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Number theory: Book 9 Proposition 21


Ἐὰν ἄρτιοι ἀριθμοὶ ὁποσοιοῦν συντεθῶσιν, ὁ ὅλος ἄρτιός ἐστιν. Συγκείσθωσαν γὰρ ἄρτιοι ἀριθμοὶ ὁποσοιοῦν οἱ ΑΒ, ΒΓ, ΓΔ, ΔΕ: λέγω, ὅτι ὅλος ὁ ΑΕ ἄρτιός ἐστιν. Ἐπεὶ γὰρ ἕκαστος τῶν ΑΒ, ΒΓ, ΓΔ, ΔΕ ἄρτιός ἐστιν, ἔχει μέρος ἥμισυ: ὥστε καὶ ὅλος ὁ ΑΕ ἔχει μέρος ἥμισυ. ἄρτιος δὲ ἀριθμός ἐστιν ὁ δίχα διαιρούμενος: ἄρτιος ἄρα ἐστὶν ὁ ΑΕ: ὅπερ ἔδει δεῖξαι.

If as many even numbers as we please be added together, the whole is even. For let as many even numbers as we please, AB, BC, CD, DE, be added together; I say that the whole AE is even. For, since each of the numbers AB, BC, CD, DE is even, it has a half part; [VII. Def. 6] so that the whole AE also has a half part.