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ἐὰν τετράγωνος τετράγωνον μετρῇ, καὶ πλευρὰ τὴν πλευρὰν μετρήσει: καὶ ἐὰν πλευρὰ τὴν πλευρὰν μετρῇ, καὶ τετράγωνος τὸν τετράγωνον μετρήσει.

ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ
5α, Β, πλευραὶ δὲ αὐτῶν ἔστωσαν οἱ Γ, Δ, δὲ Α τὸν Β μετρείτω: λέγω, ὅτι καὶ Γ τὸν Δ μετρεῖ.
5

Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω: οἱ Α, Ε, Β ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ
10λόγῳ. καὶ ἐπεὶ οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ Α τὸν Β, μετρεῖ ἄρα καὶ Α τὸν Ε. καί ἐστιν ὡς Α πρὸς τὸν Ε, οὕτως Γ πρὸς τὸν Δ: μετρεῖ ἄρα καὶ Γ τὸν Δ.

πάλιν δὴ Γ τὸν Δ μετρείτω: λέγω, ὅτι καὶ Α τὸν Β μετρεῖ.
15

τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, ὅτι οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ. καὶ ἐπεί ἐστιν ὡς Γ πρὸς τὸν Δ, οὕτως Α πρὸς τὸν Ε, μετρεῖ δὲ Γ τὸν Δ, μετρεῖ ἄρα καὶ Α τὸν Ε. καί εἰσιν οἱ Α, Ε, Β ἑξῆς ἀνάλογον: μετρεῖ ἄρα
20καὶ Α τὸν Β.

ἐὰν ἄρα τετράγωνος τετράγωνον μετρῇ, καὶ πλευρὰ τὴν πλευρὰν μετρήσει: καὶ ἐὰν πλευρὰ τὴν πλευρὰν μετρῇ, καὶ τετράγωνος τὸν τετράγωνον μετρήσει: ὅπερ ἔδει δεῖξαι.

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