§22## κβ΄.
## 22.
Ἐν τοῖς κυρτοῖς ἐνόπτροις ἀπὸ τῶν ἐλασσόνων ἐνόπτρων ἐλάσσονα φαίνεται τὰ εἴδωλα.
In convex mirrors, from smaller mirrors the images appear smaller.
ἔστω σφαῖρα μείζων μὲν ἡ ΑΓ, ἐλάσσων δὲ ἡ ΕΛ περὶ τὸ αὐτὸ κέντρον τὸ Θ, ὄμμα δὲ τὸ Β, καὶ ἐπεζεύχθω ἡ ΒΑΘ, καὶ ἀπὸ τῆς σφαίρας ἀνακεκλάσθω ὄψις ἡ ΒΓ∠.
Let the greater sphere be AG, and the smaller EL about the same center Th, and the eye B, and let BATh be joined, and from the sphere let the visual ray BGD be reflected.
λέγω, ὅτι ἡ ἀνακλασθησομένη ὄψις ἀπὸ τῆς ἐλάσσονος σφαίρας ἐπὶ τὸ ∠ οὔτε διὰ τοῦ Γ πεσεῖται οὔτε ἐκτὸς τοῦ πιπτέτω γὰρ πρότερον, εἰ δυνατόν, διὰ τοῦ Γ καὶ ἀνακεκλάσθω ἀπὸ τῆς ἐλάσσονος σφαίρας ἐπὶ τὸ ∠ καὶ ἔστω ἡ ΒΕ∠, καὶ ἐπεζεύχθω ἀπὸ τοῦ Θ ἐπὶ τὸ Γ καὶ ἐκβεβλήσθω ἐπὶ τὸ Κ. δίχα δὴ τεμεῖ ἡ ΘΓΚ τὴν ὑπὸ τῶν ΒΓ∠ γωνίαν διὰ τὸ τὴν ΒΓ∠ ἴσας ποιεῖν γωνίας πρὸς τῇ περιφερείᾳ διὰ τὴν ἀνάκλασιν.
I say that the visual ray which is going to be reflected from the smaller sphere to D will fall neither through G nor outside of it. For let it fall first, if possible, through G and let it be reflected from the smaller sphere to D, and let it be BED, and let a line be joined from Th to G and extended to K. Therefore ThGK will bisect the angle BGD because BGD makes equal angles with the circumference because of the reflection.
διὰ τὰ αὐτὰ δὲ καὶ ἡ ἀπὸ τοῦ Θ ἐπὶ τὸ Ε ἐπιζευγνυμένη καὶ ἐκβληθεῖσα δίχα τεμεῖ τὴν ὑπὸ ΒΕ∠.
For the same reasons also the line joined from Th to E and extended will bisect the angle BED.
τεμνέτω καὶ ἔστω ἡ ΘΕΖ. ἐπεὶ μείζων ἐστὶν ἡ περιεχομένη ὑπὸ τῶν ΒΓ∠ τῆς ὑπὸ ΒΕ∠, καὶ ἡ ἡμίσεια τῆς ἡμισείας μείζων ἐστὶν ἡ ὑπὸ ΒΓΚ τῆς ὑπὸ ΒΕΖ. ἔστι δὲ καὶ ἐλάσσων· ὅπερ ἀδύνατον.
Let it bisect it and let it be ThEZ. Since the angle contained by BGD is greater than that by BED, the half is also greater than the half, namely BGK is greater than BEZ. But it is also smaller; which is impossible.
οὐκ ἄρα ἥξει διὰ τοῦ Γ ἡ ἀνακλωμένη ὄψις ἀπὸ τῆς ἐλάσσονος σφαίρας.
Therefore, the reflected visual ray from the smaller sphere will not come through G.
ὑποκείσθω δὲ πάλιν τὰ αὐτά, καὶ ἡ ἀπὸ τῆς ἐλάσσονος σφαίρας ἀνακλωμένη ὄψις ἡ ΒΕ∠ ἐκτὸς πιπτέτω τοῦ Γ, καὶ τεμνέτω ἡ ΒΕ τὴν μείζονα σφαῖραν κατὰ τὸ Ζ. ἡ δὴ ἀπὸ τοῦ Ζ ἀνακλωμένη ὄψις ἡ ΒΖΚ οὐ συμπεσεῖται τῇ Γ∠ τοῦτο γὰρ δέδεικται.
And let the same things be assumed again, and let the visual ray BED reflected from the smaller sphere fall outside of G, and let BE cut the greater sphere at Z. Therefore, the visual ray BZK reflected from Z will not meet GD; for this has been shown.
τῇ ἄρα Ε∠ συμπιπτέτω κατὰ τὸ Κ. ἡ ἄρα ΒΖΚ ὄψις ἀνακλωμένη ἀπὸ τοῦ μείζονος ἐνόπτρου ὁρᾷ τὸ Κ, καὶ ἡ αὐτὴ ἡ ΒΕΚ ἀνακλωμένη ἀπὸ τοῦ ἐλάσσονος ἐνόπτρου ὁρᾷ τὸ αὐτὸ Κ· τοῦτο δὲ ἐπάνω ἐδείχθηἀδύνατον.
Therefore, let it meet ED at K. Therefore, the visual ray BZK reflected from the greater mirror sees K, and the same visual ray BEK reflected from the smaller mirror sees the same K; but this was shown above to be impossible.
μεταξὺ ἄρα πεσεῖται τῶν Γ Α ἡ ἀνακλωμένη ὄψις ἀπὸ τοῦ ἐλάσσονος ἐνόπτρου ἐπὶ τὸ ∠.
Therefore, the reflected visual ray from the smaller mirror to D will fall between G and A.
ὁμοίως δὲ δειχθήσεται καὶ ἡ ἀπὸ τοῦ ἑτέρου μέρους τὸ αὐτὸ ποιοῦσα.
And similarly it will be shown that the ray from the other part does the same.
ὑπὸ ἐλάσσονος ἄρα γωνίας θεωρεῖται τῆς πρὸς τῷ Β γιγνονένης ἀπὸ τοῦ ἐλάσσονος ἐνόπτρου ἤπερ ἀπὸ τοῦ μείζονος.
Therefore, the object is viewed under a smaller angle from the smaller mirror, namely that formed at B, than from the greater.
ἔλασσον ἄρα φαίνεται τὸ εἴδωλον ἀπὸ τοῦ ἐλάσσονος ἐνόπτρου.
Therefore, the image appears smaller from the smaller mirror.
§23## κγ΄.
## 23.
Ἐν τοῖς κυρτοῖς ἐνόπτροις τὰ εἴδωλα κυρτὰ φαόνεται.
In convex mirrors, the images appear convex.
ἔστω κυρτὸν ἔνοπτρον τὸ ΑΓ, ὄμμα δὲ τὸ Ε, ὄψεις δὲ ἀνακλώμεναι αἱ ΕΑ, ΕΓ ἐπὶ τὰ ∠, Β, ἡ δὲ ΖΕ ἀνακλωμένη διʼ ἑαυτῆς ἐπὶ τὸ Ε. οὐκοῦν τῶν ὄψεων μέγισται μέν εἰσι τῷ μήκει αἱ πορρωτάτω, ἐλάχισται δὲ αἱ κατὰ μέσον, ὥσπερ ἡ ΕΖ. φαίνεται ἄρα τοῦ ἐνόπτρου ἔγγιον μᾶλλον τὸ Ε, πορρωτάτω δὲ τὸ Β καὶ τὸ ∠.
Let the convex mirror be AG, the eye E, and the visual rays EA, EG reflected to D, B, and let ZE be reflected on itself to E. Therefore, of the visual rays, those furthest away are greatest in length, and those in the middle, like EZ, are smallest. Therefore, E [the central part of the image] appears closer to E [the eye], and B and D furthest.
ὥστε ὅλον κυρτὸν φαίνεται.
So the whole appears convex.
§24## κδ΄.
## 24.
Ἐν τοῖς κοίλοις ἐνόπτροις ἐὰν ἐπὶ τοῦ κέντρου τὸ ὄμμα τεθῇ, αὐτὸ μόνον φαίνεται τὸ ὄμμα.
In concave mirrors, if the eye is placed at the center, only the eye itself appears.
ἔστω κοῖλον ἔνοπτρον τὸ ΑΓ ∠, κέντρον δὲ αὐτοῦ τὸ Β, ὄψεις δὲ αἱ ΒΑ, ΒΓ, Β∠.
Let the concave mirror be AGD, its center B, and the visual rays BA, BG, BD.
οὐκοῦν ἴση ἡ Ε γωνία τῇ Ζ. ἥξει ἄρα ἀνακλωμένη ἡ ΒΓ ὄψις ἐπὶ τὸ Β. ὁμοίως δὲ καὶ αἱ λοιπαί.
Therefore, the angle E is equal to Z. Therefore, the reflected visual ray BG will come to B. And similarly also the rest.
αὐτὸ μόνον ἄρα ὁρᾶται τὸ Β.
Therefore, only B itself is seen.