for genesis and evanishment taken absolutely are absolute opposites, and any specific genesis and evanishment are specifically opposite to each other; and so, since nothing can be changing in opposite directions at the same time, a change in neither direction can be continuous, but there must be an interval of time between any two stretches of it. Nor does it make any difference whether you count these contradictory changes amongst ‘ contraries * or not, for in any case, as long as changes in the two directions cannot coexist, this point does not affect the argument. Nor does it matter if there is no necessity for the thing to come to rest in the con- tradictory state, or if there is no state of rest opposed to the change: evanishment is passing into non- existence, and it may be true that the non-existent is not in a state of rest. All that matters is that there should be an interval of time; for then the change is not continuous, just as in the other cases of change what mattered was not the contrariety between them but that they could not both occur in the same thing at the same time. Nor need we be troubled by one and the same thing being the contrary of more things than one, for instance one motion being opposed to ‘ station’ and also to the opposite motion. We may be content to take it that a motion may be opposed, under a certain aspect, either to the contrary motion or to absence of motion, just as equality or the mean may be contrasted either with what is short of it or what it is short of, and that neither opposite movements nor opposite changes can coexist in the same subject. Moreover, in the case of genesis and evanishment, it would surely be too monstrous to suppose that a thing must vanish ARISTOTLE 261 b γενόμενον εὐθὺς ἀνάγκη φθαρῆναι καὶ μηδένα 25 χρόνον διαμεῖναι" ὥστε ἐκ τούτων ἂν ἡ πίστις γένοιτο ταῖς ἄλλαις" φυσικὸν γὰρ τὸ ὁμοίως ἔχειν ἐν πάσαις.
CHAPTER VIII ARGUMENT We shall now show that everlasting, uniform and un- interrupted motion does exist and that it is of necessity All local motion is either circular or rectilinear or a combination of these; therefore if either of these is incapable of being continuous no combination of them can be so. Rectilinear motion cannot be continuous (i.e. uniform, un- interrupted and everlasting), because the mobile that goes on moving when it reaches the end of the line divides its motion into two specifically different ones by turning back and re- versing the direction. Motion on a continuous path is con- tinuous as long as the mobile does not stop moving. But if it does stop and then go on again, the point at which tt does so marks the end of one movement and the beginning of another. Motion is hindered by the mobile pausing at, but not by its passing through, a point on its path. The pause makes the point function dually as the end of one movement and the beginning of another; the mobile ‘ arrives there’ at the end of the first and ‘departs thence’ at the beginning of the second movement, but not during one continuous movement.
This analysis does not apply to a point at which the mobile turns back, for no pause 15 needed to give this point the double function of the end of, say, a forward and the beginning of a backward movement; and the mobile ‘ arrives there’ at the end of the first and ‘ departs thence’ at the beginning of the second movement just as much as in the former case, although the arrival and departure are here simultaneous (Ὁ 28-263 41). [The difficulties suggested by the simultaneous arrival and departure are dealt with at 263 b 9 sqq 1 PHYSICS, VIII. vir.—-vwi1t.
at the very instant of coming into existence and not endure any time at all. This would plead, on the principle of analogy that runs through nature, for the belief that it is so in all the other cases.
CHAPTER VIII ARGUMENT (continued) Conclusion: that rectilinear motion cannot go on for ever without interruption (a 1-8).
The parallelism between time and space which solves Zeno’s Achilles and dichotomy dilemmas, as presented in Bk. WI. chs. ui. and ix., does not touch the question of the passage of time it- self: this is now shown to be a case of the distinction between the potential and the actual: it is impossible to count (or to go through) an illimitable number of actual points (or periods) of time but not impossible to count or go through a period whose potential divisibility is illimitable, if its actual extent and the actual number of its parts are limited (a 3-b 9).
The instant which divides the past and future time per- Jorms a double function: end of the past and beginning of the future. But the point that divides states that counteract each other—so that the object is always in ether the one or the other—has not a dual function, for to end the one is to begin the other; therefore at that point of time it is the other, and has done being the first. So that instant belongs to the future and not to the past (b 9-264 a 6).
Other considerations which also lead to the conclusion that no rectilinear movement can be everlasting, uniform and uninterrupted: (1) anything that moves continuously is making for its goal from the beginning, but a mobile which reverses the direction of its motion, moves away from what it was formerly moving towards; if then reverse movements were continuous, 1t would still be moving towards what it was now moving away from, which is impossible. (2) Any- thing capable of moving is capable of resting from motion, 8 So 35 ARISTOTLE ARGUMENT (continued) and as a thing cannot move in opposite directions simul- taneously it must rest from motion in one direction while moving in the opposite one.
Another example of (1) in respect to change of quality Continuity of tume does not involve continuity of modi- fication: for without pausing, a mobile can reverse the direction of its motion and thereby break the continuity of the modification (b 6-9).
Ὅτι δ᾽ ἐνδέχεται εἶναί τινα ἄπειρον, μίαν οὖσαν καὶ συνεχῆ, καὶ αὕτη ἐστὶν ἡ κύκλῳ, λέγωμεν νῦν.
Πᾶν γὰρ κινεῦται τὸ φερόμενον ἢ κύκλῳ ἢ εὐθεῖαν ἢ μικτήν: ὥστ᾽ εἰ μηδ᾽ ἐκείνων ἡ ἑτέρα συνεχής, οὐδὲ τὴν ἐξ ἀμφοῖν οἷόν τε εἶναι συγ- κειμένην. ὅτι δὲ τὸ φερόμενον τὴν εὐθεῖαν καὶ πεπερασμένην οὐ φέρεται συνεχῶς, δῆλον. ἀνα- κάμπτει yap: τὸ δ᾽ ἀνακάμπτον τὴν εὐθεῖαν τὰς ἐναντίας κινεῖται κινήσεις" ἐναντία γὰρ κατὰ τόπον ἡ ἄνω τῇ κάτω, καὶ ἡ εἰς τὸ πρόσθεν τῇ εἰς τοὔπι- σθεν, καὶ ἡ εἰς ἀριστερὰ τῇ εἰς δεξιά: τόπου γὰρ ἐναντιώσεις αὗται. τίς δ᾽ ἐστὶν ἡ μία καὶ συνεχὴς > In speaking of a combination of the circle and the straight line Aristotle no doubt had a spiral in his mind (cf. 347 a 1). He never deals expressly with plane curves other than the circle, but he would have had no difficulty in explaining many of them, for example, as the track of a body rotating round a point moving on a straight line in the plane of its rotation (the movement of a point on the circumference of the wheel of a carriage) or as the movement of a point on the circumference of a circle the centre of which was itself moving round the circumference of another circle (a point on an epicycle) or as a circle seen obliquely; and so on. (Cf. chap. ix. init. where Aristotle says that any track of PHYSICS, VIII. vii.
ARGUMENT (continued) Continuity of rotatory motion involves no impossible con- sequences, but any change between opposites must either stop altogether or else reverse its direction when the mutabile reaches either extreme. It cannot therefore go on for ever without interruption (Ὁ 9-265 ἃ 1). This disproves the view that all sensible things are in continuous motion (a 1-10).
Rotatory locomotion then is the only change which can be continuous, i.e. everlasting, uniform and uninterrupted We are now to show that there actually is, in nature, a motion ever-enduring, uniform, and uninterrupted; and that its nature is that of rotation.
All local motion is circular or rectilinear or a combination of the two,® so that if either of these cannot be continuous, neither can any combination of them be so.* Now it is obvious that the motion of a body moving on a finite straight line cannot be continuous. For to go on when it has come to the end, it must turn back, and to go back along the same line is to make the contrary motion, not to go on with the same one; for upward movement is contrary to downward movement, movement forward to movement backward, movement to right to movement to left, these being the pairs of con- traries in place. But we have already satisfied ourlocomotion must be either rectilinear or circular or a compound of these two. Plato, Parm. 145 8 καὶ σχήματος δή τινος...μετέχοι ἂν τὸ ἕν, ἤτοι εὐθέος ἢ στρογγύλου ἤ τινὸς μεικτοῦ ἐξ dudotv.—C.]
¢ {Accordingly the claim of any compound motion to be continuous can be disproved by showing that rectilinear motion cannot be so: and this we proceed to do. Motion on a finite straight line is alone considered because, accord- ing to Aristotle, no actual infinite straight line exists.—C.]
VOL. II 2B 369 ARISTOTLE 262a κίνησις διώρισται πρότερον, ὅτι ἡ TOU ἑνὸς Kal ἐν ἑνὶ χρόνῳ Kal ἐν ἀδιαφόρῳ κατ᾽ εἶδος (τρία yap Hr— , Ps ἣ TO τε κινούμενον, οἷον ἄνθρωπος ἢ θεός, καὶ ὅτε, οἷον χρόνος, Kal τρίτον TO ἐν ᾧ" τοῦτο δ᾽ ἐστὶ τόπος or ἢ πάθος ἢ εἶδος ἢ μέγεθος)": τὰ δ᾽ ἐναντία διαφέρει τῷ εἴδει, καὶ οὐχ ἕν: τόπου δ᾽ αἱ εἰρημέναι δια- φοραί. σημεῖον δ᾽ ὅτι ἐναντία κίνησις ἡ ἀπὸ τοῦ Α πρὸς τὸ B τῇ ἀπὸ τοῦ B πρὸς τὸ A, ὅτι ἱστᾶσι καὶ παύουσιν ἀλλήλας, ἐὰν ἅμα γίγνωνται: καὶ ἐπὶ κύκλου ὡσαύτως, οἷον ἡ ἀπὸ τοῦ A ἐπὶ τὸ B 10 τῇ ἀπὸ τοῦ A ἐπὶ τὸ T+ ἱστᾶσι yap, κἂν συνεχεῖς ὦσι καὶ μὴ γίγνηται ἀνάκαμψις, διὰ τὸ τὰ ἐναντία φθείρειν καὶ κωλύειν ἄλληλα: ἀλλ᾽ οὐχ ἡ εἰς τὸ πλάγιον τῇ ἄνω. Μ tA \ A a 9 / Ὄ on adtora δὲ φανερὸν ὅτι ἀδύνατον εἶναι συνεχῆ τὴν ἐπὶ τῆς εὐθείας κίνησιν, ὅτι ἀνακάμπτον 15 ἀναγκαῖον στῆναι, οὐ μόνον ἐπ᾽ εὐθείας, ἀλλὰ κἂν κύκλον φέρηται: οὐ γὰρ ταὐτὸν κύκλῳ φέρεσθαι καὶ κύκλον: ἔστι γὰρ OTE μὲν συνείρειν κινούμενον, ὁτὲ δ᾽ ἐπὶ τὸ αὐτὸ ἐλθὸν ὅθεν ὡρμήθη ἀνακάμψαι >{Or ‘in a field that is an indivisible species.” Cf. 227 Ὁ 29 καὶ ἐν ᾧ γὰρ ὃν δεῖ εἶναι καὶ ἄτομον (οἷον τὸ eléos).—C. | ὃ. πλάγιος, as usually in Aristotle, ‘sideways’ or ‘ across,’ not ‘ oblique.’ 4 {So they do not annihilate one another.—C.] This passage amounts to an express recognition of the principle of ‘virtual velocities,’ and implicitly to the conception of ‘resultant’ movements.
ARISTOTLE 3 2628 πάλιν. ὅτι δ᾽ ἀνάγκη ἵστασθαι, ἡ πίστις οὐ μόνον ἐπὶ τῆς αἰσθήσεως ἀλλὰ καὶ ἐπὶ τοῦ λόγου. ἀρχὴ 20 δὲ ἧδε: τριῶν γὰρ ὄντων ἀρχῆς μέσου τελευτῆς, τὸ μέσον πρὸς ἑκάτερον ἄμφω ἐστί, καὶ τῷ μὲν ἀριθμῷ ἕν, τῷ λόγῳ δὲ δύο. ἔτι δὲ ἄλλο ἐστὶ τὸ δυνάμει καὶ τὸ ἐνεργείᾳ. ὥστε τῆς εὐθείας τῶν evTos τῶν ἄκρων ὁτιοῦν σημεῖον δυνάμει μὲν ἔστι μέσον, ἐνεργείᾳ δ᾽ οὐκ ἔστιν, ἐὰν μὴ διέλῃ 4ᾳ ταύτην καὶ ἐπιστὰν πάλιν ἄρξηται κινεῖσθαι: οὕτω δὲ τὸ μέσον ἀρχὴ γίγνεται καὶ τελευτή, ἀρχὴ μὲν τῆς ὕστερον, τελευτὴ δὲ τῆς πρώτης" λέγω δ᾽ οἷον ἐὰν φερόμενον τὸ A στῇ ἐπὶ τοῦ B ‘Kat πάλιν φέρηται ἐπὶ τὸ Τ᾽. ὅταν δὲ συνεχῶς φέρηται, οὔτε γεγονέναι οὔτε ἀπογεγονέναι οἷόν τε TO A κατὰ τὸ 80 Β σημεῖον, ἀλλὰ μόνον εἶναι ἐν τῷ νῦν, ἐν χρόνῳ δ᾽ οὐδενὶ πλὴν οὗ τὸ νῦν διαίρεσίς ἐστιν ἐν τῷ ὅλῳ. εἰ δὲ γεγονέναι τις θήσει καὶ ἀπογεγονέναι, ἀεὶ α The point B divides the track into two parts, but these parts are contiguous, and therefore the continuity of the track ‘isnotbroken. Ifthe mobile pauses at B, though the two parts of the track are contiguous, the two parts of the movement are not; for they are separated by the pause in time, and therefore the continuity of the movement is destroyed. If on the other hand the mobile goes straight on without pausing, then not only are the two parts of the track contiguous one to the other, but so also are the two parts of the time and the two parts of the movement, and there is no break in the continuity of any of them. Therefore though any midway point can be made into a terminus by the mobile pausing there, it is not one by nature. It will be shown later (262 Ὁ 22 sqq.) that the end of the track is by nature a terminus in the movement (though not necessarily in the time, for the mobile need not pause there, but may return immediately on its track), for the forward movement must cease at the end of the track whether it gives place to PHYSICS, VIII. vurz.
We may convince ourselves that reversal of a move- ment involves stopping it, not only by observation, but by reasoning, starting as follows. Take point A as the beginning, point C as the end, and a point B A B C A B C between them. This ‘point between,’ as soon as we take it, divides AC into two, and itself constitutes an end with respect to A and a beginning with respect to C, and thus, while only single in place, it is double in function. We shall see that the distinc- tion between potentiality and actuality also comes into play here, and so, whereas any point between the extremities may be made to function dually in the sense explained, it does not actually function unless the mobile actually divides the line by stopping and beginning to move again. Else there were one movement, not two, for it is just this that erects the “point between’ into a beginning and an end, the beginning of the second and end of the first movement—I mean just the fact of the mobile stopping at B and then going on again to C.* But if the movement be continuous, we must note that we cannot with strict propriety say either that the mobile ‘has come’ to B or that it “has left’ it, but only that it ‘is there’ at an in- stantaneous “ now,’ and not zm any space or period of time at all—except in the sense that the ‘now’ was included or embraced in the whole period of the movement of which it marks a potential division. But if anyone should say that it has ‘arrived’ at every potential division in succession and ‘departed’ from the reverse movement or to a state of rest (see 228 Ὁ sqqg. and ARISTOTLE 262» στήσεται TO A φερόμενον. ἀδύνατον yap τὸ A 1 Φ ἅμα γεγονέναι τε ἐπὶ τοῦ Β καὶ ἀπογεγονέναι" ἐν ἄλλῳ dpa σημείῳ χρόνου: χρόνος dpa ἔσται ὁ ἐν μέσῳ: ὥστε ἠρεμήσει TO A ἐπὶ τοῦ B, ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων σημείων: ὃ γὰρ αὐτὸς λόγος καὶ ἐπὶ πάντων. ὅταν δὲ χρήσηται τὸ φερόμενον Α τῷ Β μέσῳ καὶ τελευτῇ καὶ ἀρχῇ, ἀνάγκη στῆναι, διὰ τὸ δύο ποιεῖν ὥσπερ ἂν εἰ καὶ νοήσειεν. ἀλλ᾽ ἀπὸ μὲν τοῦ A σημείου ἀπογέγονε τῆς ἀρχῆς, ἐπὶ δὲ τοῦ I γέγονεν, ὅταν τελευτήσῃ καὶ στῇ, διὸ καὶ πρὸς τὴν ἀπορίαν τοῦτο λεκτέον. ἔχει yap ἀπορίαν τήνδε' εἰ yap εἴη ἡ TO E τῇ Z ἴση, καὶ TO A φέροιτο συνεχῶς ἀπὸ τοῦ ἄκρου πρὸς τὸ Γ, ἅμα δ᾽ εἴη τὸ A ἐπὶ τῷ Β σημείῳ καὶ τὸ A φέροιτο ἀπὸ τῆς ZL ἄκρας πρὸς τὸ H ὁμαλῶς καὶ τῷ αὐτῷ τάχει τῷ A, τὸ A ἔμπροσθεν ἥξει ἐπὶ τὸ Ἡ ἢ τὸ A ἐπὶ τὸ I: τὸ yap πρότερον ὁρμῆσαν @ {A is here the mobile which starts from the point A.
ὃ [Or ‘ But when the moving A does treat the intermediate point B both as the end-point of one movement and the starting-point of another, then A must come to a stand, because it is making distinct use of the two aspects of B which can be distinguished in thought. On the other hand it has “departed from”’ the point A—the beginning (of the finite line), and it “ has arrived ’’ at the point C (the end of the finite line), when it stops at the end of its course.’ This last sentence distinguishes the two ends of the finite line as actual starting- and finishing-poits (since the line is actually limited by them) from intermediate points, such as B, which PHYSICS, VIIT. vu.
PHYSICS, VIIT. vu.
it, he will have to assert that as it moved it was continually coming to a stand. For it cannot ‘have arrived’ at a point (which implies that it is there) and ‘have departed’ from it (which implies that it is not there) at the same point of time. So there are two points of time concerned, with a period of time between them; and consequently A? will be at rest at B and equally at every other point, for it is the same case with them all. It would stop everywhere, then, if there was an actual division everywhere. ὃ But only when it actually does divide its course at a ‘point between’ does it make that point both an end and a beginning, and in that case it does actually stop there and must do so in the very act of realizing the conceptual duality. In such a case it ‘has left’ its starting-point as soon as the movement has begun and ‘has arrived’ at C when it ‘makes an end’ there and stops. This reasoning will solve a problem that suggests itself here. It may be said: “Let the line from E equal the line from Z; and let A move continuously from the extreme point E B E | ς Ζ Η towards C; and at the time when A is αὐ the point B let D be moving from the extreme point Z towards H uniformly and at the same rate as A: then D will reach H before A reaches C, for the one that are only potential starting- or finishing-points unless or until A actually pauses at them.—C.]
ARISTOTLE 262b15 καὶ ἀπελθὸν πρότερον ἐλθεῖν ἀνάγκη. οὐκ ἄρα ἅμα γέγονε τὸ A ἐπὶ τὸ B καὶ ἀπογέγονεν ἀπ᾽ αὐτοῦ, ὁ ὑστερίζει: εἰ γὰρ ἅμα, οὐχ ὕστεριεῖ, a ἀνάγκη ἔσται ἵστασθαι. οὐκ ἄρα θετέον, ὅτε τὸ A ἐγένετο κατὰ τὸ B, τὸ A ἅμα κινεῖσθαι ἀπὸ τοῦ Z ἄκρου" εἶ yap ἔσται γεγονὸς τὸ A ἐπὶ τοῦ B, 90 ἔσται καὶ TO ἀπογενέσθαι, Kal οὐχ ἅμα" ἀλλ᾽ ἦν ἐν τομῇ χρόνου καὶ οὐκ ἐν χρόνῳ. ἐνταῦθα μὲν οὖν ἀδύνατον οὕτω λέγειν ἐπὶ τῆς συνεχοῦς" ἐπὶ δὲ τοῦ ἀνακάμπτοντος ἀνάγκη λέγειν οὕτως. εἰ γὰρ ἡ τὸ Ἡ φέροιτο πρὸς τὸ A καὶ πάλιν ἀνακάμψασα 2 κάτω φέροιτο, τῷ ἄκρῳ ep οὗ A τελευτῇ καὶ ἀρχῇ κέχρηται--τῷ évi σημείῳ ws δύο": διὸ στῆναι ἀνάγκη, καὶ οὐχ ἅμα γέγονεν ἐπὶ τῷ Δ καὶ ἀπ- ελήλυθεν ἀπὸ τοῦ A+ ἐκεῖ yap ἂν ἅμα εἴη καὶ οὐκ εἴη ἐν τῷ αὐτῷ νῦν. ἀλλὰ μὴν τήν γε πάλαι λύσιν οὐ λεκτέον: οὐ γὰρ ἐνδέχεται λέγειν ὅτι 1 [δύο: fort. δυσί, Cf. 968 ἃ 94 and Simplic. 1286, 32.—C.] α [Alexander (Simplic. 1285. 14) explains that D is supposed to ‘start and get away first’ from a point on its own line ZH corresponding to the point B on the other line EC. D gets ahead because it is not supposed to ‘ arrive’ at this point at one moment and ‘leave’ it at another, as A is supposed to arrive at and leave B and so to lose time there.— C.} > [Viz. ‘A has arrived at, or has left, an intermediate point.’—C.] ¢ This appears to follow from the foregoing argument. But in the next sentence Aristotle shows that this is not so, and that the terms ‘ arrive’ and ‘depart’ can be used in this case: for though it is true that the end of the line is an indivisible point in space, and that the mobile is only there at an indivisible instant in time (if it reverses its motion there PHYSICS, VIII, vir.
PHYSICS, VIII, vir.
starts and gets away first must arrive first.”"* This argument implies that A has ‘arrived’ at B at one instant and has ‘left’ B at another: that is the reason why A gets left behind. If the arrival and departure coincide at the same instant, A will not be left behind; we have to suppose that A comes to a stand at B. So the fallacy lies in the supposition that at the same time that D was moving from Z, A ‘arrwed’ at B; for if A is to be said to ‘ have arrived’ at B, it will also have to ‘leave’ B and the two events will not be simultaneous; but the truth is that A is ‘at B’ only at a sectional point (or potential division) of time and does not spend any time there. In this case, then, where the motion is continuous, we ought not to use these expressions. ὃ On the other hand they must be used in the case of a mobile that turns back on its course. For suppose a mobile H moves as far as D and then turns back and moves down again: then it has made the one point at which it turned function both as a beginning and an end, and therefore as two. It must therefore have stopped there; it cannot have arrived at it and have departed from it simultaneously, since that would involve being there and not being there at the same instant. The argument used to solve the difficulty just above does not apply here: we cannot say now, as we did then, that the mobile without pausing)—so that on these two counts there is nothing to distinguish it from a midway point—yet the end of the line is by nature a terminus in the movement and the midway point is not: for the forward motion must stop at the end of the line, but need not stop at a midway point (see p. 372 note a). The mobile can only, strictly speaking, be said to ‘arrive at’ or ‘depart from’ a terminus; but as we have just seen there is no need for it to pause at a terminus.
σι ARISTOTLE ἐστὶ κατὰ τὸ A ἡ τὸ H ἐν τομῇ, οὐ γέγονε δὲ οὐδὲ ἀπογέγονεν: ἀνάγκη yap ἐπὶ τέλος ἐλθεῖν τὸ ἐνεργείᾳ ὄν, μὴ δυνάμει. τὸ μὲν οὖν ἐν μέσῳ δυνάμει ἐστί, τοῦτο δ᾽ ἐνεργείᾳ, καὶ τελευτὴ μὲν κάτωθεν, ἀρχὴ δὲ ἄνωθεν: καὶ τῶν κινήσεων ἄρα ὡσαύτως.
Ανάγκη dpa στῆναι τὸ ἀνακάμπτον ἐπὶ τῆς εὐθείας. οὐκ ἄρα ἐνδέχεται συνεχῆ κίνησιν εἶναι ἀίδιον' ἐπὶ τῆς εὐθείας.
Τὸν αὐτὸν δὲ τρόπον ἀπαντητέον καὶ πρὸς τοὺς ἐρωτῶντας τὸν Ζήνωνος λόγον, [καὶ ἀξιοῦντας] εἰ ἀεὶ τὸ ἥμισυ διιέναι δεῖ, ταῦτα δ᾽ ἄπειρα, τὰ ὃ ἄπειρα ἀδύνατον διεξελθεῖν, ἢ ὡς τὸν αὐτὸν τοῦτον λόγον τινὲς ἄλλως ἐρωτῶσιν, ἀξιοῦντες dua τῷ κινεῖσθαι τὴν ἡμίσειαν πρότερον ἀριθμεῖν καθ᾽ Ψ Ὁ / 3 oe Ld εκαστον γιγνόμενον TO NULOV, ὥστε διελθόντος 10 τὴν ὅλην ἄπειρον συμβαΐνει ἠριθμηκέναι ἀριθμόν" τοῦτο δ᾽ ὁμολογουμένως ἐστὶν ἀδύνατον. ἐν μὲν οὖν τοῖς πρώτοις λόγοις τοῖς περὶ κινήσεως ἐλύο- μεν διὰ τοῦ τὸν χρόνον ἄπειρα ἔχειν ἐν αὑτῷ: οὐδὲν γὰρ ἄτοπον εἰ ἐν ἀπείρῳ χρόνῳ ἄπειρα διέρχεταί τις, ὁμοίως δὲ τὸ ἄπειρον ἔν τε τῷ μήκει ὑπάρχει 1 [εἶναι ἀίδιον E?K, Simplic. 1287. 18 (lemma) ἃ: ἀΐδιον εἶναι FHI, Simplic. aid. A: ἴδιον (sic) post εὐθείας EX, Fort. εἶναι καὶ ἀίδιον: cf. Simplic. 1288. 28 (paraphr.) οὐκ dpa ἐνδέχεται τὴν ἐπὶ τῆς εὐθείας κίνησιν συνεχῆ εἶναι καὶ aidior.—C. | 2 [καὶ ἀξιοῦντας omitted by the Oxf. Trans. as probably ‘a gloss introduced under the influence of ἀξιοῦντες, 1.7. The words have no construction unless ef and δεῖ are omitted, with K.—C.]
PHYSICS, VIII. νι.
is only at the point D ‘at’ a sectional point of time, but has never ‘arrived at it’ or ‘ departed from it,’ for the ‘end’ which it comes to must be an end in actuality, not in potentiality only. So a ‘point between’ the extremities of a continuous line is only potentially a beginning and an end, but this one is actual; it is both the finishing-point as regarded from below and the starting-point as regarded from above, and so also the end-point of one motion and the beginning-point of the other.
The conclusion is that the motion of the mobile that turns back upon a straight line must stop. It is impossible, therefore, that there should be, on a straight line, continuous movement that is everlasting.
This reasoning, further, enables us to meet those who, in the terms of Zeno’s argument, ask whether it is true that you must always go half-way to a point before you get there, and there is always a half-way point between the last half-way point that you have reached and the point itself that you are making for, and so you can never get there, because you would have to pass through an infinite number of points. Or, as others put it: If you count the first half of the journey and then the half of what is left and so on, you would have to count an infinite series of numbers before you got to the end of the journey; which is admitted to be impossible. It is true that in our previous studies concerning movement ὦ we solved this puzzle by pointing out that since time, just as much as space, is divisible without limit and with respect to this capacity is illimitable, there is no contradiction in a man passing through an infinite number of points in a time which is ‘ infinite’ in precisely the same sense as the distance to be tra- 20 25 30 ARISTOTLE καὶ ev τῷ χρόνῳ. ἀλλ᾽ αὕτη ἡ λύσις πρὸς μὲν τὸν ἐρωτῶντα ἱκανῶς ἔχει---ἠρωτᾶτο γὰρ εἰ ἐν πεπερα- σμένῳ ἄπειρα ἐνδέχεται διεξελθεῖν 7) ἀριθμῆσαι--- πρὸς δὲ τὸ πρᾶγμα καὶ τὴν ἀλήθειαν. οὐχ ἱκανῶς. ἂν γάρ τις, ἀφέμενος τοῦ μήκους καὶ τοῦ ἐρωτᾷν εἰ ἐν πεπερασμένῳ χρόνῳ ἐνδέχεται ἄπειρα δι- εξελθεῖν, πυνθάνηται ἐπ᾽ αὐτοῦ τοῦ χρόνου ταῦτα (ἔχει γὰρ ὁ χρόνος ἀπείρους διαιρέσεις), οὐκέτι ἱκανὴ ἔσται αὕτη ἡ λύσις, ἀλλὰ τὸ ἀληθὲς λεκτέον, ὅπερ εἴπομεν ἐν τοῖς ἄρτι λόγοις. ἂν γάρ τις τὴν συνεχῆ διαιρῇ εἰς δύο ἡμίση, οὗτος τῷ ἑνὶ σημείῳ ὡς δυσὶ χρῆταυ--ποιεῖ γὰρ αὐτὸ ἀρχὴν. καὶ τελευτήν --οοὕτω δὲ ποιεῖ ὅ τε ἀριθμῶν καὶ ὃ εἰς τὰ ἡμίση διαιρῶν. οὕτω δὲ διαιροῦντος, οὐκ ἔσται συνεχὴς οὔθ᾽ ἡ γραμμὴ οὔθ᾽ ἡ κίνησις" ἡ γὰρ συνεχὴς κίνησις συνεχοῦς ἐστιν, ἐν δὲ τῷ συνεχεῖ ἔστι μὲν ἄπειρα ἡμίση, ἀλλ᾽ οὐκ ἐντελεχείᾳ ἀλλὰ δυνάμει. ἂν δὲ ποιῇ ἐντελεχείᾳ, οὐ ποιήσει -συνεχῆ ἀλλὰ στήσει" ὅπερ ἐπὶ τοῦ ἀριθμοῦντος τὰ ἡμίση φανερόν ἐστιν ὅτι συμβαίνει" τὸ γὰρ ἕν σημεῖον ἀνάγκη αὐτῷ ἀριθμεῖν δύο" τοῦ μὲν. γὰρ ἑτέρου τελευτὴ ἡμίσεος, τοῦ δ᾽ ἑτέρου ἀρχὴ ἔσται, ἂν μὴ μίαν api τὴν συνεχῆ ἀλλὰ δύο ἡμισείας. ὥστε ¢ [The solution given earlier got over the difficulty of traversing an infinite number of points by appealing to the infinite divisibility of the stretch of time taken. But the objector may shift his ground and say: Leave the distance traversed out of account and take merely a moving body whose motion occupies a finite time. How can it get to the end of the time, if it has to get through an infinite number of subdivisions of that time?—C.] Ὁ It is the grammar only, not the sense, that is doubtful. I follow Simplic. 1291. 14 sqq. If this is right, Aristotle uses ἀριθμεῖν for * taking the two functions of the point into con- sideration ’ in 263 b 1, and for ‘enumerating’ the lines in b 2.
PHYSICS, VIII. vu.
versed is. But this solution, though adequate as a reply to the question (which was, whether it is possible in a finite time to go through or to count an infinite number of points), does not really settle the under- lying truth or get at realities. For what if a man, dropping the element of distance and the question of the possibility of traversing an infinite number of distances in a finite time, were to confine his question to the time only; for this contains an illimitable number of divisions? 2 It would then be no solution to say thatthereis no limit to the divisibility of time itself, but we should have to fall back upon the truth we have just arrived at. For whoever divides the continuous into two halves thereby confers a double function upon the point of division, for he makes it both a beginning and an end. And that is just what the counting man, or the dividing man whose half-sections he counts, is doing; and by the very act of division both the line and the movement cease to be con- tinuous; for the movement is not continuous unless the mobile and the time and the track with which it is concerned are continuous. And though it be true that there is no limit to the potential dicho- tomy of any continuum, it is not true that it is actually dichotomized to infinity. But to make an actual bisection is to effect a motion that is not continuous but interrupted, as is patent in the case of one who counts the segments; for he must take the bisecting point twice, once as an end and once as a beginning (which we have seen to involve an interruption of continuity)—I mean if he does not count the continuous line as one, but the separated halves as two.” Accordingly, if we are asked whether ARISTOTLE 268} λεκτέον πρὸς τὸν ἐρωτῶντα εἰ ἐνδέχεται ἄπειρα 5 10 15 διεξελθεῖν ἢ ἐν χρόνῳ ἢ ἐν μήκει, ὅτι ἔστιν ὥς, ἔστι ὡς οὔ: ἐντελεχείᾳ μὲν γὰρ ὄντα οὐκ ἐνδέχεται, δυνάμει δὲ ἐνδέχεται. ὃ yap συνεχῶς κινούμενος κατὰ συμβεβηκὸς ἄπειρα διελήλυθεν, ἁπλῷ ὃ᾽ οὔ- βέβ \ ~ “~ ” ς ὕ: συμβέβηκε γὰρ τῇ γραμμῇ ἄπειρα ἡμίσεα εἶναι, ἡ δ᾽ οὐσία ἐστὶν ἑτέρα καὶ τὸ εἶναι.
HAov δὲ Kat ὅτι ἐὰν μή τις ποιῇ τοῦ χρόνου TO διαιροῦν σημεῖον τὸ πρότερον καὶ ὕστερον ἀεὶ τοῦ ὑστέρου τῷ πράγματι, ἔσται ἅμα τὸ αὐτὸ ὃν Ἁ al Kal οὐκ ὄν, Kal ὅτε γέγονεν οὐκ ὄν. τὸ σημεῖον μὲν οὖν ἀμφοῖν κοινόν, καὶ τοῦ προτέρου καὶ τοῦ ὑστέρου, καὶ ταὐτὸν Kal ἕν ἀριθμῷ, λόγῳ δ᾽ οὐ ταὐτόν (τοῦ μὲν γὰρ τελευτή, τοῦ δὲ ἀρχή)" τῷ δὲ πράγματι ἀεὶ τοῦ ὑστέρου πάθους ἐστίν. χρόvos ep @ ΑΓΒ, πρᾶγμα ἐφ᾽ @ Δ’ τοῦτο ἐν μὲν τῷ A χρόνῳ λευκόν, ἐν δὲ τῷ Β οὐ λευκόν. ἐν τῷ T ἄρα λευκὸν καὶ οὐ λευκόν: ἐν ὁτῳοῦν yap τοῦ A α Thus ἃ movement which is uninterrupted is a single whole and has not the characteristics of multiplicity which would belong to it if it were interrupted and divided into segments. And we must distinguish between the sum of (rival) potenti- alities which include every alternative originally open to a thing (6.0. either unity or multiplicity but not both at once in the same respect) and those potentialities which actually are realized when the choice between alternatives has been made.
PHYSICS, VIII. vin.
PHYSICS, VIII. vin.
it is possible to go through an unlimited number of points, whether in a period of time or in a length, we must answer that in one sense it is possible but in another not. Ifthe points are actual it is impossible, but if they are potential it is possible. For one who moves continuously traverses an illimitable number of points only in an accidental, not in an unqualified, sense; it is an accidental characteristic of the line that it is an illimitable number of half-lengths; its essential nature is something different.?
It is also evident that, when speaking of the subject of motion or change, unless we assign the instant that divides past and future time to the state into which that object turns and in which it will be for the future rather than to that which it turns out of and in which it was in the past, we shall have to say that the same thing both exists and does not exist at the same instant, and when it has become something it is not that something which it has become. It is true that in continuous time the point is common to the past and future and is one and the same numeri- cally, though not in function, being the end of the one and the beginning of the other; but as regards the subject of change it always belongs to the future and not to the past state of that subject. For suppose the time is represented by A and B, and the dividing ‘now’ by C, and call the thing that suffers change D, and suppose that D is white during the whole of A and not-white during the whole of: then at the instant C it will be both white and not-white; for if it really is white during the whole of A, it must ARISTOTLE 268 b λευκὸν ἀληθὲς εἰπεῖν, εἰ πάντα τὸν ypdvov τοῦτον ἣν λευκόν, καὶ ev τῷ B μὴ λευκόν, τὸ δὲ Τ' ἐν 20 ἀμφοῖν. οὐκ apa δοτέον ἐν παντί, ἀλλὰ πλὴν τοῦ τελευταίου νῦν ἐφ᾽ οὗ τὸ Γ΄’ τοῦτο δ᾽ ἤδη τοῦ ὑστέρου"- Kat εἰ ἐγίγνετο ov* λευκὸν καὶ εἰ ἐφθείpero λευκὸν ἐν τῷ A παντί, γέγονεν ἢ ἔφθαρται ἐν τῷ Τ΄. ὥστε λευκὸν" ἢ μὴ λευκὸν ἐν ἐκείνῳ πρῶτον ἀληθὲς εἰπεῖν, ἢ ὅτε γέγονεν οὐκ ἔσται 95 καὶ ὅτε ἔφθαρται ἔσται, ἢ ἅμα λευκὸν καὶ οὐ ΑἉ λευκὸν καὶ ὅλως ὃν καὶ μὴ ὃν ἀνάγκη εἶναι.
Ki δ᾽ 6 ἂν ἢ πρότερον μὴ ὄν, ἀνάγκη γίγνεσθαι ὄν, καὶ ὅτε γίγνεται μὴ ἔστιν, οὐχ οἷόν τε εἰς ἀτόμους χρόνους διαιρεῖσθαι τὸν χρόνον. εἰ γὰρ ἐν τῷ Α χρόνῳ τὸ Δ ἐγίγνετο λευκόν, γέγονε δ᾽ 80 ἅμα καὶ ἔστιν ἐν ἑτέρῳ ἀτόμῳ χρόνῳ ἐχομένῳ δὲ ἐν τῷ B, εἰ ἐν τῷ A ἐγίγνετο, οὐκ ἦν, ἐν δὲ τῷ B ἐστί, γένεσιν δεῖ τινα εἷναι μεταξύ, wore καὶ ie ’ δ \ a 264a χρόνον ἐν ᾧ ἐγίγνετο. οὐ yap ὁ αὐτὸς ἔσται λόγος καὶ τοῖς μὴ ἄτομα λέγουσιν, ἀλλ᾽ αὐτοῦ 1 [τοῦ ὑστέρου Oxf. Trans. coll. Philop. 845. 31, Simplic.
2 [οὐ om. I and apparently Simplic. 1295. 26.—C,] 3 [ὥστε el ἣν λευκόν E.—C.]
ῃ ἴα this meaning of ἐφθείρετο λευκόν see note on 954 ἃ