## Book IX, Proposition 30

If an odd number measure an even number, it will also measure the half of it.

 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.