§15.5Διὰ τί τοῦ ἡλίου ὁμοτόνως φερομένου ἐν τῷ ἴσῳ χρόνῳ, οὐχ ἡ αὐτὴ αὔξησις καὶ φθίσις τῶν σκιῶν;
Why, although the sun moves uniformly in equal time, is there not the same increase and decrease of shadows?
ἢ ὅτι ἴσαι γίνονται αἱ γωνίαι πρὸς τὰ ὁρώμενα, αἱ ἀπὸ τῶν ἀκτίνων ὑπὸ ταῖς ἴσαις περιφερείαις;
Is it because the angles made towards the objects seen, which are formed by the rays under equal circumferences, become equal?
εἰ δ’ αὗται καὶ ἐμβαλλόμεναι ποιοῦσιν ἀκτῖνας ἐν τῷ τριγώνῳ, ὅπερ ἔχεται ὑπό τε τῆς πρώτης ἀκτῖνος καὶ τοῦ ὁρωμένου καὶ τῆς σκιᾶς.
And if these, being extended, make rays in the triangle which is contained by the first ray, the object seen, and the shadow [what then]?
εἰ δ’ αἱ γωνίαι ἴσαι, ἀνάγκη τὴν πορρωτέρω γραμμὴν τοῦ ὁρωμένου μείζω εἶναι τῆς ἐγγυτέρω· τοῦτο γὰρ ἴσμεν.
If the angles are equal, it is necessary for the line further from the object seen to be greater than the nearer; for this we know.
διῃρήσθω οὖν ἡ περιφέρεια εἰς ἴσα ὁσαοῦν πλήθει, ὁράσθω δὲ τὸ Θ. ὅταν οὖν ὁ ἥλιος ἐπὶ τοῦ Α προσλαβὼν τὸ Θ ποιήσῃ τινὰ σκιὰν ἐν τῷ Θ Α, ἀνάγκη δὴ τὴν ἀκτῖνα ἐπὶ τὸ Α πίπτειν.
Let the circumference then be divided into any number of equal parts, and let Θ be the object seen. When therefore the sun is at A, and taking in Θ, makes a certain shadow on Θ A, it is necessary indeed for the ray to fall upon A.
ὅταν δ’ ἔλθῃ ἐπὶ τὸ Β, ἡ ἀπὸ τοῦ Β ἀκτὶς ἐντὸς τῆς Θ Α πεσεῖται, καὶ ὅταν πάλιν ἐπὶ τὸ Γ μεταβῇ, ὡσαύτως· εἰ δὲ μή, εὐθεῖα εὐθείας διχῇ ἅψεται.
But when it comes to B, the ray from B will fall inside Θ A, and when it moves again to Γ, likewise; otherwise, a straight line will touch another straight line in two points.
ἐπεὶ οὖν ἴση ἡ Α Β τῇ Β Γ, καὶ αἱ γωνίαι αἱ ὑπὸ ταύτης αἱ πρὸς τῷ Δ ἴσαι ἔσονται· πρὸς τῷ κέντρῳ γάρ.
Since therefore A B is equal to B Γ, the angles subtended by them at Δ will also be equal; for they are at the center.
εἰ δὲ τῇ τοῦ Δ, καὶ ἐν τῷ τριγώνῳ· κατὰ κορυφὴν γὰρ ταύταις.
And if those at Δ are equal, so are those in the triangle; for they are vertically opposite to them.
ὥστ’ ἐπεὶ εἰς ἴσα διαιρεῖται ἡ γωνία, μείζων ἔσται ἡ Δ Ε τῆς Ε Ζ τῇ Δ Θ. ὁμοίως δὲ καὶ αἱ ἄλλαι ἂς ποιοῦσιν αἱ ἀπὸ τῆς περιφερείας ἀκτῖνες.
So that, since the angle is divided into equal parts, Δ Ε will be greater than Ε Ζ with respect to Δ Θ. And likewise also the others which the rays from the circumference make.
ἅμα δὲ δῆλον καὶ ὅτι κατὰ μεσημβρίαν ἐλαχίστην ἀναγκαῖον εἶναι τὴν σκιάν, καὶ ὅτι αἱ ἐπιδόσεις τότε ἐλάχσται.
And at the same time it is also clear that at midday the shadow is necessarily shortest, and that the increases then are least.
μάλιστα γὰρ καθ’ ἡμᾶς ὁ ἥλιος τῆς μεσημβρίας ἐστίν, καὶ πνῖγος γίνεται διά τε τὴν εἰρημένην αἰτίαν καὶ ὅτι ἀπνεύματος· ὅταν γὰρ διακρίνῃ τὸν πρὸς τῇ γῇ ἀέρα, πνεῦμα γίνεται.
For the sun is most directly over us at midday, and stifling heat occurs both for the reason mentioned and because there is no wind; for when it dissolves the air near the earth, wind is produced.
εἰ οὖν ἅμα ἐν ἀμφοτέροις τοῖς ἡμισφαιρίοις, εἰκότως ἂν αἱ μέσαι νύκτες καὶ ἡ μεσημβρία ἀπνεύματοι εἶεν.
If therefore this happens at the same time in both hemispheres, it is reasonable that midnight and midday should be windless.
§15.6Διὰ τί ὁ ἥλιος διὰ τῶν τετραπλεύρων διέχων οὐκ εὐθύγραμμα ποιεῖ τὰ σχήματα, ἀλλὰ κύκλους, οἷον ἐν ταῖς ῥίψεσιν;
Why does the sun, when passing through quadrilateral openings, not make rectilinear shapes, but circles, as in the case of wicker-screens?
ἢ ὅτι ἡ τῶν ὄψεων ἔκπτωσις κῶνός ἐστιν, τοῦ δὲ κώνου κύκλος ἡ βάσις, ὥστε πρὸς ὃ ἂν προσπίπτωσιν αἱ τοῦ ἡλίου ἀκτῖνες, κυκλοτερεῖς φαίνονται.
Is it because the emission of vision is a cone, and the base of a cone is a circle, so that whatever the rays of the sun fall upon, they appear circular?
ἀναγκαῖον μὲν γάρ ἐστι καὶ τὸ ὑπὸ τοῦ ἡλίου σχῆμα ὑπ’ εὐθειῶν περιέχεσθαι, εἴπερ αἱ ἀκτῖνες εὐθεῖαι.
For it is necessary that the shape made by the sun also be contained by straight lines, if indeed the rays are straight.
ὅταν γὰρ εὐθεῖαι πρὸς εὐθεῖαν προσπίπτωσιν, εὐθύγραμμον ποιοῦσιν.
For when straight lines fall upon a straight line, they make a rectilinear shape.
ἐπὶ δὲ τῶν ἀκτίνων συμβαίνει τοῦτο· πρὸς εὐθεῖαν γὰρ προσπίπτουσι τὴν τοῦ ῥιπὸς γραμμήν, ᾗ διαλάμπουσι, καὶ αὗται εὐθεῖαί εἰσιν, ὥστε πρὸς εὐθεῖαν ἔσται ἡ ἔκπτωσις.
And in the case of the rays this happens; for they fall upon a straight line, the line of the wicker-screen through which they shine, and they themselves are straight, so that the emission will be towards a straight line.
ἀλλὰ διὰ τὸ ἀσθενεῖς εἶναι τὰς ἀποσχιζομένας ἀπὸ τῶν ὄψεων πρὸς τὰ ἄκρα τῶν εὐθειῶν, οὐχ ὁρᾶται τὰ ἐν ταῖς γωνίαις· ἀλλ’ ὅσον μὲν τῆς εὐθείας ἐνυπάρχει ἐν τῷ κώνῳ, ποιεῖ αὐτήν, τὸ δὲ λοιπὸν οὐ ποιεῖ, ἀλλὰ λανθάνουσιν αἱ ὄψεις ἐπιπίπτουσαι.
But because those branching off from the vision towards the extremities of the straight lines are weak, the things in the corners are not seen; but as much of the straight line as is present within the cone, it makes [visible], while the rest it does not, but the visions falling upon it escape notice.
πολλὰ γὰρ οὐχ ὁρῶμεν ἐφ’ ἃ διικνεῖται ἡ ὄψις, οἷον τὰ ἐν τῷ σκότει.
For we do not see many things which our vision reaches, such as things in the dark.
ὅμοιον δὲ τούτῳ καὶ τὸ τὸ τετράγωνον πολυγωνοειδὲς φαίνεσθαι, ἐὰν δὲ πλέον ἀφιστῇ, κύκλον.
And similar to this is the fact that a square appears polygonal, and if it is removed further, a circle.
ὄντος γὰρ κώνου τῆς τῶν ὄψεων ἐκπτώσεως, ἀφισταμένου τοῦ σχήματος εἰς τὸ πόρρω αἱ μὲν εἰς τὰς γωνίας ἀποσχιζόμεναι τῶν ὄψεων διὰ τὸ ἀσθενεῖς εἶναι καὶ ὀλίγαι οὐχ ὁρῶσι, πλέονος τοῦ ἀποστήματος γινομένου, αἱ δὲ εἰς τὸ μέσον προσπίπτουσαι, ἀθρόαι καὶ ἰσχυραὶ οὖσαι, διαμένουσιν.
For since the emission of vision is a cone, as the shape is removed into the distance, because the distance becomes greater, those of the visions branching off into the corners do not see, owing to being weak and few, while those falling on the center, being dense and strong, persist.
ἐγγὺς μὲν οὖν ὄντος τοῦ σχήματος δύναται καὶ τὰ ἐν ταῖς γωνίαις ὁρᾶν, πλεῖον δὲ αὐτῶν γένομένου ἀδυνατοῦσιν.
So when the shape is near, it is possible to see even the things in the corners, but when the distance becomes greater, they are unable to do so.
διὸ καὶ ἡ περιφερὴς ἀπαγομένη εὐθεῖα φαίνεται.
Which is also why a curved line, when drawn away, appears straight.
καὶ ἡ σελήνη ὑπὸ εὐθειῶν περιέχεσθαι δοκεῖ τῇ ὀγδόῃ, ἐὰν μὴ κατὰ τὸ πλάτος, ἀλλὰ κατὰ τὴν περιέχουσαν γραμμὴν αἱ ὄψεις προσπίπτωσιν.
And the moon seems to be contained by straight lines on the eighth day, if the visions fall not upon its breadth but upon its bounding line.
ἐγγὺς μὲν γὰρ οὔσης τῆς περιφερείας δύνανται διακρίνειν αἱ ὄψεις ὅσῳ ἐγγύτερόν ἐστι θάτερον θατέρου μέρους τῆς περιφερείας· πόρρω δὲ γινομένης οὐ διαισθάνεται, ἀλλὰ δοκεῖ αὐτῇ ἐξ ἴσου εἶναι.
For when the curve is near, the visions are able to distinguish how much nearer one part of the curve is than another; but when it becomes far, it does not perceive it, but it seems to it to be equidistant.
διὸ καὶ εὐθεῖα φαίνεται.
Which is why it also appears straight.