§8.prop.13ἐὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, καὶ πολλαπλασιάσας ἕκαστος ἑαυτὸν ποιῇ τινα, οἱ γενόμενοι ἐξ αὐτῶν ἀνάλογον ἔσονται· καὶ ἐὰν οἱ ἐξ ἀρχῆς τοὺς γενομένους πολλαπλασιάσαντες ποιῶσί τινας, καὶ αὐτοὶ ἀνάλογον ἔσονται.
If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the numbers arisen from them will be in proportion; and, if the original numbers by multiplying the numbers arisen [from them] make some numbers, these also will be in proportion.
ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ Α, Β, Γ, ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Β πρὸς τὸν Γ, καὶ οἱ Α, Β, Γ ἑαυτοὺς μὲν πολλαπλασιάσαντες τοὺς Δ, Ε, Ζ ποιείτωσαν, τοὺς δὲ Δ, Ε, Ζ πολλαπλασιάσαντες τοὺς Η, Θ, Κ ποιείτωσαν· λέγω, ὅτι οἵ τε Δ, Ε, Ζ καὶ οἱ Η, Θ, Κ ἑξῆς ἀνάλογόν εἰσιν.
Let there be as many numbers as we please in continued proportion, Α, Β, Γ, as Α is to Β, so Β to Γ, and let Α, Β, Γ by multiplying themselves make Δ, Ε, Ζ, and by multiplying Δ, Ε, Ζ let them make Η, Θ, Κ; I say that both Δ, Ε, Ζ and Η, Θ, Κ are in continued proportion.
ὁ μὲν γὰρ Α τὸν Β πολλαπλασιάσας τὸν Λ ποιείτω, ἑκάτερος δὲ τῶν Α, Β τὸν Λ πολλαπλασιάσας ἑκάτερον τῶν Μ, Ν ποιείτω.
For let Α by multiplying Β make Λ, and let each of Α, Β by multiplying Λ make each of Μ, Ν.
καὶ πάλιν ὁ μὲν Β τὸν Γ πολλαπλασιάσας τὸν Ξ ποιείτω, ἑκάτερος δὲ τῶν Β, Γ τὸν Ξ πολλαπλασιάσας ἑκάτερον τῶν Ο, Π ποιείτω.
And again let Β by multiplying Γ make Ξ, and let each of Β, Γ by multiplying Ξ make each of Ο, Π.
ὁμοίως δὴ τοῖς ἐπάνω δείξομεν, ὅτι οἱ Δ, Λ, Ε καὶ οἱ Η, Μ, Ν, Θ ἑξῆς εἰσιν ἀνάλογον ἐν τῷ τοῦ Α πρὸς τὸν Β λόγῳ, καὶ ἔτι οἱ Ε, Ξ, Ζ καὶ οἱ Θ, Ο, Π, Κ ἑξῆς εἰσιν ἀνάλογον ἐν τῷ τοῦ Β πρὸς τὸν Γ λόγῳ.
Similarly to what was shown above, we shall prove that Δ, Λ, Ε and Η, Μ, Ν, Θ are in continued proportion in the ratio of Α to Β, and further Ε, Ξ, Ζ and Θ, Ο, Π, Κ are in continued proportion in the ratio of Β to Γ.
καί ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Β πρὸς τὸν Γ· καὶ οἱ Δ, Λ, Ε ἄρα τοῖς Ε, Ξ, Ζ ἐν τῷ αὐτῷ λόγῳ εἰσὶ καὶ ἔτι οἱ Η, Μ, Ν, Θ τοῖς Θ, Ο, Π, Κ. καί ἐστιν ἴσον τὸ μὲν τῶν Δ, Λ, Ε πλῆθος τῷ τῶν Ε, Ξ, Ζ πλήθει, τὸ δὲ τῶν Η, Μ, Ν, Θ τῷ τῶν Θ, Ο, Π, Κ·
And as Α is to Β, so is Β to Γ; therefore Δ, Λ, Ε are in the same ratio with Ε, Ξ, Ζ, and further Η, Μ, Ν, Θ with Θ, Ο, Π, Κ.
διʼ ἴσου ἄρα ἐστὶν ὡς μὲν ὁ Δ πρὸς τὸν Ε, οὕτως ὁ Ε πρὸς τὸν Ζ, ὡς δὲ ὁ Η πρὸς τὸν Θ, οὕτως ὁ Θ πρὸς τὸν Κ· ὅπερ ἔδει δεῖξαι.
And the multitude of Δ, Λ, Ε is equal to the multitude of Ε, Ξ, Ζ, and that of Η, Μ, Ν, Θ to that of Θ, Ο, Π, Κ; therefore, ex aequali, as Δ is to Ε, so is Ε to Ζ, and as Η is to Θ, so is Θ to Κ; which was to be proved.
§8.prop.14ἐὰν τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει.
If a square number measure a square number, the side will also measure the side; and, if the side measure the side, the square will also measure the square.
ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ α, Β, πλευραὶ δὲ αὐτῶν ἔστωσαν οἱ Γ, Δ, ὁ δὲ Α τὸν Β μετρείτω· λέγω, ὅτι καὶ ὁ Γ τὸν Δ μετρεῖ.
Let Α, Β be square numbers, and let their sides be Γ, Δ, and let Α measure Β; I say that Γ also measures Δ.
ὁ Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω· οἱ Α, Ε, Β ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ.
For let Γ by multiplying Δ make Ε; therefore Α, Ε, Β are in continued proportion in the ratio of Γ to Δ.
καὶ ἐπεὶ οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ ὁ Α τὸν Β, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε. καί ἐστιν ὡς ὁ Α πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν Δ· μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ.
πάλιν δὴ ὁ Γ τὸν Δ μετρείτω· λέγω, ὅτι καὶ ὁ Α τὸν Β μετρεῖ.
And since Α, Ε, Β are in continued proportion, and Α measures Β, therefore Α also measures Ε. And as Α is to Ε, so is Γ to Δ; therefore Γ also measures Δ. Next, let Γ measure Δ; I say that Α also measures Β.
τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, ὅτι οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ.
For with the same construction, we shall similarly prove that Α, Ε, Β are in continued proportion in the ratio of Γ to Δ.
καὶ ἐπεί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Ε, μετρεῖ δὲ ὁ Γ τὸν Δ, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε. καί εἰσιν οἱ Α, Ε, Β ἑξῆς ἀνάλογον· μετρεῖ ἄρα καὶ ὁ Α τὸν Β.
ἐὰν ἄρα τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ τὴν πλευρὰν μετρήσει· καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει· ὅπερ ἔδει δεῖξαι.
And since as Γ is to Δ, so is Α to Ε, and Γ measures Δ, therefore Α also measures Ε. And Α, Ε, Β are in continued proportion; therefore Α also measures Β. Therefore, if a square number measure a square number, the side will also measure the side; and, if the side measure the side, the square will also measure the square; which was to be proved.