§16.9Διὰ τί τὰ μόρια τῶν φυτῶν καὶ τῶν ζῴων, ὅσα μὴ ὀργανικά, πάντα περιφερῆ, τῶν μὲν φυτῶν τὸ στέλεχος καὶ οἱ πτόρθοι, τῶν δὲ ζῴων κνῆμαι, μηροί, βραχίονες, θώραξ τρίγωνον δὲ οὐδὲ πολύγωνον οὔτε ὅλον οὔτε μόριόν ἐστιν;
Why are all the non-organic parts of plants and animals circular, such as the stem and shoots of plants, and the shins, thighs, arms, and chest of animals, while there is no triangle or polygon, either as a whole or as a part?
πότερον, ὥσπερ Ἀρχύτας ἔλεγεν, διὰ τὸ ἐν τῇ κινήσει τῇ φυσικῇ ἐνεῖναι τὴν τοῦ ἴσου ἀναλογίαν (κινεῖσθαι γὰρ ἀνάλογον πάντα), ταύτην δὲ μόνην εἰς αὑτὴν ἀνακάμπτειν, ὥστε κύκλους ποιεῖν καὶ στρογγύλα, ὅταν ἐγγένηται;
Is it, as Archytas used to say, because the proportion of equality is present in natural motion (for all things move proportionally), and this alone turns back upon itself, so as to produce circles and rounded shapes when it occurs?
§16.10Διὰ τί ἐν τοῖς ἐσχάτοις ἀεὶ γίνεται περιφερῆ;
Why do the extremities always become circular?
ἢ ὅτι ἡ φύσις ἐκ τῶν ἐνδεχομένων πάντα ποιεῖ ὡς δυνατὸν ἄριστα καὶ κάλλιστα, τὸ δὲ σχῆμα τοῦτο κάλλιστον, τὸ αὐτὸ αὑτῷ ὁμοιότατον.
Or is it because nature makes all things from what is possible as best and most beautiful as possible, and this shape is the most beautiful, being most similar to itself?
§16.11Διὰ τί, ἐὰν κύκλος ῥιφθῇ, τὸ μὲν πρῶτον εὐθεῖαν γράφει, παυόμενος δὲ ἕλικα, ἕως ἂν πέσῃ;
Why is it that, if a hoop is thrown, it at first describes a straight line, but as it ceases, it describes a spiral until it falls?
ἢ εὐθεῖαν μὲν τὸ πρῶτον ὅτι ὁμοίως ἔνθεν καὶ ἔνθεν ὁ ἀὴρ ἀπορθοῖ;
Or is it at first straight because the air keeps it upright equally on this side and on that?
ἴσης οὖν οὔσης τῆς ῥοπῆς ἔνθεν καὶ ἔνθεν, ἀνάγκη καὶ τὴν γραμμὴν τοιαύτην εἶναι, ἣ ἴσον διαιρεῖ τὸν τόπον ἔνθεν καὶ ἔνθεν· τοιαύτη δέ ἐστιν εὐθεῖα.
When, therefore, the inclination is equal on this side and on that, the line also must be of such a kind as divides the space equally on this side and on that; and such a line is a straight line.
ὅταν δὲ βρίσῃ ἐπὶ θάτερον μέρος δι’ ἀνωμαλίαν τοῦ περιισταμένου ἀέρος, οὐκέτι ἴσην γράφει τό τε ἐντὸς καὶ τὸ ἐκτὸς μέρος, ἀλλ’ ἀνάγκῃ περιφερῆ.
But when it inclines to one side owing to the irregularity of the surrounding air, it no longer describes the inner and the outer part equally, but by necessity a circular one.
§16.12Διὰ τί τοῖς ἄνισον τὸ βάθος ἔχουσι μεγέθεσιν, ἐάν τις κουφότερον κινῇ τῶν μερῶν, κύκλῳ περιφέρεται τὸ βαλλόμενον, οἷον τοῖς μεμολιβδωμένοις ἀστραγάλοις συμβαίνει, ἐάν τις βάλλῃ τὸ κουφότερον πρὸς αὑτὸν στρέψας μέρος;
Why is it that, in the case of magnitudes having unequal thickness, if one moves the lighter of the parts, the thrown object is carried in a circle, as happens with leaded knucklebones when one throws them having turned the lighter part towards oneself?
ἢ ὅτι τὸ βαρύτερον ἀδύνατον ἰσοδρομεῖν τῷ κουφοτέρῳ ἀπὸ τῆς αὐτῆς ἰσχύος ῥιφθέν;
Is it because the heavier part cannot run at equal speed with the lighter when thrown with the same force?
ἐπεὶ δὲ ἀνάγκη μὲν κινεῖσθαι, ἐξ ἴσου δὲ καὶ ἐπ’ εὐθείας ἀδύνατον, ἀνάγκη εἰς τὸ ἐντὸς φερόμενον κύκλῳ φέρεσθαι· οἷον εἰ ὅλως τι ἦν αὐτοῦ ἀκίνητον διὰ βάρος ἐν μέσῳ, τὸ μὲν πρὸς τῷ ἀφιέντι εἰς τὸ πρόσθεν ἂν ἐκινήθη αὐτοῦ μέρος, τὸ δὲ ὑπ’ ἐκεῖνα πρὸς τὸν ἀφιέντα.
And since it is necessary to move, but impossible to do so equally and in a straight line, it must be carried in a circle, being borne towards the inside; just as if some part of it in the middle were entirely motionless owing to weight, the part of it near the thrower would move forward, and the part below that towards the thrower.
ἐπεὶ δὲ κινεῖται μὲν τὸ πᾶν, ἔχει δὲ ἐν μέσῳ τὸ βάρος φερόμενον, ἀνάγκη ταὐτὸ τοῦτο ποιεῖν.
Since, then, the whole moves, and has the weight carried in the middle, it must do this very same thing.
§16.13Διὰ τί τὰ φερόμενα ὅταν ἀντιπέσῃ, ἀφάλλεται εἰς τοὐναντίον ἢ πέφυκεν φέρεσθαι, καὶ πρὸς ὁμοίας γωνίας;
Why is it that thrown objects, when they strike against something, rebound in the opposite direction to that in which they naturally travel, and at equal angles?
ἢ ὅτι οὐ μόνον ἐκείνην φέρεται τὴν φορὰν ἣν φέρεται κατὰ τὸ οἰκεῖον μέρος, ἀλλὰ καὶ τὴν ὑπὸ τοῦ ἀφιέντος γινομένην;
Is it because they are carried not only with that proper motion which they have in virtue of their own part, but also with that produced by the thrower?
ἡ μὲν οὖν οἰκεία παύεται, ὅταν εἰς τὸν οἰκεῖον ἔλθῃ τόπον (ἅπαν φὰρ ἠρεμεῖ ἐλθὸν εἰς ὃν φέρεται τόπον κατὰ φύσιν), καθ’ ἣν δ’ ἔχει ἀλλοτρίαν, ἀνάγκη ἔτι κινεῖσθαι, οὐκ εἰς τὸ πρόσθεν δὲ διὰ τὸ κωλύεσθαι, ἀλλ’ ἢ εἰς τὸ πλάγιον ἢ εἰς τὸ ὀρθόν.
Now the proper motion ceases when they come to their proper place (for everything rests when it has come to the place to which it is naturally carried), but in virtue of the external motion they have, they must still move, yet not forward because they are blocked, but either obliquely or vertically.
ἅπαντα δὲ ἀποπηδᾷ πρὸς ὁμοίας γωνίας διὰ τὸ φέρεσθαι μὲν ἐνταῦθα οὗ ἡ κίνησις φέρει, ἣν ἐποίησεν ὁ ἀφείς· ἐκεῖ δὲ πρὸς ὀξεῖαν ἢ πρὸς ὀρθὴν φέρεσθαι συμβαίνει.
And all things rebound at equal angles because they are carried to where the motion produced by the thrower carries them; and there it happens to be carried at an acute or a right angle.
ἐπεὶ οὖν τὸ ἀντικροῦσαν κωλύει τὴν εἰς εὐθὺ κίνησιν, ὁμοίως κωλύει τὸ φερόμενον καὶ τὴν φορὰν αὐτοῦ.
Since, then, that which strikes against them blocks their straight motion, it likewise blocks the moving object and its motion.
ὥσπερ οὖν ἐν τοῖς κατόπτροις τὸ ἄκρον τῆς εὐθείας οὗ ξυνέπεσεν ἡ ὄψις φαίνεται, καὶ ἐν τοῖς φερομένοις οὕτω τὸ ἐναντίον γίνεται· τοσαύτην γὰρ γωνίαν ἀπέωσται ὅση γίνεται ἡ κατὰ κορυφήν.
Just as, therefore, in mirrors the end of the straight line is seen where the sight falls, so also in thrown objects the opposite occurs; for it is repelled at an angle as large as the vertically opposite angle.
δεῖ γὰρ νοῆσαι μετακινουμένην τὴν γωνίαν καὶ τὴν φοράν.
For one must conceive the angle and the motion as being transposed.
τούτου δὲ γενομένου φανερὸν ὅτι πρὸς ὁμοίας γωνίας ἀνάγκη ἀφάλλεσθαι.
This being so, it is clear that they must rebound at equal angles.