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Continued proportions in number theory: Book 8 Proposition 16


Ἐὰν τετράγωνος ἀριθμὸς τετράγωνον ἀριθμὸν μὴ μετρῇ, οὐδὲ ἡ πλευρὰ τὴν πλευρὰν μετρήσει: κἂν ἡ πλευρὰ τὴν πλευρὰν μὴ μετρῇ, οὐδὲ ὁ τετράγωνος τὸν τετράγωνον μετρήσει. εστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, πλευραὶ δὲ αὐτῶν ἔστωσαν οἱ Γ, Δ, καὶ μὴ μετρείτω ὁ Α τὸν Β: λέγω, ὅτι οὐδὲ ὁ Γ τὸν Δ μετρεῖ. Εἰ γὰρ μετρεῖ ὁ Γ τὸν Δ, μετρήσει καὶ ὁ Α τὸν Β. οὐ μετρεῖ δὲ ὁ Α τὸν Β: οὐδὲ ἄρα ὁ Γ τὸν Δ μετρήσει. Μὴ μετρείτω [δὴ] πάλιν ὁ Γ τὸν Δ: λέγω, ὅτι οὐδὲ ὁ Α τὸν Β μετρήσει. Εἰ γὰρ μετρεῖ ὁ Α τὸν Β, μετρήσει καὶ ὁ Γ τὸν Δ. οὐ μετρεῖ δὲ ὁ Γ τὸν Δ: οὐδ' ἄρα ὁ Α τὸν Β μετρήσει: ὅπερ ἔδει δεῖξαι.

If a square number do not measure a square number, neither will the side measure the side; and, if the side do not measure the side, neither will the square measure the square. Let A, B be square numbers, and let C, D be their sides; and let A not measure B; I say that neither does C measure D. For, if C measures D, A will also measure B. [VIII. 14] But A does not measure B; therefore neither will 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. 14]