15 γ΄. Έστωσαν ἴσαι εὐθεῖαι αἱ Α Β, μείζων δὲ ἡ Γ∠ τῆς Ν ~~ἐλάσσων οὖσα ἑκατέρας τῶν Α Β~~, καὶ πεποιήσθω ὡς μὲν ἡ Α πρὸς τὴν Γ∠, ἡ Γ∠ πρὸς τὴν ΕΖ, καὶ ἡ ΕΖ πρὸς τὴν ΗΘ, ὡς δὲ ἡ Β πρὸς τὴν Ν, ἡ Ν πρὸς τὴν Π καὶ ἡ Π πρὸς τὴν Ρ· λέγω ὅτι ἡ Ρ ἐλάσσον ἐστὶν τῆς ΗΘ.

Ἐπεὶ γὰρ μείζων ἐστὶν ἡ Γ∠ τῆς Ν, κείσθω τῇ Ν ἴση ἡ ΓΚ ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν ΓΚ, ἡ Β πρὸς τὴν Ν. καὶ ἐπεὶ ὡς ἡ Α πρὸς τὴν Γ∠, ἡ Γ∠ πρὸς ΕΖ, γεγενήσθω ὡς ἡ Α πρὸς τὴν ΓΚ οὕτως ἡ ΓΚ πρὸς ΕΛ. ἔστι δὲ καὶ ὡς ἡ Β πρὸς Ν, ἡ Ν πρὸς Π, καὶ ἴση ἐστὶν ἡ μὲν Α τῇ Β, ἡ δὲ ΓΚ τῇ Ν· διʼ ἴσου ἄρα καὶ ὡς ἡ Α πρὸς τὴν ΕΛ, ἡ Β πρὸς τὴν Π· ἴση ἄρα ἡ ΕΛ τῇ Π. διὰ τὰ αὐτὰ δή ἐπεί ἐστιν ὡς ἡ Α πρὸς τὴν Γ∠, οὕτως ἡ ΕΖ πρὸς ΗΘ, ἔσται ἄρα καὶ ὡς ἡ Α πρὸς τὴν ΓΚ, ἡ ΓΚ πρὸς τὴν ΕΛ, καὶ ἡ ΕΛ πρὸς ἐλάσσονα τῆς ΗΘ. ἔστω πρὸς τὴν ΗΜ· ἐπεὶ οὖν ἐστιν ὡς μὲν ἡ ΓΚ πρὸς τὴν ΕΛ, ἡ ΕΛ πρὸς τὴν ΗΜ, ὡς δὲ ἡ Ν πρὸς τὴν Π, οὕτως ἡ Π πρὸς τὴν Ρ, καὶ ἴση ἐστὶν ἡ ΓΚ τῇ Ν, ἡ δὲ ΕΛ τῇ Π, ἴση ἄρα καὶ ἡ Ρ τῇ ΗΜ· ἐλάσσων ἄρα ἡ P τῆς ΗΘ.