Number theory: Book 9 Proposition 30
Translations
Ἐὰν περισσὸς ἀριθμὸς ἄρτιον ἀριθμὸν μετρῇ, καὶ τὸν ἥμισυν αὐτοῦ μετρήσει. Περισσὸς γὰρ ἀριθμὸς ὁ Α ἄρτιον τὸν Β μετρείτω: λέγω, ὅτι καὶ τὸν ἥμισυν αὐτοῦ μετρήσει. Ἐπεὶ γὰρ ὁ Α τὸν Β μετρεῖ, μετρείτω αὐτὸν κατὰ τὸν Γ: λέγω, ὅτι ὁ Γ οὐκ ἔστι περισσός. εἰ γὰρ δυνατόν, ἔστω. καὶ ἐπεὶ ὁ Α τὸν Β μετρεῖ κατὰ τὸν Γ, ὁ Α ἄρα τὸν Γ πολλαπλασιάσας τὸν Β πεποίηκεν. ὁ Β ἄρα σύγκειται ἐκ περισσῶν ἀριθμῶν, ὧν τὸ πλῆθος περισσόν ἐστιν. ὁ Β ἄρα περισσός ἐστιν: ὅπερ ἄτοπον: ὑπόκειται γὰρ ἄρτιος. οὐκ ἄρα ὁ Γ περισσός ἐστιν: ἄρτιος ἄρα ἐστὶν ὁ Γ. ὥστε ὁ Α τὸν Β μετρεῖ ἀρτιάκις. διὰ δὴ τοῦτο καὶ τὸν ἥμισυν αὐτοῦ μετρήσει: ὅπερ ἔδει δεῖξαι.
If an odd number measure an even number, it will also measure the half of it. For let the odd number A measure the even number B; I say that it will also measure the half of it. For, since A measures B, let it measure it according to C; I say that C is not odd. For, if possible, let it be so. Then, since A measures B according to C, therefore A by multiplying C has made B. Therefore B is made up of odd numbers the multitude of which is odd. Therefore B is odd: [IX. 23] which is absurd, for by hypothesis it is even. Therefore C is not odd; therefore C is even. Thus A measures B an even number of times.