§8.prop.19#1δύο ὁμοίων στερεῶν ἀριθμῶν δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί· καὶ ὁ στερεὸς πρὸς τὸν ὅμοιον στερεὸν τριπλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος πλευρὰ πρὸς τὴν ὁμόλογον πλευράν.
Between two similar solid numbers there fall two mean proportional numbers; and the solid number has to the similar solid number a ratio triplicate of that which the corresponding side has to the corresponding side.
ἔστωσαν δύο ὅμοιοι στερεοὶ οἱ Α, Β, καὶ τοῦ μὲν Α πλευραὶ ἔστωσαν οἱ Γ, Δ, Ε, τοῦ δὲ Β οἱ Ζ, Η, Θ. καὶ ἐπεὶ ὅμοιοι στερεοί εἰσιν οἱ ἀνάλογον ἔχοντες τὰς πλευράς, ἔστιν ἄρα ὡς μὲν ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ζ πρὸς τὸν Η, ὡς δὲ ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Θ. λέγω, ὅτι τῶν Α, Β δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί, καὶ ὁ Α πρὸς τὸν Β τριπλασίονα λόγον ἔχει ἤπερ ὁ Γ πρὸς τὸν Ζ καὶ ὁ Δ πρὸς τὸν Η καὶ ἔτι ὁ Ε πρὸς τὸν Θ.
ὁ Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Κ ποιείτω, ὁ δὲ Ζ τὸν Η πολλαπλασιάσας τὸν Λ ποιείτω.
¦5 Let Α, Β be two similar solid numbers, and let Γ, Δ, Ε be the sides of Α, and Ζ, Η, Θ those of Β. And since similar solid numbers are those which have their sides proportional, therefore, as Γ is to Δ, so is Ζ to Η, and as Δ is to Ε, so is Η to Θ. I say that between Α, Β there fall two mean proportional numbers, and Α has to Β a ratio triplicate of that which Γ has to Ζ, and Δ to Η, and further Ε to Θ. For let Γ by multiplying Δ make Κ, and let Ζ by multiplying Η make Λ.
καὶ ἐπεὶ οἱ Γ, Δ τοῖς Ζ, Η ἐν τῷ αὐτῷ λόγῳ εἰσίν, καὶ ἐκ μὲν τῶν Γ, Δ ἐστιν ὁ Κ, ἐκ δὲ τῶν Ζ, Η ὁ Λ, οἱ Κ, Λ ὅμοιοι ἐπίπεδοί εἰσιν ἀριθμοί· τῶν Κ, Λ ἄρα εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός.
And since Γ, Δ are in the same ratio with Ζ, Η, and Κ is the product of Γ, Δ, and Λ the product of Ζ, Η, therefore Κ, Λ are similar plane numbers; therefore between Κ, Λ there is one mean proportional number.
ἔστω ὁ Μ. ὁ Μ ἄρα ἐστὶν ὁ ἐκ τῶν Δ, Ζ, ὡς ἐν τῷ πρὸ τούτου θεωρήματι ἐδείχθη.
Let it be Μ. Therefore Μ is the product of Δ, Ζ, as was proved in the theorem preceding this.
καὶ ἐπεὶ ὁ Δ τὸν μὲν Γ πολλαπλασιάσας τὸν Κ πεποίηκεν, τὸν δὲ Ζ πολλαπλασιάσας τὸν Μ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Ζ, οὕτως ὁ Κ πρὸς τὸν Μ. ἀλλʼ ὡς ὁ Κ πρὸς τὸν Μ, ὁ Μ πρὸς τὸν Λ. οἱ Κ, Μ, Λ ἄρα ἑξῆς εἰσιν ἀνάλογον ἐν τῷ τοῦ Γ πρὸς τὸν Ζ λόγῳ.
And since Δ by multiplying Γ has made Κ, and by multiplying Ζ has made Μ, therefore, as Γ is to Ζ, so is Κ to Μ. But, as Κ is to Μ, so is Μ to Λ; therefore Κ, Μ, Λ are in continued proportion in the ratio of Γ to Ζ.
καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ζ πρὸς τὸν Η, ἐναλλὰξ ἄρα ἐστὶν ὡς ὁ Γ πρὸς τὸν Ζ, οὕτως ὁ Δ πρὸς τὸν Η. διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Δ πρὸς τὸν Η, οὕτως ὁ Ε πρὸς τὸν Θ. οἱ Κ, Μ, Λ ἄρα ἑξῆς εἰσιν ἀνάλογον ἔν τε τῷ τοῦ Γ πρὸς τὸν Ζ λόγῳ καὶ τῷ τοῦ Δ πρὸς τὸν Η καὶ ἔτι τῷ τοῦ Ε πρὸς τὸν Θ. ἑκάτερος δὴ τῶν Ε, Θ τὸν Μ πολλαπλασιάσας ἑκάτερον τῶν Ν, Ξ ποιείτω.
And since, as Γ is to Δ, so is Ζ to Η, therefore, alternately, as Γ is to Ζ, so is Δ to Η. For the same reason also, as Δ is to Η, so is Ε to Θ. Therefore Κ, Μ, Λ are in continued proportion both in the ratio of Γ to Ζ, and in that of Δ to Η, and further in that of Ε to Θ. Now let each of the numbers Ε, Θ by multiplying Μ make each of the numbers Ν, Ξ.
καὶ ἐπεὶ στερεός ἐστιν ὁ Α, πλευραὶ δὲ αὐτοῦ εἰσιν οἱ Γ, Δ, Ε, ὁ Ε ἄρα τὸν ἐκ τῶν Γ, Δ πολλαπλασιάσας τὸν Α πεποίηκεν.
And since Α is a solid number, and Γ, Δ, Ε are its sides, therefore Ε by multiplying the product of Γ, Δ has made Α.
ὁ δὲ ἐκ τῶν Γ, Δ ἐστιν ὁ Κ· ὁ Ε ἄρα τὸν Κ πολλαπλασιάσας τὸν Α πεποίηκεν.
But the product of Γ, Δ is Κ; therefore Ε by multiplying Κ has made Α.
διὰ τὰ αὐτὰ δὴ καὶ ὁ Θ τὸν Λ πολλαπλασιάσας τὸν Β πεποίηκεν.
For the same reason also Θ by multiplying Λ has made Β.