Continued proportions in number theory: Book 8 Proposition 14
Translations
Ἐὰν τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ τὴν πλευρὰν μετρήσει: καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει. Ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, πλευραὶ δὲ αὐτῶν ἔστωσαν οἱ Γ, Δ, ὁ δὲ Α τὸν Β μετρείτω: λέγω, ὅτι καὶ ὁ Γ τὸν Δ μετρεῖ. Ὁ Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω: οἱ Α, Ε, Β ἄρα ἑξῆς ἀνάλογόν εἰσιν ἐν τῷ τοῦ Γ πρὸς τὸν Δ λόγῳ. καὶ ἐπεὶ οἱ Α, Ε, Β ἑξῆς ἀνάλογόν εἰσιν, καὶ μετρεῖ ὁ Α τὸν Β, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε. καί ἐστιν ὡς ὁ Α πρὸς τὸν Ε, οὕτως ὁ Γ πρὸς τὸν Δ: μετρεῖ ἄρα καὶ ὁ Γ τὸν Δ. Πάλιν δὴ ὁ Γ τὸν Δ μετρείτω: λέγω, ὅτι καὶ ὁ Α τὸν Β μετρεῖ. 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 ὁ Γ πρὸς τὸν Δ, οὕτως ὁ Α πρὸς τὸν Ε, μετρεῖ δὲ ὁ Γ τὸν Δ, μετρεῖ ἄρα καὶ ὁ Α τὸν Ε. καί εἰσιν οἱ Α, Ε, Β ἑξῆς ἀνάλογον: μετρεῖ ἄρα καὶ ὁ Α τὸν Β. Ἐὰν ἄρα τετράγωνος τετράγωνον μετρῇ, καὶ ἡ πλευρὰ τὴν πλευρὰν μετρήσει: καὶ ἐὰν ἡ πλευρὰ τὴν πλευρὰν μετρῇ, καὶ ὁ τετράγωνος τὸν τετράγωνον μετρήσει: ὅπερ ἔδει δεῖξαι.
If a square measure a square, the side will also measure the side; and, if the side measure the side, the square will also measure the square. Let A, B be square numbers, let C, D be their sides, and let A measure B; I say that C also measures D. For let C by multiplying D make E; therefore A, E, B are continuously proportional in the ratio of C to D. [VIII. 11] And, since A, E, B are continuously proportional, and A measures B, therefore A also measures E. [VIII. 7] And, as A is to E, so is C to D; therefore also C measures D. [VII. Def. 20] Again, let C measure D; I say that A also measures B. For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D. And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20] And A, E, B are continuously proportional; therefore A also measures B.