§15.7Διὰ τί τῆς σελήνης σφαιροειδοῦς οὔσης εὐθεῖαν ὁρῶμεν, ὅταν ᾖ διχότομος;
Why, although the moon is spherical, do we see a straight line when it is halved?
ἢ ὅτι ἐν τῷ αὐτῷ ἐπιπέδῳ ἡ ὄψις γίνεται καὶ ἡ τοῦ κύκλου περιφέρεια, ἣν ὁ ἥλιος ποιεῖ προσβάλλων τῇ σελήνῃ;
Is it because the vision and the circumference of the circle, which the sun makes by falling upon the moon, come to be in the same plane?
ὅτε δὲ τοῦτο γένοιτο, εὐθεῖα γραμμὴ ἐφαίνετο ὁ ἥλιος.
And when this happened, the sun appeared as a straight line.
ἐπεὶ γὰρ ἀνάγκη τὸ προσβάλλον τὰς ὄψεις πρὸς τὴν σφαῖραν κύκλον ὁρᾶν, ἡ δὲ σελήνη σφαιροειδής, καὶ ὁ ἥλιος ὁρᾷ αὐτήν, κύκλος ἂν εἴη ὁ ὑπὸ τοῦ ἡλίου γινόμενος.
For since it is necessary that that which projects its vision toward a sphere sees a circle, and the moon is spherical, and the sun sees it, the boundary produced by the sun would be a circle.
οὗτος οὖν ὅταν μὲν ἐξ ἐναντίας ἡμῖν γένηται, ὅλος φαίνεται καὶ δοκεῖ πανσέληνος εἶναι· ὅταν δὲ παραλλάττῃ διὰ τὴν τοῦ ἡλίου μετάβασιν, ἡ περιφέρεια αὐτοῦ κατὰ τὴν ὄψιν γίνεται, ὥστε εὐθεῖα φαίνεται.
Therefore, when this is directly opposite to us, the whole of it is seen and it seems to be a full moon; but when it shifts owing to the movement of the sun, its circumference comes to be in the line of vision, so that it appears straight.
τὸ δὲ ἕτερον μέρος περιφερές, ὅτι ἐξ ἐναντίας κεῖται τῇ ὄψει ἡμισφαίριον.
But the other part is curved, because the hemisphere lies directly opposite to the vision.
τὸ δὲ τοιοῦτο ἐφαίνετο ἡμικύκλιον.
And such a shape appeared as a semicircle.
ἀεὶ γὰρ ἡ σελήνη κατ’ ἀντικρύ ἐστιν τῆς ὄψεως.
For the moon is always directly opposite to our vision.
ἀλλ’ ὅταν ὁ ἥλιος ἐπιβάλλῃ, οὐχ ὁρῶμεν, καὶ ἀναπληροῦται μετὰ τὴν ὀγδόην ἐκ τοῦ μέσου, ὅτι ἐπιπαρεξιὼν ὁ ἥλιος ἐκκλινέστερον ἡμῖν ποιεῖ τὸν κύκλον.
But when the sun falls upon it, we do not see it, and after the eighth day it is filled up from the center, because the sun, passing along the side, makes the circle more inclined toward us.
οὕτω δὲ τιθέμενος πρὸς τὴν ὄψιν ὁ κύκλος κώνου τομῇ ἐμφερὴς ἐγένετο.
And the circle, being thus placed in relation to the vision, became similar to the section of a cone.
μηνοειδὴς δὲ φαίνεται, ὅταν ὁ ἥλιος μεταβῇ.
And it appears crescent-shaped when the sun moves away.
ὅταν γὰρ κατὰ τὰ ἔσχατα σημεῖα, καθ’ ἃ διχότομος φαίνεται, ὁ κύκλος ὁ τοῦ ἡλίου γένηται, περιφέρεια φαίνεται ἡ τοῦ κύκλου.
For when the circle of the sun comes to be at the extreme points, at which it appears halved, the circumference of the circle is seen.
οὐ γὰρ ἔτι κατ’ εὐθεῖάν ἐστι τῇ ὄψει, ἀλλὰ παραλλάττει.
For it is no longer in a straight line with the vision, but shifts.
τούτου δὲ γινομένου, καὶ διὰ τῶν αὐτῶν σημείων τοῦ κύκλου ὄντος, ἀνάγκη μηνοειδῆ φαίνεσθαι.
And when this happens, and since the circle passes through the same points, it is necessary for it to appear crescent-shaped.
μέρος γάρ τι τοῦ κύκλου κατὰ τὴν ὄψιν εὐθύς ἐστιν, τοῦ προτέρου ἐξ ἐναντίας ὄντος ὥστε τοῦ λαμπροῦ ἀποτέμνεται· εἶθ’ οὕτως καὶ τὰ ἄκρα μένουσιν ἐν τῷ αὐτῷ, ὥστε ἀνάγκη μηνοειδῆ φαίνεσθαι.
For some part of the circle is straight in relation to the vision, while the former part is directly opposite, so that it is cut off from the bright part; and then, in this way, the extremities also remain in the same place, so that it is necessary for it to appear crescent-shaped.
μᾶλλον δὲ καὶ ἧττον διὰ τὴν τοῦ ἡλίου κίνησιν.
And it does so more or less owing to the motion of the sun.
μεταβαίνοντος γὰρ τοῦ ἡλίου καὶ ὁ κύκλος ὃν ὁρᾷ ἐπιστρέφεται, ἐν τοῖς αὐτοῖς σημείοις ὤν· ἀπείρους γὰρ ἐγκλίσεις ἐγχωρεῖ αὐτὸν κλιθῆναι, εἴπερ γραφῆναι τοὺς μεγίστους κύκλους διὰ τῶν αὐτῶν σημείων ἀπείρους ἐνδέχεται.
For as the sun moves, the circle which it sees is also rotated, while remaining in the same points; for it is possible for it to be inclined at infinite inclinations, if indeed it is possible for infinite greatest circles to be described through the same points.
§15.8Διὰ τί ὁ ἥλιος καὶ ἡ σελήνη σφαιροειδῆ ὄντα ἐπίπεδα φαίνεται;
Why do the sun and the moon, being spherical, appear flat?
ἢ ὅτι πάντων ὅσων τὸ ἀπόστημα ἄδηλον, ὅτε πλεῖον ἢ ἔλαττον ἀφέστηκεν, ἐξ ἴσου φαίνονται;
Is it because all things whose distance is unclear, whether they are removed more or less, appear equidistant?
ὥστε καὶ ἐφ’ ἑνὸς μὲν μόρια δ’ ἔχοντος, ἂν μὴ τῇ χρόᾳ διαφέρῃ, ἀνάγκη τά μόρια ἐξ ἴσου φαίνεσθαι, τὸ δ’ ἐξ ἴσου ὁμαλὲς καὶ ἐπί- §15.9Διὰ τί τὰς σκιὰς ποιεῖ ὁ ἥλιος ἀνίσχων καὶ δύνων μακράς, αἰρόμενος δὲ ἐλάττους, ἐπὶ τῆς μεσημβρίας δ’ ἐλαχίστας;
So that even in the case of a single object having parts, if it does not differ in color, it is necessary for its parts to appear equidistant, and that which is equidistant is smooth and flat- Why does the sun make shadows long when rising and setting, but shorter when rising higher, and shortest at midday?
ἢ ὅτι ἀνίσχων τὸ μὲν πρῶτον παράλληλον ποιήσει τὴν σκιὰν τῇ γῇ, καὶ ἄπειρον ὡς ἄνισον ὑπερτείνει, ἔπειτα μακράν· ἀεὶ δ’ ἐλάττω διὰ τὸ ἀεὶ τὴν ἀπὸ τοῦ ἀνωτέρου σημείου εὐθεῖαν ἐντὸς πίπτειν.
Is it because when rising it will at first make the shadow parallel to the earth, and extends it to infinity as being unequal, and then makes it long; but always smaller because the straight line from the higher point always falls inside?
γνώμων τὸ Α Β, ἥλιος οὗ τὸ Γ καὶ οὗ τὸ Δ·
Let the gnomon be A B, and let the sun be at Γ and at Δ; and the ray from Γ, on which is Γ Ζ, will be further outward than Γ Ε.
ἡ δὲ ἀπὸ τοῦ Γ ἀκτίς, ἐφ’ ἧς τὸ Γ Ζ, ἐξωτέρω ἔσται τῆς Γ Ε. ἔστι δὲ σκιὰ ἡ μὲν Β Ε ἀνωτέρω ὄντος τοῦ ἡλίου, ἡ δὲ Β Ζ κατωτάτω· ἐλαχίστη δέ, ὅσῳ ἀνωτάτω, ᾗ καὶ ὑπὲρ τῆς κεφαλῆς.
And the shadow is B E when the sun is higher, and B Z when it is lowest; and it is shortest when the sun is highest, in which case it is also directly overhead.