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ἅπας ἀριθμὸς παντὸς ἀριθμοῦ ἐλάσσων τοῦ μείζονος ἤτοι μέρος ἐστὶν μέρη.

ἔστωσαν δύο ἀριθμοὶ οἱ Α, ΒΓ, καὶ ἔστω ἐλάσσων ΒΓ: λέγω, ὅτι ΒΓ τοῦ Α ἤτοι μέρος ἐστὶν μέρη.

οἱ Α, ΒΓ γὰρ ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν οὔ. ἔστωσαν πρότερον οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους. διαιρεθέντος δὴ τοῦ ΒΓ εἰς τὰς ἐν αὐτῷ μονάδας ἔσται ἑκάστη μονὰς τῶν ἐν τῷ ΒΓ μέρος τι τοῦ Α: ὥστε μέρη ἐστὶν ΒΓ τοῦ Α.

μὴ ἔστωσαν δὴ οἱ Α, ΒΓ πρῶτοι πρὸς ἀλλήλους: δὴ ΒΓ τὸν Α ἤτοι μετρεῖ οὐ μετρεῖ. εἰ μὲν οὖν ΒΓ τὸν Α μετρεῖ, μέρος ἐστὶν ΒΓ τοῦ Α. εἰ δὲ οὔ, εἰλήφθω τῶν Α, ΒΓ μέγιστον κοινὸν μέτρον Δ, καὶ διῃρήσθω ΒΓ
15εἰς τοὺς τῷ Δ ἴσους τοὺς ΒΕ, ΕΖ, ΖΓ. καὶ ἐπεὶ Δ τὸν Α μετρεῖ, μέρος ἐστὶν Δ τοῦ Α: ἴσος δὲ Δ ἑκάστῳ τῶν ΒΕ, ΕΖ, ΖΓ: καὶ ἕκαστος ἄρα τῶν ΒΕ, ΕΖ, ΖΓ τοῦ Α μέρος ἐστίν: ὥστε μέρη ἐστὶν ΒΓ τοῦ Α.

ἅπας ἄρα ἀριθμὸς παντὸς ἀριθμοῦ ἐλάσσων τοῦ
20μείζονος ἤτοι μέρος ἐστὶν μέρη: ὅπερ ἔδει δεῖξαι.

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