§8.8#2διὸ
καὶ πρὸς τὴν ἀπορίαν τοῦτο λεκτέον· ἔχει γὰρ ἀπορίαν τήνδε.
Therefore, this must also be said in reply to the difficulty; for it involves the following difficulty.
εἰ γὰρ εἴη ἡ τὸ Ε τῇ Ζ ἴση καὶ τὸ Α φέροιτο συνεχῶς ἀπὸ τοῦ ἄκρου πρὸς τὸ Γ, ἄμα δʼ εἴη τὸ Α ἐπὶ τῷ Β σημείῳ, καὶ τὸ Δ φέροιτο ἀπὸ τῆς Ζ ἄκρας πρὸς τὸ Η ὁμαλῶς καὶ τῷ αὐτῷ τάχει τῷ Α, τὸ Δ ἔμπροσθεν ἤξει ἐπὶ τὸ Η ἢ τὸ Α ἐπὶ τὸ Γ·
For if line E were equal to line Z, and A were carried continuously from the extremity toward C, and at the same time A were at point B, and D were carried from the extremity Z toward H uniformly and with the same speed as A, D will arrive at H before A [arrives] at C.
τὸ γὰρ πρότερον ὁρμῆσαν καὶ ἀπελθὸν πρότερον ἐλθεῖν ἀνάγκη.
For that which has started and departed earlier must necessarily arrive earlier.
οὐ γὰρ ἅμα γέγονε τὸ Α ἐπὶ τῷ Β καὶ ἀπογέγονεν ἀπʼ αὐτοῦ, διὸ ὑστερίζει.
For A has not at the same time arrived at B and departed from it, which is why it lags behind.
εἰ γὰρ ἅμα, οὐχ ὑστεριεῖ, ἀλλʼ ἀνάγκη ἔσται ἵστασθαι.
For if [they are] at the same time, it will not lag behind, but it will be necessary for it to stop.
οὐκ ἄρα θετέον, ὅτε τὸ Α ἐγένετο κατὰ τὸ Β. τὸ Δ ἅμα κινεῖσθαι ἀπὸ τοῦ ἄκρου (εἰ γὰρ ἔσται γεγονὸς τὸ Α ἐπὶ τοῦ Β, ἔσται καὶ τὸ ἀπογενέσθαι, καὶ οὐχ ἅμα, ἀλλʼ ἦν ἐν τομῇ χρόνου καὶ οὐκ ἐν χρόνῳ.
Therefore, it must not be posited that, when A is at B, D starts to move at the same time from the extremity. (For if A is to have arrived at B, there will also be its having departed, and not at the same time; but it was in a division of time and not in time.
ἐνταῦθα μὲν οὖν ἀδύνατον οὕτως λέγειν ἐπὶ τῆς συνεχοῦς· ἐπὶ δὲ τοῦ ἀνακάμπτοντος ἀνάγκη λέγειν οὕτως.
Thus, on the one hand, in this case it is impossible to speak in this way of continuous motion; but on the other hand, in the case of that which turns back, it is necessary to speak in this way.
εἰ γὰρ ἡ τὸ Η φέροιτο πρὸς τὸ Δ καὶ πάλιν ἀνακάμψασα κάτω φέροιτο, τῷ ἄκρῳ ἐφʼ οὗ Δ τελευτῇ καὶ ἀρχῇ κέχρηται, τῷ ἑνὶ σημείῳ ὡς δύο διὸ στῆναι ἀνάγκη·
For if H is carried toward D and, having turned back again, is carried downward, it has used the extremity on which D [lies] as both an end and a beginning, the single point as two; therefore it must necessarily stop.
καὶ οὐχ ἅμα γέγονεν ἐπὶ τῷ Δ καὶ ἀπελήλυθεν ἀπὸ τοῦ Δ· ἐκεῖ γὰρ ἂν ἅμα εἴη καὶ οὐκ εἴη ἐν τῷ αὐτῷ νῦν.
And it has not at the same time arrived at D and departed from D; for in that case it would at the same time be and not be in the same 'now'.
ἀλλὰ μὴν τήν γε πάλαι λύσιν οὐ λεκτέον· οὐ γὰρ ἐνδέχεται λέγειν ὅτι ἐστὶν κατὰ τὸ Δ ἡ τὸ Η ἐν τομῇ, οὐ γέγονε δὲ οὐδʼ ἀπογέγονεν.
But indeed, the former solution must not be stated; for it is not possible to say that H is at D in a division [of time], but has neither arrived nor departed.
ἀνάγκη γὰρ ἐπὶ τέλος ἐλθεῖν τὸ ἐνεργείᾳ ὄν, μὴ δυνάμει.
For that which is in actuality, not in potentiality, must necessarily arrive at an end.
τὰ μὲν οὖν ἐν μέσῳ δυνάμει ἔστι, τοῦτο δʼ ἐνεργείᾳ, καὶ τελευτὴ μὲν κάτωθεν, ἀρχὴ δὲ ἄνωθεν· καὶ τῶν κινήσεων ἄρα ὡσαύτως.
The points in the middle, then, are so in potentiality, but this one is so in actuality, being an end from below and a beginning from above; therefore, it is the same for the motions too.
ἀνάγκη ἄρα στῆναι τὸ ἀνακάμπτον ἐπὶ τῆς εὐθείας.
Therefore, that which turns back on a straight line must necessarily stop.
οὐκ ἄρα ἐνδέχεται συνεχῆ κίνησιν εἶναι ἐπὶ τῆς εὐθείας ἀΐδιον.
Therefore, it is not possible for there to be a continuous eternal motion on a straight line.
τὸν αὐτὸν δὲ τρόπον ἀπαντητέον καὶ πρὸς τοὺς ἐρωτῶντας τὸν Ζήνωνος λόγον, εἰ ἀεὶ τὸ ἥμισυ διιέναι δεῖ, ταῦτα δʼ ἄπειρα, τὰ δʼ ἄπειρα ἀδύνατον διεξελθεῖν, ἢ ὡς τὸν αὐτὸν τοῦτον λόγον τινὲς ἄλλως ἐρωτῶσιν, ἀξιοῦντες ἅμα τῷ κινεῖσθαι τὴν ἡμίσειαν πρότερον ἀριθμεῖν καθʼ ἕκαστον γιγνόμενον τὸ ἥμισυ, ὥστε διελθόντος τὴν ὅλην ἄπειρον συμβαίνει ἠριθμηκέναι ἀριθμόν· τοῦτο δʼ ὁμολογουμένως ἐστὶν ἀδύνατον.
In the same way, one must also reply to those who ask Zeno's argument, whether it is always necessary to traverse the half, and these are infinite, and it is impossible to traverse infinite things, or as some ask this same argument in another way, demanding that, at the same time as the motion, one must first count each half as it arises, so that when the whole [line] has been traversed, it turns out that one has counted an infinite number; but this is confessedly impossible.
ἐν μὲν οὖν τοῖς πρώτοις λόγοις τοῖς περὶ κινήσεως
ἐλύομεν 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 καὶ ἐν τῷ χρόνῳ.
Now, in our first discussions concerning motion, we solved it by the fact that time has infinite things in itself; for it is nothing absurd if someone traverses infinite things in infinite time; for the infinite belongs similarly to length and to time.
ἀλλʼ αὕτη ἡ λύσις πρὸς μὲν τὸν ἐρωτῶντα ἱκανῶς ἔχει (ἠρωτᾶτο γὰρ εἰ ἐν πεπερασμένῳ ἄπειρα ἐνδέχεται διεξελθεῖν ἢ ἀριθμῆσαι), πρὸς δὲ τὸ πρᾶγμα καὶ τὴν ἀλήθειαν οὐχ ἱκανῶς·
But this solution is sufficient in relation to the questioner (for the question was whether it is possible to traverse or count infinite things in a finite [time or length]), but it is not sufficient in relation to the fact and the truth.
ἂν γάρ τις ἀφέμενος τοῦ μήκους καὶ τοῦ ἐρωτᾶν εἰ ἐν πεπερασμένῳ χρόνῳ ἐνδέχεται ἄπειρα διεξελθεῖν, πυνθάνηται ἐπʼ 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.
For if someone, letting go of length and of asking whether it is possible to traverse infinite things in a finite time, inquires these same things of time itself (for time has infinite divisions), this solution will no longer be sufficient, but the truth must be stated, which we expressed in our recent discussions.
ἐὰν γάρ τις τὴν συνεχῆ διαιρῇ εἰς δύο ἡμίση, οὗτος τῷ ἑνὶ σημείῳ ὡς δυσὶ χρῆται· ποιεῖ γὰρ αὐτὸ ἀρχὴν καὶ τελευτήν.
For if someone divides a continuous line into two halves, he uses the single point as two; for he makes it a beginning and an end.
οὕτω δὲ ποιεῖ ὅ τε ἀριθμῶν καὶ ὁ εἰς τὰ ἡμίση διαιρῶν.
And both the one who counts and the one who divides into halves do this.
οὕτω δὲ διαιροῦντος οὐκ ἔσται συνεχὴς οὔθʼ ἡ γραμμὴ οὔθʼ ἡ κίνησις· ἡ γὰρ συνεχὴς κίνησις συνεχοῦς ἐστιν, ἐν δὲ τῷ συνεχεῖ ἔνεστι μὲν ἄπειρα ἡμίση, ἀλλʼ οὐκ ἐντελεχείᾳ ἀλλὰ δυνάμει.
But if he divides in this way, neither the line nor the motion will be continuous; for continuous motion is of a continuous thing, and in that which is continuous there are indeed infinite halves, but not in actuality but in potentiality.
ἂν δὲ ποιῇ ἐντελεχείᾳ, οὐ ποιήσει συνεχῆ, ἀλλὰ στήσει, ὅπερ ἐπὶ τοῦ ἀριθμοῦντος τὰ ἡμίσεα φανερόν ἐστιν ὅτι συμβαίνει· τὸ γὰρ ἓν σημεῖον ἀνάγκη αὐτῷ ἀριθμεῖν δύο τοῦ μὲν γὰρ ἑτέρου τελευτὴ ἡμίσεος τοῦ δʼ ἑτέρου ἀρχὴ ἔσται, ἂν μὴ μίαν ἀριθμῇ τὴν συνεχῆ. ἀλλὰ δύο ἡμισείας.
And if he makes them in actuality, he will not make the motion continuous, but will stop it, which is clear to occur in the case of the one who counts the halves; for he must necessarily count the single point as two: for it will be the end of one half and the beginning of the other, unless he counts it not as one continuous line but as two halves.
ὥστε λεκτέον πρὸς τὸν ἐρωτῶντα εἰ ἐνδέχεται ἄπειρα διεξελθεῖν ἢ ἐν χρόνῳ ἢ ἐν μήκει, ὅτι ἔστιν ὡς, ἔστιν δʼ ὡς οὔ.
So it must be said to the one who asks whether it is possible to traverse infinite things either in time or in length, that in one way it is, but in another way it is not.
ἐντελεχείᾳ μὲν γὰρ ὄντα οὐκ ἐνδέχεται, δυνάμει δὲ ἐνδέχεται· ὁ γὰρ συνεχῶς κινούμενος κατὰ συμβεβηκὸς ἄπειρα διελήλυθεν, ἀπλῶς δʼ οὔ· συμβέβηκε γὰρ τῇ γραμμῇ ἄπειρα ἡμίσεα εἶναι, ἡ δʼ οὐσία ἐστὶν ἑτέρα καὶ τὸ εἶναι.
For on the one hand, it is not possible if they are in actuality, but on the other hand, it is possible if they are in potentiality; for the one who moves continuously has traversed infinite things accidentally, but not absolutely; for it is accidental to the line to be infinite halves, but its essence and being are other.