§1.6#2ἔτι δὲ καὶ ἐγχοωρεῖ σύμμετρα λαβεῖν·
Furthermore, it is also possible to take commensurable ones; for it makes no difference whether we begin from the weight or from the magnitude.
οὐδὲν γὰρ διαφέρει ἄρχεσθαι ἀπὸ τοῦ βάρους ἢ ἀπὸ τοῦ μεγέθους, σἵον ἂν ληφθῇ σύμμετρον βάρος τῷ Γ τὸ ἐφʼ ᾧ τὸ Ε, καὶ ἀπὸ τοῦ ἀπείρου ἀφαιρεθῇ τὸ ἔχον τὸ ἐφʼ ᾧ τὸ E βάρος,οίον τὸ ΒΔ, εἶτα ὡς τὸ βάρος πρὸς τὸ βάρος, τὸ ΒΔ πρὸς ἄλλο γένηται μέγεθος, οἷον, πρὸς τὸ ΒΖ· ἐνδέχεται γὰρ ἀπείρου ὄντος τοῦ μεγέθους ὑποσονοῶν ἀφαιρεθῆναι·
For instance, if a weight commensurable with Γ is taken, say E, and there is subtracted from the infinite [body] that which has the weight E, say BΔ, and then as weight is to weight, so BΔ is made to another magnitude, say BZ; for since the magnitude is infinite, it is possible to subtract any amount whatsoever.
τούτων γὰρ ληφθέντων σύμμετρα ἔσται καὶ τὰ μεγέθη καὶ τὰ βάρη ἀλλήλοις.
For when these are taken, both the magnitudes and the weights will be commensurable with one another.
οὐδὲ δὴ τὸ μέγεθος ὁμοιοβαρὲς εἶναι ἢ ἀνομοιοβαρὲς οὐδὲν διοίσει πρὸς τὴν ἀπόδειξιν· ἀεὶ γὰρ ἔσται λαβεῖν· ἰσοβαρῆ σώματα τῷ ΒΔ, ἀπὸ τοῦ ἀπείρου ὑποσαοῶν ἢ ἀφαιροῦντας ἤ προστιθέντας.
Nor indeed will it make any difference for the proof whether the magnitude is of uniform weight or non-uniform weight; for it will always be possible to take bodies of equal weight to BΔ, by subtracting or adding any number whatsoever of them from the infinite.
ὥστεδῆλον ἐκ τῶν εἰρημένων ὅτιούκ ἔσται τοῦ ἀπείρου σοώματος πεπερασμένον τὸ βάρος.
Thus it is clear from what has been said that the weight of the infinite body cannot be finite.
ἄπειρον ἄρα.
Therefore it is infinite.
εἰ τοίνυν τοῦτʼ ἀδύνατον, καὶ τὸ ἄπειρόν τι εἶναι σῶμα ὀδύνατον.
If, therefore, this is impossible, it is also impossible for any body to be infinite.
ἀλλὰ μὴν ὅτι ἄπειρον εἶναι βάρος ἀδύνατον, ἐκ τῶνδε φανερόν.
But indeed, that it is impossible for weight to be infinite is clear from the following.
εἰ γὰρ τὸ τοσονδὶ βάρος τὴν τοσήνδε ἐν τῷδε τῷ χρόνῳ κινεῖται, τὸ τοσούτον καὶ ἔτι ἐν ἐλάττονι, καὶ τὴν ἀναλογίαν ἦν τὰ βάρη ἔχει, οἱ χρόνοι ἀνάπαλιν ἕξουσιν, οἷον εἰ τὸ ἥμισυ βάρος ἐν τῷδε, τὸ διοπλάσιον ἐν ἡμίσει τούτου.
For if a weight of such-and-such size moves through such-and-such a distance in this time, a weight of so much and more [will move] in less time, and the times will have the inverse ratio to that which the weights have; for instance, if half the weight moves in this time, double the weight [will move] in half of this.
ἔτι τὸ πεπερασμένον βάρος ἅπασαν πεπερασμένην δίεισιν ἔν τινι χρόνῳ πεπερασμένῳ.
Furthermore, a finite weight traverses any finite distance in some finite time.
ἀνάγκη ἄρα ἐκ τούτου, εἴ τι ἔστιν ἄπειμρον βάρος, κινεῖσθαι μὲν ἡ τοσόνδε ὅσον τὸ πεπερασμένον, καὶ ἔτι μὴ κινεῖσθαι δέ, ᾖ ἀνάλογον μὲν δεῖ κατὰ τὰς ὑπεροχὰς κινεῖσθαι, ἐναντίως δὲ τὸ μεῖζον ἐν τῷ ἐλάττονι.
It is necessary therefore from this, if there is any infinite weight, on the one hand that it should move as much as the finite, and on the other hand that it should not move, insofar as it must move proportionally according to the excesses, but conversely the greater in less time.
λόγος δʼ ούθείς ἐστι τοῦ ἀπείρου πρὸς τὸ πεπερασμένον, τοῦ δʼ ἐλάττονος χρόνου πρὸς τὸν μείζω πεπερασμένον·
But there is no ratio of the infinite to the finite, whereas there is a ratio of the less finite time to the greater.
ἀλλʼ ἀεὶ ἐν ἐλάττονι.
But [the infinite] will always move in less time.
ἐλάχιστος δʼ οὐκ ἔστιν.
Yet there is no least time.
οὐδʼ εἰ ἦν, ὄφελος ἄν ἦν· ἄλλο γάρ ἄν τι πεπερασμένον ἐλήφθη ἐν τῷ αὐτῷ λόγῳ, ἐν ᾧ τὸ ἄπειρον πρὸς ἕτερον μεῖζον, ὥστʼ ἐν ἴσῳ χρόνῳ τὴν ἴσην ἂν ἐκινεῖτο τὸ ἄπειρον τῷ πεπερασμένῳ.
Nor, if there were, would it be of any use; for some other finite weight could be taken in the same ratio as that which the infinite has to some other greater weight, so that the infinite would move through the same distance in an equal time to the finite.
ἀλλʼ ἀδύνατον.
But this is impossible.
ἀλλὰ μὴν ἀνάγκη γε, εἴπερ ἐν ὁπηλικῳοὕν χρόνῳ πεπερασμένῳ δὲ κινεῖται τὸ ἄπειρον, καὶ ἄλλο ἐν τῷ αὐτῷ τούτῳ πεπερασμένον βάρος κι νεῖσθαί τινα πεπερασμένην.
But indeed it is necessary, if the infinite moves in some finite time, however small, that some other finite weight should also move through some finite distance in this same time.
ἀδύνατον ἄρα ἄπειρον εἶναι βάρος, ὁμοίως δὲ καὶ κουφότητα.
Therefore, it is impossible for there to be an infinite weight, and similarly for lightness.
καὶ σώματα ἄρʼ ἄπειρον βάρος ἔχοντα καὶ κουφότητα ἀδύνατον.
Consequently, it is also impossible for there to be bodies having infinite weight or lightness.
Ὅτι μὲν οὖν οὐκ ἔστιν ἄπειρον σῶμα, δῆλον διά τε τῶν κατὰ μέρος θεωροῦσι τοτον τὸν τρόπον, καὶ καθόλου σκοπουμένοις μὴ μόνον κατὰ τούς λόγους τούς ἐν τοῖς περὶ τὰς ἀρχὰς εἰρημένοις ἡμῖν (διωρίσθη γὰρ κἀκεῖ καθόλου πρότερον περὶ ἀπείρου πῶς ἔστι καὶ πῶς οὐκ ἔστιν) ἀλλὰ καὶ νῦν ἄλλον τρόπον.
That there is no infinite body, then, is clear both to those who examine the matter in detail in this way, and to those who view it universally, not only according to the arguments stated by us in our discussions on principles (for there too it was previously defined universally concerning the infinite, how it exists and how it does not), but also now in another way.
μετὰ δὲ ταῦτʼ ἐπισκεπτέον κἂν εἰ μὴ ἄπειρον μὲν τὸ σῶμα τὸ πᾶν, οὐ μὴν ἀλλά τοσοῦτόν γε ὥστʼ εἶναι πλείους οὐρανούς· τάχα γὰρ ἄν τις τοῦτʼ ἀπορήσειεν, ὅτι καθάπερ ὁ περὶ ἡμᾶς κόσμος συνέστηκεν, οὐδὲν κωλύει καὶ ἑτέρους εἶναι πλείους μὲν ἑνός, μὴ μέντοι γε ἀπείρους.
After these things, we must investigate whether, even if the body of the whole is not infinite, it is nevertheless of such a size that there are several heavens; for perhaps someone might raise this difficulty, that just as the world around us is constituted, nothing prevents there being others also, more than one, though not indeed infinite.
πρῶτον δʼ εἴπωμεν καθόλου περὶ τοῦ ἀπείρου.
But first, let us speak universally about the infinite.