Book IX, Proposition 5

If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.

Ἐὰν κύβος ἀριθμὸς ἀριθμόν τινα πολλαπλασιάσας κύβον ποιῇ, καὶ ὁ πολλαπλασιασθεὶς κύβος ἔσται. Κύβος γὰρ ἀριθμὸς ὁ Α ἀριθμόν τινα τὸν Β πολλαπλασιάσας κύβον τὸν Γ ποιείτω: λέγω, ὅτι ὁ Β κύβος ἐστίν. Ὁ γὰρ Α ἑαυτὸν πολλαπλασιάσας τὸν Δ ποιείτω: κύβος ἄρα ἐστίν ὁ Δ. καὶ ἐπεὶ ὁ Α ἑαυτὸν μὲν πολλαπλασιάσας τὸν Δ πεποίηκεν, τὸν δὲ Β πολλαπλασιάσας τὸν Γ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, ὁ Δ πρὸς τὸν Γ. καὶ ἐπεὶ οἱ Δ, Γ κύβοι εἰσίν, ὅμοιοι στερεοί εἰσιν. τῶν Δ, Γ ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί. καί ἐστιν ὡς ὁ Δ πρὸς τὸν Γ, οὕτως ὁ Α πρὸς τὸν Β: καὶ τῶν Α, Β ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί. καί ἐστι κύβος ὁ Α: κύβος ἄρα ἐστὶ καὶ ὁ Β: ὅπερ ἔδει δεῖξαι.

If a cube number by multiplying any number make a cube number, the multiplied number will also be cube. For let the cube number A by multiplying any number B make the cube number C; I say that B is cube. For let A by multiplying itself make D; therefore D is cube. [IX. 3] Now, since A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17] And since D, C are cube, they are similar solid numbers. Therefore two mean proportional numbers fall between D, C. [VIII. 19] And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also. [VIII. 8]

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