§8.2Εἰ μὲνδὴ τάχιστα τὰ τοιαῦτα, διά τε τὸ μικρῷ ἄπτεσθαι τοῦ ἐπιπέδου, ὥσπερ ὁ κύκλος κατὰ στιγμήν, καὶ διὰ τὸ μὴ προσκόπτειν· ἀφέστηκε γὰρ τῆς γῆς ἡ γωνία.
If indeed such things are moved most quickly, it is both because they touch the flat surface only slightly, as a circle does at a point, and because they do not bump against things; for its angle is kept away from the ground.
Καὶ ἔτι ᾦ ἂν ἀπαντήσῃ σώματι, πάλιν τούτου κατὰ μικρὸν ἅπτεται.
And further, whatever body it meets, it touches it again only slightly.
Εἰ δ’ εὐθύγραμμον ἦν, τῇ εὐθείᾳ ἐπὶ πολὺ ἥπτετο ἂν τοῦ ἐπιπέδου.
If it were rectilinear, it would touch the flat surface for a long distance along its straight line.
§8.3Ἕτι ᾖ ῥέπει ἐπὶ τὸ βάρος, ταύτῃ κινεῖ ὁ κινῶν.
Furthermore, in the direction in which it inclines by its weight, in that direction the mover moves it.
Ὄταν μὲν γὰρ πρὸς ὄρθιον ἡ διάμετρος ᾖ τοῦ κύκλου τῷ ἐπιπέδῳ, ἁπτομένου τοῦ κύκλου κατὰ στιγμνὴν τοῦ ἐπιπέδου, ἴσὸν τὸ βάρος ἐπ’ ἀμφότερα διαλαμβάνει ἡ διάμετρος· ὅταν δὲ κινῆται, εὐθύς πλέον ἐφ’ ᾧ κινεῖται, ὥσπερ ῥέπον.
For when the diameter of the circle is perpendicular to the flat surface, while the circle touches the flat surface at a point, the diameter divides the weight equally on both sides; but when it is moved, immediately there is more weight in the direction in which it is moved, as if it were inclining.
§8.4Ἐντεῦθεν εὐκινητότερον τῷ ὠθοῦντι εἰς τοὔμπροσθεν· ἐφ’ ὃ γὰρ ῥέπει ἕκαστον, εὐκίνητόν ἐστιν, εἴπερ καὶτὸ ἐπὶ τὸ ἐναντίον τῆς ῥοπῆς δυσκίνητον.
Hence it is easier to move for the one who pushes it forward; for in the direction in which each thing inclines, it is easy to move, if indeed that which is in the direction opposite to the inclination is hard to move.
Ἔτι λέγουσί τινες ὅτι καὶ ἡ γραμμὴ ἡ τοῦ κύκλου ἐν φορᾷ ἐστὶν ἀεί, ὥσπερ τὰ μένοντα, διὰ τὸ ἀντερείδειν, οἷον καὶ τοῖς μείζοσι κύκλοις ὑπάρχει πρὸς τοὺς ἐλάττονας.
Further, some say that the line of the circle is always in motion, just like things that remain stationary, because of its resistance, as is also the case with larger circles compared to smaller ones.
§8.5Θᾶττον γὰρ ὑπὸ τῆς ἴσης ἰσχύος κινοῦνται οἱ μείζους καὶ τὰ βάρη κινοῦσι, διὰ τὸ ῥοπήν τινα ἔχειν τὴν γωνίαν τὴν τοῦ μείζονος κύκλου πρὸς τὴν τοῦ ἐλάττονος, καὶ εἶναι ὅπερ ἡ διάμετρος πρὸς τὴν διάμετρον.
For larger circles are moved faster by the same force and they move weights, because the angle of the larger circle has a certain inclination compared to that of the smaller, and is in the same relation as diameter to diameter.
Ἀλλὰ μὴν πᾶς κύκλος μείζων πρὸς ἐλάττονα· ἄπειροι γὰρ οἱ ἐλάττονες.
But indeed, every circle is larger in comparison to a smaller one; for the smaller ones are infinite in number.
§8.6Εἰ δὲ καὶ πρὸς ἕτερον ἔχει ῥοπὴν ὁ κύκλος, ὁμοίως δὲ εὐκίνητος, καὶ ἄλλην ἂν ἔχοι ῥοπὴν ὁ κύκλος καὶ τὰ ὑπὸ κύκλου κινούμενα, κἂν μὴ τῇ ἁψῖδι ἅπτηται τοῦ ἐπιπέδου, ἀλλ’ ἢπαρὰ τὸ ἐπίπεδον, ἢ ὡςαἱ τροχιλέαι· καὶ γὰρ οὕτως ἔχοντα ῥᾷστα κινοῦνται καὶ κινοῦσι τὸ βάρος.
And if the circle has an inclination also towards another thing, and is likewise easy to move, the circle and the things moved by the circle would also have another inclination, even if it does not touch the flat surface with its circumference, but either parallel to the flat surface, or like pulleys; for even when they are in this state, they are moved and move the weight most easily.
Ἥ οὐ τῷ κατὰ μικρὸν ἅπτεσθαι καὶ προσκρούειν, ἀλλὰ δι’ ἄλλην αἰτίαν.
Or is it not because of touching and knocking against things slightly, but due to another cause?
§8.7Αὕτη δέ ἐστιν ἡ εἰρημένη πρότερον, ὅτι ἐκ δύο φορῶν γεγένηται ὁ κύκλος, ὥστε μίαν αὐτῶν αἰεὶ ἔχειν ῥοπήν, καὶ οἶον φερόμενον αὐτὸν αἰεὶ κινοῦσιν οἱ κινοῦντες, ὅταν κινῶσι κατὰ τὴν περιφέρειαν ὁπωσοῦν, Φερομένην γὰρ αὐτὴν κινοῦσιν·
And this is the one mentioned before, namely that the circle is generated from two motions, so that it always has an inclination along one of them, and those who move it always move it as if it were already in motion, whenever they move it along the circumference in any way whatsoever; for they move it while it is in motion.
τὴν μὲν γὰρ εἰς τὸ πλάγιον αὐτοῦ κίνησιν ὠθεῖ τὸ κινοῦν, τὴν δὲ ἐπὶ τῆς διαμέτρου αύτὸς κινεῖται.
For the mover pushes its motion to the side, but it moves by itself along the diameter.