§8.prop.5οἱ ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσι τὸν συγκείμενον ἐκ τῶν πλευρῶν.
Plane numbers have to one another the ratio compounded of their sides.
ἔστωσαν ἐπίπεδοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ μὲν Α πλευραὶ ἔστωσαν οἱ Γ, Δ ἀριθμοί, τοῦ δὲ Β οἱ Ε, Ζ· λέγω, ὅτι ὁ Α πρὸς τὸν Β λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν.
Let there be plane numbers Α, Β, and let the sides of Α be the numbers Γ, Δ, and those of Β, Ε, Ζ; I say that Α has to Β the ratio compounded of their sides.
λόγων γὰρ δοθέντων τοῦ τε ὃν ἔχει ὁ Γ πρὸς τὸν Ε καὶ ὁ Δ πρὸς τὸν Ζ εἰλήφθωσαν ἀριθμοὶ ἑξῆς ἐλάχιστοι ἐν τοῖς ΓΕ, ΔΖ λόγοις, οἱ Η, Θ, Κ, ὥστε εἶναι ὡς μὲν τὸν Γ πρὸς τὸν Ε, οὕτως τὸν Η πρὸς τὸν θ, ὡς δὲ τὸν Δ πρὸς τὸν Ζ, οὕτως τὸν Θ πρὸς τὸν Κ. καὶ ὁ Δ τὸν Ε πολλαπλασιάσας τὸν Λ ποιείτω.
For, the ratios being given which Γ has to Ε and Δ to Ζ, let there be taken the least numbers Η, Θ, Κ continuously in the ratios ΓΕ, ΔΖ, so that, as Γ is to Ε, so is Η to Θ, and as Δ is to Ζ, so is Θ to Κ. And let Δ by multiplying Ε make Λ.
καὶ ἐπεὶ ὁ Δ τὸν μὲν Γ πολλαπλασιάσας τὸν Α πεποίηκεν, τὸν δὲ Ε πολλαπλασιάσας τὸν Λ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Α πρὸς τὸν Λ. ὡς δὲ ὁ Γ πρὸς τὸν Ε, οὕτως ὁ Η πρὸς τὸν Θ· καὶ ὡς ἄρα ὁ Η πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Λ. πάλιν, ἐπεὶ ὁ Ε τὸν Δ πολλαπλασιάσας τὸν Λ πεποίηκεν, ἀλλὰ μὴν καὶ τὸν Ζ πολλαπλασιάσας τὸν Β πεποίηκεν, ἔστιν ἄρα ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Λ πρὸς τὸν Β. ἀλλʼ ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Θ πρὸς τὸν Κ· καὶ ὡς ἄρα ὁ Θ πρὸς τὸν Κ, οὕτως ὁ Λ πρὸς τὸν Β. ἐδείχθη δὲ καὶ ὡς ὁ Η πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Λ· διʼ ἴσου ἄρα ἐστὶν ὡς ὁ Η πρὸς τὸν Κ, ὁ Α πρὸς τὸν Β, ὁ δὲ Η πρὸς τὸν Κ λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν·
And since Δ by multiplying Γ has made Α, and by multiplying Ε has made Λ, therefore, as Γ is to Ε, so is Α to Λ. But as Γ is to Ε, so is Η to Θ; therefore also, as Η is to Θ, so is Α to Λ. Again, since Ε by multiplying Δ has made Λ, but indeed also by multiplying Ζ has made Β, therefore, as Δ is to Ζ, so is Λ to Β. But as Δ is to Ζ, so is Θ to Κ; therefore also, as Θ is to Κ, so is Λ to Β. And it was also proved that, as Η is to Θ, so is Α to Λ; therefore, ex aequali, as Η is to Κ, so is Α to Β.
καὶ ὁ Α ἄρα πρὸς τὸν Β λόγον ἔχει τὸν συγκείμενον ἐκ τῶν πλευρῶν· ὅπερ ἔδει δεῖξαι.
But Η has to Κ the ratio compounded of their sides; therefore Α also has to Β the ratio compounded of their sides; which was to be proved.
§8.prop.6ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, ὁ δὲ πρῶτος τὸν δεύτερον μὴ μετρῇ, οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει.
If there be any number of numbers in continued proportion, and the first do not measure the second, neither will any other measure any other.
ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, δ, Ε, ὁ δὲ Α τὸν Β μὴ μετρείτω· λέγω, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει.
Let there be any number of numbers in continued proportion, Α, Β, Γ, Δ, Ε, and let Α not measure Β; I say that neither will any other measure any other.
ὅτι μὲν οὖν οἱ Α, Β, Γ, Δ, Ε ἑξῆς ἀλλήλους οὐ μετροῦσιν, φανερόν· οὐδὲ γὰρ ὁ Α τὸν Β μετρεῖ.
Now that Α, Β, Γ, Δ, Ε do not measure one another consecutively is manifest; for neither does Α measure Β.
λέγω δή, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει.
I say indeed that neither will any other measure any other.
εἰ γὰρ δυνατόν, μετρείτω ὁ Α τὸν Γ. καὶ ὅσοι εἰσὶν οἱ Α, Β, Γ, τοσοῦτοι εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Β, Γ οἱ Ζ, Η, Θ. καὶ ἐπεὶ οἱ Ζ, Η, Θ ἐν τῷ αὐτῷ λόγῳ εἰσὶ τοῖς Α, Β, Γ, καί ἐστιν ἴσον τὸ πλῆθος τῶν Α, Β, Γ τῷ πλήθει τῶν Ζ, Η, Θ, διʼ ἴσου ἄρα ἐστὶν ὡς ὁ Α πρὸς τὸν Γ, οὕτως ὁ Ζ πρὸς τὸν Θ. καὶ ἐπεί ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως ὁ Ζ πρὸς τὸν Η, οὐ μετρεῖ δὲ ὁ Α τὸν Β, οὐ μετρεῖ ἄρα οὐδὲ ὁ Ζ τὸν Η· οὐκ ἄρα μονάς ἐστιν ὁ Ζ· ἡ γὰρ μονὰς πάντα ἀριθμὸν μετρεῖ.
For, if possible, let Α measure Γ. And as many as are Α, Β, Γ, let so many least numbers of those which have the same ratio as Α, Β, Γ be taken, namely Ζ, Η, Θ. And since Ζ, Η, Θ are in the same ratio as Α, Β, Γ, and the multitude of Α, Β, Γ is equal to the multitude of Ζ, Η, Θ, therefore, ex aequali, as Α is to Γ, so is Ζ to Θ. And since as Α is to Β, so is Ζ to Η, and Α does not measure Β, therefore Ζ also does not measure Η; therefore Ζ is not a unit, for the unit measures every number.
καί εἰσιν οἱ Ζ, Θ πρῶτοι πρὸς ἀλλήλους.
And Ζ, Θ are prime to one another.
καί ἐστιν ὡς ὁ Ζ πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Γ· οὐδὲ ὁ Α ἄρα τὸν Γ μετρεῖ.
And as Ζ is to Θ, so is Α to Γ; therefore Α also does not measure Γ.
ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει· ὅπερ ἔδει δεῖξαι.
Similarly indeed we shall prove that neither will any other measure any other; which was to be proved.