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ἐὰν ὦσιν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον, οἱ δὲ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους ὦσιν, οὐκ ἔσται ὡς πρῶτος πρὸς τὸν δεύτερον, οὕτως ἔσχατος πρὸς ἄλλον τινά.
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ἔστωσαν ὁσοιδηποτοῦν ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, Δ, οἱ δὲ ἄκροι αὐτῶν οἱ Α, Δ πρῶτοι πρὸς ἀλλήλους ἔστωσαν: λέγω, ὅτι οὐκ ἔστιν ὡς Α πρὸς τὸν Β, οὕτως Δ πρὸς ἄλλον τινά.

εἰ γὰρ δυνατόν, ἔστω ὡς Α πρὸς τὸν Β, οὕτως Δ
10πρὸς τὸν Ε: ἐναλλὰξ ἄρα ἐστὶν ὡς Α πρὸς τὸν Δ, Β πρὸς τὸν Ε. οἱ δὲ Α, Δ πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι ἀριθμοὶ μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις τε ἡγούμενος τὸν ἡγούμενον καὶ ἑπόμενος τὸν ἑπόμενον. μετρεῖ ἄρα Α τὸν Β. καί ἐστιν
15ὡς Α πρὸς τὸν Β, Β πρὸς τὸν Γ. καὶ Β ἄρα τὸν Γ
15μετρεῖ: ὥστε καὶ Α τὸν Γ μετρεῖ. καὶ ἐπεί ἐστιν ὡς Β πρὸς τὸν Γ, Γ πρὸς τὸν Δ, μετρεῖ δὲ Β τὸν Γ, μετρεῖ ἄρα καὶ Γ τὸν Δ. ἀλλ᾽ Α τὸν Γ ἐμέτρει: ὥστε Α καὶ τὸν Δ μετρεῖ. μετρεῖ δὲ καὶ ἑαυτόν. Α ἄρα τοὺς
20α, Δ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα ἔσται ὡς Α πρὸς τὸν Β, οὕτως Δ πρὸς ἄλλον τινά: ὅπερ ἔδει δεῖξαι.

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