§1.6#16 Αλλὰ μὴν οὐδὲ τὸ ἐπὶ τὸ μέσον οὐδὲ τὸ ἀπὸ τοῦ μέσου φερόμενον ἄπειρον ἔσται· ἐναντίαι γὰρ αἱ φοραὶ ἡ ἄνω καὶ ἡ κάτω, αἱ δʼ ἐναντίαι εἰς ἐναντίους τόπους.
6 But indeed, neither that which moves toward the center nor that which moves away from the center will be infinite; for the upward and downward movements are contrary, and contrary movements are to contrary places.
τῶν δʼ ἐναντίων εἰ θάτερον ὥρισται, καὶ θάτερον ὡρισμένον ἔσται.
And of contraries, if one is defined, the other must also be defined.
τὸ δὲ μέσον ὥρισται· εἰ γὰρ ὁποθενοῦν φέροιτο κάτω τὸ ὑφιστάμενον, οὖκ ἐνδέχεται πορροώτερον ἐλθεῖν τοῦ μέσου.
But the center is defined; for from wherever that which sinks moves downward, it is impossible for it to go further than the center.
ὡρισμένου οὖν τοῦ μέσου καὶ τὸν ἄνω τόπον ἀνάγκη ὡρίσθαι.
Therefore, since the center is defined, the upward place must also of necessity be defined.
εἰ δʼ οἱ τόποι ὡρισμένοι καὶ πεπερασμένοι, καὶ τὰ σώματα ἔσται πεπερασμένα.
And if the places are defined and finite, the bodies must also be finite.
ἔτι εἰ τὸ ἄνω καὶ κάτω ὥρισται, καὶ τὸ μεταξύ ἀνάγκη ὡρίσθαι.
Furthermore, if the upward and downward are defined, the intermediate must also of necessity be defined.
εἰ γὰρ μὴ ὥρισται, ἄπειρος ἂν εἴη κίνησις· τοῦτο δʼ ὅτι ἀδύνατον, δέδεικται πρότερον.
For if it were not defined, movement would be infinite; and that this is impossible has been shown before.
ὥρισται ἄρα τὸ μέσον, ὥστε καὶ τὸ ἐν τούτῳ σῶμα ἢ ὂν ἢ γενέσθαι δυνατόν.
Therefore, the intermediate is defined, so that the body in it, either being or capable of being there, [is also defined].
ἀλλὰ μὴν τὸ ἄνω καὶ κάτω φερόμενον σῶμα δύναται ἐν τούτῳ γενέσθαι· πέφυκε γὰρ τὸ μὲν ἀπὸ τοῷ μέσου κινεῖσθαι, τὸ δʼ ἐπὶ τὸ μέσον.
But indeed, the body that moves upward and downward is capable of being in this; for the one has a natural tendency to move from the center, and the other toward the center.
ἔκ τε δὴ τούτων φανερὸν ὅτι οὐκ ἐνδέχεται σῶμα εἶναι ἄπειρον, καὶ πρὸς τούτοις εἰ βάρος μή ἐστινἄπειρον, οὐδʼ ἂντούτων τῶν σωμάτωνοὐθὲν εἴη ἄπείρον· ἀνάγκη γὰρ τοῦ ἀπείρου σώματος ἄπειρον εἶναι καὶ τὸ βάρος.
It is clear, then, from these things that it is impossible for a body to be infinite, and in addition to these, if weight is not infinite, none of these bodies could be infinite; for of an infinite body, the weight must also of necessity be infinite.
ὁ δʼ αὐτὸς λόγος ἐστὶ καὶ ἐπὶ τοῦ κούφου· εἰ γάρ ἐστιν ἄπειρος βαρύτης, ἔστι καὶ κουφότης, ἂν ἄπειρον ἢ τὸ ἐπιπολάζον.
And the same argument holds also for the light; for if there is infinite weight, there is also infinite lightness, if that which floats on top is infinite.
δῆλον δʼ ἐκ τῶνδε.
And this is clear from the following.
ἔστω γὰρ πεπερασμένον, καὶ εἰλήφθω τὸ μὲν ἄπειρον σῶμα ἐφʼ ᾧ τὸ AB, τὸ δὲ βάρος αὐτοῷ ἐφʼ ᾧ τὸ Γ. ἀφῃρήσθω οὖν ἀπὸ τοῦ ἀπείρου πεπερασμένον μέγεθος ἐφʼ ᾧ τὸ BΔ·
For let [the weight] be finite, and let the infinite body be that designated by AB, and its weight by Γ.
καὶ τὸ βάρος αὐτοῦ ἔστω ἔφʼ ᾧ τὸ E. τὸ δὴ E τοῦ Γ ἔλαττον ἔσται·
Let there be subtracted, then, from the infinite body a finite magnitude BΔ; and let its weight be E.
τὸ γὰρ τοῦ ἐλάττονος βάρος ἔλαττον.
Now E will be less than Γ; for the weight of the less is less.
καταμετρείτω δὴ τὸ ἔλαττον ὑποσακισοῶν, καὶ ὡς τὸ βάρος τοὔλαττον πρὸς τὸ μεῖζον, τὸ BΔ πρὸς 273b τὸ ΒΖ γεγενήσθω· ἐνδέχεται γὰρ ἀφελεῖν τοῦ ἀπείρου ὑποσονοῦν.
Let the less weight, then, measure [the greater] some number of times, and as the less weight is to the greater, so let BΔ be made to BZ; for it is possible to subtract any magnitude whatsoever from the infinite.
εἰ τοίνυν ἀνάλογον τὰ μεγέθη τοῖς βάρεσι, τὸ δʼ ἕλαττον βάρος τοῦ ἐλάττονός ἐστι μεγέθους, καὶ τὸ μεῖζον ἂν εἴη τοῦ μείζονος.
If, therefore, the magnitudes are proportional to the weights, and the less weight is of the less magnitude, the greater would also be of the greater.
ἴσον ἄρα ἔσται τὸ τοῦ πεπερασμένου καὶ τὸ τοῦ ἀπείρου βάρος.
Therefore, the weight of the finite and that of the infinite will be equal.
ἔτι εἰ τοῦ μείζονος σώματος μεῖζον τὸ βάρος, τὸ τοῷ ΗΒ μεῖζον ἔσται βάρος ἢ τὸ τοῦ ΖΒ, ὥστε τὸ τοῦ πεπερασμένου βάρος μεῖζον ἢ τὸ τοῦ ἀπείρου.
Furthermore, if the weight of the greater body is greater, the weight of HB will be greater than that of ZB, so that the weight of the finite is greater than that of the infinite.
καὶ τῶν ἀνίσων δὲ μεγεθῶν ταὐτὸν βάρος ἔσται· ἄνισον γὰρ τῷ πεπερασμένῳ τὸ ἄπειρον.
Also, unequal magnitudes will have the same weight; for the infinite is unequal to the finite.
οὐθὲν δὲ διαφέρει τὰ βάρη σύμμετρα εἶναι ἢ ἀσύμμετραʼ καὶ γὰρ ἀσυμμέτρων ὄντων ὁ αὐτὸς ἔσται λόγος, οἷον εἰ τὸ E τρίτον ὑπερβάλλει μετροῦν τὸ Γ βάρος·
And it makes no difference whether the weights are commensurable or incommensurable; for even if they are incommensurable, the same argument will hold, as for instance if E, in measuring the weight Γ, exceeds by a third; for if three whole magnitudes BΔ are taken, the weight will be greater than that designated by Γ.
τῶν γὰρ ΒΔ μεγεθῶν τριῶν ὅλων ληφθέντων μεῖζον ἔσται τὸ βάρος ἢ τὸ ἐφʼ ᾦ Γ. ὥστε τὸ αὐτὸ ἔσται ἀδύνατον.
Consequently, the same impossibility will result.