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δύο τετραγώνων ἀριθμῶν εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός, καὶ τετράγωνος πρὸς τὸν τετράγωνον διπλασίονα λόγον ἔχει ἤπερ πλευρὰ πρὸς τὴν πλευράν.

ἔστωσαν τετράγωνοι ἀριθμοὶ οἱ Α, Β, καὶ τοῦ μὲν Α
5πλευρὰ ἔστω Γ, τοῦ δὲ Β Δ: λέγω, ὅτι τῶν Α, Β εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός, καὶ Α πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ Γ πρὸς τὸν Δ.
10

Γ γὰρ τὸν Δ πολλαπλασιάσας τὸν Ε ποιείτω. καὶ ἐπεὶ τετράγωνός ἐστιν Α, πλευρὰ δὲ αὐτοῦ ἐστιν Γ, Γ ἄρα ἑαυτὸν πολλαπλασιάσας τὸν Α πεποίηκεν. διὰ τὰ αὐτὰ δὴ καὶ Δ ἑαυτὸν πολλαπλασιάσας τὸν Β πεποίηκεν. ἐπεὶ οὖν Γ ἑκάτερον τῶν Γ, Δ πολλαπλασιάσας
15ἑκάτερον τῶν Α, Ε πεποίηκεν, ἔστιν ἄρα ὡς Γ πρὸς τὸν Δ, οὕτως Α πρὸς τὸν Ε. διὰ τὰ αὐτὰ δὴ καὶ ὡς Γ πρὸς τὸν Δ, οὕτως Ε πρὸς τὸν Β. καὶ ὡς ἄρα Α πρὸς τὸν Ε, οὕτως Ε πρὸς τὸν Β. τῶν Α, Β ἄρα εἷς μέσος ἀνάλογόν ἐστιν ἀριθμός.
20

λέγω δή, ὅτι καὶ Α πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ Γ πρὸς τὸν Δ. ἐπεὶ γὰρ τρεῖς ἀριθμοὶ ἀνάλογόν εἰσιν οἱ Α, Ε, Β, Α ἄρα πρὸς τὸν Β διπλασίονα λόγον ἔχει ἤπερ Α πρὸς τὸν Ε. ὡς δὲ Α πρὸς τὸν Ε, οὕτως Γ πρὸς τὸν Δ. Α ἄρα πρὸς τὸν Β διπλασίονα λόγον
25ἔχει ἤπερ Γ πλευρὰ πρὸς τὴν Δ: ὅπερ ἔδει δεῖξαι.

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