CatoptricsΚατοπτρικά
Hero of Alexandria Catoptrics PDF
The Catoptrics is an ancient Greek treatise on optics and mirrors, traditionally ascribed to the mathematician and engineer Hero of Alexandria, who lived in the 1st century CE. Written in Koine Greek, the work systematically applies geometric principles to analyze the behavior of light and the phenomenon of reflection. Its investigations include the reasons why mirrors reverse images laterally but not vertically, along with detailed examinations of the properties of plane, convex, and concave mirrors. The surviving text, however, is widely regarded by modern scholarship as a later compilation, likely from the 6th century CE, which may preserve some authentic Heroian material within a substantially revised framework. This composite nature has generated considerable debate regarding its precise authorship. Structured as a formal treatise, though the extant version appears fragmentary, it was composed for an audience of mathematicians, engineers, and educated patrons, characteristically blending theoretical geometry with practical application. Despite uncertainties surrounding its provenance, the Catoptrics proved to be a work of enduring influence. It was transmitted through Byzantine and Arabic scholarly traditions and engaged by later optical theorists such as Alhazen, securing its place as a significant text in the historical development of optics.
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| (50) | Ἐπειδὴ γὰρ τοῦτο ὡμολογημένον ἐστὶ παρὰ πᾶσιν, ὅτι οὐδὲν μάτην ἐργάζεται ἡ φύσις οὐδὲ ματαιοπονεῖ, ἐὰν μὴ δώσωμεν πρὸς ἴσας γωνίας γίνεσθαι τὴν ἀνάκλασιν, πρὸς ἀνίσους ματαιοπονεῖ ἡ φύσις, καὶ ἀντὶ τοῦ διὰ βραχείας περιόδου φθάσαι τὸ ὁρώμενον τὴν ὄψιν, διὰ μακρᾶς περιόδου τοῦτο φανήσεται καταλαμβάνουσα. εὑρεθήσονται γὰρ αἱ τὰς ἀνίσους γωνίας περιέχουσαι εὐθεῖαι, αἵτινες ἀπὸ τῆς ὄψεως [περιέχουσαι] †φερομένας πρὸς τὸ κάτοπτρον κἀκεῖθεν πρὸς τὸ ὁρώμενον, μείζονες οὖσαι τῶν τὰς ἴσας γωνίας περιεχουσῶν εὐθειῶν. καὶ ὅτι τοῦτο ἀληθές, δῆλον ἐντεῦθεν. Ὑποκείσθω γὰρ τὸ κάτοπτρον εὐθεῖά τις ἡ ΑΒ, καὶ ἔστω τὸ μὲν ὁρῶν Γ, τὸ δ’ ὁρώμενον τὸ Δ, τὸ δὲ Ε σημεῖον τοῦ κατόπτρου, ἐν ᾧ προσπίπτουσα ἡ ὄψις ἀνακλᾶται πρὸς τὸ ὁρώμενον, ἔστω, καὶ ἐπεζεύχθω ἡ ΓΕ, ΕΔ. λέγω ὅτι ἡ ὑπὸ ΑΕΓ γωνία ἴση ἐστὶ τῇ ὑπὸ ΔΕΒ. εἰ γὰρ μὴ ἔστιν ἴση, ἔστω ἕτερον σημεῖον τοῦ κατόπτρου, ἐν ᾧ προσπίπτουσα ἡ ὄψις πρὸς ἀνίσους γωνίας ἀνακλᾶται, τὸ Ζ, καὶ ἐπεζεύχθω ἡ ΓΖ, ΖΔ. δῆλον ὅτι ἡ ὑπὸ ΓΖΑ γωνία μείζων ἐστὶ τῆς ὑπὸ ΔΖΕ γωνίας. λέγω ὅτι αἱ ΓΖ, ΖΔ εὐθεῖαι, αἵτινες τὰς ἀνίσους γωνίας περιέχουσιν ὑποκειμένης τῆς ΑΒ εὐθείας, μείζονές εἰσι τῶν ΓΕ, ΕΔ εὐθειῶν, αἵτινες τὰς ἴσας γωνίας περιέχουσι μετὰ τῆς ΑΒ. ἤχθω γὰρ κάθετος ἀπὸ τοῦ Δ ἐπὶ τὴν ΑΒ κατὰ τὸ Η σημεῖον καὶ ἐκβεβλήσθω ἐπ’ εὐθείας ὡς ἐπὶ τὸ Θ. φανερὸν δὴ ὅτι αἱ πρὸς τῷ Η γωνίαι ἴσαι εἰσίν. ὀρθαὶ γάρ εἰσι. καὶ ἔστω ἡ ΔΗ τῇ ΗΘ ἴση, καὶ ἐπεζεύχθω ἡ ΘΖ καὶ ἡ ΘΕ. αὕτη μὲν ἡ κατασκευή. ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΗ τῇ ΗΘ, ἀλλὰ καὶ ἡ ὑπὸ ΔΗΕ γωνία τῇ ὑπὸ ΘΗΕ γωνίᾳ ἴση ἐστί, κοινὴ δὲ πλευρὰ τῶν δύο τριγώνων ἡ ΗΕ, καὶ βάσις ἡ ΘΕ βάσει τῇ ΕΔ ἴση ἐστί, καὶ τὸ ΗΘΕ τρίγωνον τῷ ΔΗΕ τριγώνῳ ἴσον ἐστί, καὶ 〈αἱ〉 λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις εἰσὶν ἴσαι, ὑφ’ ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν. ἴση ἄρα ἡ ΘΕ τῇ ΕΔ. πάλιν ἐπειδὴ τῇ ΗΘ ἴση ἐστὶν ἡ ΗΔ καὶ γωνία ἡ ὑπὸ ΔΗΖ γωνίᾳ τῇ ὑπὸ ΘΗΖ ἴση ἐστί, κοινὴ δὲ ἡ ΗΖ τῶν δύο τριγώνων τῶν ΔΗΖ καὶ ΘΗΖ, καὶ βάσις ἄρα ἡ ΘΖ βάσει τῇ ΖΔ ἴση ἐστί, καὶ τὸ ΖΗΔ τρίγωνον τῷ ΘΗΖ τριγώνῳ ἴσον ἐστίν. ἴση ἄρα ἐστὶν ἡ ΘΖ τῇ ΖΔ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΘΕ τῇ ΕΔ, κοινὴ προσκείσθω ἡ ΕΓ. δύο ἄρα αἱ ΓΕ, ΕΔ δυσὶ ταῖς ΓΕ, ΕΘ ἴσαι εἰσίν. ὅλη ἄρα ἡ ΓΘ δυσὶ ταῖς ΓΕ, ΕΔ ἴση ἐστί. καὶ ἐπεὶ παντὸς τριγώνου αἱ δύο πλευραὶ τῆς λοιπῆς μείζονές εἰσι πάντῃ μεταλαμβανόμεναι, τριγώνου ἄρα τοῦ ΘΖΓ αἱ δύο πλευραὶ αἱ ΘΖ, ΖΓ μιᾶς τῆς ΓΘ μείζονές εἰσιν. ἀλλ’ ἡ ΓΘ ἴση ἐστὶ ταῖς ΓΕ, ΕΔ. αἱ ΘΖ, ΖΓ ἄρα μείζονές εἰσι τῶν ΓΕ, ΕΔ. ἀλλ’ ἡ ΘΖ τῇ ΖΔ ἐστὶν ἴση. αἱ ΖΓ, ΖΔ ἄρα τῶν ΓΕ, ΕΔ μείζονές εἰσι. καί εἰσιν αἱ ΓΖ, ΖΔ αἱ τὰς ἀνίσους γωνίας περιέχουσαι· αἱ ἄρα τὰς ἀνίσους γωνίας περιέ χουσαι μείζονές εἰσι τῶν τὰς ἴσας γωνίας περιεχουσῶν. |