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Book 9 Proposition 29



Ἐὰν περισσὸς ἀριθμὸς περισσὸν ἀριθμὸν πολλαπλασιάσας ποιῇ τινα, ὁ γενόμενος περισσὸς ἔσται. Περισσὸς γὰρ ἀριθμὸς ὁ Α περισσὸν τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω: λέγω, ὅτι ὁ Γ περισσός ἐστιν. Ἐπεὶ γὰρ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ Γ ἄρα σύγκειται ἐκ τοσούτων ἴσων τῷ Β, ὅσαι εἰσὶν ἐν τῷ Α μονάδες. καί ἐστιν ἑκάτερος τῶν Α, Β περισσός: ὁ Γ ἄρα σύγκειται ἐκ περισσῶν ἀριθμῶν, ὧν τὸ πλῆθος περισσόν ἐστιν. ὥστε ὁ Γ περισσός ἐστιν: ὅπερ ἔδει δεῖξαι.

If an odd number by multiplying an odd number make some number, the product will be odd. For let the odd number A by multiplying the odd number B make C; I say that C is odd. For, since A by multiplying B has made C, therefore C is made up of as many numbers equal to B as there are units in A. [VII. Def. 15] And each of the numbers A, B is odd; therefore C is made up of odd numbers the multitude of which is odd.