§6.9Ζήνων δὲ παραλογίζεται· εἰ γὰρ αἰεί, φησίν, ἠρεμεῖ πᾶν ὅταν ᾖ κατὰ τὸ ἴσον, ἔστιν δʼ αἰεὶ τὸ φερόμενον ἐν τῷ νῦν, ἀκίνητον τὴν φερομένην εἶναι ὀϊστόν.
Zeno reasons fallaciously; for if, he says, everything is always at rest when it is in a space equal to itself, and that which is in motion is always in the now, the moving arrow must be motionless.
τοῦτο δʼ ἐστὶ ψεῦδος· οὐ γὰρ σύγκειται ὁ χρόνος ἐκ τῶν νῦν τῶν ἀδιαιρέτων, ὥσπερ οὐδʼ ἄλλο μέγεθος οὐδέν.
But this is false; for time is not composed of indivisible 'nows', just as no other magnitude is either.
τέτταρες δʼ εἰσὶν οἱ λόγοι περὶ κινήσεως Ζήνωνος οἱ παρέχοντες τὰς δυσκολίας τοῖς λύουσιν, πρῶτος μὲν ὁ περὶ τοῦ μὴ κινεῖσθαι διὰ τὸ πρότερον εἰς τὸ ἥμισυ δεῖν ἀφικέσθαι τὸ φερόμενον ἢ πρὸς τὸ τέλος, περὶ οὗ διείλομεν ἐν τοῖς πρότερον λόγοις.
And there are four arguments of Zeno about motion which provide difficulties for those who try to solve them. First is the one about there being no motion because that which is in motion must arrive at the half-way stage before it arrives at the end, which we have distinguished in our previous discussions.
δεύτερος δʼ ὁ καλούμενος Ἀχιλλεύς·
Second is the so-called 'Achilles'.
ἔστι δʼ οὗτος, ὅτι τὸ βραδύτατον οὐδέποτε καταληφθήσεται θέον ὑπὸ τοῦ ταχίστου· ἔμπροσθεν γὰρ ἀναγκαῖον ἐλθεῖν τὸ διῶκου ὅθεν ὥρμησεν τὸ φεῦγον, ὥστε ἀεί τι προέχεινν ἀναγκαῖον τὸ βραδύτερον.
This is that the slowest runner will never be overtaken by the fastest; for the pursuer must first reach the point from which the pursued started, so that the slower must always be some distance ahead.
ἔστιν δὲ καὶ οὗτος ὁ αὐτὸς λόγος τῷ διχοτομεῖν, διαφέρει δʼ ἐν τῷ διαιρεῖν μὴ δίχα τὸ προσλαμβανόμενον μέγεθος.
And this is the same argument as the bisection, but it differs in not dividing the magnitude added by halves.
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 διαιρουμένου πως τοῦ μεγέθους· ἀλλὰ πρόσκειται ἐν τούτῳ ὅτι οὐδὲ τὸ τάχιστον τετραγῳδημένον ἐν τῷ διώκειν τὸ βραδύτατον), ὥστʼ ἀνάγκη καὶ τὴν λύσιν εἶναι τὴν αὐτήν.
That the slower is not overtaken, then, is a consequence of the argument, but it arises from the same cause as the bisection (for in both cases it happens that the goal is not reached because the magnitude is divided in a certain way; but there is added in this argument that not even the fastest is tragically exaggerated in its pursuit of the slowest), so that the solution must also be the same.
τὸ δʼ ἀξιοῦν ὅτι τὸ προέχον οὐ καταλαμβάνεται, ψεῦδος· ὅτε γὰρ προέχει, οὐ καταλαμβάνεται· ἀλλʼ ὅμως καταλαμβάνεται, εἴπερ δώσει διεξιέναι τὴν πεπερασμένην.
But to claim that that which is ahead is not overtaken is false; for when it is ahead, it is not overtaken, but nevertheless it is overtaken, if one is allowed to pass through a finite distance.
οὗτοι μὲν οὖν οἱ δύο λόγοι, τρίτος δʼ ὁ νῦν ῥηθείς, ὅτι ἡ ὀϊστὸς φερομένη ἕστηκεν.
These then are two arguments, and the third is the one just mentioned, that the moving arrow is stationary.
συμβαίνει δὲ παρὰ τὸ λαμβάνειν τὸν χρόνον συγκεῖσθαι ἐκ τῶν νῦν· μὴ διδομένου γὰρ τούτου οὐκ ἔσται ὁ συλλογισμός.
This arises from assuming that time is composed of 'nows'; for if this is not granted, the syllogism will not hold.
τέταρτος δʼ ὁ περὶ τῶν ἐν τῷ σταδίῳ κινουμένων ἐξ
ἐναντίας ἴσων ὄγκαων παῤ ἴσους, τῶν μὲν ἀπὸ τέλους τοῦ σταδίου τῶν δʼ ἀπὸ μέσου, ἴσῳ τάχει, ἐν ᾧ συμβαίνειν οἴεται ἴσον εἶναι χρόνον τῷ διπλασίῳ τὸν ἥμισυν.
The fourth is the one concerning the equal bodies moving in the stadium from opposite directions past equal bodies, some from the end of the stadium, others from the middle, with equal speed, in which he thinks it follows that half the time is equal to its double.
ἔστι δʼ ὁ παραλογισμὸς ἐν τῷ τὸ μὲν παρὰ κινούμενον τὸ δὲ παῤ ρεμοῦν τὸ ἴσον μέγεθος ἀξιοῦν τῷ ἴσῳ τάχει τὸν ἴσον φέρεσθαι χρόνον·
And the fallacy lies in claiming that an equal magnitude with equal speed takes an equal time to move past that which is in motion as past that which is at rest.
τοῦτο δʼ ἐστὶ ψεῦδος, οἷον ἔστωσαν οἱ ἑστῶτες ἴσοι ὄγκοι ἐφʼ ὧν τὰ AA, οἱ δʼ ἐφʼ ὧν τὰ BB ἀρχόμενοι ἀπὸ τοῦ μέσου, ἴσοι τὸν ἀριθμὸν τούτοις ὄντες καὶ τὸ μέγεθος, οἱ δʼ ἐφʼ ὧν τὰ ΓΓ ἀπὸ τοῦ ἐσχάτου, ἴσοι τὸν ἀριθμὸν ὄντες τούτοις καὶ τὸ μέγεθος, καὶ ἰσοταχεῖς τοῖς B. συμβαίνει δὴ τὸ πρῶτον B ἅμα ἐπὶ τῷ ἐσχάτῳ εἶναι καὶ τὸ πρῶτον Γ, παῤ ἄλληλα κινουμένων.
But this is false. For example, let the stationary equal bodies be A A, and those starting from the middle be B B, equal in number and in magnitude to them, and those starting from the end be C C, equal in number and in magnitude to these, and having equal speed to the B's. It follows then that the first B is at the end at the same time as the first C, as they move past one another.
συμβαίνει δὲ τὸ Γ παρὰ πάντα διεξεληλυθέναι, τὸ δὲ B παρὰ τὰ ἡμίση· ὥστε ἥμισυν εἶναι τὸν χρόνον· ἴσον γὰρ ἑκάτερόν ἐστιν παῤ ἕκαστον.
And it follows that C has passed all [the A's], while B has passed half of them; so that the time is half; for each is equal past each.
ἅμα δὲ συμβαίνει τὸ πρῶτον B παρὰ πάντα τὰ Γ παρεληλυθέναι· ἅμα γὰρ ἔσται τὸ πρῶτον Γ καὶ τὸ πρῶτον B ἐπὶ τοῖς ἐναντίοις ἐσχάτοις, διὰ τὸ ἀμφότερα ἴσον χρόνον παρὰ τὰ Α γίγνεσθαι.
But at the same time it follows that the first B has passed all the C's; for the first C and the first B will be at opposite ends at the same time, because both take an equal time in passing past the A's.
ὁ μὲν οὖν λόγος οὗτός ἐστιν, συμβαίνει δὲ παρὰ τὸ εἰρημένον ψεῦδος.
The argument then is this, and it arises from the aforementioned falsehood.
οὐδὲ δὴ κατὰ τὴν ἐν τῇ ὀντιφάσει μεταβολὴν οὐθὲν 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 ὁποτερρῳοῦν ἐστιν, οὐ λεχθήσεται λευκὸν ἢ οὐ λευκόν· λευκὸν γὰρ λέγομεν ἢ οὐ λευκὸν οὐ τῷ ὅλον εἶναι τοιοῦτον, ἀλλὰ τῷ τὰ πλεῖστα ἢ τὰ κυριώτατα έρη· οὐ ταὐτὸ δʼ ἐστὶν μὴ εἶναί τε ἐν τούτῳ καὶ μὴ εἶναι ἐν τούτῳ ὅλον.
Nor indeed in respect of change in contradiction will there be any impossibility for us, as for example if that which is changing from non-white to white is in neither, that it will therefore be neither white nor non-white; for if it is not as a whole in either of them, it will not therefore fail to be called white or non-white; for we call a thing white or non-white not because it is as a whole of such a character, but because most or its most important parts are so; and it is not the same thing not to be in this and not to be as a whole in this.
ὁμοίως δὲ καὶ ἐπὶ τοῦ ὄντος καὶ ἐπὶ τοῦ μὴ ὄντος καὶ τῶν ἄλλων τῶν κατʼ ἀντίφασιν· ἔσται μὲν γὰρ ἐξ ἀνάγκης ἐν θατέρῳ τῶν ἀντικειμένων, ἐν οὐδετέρῳ δʼ ὅλον αἰεί.
And likewise also in the case of being and non-being and other things in contradiction; for it will of necessity be in one of the opposites, but never as a whole in either of them.
πάλιν δʼ ἐπὶ τοῦ κύκλου καὶ ἐπὶ τῆς σφαίρας καὶ ὅλως τῶν ἐν αὑτοῖς κινουμένων, ὅτι συμβήσεται αὐτὰ ἠρεμεῖν· ἐν γὰρ τῷ αὐτῷ τόπῳ χρόνον τινὰ ἔσται καὶ αὐτὰ καὶ τὰ μέρη, ὥστε ἠρεμήσει ἅμα καὶ κινήσεται.
Again, on the circle and the sphere and in general things that move in themselves, that it will follow that they are at rest; for both themselves and their parts will be in the same place for some time, so that they will be at rest and in motion at the same time.
πρῶτον μὲν γὰρ τὰ μέρη οὐκ ἔστιν ἐν τῷ αὐτῷ οὐθένα χρόνον, εἶτα καὶ τὸ ὅλον μεταβάλλει αἰεὶ εἰς ἕτερον· οὐ γὰρ ἡ αὐτή ἐστιν ἡ ἀπὸ τοῦ Α λαμβανομένη περιφέρεια καὶ ἡ ἀπὸ τοῦ Β καὶ τοῦ Γ καὶ τῶν ἄλλων ἑκάστου σημείων, πλὴν ὡς ὁ μουσικὸς ἄνθρωπος καὶ ἄνθρωπος, ὅτι συμβέβηκεν.
For first of all, the parts are not in the same place for any time, and secondly, the whole also is always changing into another; for the circumference taken from A is not the same as that from B and from C and from each of the other points, except as the musical man and the man are the same, because it is accidental.
ὥστε μεταβάλλει αἰεὶ ἡ ἑτέρα εἰς τὴν ἑτέραν, καὶ οὐδέποτε ἡρεμήσει.
So that one is always changing into the other, and it will never be at rest.
τὸν αὐτὸν δὲ τρόπον καὶ ἐπὶ τῆς σφαίρας καὶ ἐηὶ τῶν ἄλλων τῶν ἐν αὐτοῖς κινουμένων.
And in the same way also for the sphere and for other things that move in themselves.