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.