§1.7#17 Ἀνάγκη δὴ σῶμα πᾶν ἤτοι ἄπειρον εἶναι ἢ πεπερασμένον, καὶ εἰ ἄπειρον, ἤτοι ἀνομοιομερὲς ἅπαν ἢ ὁμοιομερές, κἂν εἰ ἀνομοιομερές, ἤτοι ἐκ πεπερασμένων εἰδῶν ἢ ἐξ ἀπείρων.
7 It is necessary that every body is either infinite or finite, and if infinite, either entirely non-uniform or uniform; and even if it is non-uniform, it must consist either of finite kinds or of infinite kinds.
ὅτι μὲν τοίνυν οὐχ οἷόν τε ἐξ ἀπείρων, φανερόν, εἴ τις ἡμῖν ἐάσει μένειν τὰς πρώτας ὑποθέσεις· πεπερασμένων γὰρ τῶν πρώτων κινήσεων οὐσῶν ἀνάγκη καὶ 274b τὰς ἰδέας τῶν ἁπλῶν σωμάτων εἶναι πεπερασμένας.
Now, that it cannot consist of infinite kinds is clear, if one allows our primary assumptions to remain valid; for since the primary movements are finite, it is necessary that 274b the forms of simple bodies are also finite.
ἁπλῆ μὲν γὰρ ἡ τοῦ ἁπλοῦ σώματος κίνησις, αἱ δʼ ἀπίαι πεπερασμέναι κινήσεις εἰσίν· ἀνάγκη δὲ ἀεὶ κίνησιν ἔχειν σῶμα πἄν φυσικόν.
For the movement of a simple body is simple, and the simple movements are finite; and it is necessary that every natural body always has a movement.
ἀλλὰ μὴν εἴπερ γε ἐκ πεπερασμένων ἔσται τὸ ἄπειρον, ἀνάγκη καὶ τῶν μορίων ἕκαστον εἶναι ἄπειρον, λέγω δʼ οἷον τὸ ὅδωρ ἢ τὸ πῦρ.
But indeed, if the infinite is to consist of finite kinds, it is necessary that each of its parts is also infinite—I mean, for instance, the water or the fire.
ἀλλʼ ἀδύνατον· δέδεικται γὰρ ὅτι οὕτε βάρος οὔτε κουφότης ἐστὶν ἄπειρος.
But this is impossible; for it has been shown that neither weight nor lightness is infinite.
ἔτι ἀναγκαῖον ἀπείρους τῷ μεγέθει εἶναι καὶ τούς τόπους αὐτῶν, ὥστε καὶ τὰς κινήσεις ἀπείρους εἶναι πάντων.
Furthermore, it is necessary that their places also are infinite in magnitude, so that the movements of all of them are also infinite.
τοῦτο δʼ ἐδύνατον, εἰ θήσομεν ἀληθεῖς εἶναι τὰς πρώτας ὑποθέσεις καὶ μήτε τὸ κάτω φερόμενον εἰς ἄπειρον ἐνδέχεσθαι φέρεσθαι μήτε τὸ ἄνω κατὰ τὸν αὐτὸν λόγον.
But this is impossible, if we assume our primary assumptions to be true, and that neither that which is carried downward can be carried to infinity, nor that which is carried upward according to the same ratio.
ἀδύνατον γὰρ γίνεσθαι ἂ μὴ ἐνδέχεται γενέσθαι, ὁμοίως ἐπὶ τοῦ τοιόνδε καὶ τοσόνδε καὶ τοῦ ποῦ.
For it is impossible for those things to come to be which cannot come to be, and similarly in the case of quality, quantity, and place.
λέγω δʼ, εἰ ἀδύνατον γενέσθαι λευκὸν ἢ πηχυαῖον ἢ ἐν Αἰγύπτῳ, καὶ γίνεσθαί τι τούτων ἀδύνατον.
I mean, if it is impossible to become white, or a cubit long, or to be in Egypt, it is also impossible for any of these to be coming to be.
ἀδύνατον ἄρα καὶ φέρεσθαι ἐκεῖ οὗ μηθὲν δυνατὸν ἀφικέσθαι φερόμενον.
Therefore, it is also impossible to be carried to a place where nothing that is carried is capable of arriving.
ἔτι εἰ καὶ διεσπασμένον ἐστίν, οὐδὲν ἧττον ἐνδέχοιτʼ ἂν τὸ ἐξ ἀπάντων πῦρ ἄπειρον εἶναι.
Furthermore, even if it is dispersed, it might nonetheless be possible for the fire composed of all [the parts] to be infinite.
ἀλλὰ σῶμα ἦν τὸ πάντη διάστασιν ἔχον· ὥστε πῶς οἷόν τε πλείω μὲν ἀνόμοια, ἕκαστον δʼ αὐτῶν ἄπειρον εἶναι;
But a body was defined as that which has dimension in every direction; so how is it possible for there to be several dissimilar bodies, each of them being infinite?
πάντῃ γὰρ ἕκαστον δεῖ ἄπειρον εἶναι.
For each must be infinite in every direction.
ἀλλὰ μὴν οὐδὲ πᾶν ὁμοιομερὲς ἐνδέχεται τὸ ἄπειρον εἶναι.
But indeed, neither is it possible for the infinite to be entirely uniform.
πρῶτον μὲν γὰρ οὐκ ἔστιν ἄλλη παρὰ ταύτας κίνησις.
For in the first place, there is no other movement besides these.
ἕξει οὖν μίαν τούτων.
It will therefore have one of these.
εἰ δὲ τοῦτο, συμβήσεται ἢ βάρος ἄπειρον ἢ κουφότητα εἶναι ἄπειρον.
And if so, it will result that either weight is infinite or lightness is infinite.
ἀλλὰ μὴν οὐδʼ οἷόν τε τὸ κύκλῳ σῶμα φερόμενον ἄπειρον.
But indeed, neither is it possible for the body carried in a circle to be infinite.
ἀδύνατον γὰρ τὸ ἄπειρον φέρεσθαι κύκλῳʼ οὐθὲν γὰρ διαφέρει τοῦτολέγειν ἢ τὸ τὸν οὐρανὸν φάναι ἄπειρονεἶναι,τοῦτο δὲ δέδεικται ὅτι οὐ δυνατόν.
For it is impossible for the infinite to be carried in a circle; for saying this makes no difference from saying that the heaven is infinite, and this has been shown to be impossible.
ἀλλὰ μὴν οὐδʼ ὅλως γε τὸ ἄπειρον ἐνδέχεται κινεῖσθαι.
But indeed, neither is it possible at all for the infinite to move.
ἢ γὰρ κατὰ φύσιν κινηθήσεται ἢ βίᾳʼ καὶ εἰ βίᾳ, ἔστιν αὐτῷ καὶ ἡ κατὰ φύσιν, ὥστε καὶ τόπος ἄλλος ἴδιος εἰς ὃν οἰσθήσεται.
For it will move either according to nature or by force; and if by force, it has also its motion according to nature, so that there will also be another proper place into which it will be carried.
τοῦτο δʼ αδύνατον.
But this is impossible.
Ὅτι δʼ ὅλως ἀδύνατον ἄπειρον ὑπὸ πεπερασμένου παθεῖν τι ἢ ποιῆσαι τὸ πεπερασμένον, ἐκ τῶνδε φανερόν. ἔστω 275a γὰρ ἄπειρον ἐφʼ οὖ Α, πεπερασμένον ἐφʼ οὖ Β, χρόνος ἐν ᾧ ἐκίνησέ τι ἢ ἐκινήθη Γ. εἰ δὴ ὑπὸ τοῦ Β τὸ Α ἐθερμάνθη ἢ ὥσθη ἢ ἄλλο τι ἔπαθεν ἢ καὶ ὁτιοῦν ἐκινήθη ἐν τῷ χρόνῳ ἐφʼ οὖ Γ, ἔστω τὸ Δ τοῦ Β ἔλαττον, καὶ τὸ ἔλαττον ἐν τῷ ἴσῳ χρόνῳ ἔλαττον κινείτω· ἔστω δὲ τὸ ἐφʼ ᾧ E ὑπὸ τοῦ Δ ἠλλοιωμένον.
And that it is entirely impossible for the infinite to be acted upon by the finite, or for the infinite to act upon the finite, is clear from the following. 275a For let the infinite be A, the finite B, and the time in which it caused movement or was moved be Γ. If indeed A was heated or pushed by B, or underwent some other affection, or was moved in any way whatsoever in the time Γ, let Δ be less than B, and let the less in the same time cause less movement; and let that which has been altered by Δ be E.
ὃ δή ἐστι τὸ Δ πρὸς τὸ Β, τὸ E ἔσται πρὸς πεπερασμένον τι.
Then whatever ratio Δ has to B, E will have that same ratio to some finite quantity.
ἔστω δὴ τὸ μὲν ἴσον ἐν. ἴσῳ χρόνῳ ἴσον ἀλλοιοῦν, τὸ δʼ ἔλαττον ἐν τῷ ἴσῳ ἔλαττον, τὸ δὲ μεῖζον μεῖζον, τοσοῦτον δὲ ὅσον ἀνάλογον ἔσται ὅπερ τὸ μεῖζον πρὸς τὸ ἔλαττον.
For let that which is equal alter an equal amount in an equal time, and that which is less alter less in the equal time, and that which is greater alter greater, and as much greater as will be proportional to the ratio of the greater to the less.
οὐκ ἄρα τὸ ἄπειρον ὑπʼ οὐδενὸς πεπερασμένου κινήσεται ἐν οὐθενὶ χρόνῳ· ἔλαττον γὰρ ἄλλο ἐν τῷ ἴσῳ ὑπὸ ἐλάττονος κινηθήσεται, πρὸς ὃ τὸ ἀνάλογον πεπερασμένον ἔσται· τὸ γὰρ ἄπειρον πρὸς τὸ πεπερασμένον ἐν οὐθενὶ λόγῳ ἐστίν.
Therefore, the infinite will not be moved by any finite thing in any time; for another less thing will be moved in the same time by a less cause, to which the proportional [moved object] will be finite, since there is no ratio of the infinite to the finite.
ἀλλὰ μὴν οὐδὲ τὸ ἄπειρον ἐν οὐθενὶ χρόνῳ κινήσει τὸ πεπερασμένον.
But indeed, neither will the infinite in any time move the finite.
ἔστω γὰρ ἐφʼ ᾧ τὸ Α ἄπειρον, καὶ τὸ Β πεπερασμένον, χρόνος ἐν ᾧ τὸ Γ. οὐκοῦν τὸ Δ ἐν τῷ Γ ἔλαττον τοῦ Β κινήσει·
For let A be infinite, and B finite, and the time Γ. Therefore, Δ in the time Γ will move something less than B; let this be Ζ.
ἔστω τὸ Ζ. ὃ δή ἐστι τὸ ΒΖ ὅλον πρὸς τὸ Ζ, τὸ E ἔχον τὸν λόγον τοῦτον ἔστω πρὸς τὸ Δ. κινήσει ἄρα τὸ E τὸ ΒΖ ἐν τῷ Γ. τὸ πεπερασμένον τοίνυν καὶ τὸ ἄπειρον ἐν ἴσῳ χρόνῳ ἀλλοιώσει.
Then whatever ratio the whole ΒΖ has to Ζ, let Ε have this same ratio to Δ. Therefore Ε will move ΒΖ in the time Γ. Therefore, the finite and the infinite will alter [something] in an equal time.
ἀλλʼ ἀδύνατον· ἐν ἐλάττονι γὰρ τὸ μεῖζον ὑπέκειτο.
But this is impossible; for it was assumed that the greater [does so] in less time.
ἀλλʼ ἀεὶ ὁ ληφθεὶς χρόνος ταὐτὸ ποιήσει, ὥστʼ οὐκ ἔσται χρόνος οὐθεὶς ἐν ᾧ κινήσει.
But any time taken will always produce the same result, so that there will be no time in which it will cause movement.