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ἐὰν ἀριθμός ἀριθμοῦ μέρος , καὶ ἕτερος ἑτέρου τὸ αὐτὸ μέρος , καὶ συναμφότερος συναμφοτέρου τὸ αὐτὸ μέρος ἔσται, ὅπερ εἷς τοῦ ἑνός.

ἀριθμὸς γὰρ Α ἀριθμοῦ τοῦ ΒΓ
5μέρος ἔστω, καὶ ἕτερος Δ ἑτέρου τοῦ ΕΖ τὸ αὐτὸ μέρος, ὅπερ Α τοῦ ΒΓ: λέγω, ὅτι καὶ συναμφότερος Α, Δ συναμφοτέρου τοῦ ΒΓ, ΕΖ τὸ αὐτὸ μέρος ἐστίν, ὅπερ Α τοῦ ΒΓ.
10

ἐπεὶ γάρ, μέρος ἐστὶν Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ καὶ Δ τοῦ ΕΖ, ὅσοι ἄρα εἰσὶν ἐν τῷ ΒΓ ἀριθμοὶ ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἀριθμοὶ ἴσοι τῷ Δ. διῃρήσθω μὲν ΒΓ εἰς τοὺς τῷ Α ἴσους τοὺς ΒΗ, ΗΓ, δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους τοὺς ΕΘ, ΘΖ: ἔσται
15δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ. καὶ ἐπεὶ ἴσος ἐστὶν μὲν ΒΗ τῷ Α, δὲ ΕΘ τῷ Δ, καὶ οἱ ΒΗ, ΕΘ ἄρα τοῖς Α, Δ ἴσοι. διὰ τὰ αὐτὰ δὴ καὶ οἱ ΗΓ, ΘΖ τοῖς Α, Δ. ὅσοι ἄρα εἰσὶν ἐν τῷ ΒΓ ἀριθμοὶ ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τοῖς ΒΓ, ΕΖ ἴσοι τοῖς
20α, Δ. ὁσαπλασίων ἄρα ἐστὶν ΒΓ τοῦ Α, τοσαυταπλασίων ἐστὶ καὶ συναμφότερος ΒΓ, ΕΖ συναμφοτέρου τοῦ Α, Δ. ἄρα μέρος ἐστὶν Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος Α, Δ συναμφοτέρου τοῦ ΒΓ, ΕΖ: ὅπερ ἔδει δεῖξαι.

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