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ἐὰν ὦσιν ὁποσοιοῦν ἀριθμοὶ καὶ ἄλλοι αὐτοῖς ἴσοι τὸ πλῆθος σύνδυο λαμβανόμενοι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ δι᾽ ἴσου ἐν τῷ αὐτῷ λόγῳ ἔσονται.

ἔστωσαν ὁποσοιοῦν ἀριθμοὶ οἱ Α, Β, Γ καὶ ἄλλοι
5αὐτοῖς ἴσοι τὸ πλῆθος σύνδυο λαμβανόμενοι ἐν τῷ αὐτῷ λόγῳ οἱ Δ, Ε, Ζ, ὡς μὲν Α πρὸς τὸν Β, οὕτως Δ πρὸς τὸν Ε, ὡς δὲ Β πρὸς τὸν Γ, οὕτως Ε πρὸς τὸν Ζ: λέγω, ὅτι καὶ δι᾽ ἴσου ἐστὶν ὡς Α πρὸς τὸν Γ, οὕτως Δ πρὸς τὸν Ζ.
10

ἐπεὶ γάρ ἐστιν ὡς Α πρὸς τὸν Β, οὕτως Δ πρὸς τὸν Ε, ἐναλλὰξ ἄρα ἐστὶν ὡς Α πρὸς τὸν Δ, οὕτως Β πρὸς τὸν Ε. πάλιν, ἐπεί ἐστιν ὡς Β πρὸς τὸν Γ, οὕτως Ε πρὸς τὸν Ζ, ἐναλλὰξ ἄρα ἐστὶν ὡς Β πρὸς τὸν Ε, οὕτως Γ πρὸς τὸν Ζ. ὡς δὲ Β πρὸς τὸν Ε, οὕτως Α πρὸς τὸν
15δ: καὶ ὡς ἄρα Α πρὸς τὸν Δ, οὕτως Γ πρὸς τὸν Ζ: ἐναλλὰξ ἄρα ἐστὶν ὡς Α πρὸς τὸν Γ, οὕτως Δ πρὸς τὸν Ζ: ὅπερ ἔδει δεῖξαι.

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