§6.7Ἐπεὶ δὲ πᾶν τὸ κινούμενον ἐν χρόνῳ κινεῖται, καὶ ἐν τῷ πλείονι μεῖζον μέγεθος, ἐν τῷ ἀπείρῳ χρόνῳ ἀδύνατόν ἐστιν πεπερασμένην κινεῖσθαι, μὴ τὴν αὐτὴν αἰεὶ καὶ τῶν ἐκείνης τι κινούμενον, ἀλλʼ ἐν ἅπαντι ἅπασαν, ὅτι μὲν οὖν εἴ τι ἰσοταχῶς κινοῖτο, ἀνάγκη τὸ πεπερασμένον ἐν πεπερασμένῳ κινεῖσθαι, δῆλον (ληφθέντος γὰρ μορίου ὃ καταμετρήσει τὴν ὅλην, ἐν ἴσοις χρόνοις τοσούτοις ὅσα τὰ μόριά ἐστιν, τὴν ὅλην κεκίνηται, ὥστʼ ἐπεὶ ταῦτα πεπέρανται καὶ τῷ πόσον ἕκαστον καὶ τῷ ποσάκις ἅπαντα, καὶ ὁ χρόνος ἂν εἴη πεπερασμένος· τοσαυτάκις γὰρ ἔσται τοσοῦτος, ὅσος ὁ τοῦ μορίου χρόνος πολλαπλασιασθεὶς τῷ πλήθει τῶν μορίων)· ἀλλὰ δὴ κἂν μὴ ἰσοταχῶς, διαφέρει οὐθέν.
Since everything that moves moves in time, and in a greater [time] [moves] a greater magnitude, it is impossible to move [over] a finite [magnitude] in infinite time—not by moving over the same [magnitude] repeatedly, or some part of it, but [by moving over] the whole of it in the whole [time]. That, therefore, if something should move with equal speed, it must move [over] the finite in a finite [time], is manifest (for a part having been taken which will measure the whole, it has moved [over] the whole in as many equal times as there are parts; so that since these are finite both in the quantity of each and in the number of times of all, the time also would be finite; for the time will be so many times as large, being equal to the time of the part multiplied by the multitude of the parts); but indeed even if it does not move with equal speed, it makes no difference.
ἔστω γὰρ ἐφʼ ἧς τὸ ΑΒ διάστημα πεπερασμένον, ὃ κεκίνηται ἐν τῷ ἀπείρῳ, καὶ ὁ χρόνος ἄπειρος ἐφʼ οὗ τὸ ΓΔ. εἰ δὴ ἀνάγκη πρότερον ἕτερον ἑτέρου κεκινῆσθαι (τοῦτο δὲ δῆλον, ὅτι τοῦ χρόνου ἐν τῷ προτέρῳ καὶ ὑστέρῳ ἕτερον κεκίνηται·
For let the finite distance be AB, over which it has moved in infinite time, and let the infinite time be ΓΔ. If indeed it is necessary that one part must have moved prior to another (and this is clear, that in the prior and posterior of time, a different part has moved; for always in a greater [time] a different part will have been moved, whether it changes with equal speed or not, and whether the motion intensifies or slackens or remains [the same], none the less), let there be taken some part of the distance AB, namely AE, which will measure AB.
ἀεὶ γὰρ ἐν τῷ πλείονι ἕτερον ἔσται κεκινημένον, ἐάν τε ἰσοταχῶς ἐάν τε μὴ ἰσοταχῶς μεταβάλλῃ, καὶ ἐάν τε ἐπιτείνῃ ἡ κίνησις ἐάν τε ἀνιῇ ἐάν τε μένῃ, οὐθὲν ἧττον), εἰλήφθω δή τι τοῦ AB διαστήματος, τὸ AE, ὃ καταμετρήσει τὴν AB. τοῦτο δὴ τοῦ ἀπείρου ἔν τινι ἐγένετο χρόνῳ· ἐν ἀπείρῳ γὰρ οὐχ οἶόν τε· τὸ γὰρ ἅπαν ἐν ἀπείρῳ.
This indeed took place in some [finite] time of the infinite [time]; for in infinite [time] it is not possible, since the whole is in infinite [time].
καὶ πάλιν ἕτερον δὴ ἐὰν λάβω ὅσον τὸ AE, ἀνάγκη ἐν πεπερασμένῳ χρόνῳ· τὸ γὰρ ἅπαν ἐν ἀπείρῳ.
And again, if I take another part as large as AE, it must be in a finite time; for the whole is in infinite [time].
καὶ οὕτω δὴ λαμβάνων, ἐπειδὴ τοῦ μὲν ἀπείρου οὐθὲν ἔστι μόριον ὃ καταμετ ρήσει(ἀδύνατον γὰρ τὸ ἄπειρον εἶναι ἐκ πεπερασμένων καὶ ἴσων καὶ ἀνίσων, διὰ τὸ καταμετρηθήσεσθαι τὰ πεπερασμένα πλήθει καὶ μεγέθει ὑπό τινος ἑνός, ἐάν τε ἴσα ᾖ ἐάν τε ἄνικα, ὡρισμένα δὲ τῷ μεγέθει, οὐθὲν ἧττον), τὸ δὲ διάστημα τὸ πεπερασμένον ποσοῖς τοῖς AE μετρεῖται, ἐν πεπερασμένῳ ἂν χρόνῳ τὸ AB κινοῖτο (ὡσαύτως δὲ καὶ ἐπὶ ἠρεμήσεως)· ὥστε οὔτε γίγνεσθαι οὔτε φθείρεσθαι οἷόν τε ἀεί τι τὸ αὐτὸ καὶ ἕν.
And taking [parts] in this way, since of the infinite there is no part which will measure it (for it is impossible for the infinite to consist of finite parts, whether equal or unequal, because things that are finite in multitude and in magnitude would be measured by some single thing, whether they are equal or unequal, provided they are limited in magnitude, it makes no difference), but the finite distance is measured by a certain number of AE, the AB would be moved over in a finite time (and likewise also in the case of coming to rest); so that it is not possible for one and the same thing to be always coming to be or being destroyed.
ὁ αὐτὸς δὲ λόγος καὶ ὅτι οὐδʼ ἐν πεπερασμένῳ χρόνῳ ἄπειρον οἶόν τε κινεῖσθαι οὐδʼ ἠρεμίζεσθαι, οὔθʼ ὁμαλῶς κινούμενον οὔτʼ ἀνωμάλως.
And the same argument [shows] also that neither is it possible in a finite time for an infinite [magnitude] to be moved or to come to rest, whether moving uniformly or non-uniformly.
ληφθέντος γάρ τινος μέρους ὃ ἀναμετρήσει τὸν ὅλον χρόνον, ἐν τούτῳ ποσόν τι διέξεισιν τοῦ μεγέθους καὶ οὐχ ὅλον (ἐν γὰρ τῷ παντὶ τὸ ὅλον, καὶ πάλιν ἐν τῷ ἴσῳ ἄλλο, καὶ ἐν ἑκάστῳ ὁμοίως, εἴτε ἴσον εἴτε ἄνισον τῷ ἐξ ἀρχῆς· διαφέρει γὰρ οὐδέν, εἰ μόνον πεπερασμένον ἕκαστον·
For a certain part having been taken which will measure the whole time, in this [part] it will go through some finite amount of the magnitude and not the whole (for in the whole [time] [it goes through] the whole); and again in an equal [time] another [finite part], and in each [part of time] likewise, whether equal or unequal to the first; for it makes no difference, if only each is finite.
δῆλον γὰρ ὡς ἀναιρουμένου τοῦ χρόνου τὸ ἄπειρον οὐκ ἀναιρεθήσεται, πεπερασμένης τῆς ἀφαιρέσεως γιγνομένης καὶ τῷ ποσῷ καὶ τῷ ποσάκις· ὥστʼ οὐ δίεισιν ἐν πεπερασμένῳ χρόνῳ τὸ ἄπειρον.
For it is clear that as the time is taken away, the infinite will not be taken away, since the removal is finite both in quantity and in number of times; so that it will not go through the infinite in a finite time.
οὐδέν τε διαφέρει τὸ μέγεθος ἐπὶ θάτερα ἢ ἐπʼ ἀμφότερα εἶναι ἄπειρον· ὁ γὰρ αὐτὸς ἔσται λόγος.
And it makes no difference whether the magnitude is infinite in one direction or in both; for the argument will be the same.
ἀποδεδειγμένων δὲ τούτων φανερὸν ὅτι οὐδὲ τὸ πεπερασμένον μέγεθος τὸ ἄπειρον ἐνδέχεται διελθεῖν ἐν πεπερασμένῳ διὰ τὴν αὐτὴν αἰτίαν· ἐν γὰρ τῷ μορίῳ τοῦ χρόνου πεπερασμένον δίεισι, καὶ ἐν ἑκάστῳ ὡσαύτως, ὥστʼ ἐν τῷ παντὶ πεπερασμένον.
And these things having been demonstrated, it is manifest that neither is it possible for a finite magnitude to traverse the infinite in a finite [time], for the same cause; for in a part of the time it will traverse a finite [magnitude], and in each [part] likewise, so that in the whole [time] it [will traverse] a finite [magnitude].
ἐπεὶ δὲ τὸ πεπερασμένον οὐ δίεισι τὸ ἄπειρον ἐν πεπερασμένῳ χρόνῳ, δῆλον ὡς οὐδὲ τὸ ἄπειρον τὸ πεπερασμένον·
And since that which is finite does not traverse the infinite in a finite time, it is clear that neither does the infinite [traverse] the finite.
εἰ γὰρ τὸ ἄπειρον τὸ πεπερασμένον, ἀνάγκη καὶ τὸ πεπερασμένον διιέναι τὸ ἄπειρον.
For if the infinite [traverses] the finite, it is necessary that the finite also traverses the infinite.
οὐδὲν γὰρ διαφέρει ὁποτερονοῦν εἶναι τὸ κινούμενον· ἀμφοτέρως γὰρ τὸ πεπερασμένον δίεισι τὸ ἄπειρον.
For it makes no difference which of the two is that which moves; for in both ways the finite traverses the infinite.
ὅταν γὰρ κινῆται τὸ ἄπειρον ἐφʼ ᾧ τὸ A, ἔσται τι αὐτοῦ κατὰ τὸ B τὸ πεπερασμένον, οἶον τὸ ΓΔ, καὶ πάλιν ἄλλο καὶ ἄλλο, καὶ αἰεὶ οὕτως.
For when the infinite moves, on which [is] A, there will be some part of it alongside the finite B, say ΓΔ, and again another and another, and always so.
ὥσθʼ ἄμα συμβήσεται τὸ ἄπειρον κεκινῆσθαι τὸ πεπερασμένον καὶ τὸ πεπερασμένον διεληλυθέναι τὸ ἄπειρον· οὐδὲ γὰρ ἴσως δυνατὸν ἄλλως τὸ ἄπειρον κινηθῆναι τὸ πεπερασμένον ἢ τῷ τὸ πεπερασμένον διιέναι τὸ ἄπειρον, ἢ φερόμενον ἢ ἀναμετροῦν.
Consequently, it will happen at the same time that the infinite has moved [past] the finite and the finite has traversed the infinite; for indeed it is perhaps not possible otherwise for the infinite to have moved [past] the finite than by the finite traversing the infinite, either by being carried along [past it] or by measuring it.
ὥστʼ ἐπεὶ τοῦτʼ ἀδύνατον, οὐκ ἂν διίοι τὸ ἄπειρον τὸ πεπερασμένον.
So that since this is impossible, the infinite would not traverse the finite.
ἀλλὰ μὴν οὐδὲ τὸ ἄπειρον ἐν πεπερασμένω χρόνῳ τὸ ἄπειρον δίεισιν· εἰ γὰρ τὸ ἄπειρον, καὶ τὸ πεπερασμένον· ἐνυπάρχει γὰρ τῷ ἀπείρῳ τὸ πεπερασμένον.
But indeed neither does the infinite traverse the infinite in a finite time; for if the infinite [does], the finite also [does]; for the finite is present in the infinite.
ἔτι δὲ καὶ τοῦ χρόνου ληφθέντος ἡ αὐτὴ ἔσται ἀπόδειξις.
Furthermore, also if the time is taken, the proof will be the same.
ἐπεὶ δʼ οὔτε τὸ πεπερασμένον τὸ ἄπειρον οὔτε τὸ ἄπειρον τὸ πεπερασμένον οὔτε τὸ ἄπειρον τὸ ἄπειρον ἐν πεπερασμένω χρόνῳ κινεῖται, φανερὸν ὅτι οὐδὲ κίνησις ἔσται ἄπειρος ἐν πεπερασμένῳ χρόνῳ· τί γὰρ διαφέρει τὴν κίνησιν ἢ τὸ μέγεθος ποιεῖν ἄπειρον;
And since neither the finite [moves over] the infinite, nor the infinite [over] the finite, nor the infinite [over] the infinite in a finite time, it is manifest that neither will there be an infinite motion in a finite time; for what difference does it make to make the motion or the magnitude infinite?
ἀνάγκη γάρ, εἰ ὁποτερονοῦν, καὶ θάτερον εἶναι ἄπειρον· πᾶσα γὰρ φορὰ ἐν τόπῳ.
For it is necessary, if either of them is, that the other also is infinite; for all locomotion is in place.