὕτω διὰ τὸ μηδέποτε ὑπολείπειν τὴν διαίρεσιν. ὥστε οὐκ ἔσται πρῶτον εἰς ὃ μετα- βέβληκεν. ὁμοίως δὲ καὶ ἐπὶ τῆς τοῦ ποσοῦ μετα- βολῆς" καὶ γὰρ αὕτη ἐν συνεχεῖ ἐστι. φανερὸν οὖν ὅτι ἐν μόνῃ τῶν κινήσεων τῇ κατὰ τὸ ποιὸν ἐνδέχεται ἀδιαίρετον καθ᾽ αὑτὸ εἶναι.
“866 Introduction to chapter iv.
> [AB is a part of space occupied by a ‘magnitude’ (ré κινούμενον μέγεθος, Simplic.), also called AB; BC is the extension of this space, over which the motion is to take place.—C.]
9 ee the argument at 236a15. Here the objection is to thinking of space as capable of being filled by a row of suc- cessive atomic magnitudes, and to regarding motion as CHAPTER VI ARGUMENT If by the * proper time’ of a change we mean the period which exactly coincides with its duration, then the change must be going on throughout any part, however small, of the proper tume of the whole change (236 Ὁ 19-32).
From this it follows that (1) that which is found to be changing must have been changing previously, (2) that PHYSICS, VI. v.—v1.
no primary part. Take the case of magnitudes.® Let AB represent a magnitude, and let it have changed from B to a‘ primary’ positionc. Now if A B ς a Se See Se BC is to be indivisible, we shall have two things with- out parts contiguous to one another (which is im- possible) ὁ; while if it is divisible, there will be some position, coming before C, into which the thing has changed, and yet another before that, and so on without limit, since the divisibility is never exhausted. So there is no first point that has been reached. And this argument applies also to change of quantity, since this change too is in something continuous. It is manifest, then, that change in respect of quality is the only change in which there can be a factor not directly ¢ divisible.
occurring all at once from one atomic place to the ‘ next.’— C ἃς Directly,’ for even in a quale there must be incidental divisibility.
CHAPTER VI ARGUMENT (continued) which has accomplished a change must have accomplished some lesser change previously, (8) that which is found to be changing must have previously accomplished some change (Ὁ 32-237 a 17), and (4) that which has accomplished a change must previously have been changing (a 17-28).
It is obvious that all this is true in the case of change of ARISTOTLE ARGUMENT (continued) magnitude, for as magnitude 1s divisible without limit, there can be no least possible change of magnitude (a 28-b 9). The same reasoning may be extended to the becoming and perishing of all continuous and divisible things, i.e. to the 236b19 ᾿᾿ξπεὶ δὲ TO μεταβάλλον ἅπαν ἐν ypdvw μετα- 20 βάλλει, λέγεται δ᾽ ἐν χρόνῳ μεταβάλλειν καὶ ὡς ἐν πρώτῳ καὶ ὡς καθ᾽ ἕτερον (οἷον ἐν τῷ ἐνιαυτῷ, ὅτι ἐν τῇ ἡμέρᾳ μεταβάλλει), ἐν ᾧ πρώτῳ χρόνῳ μεταβάλλει τὸ μεταβάλλον, ἐν ὁτῳοῦν ἀνάγκη τούτου μεταβάλλειν. δῆλον μὲν οὖν καὶ ἐκ τοῦ ὁρι- σμοῦ---τὸ yap πρῶτον οὕτως ἐλέγομεν---οὐ μὴν ἀλλὰ “Ὁ 26 καὶ ἐκ τῶνδε φανερόν. ἔστω γὰρ ἐν ᾧ πρώτῳ (4 κινεῖται τὸ κινούμενον ἐφ᾽ ᾧ XP, Kal διῃρήσθω κατὰ τὸ K (πᾶς γὰρ χρόνος διαιρετός). ἐν δὴ τῷ XK χρόνῳ ἤτοι κινεῖται ἢ οὐ κινεῖται, καὶ πάλιν ἐν τῷ KP ὡσαύτως. εἰ μὲν οὖν ἐν δετέρω t 5: μ μὴ PY xX κ Ρ κινεῦται, ἠρεμοΐη av ἐν τῷ παντί: κινεῖσθαι yap ἐν 80 μηθενὶ τῶν τούτου κινούμενον ἀδύνατον. εἰ δ᾽ ἐν θατέρῳ μόνῳ κινεῖται, οὐκ ἂν ἐν πρώτῳ κινοῖτο τῷ ΧΡ: καθ᾽ ἕτερον γὰρ ἡ κίνησις. ἀνάγκη ἄρα ἐν ὁτῳοῦν τοῦ ΧΡ κεκινῆσθαι.
PHYSICS, VI. vz.
ARGUMENT (continued) consummations of all continuous changes; and also to all temporal changes of any kind. So there is no smallest poss- ible, and no irreducible first stage of either change or be- coming (b 9-22).
Au that changes changes ‘in time’; but when we speak of the time ‘in’ which a change occurs, we may mean either the ‘primary’ or proper time coinciding with the change or a longer period including the proper time (for instance it may have happened ‘ in’ such and such a year, because that year includes the day occupied by the change). That being so, the change must be taking place during every part of the proper time which the whole change occupies. This follows from our definition (for this is what we have taken ‘the proper time’ to mean), but it can also be demonstrated as follows. Let the proper time of the movement be represented by OT, and let OT be divided at 8 (for every space of time is divisible). Then in the period OS the mobile is either moving or not; and the same with ST. If it is mov- ing in neither of these periods it is at rest throughout OT, for it is impossible for it to move in OT if it moves in neither of its parts. If it is moving in one only of the two parts, then OT is not the proper time of its movement, for it pertains to the whole OT only in virtue of pertaining to a part distinguish- able from that whole. It must then have been moving in any part of OT if OT is to be the proper time of the movement.
ARISTOTLE 286} Δεδειγμένου δὲ τούτου, φανερὸν ὅτι πᾶν τὸ κινούμενον ἀνάγκη κεκινῆσθαι πρότερον. εἰ γὰρ K A anaes ASH ΝΣ ον ΒΟ ΒΝΝΝΝΝΒΗΡΟΒΒΒΒΟΝΟΒΒΟΜΒΟΝΝΟΒΝΝΙΝΝΝΟΒΉΝ XS 35 ἐν TH XP πρώτῳ χρόνῳ τὸ KA κεκίνηται μέγεθος, ἐν τῷ ἡμίσει τὸ ὁμοταχῶς κινούμενον καὶ ἅμα 237a ἀρξάμενον τὸ ἥμισυ ἔσται κεκινημένον. εἰ δὲ τὸ ὁμοταχὲς ἐν τῷ αὐτῷ χρόνῳ κεκίνηταί τι, καὶ θάτερον ἀνάγκη ταὐτὸ κεκινῆσθαι μέγεθος: ὥστε κεκινημένον ἔσται τὸ κινούμενον. ἔτι δὲ εἰ ἐν τῷ παντὶ χρόνῳ τῷ ΧΡ κεκινῆσθαι λέγομεν, ἢ ὅλως σι ἐν ὁτῳοῦν χρόνῳ, τῷ λαβεῖν τὸ ἔσχατον αὐτοῦ νῦν (τοῦτο yap ἐστι τὸ ὁρίζον, Kat τὸ μεταξὺ τῶν viv χρόνος), κἂν ἐν τοῖς ἄλλοις ὁμοίως λέγοιτο κεκινῆσθαι. τοῦ δ᾽ ἡμίσεος ἔσχατον ἡ διαίρεσις" ὥστε Kal ἐν τῷ ἡμίσει κεκινημένον ἔσται, καὶ ὅλως ἐν ὁτῳοῦν τῶν μερῶν: ἀεὶ yap dua TH τομῇ 10 χρόνος ἐστὶν ὡρισμένος ὑπὸ τῶν νῦν. εἰ οὖν ἅπας « Even if our observations are confined to noting what has occurred by the end of a time, we shall see that something has occurred by the end of half the time.
PHYSICS, VI. νι.
On the strength of this we can show that anything found to be moving must have been moving before.
D distance eee em sty res er eee {ΡΣ ΞΞΞΞΞΈ Ξ ΞΞΞΕΙ τες Geter sce ony ee tee ee @----- time O Ss T For (1) if P has covered the distance AD in the proper time OT, then Q starting at the same time and moving with equal velocity will cover half the dis- tance in half the time. And of two moving things that move at the same pace, if one accomplishes a given distance in a given time the other must do the same. (Therefore P will have already covered a certain distance in the time OS, and so was moving in a certain period less than OT, which accordingly was not the actual first assignable period during which the mobile was in motion.) So that which is found to be moving must have been moving before. Again, (2) if what enables us to say that move- ment has taken place in the whole time OT—or generally in any period you please—is that we have taken the terminal ‘now’ (for the terminal ‘now’ is what defines a period, a period being what lies between two ‘nows’), then movement may equally be said to have taken place in the other periods (OS and ST). But the half (of OT) has its terminal point (S) in our division; accordingly movement will have taken place in the half or, generally, in any part you please; for wherever we divide the period, it will be limited by a ‘now’ coinciding with that division. And since any period of time is divis- ARISTOTLE ARISTOTLE 2818 μὲν χρόνος διαιρετός, τὸ δὲ μεταξὺ τῶν νῦν χρόνος, ἅπαν τὸ μεταβάλλον ἄπειρα ἔσται μετα- βεβληκός. ἔτι δ᾽ εἰ τὸ συνεχῶς μεταβάλλον καὶ μὴ φθαρὲν μηδὲ πεπαυμένον τῆς μεταβολῆς ἢ μετα- βάλλειν ἣ μεταβεβληκέναι ἀναγκαῖον ἐν ὁτῳοῦν, 16 ἐν δὲ τῷ νῦν οὐκ ἔστι μεταβάλλειν, ἀνάγκη μετα- βεβληκέναι καθ᾽ ἕκαστον τῶν viv ὥστ᾽ εἰ τὰ νῦν ἄπειρα, πᾶν τὸ μεταβάλλον ἄπειρα ἔσται μετα- βεβληκός. Οὐ μόνον δὲ τὸ μεταβάλλον ἀνάγκη μεταβεβλη- κέναι, ἀλλὰ Kat τὸ μεταβεβληκὸς ἀνάγκη μετα- βάλλειν πρότερον. ἅπαν γὰρ τὸ ἔκ τινος εἴς τι 20 μεταβεβληκὸς ἐν χρόνῳ μεταβέβληκεν. ἔστω γὰρ ev τῷ viv ἐκ τοῦ εἷς τὸ μεταβεβληκός. οὐκοῦν ἐν μὲν τῷ αὐτῷ νῦν ἐν ᾧ ἐστιν ἐν τῷ Α, ov μεταβέβληκεν (ἅμα yap ἂν εἴη ἐν τῷ A καὶ τῷ Β): τὸ γὰρ μεταβεβληκός, ὅτε μεταβέβληκεν, ὅτι οὐκ ἔστιν ἐν TovT@, δέδεικται πρότερον. εἰ δ᾽ ἐν ἄλλῳ, μεταξὺ ἔσται χρόνος" οὐ yap ἦν ἐχόμενα τὰ νῦν. ἐπεὶ οὖν ἐν χρόνῳ μεταβέβληκε, χρόνος δ᾽ ἅπας διαιρετός, ἐν τῷ ἡμίσει ἄλλο ἔσται μετα- βεβληκός, καὶ πάλιν ἐν τῷ ἐκείνου ἡμίσει ἄλλο, καὶ ἀεὶ οὕτως: ὥστε μεταβάλλοι ἂν πρότερον. Ere δ᾽ ἐπὶ τοῦ μεγέθους φανερώτερον τὸ λεχθέν, 1 [τούτῳ codd.: Simplic. 994. 30 has ταὐτῷ (ACM) or ἐνταῦθα (aF). If τούτῳ is sound, perhaps we should read ὅτε μεταβέβληκεν, «ἐξ οὗ μεταβέβληκεν», dri...—C.]
PHYSICS, VI. νι.
ible, and what lies between the two ‘ nows’ is time, it follows that anything that changes at all has already completed an unlimited succession of changes. Again (3) anything that is continuously changing and has neither perished nor ceased from changing, must at any point either be changing or have changed, and since we have proved that there is no such thing as changing at a ‘now,’ it follows that at any ‘now’ the changing thing must ‘ have changed,’ so that if the nows are without limit, the mobile will have accomplished unlimited changes. And not only must that which is in process of change have previously accomplished changes, but that which has accomplished a change must previ- ously have been in the process of changing. For if a thing has changed from this to that, it has occupied time in doing so. For suppose a thing has accom- plished the change from A to B at aninstant. Then the instant at which it has accomplished the change is not the same as that at which it was in A (other- wise it would be in A and B at the same time); for it has already been proved? that when a thing has changed, it is not where it was before it had changed. But if the ‘now’ of A and the ‘now’ of B are not the same, there must be a space of time between them, for one ‘ now’ is not contiguous with another one. Since, then, the change has occupied time and time is always divisible, in half the time occupied by the whole change a certain other change will have been accomplished, and again another in the half of that, and so on without limit. The subject, then, has been changing before it has changed. Moreover, it is helpful to note that all this is more obvious in the case of magnitude, because the ARISTOTLE nn διὰ τὸ συνεχὲς εἶναι τὸ μέγεθος ἐν ᾧ μεταβάλλει τὸ μεταβάλλον. ἔστω γάρ τι μεταβεβληκὸς ἐκ τοῦ D εἰς TO A. οὐκοῦν εἰ μὲν ἀδιαίρετόν ἐστι τὸ ΓΔ, ἀμερὲς ἀμεροῦς ἔσται ἐχόμενον. ἐπεὶ δὲ τοῦτο ἀδύνατον, ἀνάγκη μέγεθος εἶναι τὸ μεταξὺ καὶ εἰς ἄπειρα διαιρετόν' ὥστ᾽ εἰς ἐκεῖνα μεταβάλλει 86 πρότερον. ἀνάγκη ἄρα πᾶν τὸ μεταβεβληκὸς μεταβάλλειν πρότερον" ἡ γὰρ αὐτὴ ἀπόδειξις καὶ 237 b ἐν τοῖς μὴ συνεχέσιν, οἷον ἔν τε τοῖς ἐναντίοις καὶ ἐν ἀντιφάσει" ληψόμεθα γὰρ τὸν χρόνον ἐν ᾧ μεταβέβληκε, καὶ πάλιν ταὐτὰ ἐροῦμεν. “Ὥστε ἀνάγκη τὸ μεταβεβληκὸς μεταβάλλειν καὶ 5 τὸ μεταβάλλον μεταβεβληκέναι, καὶ ἔστι τοῦ μὲν μεταβάλλειν τὸ μεταβεβληκέναι πρότερον, τοῦ δὲ μεταβεβληκέναι τὸ μεταβάλλειν" καὶ οὐδέποτε ληφθήσεται τὸ πρῶτον. αἴτιον δὲ τούτου τὸ μὴ εἶναι ἀμερὲς ἀμεροῦς ἐχόμενον" ἐπ᾽ ἄπειρον γὰρ ἡ διαίρεσις, καθάπερ ἐπὶ τῶν αὐξανομένων καὶ καθαιρουμένων γραμμῶν. τὸ Φανερὸν οὖν ὅτι καὶ τὸ γεγονὸς ἀνάγκη γίγνεσθαι πρότερον καὶ ἱ τὸ γιγνόμενον γεγονέναι, ὅσα διαιρετὰ καὶ συνεχῆ---οὐ μέντοι ἀεὶ ὃ γίγνεται, ἀλλ᾽ ἄλλο α 4,4. the distance from A has extended from C to D.
Ὁ (Literally, * divisible into an unlimited number of parts. So the thing is in process of changing into those (innumer- able parts) before (it completes its change at D).’—C.]
¢ The continuity of contradictory changes such as coming into and out of existence is explained below, 237 Ὁ 13 sqq.
4 [Simplicius 996. 19 explains the process meant: Bisect a line and keep one half undivided; bisect the other half and add one of its halves to your original undivided half, and so on.—C. 4 [Aristotle has already said at a 35 that the same reasoning apples to changes between contraries and between contra- PHYSICS, VI. νι.
magnitude over which the change takes place is continuous. For suppose a thing has changed from C to D.* Then if CD were indivisible, two things which have no parts would be consecutive, and since this is impossible the space between must be a magnitude and therefore divisible without limit. So the subject effects innumerable changes before it has effected any given change.? So anything that has changed must previously have been changing, for the same proof holds for changes with respect to what is not continuous,’ namely changes between contraries and between contradictories. For if we consider the time which the accomplished change has occupied, we may apply the same reasoning whatever the nature of the change.
So then that which has changed must have been changing, and that which is changing must have changed; there is a ‘having changed’ that comes before ‘changing’ and a ‘changing’ before “ having changed,’ and we shall never find a stage which is the (irreducible) first stage; the reason being that two in- divisibles can never be contiguous, but the interspace can be subdivided without limit, as a line may be divided without limit in such a way that one part is always increasing and the other decreasing.@ It is further clear ¢ that in the case of all divisible and continuous things, whatever has come to be must previously have been coming to be and what- ever is coming to be must previously have come to be, (but not all things are continuous and so what dictories (i.e. becoming and perishing) if you consider the factor of time. He now adds that the argument from the infinite divisibility of magnitude also applies to becoming and perishing of such things as are continuous and divisible magnitudes.—C. | ARISTOTLE 2871} ἐνίοτε, οἷον τῶν ἐκείνου TL, ὥσπερ τῆς οἰκίας τὸν θεμέλιον---ὁμοίως δὲ καὶ ἐπὶ τοῦ φθειρομένου καὶ ἐφθαρμένου. εὐθὺς “γὰρ ἐνυπάρχει τῷ γιγνομένῳ 1b καὶ τῷ φθειρομένῳ a ἄπειρόν τι, συνεχεῖ γε ὄντι, καὶ οὐκ ἔστιν οὔτε γίγνεσθαι μὴ γεγονός τι οὔτε γεγο- νέναι μὴ γιγνόμενόν τι. ὁμοίως δὲ καὶ ἐπὶ τοῦ φθείρεσθαι καὶ ἐπὶ τοῦ ἐφθάρθαι: ἀεὶ γὰρ ἔσται τοῦ μὲν φθείρεσθαι τὸ ἐφθάρθαι πρότερον, τοῦ δὲ ἐφθάρθαι τὸ φθείρεσθαι. φανερὸν οὖν ὅτι καὶ τὸ 80 γεγονὸς ἀνάγκη γίγνεσθαι πρότερον καὶ τὸ γιγνό- μενον γεγονέναι" πᾶν γὰρ μέγεθος καὶ πᾶς χρόνος ἀεὶ διαιρετά: ὥστ᾽ ἐν ᾧ ἂν ἧ, οὐκ ἂν εἴη ὡς πρώτῳ.
* (Viz. bodies, which are necessarily continuous and in- finitely divisible.—C.]
ὃ That is: must have accomplished some stage of its becoming.
CHAPTER VII ARGUMENT The aim of this chapter is to show (1) that a limited change or ‘motion’ could not occupy unlimited time; and as a definite process of “coming to rest’ or of ‘becoming’ or of ‘perishing’ is a limited change, none of these could occupy unlimited time; and (2) | that in a limited time there could not be unlimited ‘ motion.’
The only kind of motion which is directly dealt with in this chapter ts locomotion, for every kind of change can be represented by this kind of movement.
Prop. (1). If the motion is uniform it can be divided into a limited number of equal parts, and each of these parts will correspond to an equal period of time; so that the whole time can be divided into the same number of equal PHYSICS, VI. vz.—viz.
has already come to be may be other than that which is coming to be; it may be some independent part of it, as when the foundation has come but the house is still coming to be,) and it is the same with passing away and having passed away. For both what is coming-into-being and what is passing-away contain, in so far as they are continuous things,” an element of unlimitedness; and it is impossible for anything to be in process of coming to be unless it has already come to be something or to have come to be unless it has previously been in process of becoming something; and it is the same with pass- ing away and having passed away, for passing away youst always have been preceded by having passed away, and having passed away by passing. It is clear therefore in the case of coming to be, no less than in other changes, that what has come to be must previously have been in process of coming to be, and that what is coming to be must have come to be® previously, for every magnitude and every period of time is divisible without limit. Conse- quently whatever stage the thing may have reached, that stage can never be the irreducible first stage in the process.
CHAPTER VII ARGUMENT (continued) parts. On the assumption that these periods of time are limited in duration as well as in number, tt follows that the whole time will be limited in duration (237 b 23-34). The proposition is next demonstrated for motion that is not uniform (in this demonstration it is assumed that the velocity cannot fall below any assignable limit whatever ARISTOTLE ARGUMENT (continued) (b 34-288 a 19)). Prop. (2) is demonstrated in a similar manner, An ‘unlimited motion’ may be either (a) that which is limited traversing that which is unlimited, or 2310 Lurel δὲ πᾶν TO κινούμενον ἐν χρόνῳ κινεῖται, καὶ ἐν τῷ πλείονι μεῖζον μέγεθος, ἐν τῷ ἀπείρῳ χρόνῳ ἀδύνατόν ἐστι πεπερασμένην κινεῖσθαι---μὴ τὴν αὐτὴν ἀεὶ καὶ τῶν ἐκείνης τι κινούμενον, ἀλλ᾽ ἐν ἅπαντι ἅπασαν.
Ὅτι μὲν οὖν εἴ τι ἰσοταχῶς κινοῖτο, ἀνάγκη τὸ πεπερασμένον ἐν πεπερασμένῳ κινεῖσθαι, δῆλον. ληφθέντος γὰρ μορίου ὃ καταμετρήσει τὴν ὅλην, 80 ἐν ἴσοις χρόνοις τοσούτοις ὅσα τὰ μόριά ἐστι, τὴν ὅλην κεκίνηται. ὥστ᾽ ἐπεὶ ταῦτα πεπέρανται καὶ τῷ πόσον ἕκαστον καὶ τῷ ποσάκις ἅπαντα, καὶ 6 χρόνος ἂν εἴη πεπερασμένος" τοσαυτάκις γὰρ ἔσται τοσοῦτος ὅσος 6 τοῦ μορίου χρόνος πολλαπλασιασθεὶς τῷ πλήθει τῶν μορίων.
α [μὴ τὴν αὐτὴν xrd. seems intended to exclude perpetual rotation; ‘for the heavenly bodies, for instance, can con- tinually revolve over a finite distance during an unlimited time’ (Philop. 812. 22). In such rotation a body is (1) “moving continually with the same motion (7.e. along a circular track) and (continually) executing some part of that motion.’ The last words are somewhat obscure. They might mean (1) that the body is not to be supposed to execute one revolution and then stop, but to be always moving in some region of the track; but the following words suggest that the meaning is rather (2) that the body is executing one revolu- tion after another, each revolution being regarded as a limited part of the unlimited perpetual motion. Aristotle means that he is not denying that a body can revolve for an unlimited time on a limited track, but is denying that it takes ‘all’ unlimited time to execute ‘all’ a limited motion.—C.]
> [Literally, ‘ Consequently, since these fractions are finite, both in the sense that each is a finite quantity, and in the 2!
pa | PHYSICS, VI. vii.
ARGUMENT (continued) (b) that which is unlimited clearing that which is limited, or (c) that which is unlimited clearing that which is un- limited (a 19--Ὁ 22).
SINCE every thing that moves occupies time in moving, and moves further in a longer time, it is impossible for a limited motion to occupy unlimited time. I am not speaking of a locally recurrent or re- entrant motion in which the same limited track may be traversed an unlimited number of times,” but of the whole distance covered in the whole time. Now it is clear enough that if the motion is of uniform velocity, a limited distance must be traversed in a limited time. For if we take any fraction of the motion the whole-motion will be some multiple of that fraction and the time-occupied-by-the-whole- motion will be the same multiple of the time-occupied- by - the - fraction- of-the-motion. Consequently, since these fractions are finite both in magnitude and in number, the time also will be finite; for it will be a multiple of the time-occupied-by-the-fractional- motion, and will be equal to that period of time multiplied by the number of fractional-motions.° sense that their sum is a finite multiple of one of them, the time also will be finite; for it will be a multiple of the fractional period (occupied by the fractional motion), equal to such a period multiplied by the number of the fractional movements.’—C.] Let = be a given fraction ofthe