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οἱ πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.

ἔστωσαν πρῶτοι πρὸς ἀλλήλους ἀριθμοὶ οἱ Α, Β: λέγω, ὅτι οἱ Α, Β ἐλάχιστοί εἰσι
5τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς.

εἰ γὰρ μή, ἔσονταί τινες τῶν Α, Β ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β. ἔστωσαν οἱ Γ, Δ.

ἐπεὶ οὖν οἱ ἐλάχιστοι ἀριθμοὶ τῶν τὸν
10αὐτὸν λόγον ἐχόντων 2αὐτοῖς2 μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ἰσάκις τε μείζων τὸν μείζονα καὶ ἐλάττων τὸν ἐλάττονα, τουτέστιν τε ἡγούμενος τὸν ἡγούμενον καὶ ἑπόμενος τὸν ἑπόμενον, ἰσάκις ἄρα Γ τὸν Α μετρεῖ καὶ Δ τὸν Β.
15ὁσάκις δὴ Γ τὸν Α μετρεῖ, τοσαῦται μονάδες ἔστωσαν ἐν τῷ Ε. καὶ Δ ἄρα τὸν Β μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας. καὶ ἐπεὶ Γ τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Ε μονάδας, καὶ Ε ἄρα τὸν Α μετρεῖ κατὰ τὰς ἐν τῷ Γ μονάδας. διὰ τὰ αὐτὰ δὴ Ε καὶ τὸν Β μετρεῖ κατὰ τὰς
20ἐν τῷ Δ μονάδας. Ε ἄρα τοὺς Α, Β μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα ἔσονταί τινες τῶν Α, Β ἐλάσσονες ἀριθμοὶ ἐν τῷ αὐτῷ λόγῳ ὄντες τοῖς Α, Β. οἱ Α, Β ἄρα ἐλάχιστοί εἰσι τῶν τὸν αὐτὸν λόγον ἐχόντων αὐτοῖς: ὅπερ ἔδει δεῖξαι.

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