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Number theory: Book 9 Proposition 23


Ἐὰν περισσοὶ ἀριθμοὶ ὁποσοιοῦν συντεθῶσιν, τὸ δὲ πλῆθος αὐτῶν περισσὸν ᾖ, καὶ ὁ ὅλος περισσὸς ἔσται. Συγκείσθωσαν γὰρ ὁποσοιοῦν περισσοὶ ἀριθμοί, ὧν τὸ πλῆθος περισσὸν ἔστω, οἱ ΑΒ, ΒΓ, ΓΔ: λέγω, ὅτι καὶ ὅλος ὁ ΑΔ περισσός ἐστιν. Ἀφῃρήσθω ἀπὸ τοῦ ΓΔ μονὰς ἡ ΔΕ: λοιπὸς ἄρα ὁ ΓΕ ἄρτιός ἐστιν. ἔστι δὲ καὶ ὁ ΓΑ ἄρτιος: καὶ ὅλος ἄρα ὁ ΑΕ ἄρτιός ἐστιν. καί ἐστι μονὰς ἡ ΔΕ. περισσὸς ἄρα ἐστὶν ὁ ΑΔ: ὅπερ ἔδει δεῖξαι.

If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd. For let as many odd numbers as we please, AB, BC, CD, the multitude of which is odd, be added together; I say that the whole AD is also odd. Let the unit DE be subtracted from CD; therefore the remainder CE is even. [VII. Def. 7] But CA is also even; [IX. 22] therefore the whole AE is also even. [IX. 21] And DE is an unit.