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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.

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