Continued proportions in number theory: Book 8 Proposition 17
Translations
Ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μὴ μετρῇ, οὐδὲ ἡ πλευρὰ τὴν πλευρὰν μετρήσει: κἂν ἡ πλευρὰ τὴν πλευρὰν μὴ μετρῇ, οὐδὲ ὁ κύβος τὸν κύβον μετρήσει. Κύβος γὰρ ἀριθμὸς ὁ Α κύβον ἀριθμὸν τὸν Β μὴ μετρείτω, καὶ τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ: λέγω, ὅτι ὁ Γ τὸν Δ οὐ μετρήσει. Εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, καὶ ὁ Α τὸν Β μετρήσει. οὐ μετρεῖ δὲ ὁ Α τὸν Β: οὐδ' ἄρα ὁ Γ τὸν Δ μετρεῖ. Ἀλλὰ δὴ μὴ μετρείτω ὁ Γ τὸν Δ: λέγω, ὅτι οὐδὲ ὁ Α τὸν Β μετρήσει. Εἰ γὰρ ὁ Α τὸν Β μετρεῖ, καὶ ὁ Γ τὸν Δ μετρήσει. οὐ μετρεῖ δὲ ὁ Γ τὸν Δ: οὐδ' ἄρα ὁ Α τὸν Β μετρήσει: ὅπερ ἔδει δεῖξαι.
If a cube number do not measure a cube number, neither will the side measure the side; and, if the side do not measure the side, neither will the cube measure the cube. For let the cube number A not measure the cube number B, and let C be the side of A, and D of B; I say that C will not measure D. For if C measures D, A will also measure B. [VIII. 15] But A does not measure B; therefore neither does C measure D. Again, let C not measure D; I say that neither will A measure B. For, if A measures B, C will also measure D. [VIII. 15]