motion; then m = will be the whole motion, then if ὁ is the time occupied by the fractional motion mé will be the time occupied by the whole motion. Now m and z are both by hypothesis limited, therefore Aristotle’s conclusion depends on the unproved assumption that ¢ is also limited. See ARISTOTLE ath ᾿Αλλὰ δὴ κἂν μὴ ἰσοταχῶς, διαφέρει οὐθέν. 85 ἔστω yap ἐφ᾽ ἧς τὸ ΑΒ' διάστημα πεπερασμένον ὃ 8888 Κεκίνηται ἐν τῷ ἀπείρῳ, καὶ ὃ χρόνος ἄπειρος ἐφ᾽ οὗ τὸ ΓΔ. εἰ δὴ ἀνάγκη πρότερον ἕτερον ἑτέρου κεκινῆσθαν--τοῦτο δὲ δῆλον, ὅτι τοῦ χρόνου ἐν τῷ προτέρῳ καὶ ὑστέρῳ ἕτερον κεκίνηται: ἀεὶ γὰρ ἐν τῷ πλείονι ἕτερον ἔσται κεκινημένον, ἐάν τε ὅ ἰσοταχῶς ἐάν τε μὴ ἰσοταχῶς μεταβάλλῃ, καὶ ἐάν τε ἐπιτείνῃ ἡ κίνησις, ἐάν τε ἀνίῃ, ἐάν τε μένῃ, οὐθὲν ἧττον---εἰλήφθω δή τι τοῦ ΔΒ διαστήματος, AE B τὸ AE, ὃ καταμετρήσει τὴν AB. τοῦτο δὴ τοῦ ἀπείρου ἔν τινι ἐγένετο χρόνῳ" ἐν ἀπείρῳ γὰρ οὐχ 10 οἷόν τε, τὸ γὰρ ἅπαν ἐν ἀπείρῳ. καὶ πάλιν ἕτερον δὴ ἐὰν λάβω ὅσον τὸ AE, ἀνάγκη ἐν πεπερασμένῳ χρόνῳ: τὸ γὰρ ἅπαν ἐν ἀπείρῳ. καὶ οὕτω δὴ λαμβάνων, ἐπειδὴ τοῦ μὲν ἀπείρου οὐθέν ἐστι μό- ριον ὃ καταμετρήσει (ἀδύνατον γὰρ τὸ ἄπειρον εἶναι ἐκ πεπερασμένων καὶ ἴσων καὶ ἀνίσων, διὰ τὸ καταμετρηθήσεσθαι τὰ πεπερασμένα πλήθει Kal μεγέθει ὑπό τινος ἑνός, ἐάν τε ἴσα ἢ ἐάν τε πποονα 1 Ve AB Bonitz, ef. Simplic. 1000. 19: τὸ A καὶ τὸ B codd.
* Unlimited time would be better represented by ‘ the line from 0,’ without giving T as the other limit of what is by hypothesis unlimited.
> Assuming that the velocity cannot fall below any assign- able limit whatever.
¢ 4.é. less than the unlimited time which it takes to traverse PHYSICS, VI. vu.
But if the motion were not of uniform velocity it would make no difference. For let the limited distance supposed to be covered in an unlimited time be re- presented by AD and the unlimited time by OT.* Then one part of the distance must have been traversed before another part (this is clear because the distance traversed in the earlier part of the time is different from the distance traversed in the later, for with every increase in the time occupied a different amount of the distance will have been traversed, no matter whether the velocity be uniform or not, and none the less though the rate of motion be intensified or relaxed ὃ or constant). That being so, take a part AC of the whole distance AD and A Cc D A Cc D let AC be an exact measure of AD. ‘Then this part of the motion will occupy a limited stretch of the supposed unlimited time, for since the whole AD does but occupy an unlimited time, a fraction of it must occupy less.° But again, if I take another part of AD equal to AC that too must for the same reason occupy a limited time. Andif I go on doing this, since there are no definite periods of time which, added together, will make up an un- limited time (for the unlimited cannot be made up of limited items, whether these items are equal to one another or not, because there will always be some unit which will be an exact measure of a limited aggregate of magnitudes, no matter whether they be equal to each other or not, so long as they the whole distance (because it cannot take as long to traverse a part as it does to traverse the whole). See Ὁ. 112 note ὁ.
VOL, II M 161 ARISTOTLE 2384 ἄνισα, ὡρισμένα δὲ τῷ μεγέθει, οὐθὲν ἧττον), τὸ δὲ διάστημα τὸ πεπερασμένον ποσοῖς τοῖς AE μετρεῖται, ἐν πεπερασμένῳ ἂν χρόνῳ τὸ ΑΒ κινοῖτο. ὡσαύτως δὲ καὶ ἐπὶ ἠρεμήσεως.
Ὥ " γνεσθαι οὔτε φθείρεσθαι οἷόν τε στε οὔτε γίγνεσ ρ ἀεί τι τὸ αὐτὸ καὶ ἕν.
2 Ὃ αὐτὸς δὲ λόγος καὶ ὅτι οὐδ᾽ ἐν πεπερασμένῳ χρόνῳ ἄπειρον οἷόν τε κινεῖσθαι οὐδ᾽ ἠρεμίζεσθαι οὔθ᾽ ὁμαλῶς κινούμενον οὔτ᾽ ἀνωμάλως. λη- φθέντος γάρ τινος μέρους ὃ ἀναμετρήσει τὸν ὅλον χρόνον, ἐν τούτῳ ποσόν τι διέξεισι τοῦ μεγέθους καὶ οὐχ ὅλον (ἐν γὰρ τῷ παντὶ τὸ ὅλον), καὶ πάλιν ἐν 2 τῷ ἴσῳ ἄλλο, καὶ ἐν ἑκάστῳ ὁμοίως, εἴτε ἴσον εἴτε ἄνισον τῷ ἐξ ἀρχῆς" διαφέρει γὰρ οὐδέν, εἰ μόνον πεπερασμένον τι ἕκαστον" δῆλον γὰρ ὡς ἀναιρου- μένου τοῦ χρόνου τὸ ἄπειρον οὐκ ἀναιρεθήσεται, πεπερασμένης τῆς ἀφαιρέσεως γιγνομένης καὶ τῷ ποσῷ καὶ τῷ ποσάκις. ὥστ᾽ οὐ δίεισιν ἐν πεπε- 80 ρασμένῳ χρόνῳ τὸ ἄπειρον. οὐδέν τε διαφέρει τὸ μέγεθος ἐπὶ θάτερα ἢ ἐπ᾽ ἀμφότερα εἶναι ἄπειρον" ὁ γὰρ αὐτὸς ἔσται λόγος.
᾿Αποδεδειγμένων δὲ τούτων, φανερὸν ὅτι οὐδὲ τὸ πεπερασμένον μέγεθος τὸ ἄπειρον ἐνδέχεται δι- ελθεῖν ἐν πεπερασμένῳ χρόνῳ, διὰ τὴν αὐτὴν 86 αὐτίαν" ἐν γὰρ τῷ μορίῳ τοῦ χρόνου πεπερασμένον * [Defined at 230 a 4 as ‘ the motion to the goal at which the thing is at a standstill.’-—C.]
> [As usual, Aristotle adds the application of the principle, established in the case of change, to the case of coming-into- being and perishing: ‘ One and the same thing (being finite) cannot be in process of coming into being, or in process of perishing, for ever.’—C.] The creation or annihilation of a PHYSICS, VI. vu.
PHYSICS, VI. vu.
are finite in magnitude) and since, on the other hand, a certain multiple of AC will cover the whole limited distance AD, it follows that AD will be traversed in a limited period of time. And the case is the same with the process of being brought to rest.® As a consequence ὃ one and the same definite pro- cess of coming into being or passing out of it cannot possibly occupy an unlimited time.
By the same reasoning it follows conversely that, whether the motion be uniform or no, an illimitable process of moving or of coming to rest cannot be accomplished in a limited time. For if we take a definite fraction of the whole time, that fraction will allow a certain definite stretch, but not the whole, of the magnitude to be traversed (for it is only in the whole time that the whole magnitude is covered), and again in another equal fraction of the time, another definite stretch of the magnitude, andsoon. Whether each successive stretch is equal to the first or not makes no difference, so long as each one is limited; for it is evident that when the limited time is ex- hausted the unlimited magnitude will not be, for the subtractions from it are limited both in the how much and in the how many. Consequently the unlimited magnitude will not be traversed in a limited time. Nor does it matter whether the magnitude is unlimited in one direction only or in both, for the reasoning holds for either.
And from what has now been proved it follows that a limited magnitude could not traverse an un- limited magnitude in a limited time. And this for the same reason as before. For in a fraction of the definite thing is a limited change and therefore the above reasoning would apply here also.
ARISTOTLE 238a δίεισι, Kal ἐν ἑκάστῳ ὡσαύτως" ὥστ᾽ ἐν τῷ παντὶ πεπερασμένον. Ἔ A de A 4 2 δέ \ Ψ πεὶ δὲ τὸ πεπερασμένον οὐ δίεισι τὸ ἄπειρον 2880 ἐν πεπερασμένῳ χρόνῳ, δῆλον ὡς οὐδὲ TO ἄπειρον TO πεπερασμένον. εἰ yap TO ἄπειρον τὸ πεπε- \ ρασμένον, ἀνάγκη Kal TO πεπερασμένον διιέναι TO ἄπειρον. οὐδὲν γὰρ διαφέρει ὁποτερονοῦν εἶναι τὸ 3 5 κινούμενον: ἀμφοτέρως γὰρ τὸ πεπερασμένον δίεισι ΓΑ Β τὸ ἄπειρον. ὅταν γὰρ κινῆται τὸ ἄπειρον ἐφ᾽ ᾧ .
τὸ A, ἔσται τι αὐτοῦ κατὰ τὸ Β πεπερασμένον, οἷον τὸ ΓΔ, καὶ πάλιν ἄλλο Kat ἄλλο, καὶ ἀεὶ Ψ οὕτως. ὥστε ἅμα συμβήσεται τὸ ἄπειρον κε- κινῆσθαι τὸ πεπερασμένον καὶ τὸ πεπερασμένον 10 διεληλυθέναι τὸ ἄπειρον. οὐδὲ γὰρ ἴσως δυνατὸν ἄλλως τὸ ἄπειρον κινηθῆναι τὸ πεπερασμένον ἢ τῷ τὸ πεπερασμένον διιέναι TO ἄπειρον, ἢ φερόα Cannot ‘clear’ it (διελθεῖν), that is to say, cannot have all got past it, so that none of it is on the same side of the limited stationary magnitude as all of it was to begin with. The importance of the διελθεῖν is significant in relation to the misunderstanding in Book VII. chapter i.
> [Imagine an army forming a column marching past a saluting-base occupied by a general and his staff. Then imagine it 1s the saluting-base that moves carrying the general and his staff along the extent of the stationary column.
PHYSICS, VI. vn.
time it would traverse only a limited magnitude, and so too in each successive fractional stretch of the time, and so only a limited magnitude in the whole time. And since (as we have just seen) a_ limited magnitude could not traverse an unlimited in a limited time, evidently neither could an unlemrted magnitude clear? a limited in a limited time. For if the unlimited could clear the limited, the limited could traverse the unlimited; for it makes no differ- ence which of the two we think of as moving, for either case involves the traversing of the unlimited by the el ll el ll limited. Let the line a represent an unlimited and the line 6 a limited magnitude. Then when a is moving, there will be a limited stretch of it, CD, over against the limited magnitude 6 and then another and another and so on for ever. Consequently the two ways of regarding the process will come to the same thing: we may think either of the unlimited clearing the limited, or the limited traversing the unlimited in the opposite direction.? For we may say that the unlimited could not move over the limited in any other way than by the limited travers- ing the unlimited, that is either travelling along it or measuring it off into parts each equal to its own These are merely two different ways of obtaining one and the same result.—C. ] ARISTOTLE 238b μενον ἢ ἀναμετροῦν. ὥστ᾽ ἐπεὶ τοῦτ᾽ ἀδύνατον, οὐκ ἂν διίοι τὸ ἄπειρον τὸ πεπερασμένον.
᾿Αλλὰ μὴν οὐδὲ τὸ ἄπειρον ἐν πεπερασμένῳ χρόνῳ τὸ ἄπειρον δίεισιν. εἰ γὰρ τὸ ἄπειρον, καὶ 15 τὸ πεπερασμένον: ἐνυπάρχει yap τῷ ἀπείρῳ τὸ πεπερασμένον. ἔτι δὲ καὶ τοῦ χρόνου ληφθέντος ἡ αὐτὴ ἔσται ἀπόδειξις.
Ἐπεὶ δ᾽ οὔτε τὸ πεπερασμένον τὸ ἄπειρον δίεισιν οὔτε τὸ ἄπειρον τὸ πεπερασμένον οὔτε τὸ ἄπειρον τὸ ἄπειρον' ἐν πεπερασμένῳ χρόνῳ κινεῖται, φα- νερὸν ὅτι οὐδὲ κίνησις ἔσται ἄπειρος ἐν πεπε- 20 ρασμένῳ χρόνῳ. τί γὰρ διαφέρει τὴν κίνησιν ἢ τὸ μέγεθος ποιεῖν ἄπειρον; ἀνάγκη γάρ, εἰ ὅποτε- ρονοῦν, καὶ θάτερον εἶναι ἄπειρον. πᾶσα yap φορὰ ἐν τόπῳ.
1 [The second τὸ ἄπειρον is omitted in E. This may be due to an easy slip of the pen or the writer of E may have understood. the construction to be: ‘ Since in a limited time neither can the limited traverse an unlimited space nor does the unlimited move over either a limited or an unlimited space...’ The (unfortunately corrupt) paraphrase of Philoponus 871. 21 suggests that he took the passage so and read ἐπεὶ δ᾽ οὔτε τὸ πεπερασμένον τὸ ἄπειρον δίεισιν, οὔτε τὸ ἄπειρον οὔτε (7d?) πεπερασμένον οὔτε (7d?) ἄπειρον ἐν πεπερα- σμένῳ χρόνῳ. In any case the meaning is unaffected.—C.]
ig our illustration we might imagine either the general and his staff riding along the column, or the length of the saluting-base being used as a measure of the column’s length.—C. | PHYSICS, VI. vit.
length.* And since this is impossible it is imposs- ible that the unlimited should clear the limited.
Nor is it possible in a limited time for an unlimited magnitude to clear an unlimited one; for if it were, then a fortiort it would be possible for it to clear a limited one, since unlimited magnitude would in- clude limited magnitude. By like reasoning it can be shown that a limited mobile cannot occupy an unlimited time in measuring up a limited distance.?
Since then it is impossible that in a limited time either the limited should traverse the unlimited, or the unlimited clear the limited, or the unlimited move over the unlimited,’ it is evident that in a limited time there could not be unlimited motion: for what difference does it make whether we suppose the movement or the distance to be unlimited, for if either of the two is unlimited so must the other be, for all locomotion is in space? ὦ > [Or, * the same point (just stated) can be proved if we take the time ’ (and argue, on the same lines as at 238 a 20 ff, by dividing the time—rather than the magnitude—into a finite number of parts).—C.]
¢ This does not mean that a limited mobile could not come to be over against different stretches of an illimitable distance, or that different stretches of an illimitable mobile could not come to be over against the same limited object, in a limited time.
ὦ [Throughout this chapter Aristotle has really been thinking and speaking of locomotion only, and in any case he regards locomotion as the fundamental kind of change involved in the other kinds.—C. ] ARISTOTLE CHAPTER VIII ARGUMENT [Since what is coming to a stand must still be in motion, it follows that conclusions already established about things in motion apply to things that arecoming to a stand (238 Ὁ 23- (1) Coming to a stand must occupy a period of time (2) It must be happening in any part, however small, of the proper time occupied by the whole process (b 30-36); and hence, (9) Within the limits of the proper time occupied by the whole process, there is no portion of time, however small, which can mark an wrreducible earliest stage of the process easpe3 "Kiret δὲ πᾶν ἢ κινεῖται ἢ ἠρεμεῖ τὸ πεφυκὸς ὅτε πέφυκε καὶ οὗ καὶ ὥς, ἀνάγκη τὸ ἱστάμενον ὅτε os ἵσταται κινεῖσθαι: εἰ γὰρ μὴ κινεῖται, ἠρεμήσει" ἀλλ᾽ οὐκ ἐνδέχεται ἠρεμίζεσθαι τὸ ἠρεμοῦν.
Τούτου δ᾽ ἀποδεδειγμένου, φανερὸν ὅτι καὶ ἐν χρόνῳ ἵστασθαι ἀνάγκη. τὸ γὰρ κινούμενον ἐν χρόνῳ κινεῖται, τὸ δ᾽ ἱστάμενον δέδεικται κινού- μενον: ὥστε ἀνάγκη ἐν χρόνῳ ἵστασθαι. ἔτι de® 80. τὸ μὲν θᾶττον καὶ βραδύτερον ἐν χρόνῳ λέγομεν, ἵστασθαι δ᾽ ἔστι θᾶττον καὶ βραδύτερον.
Ἔν ᾧ δὲ χρόνῳ πρώτῳ τὸ ἱστάμενον ἵσταται, ἐν ὁτῳοῦν ἀνάγκη τούτου ἵστασθαι. διαιρεθέντος 1 [ἔτι δὲ E: ἔτι δ᾽ εἰ cett. As Themistius (198. 7) and Simplicius (1007. 11) saw, this sentence contains a further proof that coming to rest takes time. It has no connexion with the next.—C. ] δ [i.e. the ‘ proper time’ of the process as defined chapter Vi. init.—C, | PHYSICS, VI. vir.
CHAPTER VIII ARGUMENT (continued) Similarly there 1s no irreducible earliest stage in a period of ‘ being at rest’ (a 10-23).
Finally, if we take the period of time properly occupied by a movement, the moving thing cannot, during any part of that period, be situated so as exactly to ‘cover’ any stationary object. It can, and must, be so situated at any indiviscble instant; but an indivisible instant, as we have seen, is not a period of time, and neither rest nor motion can occur except during a period of tume (a 28-- 4).
This last conclusion is established specially with a view to the refutation of Zeno’s argument at the beginning of the next chapter.—C. | Now since anything which is naturally capable of being in motion or at rest can only move or rest when and where and how its nature allows, it follows that when a mobile is being brought to rest it must be in motion”; for if it is not in motion it will be at rest, and that which is at rest cannot be in process of being brought to rest.
This being so, it is evident that being brought to rest is an experience that occupies time; for anything in motion is moving in time, and what is being brought to rest, as has been shown, is in motion and therefore its being brought to rest occupies time. Again the terms ‘faster’ and ‘slower’ are applied exclusively to what occupies time, and we do apply them to the process of being brought to rest.
If we say that such and such a period is, as such, the period during which the mobile is being brought to rest,’ it must be in process of being brought to rest during any and every part of that period. For ARISTOTLE 238b yap τοῦ χρόνου, εἰ μὲν ἐν μηδετέρῳ τῶν μερῶν ἵσταται, οὐδὲ ἐν τῷ ὅλῳ, ὥστ᾽ οὐκ ἂν ἵσταυιτο τὸ ἱστάμενον" εἰ δ᾽ ἐν θατέρῳ: οὐκ ἂν ἐν πρώτῳ ὅλῳ Α 35 ἵσταιτο, καθ᾽ ἕτερον γὰρ ἐν τούτῳ ἵσταται" καθ- ἅπερ ἐλέχθη Kat ἐπὶ τοῦ κινουμένου πρότερον. 239a ὥσπερ δὲ τὸ κινούμενον οὐκ ἔστιν ἐν ᾧ πρώτῳ κινεῖται, οὕτως οὐδ᾽ ἐν ᾧ ἵσταται TO ἱστάμενον" A on a ΄- οὔτε γὰρ τοῦ κινεῖσθαι οὔτε τοῦ ἵστασθαι ἔστι τι “~ td a nN πρῶτον. ἔστω yap ἐν ᾧ πρώτῳ ἵσταται ἐφ᾽ ᾧ τὸ ΑΒ. τοῦτο δὴ ἀμερὲς μὲν οὐκ ἐνδέχεται εἶναι" 5 κίνησις yap οὐκ ἔστιν ἐν TH ἀμερεῖ, διὰ TO κεκινῆ- σθαΐί τι αὐτοῦ" τὸ δ᾽ ἱστάμενον δέδεικται κινούμενον. ἀλλὰ μὴν εἰ διαιρετόν ἐστιν, ἐν ὁτῳοῦν αὐτοῦ τῶν μερῶν ἵσταται: τοῦτο yap δέδεικται πρότερον, ὅτι ἐν ᾧ πρώτῳ ἵσταται, ἐν ὁτῳοῦν τῶν ἐκείνου ἵστα- ται. ἐπεὶ οὖν χρόνος ἐστὶν ἐν ᾧ πρώτῳ ἵσταται, 10 καὶ οὐκ ἄτομον, ἅπας δὲ χρόνος εἰς ἄπειρα μεριστός, οὐκ ἔσται ἐν ᾧ πρώτῳ ἵσταται.
α [καθ᾽ ἕτερον, as opposed to ἐν πρώτῳ, was used at 286} 21 of a longer period of which the proper time is only a part, here of a shorter period which is part of the proper time. The last words refer to the argument at 236 b 23-32.—C.] * Ὁ [As shown in chapter vi., 236 Ὁ 32 ff. ἐν ᾧ πρώτῳ here means no earliest part within the limits of the proper time (ἐν ᾧ πρώτῳ) as used in the last paragraph.—C.]
¢ [Literally, ‘ because of its having accomplished motion over some part of it,’ 1.6. whatever distance we take, a thing in motion over that distance has always moved over some part of it. But if the distance has parts, it is not indivisible. The reading of E, διὰ τὸ κεκινῆσθαι ἄν τι αὐτοῦ, would yield the same sense.—C. ] PHYSICS, VI. vir.
if we divide this period into two parts, then if the mobile is not in process of being brought to rest in either of them, neither is it so in both together, and therefore it would not be (as by hypothesis it is) in process of being brought to rest at all. Whereas if it were being brought to rest during one only of the parts the whole period would not be the proper time of the process, which would take place ‘in’ that period only in virtue of its taking place during one part of it, as has already been shown in the case of moving things in general.4 And just as there is no first (irreducible) period during which a moving thing can be said to be in motion,’ so likewise there is no first (irreducible) period during which a thing that is being brought to rest can be said to be undergoing the process; for there is no irreducible earliest stage of the process in either case. For if there were such a period, let it be OT. Then OT cannot be in- divisible; for no motion can occur in an indivisible instant, since it is always possible for a part of the motion to take place in a part of the time,° and we have shown that what is being brought to rest is in motion. If, on the other hand, OT is divisible, the process of arresting must be in progress in every part of it, for we have just shown that the process of arresting must be going on in every part of the proper time of that process. Since, then, the proper time occupied by a process of coming to rest is a stretch of time, not an indivisible instant, and any stretch of time is divisible without limit, there will be no assignable period of time (such as OT) which can mark an irreducible ‘earliest stage’ of the process.
ARISTOTLE 239 a Οὐδὲ δὴ τὸ ἠρεμοῦν ὅτε πρῶτον ἠρέμησεν. ἔστιν. ἐν ἀμερεῖ μὲν γὰρ οὐκ ἠρέμησε, διὰ τὸ μὴ εἶναι κίνησιν ἐν ἀτόμῳ" ἐν ᾧ δὲ τὸ ἠρεμεῖν, καὶ τὸ κινεῖσθαι" τότε γὰρ ἔφαμεν ἠρεμεῖν, ὅτε καὶ ἐν ᾧ πεφυκὸς κινεῖσθαι μὴ κινεῖται τὸ πεφυκός. ἔτι δὲ 15 καὶ τότε λέγομεν ἠρεμεῖν, ὅταν ὁμοίως ἔχῃ νῦν καὶ πρότερον, ὡς οὐχ evi τινι κρίνοντες ἀλλὰ δυοῖν τοῖν ἐλαχίστοιν" ὥστ᾽ οὐκ ἔσται ἐν ᾧ ἠρεμεῖ ἀμερές. εἰ δὲ μεριστόν, χρόνος ἂν εἴη, καὶ ἐν ὁτῳοῦν αὐτοῦ τῶν μερῶν ἠρεμήσει: τὸν αὐτὸν γὰρ τρόπον δει- 90 χθήσεται ὃν καὶ ἐπὶ τῶν “πρότερον. ὥστ᾽ οὐδὲν ἔσται πρῶτον" τούτου δ᾽ αἴτιον ὅτι ἠρεμεῖ μὲν καὶ κινεῖται πᾶν ἐν χρόνῳ, χρόνος δ᾽ οὐκ ἔστι πρῶτος οὐδὲ “μέγεθος οὐδ᾽ ὅλως συνεχὲς οὐδέν: ἅπαν γὰρ εἰς ἄπειρα μεριστόν.
, πεὶ δὲ πᾶν τὸ κινούμενον ἐν χρόνῳ κινεῖται καὶ ἔκ τινος εἴς τι μεταβάλλει, ἐν ᾧ χρόνῳ κινεῦται *[Or ἐν ᾧ ᾧ may be taken, as by the Oxford Trans., as the equivalent to ὅτε ‘added simply for the sake of introducing the exact expression used immediately before.’—C.
> Philoponus (815. 20) is certainly right in taking ἐλάχιστον as equivalent to πέρας. [I think Philoponus 1s certainly wrong. The phrase means ‘ by two as the minimum’; ef. 1181 a 15, ἔστι δὲ τὸ ἴσον ἐν ἐλαχίστοις δυσίν. ‘We mark off (determine, distinguish) a condition of rest, not by any single point but by not less than two.’—C.]
¢ [κατά τι here (as in κατὰ τὸ B at 238 Ὁ 6) means being situated over against some definite stationary object or measure of distance. πρῶτον (as in the phrase ἐν ᾧ πρώτῳ χρόνῳ) means the exact correspondence with this object or distance as opposed to the loose sense of being over against some part of the object. Aristotle is thinking of his moving thing as moving past something which would exactly define its position at a given instant—e.g. the piece of road exactly occupied by an army. If the army occupies that place for PHYSICS, VI. vit.
PHYSICS, VI. vit.
In like manner there is no irreducible first period during which the thing at rest has been resting. For on the one hand it could not be resting in an indivisible instant, because it could not be moving in an in- divisible instant, and motion must be possible in any time in which rest is possible; for we defined the state of rest as the state of a subject which is naturally capable of motion and exists in a time and medium in which * motion is naturally possible, but which is not moving. Moreover, we say that a thing is at rest when it has not changed its state between now and some previous instant; so that we do not judge rest by reference to one limit,’? but by reference to two. So there can be no indivisible instant in which it is at rest. On the other hand, if what we are speaking of is divisible, it must be a period of time, and in every part of that period the mobile is in a state of “being at rest,’ as demonstrated above. Consequently there can be no period whatever, of which (and of no shorter one) it can be said that in it the mobile was first at rest. The rationale of all this is that all motion and rest occurs in time, and there is no smallest or irreducible first component of time or of dimension or of anything that is continuous, for all such are divisible without limit.
And since whatever is in motion moves in a period of time and changes from one position to another, it is impossible that the mobile should in its entirety be exactly over against any definite (stationary) thing ὁ during the period occupied by its motion— any period of time, however short, during that period it is at rest, not moving, but at any indivisible instant it must be over against some such place.—C. | ARISTOTLE 239a2% καθ᾽ αὑτὸ Kal μὴ τῷ ἐν ἐκείνου τινί, ἀδύνατον τότε κατά τι εἶναι πρῶτον TO κινούμενον. TO yap ἠρεμεῖν ἐστι TO ἐν TH αὐτῷ εἶναι χρόνον τινὰ Kat αὐτὸ καὶ τῶν μερῶν ἕκαστον: οὕτω γὰρ λέγομεν ἠρεμεῖν, ὅταν ἐν ἄλλῳ καὶ ἄλλῳ τῶν viv ἀληθὲς ἢ εἰπεῖν ὅτι ἐν τῷ αὐτῷ καὶ αὐτὸ καὶ τὰ μέρη. εἰ 4 ~ F U κι 80 δὲ τοῦτ᾽ ἔστι τὸ ἠρεμεῖν, οὐκ ἐνδέχεται τὸ μετα- βάλλον κατά τι εἷναι ὅλον κατὰ τὸν πρῶτον χρόνον" ὁ γὰρ χρόνος διαιρετὸς ἅπας, ὥστε ἐν ἄλλῳ καὶ ἄλλῳ αὐτοῦ μέρει ἀληθὲς ἔσται εἰπεῖν ὅτι ἐν ταὐτῷ ἐστι καὶ αὐτὸ καὶ τὰ μέρη. εἰ γὰρ μὴ οὕτως ἀλλ᾽ ἐν ἑνὶ μόνῳ τῶν νῦν, οὐκ ἔσται χρόνον οὐθένα gs Κατά τι, ἀλλὰ κατὰ TO πέρας τοῦ χρόνου. ἐν δὲ 239bT@ νῦν ἔστι μὲν ἀεὶ κατά TL μένον, οὐ μέντοι ἠρεμεῖ---οὔτε yap κινεῖσθαι οὔτε ἠρεμεῖν ἔστιν ἐν τῷ νῦν--ἀλλὰ μὴ κινεῖσθαι μὲν ἀληθὲς ἐν τῷ νῦν 4 καὶ εἶναι κατά TL, ἐν χρόνῳ δ᾽ οὐκ ἐνδέχεται εἶναι κατὰ TO ἠρεμοῦν: συμβαίνει yap τὸ φερόμενον ἠρεμεῖν.
1 [τῷ ἐν ἐκείνου τινί EHI, Simplic. 1010. 2 and 27 (lemma): τῶν ἐν ἐκείνου τινί cett. Cf. Simplic. 1009. 29 (paraphr.) τῷ ἔν τινι τῶν ἐκείνου, which suggests that, as the Oxf. Trans. conjectures, the original text had τῷ ἐν τῶν ἐκείνου rwi.—C.]
α fi.e. the * proper time’ as already defined.—C.] ὃ That is to say that the mobile and the stationary object shall exactly fit, so that no part of either is left ‘ uncovered ’ by the other.
PHYSICS, VI. vit.
occupied, that is to say, in the proper sense, not in the sense that the motion falls within some part of the period in question.? For if a mobile is, in its entirety and all its parts, in the same place during a certain period of time, it is then at rest and not in motion; for the definition of being at rest is that it is true to say of the resting thing that from one ‘now’ to another both it and all its parts remain where they were. So if this is what being at rest means, it is impossible that a thing which is moving shall in its entirety exactly ‘cover’® a definite stationary thing during any part of the time properly occupied by its motion. For since time is divisible without limit and during every part of the proper time of the motion the mobile must be in motion, if it could be stationary in any part of it, however small, it could be stationary in such parts successively and so in the whole period. If the assertion refers not to an interval between two “ nows’ but to one single ‘now, then the moving thing will not be so situated during any period of time at all but only at a limit of suchaperiod. Nowitis true that at any particular instant the moving thing is always situated over against some stationary thing, but it is not ‘at rest,’ for in the indivisible instant there is neither rest nor motion. Rather, while it is true to say of the moving thing that at the indivisible instant it “ does not move ’ and is over against some definite thing, it cannot during any period of time be over against something that is at rest; for if it were it would be both moving and resting.
ARISTOTLE CHAPTER IX INTRODUCTORY NOTE Zeno’s four celebrated Dilemmas, and two others, are dealt with in this chapter. The forms in which they are given here are presumably those which were currently accepted in Aristotle’s time, and his refutations are satis- factory as far as they go.
Zeno’s arguments are designed to prove that whether the divisibility of time and distance is limitable or not, motion is alike impossible.
The Ist and 2nd are dilemmas on the supposition that there is no limit to the divisibility of time and distance, the 3rd and 4th on the supposition that they are divisible into indivisible atoms.
The 1st of the other two dilemmas attempts to show that change of quality, and the 2nd that rotation, is impossible.
They all depend on the equivocal use ofterms. Different (and sometimes wholly unconnected) meanings are given to the same term in the course of the argument, and con- clusions established for one meaning (and excluded from the other) are then transferred to that other. This is done so elusively that it easily escapes detection. If the same term were always used in the same sense, there would be no dilemma. As soon as it is clear how the terms are pie in each case the problems raised present no great diffi- culty.
Aristotle’s criticisms are readily followed, except that of Zeno’s 4th dilemma, the significance of which may be easily overlooked. The argument appears to be this: if motion, time, and distance consist of indivisible atoms, it will always require an equal time to traverse an equal distance and there can be no differences of velocity, as one atom of time and one atom of distance must always correspond to one atom of motion; for if erther corre- sponded to more than one, it (the atom of time or distance) would be divisible, because one atom of motion would PHYSICS, VI. rx.
CHAPTER IX INTRODUCTORY NOTE (continued) correspond to less than an atom of time or distance; and if one atom of motion corresponded to more than one of time or distance, then the atom of motion would be divisible for the same reason.
Now suppose that two sets of equal bodies (B’s and C’s) move past a set of equal stationary bodies (A’s) with equal speed but in opposite directions; it is evident that in the same time a B will pass one A and two C’s. But on our hypothesis (1) if it takes one unit of time for a B to pass one A it will take one unit for it to pass one C. (2) If it takes one unit to pass one C it will take two units to pass two C’s and (3) it always takes an equal time to pass an equal number of C’s. But we have just seen a B pass one A and two C’s in one and the same unit, so that (by 1) it takes one unit, and (by 2) two units of time to pass two C’s, and (by 8) these two periods must be equal. Thus if we accept the hypotheses, we must accept the conclusion that one unit of time is equal to two, which is absurd.
Aristotle’s criticism does not attack the method of deduction but the hypothesis itself, which precludes the possibility of passing different objects at different speeds, ‘ for,’ he says, ‘ the assumption that a moving object takes the same time in passing another object whether that other is stationary or in motion, is false.’
But there are deeper problems underlying Zeno’s para- doxes which challenge belief in the reality, not only of motion, but of time, distance, or any continuum, and it is these which have chiefly engaged the attention of modern writers (see M. Noél’s article “‘ Le mouvement et les argu- ments de Zénon d’Elée,” in the Revue de Métaphysique et de Morale, i. pp. 107-125). With the problems underlying the 3rd and 4th dilemmas Aristotle is not here concerned, for he rejects the assumptions on which they rest, and after saying this he gives them no further consideration.
VOL. II N 177 ARISTOTLE INTRODUCTORY NOTE (continued) But it is his business to deal with any problem involved in assumptions which he accepts, e.g. the Dichotomy, and his final answer to this is not given till Bk. VIJI. chap. viii. What he says is briefly this: that indivisible boundaries as such present no obstruction, and make no difference to the possibility of reaching the end of a limited continuum, so that it makes no difference how many there may be. But if they have to be counted, or in any other way require individual attention, they do cause obstruction, and the more of them there are the greater the obstruction.
Now the illimitable set of potential boundaries by which a continuum is inherently divisible cannot in their totality ARGUMENT Zeno’s contention that ‘ the flying arrow is not moving ’ depends on the assumption that the time of its flight is made up of indivisible instants in each of which it is at rest. This assumption has been shown to be false (239 Ὁ 5~9).
Zeno’s four arguments against motion being a reality, are each in turn examined and refuted.
(1) The Dichotomy.—That a moving object will never reach any given point, because however near it may be, it must always first accomplish a half-way stage, and then the half-way stage of what is left and so on, and this series has no end. Therefore, the object can never reach the end of any given distance. This has already been refuted (b 9-14).
(2) The Achilles.—That the swiftest racer can never over- take the slowest, if the slowest is given any start at all; be- cause the slowest will have passed beyond his starting-point when the swiftest reaches it, and beyond the point he has then reached when the swiftest reaches itand soon adinfin. This rests on the same fallacy as (1) (Ὁ 14-80).
PHYSICS, VI. rx.
INTRODUCTORY NOTE (continued) demand individual attention, for they are only defined generically and have not reached final and complete actual- ization as unique individuals, each distinguishable from any other member of its set. But the boundaries which are actually made by the dichotomy do reach final and complete actualization, and must be recognized as such; their number, however, depends on the amount of dicho- tomy which is actually completed, and as illimitable dichotomy can never be completed, the boundaries which are ever actually made by it cannot be illimitable; and as there is no impossibility in dealing with any limited number however great, there is no impossibility of reaching the end of a limited continuum.
ARGUMENT (continued) (3) The Flying Arrow.—That it is impossible for a thing to be moving during a period of time, because it is impossible Sor it to be moving at an indivisible instant. This assumes that a period of time is made up of indivisible instants, which cannot be granted (b 30-33).
(4) The Stadium.—That half a given period of time is equal to the whole of it; because equal motions must occupy equal times, and yet the time occupied in passing the same number of equal objects varies according as the objects are moving or stationary. The fallacy lies in the assumption that a moving body passes moving and stationary objects with equal velocity (Ὁ 33-240 a 18).
Two further fallacies about movement are refuted: