§4.8#2ἔτι δὲ καὶ ἐκ τῶνδε φανερὸν τὸ λεγόμενον.
Moreover, what is said is also clear from the following.
ὁρῶμεν γὰρ τὸ αὐτὸ βάρος καὶ σῶμα θᾶττον φερόμενον διὰ δύο αἰτίας, ἢ τῷ διαφέρειν τὸ δι’ οὗ, οἷον διʼ ὕδατος ἢ γῆς ἢ διʼ ὕδατος ἢ ἀέρος, ἢ τῷ διαφέρειν τὸ φερόμενον, ἐὰν τἆλλα ταὐτὰ ὑπάρχῃ, διὰ τὴν ὑπεροχὴν τοῦ βάρους ἢ τῆς κουφότητος.
For we see that the same weight and body is moved faster for two causes: either because the medium through which it is carried differs—for instance, through water or earth, or through water or air—or because the body carried differs, if the other conditions remain the same, due to the excess of weight or lightness.
τὸ μὲν οὖν διʼ οὗ φέρεται αἴτιον, ὅτι ἐμποδίζει μάλιστα μὲν ἀντιφερόμενον, ἔπειτα καὶ μένον· μᾶλλον δὲ τὸ μὴ εὐδιαίρετον· τοιοῦτο δὲ τὸ παχύτερον.
Now, the medium through which it is carried is a cause because it impedes especially when it is moving in the opposite direction, and next even when it is remaining at rest; and more so that which is not easily divided, and such is that which is thicker.
τὸ δὴ ἐφʼ οὗ Α οἰσθήσεται διὰ τοῦ Β τὸν ἐφʼ ᾧ Γ χρόνον, διὰ δὲ τοῦ Δ λεπτοτέρου ὄντος τὸν ἐφʼ ᾧ Ε, εἰ ἴσον τὸ μῆκος τὸ τοῦ Β τῷ Δ, κατὰ τὴν ἀναλογίαν τοῦ ἐμποδίζοντος σώματος.
Therefore, A will be carried through B in time G, and through D, which is thinner, in time E, if the length of B is equal to D, according to the proportion of the impeding body.
ἔστω γὰρ τὸ μὲν Β ὕδωρ, τὸ δὲ Δ ἀήρ· ὅσῳ δὴ λεπτότερον ἀὴρ ὕδατος καὶ ἀσωματώτερον, τοσούτῳ θᾶττον τὸ Α διὰ τοῦ Δ οἰσθήσεται ἢ διὰ τοῦ Β. ἐχέτω δὴ τὸν αὐτὸν λόγον ὅνπερ διέστηκεν ἀὴρ πρὸς ὕδωρ, τὸ τάχος πρὸς τὸ τάχος.
For let B be water, and D be air; then, by as much as air is thinner and more incorporeal than water, by so much faster will A be carried through D than through B. Let, then, the speed have the same ratio to the speed as air differs from water.
ὥστε εἰ διπλασίως λεπτόν, ἐν διπλασίῳ χρόνῳ τὴν τὸ Β δί.
So that if it is twice as thin, it will traverse the distance of B in twice the time than that of D, and the time G will be twice the time E.
εισιν ἢ τὴν τὸ Δ, καὶ ἔσται ὁ ἐφʼ Γ χρόνος διπλάσιος τοῦ ἐφʼ ᾧ Ε. καὶ ἀεὶ δὴ ὅσῳ ἂν ᾖ ἀσωματώτερον καὶ ἧττον ἐμποδιστικὸν καὶ εὐδιαιρετώτερον διʼ οὗ φέρεται, θᾶττον οἰσθήσεται.
And always, by as much as the medium through which it is carried is more incorporeal, less impeding, and more easily divided, the faster it will be carried.
τὸ δὲ κενὸν οὐδένα ἔχει λόγον ᾧ ὑπερέχεται ὑπὸ τοῦ σώματος, ὥσπερ οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν.
But the void has no ratio by which it is exceeded by a body, just as nothing has no ratio to a number.
εἰ γὰρ τὰ τέτταρα τῶν τριῶν ὑπερέχει ἑνί, πλείονι δὲ τοῖν δυοῖν, καὶ ἔτι πλείονι τοῦ ἑνὸς ἢ τοῖν δυοῖν, τοῦ δὲ μηδενὸς οὐκέτι ἔχει λόγον ᾧ ὑπερέχει· ἀνάγκη γὰρ τὸ ὑπερέχον διαιρεῖσθαι εἴς τε τὴν ὑπεροχὴν καὶ τὸ ὑπερεχόμενον, ὥστε ἔσται τὰ τέτταρα ὅσῳ τε ὑπερέχει καὶ οὐδέν.
For if four exceeds three by one, and two by more, and still more one than two, yet to nothing it no longer has a ratio by which it exceeds; for that which exceeds must be divided into the excess and that which is exceeded, so that four will be as much as it exceeds and nothing.
διὸ οὐδὲ γραμμὴ στιγμῆς ὑπερέχει, εἰ μὴ σύγκειται ἐκ στιγμῶν.
For this reason, neither does a line exceed a point, unless it is composed of points.
ὁμοίως δὲ καὶ τὸ κενὸν πρὸς τὸ πλῆρες οὐδένα οἷόν τε ἔχειν λόγον, ὥστε οὐδὲ τὴν κίνησιν, ἀλλʼ εἰ διὰ τοῦ λεπτοτάτου ἐν τοσῳδὶ τὴν τοσήνδε φέρεται, διὰ τοῦ κενοῦ παντὸς ὑπερβάλλει λόγου.
Likewise, the void cannot have any ratio to the full, so that neither can the motion; but if it is carried through the thinnest medium over such a distance in such a time, through the void it exceeds every ratio.
ἔστω γὰρ
τὸ Ζ κενόν, ἴσον δὲ τοῖς Β καὶ Δ. τὸ δὴ Α εἰ δίεισι καὶ κινηθήσεται ἐν τινὶ μὲν χρόνῳ, τῷ ἐφʼ οὗ Η, ἐν ἐλάττονι δὲ τοῦ ἐφʼ οὗ Ε, τοῦτον ἕξει τὸν λόγον τὸ κενὸν πρὸς τὸ πλῆρες.
For let Z be a void, equal to B and D. Now, if A traverses and is moved in some time, namely H, and in a time less than E, the void will have this ratio to the full.
ἀλλʼ ἐν τοσούτῳ χρόνῳ ὅσος ἐφʼ οὗ τὸ Η, τοῦ Δ τὸ Α δίεισι τὴν τὸ Θ. δίεισι δέ γε κἂν ᾖ τι λεπτότητι διαφέρον τοῦ ἀέρος ἐφʼ ᾧ τὸ ταύτην τὴν ἀναλογίαν ἣν ἔχει ὁ χρόνος ἐφʼ Ε πρὸς τὸν ἐφʼ Η. ἂν γὰρ τοσούτῳ λεπτότερον τὸ ἐφʼ σῶμα τοῦ Δ, ὅσῳ ὑπερέχει τὸ Ε τοῦ Η, ἀντεστραμμένως δίεισι τῷ τάχει ἐν τῷ τοσούτῳ ὅσον τὸ Η, τὴν τὸ τὸ ἐφʼ οὗ Α, ἐὰν φέρηται.
But in as much time as H, A will traverse the distance of D equal to Th. And indeed, it will also traverse it if there is some body differing in thinness from air (let it be [I]), having this proportion which the time E has to the time H. For if that [thin] body is thinner than D by as much as E exceeds H, it will, in inverse proportion to its speed, traverse in a time as much as H the distance [Z] of A, if it is carried.
ἐὰν τοίνυν μηδὲν ᾖ σῶμα ἐν τῷ Ζ, ἔτι θᾶττον.
If, then, there is no body in Z, it will be still faster.
ἀλλʼ ᾖν ἐν τῷ Η. ὥστʼ ἐν ἴσῳ χρόνῳ δίεισι πλῆρές τε ὂν καὶ κενόν.
But it was in H.
ἀλλʼ ἀδύνατον.
So that in an equal time it will traverse both the full and the void.
φανερὸν τοίνυν ὅτι, εἰ ἔστι χρόνος ἐν ᾧ τοῦ κενοῦ ὁτιοῦν οἰσθήσεται, συμβήσεται τοῦτο τὸ ἀδύνατον·
But this is impossible.
ἐν ἴσῳ γὰρ ληφθήσεται πλῆρές τε ὂν διεξιέναι τι καὶ κενόν· ἔσται γάρ τι ἀνάλογον σῶμα ἕτερον πρὸς ἕτερον ὡς χρόνος πρὸς χρόνον.
It is clear, then, that if there is any time in which it is carried through any part of the void, this impossible thing will happen; for it will be taken to traverse a certain distance both in the full and in the void in an equal time; for there will be some body in proportion to another body as time is to time.
ὡς δʼ ἐν κεφαλαίῳ εἰπεῖν, δῆλον τὸ τοῦ συμβαίνοντος αἴτιον, ὅτι κινήσεως μὲν πρὸς κίνησιν πάσης ἔστι λόγος (ἐν χρόνῳ γάρ ἐστι, χρόνου δὲ παντὸς ἔστι πρὸς χρόνον, πεπερασμένων ἀμφοῖν, κενοῦ δὲ πρὸς πλῆρες οὐκ ἔστιν.
To speak in summary, the cause of what happens is clear: that of every motion to another motion there is a ratio (for it is in time, and of every time to another time there is a ratio, both being finite), but of the void to the full there is no ratio.