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ἐὰν ἀριθμὸς ἀριθμοῦ μέρος , καὶ ἕτερος ἑτέρου τὸ αὐτὸ μέρος , καὶ ἐναλλάξ, μέρος ἐστὶν μέρη πρῶτος τοῦ τρίτου, τὸ αὐτὸ μέρος ἔσται τὰ αὐτὰ μέρη καὶ δεύτερος τοῦ τετάρτου.
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ἀριθμὸς γὰρ Α ἀριθμοῦ τοῦ ΒΓ μέρος ἔστω, καὶ ἕτερος Δ ἑτέρου τοῦ ΕΖ τὸ αὐτὸ μέρος, ὅπερ Α τοῦ ΒΓ: λέγω, ὅτι καὶ ἐναλλάξ, μέρος ἐστὶν Α τοῦ Δ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ΒΓ τοῦ ΕΖ μέρη.

ἐπεὶ γὰρ μέρος ἐστὶν Α τοῦ ΒΓ, τὸ αὐτὸ μέρος ἐστὶ
10καὶ Δ τοῦ ΕΖ, ὅσοι ἄρα εἰσὶν ἐν τῷ ΒΓ ἀριθμοὶ ἴσοι τῷ Α, τοσοῦτοί εἰσι καὶ ἐν τῷ ΕΖ ἴσοι τῷ Δ. διῃρήσθω μὲν ΒΓ εἰς τοὺς τῷ Α ἴσους τοὺς ΒΗ, ΗΓ, δὲ ΕΖ εἰς τοὺς τῷ Δ ἴσους
15τοὺς ΕΘ, ΘΖ: ἔσται δὴ ἴσον τὸ πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ.

καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΒΗ, ΗΓ ἀριθμοὶ ἀλλήλοις, εἰσὶ δὲ καὶ οἱ ΕΘ, ΘΖ ἀριθμοὶ ἴσοι ἀλλήλοις, καί ἐστιν ἴσον
20τὸ πλῆθος τῶν ΒΗ, ΗΓ τῷ πλήθει τῶν ΕΘ, ΘΖ, ἄρα μέρος ἐστὶν ΒΗ τοῦ ΕΘ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ΗΓ τοῦ ΘΖ τὰ αὐτὰ μέρη: ὥστε καὶ μέρος ἐστὶν ΒΗ τοῦ ΕΘ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ συναμφότερος ΒΓ συναμφοτέρου τοῦ ΕΖ τὰ αὐτὰ
25μέρη. ἴσος δὲ μὲν ΒΗ τῷ Α, δὲ ΕΘ τῷ Δ: ἄρα μέρος ἐστὶν Α τοῦ Δ μέρη, τὸ αὐτὸ μέρος ἐστὶ καὶ ΒΓ τοῦ ΕΖ τὰ αὐτὰ μέρη: ὅπερ ἔδει δεῖξαι.

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