Classification of incommensurables: Book 10 Proposition 24
Translations
Τὸ ὑπὸ μέσων μήκει συμμέτρων εὐθειῶν κατά τινα τῶν εἰρημένων τρόπων περιεχόμενον ὀρθογώνιον μέσον ἐστίν. Ὑπὸ γὰρ μέσων μήκει συμμέτρων εὐθειῶν τῶν ΑΒ, ΒΓ περιεχέσθω ὀρθογώνιον τὸ ΑΓ: λέγω, ὅτι τὸ ΑΓ μέσον ἐστίν. Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔ: μέσον ἄρα ἐστὶ τὸ ΑΔ. καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, ἴση δὲ ἡ ΑΒ τῇ ΒΔ, σύμμετρος ἄρα ἐστὶ καὶ ἡ ΔΒ τῇ ΒΓ μήκει: ὥστε καὶ τὸ ΔΑ τῷ ΑΓ σύμμετρόν ἐστιν. μέσον δὲ τὸ ΔΑ: μέσον ἄρα καὶ τὸ ΑΓ: ὅπερ ἔδει δεῖξαι.
The rectangle contained by medial straight lines commensurable in length is medial. For let the rectangle AC be contained by the medial straight lines AB, BC which are commensurable in length; I say that AC is medial. For on AB let the square AD be described; therefore AD is medial. And, since AB is commensurable in length with BC, while AB is equal to BD, therefore DB is also commensurable in length with BC; so that DA is also commensurable with AC. [VI. 1, X. 11]