§6.2#1Ἐπεὶ δὲ πᾶν μέγεθος εἰς μεγέθη διαιρετόν (δέδεικται γὰρ ὅτι ἀδύνατον ἐξ ἀτόμων εἶναί τι συνεχές, μέγεθος δʼ ἐστὶν ἅπαν συνεχές), ἀνάγκη τὸ θᾶττον ἐν τῷ ἴσῳ χρόνῳ μεῖζον καὶ ἐν τῶ ἐλάττονι ἴσον καὶ ἐν τῷ ἐλάττονι πλεῖον κινεῖσθαι, καθάπερ ὁρίζονταί τινες τὸ θᾶττον.
Since every magnitude is divisible into magnitudes (for it has been demonstrated that it is impossible for anything continuous to be composed of indivisibles, and every magnitude is continuous), it is necessary that the faster moves over a greater magnitude in an equal time, over an equal magnitude in less time, and over a greater magnitude in less time, just as some define the faster.
ἔστω γὰρ τὸ ἐφʼ ὧ A τοῦ ἐφʼ ᾧ B θᾶττον.
For let A be faster than B.
ἐπεὶ τοίνυν θᾶττόν ἐστιν τὸ πρότερον μεταβάλλον, ἐν ᾧ χρόνῳ τὸ Α μεταβέβληκεν ἀπὸ τοῦ Γ εἰς τὸ Δ, οἷον τῷ ZH, ἐν τούτῳ τὸ B οὔπω ἔσται πρὸς τῷ Δ, ἀλλʼ ἀπολείψει, ὥστε ἐν τῷ ἴσῳ χρόνῳ πλεῖον δίεισιν τὸ θᾶττον.
Since, then, the faster is that which changes earlier, when A has changed from C to D in a certain time, for example, in ZH, in this same time B will not yet be at D, but will fall short of it, so that in an equal time the faster traverses a greater magnitude.
ἀλλὰ μὴν καὶ ἐν τῷ ἐλάττονι πλεῖον·
Moreover, it also traverses a greater magnitude in less time.
ἐν ᾧ γὰρ τὸ A γεγένηται πρὸς τῷ Δ, τὸ B ἔστω πρὸς τῷ E τὸ βραδύτερον ὄν.
For in the time in which A has reached D, let B, which is the slower, be at E.
οὐκοῦν ἐπεὶ τὸ A πρὸς τῷ Δ γεγένηται ἐν ἅπαντι τῷ ZH χρόνῳ, πρὸς τῶ Θ ἔσται ἐν ἐλάττονι τούτου·
Then, since A has reached D in the whole time ZH, it will be at Th in less time than this; and let this time be ZK.
καὶ ἔστω ἐν τῶ ZK. τὸ μὲν οὖν ΓΘ, ὃ διελήλυθε τὸ Α, μεῖζόν ἐστι τοῦ ΓΕ, ὁ δὲ χρόνος ὁ ΖΚ ἐλάττων τοῦ παντὸς τοῦ ΖΗ. ὥστε ἐν ἐλάττονι μεῖζον δίεισιν.
Now CTh, which A has traversed, is greater than CE, while the time ZK is less than the whole time ZH. So in less time it traverses a greater magnitude.
φανερὸν δὲ ἐκ τούτων καὶ ὅτι τὸ θᾶττον ἐν ἐλάττονι χρόνῳ δίεισιν τὸ ἴσον.
It is also manifest from this that the faster traverses an equal magnitude in less time.
ἐπεὶ γὰρ τὴν μείζω ἐν ἐλάττονι διέρχεται τοῦ βραδυτέρου, αὐτὸ δὲ καθʼ αὑτὸ λαμβανόμενον ἐν πλείονι χρόνῳ τὴν μείζω τῆς ἐλάττονος, οἷον τὴν ΛΜ τῆς ΛΞ, πλείων ἂν εἴη ὁ χρόνος ὁ ΠΡ, ἐν ᾧ τὴν ΛΜ διέρχεται, ἢ ὁ ΠΣ, ἐν ᾧ τὴν ΛΞ. ὥστε εἰ ὁ ΠΡ χρόνος ἐλάττων ἐστὶν τοῦ Χ, ἐν ᾧ τὸ βραδύτερον διέρχεται τὴν ΛΞ, καὶ ὁ ΠΣ ἐλάττων ἔσται τοῦ ἐφʼ ὧ X· τοῦ γὰρ ΠΡ ἐλάττων, τὸ δὲ τοῦ ἐλάττονος ἔλαττον καὶ αὐτὸ ἔλαττον, ὥστε ἐν ἐλάττονι κινήσεται τὸ ἴσον.
For since it traverses the greater magnitude in less time than the slower, and when taken by itself, it takes more time for a greater magnitude than for a less (for example, PR for LM is more than PS for LΞ, if LM is greater than LΞ), so that if the time PR is less than X, in which the slower traverses LΞ, PS will also be less than X; for it is less than PR, and that which is less than a less is itself also less, so that it will move over the equal magnitude in less time.
ἔτι δʼ εἰ πᾶν ἀνάγκη ἢ ἐν ἴσῳ ἢ ἐν ἐλάττονι ἢ ἐν πλείονι κινεῖσθαι, καὶ τὸ μὲν ἐν πλείονι βραδύτερον, τὸ δʼ ἐν ἴσῳ ἰσοταχές, τὸ δὲ θᾶττον οὔτε ἰσοταχὲς οὔτε βραδύτερον, οὔτʼ ἂν ἐν ἴσῳ οὔτʼ ἐν πλείονι κινοῖτο τὸ θᾶττον.
Furthermore, since it is necessary that everything moves either in an equal, or in less, or in more time, and that which moves in more time is slower, while that which moves in equal time is of equal speed, and the faster is neither of equal speed nor slower, the faster would move neither in equal nor in more time.
λείπεται οὖν ἐν ἐλάττονι, ὥστʼ ἀνάγκη καὶ τὸ ἴσον μέγεθος ἐν ἐλάττονι χρόνῳ διιέναι τὸ θᾶττον.
There remains, then, only the less time, so that it is necessary that the faster also traverses an equal magnitude in less time.
ἐπεὶ δὲ πᾶσα μὲν κίνησις ἐν χρόνῳ καὶ ἐν
ἀπόντι χρόνῳ δυνατὸν κινηθῆναι, πᾶν δὲ τὸ κινούμενον ἐνδέχεται καὶ θᾶττον κινεῖσθαι καὶ βραδύτερον, ἐν ἅπαντι χρόνῳ ἔσται τὸ θᾶττον κινεῖσθαι καὶ βραδύτερον.
And since every motion is in time, and it is possible to move in any time, and everything that is in motion can move both faster and slower, in every time there will be faster and slower motion.
τούτων δʼ ὄντων ἀνάγκη καὶ τὸν χρόνον συνεχῆ εἶναι.
This being so, it is necessary that time also is continuous.
λέγω δὲ συνεχὲς τὸ διαιρετὸν εἰς αἰεὶ διαιρετά· τούτου γὰρ ὑποκειμένου τοῦ συνεχοῦς, ἀνάγκη συνεχῆ εἶναι τὸν χρόνον.
By continuous I mean that which is divisible into always divisibles; for if this definition of the continuous is assumed, it is necessary that time is continuous.
ἐπεὶ γὰρ δέδεικται ὅτι τὸ θᾶττον ἐν ἐλάττονι χρόνῳ δίεισιν τὸ ἴσον, ἔστω τὸ μὲν ἐφʼ ᾧ Α θᾶττον, τὸ δʼ ἐφʼ ᾧ B βραδύτερον, καὶ κεκινήσθω τὸ βραδύτερον τὸ ἐφʼ ΓΔ μέγεθος ἐν τῷ ΖΗ χρῳ.
For since it has been demonstrated that the faster traverses an equal magnitude in less time, let A be faster and B slower, and let the slower have moved over the magnitude CD in the time ZH.
δῆλον τοίνυν ὅτι τὸ θᾶττον ἐν ἐλάττονι τούτου κινήσεται τὸ αὐτὸ μέγεθος· καὶ κεκινήσθω ἐν τῶ ΖΘ. πάλιν δʼ ἐπεὶ τὸ θᾶττον ἐν τῷ ΖΘ διελήλυθεν τὴν ὅλην τὴν ΓΔ, τὸ βραδύτερον ἐν τῷ αὐτῷ χρόνῳ τὴν ἐλάττω δίεισιν· ἔστω οὖν ἐφʼ ἧς ΓΚ. ἐπεὶ δὲ τὸ βραδύτερον τὸ B ἐν τῷ ΖΘ χρόνῳ τὴν ΓΚ διελήλυθεν, τὸ θᾶττον ἐν ἐλάττονι δίεισιν, ὥστε πάλιν διαιρεθήσεται ὁ ΖΘ χρόνος.
It is manifest, then, that the faster will move over the same magnitude in less time than this; and let it have moved in the time ZTh. Again, since the faster has traversed the whole CD in the time ZTh, the slower in the same time will traverse a less magnitude; let this be CK. And since the slower B in the time ZTh has traversed CK, the faster will traverse it in less time, so that again the time ZTh will be divided.
τούτου δὲ διαιρουμένου καὶ τὸ ΓΚ μέγεθος διαιρεθήσεται κατὰ τὸν αὐτὸν λόγον.
And when this is divided, the magnitude CK will also be divided in the same ratio.
εἰ δὲ τὸ μέγεθος, καὶ ὁ χρόνος.
And if the magnitude is divided, the time will be also.
καὶ ἀεὶ τοῦτʼ ἔσται μεταλαμβάνουσιν ἀπὸ τοῦ θάττονος τὸ βραδύτερον καὶ ἀπὸ τοῦ βραδυτέρου τὸ θᾶττον, καὶ τῷ ἀποδεδειγμένῳ χρωμένοις· διαιρήσει γὰρ τὸ μὲν θᾶττον τὸν χρόνον, τὸ δὲ βραδύτερον τὸ μῆκος.
And this will always go on if we take alternately from the faster to the slower and from the slower to the faster, and make use of what has been demonstrated; for the faster will divide the time, and the slower the length.
εἰ οὖν αἰεὶ μὲν ἀντιστρέφειν ἀληθές, ἀντιστρεφομένου δὲ αἰεὶ γίγνεται διαίρεσις, φανερὸν ὅτι πᾶς χρόνος ἔσται συνεχής.
If, then, this alternation is always true, and upon the alternation there always occurs a division, it is manifest that all time will be continuous.
ἅμα δὲ δῆλον καὶ ὅτι μέγεθος ἅπαν ἐστὶ συνεχές· τὰς αὐτὰς γὰρ καὶ τὰς ἴσας διαιρέσεις ὁ χρόνος διαιρεῖται καὶ τὸ μέγεθος.
At the same time it is also clear that every magnitude is continuous; for time and magnitude are divided by the same and equal divisions.