§24.10Τὸ γὰρ αὐτὸ τῷ αὐτῷ τάχει φερόμενον ἴσην πέφυκε διεξιέναι· τῷ αὐτῷ δὲ τάχει ἴσην ἐστὶ κινεῖν ἀμφοτεράκις.
For the same thing carried at the same velocity is naturally fitted to traverse an equal distance; and to move equally in both cases is to have the same velocity.
Ἀρχὴ δὲ ληπτέα ἥδὲ περὶ τῆς αἰτίας αὐτῶν,ὅτι ἡ αὐτὴ δύναμιςκαὶ ἴση τὸ μὲν βραδύτερον κινεῖ μέγεθος, τὸ δὲ ταχύτερον.
But we must assume this as a principle concerning their cause: that the same and equal power moves one magnitude more slowly, and another more quickly.
§24.11Εἰ δή τι εἴη ὃ μὴ πέφυκεν ὑφ’ ἑαυτοῦ κινεῖσθαι, ἐὰντοῦτο ἅμα καὶ αὐτὸ κινῇ τὸ πεφυκὸς κινεῖσθαι, βραδύτερον κινηθήσεται ἢ εἰ αὐτὴ καθ’ αὐτὴν ἐκινεῖτο.
If then there should be something which is not naturally fitted to be moved by itself, and if this should be moved at the same time and by that which is naturally fitted to be moved, it will be moved more slowly than if it were moved by itself.
Καὶ ἐὰν μὲν πεφυκὸς ᾖ κινεῖσθαι, μὴσυγκινῆται δὲ μηθέν, ὡσαύτως ἕξει.
And if it is naturally fitted to be moved, and nothing is moved with it, it will be in the same state.
Καὶ ἀδύνατον δὴ κινεῖσθαι πλέον ἢ τὸ κινοῦν· οὐ γὰρ τὴν αὐτοῦ κινεῖται κίνησιν, ἀλλὰ τὴν τοῦ κινοῦντος.
And indeed, it is impossible to be moved more than the mover; for it is not moved with its own motion, but with that of the mover.
§24.12Εἴη δὴ κύκλος ὁ μὲν μείζων τὸ Α, ὁ δὲ ἐλάττων ἐφ’ ᾧ Β. Εἰ ὠθοίη δ’ ὁ ἐλάττων τὸν μείζω, μὴ κυλιομένου αὐτοῦ, φανερὸν ὅτι τοσοῦτον δίεισι τῆς εὐθείας ὁ μείζων, ὅσον ἐώσθη ὑπὸ τοῦ ἐλάττονος.
Let the larger circle be A, and the smaller be B. Now, if the smaller were to push the larger without itself rolling, it is clear that the larger will traverse as much of the straight line as it was pushed by the smaller.
Τοσοῦτον δέ γε ἐώσθη ὅσον ὁ μικρὸς ἐκινήθη, Ἴσην ἄρα τῆς εὐθείας διεληλύθασιν.
And it was pushed just as much as the small one was moved; therefore, they have traversed an equal distance of the straight line.
§24.13Ἀνάγκη τοίνυν καὶ εἰ κυλιόμενος ὁ ἐλάττων τὸν μείζω ὠθοίη, κυλισθῆναι μὲν ἄμα τῇ ὥσει, τοσοῦτον δ’ ὅσον ὁ ἐλάττων ἐκυλίσθη, εἰ μηθὲν αὐτὸς τῇ αὐτῇ κινήσει κινεῖται.
It is necessary, therefore, even if the smaller rolls and pushes the larger, that the larger should roll at the same time as the pushing, and as much as the smaller has rolled, if the larger itself moves nothing in that same motion by its own power.
Ὡς γὰρ καὶ ὅσον ἐκίνει, τοσοῦτον κεκινῆσθαι ἀνάγκη τὸ κινούμενον ὑπ’ ἐκείνου.
For as much as the mover was moving, so much must the thing moved by it have been moved.
Ἀλλὰ μὴν ὅ τε κύκλος τοσοῦτον ἐκίνησε τὸ αὐτό, κύκλῳ τε καὶ ποδιαίαν (ἔστω γὰρ τοσοῦτον ὃ ἐκινήθη),καὶ ὁ μέγας ἅρα τοσοῦτον ἐκινήθη.
Indeed, the small circle has moved itself by that much, both circularly and by one foot (for let the distance moved be so much); therefore, the large one also was moved by that much.
§24.14Ὁμοίως δὲ κἂν ὁ μέγας τὸν μικρὸν κινήσῃ, ἔσται κεκινημένος ὁ μικρὸς ὡς καὶ ὁ μείζων.
Similarly, even if the large one moves the small one, the small one will have been moved just as the larger was.
Καθ’ αὐτὸν μὲν δὴ κινηθεὶς ὁποτεροσοῦν, ἐάν τε ταχὺ ἐάν τε βραδέως· τῷ αὐτῷ δὲ τάχει εὐθὺς ὅσην ὁ μείζων πέφυκεν ἐξελιχθῆναι γραμμήν.
And indeed, if either moves by itself, whether quickly or slowly, [it rolls out its own line]; but at the same velocity, they immediately roll out as much line as the larger is naturally fitted to unroll.
Ὅπερ καὶ ποιεῖ τὴν ἀπορίαν, ὅτι οὐκέτι ὁμοίως ποιοῦσιν ὅταν συναρμοσθῶσιν.
This is indeed what produces the puzzle, that they no longer do the same when they are fitted together.
§24.15Τὸ δ’ ἔστιν, εἰ ὁ ἕτερος ὑπὸ τοῦ ἑτέρου κινεῖται οὺχ ἢν πέφυκεν, οὐδὲ τὴν αὐτοῦ κίνησιν.
But the truth is that, if one is moved by the other not as it is naturally fitted, nor with its own motion.
Οὐθὲν γὰρ διαφέρει περιθεῖναι καὶ ἐναρμόσαι ἢ προσθεῖναι ὁποτερονοῦν ὁποτέρῳ· ὁμοίως γάρ, ὅταν ὁ μὲν κινῇ ὁ δὲ κινῆται ὑπὸ τούτου, ὅσον ἄν κινῇ ἅτερος, τοσοῦτον κινηθήσεται ἅτερος.
For it makes no difference whether one places one around and fits it to the other, or places them alongside each other; for similarly, when one moves and the other is moved by it, as much as one moves, so much will the other be moved.
§24.16Ὅταν μὲν οὖν προσκείμενον κινῇ ἢ προσκρεμάμενον, οὐκ ἀεὶ κυλίει τις· ὅταν δὲ περὶ τὸ αὐτὸ κέντρον τεθῶσιν, ἀνάγκη κυλίεσθαι ἀεὶ τὸν ἕτερον ὑπὸ τοῦ ἑτέρου.
Therefore, when one moves something placed alongside or suspended, one does not always roll it; but when they are placed about the same center, it is necessary that one is always rolled by the other.
Ἀλλ’ οὐθὲν ἧττον οὐ τὴν αὑτοῦ κίνησιν ἅτερος κινεῖται, ἀλλ’ ὥσπερ ἂν εἰ μηδεμίαν εἶχε κίνησιν.
Nevertheless, the other does not move with its own motion, but just as if it had no motion.
Κἂν ἔχῃ. μὴ χρῆται δ’ αὐτῇ, ταὐτὸ συμβαίνει.
And even if it has one but does not use it, the same thing happens.
§24.17Ὅταν μὲν οὖν ὁ μέγας κινῇ ἐνδεδεμένον τὸν μικρόν, ὁμικρὸς κινεῖται ὅσηνπερ οὗτος· ὅταν δὲ ὁ μικρός, πάλιν ὁ μέγας ὅσην οὗτος.
Thus, when the large one moves the small one bound to it, the small one moves just as much as this one; and when the small one moves it, again the large one moves as much as this one.
Χωριζόμενος δὲ ἑκάτερος αὐτὸν κινεῖ αὐτός, Ὅτι δὲ τοῦ αὐτοῦ κέντρου ὄντος καὶ κινοῦντος τῷ αὐτῷ τάχει συμβαίνει ἄνισον διεξιέναι αὑτοὺς γραμμήν, παραλογίζεται ὁ ἀπορῶν σοφιστκῶς.
But when separated, each moves by itself. But as to why, when the center is the same and moving at the same velocity, it happens that they traverse unequal lines, the person who is puzzled is reasoning sophistically.
§24.18Τὸ αὐτὸ μὲν γάρ ἐστι κέντρον ἀμφοῖν, ἀλλὰ κατὰ συμβεβηκός,ὡς μουσικὸν καὶ λευκόν· τὸ γὰρ εἶναι ἑκατέρου κέντρου τῶν κύκλων οὐ τῷ αὐτῷ χρῆται.
For the center is indeed the same for both, but accidentally, just as "musical" and "white" are the same; for being the center of each of the circles does not employ the same essence.
Ὅταν μὲν οὖν ὁ κινῶν ᾖ ὁ μικρός, ὡς ἐκείνου κέντρον καὶ ἀρχή, ὅταν δὲ ὁ μέγας, ὡς ἐκείνου.
Therefore, when the mover is the small one, it acts as the center and principle of that one; and when the mover is the large one, as the center of that one.
Οὔκουν τὸ αὐτὸ κινεῖ ἀπλῶς, ἀλλ’ ἔστιν ὥς.
Thus, it is not simply the same thing that moves, but only in a certain sense.