Book IX, Proposition 24

If from an even number an even number be subtracted, the remainder will be even.

Ἐὰν ἀπὸ ἀρτίου ἀριθμοῦ ἄρτιος ἀφαιρεθῇ, ὁ λοιπὸς ἄρτιος ἔσται. Ἀπὸ γὰρ ἀρτίου τοῦ ΑΒ ἄρτιος ἀφῃρήσθω ὁ ΒΓ: λέγω, ὅτι ὁ λοιπὸς ὁ ΓΑ ἄρτιός ἐστιν. Ἐπεὶ γὰρ ὁ ΑΒ ἄρτιός ἐστιν, ἔχει μέρος ἥμισυ. διὰ τὰ αὐτὰ δὴ καὶ ὁ ΒΓ ἔχει μέρος ἥμισυ: ὥστε καὶ λοιπὸς [ ὁ ΓΑ ἔχει μέρος ἥμισυ ] ἄρτιος [ ἄρα ] ἐστὶν ὁ ΑΓ: ὅπερ ἔδει δεῖξαι.

If from an even number an even number be subtracted, the remainder will be even. For from the even number AB let the even number BC be subtracted: I say that the remainder CA is even. For, since AB is even, it has a half part. [VII. Def. 6]

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