Book VIII, Proposition 15

If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.

Ἐὰν κύβος ἀριθμὸς κύβον ἀριθμὸν μετρῇ, καὶ ἡ πλευρὰ τὴν πλευρὰν μετρήσει: καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ ὁ κύβος τὸν κύβον μετρήσει. Κύβος γὰρ ἀριθμὸς ὁ Α κύβον τὸν Β μετρείτω, καὶ τοῦ μὲν Α πλευρὰ ἔστω ὁ Γ, τοῦ δὲ Β ὁ Δ: λέγω, ὅτι ὁ Γ τὸν Δ μετρεῖ. Ὁ Γ γὰρ ἑαυτὸν πολλαπλασιάσας τὸν Ε ποιείτω, ὁ δὲ Δ ἑαυτὸν πολλαπλασιάσας τὸν Η ποιείτω, καὶ ἔτι ὁ Γ τὸν Δ πολλαπλασιάσας τὸν Ζ [ ποιείτω ], ἑκάτερος δὲ τῶν Γ, Δ τὸν Ζ πολλαπλασιάσας ἑκάτερον τῶν Θ, Κ ποιείτω. φανερὸν δή, ὅτι οἱ Ε, Ζ, Η καὶ οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ. καὶ ἐπεὶ οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ ὁ Α τὸν Β, μετρεῖ ἄρα καὶ τὸν Θ. καί ἐστιν ὡς ὁ Α πρὸς τὸν Θ, οὕτως ὁ Γ πρὸς τὸν Δ: μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ. Ἀλλὰ δὴ μετρείτω ὁ Γ τὸν Δ: λέγω, ὅτι καὶ ὁ Α τὸν Β μετρήσει. Τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δὴ δείξομεν, ὅτι οἱ Α, Θ, Κ, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ. καὶ ἐπεὶ ὁ Γ τὸν Δ μετρεῖ, καί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Θ, καὶ ὁ Α ἄρα τὸν Θ μετρεῖ: ὥστε καὶ τὸν Β μετρεῖ ὁ Α: ὅπερ ἔδει δεῖξαι.

If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube. For let the cube number A measure the cube B, and let C be the side of A and D of B; I say that C measures D. For let C by multiplying itself make E, and let D by multiplying itself make G; further, let C by multiplying D make F, and let C, D by multiplying F make H, K respectively. Now it is manifest that E, F, G and A, H, K, B are continuously proportional in the ratio of C to D. [VIII. 11, 12] And, since A, H, K, B are continuously proportional, and A measures B, therefore it also measures H. [VIII. 7] And, as A is to H, so is C to D; therefore C also measures D. [VII. Def. 20] Next, let C measure D; I say that A will also measure B. For, with the same construction, we can prove in a similar manner that A, H, K, B are continuously proportional in the ratio of C to D.

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