§1#2ἢ γὰρ ἀριθμῷ συμβαίνει τὴν αὐτὴν κίνησιν εἶναι ἢ γένει ἢ εἴδει.
For it happens that a motion is the same either in number, or in genus, or in species.
ἀριθμῷ μὲν οὖν λέγω τὴν αὐτὴν κίνησιν τὴν ἐκ τοῦ αὐτοῦ εἰς τὸ αὐτὸ τῷ ἀριθμῷ ἐν τῷ αὐτῷ χρόνῳ τῷ ἀριθμῷ, οἷον ἐκ τοῦδε τοῦ λευκοῦ, ὅ ἐστιν ἓν τῷ ἀριθμῷ, εἰς τόδε τὸ μέλαν κατὰ τόνδε τὸν χρόνον, ἕνα ὄντα τῷ ἀριθμῷ·
Now, I call a motion the same in number which is from the same in number to the same in number in the same time in number, as, for example, from this white, which is one in number, to this black in this time, which is one in number; for if it were in another time, it would no longer be one in number, but in species.
εἰ γὰρ κατʼ ἄλλον, οὐκέτι μία ἔσται τῷ ἀριθμῷ ἀλλὰ τῷ εἴδει. γένει δʼ ἡ αὐτὴ κίνησις ἡ ἐν τῇ αὐτῇ κατηγορίᾳ τῆς οὐσίας ἢ τοῦ γένους, εἴδει δὲ ἡ ἐκ τοῦ αὐτοῦ τῷ εἴδει εἰς τὸ αὐτὸ τῷ εἴδει, οἷον ἡ ἐκ τοῦ λευκοῦ εἰς τὸ μέλαν ἢ ἐκ τοῦ ἀγαθοῦ εἰς τὸ κακόν.
A motion is the same in genus which is in the same category of substance or of genus; and same in species which is from the same in species to the same in species, as from white to black, or from good to bad.
ταῦτα δʼ εἴρηται καὶ ἐν τοῖς πρότερον.
These things have also been said in our former books.
εἰλήφθω τοίνυν ἡ τοῦ Α κίνησις καὶ ἔστω
ἐφʼ οὗ τὸ Ε, καὶ ἡ τοῦ Β ἐφʼ οὗ τὸ Ζ, καὶ ἡ τοῦ ΓΔ ἐφʼ οὗ τὸ ΗΘ, καὶ ὁ χρόνος ἐν ᾧ κινεῖται τὸ Α ὁ Κ. ὡρισμένης δὴ τῆς κινήσεως τοῦ Α, ὡρισμένος ἔσται καὶ ὁ χρόνος καὶ οὐκ ἄπειρος ὁ Κ. ἀλλʼ ἐν τῷ αὐτῷ χρόνῳ ἐκινεῖτο τὸ Α καὶ τὸ Β καὶ τῶν λοιπῶν ἕκαστον.
Let, then, the motion of A be taken, and let it be represented by E, and that of B by Z, and that of CD by HΘ, and the time in which A is in motion, K. Now, since the motion of A is limited, the time K also will be limited and not infinite. But in the same time A was in motion, and B, and each of the rest.
συμβαίνει τοίνυν τὴν κίνησιν τὴν ΕΖΗΘ ἄπειρον οὖσαν ἐν ὡρισμένῳ χρόνῳ κινεῖσθαι τῷ Κ· ἐν ᾧ γὰρ τὸ Α ἐκινεῖτο, καὶ τὰ τῷ Α ἐφεξῆς ἅπαντα ἐκινεῖτο ἄπειρα ὄντα.
It happens, therefore, that the motion EZHΘ, which is infinite, is in motion in a limited time K; for in the time in which A was in motion, all the things consecutive to A, being infinite, were also in motion.
ὥστʼ ἐν τῷ αὐτῷ κινεῖται.
Consequently, they are in motion in the same [time].
καὶ γὰρ ἤτοι ἴση ἡ κίνησις ἔσται τῇ τοῦ Α, ἢ μείζων.
For indeed, the motion will be either equal to that of A, or greater.
διαφέρει δὲ οὐθέν· πάντως γὰρ τὴν ἄπειρον κίνησιν ἐν πεπερασμένῳ χρόνῳ συμβαίνει κινεῖσθαι, τοῦτο δʼ ἀδύνατον.
But it makes no difference; for in any case it happens that the infinite motion is in motion in a finite time, which is impossible.
οὕτω μὲν οὖν δόξειεν ἂν δείκνυσθαι τὸ ἐξ ἀρχῆς, οὐ μὴν δεῖκνυταί γε διὰ τὸ μηθὲν ἄτοπον συμβαίνειν· ἐνδέχεται γὰρ ἐν πεπερασμένῳ χρόνῳ κίνησιν ἄπειρον εἶναι, μὴ τὴν αὐτὴν δὲ ἀλλʼ ἑτέραν καὶ ἑτέραν πολλῶν κινουμένων καὶ ἀπείρων, ὅπερ συμβαίνει καὶ τοῖς νῦν.
In this way, then, the original thesis might seem to be proved, but indeed it is not proved, because no absurdity results; for it is possible for an infinite motion to exist in a finite time, not indeed the same motion, but different motions of many and infinite things that are in motion, which is indeed what happens in the present case.
ἀλλʼ εἰ τὸ κινούμενον πρώτως σωματικὴν κίνησιν ἀνάγκη ἅπτεσθαι ἢ συνεχὲς εἶναι τῷ κινοῦντι, καθάπερ ὁρῶμεν ἐπὶ πάντων τοῦτο συμβαῖνον (ἔσται γὰρ ἐξ ἀπάντων ἓν τὸ πᾶν ἢ συνεχές), τὸ δὴ ἐνδεχόμενον εἰλήφθω, καὶ ἔστω τὸ μὲν μέγεθος ἢ τὸ συνεχὲς ἐφʼ οὗ τὸ ΑΒΓΔ, ἡ δὲ τούτου κίνησις ἡ ΕΖΗΘ. διαφέρει δʼ οὐθὲν ἢ πεπερασμένον ἢ ἄπειρον·
But if that which is in motion primarily in a bodily motion must touch or be continuous with the mover, just as we see this happening in all cases (for the whole will be one or continuous out of all), let indeed the possible case be assumed, and let the magnitude or the continuous be ABCD, and its motion EZHΘ.
ὁμοίως γὰρ ἐν πεπερασμένῳ τῷ Κ κινηθήσεται ⟨ἄπειρον⟩ ἢ ἄπειρον ἢ πεπερασμένον.
And it makes no difference whether it is finite or infinite; for likewise, in the finite time K, [an infinite motion] will be in motion either in an infinite or in a finite magnitude.
τούτων δʼ ἑκάτερον τῶν ἀδυνάτων.
And each of these is among the impossible things.
φανερὸν οὖν ὅτι στήσεταί ποτε καὶ οὐκ εἰς ἄπειρον πρόεισιν τὸ ἀεὶ ὑφʼ ἑτέρου, ἀλλʼ ἔσται τι ὁ πρῶτον κινηθήσεται.
It is clear, therefore, that the process of being moved always by another will stop at some time and will not proceed to infinity, but there will be something which will first be moved.
μηδὲν δὲ διαφερέτω τὸ ὑποτεθέντος οῦτείκνυσθαι· τοῦ γὰρ ἐνδεχομένου τεθέντος οὐδὲν ἄτοπον ἔδει συμβαίνειν.
And let it make no difference that this is proved from a hypothesis; for when a possibility has been assumed, no absurdity ought to result.