Classification of incommensurables: Book 10 Proposition 19
Translations
Τὸ ὑπὸ ῥητῶν μήκει συμμέτρων κατά τινα τῶν προειρημένων τρόπων εὐθειῶν περιεχόμενον ὀρθογώνιον ῥητόν ἐστιν. Ὑπὸ γὰρ ῥητῶν μήκει συμμέτρων εὐθειῶν τῶν ΑΒ, ΒΓ ὀρθογώνιον περιεχέσθω τὸ ΑΓ: λέγω, ὅτι ῥητόν ἐστι τὸ ΑΓ. Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔ: ῥητὸν ἄρα ἐστὶ τὸ ΑΔ. καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, ἴση δέ ἐστιν ἡ ΑΒ τῇ ΒΔ, σύμμετρος ἄρα ἐστὶν ἡ ΒΔ τῇ ΒΓ μήκει. καί ἐστιν ὡς ἡ ΒΔ πρὸς τὴν ΒΓ, οὕτως τὸ ΔΑ πρὸς τὸ ΑΓ. σύμμετρον ἄρα ἐστὶ τὸ ΔΑ τῷ ΑΓ. ῥητὸν δὲ τὸ ΔΑ: ῥητὸν ἄρα ἐστὶ καὶ τὸ ΑΓ. Τὸ ἄρα ὑπὸ ῥητῶν μήκει συμμέτρων, καὶ τὰ ἑξῆς.
The rectangle contained by rational straight lines commensurable in length is rational. For let the rectangle AC be contained by the rational straight lines AB, BC commensurable in length; I say that AC is rational. For on AB let the square AD be described; therefore AD is rational. [X. Def. 4] And, since AB is commensurable in length with BC, while AB is equal to BD, therefore BD is commensurable in length with BC. And, as BD is to BC, so is DA to AC. [VI. 1] Therefore DA is commensurable with AC. [X. 11] But DA is rational; therefore AC is also rational. [X. Def. 4]