§15.1Διὰ τί περὶ τοὺς αἰγιαλοὺς αἱ καλούμεναι κρόκαι στρογγύλαι εἰσίν, ἐκ μακρῶν τῶν λίθων καὶ ὁστράκων τὸ ἐξ ὑπαρχῆς ὄντων;
Why are the so-called pebbles on beaches round, although they were originally long stones and shells?
Ἥ διότι τὰ πλεῖον ἀπέχοντα τοῦ μέσου ἐν ταῖς κινήσεσι θᾶττον φέρεται, Τὸ μὲν γὰρ μέσον γίνεται κένσθαιτρον, τὸ δὲ διάστημα ἡ ἐκ τοῦ κέντρον.
Or is it because, in movements, those parts which are further away from the center are carried more quickly? For the middle becomes the center, and the distance is the radius from the center.
Ἀεὶ δὲ ἡ μείζων ἀπὸ τῆς ἴσης κινήσεως μείζω γράφει κύκλον.
And the larger radius always describes a larger circle from an equal movement.
§15.2Τὸ δ’ ἐν ἴσῳ χρόνῳ μείζω διεξιὸν θᾶττον φέρεται.
And that which traverses a greater distance in an equal time is carried more quickly.
Τὰ δὲ φερόμενα θᾶττον ἐκ τοῦ ἴσου ἀποστήματος σφοδρότερον τύπτει.
Furthermore, those things which are carried more quickly strike more violently from an equal distance.
Τὰ δὲ τύπτοντα μᾶλλον καὶ αὐτὰ τύπτεται μᾶλλον, Ὥστε ἀνάγκη θραύεσθαι αἰεὶ τὰ πλέον ἀπέχοντα τοῦ μέσου.
And those things which strike are themselves struck more violently. Consequently, it is necessary that those parts which are further away from the center are always broken.
Τοῦτο δὲ πάσχοντα ἀνάγκη γίνεσθαι περιφερῆ.
And by undergoing this, they must of necessity become round.
§15.3Ταῖς δὲ κρόκαις διὰ τὴν τῆς θαλάττης κίνησιν, διὰ τὸ μετὰ τῆς θαλάττης κινεῖσθαι, συμβαίνει ἀεὶ ἐν κινήσει εἶναι καὶ κυλιομέναις προσκόπτειν.
For pebbles, because of the movement of the sea and because they are moved along with the sea, it happens that they are always in motion and collide as they roll.
Τοῦτο δὲ ἀνάγκη μάλιστα συμβαίνειν αὐτοῖς τοῖς ἄκροις.
And this must of necessity happen most of all to their very extremities.