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Number theory: Book 9 Proposition 7


Ἐὰν σύνθετος ἀριθμὸς ἀριθμόν τινα πολλαπλασιάσας ποιῇ τινα, ὁ γενόμενος στερεὸς ἔσται. Σύνθετος γὰρ ἀριθμὸς ὁ Α ἀριθμόν τινα τὸν Β πολλαπλασιάσας τὸν Γ ποιείτω: λέγω, ὅτι ὁ Γ στερεός ἐστιν. Ἐπεὶ γὰρ ὁ Α σύνθετός ἐστιν, ὑπὸ ἀριθμοῦ τινος μετρηθήσεται. μετρείσθω ὑπὸ τοῦ Δ, καὶ ὁσάκις ὁ Δ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ε. ἐπεὶ οὖν ὁ Δ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, ὁ Ε ἄρα τὸν Δ πολλαπλασιάσας τὸν Α πεποίηκεν. καὶ ἐπεὶ ὁ Α τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ὁ δὲ Α ἐστιν ὁ ἐκ τῶν Δ, Ε, ὁ ἄρα ἐκ τῶν Δ, Ε τὸν Β πολλαπλασιάσας τὸν Γ πεποίηκεν. ὁ Γ ἄρα στερεός ἐστιν, πλευραὶ δὲ αὐτοῦ εἰσιν οἱ Δ, Ε, Β: ὅπερ ἔδει δεῖξαι.

If a composite number by multiplying any number make some number, the product will be solid. For let the composite number A by multiplying any number B make C; I say that C is solid. For, since A is composite, it will be measured by some number. [VII. Def. 13] Let it be measured by D; and, as many times as D measures A, so many units let there be in E. Since then D measures A according to the units in E, therefore E by multiplying D has made A. [VII. Def. 15] And, since A by multiplying B has made C, and A is the product of D, E, therefore the product of D, E by multiplying B has made C.