It is sometimes argued that a change between contradictory conditions (e.g. from not-white to white) is impossible be- cause a thing must always be either white or not-white, and during the process of changing from one to the other it would be neither white nor not-white. This assumes that ‘ white’ ARISTOTLE ARGUMENT (continued) means ἡ pure white’ and ‘ not-white’ means ἡ without any trace of white,’ which is not the customary use of the terms 239b5 Ζήνων δὲ mapadoyileras: εἰ yap adel, φησίν, ἠρε- pet πᾶν ὅταν᾽' ἢ κατὰ τὸ ἴσον, ἔστι δ᾽ ἀεὶ τὸ φερό- μενον ἐν τῷ νῦν, ἀκίνητον τὴν φερομένην εἶναι ὀιστόν. τοῦτο δ᾽ ἐστὶ ψεῦδος" οὐ γὰρ σύγκειται ὁ χρόνος ἐκ τῶν νῦν τῶν ἀδιαιρέτων, ὥσπερ οὐδ ἄλλο μέγεθος οὐδέν.
10 Térrapes δ᾽ εἰσὶ λόγοι περὶ κινήσεως Ζήνωνος ot παρέχοντες τὰς δυσκολίας τοῖς λύουσι--πρῶτος μὲν ὃ περὶ τοῦ μὴ κινεῖσθαι διὰ τὸ πρότερον εἰς τὸ ἥμισυ δεῖν ἀφικέσθαι TO φερόμενον ἣ πρὸς TO τέλος, περὶ οὗ διείλομεν ἐν τοῖς πρότερον λόγοις.
Δεύτερος δὲ 6 καλούμενος ᾿Αχιλλεύς. ἔστι ὃ 1 [ἠρεμεῖ πᾶν ἢ κινεῖται ὅταν codd.; Philop. 816. 30; Simpl. 1011. 27. Zeller ejected ἢ κινεῖται on the ground that Zeno’s premiss was the definition of rest. This is supported by Themistius’s paraphrase (199. 4) ef yap ἠρεμεῖ, φησίν, ἅπαντα ὅταν ἢ κατὰ τὸ ἴσον αὑτῷ διάστημα. A possible reading would be: εἰ γὰρ ἀεί, φησίν, ἠρεμεῖ [πᾶν ἢ κινεῖται ζκαὶ μὴ κινεῖται ὅταν κτλ. ‘If everything is either at rest or in motion and is not in motion when it is over against something of equal dimensions, while a moving thing is always so (7.6. κατὰ τὸ ἴσον) at the moment, the flying arrow is motionless.’—C.]
2 [ἐν τῷ viv τῷ κατὰ τὸ ἴσον rcF, i.e. ‘at every moment the moving thing is occupying the moment (of the time occupied by its whole movement) which corresponds to the space equal to its own dimensions.’ The time is supposed to be made up of a row of successive indivisible moments corresponding, one to one, with the row of successive positions occupied by the body. τῷ κατὰ τὸ ἴσον was probably added in this uss. to make it clear that ἐν τῷ νῦν means this. Cf. Philop. 817. 6, who says that after ἔστι δὲ τὸ φερόμενον ἐν τῷ νῦν we must ARISTOTLE , ARISTOTLE , 289510 οὗτος ὅτι τὸ βραδύτατον᾽ οὐδέποτε καταληφθήσεται θέον ὑπὸ τοῦ ταχίστου" ἔμπροσθεν γὰρ ἀναγκαῖον ἐλθεῖν τὸ διῶκον ὅθεν ὥρμησε τὸ φεῦγον, ὥστ det τι προέχειν ἀναγκαῖον τὸ βραδύτερον. ἔστι 20 δὲ καὶ οὗτος ὁ αὐτὸς λόγος TH διχοτομεῖν, διαφέρει δὲ ἐν TH διαιρεῖν μὴ δίχα τὸ προσλαμβανόμενον Ou \ μέγεθος. τὸ μὲν οὖν μὴ καταλαμβάνεσθαι τὸ βραδύτερον συμβέβηκεν ἐκ τοῦ λόγου, γίγνεται δὲ παρὰ ταὐτὸ τῇ διχοτομίᾳ---ἐν ἀμφοτέροις γὰρ συμβαίνει μὴ ἀφικνεῖσθαι πρὸς τὸ πέρας διαιρου- μένου πως τοῦ μεγέθους" ἀλλὰ πρόσκειται ἐν τούτῳ 2 ὅτι οὐδὲ TO τάχιστον τετραγῳδημένον ἐν τῷ διώκειν τὸ βραδύτατον"---ὥστ᾽ ἀνάγκη Kal τὴν λύσιν εἶναι τὴν αὐτήν. τὸ δ᾽ ἀξιοῦν ὅτι τὸ προ- éyov οὐ καταλαμβάνεται, ψεῦδος- ὅτε yap προέχει οὐ καταλαμβάνεται, ἀλλ᾽ ὅμως καταλαμβάνεται, εἴπερ δώσει διεξιέναι τὴν πεπερασμένην. οὗτοι 80 μὲν οὖν οἱ δύο λόγοι. Τρίτος δὲ ὃ νῦν ῥηθείς, ὅτι ἡ ὀιστὸς φερομένη ἕστηκεν. συμβαίνει δὲ mapa τὸ λαμβάνειν τὸν 1 [βραδύτατον. Cf. Ὁ 25; Them. 199. 25; Philop. 817. 15 εἰπὼν ὅτι τὸ τάχιστον οὐ καταλήψεται τὸ βραδύτατον: Simpl.
1014. 1; Oxf. Trans.: βραδύτερον codd.—C.] 2 [βραδύτατον EI: βραδύτερον FHK.—C.]
¢ The distance from the starting-point to the point at which the slower is overtaken by the swifter is easily caleu- lated, if the start allowed and the respective velocities are PHYSICS, VI. rx.
PHYSICS, VI. rx.
which purports to show that the slowest will never be overtaken in its course by the swiftest, inasmuch as, reckoning from any given instant, the pursuer, before he can catch the pursued, must reach the point from which the pursued started at that instant, and so the slower will always be some distance in advance of the swifter. But this argument is the same as the former one which depends on bisection, with the difference that the division of the magni- tudes we successively take is not a division into halves (but according to any ratio we like to assume. between the two speeds). The conclusion of the argument is that the slower cannot be overtaken by the swifter, but it is reached by following the same lines as the ‘ bisection’ argument of the first thesis; for the reason why neither supposed process lands us at the limit, is that the method of division is expressly so designed as not to get us there, only in this second thesis a declamatory intensification is introduced by representing the swiftest racer as unable to overtake the slowest. The solution then must be identical in both cases, and the claim that the thing that is ahead is not overtaken is false. It is not overtaken while it is ahead, but none the less it zs overtaken if Zeno will allow it to traverse to the end its finite distance. So much for these two theses.
The third thesis is the one just mentioned, namely that the arrow is stationary while on its flight. The demonstration rests on the assumption that time is given. This then is the definite line or distance (ἡ πεπερασμένη) which has to be covered, and if it is granted that the racers ever reach this point, 1t follows that the slower will be over- taken by the swifter. So that the problem is reduced to the question whether the racers can ever reach this particular point on their course. See;Book VIII. chapter viii., 263 a 4 ff.
ARISTOTLE 239b χρόνον συγκεῖσθαι ἐκ τῶν νῦν' μὴ διδομένου γὰρ τούτου οὐκ ἔσται ὁ συλλογισμός. Τέταρτος δ᾽ ὃ περὶ τῶν ἐν τῷ σταδίῳ κινουμένων ἐξ ἐναντίας ἴσων ὄγκων παρ᾽ ἴσους, τῶν μὲν 85 ἀπὸ τέλους τοῦ σταδίου τῶν δὲ ἀπὸ μέσου, ἴσῳ 240aTayel, ἐν ᾧ συμβαίνειν οἴεται ἴσον εἶναι χρόνον τῷ διπλασίῳ τὸν ἥμισυν. ἔστι δ᾽ ὁ παραλογισμὸς ἐν τῷ TO μὲν παρὰ κινούμενον TO δὲ παρ᾽ ἠρεμοῦν τὸ ἴσον μέγεθος ἀξιοῦν τῷ ἴσῳ τάχει τὸν ἴσον φέρεσθαι χρόνον: τοῦτο δ᾽ ἐστὶ ψεῦδος. οἷον 5 ἔστωσαν οἱ ἑστῶτες ἴσοι ὄγκοι ἐφ᾽ ὧν τὰ AA, of ep ὧν τὰ ΒΒ ἀρχόμενοι ἀπὸ τοῦ μέσου τῶν A,* ἴσοι τὸν ἀριθμὸν τούτοις ὄντες Kal τὸ μέγεθος, οἱ δ᾽ ep ὧν τὰ ΓΓ ἀπὸ τοῦ ἐσχάτου, ἴσοι τὸν ἀριῦμον ὄντες τούτοις καὶ τὸ μέγεθος, καὶ ἰσοταχεῖς tots Β. συμβαίνει δὴ τὸ πρῶτον B ἅμα ἐπὶ τῷ & 1 [τῶν A om. EHI, Ross (ef. Ὁ. 188 note @).—C.]
PHYSICS, VI. rx.
made up of ‘nows,’ and if this be not granted the inference fails.
The fourth thesis supposes a number of objects all equal with each other in dimensions, forming two equal trains and arranged so that one train stretches from one end of a racecourse to the middle of it, and the other from the middle to the other end. Then if you let the two trains, moving in opposite directions but at the same rate, pass each other, Zeno undertakes to show that half of the time they occupy in passing each other is equal to the whole of it. The fallacy lies in his assuming that a moving object takes an equal time in passing another object equal in dimensions to itself, whether that other object is stationary or in motion; which assumption is false. For this is his demonstration. Let there be a number of objects AAAA, equal in number and bulk to those that compose the two trains but stationary in the middle of the stadium. Then let the objects BBBB, in number and dimension equal io the A’s, form one of the trains stretching from the middle of the A’s in one direction; and from the inner end of the B’s let CCCC stretch in the opposite direction, being equal in number, dimension, and rate of movement to the B’s.
[JA|A LALA ΒΙΒΙΒΙΒΡ» ΙΟΤΟΊΟΙΟ Then when they cross, the first B and the first C will ARISTOTLE 240410 ἐσχάτῳ εἶναι καὶ τὸ πρῶτον VT, παρ᾽ ἄλληλα κινουμένων. συμβαίνει δὲ τὸ 1 παρὰ πάντα τὰ Β" διεξεληλυθέναι, τὸ δὲ Β παρὰ τὰ ἡμίση: ὥστε iva ἥμισυν εἶναι TOV χρόνον" ἴσον yap ἑκάτερόν ἐστι παρ᾽ ἕκαστον. ἅμα δὲ ovpPaiver τὸ πρῶτον B* παρὰ πάντα τὰ I’ παρεληλυθέναι (ἅμα γὰρ ἔσται τὸ 1s πρῶτον Τ' καὶ τὸ πρῶτον B ἐπὶ τοῖς ἐναντίοις ἐσχά- τοις), ἴσον χρόνον παρ᾽ ἕκαστον γιγνόμενον τῶν I“ ὅσονπερ τῶν A (ὥς φησὺὴ, διὰ τὸ ἀμφότερα ἴσον χρόνον κατὰ τὰ A® γίγνεσθαι. ὁ μὲν οὖν 1 [δὲ ἘΛΕΉῊΚ, Simplic. 1017. 29, Alex. (Simpl. 1019. 27), Oxf. Trans.: δὴ cett.—C.]
2 [τὰ B, ἘΜῊ]: τὰ A, FKE?, Simplic. 1018. 1, Alex. (Simpl. 1019. 28), Oxf. Trans. The argument can be reconstructed with either reading.—C.]
4 [τῶν IT: τῶν B codd. The readings and punctuation adopted provide γιγνόμενον with a subject to agree with (τὸ B); it can hardly agree with either τὸ πρῶτον Τ' or τὸ πρῶτον B to the exclusion of the other. The change of τῶν B to τῶν T is consequential. Mr. Ross would eject ἴσον χρόνον . + « ὥς φησι (See Ὁ. 188 note a).—C.]
5 [κατὰ τὰ Az κατὰ τὸ A Alex. ap. Simplic. 1019. 32: παρὰ τὰ A codd.—C.]
PHYSICS, VI. 1.
simultaneously reach the extreme A’s in contrary directions.
ΙΑἸΑΤΑΊΑΙ B B|B|Bi> <(c|cjcic| Now during this process the first C has passed all the B’s, whereas the first B has only passed half the A’s, and therefore only taken half the time; for it takes an equal time (the minimal time) for the C to pass one B as for the B to pass one A. But during this same half-time the first B has also passed all the C’s® (though the first B takes as long, says Zeno, to pass a C as an A) because measured by their pro- gress through the A’s the B’s and C’s have had the same time in which to cross each other. Such is ¢ (The translation here omits the parenthesis in the Greek: ‘(for the first Ὁ and the first B arrive at the opposite ends simultaneously).’ This has already been stated above; it 15 repeated to justify the statement just made: that ‘the first B has passed all the 0’s in the same time’ as the first C has passed all the B’s (as stated in the previous sentence).—C.]
ὃ The first C crosses all the B’s while the first B crosses half the A’s. Therefore while the first C crosses half the A’s it will have time to cross all the B’s (as it actually does, by the conditions of the problem). But it takes as long to pass anA as a Bor Ὁ, Therefore half the time is as long as the whole time.
ARISTOTLE 240a λόγος οὗτός ἐστιν, συμβαίνει δὲ παρὰ τὸ εἰρημένον ψεῦδος.
Οὐδὲ δὴ κατὰ τὴν ἐν τῇ ἀντιφάσει μεταβολὴν 20 οὐδὲν ἡμῖν ἔσται ἀδύνατον, οἷον εἰ ἐκ τοῦ μὴ λευκοῦ εἰς τὸ λευκὸν μεταβάλλει καὶ ἐν μηδετέρῳ ἐστίν, ὡς ἄρα οὔτε λευκὸν ἔσται οὔτε οὐ λευκόν. οὐ γὰρ εἰ μὴ ὅλον ἐν ὁποτερῳοῦν ἐστιν, οὐ λεχθήσεται λευκὸν ἢ οὐ λευκόν: λευκὸν γὰρ λέγομεν ἢ οὐ λευκὸν οὐ τῷ ὅλον εἶναι τοιοῦτον ἀλλὰ τῷ τὰ * {I have printed Dr. Wicksteed’s translation of this para- graph as it stands, with some verbal corrections, and given the text it implies. Mr. W. D. Ross has kindly allowed me to make use of an unpublished paper giving his interpretation of the Stadium and the readings he would adopt. I have made the following literal translation in accordance with his readings, adding a diagram which varies from the traditional one only in placing the stationary A’s outside the course in the position of the spectators.
“The fourth is the one about the equal bodies moving in the stadium past equal bodies in the opposite direction at equal speed, some (moving) from the end of the stadium, some from the midway point (i.e. the turning-point in the double course). This, he thinks, involves the conclusion that the half-time is equal to its double (the whole time). The fallacy lies in the assumption that a body, moving with equal speed, takes an equal time in passing a moving body and a body of the same size that is at rest. This is false.
A ia ΓΤ mid CT BUT} end point: For instance let the equal stationary bodies be AA, and let BB, starting from the mid-point [omit τῶν A, with ἘΠῚ], be PHYSICS, VI. rx.
his argument, but the result depends on the fallacy above mentioned.
Nor need we be troubled by any attack on the possibility of change based on the axiom that a thing ‘must either be or not be’ but cannot ‘ both be and not be’ this or that at the same time. For, it is argued, if a thing is changing, for instance, from being not- white to being white and is on its way from one to the other, you can truly assert at the same time that it is neither white nor not-white. But this is not true, for we sometimes call a thing ‘ white ’ even if it is not entirely white, and we sometimes call a thing ‘ not-white ’ even if there is some trace of white in it; we speak of it according to its prevailing conequal to them (the A’s) in number and size, and let the CC starting from the end (of the stadium) be equal to them in number and size and equal to the B’s in speed.
‘ Then it follows that the front B is opposite the (right-hand) end A (and the rear C) at the same time as the front C is opposite the (left-hand) end A (and the rear B), when they move past one another.
A mid ΓΙ 5 τ »»- end pans a es ac es ‘And it follows that the (front) C has passed all the B’s, while the (front) B has passed half that number (of bodies, viz. the A’s), so that the (B’s) time is half (the C’s time); for either takes the same time in passing each (body).
‘And it follows that at the same moment the (front) B has passed all the C’s; for the front C and the front B will arrive at the opposite end A’s simultaneously. [Onut ἴσον χρόνον...ὥς φησι: a gloss on |. 12 ἴσον γὰρ ἑκάτερόν ἐστι παρ᾽ ἕκαστον), because both the B’s and the C’s take the same time in passing the A’s (παρὰ τὰ A).’—C.]
ARISTOTLE ARISTOTLE 240225 πλεῖστα ἢ τὰ κυριώτατα μέρη; οὐ ταὐτὸ δ᾽ ἐστὶ μὴ εἶναί τε ἐν τούτῳ καὶ μὴ εἶναι ἐν τούτῳ ὅλον. ὁμοίως δὲ καὶ ἐπὶ τοῦ ὄντος καὶ ἐπὶ τοῦ μὴ ὄντος Kal τῶν ἄλλων τῶν κατ᾽ ἀντίφασιν: ἔσται μὲν yap ἐξ ἀνάγκης ἐν θατέρῳ τῶν ἀντικειμένων, ἐν οὐδ- ετέρῳ δ᾽ ὅλον ἀεί. 80. Πάλιν ἐπὶ τοῦ κύκλου καὶ ἐπὶ τῆς σφαίρας Kat ὅλως τῶν ἐν αὑτοῖς κινουμένων, ὅτι συμβήσεται αὐτὰ ἠρεμεῖν: ἐν yap TH αὐτῷ τόπῳ χρόνον τινὰ v4 ἔσται Kal αὐτὰ Kal Ta μέρη, ὥστε ἠρεμήσει ἅμα καὶ κινήσεται. πρῶτον μὲν γὰρ τὰ μέρη οὐκ ἔστιν 240b ἐν TH αὐτῷ οὐθένα χρόνον, εἶτα καὶ τὸ ὅλον μετα- βάλλει ἀεὶ εἰς ἕτερον: οὐ yap ἡ αὐτή ἐστιν ἡ ἀπὸ τοῦ A λαμβανομένη περιφέρεια καὶ ἡ ἀπὸ τοῦ B καὶ τοῦ Τ' καὶ τῶν ἄλλων ἑκάστου σημείων, πλὴν (ὡς 6 μουσικὸς ἄνθρωπος καὶ ἄνθρωπος, ὅτι συμ- 5 βέβηκεν. ὥστε μεταβάλλει ἀεὶ ἡ ἑτέρα εἰς τὴν ἑτέραν, καὶ οὐδέποτε ἠρεμήσει. τὸν αὐτὸν δὲ τρόπον καὶ ἐπὶ τῆς σφαίρας Kal ἐπὶ τῶν ἄλλων τῶν ἐν αὑτοῖς κινουμένων. 1 [Perhaps we should read, with Simplic. 1021. 22 ὁμοίως δὲ καὶ ἐπὶ τοῦ ὄντος καὶ uy ὄντος. The mss. and Philop. 818. 10 betray some confusion.—C.]
PHYSICS, VI. ix.
dition or the conditions of its most significant parts or aspects. For to say that a thing is not in a certain condition ‘at all’ and to say that it is not ‘altogether,’ in it are two different things. And so, too, in the case of being or not being or any other pair of contradictory opposites. For during the whole pro- cess of changing it must be prevailingly one or the other and can never be exclusively either.@ Again it is said that a rotating circle or sphere or anything else that moves within its own dimensions is stationary because in itself and all its parts it will remain in the same place for the given time: so it will be in motion and at rest at the same time. But in the first place its parts are not in the same place during any space of time, and in the second place the whole is also continuously changing to a different (rotational) position ®; for the circumference measured round from A to A again is not identical with the cir- cumference as measured from B to B or from C to C or any other point, except by accidental concomitance, as the cultivated person is aman. Thus one circum- ference is ever succeeding another and it will never be at rest. So, too, with the sphere, and any other body that moves within fixed dimensions.
@ It is more proper to say that during the change, it is ‘always partially both but never wholly either than to say that it is “‘ always neither.”’ —Themistius.
> See Book IV., chapter v., Introduction and text, also chapter iv., especially note a, Vol. I. p. 211.
ARISTOTLE CHAPTER X ARGUMENT An indivisible (in the sense of that which is without parts) can only be wn motion incidentally, that is to say by being situated in something which is in motion, as things which are in a boat move when the boat moves; but the motion of the whole does not necessarily produce equal motions in every part; it cannot therefore be called a single motion (240 Ὁ 8-- 17). Several proofs are given: (1) An indivisible cannot on its own account be in process of changing, for while a thing is in the act of changing it must be partially what it is changing to, and partially what it is changing from; but an indivisible cannot be partially this and partially that, and when anything 1s wholly this or that it 1s not changing (Ὁ 17-80). So that only if time were made up of ‘nows’ and the mobile could be in a wholly different condition in each, could it change without ever being in the act of changing. This supposition has already been dismissed (b 30-241 a 6).
(2) Before a thing can move through a distance greater than itself, ἐξ must move through a distance equal to or less than itself; there is nothing less than an indivisible; ἡ 240bs ἀποδεδειγμένων δὲ τούτων, λέγομεν ὅτι τὸ ἀμερὲς οὐκ ἐνδέχεται κινεῖσθαι πλὴν κατὰ συμ- 10 βεβηκός, οἷον κινουμένου τοῦ σώματος ἢ τοῦ μεγέθους τῷ ἐνυπάρχειν, καθάπερ ἂν εἰ τὸ ἐν τῷ πλοίῳ κινοῖτο ὑπὸ τῆς τοῦ πλοίου φορᾶς ἢ τὸ μέρος TH τοῦ ὅλου κινήσει. (ἀμερὲς, δὲ λέγω τὸ κατὰ ποσὸν ἀδιαίρετον" καὶ γὰρ αἱ τῶν μερῶν κινήσεις ἕτεραί εἰσι κατ᾽ αὐτά τε τὰ μέρη καὶ κατὰ lich, Prantl: τῶν ἐνυπάρχειν E: τοῦ (τὸ H) ἐν ᾧ ὑπάρχει cett.—C.
PHYSICS, VI. x.
CHAPTER X ARGUMENT (continued) therefore a point could only measure up a distance by moving through a succession of ‘things equal to itself,’ i.e. points, and this would only be possible if distance were made up of points, which is not the case (a 6-14).
(9) A similar proof based on the wilimitable divisibility of No change between contraries can be unlimited: (a) genesis and extinction are limated by the opposite extremes of existence and non-existence. (Ὁ) Changes of quality by the extremes of contrasted qualities. (c) Growth and shrinkage by the limits of magnitude fixed by the nature of the subject. Locomotion is not necessarily between contraries and is not therefore necessarily limited in this way, but any actual change of position must be from an actual ‘ here’ to an actual “there, and movement through illmitable distance would be changing to a ‘there’ which does not exist; there- fore things cannot actually move all through illimitable distance. The only kind of change which is not limited in this way is rotary locomotion. This then can be going on through tllimitable time (a 26—-b 12).
WE are now in a position to show that that which has no parts cannot be in motion on its own account but only by its implication with something else, such as a moving body or changing magnitude, in which it exists, just as the movement of what the boat contains is involved in the motion of the boat, or that of a part in the movement of the whole of which it is a part. (By ‘that which has no parts’ I mean the quantitively indivisible. For the parts may have movements of their own distinct from the movement ARISTOTLE 240b15 τὴν τοῦ ὅλου κίνησιν. ἴδοι δ᾽ ἄν τις ἐπὶ τῆς σφαίρας μάλιστα τὴν διαφοράν: οὐ yap ταὐτὸν τάχος ἐστὶ τῶν τε πρὸς τῷ κέντρῳ Kal τῶν ἐκτὸς καὶ τῆς ὅλης, ὡς οὐ μιᾶς οὔσης κινήσεως). Καθάπερ οὖν εἴπομεν, οὕτω μὲν ἐνδέχεται κινεῖ- σθαι τὸ ἀμερὲς ὡς ὁ ἐν TH πλοίῳ καθήμενος τοῦ / 290 πλοίου θέοντος: καθ᾽ αὑτὸ δ᾽ οὐκ ἐνδέχεται. μετα- βαλλέτω γὰρ ἐκ τοῦ ΑΒ εἰς τὸ ΒΓ --εἴτ᾽ ἐκ μεγέ- iv > fous εἰς μέγεθος εἴτ᾽ ἐξ εἴδους εἰς εἶδος εἴτε κατ avTipacw—o δὲ χρόνος ἔστω ἐν ᾧ πρώτῳ μετὰ βάλλει ἐφ᾽ οὗ Δ. οὐκοῦν ἀνάγκη αὐτὸ καθ᾽ ὃν μεταβάλλει χρόνον ἢ ἐν τῷ AB εἶναι ἢ ἐν τῷ ΒΓ 95. ἢ τὸ μέν τι αὐτοῦ ἐν τούτῳ τὸ δ᾽ ἐν θατέρῳ" πᾶν γὰρ τὸ μεταβάλλον οὕτως εἶχεν. ἐν ἑκατέρῳ μέν οὖν οὐκ ἔσται TL αὐτοῦ: μεριστὸν yap ἂν εἴη. ἀλλὰ α (The connexion of thought (γάρ) seems to be: ‘I am speaking of what has no distinction of parts at all; for, if a thing has distinguishable parts, these may have movements of their own distinguishable from the motion of the whole, even though they all move together in a rigid whole. This cannot be so if a thing has no parts at all, and it is of such things that I will show that they can have only incidental motion.’ Aristotle is still thinking of the false assumptions underlying the Zenonian dilemmas, viz. that there are minimal (atomic =‘ indivisible’) times, spaces, movements, and bodies. In the Stadium, for instance, the moving bodies (B’s and C’s) were thought of as minimal bodies (ὄγκοι or * points ᾽ conceived as having indivisible magnitude) making minimal movements over minimal spaces in minimal times. Aristotle has already disposed of minimal spaces, times, and movements. It remains to attack the notion of a minimal (indivisible) body that can move. There is no such thing, for every body is infinitely divisible. What is strictly indivisible is such a thing as the point, which really has no PHYSICS, VI. x.
PHYSICS, VI. x.
of the whole.* This difference is most readily seen in the case of a rotating sphere, for the velocity of the parts varies according as they are nearer to the middle or further from it; and this difference amongst the parts is in each case distinguishable from the general motion of the whole, so that the movement of the whole collectively cannot be regarded as a single movement.)
So, as already said, an indivisible may move in- cidentally after the manner of the man sitting in the boat, but not primarily on its own account. For sup- pose a thing to change from AB to BC (whether the change concern magnitude or characteristic quality or the coming into being and passing out of it); and let the proper time during which the change is actually and continuously occurring be represented by T(A). Then it follows that during the whole of this period it must either be in the place or condition AB or the place or condition BC or partially in one and partially in the other, for these, as we saw,° are the only alterna- tives possible to anything that is in process of chang- ing. But it (@.e. an indivisible) cannot be partly in each, for then it would be divisible.¢ Nor can it be magnitude or parts at all (ἀμερές). It is now shown that the strictly indivisible thing cannot move except incidentally.—C. ] > [At 234 Ὁ 15, where it was stated that only the third logical alternative is really possible.—C.]
¢ To make this argument generally applicable (as it is obviously meant to be) it must be taken to mean that all the time the thing is changing it must preserve something—but not all—of what distinguishes that-out-of-which-it-is-coming from that-into-which-it-is-going, while something is being introduced of what distinguishes that-into-which-it-is-going from that-out-of-which-it-is-coming; and this cannot be the case with an indivisible, which is an unanalysable unity. [See note ὃ on p. 81.]
ARISTOTLE 240b μὴν odd ἐν τῷ BI μεταβεβληκὸς γὰρ ἔσται, ὑπό- κευται δὲ μεταβάλλειν. λείπεται δὴ αὐτὸ ἐν τῷ ΑΒ εἶναι καθ᾽ ὃν μεταβάλλει χρόνον" ἠρεμήσει ἄρα, 80 τὸ γὰρ ἐν τῷ αὐτῷ εἶναι χρόνον τινὰ ἠρεμεῖν ἦν.
“Ὥστ᾽ οὐκ ἐνδέχεται τὸ ἀμερὲς κινεῖσθαι οὐδ᾽ ὅλως μεταβάλλειν. μοναχῶς γὰρ ἂν οὕτως ἦν αὐτοῦ κίνησις, εἰ ὁ χρόνος ἦν ἐκ τῶν νῦν" ἀεὶ γὰρ 241 ἃ ἐν τῷ νῦν κεκινημένον ἂν ἦν καὶ μεταβεβληκός, ὥστε κινεῖσθαι μὲν μηδέποτε κεκινῆσθαι δ᾽ ἀεί. τοῦτο δ᾽ ὅτι ἀδύνατον, δέδεικται καὶ πρότερον" οὔτε γὰρ ὃ χρόνος ἐκ τῶν νῦν, οὔθ᾽ ἡ γραμμὴ ἐκ στιγμῶν, οὔθ᾽ ἡ κίνησις ἐκ κινημάτων' οὐθὲν γὰρ 5 ἄλλο ποιεῖ ὃ τοῦτο λέγων 7 ἢ τὴν κίνησιν ἐξ a ἀἁμερῶν, καθάπερ ἂν εἰ τὸν χρόνον ἐκ τῶν νῦν ἢ τὸ μέγεθος ἐκ στιγμῶν.
Ἔτι δὲ Kal ἐκ τῶνδε φανερὸν ὅτι οὔτε στιγμὴν οὔτ᾽ ἄλλο ἀδιαίρετον οὐθὲν ἐνδέχεται κινεῖσθαι. ἅπαν γὰρ τὸ κινούμενον ἀδύνατον “πρότερον μεῖζον κῳηθῆναι αὑτοῦ, πρὶν ἂν ἢ ἴσον ἢ ἔλαττον. εἰ δὴ 10 τοῦτο, φανερὸν ὅτι καὶ ἡ στιγμὴ ἔλαττον ἢ ἴσον κινηθήσεται πρῶτον" ἐπεὶ δ᾽ ἀδιαίρετος, ἀδύνατον ἔλαττον κινηθῆναι πρότερον" ἴσην ἄρα ἑαυτῇ. ὥστε ἔσται ἡ γραμμὴ ἐκ στιγμῶν" ἀεὶ γὰρ ἴσην KWwovμένη τὴν πᾶσαν γραμμὴν στιγμὴ καταμετρήσει.
At 281 Ὁ 18 ff., where (232 a 8) the passive form κίνημα, aving-been-moved ’ was contrasted with the usual (active) form κίνησις and explained by Simplicius as meaning the term (πέρας) of a motion.— ¢ That is to say, its ‘ movement’ would consist in measur- ing up the line by being over against a succession of con- tiguous indivisibles. But the conceptually indivisible cannot be contiguous one with another and make up a line.
‘a PHYSICS, VI. x.
in BC, for so it would have completed the change, whereas by hypothesis it is still in process of change. It remains, then, that it is in AB during the period in which it is changing. So it must be at rest, for, as we saw,* to remain in the same place or state during a period of time is to be at rest.
So it is not possible for the indivisible to move, or, more generally, to change in any way. For the only hypothesis on which it could be supposed to have any motion while never moving during any period of time would be the hypothesis that time is made up of ‘nows, for then in every ‘now’ it might be supposed to have moved or to have changed, in such a way as never to be in the process of moving but always in the state of having moved. Now this has already been shown? to be impossible; for neither is time made up of ‘nows,’ nor a line of points, nor motion of *“ having-movednesses.’ For this is what the asser- tion of motion being composed of atomic motions amounts to, just as though time could be built up out of ‘nows,’ or magnitude out of points.
Again, the following is another proof that neither a point nor any other indivisible can move. It is im- possible that anything should move through a distance greater than itself without having first moved through a space equal to or less thanitself. And this being so, it is evident that neither can a point move through any distance without having first moved through a distance less than itself or equal to it; and since it is indivis- ible, the distance it has moved through cannot be less than itself, and must therefore be equal to it. So the line would be made up of points; for the point, by having moved through a succession of things equal to itself, would have measured up the line®; and if this ARISTOTLE ARISTOTLE 2414 εἰ δὲ τοῦτο ἀδύνατον, καὶ TO κινεῖσθαι TO ἀδι- aiperov ἀδύνατον. 16 "Ere δ᾽ εἰ ἅπαν ἐν χρόνῳ κινεῖται, ἐν δὲ τῷ viv μηθέν, ἅπας δὲ χρόνος διαιρετός, εἴη ἄν τις χρόνος ἐλάττων ὁτῳοῦν τῶν κινουμένων ἢ ἐν ᾧ κινεῖται id e, ns ὅσον αὐτό (οὗτος μὲν yap ἔσται χρόνος ἐν ᾧ κινεῖται, διὰ TO πᾶν ἐν χρόνῳ κινεῖσθαι, χρόνος δὲ ~ ὔ Ἁ 20 πᾶς διαιρετὸς δέδεικται πρότερον). εἰ ἄρα στιγμὴ κινεῖται, ἔσται τις χρόνος ἐλάττων ἢ ἐν ᾧ αὐτὴ ἐκινήθη. ἀλλ᾽’ ἀδύνατον" ἐν γὰρ τῷ ἐλάττονι ἔλατ- τον ἀνάγκη κινεῖσθαι, ὥστε ἔσται διαιρετὸν τὸ ἀδιαίρετον εἰς τὸ ἔλαττον, ὥσπερ Kal ὃ χρόνος εἰς TOV χρόνον: μοναχῶς yap ἂν κινοῖτο TO ἀμερὲς καὶ 25 ἀδιαίρετον, εἰ ἦν ἐν τῷ νῦν κινεῖσθαι δυνατὸν τῷ ἀτόμῳ- τοῦ γὰρ αὐτοῦ λόγου ἐν τῷ νῦν κινεῖσθαι καὶ ἀδιαίρετόν τι κινεῖσθαι. Μεταβολὴ δ᾽ οὐκ ἔστιν οὐδεμία ἄπειρος. ἅπασα γὰρ ἦν ἔκ τινος εἴς τι καὶ ἡ ἐν ἀντιφάσει καὶ ἡ ἐν ἐναντίοις: ὥστε τῶν μὲν κατ᾽ ἀντίφασιν ἡ φάσις Ω ‘ λ καὶ ἡ ἀπόφασις πέρας, οἷον γενέσεως μὲν τὸ 30 Ov φθορᾶς δὲ τὸ μὴ Ov, τῶν δὲ ἐν τοῖς ἐναντίοις τὰ ἐναντία (ταῦτα yap ἄκρα τῆς μεταβολῆς), ware καὶ ἀλλοιώσεως πάσης" ἐξ ἐναντίων yap τινων ἢ ἀλλοίωσις. ὁμοίως δὲ καὶ αὐξήσεως καὶ φθίσεως" 1 {I have punctuated so as to indicate that ἀλλοιώσεως πάσης is povernee by πέρας ἐστὶ τὰ ἐναντία (not ἄκρα) under- stood.—C. ] α The argument seems to be that movement would be possible to an indivisible ‘only if an indivisible transit were possible, and this could only occur in an indivisible ‘now.’ But no motion can occur in a ‘now.’ Hence no transit can PHYSICS, VI. x.
is impossible, so is the movement of anything that is indivisible.
Again, since a thing must occupy a certain period of time in moving (and cannot move in a ‘now’) and every period of time is divisible, there could be a time shorter than the time that any mobile requires to pass through a space equal to itself in dimension (for this shorter time will always be a period of time in which motion takes place—not an indivisible instant—be- cause any motion occupies a period of time, and we have proved above that any period of time is divisible). If, then, a point moves, there must be a time shorter than that in which it completes a transit equalling it dimensionally. But that is impossible; for in that shorter time it would move a part of the space equal to itself. So the indivisible space would be divided so as to be smaller, in correspondence with the shorter time; for that which has no parts and is indivisible could only move if movement were possible in atomic time, for moving in the ‘ now ’ is equivalent to making an indivisible movement, 4 Nor can any change be without limit; for we have agreed that every change is from this to that, whether the * this ’ and ‘ that ’ are contrasted or contradictory. Thus the limits of changes between the contradic- tories are the positive and the negative, e.g. existence as the limit of genesis and non-existence of extinction; and in the case of contrasts the contrasted qualities in question, for such are the extreme points of the change. This applies to every form of modification, for modification must be from one quality to another contrasted with it. And it is the same with growth and be indivisible, and consequently no indivisible can move except accidentally.
5 10 ARISTOTLE αὐξήσεως μὲν yap TO πέρας τοῦ" κατὰ τὴν οἰκείαν φύσιν τελείου μεγέθους, φθίσεως δὲ ἡ τούτου ἔκστασις. ἡ δὲ φορὰ οὕτω μὲν οὐκ ἔσται πεπε- ρασμένη" οὐ γὰρ πᾶσα ἐν ἐναντίοις. ἀλλ᾽ ἐπειδὴ TO ἀδύνατον τμηθῆναι οὕτω, TH μὴ ἐνδέχεσθαι τμηθῆναι (πλεοναχῶς γὰρ λέγεται τὸ ἀδύνατον), οὐκ ἐνδέχεται [τὸ οὕτως ἀδύνατον)" τέμνεσθαι, οὐ ὅλως τὸ ἀδύνατον γενέσθαι γίγνεσθαι, οὐδὲ TO μεταβαλεῖν ἀδύνατον ἐνδέχοιτ᾽ ἂν μεταβάλλειν εἰς ὃ ἀδύνατον μεταβαλεῖν. εἰ οὖν τὸ φερόμενον μεταβάλλοι εἴς τι, καὶ δυνατὸν ἔσται μεταβαλεῖν. ὥστε οὐκ ἄπειρος ἡ κίνησις, οὐδ᾽ οἰσθήσεται τὴν ἄπειρον" ἀδύνατον γὰρ διελθεῖν αὐτήν. ὅτι μὲν οὖν οὕτως 1 [τὸ πέρας rod: πέρας τὸ τοῦ Prantl._—C.]
2 |The sentence down to τέμνεσθαι looks like a mixture of two constructions: (1) ἐπειδὴ τὸ ἀδύνατον τμηθῆναι οὕτω «λέγεται», τῷ μὴ ἐνδέχεσθαι τμηθῆναι (πλεοναχῶς...ἀδύνατον), οὐκ ἐνδέχεται τὸ οὕτως ἀδύνατον τέμνεσθαι, where the last clause is apodosis, and τὸ οὕτως ἀδύνατον is needed and in place. The sentence might be so understood by one who did not see that the apodosis really begins with οὐδὲ τὸ μεταβαλεῖν ἀδύνατον. (2) ἐπειδὴ τὸ οὕτως ἀδύνατον τμηθῆναι, τῷ μὴ ἐνδέχεσθαι τμηθῆναι (πλεοναχῶς...ἀδύνατον), οὐκ ἐνδέχεται τέμνεσθαι. This is the construction required, but in it τὸ οὕτως ἀδύνατον before τέμνεσθαι is superfluous and misplaced. Simplicius’s paraphrase (1030. 24): καὶ λέγει ὅτι τὸ οὕτως ἀδύνατον τμηθῆναι, ws μὴ ἐνδέχεσθαι τμηθῆναι, οὐκ ἐνδέχεται τέμνεσθαι, would be the most satisfactory reading, with τῷ (or possibly ὥστε, cf. οὕτως...ὥστε below, ll, 10-11 and 13) for Simplicius’s ws. I assume that τὸ οὕτως ἀδύνατον either was inserted in order to obtain con- struction (1), or arises from an attempt to correct τὸ ἀδύνατον τμηθῆναι οὕτω (where F omits τὸ and E omits οὕτω) to τὸ οὕτως ἀδύνατον τμηθῆναι.---(,} “(For ἔκστασις ef. De caelo 286 ἃ 18 ὕστερον δὲ τὸ παρὰ φύσιν τοῦ κατὰ φύσιν καὶ ἔκστασίς τίς ἐστιν ἐν τῇ γενέσει τὸ PHYSICS, VI. x.
shrinkage, for growth finds a limit in the full size admitted by the special nature of the subject con- cerned, and shrinkage in the furthest removal from that size which that nature admits.? The limits of local motion do not come under the same principle, for they are not all between contrasted limits.? But since that which cannot be divided in the sense that it is conceptually indivisible ὁ (for ‘cannot’ has more senses than one) cannot be in process of being divided, and more generally it is impossible that that which is incapable of happening @ should be in process of happening, so neither is it conceivable that a thing incapable of a certain change should be in process of undergoing the change to that which it cannot change to. If, then, the moving body is in process of changing to some place, it must be possible for it to change to that place. So the motion cannot be unlimited, nor can the mobile travel all the way through an unlimited space, for it would have to complete up to the limit that which has no limit. It is clear, then, παρὰ φύσιν τοῦ κατὰ φύσιν. Mr. Stocks (Oxf. Trans.) there translates ἔκστασις by ‘derangement.’ At 246 Ὁ 2 the natural excellences of things are called τελειώσεις (perfections, fulfilments), their defects éxordces.—C.]
Ὁ In the physical universe the centre and circumference may be contrasted as the limits of movement up and down; but the re-entrant movement of a revolving sphere seems to have no limits. The commentators add the movement of animals as another instance.
¢ [The illustration from ‘that which cannot be cut’ is probably taken from the atom of Leucippus and Democritus, which they declared to be physically incapable of being cut, being perfectly solid, but not like the mathematical point which is conceptually indivisible, having no parts.—C.]
ὦ [Here γενέσθαι means ‘ happen,’ ‘ come to pass’; hence ὅλως, which would be inappropriate if genesis were meant.
ARISTOTLE 5415 οὐκ ἔστιν ἄπειρος μεταβολὴ ὥστε μὴ ὡρίσθαι πέρασι, φανερόν. AAW’ εἰ οὕτως ἐνδέχεται ὥστε τῷ χρόνῳ εἶναι ἄπειρον THY αὐτὴν οὖσαν καὶ μίαν, σκεπτέον. μὴ 15 μιᾶς μὲν yap γιγνομένης, οὐθὲν ἴσως κωλύει, οἷον εἰ μετὰ τὴν φορὰν ἀλλοίωσις εἴη, Kal μετὰ τὴν ἀλλοίωσιν αὔξησις, καὶ πάλιν γένεσις: οὕτω yap ἀεὶ μὲν ἔσται τῷ χρόνῳ κίνησις, ἀλλ᾽ od pia διὰ τὸ μὴ εἶναι μίαν ἐξ ἁπασῶν. ὥστε δὲ γίγνεσθαι μίαν, οὐκ ἐνδέχεται ἄπειρον εἶναι τῷ χρόνῳ, πλὴν 20 μιᾶς" αὕτη δ᾽ ἐστὶν ἡ κύκλῳ φορά.
PHYSICS, VI. x.
that there can be no unlimited change in the sense of an accomplished change that has no limits.
It remains, however, to inquire whether any change, remaining one and the same in its nature, can go on without any limit in time. I take it that if you allow the nature of the change to alter, there will be no difficulty in assuming it to go on without any limit, for instance if a local movement should be followed by a change of quality, and that by an expansion, and that by a genetic change; for thus movement (in the large sense) would always be going on in time, but would not be one movement, since there would be no unity in the successive kinds of movement.* But if it is to remain one and the same in kind, there is no move- ment that can go on without limit in time save one, namely rotary locomotion.® @ See Book VIII. chapter viii. > See De caelo, i. 5.
BOOK VII INTRODUCTION [Sumpricrus, in his Introduction to this Book, remarks that the more important and relevant of the problems treated in it are discussed in more detail in Book VIII. Some ancient critics accordingly regarded Book VII. as super- fluous, and Eudemus passed it over. Themistius treats it in summary fashion. Simplicius himself conjectures that Aristotle wrote Book VII. at some earlier time and, when he had dealt with some of its topics more fully in Book VIIL., allowed it to stand as a sort of introductory study.
Chapter I. argues that whatever is in motion must be kept in motion by something, and that if there is a series of things, each member of which moves the next, the series cannot be infinite but must terminate in a First Mover.
Chapter II. The initiating cause of movement or change must be in direct touch with the thing moved or changed. This is true of locomotion, all forms of which can be re- duced to varieties of pulling and pushing, where direct contact is clearly necessary. It is also true of alteration of quality (including such alterations as are accompanied by sensation in animate beings) and also of change of quantity. Alteration of quality is further studied in Chapter III. and distinguished from other processes which involve, but are not identical with, changes of quality. Such are the shaping of material into form, and the forma- tion or loss of physical, moral, or intellectual excellences and defects.
Chapter IV. contains a long dialectical inquiry into the question, what conditions must be satisfied if two changes PHYSICS, VII. INTRODUCTION