α΄.
Τῶν δεδομένων μεγεθῶν ὁ λόγος ὁ πρὸς ἄλληλα δέδοται.
ἔστω δεδομένα μεγέθη τὰ Α, Β· λέγω, ὅτι τοῦ Α
πρὸς τὸ Β λόγος ἐστὶ δοθείς.
πρὸς τὸ Β λόγος ἐστὶ δοθείς.
ἐπεὶ γὰρ δέδοται τὸ Α, δυνατόν ἐστιν αὐτῷ ἴσον πορίσασθαι. πεπορίσθω καὶ ἔστω τὸ Γ. πάλιν, ἐπεὶ δεδομένον ἐστὶ τὸ Β, δυνατόν ἐστιν αὐτῷ ἴσον πορί- σασθαι. πεπορίσθω καὶ ἔστω τὸ Δ. ἐπεὶ οὗν ἴσον
ἐστὶ τὸ μὲν Α τῷ Γ, τὸ δὲ Β τῷ Δ, ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως τὸ Β πρὸς τὸ Δ· ἐναλλὰξ ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ πρὸς τὸ Δ. τοῦ Α ἄρα πρὸς τὸ Β λόγος ἐστὶ δοθείς· ὁ αὐτὸς γὰρ αὐτῷ πε- πόρισται ὁ τοῦ Γ πρὸς τὸ Δ.
ἐστὶ τὸ μὲν Α τῷ Γ, τὸ δὲ Β τῷ Δ, ἔστιν ἄρα ὡς τὸ Α πρὸς τὸ Γ, οὕτως τὸ Β πρὸς τὸ Δ· ἐναλλὰξ ὡς τὸ Α πρὸς τὸ Β, οὕτως τὸ πρὸς τὸ Δ. τοῦ Α ἄρα πρὸς τὸ Β λόγος ἐστὶ δοθείς· ὁ αὐτὸς γὰρ αὐτῷ πε- πόρισται ὁ τοῦ Γ πρὸς τὸ Δ.
Data. Euclid. ed. Johan Ludvig Heiberg, ed. Heinrich Menge. United States. 2016.
Sponsored by Harvard College Library.