§8.prop.4#2μὴ μετρείτω δὴ ὁ Ε τὸν Κ. καὶ εἰλήφθω ὑπὸ τῶν Ε, Κ ἐλάχιστος μετρούμενος ἀριθμὸς ὁ Μ. καὶ ὁσάκις μὲν ὁ Κ τὸν Μ μετρεῖ, τοσαυτάκις καὶ ἑκάτερος τῶν Θ, Η ἑκάτερον τῶν Ν, Ξ μετρείτω, ὁσάκις δὲ ὁ Ε τὸν Μ μετρεῖ, τοσαυτάκις καὶ ὁ Ζ τὸν Ο μετρείτω.
Indeed, let Ε not measure Κ. And let there be taken the least number Μ measured by Ε, Κ. And as many times as Κ measures Μ, so many times let each of Θ, Η measure each of Ν, Ξ; and as many times as Ε measures Μ, so many times let Ζ also measure Ο.
ἐπεὶ ἰσάκις ὁ Θ τὸν Ν μετρεῖ καὶ ὁ Η τὸν Ξ, ἔστιν ἄρα ὡς ὁ Θ πρὸς τὸν Η, οὕτως ὁ Ν πρὸς τὸν Ξ. ὡς δὲ ὁ Θ πρὸς τὸν Η, οὕτως ὁ Α πρὸς τὸν Β· καὶ ὡς ἄρα ὁ Α πρὸς τὸν Β, οὕτως ὁ Ν πρὸς τὸν ξ.
Since Θ measures Ν as many times as Η measures Ξ, therefore, as Θ is to Η, so is Ν to Ξ. But as Θ is to Η, so is Α to Β; therefore also, as Α is to Β, so is Ν to Ξ.
διὰ τὰ αὐτὰ δὴ καὶ ὡς ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Ξ πρὸς τὸν Μ. πάλιν, ἐπεὶ ἰσάκις ὁ Ε τὸν Μ μετρεῖ καὶ ὁ Ζ τὸν Ο, ἔστιν ἄρα ὡς ὁ Ε πρὸς τὸν Ζ, οὕτως ὁ Μ πρὸς τὸν Ο· οἱ Ν, Ξ, Μ, Ο ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τοῖς τοῦ τε Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι τοῦ Ε πρὸς τὸν Ζ λόγοις.
For the same reasons indeed, as Γ is to Δ, so is Ξ to Μ. Again, since Ε measures Μ as many times as Ζ measures Ο, therefore, as Ε is to Ζ, so is Μ to Ο; therefore Ν, Ξ, Μ, Ο are in continued proportion in the ratios of Α to Β, of Γ to Δ, and further of Ε to Ζ.
λέγω δή, ὅτι καὶ ἐλάχιστοι ἐν τοῖς ΑΒ, ΓΔ, ΕΖ λόγοις.
I say indeed that they are also the least in the ratios of ΑΒ, ΓΔ, ΕΖ.
εἰ γὰρ μή, ἔσονταί τινες τῶν Ν, Ξ, Μ, Ο ἐλάσσονες ἀριθμοὶ ἑξῆς ἀνάλογον ἐν τοῖς ΑΒ, ΓΔ, ΕΖ λόγοις.
For if not, there will be some numbers less than Ν, Ξ, Μ, Ο in continued proportion in the ratios of ΑΒ, ΓΔ, ΕΖ.
ἔστωσαν οἱ Π, Ρ, Σ, Τ. καὶ ἐπεί ἐστιν ὡς ὁ Π πρὸς τὸν Ρ, οὕτως ὁ Α πρὸς τὸν Β, οἱ δὲ Α, Β ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς ἰσάκις ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον, ὁ Β ἄρα τὸν Ρ μετρεῖ.
Let them be Π, Ρ, Σ, Τ. And since as Π is to Ρ, so is Α to Β, and Α, Β are the least, and the least numbers measure those which have the same ratio as them an equal number of times, the antecedent the antecedent and the consequent the consequent, therefore Β measures Ρ.
διὰ τὰ αὐτὰ δὴ καὶ ὁ Γ τὸν Ρ μετρεῖ· οἱ Β, Γ ἄρα τὸν Ρ μετροῦσιν.
For the same reasons indeed, Γ also measures Ρ; therefore Β, Γ measure Ρ.
καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Β, Γ μετρούμενος τὸν Ρ μετρήσει.
Therefore the least number measured by Β, Γ will also measure Ρ.
ἐλάχιστος δὲ ὑπὸ τῶν Β, Γ μετρούμενός ἐστιν ὁ Η· ὁ Η ἄρα τὸν Ρ μετρεῖ.
But the least number measured by Β, Γ is Η; therefore Η measures Ρ.
καί ἐστιν ὡς ὁ Η πρὸς τὸν Ρ, οὕτως ὁ Κ πρὸς τὸν Σ· καὶ ὁ Κ ἄρα τὸν Σ μετρεῖ.
And as Η is to Ρ, so is Κ to Σ; therefore Κ also measures Σ.
μετρεῖ δὲ καὶ ὁ Ε τὸν Σ· οἱ Ε, Κ ἄρα τὸν Σ μετροῦσιν.
But Ε also measures Σ; therefore Ε, Κ measure Σ.
καὶ ὁ ἐλάχιστος ἄρα ὑπὸ τῶν Ε, Κ μετρούμενος τὸν Σ μετρήσει.
Therefore the least number measured by Ε, Κ will also measure Σ.
ἐλάχιστος δὲ ὑπὸ τῶν Ε, Κ μετρούμενός ἐστιν ὁ Μ· ὁ Μ ἄρα τὸν Σ μετρεῖ ὁ μείζων τὸν ἐλάσσονα· ὅπερ ἐστὶν ἀδύνατον.
But the least number measured by Ε, Κ is Μ; therefore Μ measures Σ, the greater the less; which is impossible.
οὐκ ἄρα ἔσονταί τινες τῶν Ν, Ξ, Μ, Ο ἐλάσσονες ἀριθμοὶ ἑξῆς ἀνάλογον ἔν τε τοῖς τοῦ Α πρὸς τὸν Β καὶ τοῦ Γ πρὸς τὸν Δ καὶ ἔτι τοῦ Ε πρὸς τὸν Ζ λόγοις· οἱ Ν, Ξ, Μ, Ο ἄρα ἑξῆς ἀνάλογον ἐλάχιστοί εἰσιν ἐν τοῖς ΑΒ, ΓΔ, ΕΖ λόγοις· ὅπερ ἔδει δεῖξαι.
Therefore there will not be any numbers less than Ν, Ξ, Μ, Ο in continued proportion in the ratios of Α to Β, of Γ to Δ, and further of Ε to Ζ; therefore Ν, Ξ, Μ, Ο are the least in continued proportion in the ratios of ΑΒ, ΓΔ, ΕΖ; which was to be proved.