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ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, δὲ πρῶτος τὸν δεύτερον μὴ μετρῇ, οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει.

ἔστωσαν ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ,
5δ, Ε, δὲ Α τὸν Β μὴ μετρείτω: λέγω, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει.

ὅτι μὲν οὖν οἱ Α, Β, Γ, Δ, Ε ἑξῆς ἀλλήλους οὐ μετροῦσιν,
10φανερόν: οὐδὲ γὰρ Α τὸν Β μετρεῖ. λέγω δή, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει. εἰ γὰρ δυνατόν, μετρείτω Α τὸν Γ. καὶ ὅσοι εἰσὶν οἱ Α, Β, Γ, τοσοῦτοι εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ
15τῶν 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,
15Γ, καί ἐστιν ἴσον τὸ πλῆθος τῶν Α, Β, Γ τῷ πλήθει τῶν Ζ, Η, Θ, δι᾽ ἴσου ἄρα ἐστὶν ὡς Α πρὸς τὸν Γ, οὕτως Ζ πρὸς τὸν Θ. καὶ ἐπεί ἐστιν ὡς Α πρὸς τὸν Β, οὕτως Ζ
20πρὸς τὸν Η, οὐ μετρεῖ δὲ Α τὸν Β, οὐ μετρεῖ ἄρα οὐδὲ Ζ τὸν Η: οὐκ ἄρα μονάς ἐστιν Ζ: γὰρ μονὰς πάντα ἀριθμὸν μετρεῖ. καί εἰσιν οἱ Ζ, Θ πρῶτοι πρὸς ἀλλήλους οὐδὲ Ζ ἄρα τὸν Θ μετρεῖ. καί ἐστιν ὡς Ζ πρὸς τὸν Θ, οὕτως Α πρὸς τὸν Γ: οὐδὲ Α ἄρα τὸν Γ μετρεῖ. ὁμοίως δὴ
25δείξομεν, ὅτι οὐδὲ ἄλλος οὐδεὶς οὐδένα μετρήσει: ὅπερ ἔδει δεῖξαι.

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load focus English (Thomas L. Heath, Sir Thomas Little Heath, 1956)
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