§6## Ϛ΄.
## 6.
Ἐν τοῖς κοίλοις ἐνόπτροις ἐὰν ἀνὰ μέσον τοῦ κέντρου καὶ τῆς περιφερείας θῇς τὸ ὄμμα, ὁτὲ μὲν συμπεσοῦνται αἱ ὄψεις ἀνακλώμεναι, ὁτὲ δὲ οὐ συμπεσοῦνται.
In concave mirrors, if you place the eye between the center and the circumference, the reflected visual rays will sometimes meet, and sometimes they will not meet.
ἔστω ἔνοπτρον κοῖλον τὸ ΑΓ, κέντρον δὲ αὐτοῦ τὸ ∠, ὄμμα δὲ κείσθω τὸ Β μεταξὺ τοῦ κέντρου καὶ τῆς περιφερείας, ὄψεις δὲ αἱ ΒΑ, ΒΓ ἀνακλώμεναι ἐπὶ τὰ Η, Ζ, καὶ ἐκβεβλήσθωσαν αἱ ὄψεις ἕως τοῦ ἐνόπτρου αἱ ΑΘ, ΓΚ. ἡ ΑΘ δὴ τῆς ΓΚΘ ἢ μείζων ἐστὶν ἢ ἴση ἢ ἐλάσσων.
Let the concave mirror be AC, its center D, and let the eye B be placed between the center and the circumference, and let the reflected visual rays be BA, BC towards H, Z, and let the visual rays be extended to the mirror as ATH, GK. Indeed, ATH is either greater than, equal to, or less than GK.
εἰ μὲν οὖν ἴση ἐστὶν ἡ ΑΘ ὄψις τῇ ΓΚ ὄψει, ἴση ἐστὶ καὶ ἡ ΑΓΘ περιφέρεια τῇ ΓΘΚ περιφερείᾳ.
If, therefore, the visual ray ATH is equal to the visual ray GK, the circumference ACTH is also equal to the circumference CGTK.
ὥστε καὶ ἡ Μ γωνία τῇ Ξ· αἱ γὰρ τῶν ἴσων περιφερειῶν γωνίαι ἴσαι εἰσὶν ἀλλήλαις.
So that the angle M is also equal to X; for the angles of equal circumferences are equal to one another.
καὶ αἱ Μ, Λ γωνίαι ἄρα ταῖς Ν, Ξ εἰσιν ἴσαι διὰ τὴν ἀνάκλασιν.
Therefore, the angles M, L are also equal to N, X because of the reflection.
καὶ λοιπὴ ἄρα ἡ Ο τῇ Π ἴση ἐστίν.
Therefore, the remaining O is also equal to P.
μείζων ἄρα ἡ Ρ τῆς Ο. ἐπεὶ γὰρ ἡ Ρ γωνία τῆς Π μείζων ἐστὶ διὰ τὸ ἐκτὸς εἶναι, ἡ δὲ Π τῇ Ο ἴση, καὶ ἡ Ρ ἄρα τῆς Ο μείζων ἐστίν.
Therefore, R is greater than O. For since angle R is greater than P because of its being external, and P is equal to O, then R is also greater than O.
κοινὴ προσκείσθω ἡ ὑπὸ ΟΡΖ. συμπεσοῦνται ἄρα αἱ ΓΖ, ΑΗ ὡς ἐπὶ τὰ Η, Ζ. τὸ δʼ αὐτὸ ἔσται, κἂν μείζων ἡ ΑΘ ὄψις τῆς ΓΚ· μείζονες γὰρ ἔσονται αἱ Λ, Μ γωνίαι τῶν Ν, Ξ, ἡ δὲ Π τῆς Ο μείζων ἔσται καὶ ἡ Ρ τῆς Ο. ἐὰν δὲ ἡ ΑΘ εὐθεῖα ἐλάσσων τῆς ΓΚ, διὰ τὰ αὐτὰ μείζων ἔσται ἡ Ο γωνία τῆς Π. ἔστι δὲ καὶ ἡ τῆς Π μείζων.
Let the common angle ORZ be added. Therefore, GZ, AH will meet towards H, Z. And the same will hold even if the visual ray ATH is greater than GK; for the angles L, M will be greater than N, X, and P will be greater than O, and R will be greater than O. But if the straight line ATH is less than GK, for the same reasons angle O will be greater than P. And [angle R] is also greater than P.
οὐδὲν ἄρα κωλύει ἴσην εἶναι τὴν Ρ τῇ Ο ἢ ἐλάσσονα τῆς Ο, καὶ μὴ συμπίπτειν τὴν ΑΗ τῇ ΓΖ. φανερὸν δέ, ὅτι, κἄν τε μείζων ᾖ ἡ ΑΘ περιφέρεια τῆς ΓΚ, ἐάν τε ἴση, ἡ σύμπτωσις τῶν ἀνακλάσεων οὔτε ἐπὶ τῆς περιφερείας τοῦ κύκλου οὔτε ἐκτὸς οὐ μὴ γίνηται, ἀλλʼ ἐντὸς μόνον.
Therefore, nothing prevents R from being equal to O or less than O, and AH from not meeting GZ. And it is clear that, whether the circumference ATH is greater than GK or equal, the meeting of the reflections will by no means occur either on the circumference of the circle or outside, but only inside.
§7## ζ΄.
## 7.
Τὰ ὕψη καὶ τὰ βάθη ἀπὸ τῶν ἐπιπέδων ἐνόπτρων ἀνεστραμμένα φαίνεται.
Heights and depths appear inverted by plane mirrors.
ἔστω ὕψος μὲν τὸ ΑΕ, ἔνοπτρον δὲ ἐπίπεδον τὸ ΑΛ, ὄμμα δὲ τὸ Β, ὄψεις δὲ αἱ ΒΓ, Β∠ ἀνακλώμεναι ἐπὶ τὰ Ε, Κ. οὐκοῦν φαίνεται ἐκβληθεισῶν τῶν ὄψεων ἐπʼ εὐθείας τὸ μὲν Ε τὸ ἄνω ἐπὶ τοῦ Θ κάτω ὄντος, τὸ δὲ Κ κάτω ὄν ἐπὶ τοῦ Ζ τοῦ ἄνω ὄντος.
Let the height be AE, the plane mirror AL, the eye B, and the reflected visual rays BC, BD towards E, K. Therefore, when the visual rays are extended in a straight line, E, which is above, appears on T, which is below, and K, which is below, on Z, which is above.
ὥστε ἀνεστραμμένα ἐστὶ τῇ φαντασίᾳ.
So that they are inverted in the visual image.
ἔστω πάλιν βάθος μὲν τὸ ΕΑ, ἔνοπτρον δὲ ἐπίπεδον τὸ ΑΓ, ὄμμα δὲ τὸ ∠, ὄψεις δὲ αἱ ∠Γ, ∠Β ἀνακλώμεναι ἐπὶ τὰ Ε, Ζ. ὁμοίως τῶν ὄψεων ἐκβληθεισῶν ἐπὶ τὰ Θ, Κ φανεῖται τὸ μὲν Ε κάτω ὄν ἐπὶ τοῦ Θ ἄνω ὄντος, τὸ δὲ Ζ ἄνω ὄν ἐπὶ τοῦ Κ κάτω ὄντος.
Let again the depth be EA, the plane mirror AC, the eye D, and the reflected visual rays DC, DB towards E, Z. Similarly, when the visual rays are extended to T, K, E, which is below, will appear on T, which is above, and Z, which is above, on K, which is below.