Book IX, Proposition 33

If a number have its half odd, it is even-times odd only.

 Ἐὰν ἀριθμὸς τὸν ἥμισυν ἔχῃ περισσόν, ἀρτιάκις περισσός ἐστι μόνον. Ἀριθμὸς γὰρ ὁ Α τὸν ἥμισυν ἐχέτω περισσόν: λέγω, ὅτι ὁ Α ἀρτιάκις περισσός ἐστι μόνον. Ὅτι μὲν οὖν ἀρτιάκις περισσός ἐστιν, φανερόν: ὁ γὰρ ἥμισυς αὐτοῦ περισσὸς ὢν μετρεῖ αὐτὸν ἀρτιάκις. λέγω δή, ὅτι καὶ μόνον. εἰ γὰρ ἔσται ὁ Α καὶ ἀρτιάκις ἄρτιος, μετρηθήσεται ὑπὸ ἀρτίου κατὰ ἄρτιον ἀριθμόν: ὥστε καὶ ὁ ἥμισυς αὐτοῦ μετρηθήσεται ὑπὸ ἀρτίου ἀριθμοῦ περισσὸς ὤν: ὅπερ ἐστὶν ἄτοπον. ὁ Α ἄρα ἀρτιάκις περισσός ἐστι μόνον: ὅπερ ἔδει δεῖξαι. If a number have its half odd, it is even-times odd only. For let the number A have its half odd; I say that A is even-times odd only. Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times. [VII. Def. 9] I say next that it is also even-times odd only. For, if A is even-times even also, it will be measured by an even number according to an even number; [VII. Def. 8] so that the half of it will also be measured by an even number though it is odd: which is absurd.