§6.3γ΄.
3.
Εἰς τρεῖς εὐθείας τὰς ΑΒ, ΓΑ, ∠Α διήχθωσαν δύο εὐθεῖαι αἱ ΘΕ. Θ∠· ὅτι ἐστίν, ὡς τὸ ὑπὸ ΘΕ ΗΖ πρὸς τὸ ὑπὸ ΘΗ, ΖΕ, οὕτως τὸ ὑπὸ ΘΒ, ∠Γ πρὸς τὸ ὑπὸ Θ∠, ΒΓ.
ἤχθω διὰ μὲν τοῦ Θ τῇ ΖΓΑ παράλληλος ἡ ΚΛ, καὶ αἱ ∠Α, ΑΒ συμπιπτέτωσαν αὐτῇ κατὰ τὰ Κ, Λ σημεῖα, διὰ δὲ τοῦ Λ τῇ ∠Α παράλληλος ἡ ΛΜ καὶ συμπιπτέ τω τῇ ΕΘ ἐπὶ τὸ Μ.
ἐπεὶ οὖν ἐστιν, ὡς μὲν ἡ ΕΖ πρὸς τὴν ΖΑ, οὕτως ἡ ΕΘ πρὸς τὴν ΘΛ, ὡς δὲ ἡ Α πρὸς τὴν ΖΗ, οὕτως ἡ ΘΛ πρὸς τὴν ΘΜ καὶ γὰρ ἡ Θ Κ πρὸς τὴν ΘΗ ἐν παραλλήλῳ· διίσου ἄρα ἐστίν, ὡς ἡ ΕΖ πρὸς τὴν ΖΗ, οὕτως ἡ ΕΘ πρὸς τὴν ΘΜ. τὸ ἄρα ὑπὸ τῶν ΘΕ, ΗΖ ἴσον ἐστὶν τῷ ὑπὸ τῶν ΕΖ, ΘΜ. ἄλλο δέ τι τυχὸν τὸ ὑπὸ τῶν ΕΖ, ΘΗ ἔστιν ἄρα, ὡς τὸ ὑπὸ τῶν ΕΘ, ΗΖ πρὸς τὸ ὑπὸ τῶν ΕΖ, ΗΘ, οὕτως τὸ ὑπὸ ΕΖ, ΘΜ πρὸς τὸ ὑπὸ ΕΖ, ΗΘ, τουτέστιν ἡ ΘΜ πρὸς ΘΗ, τουτέστιν ἡ ΛΘ πρὸς τὴν ΘΚ. κατὰ τὰ αὐτὰ καί, ὡς ἡ ΚΘ πρὸς τὴν ΘΛ, οὕτως τὸ ὑπὸ Θ∠, ΒΓ πρὸς τὸ ὑπὸ ΘΒ, Γ∠ ἀνάπαλιν ἄρα γίνεται, ὡςἡ ΛΘπρὸς τὴν ΘΚ, οὕτως τὸ ὑπὸ ΘΒ, Γ∠ πρὸς τὸ ὑπὸ Θ∠, ΒΓ. ὡς δὲ ἡ ΛΘ πρὸς τὴν Θ Κ, οὕτως ἐδείχθη τὸ ὑπὸ ΕΘ, Η πρὸς τὸ ὑπὸ ΕΖ, ΗΘ καὶ ὡς ἄρα τὸ ὑπὸ ΕΘ, ΗΖ πρὸς τὸ ὑπὸ ΕΖ, ΗΘ, οὕτως τὸ ὑπὸ ΘΒ, Γ∠ πρὸς τὸ ὑπὸ Θ∠, ΒΓ.
Διὰ δὲ τοῦ συνημμένου οὕτως·
Let there be drawn two straight lines ΘΕ, Θ∠ across three straight lines ΑΒ, ΓΑ, ∠Α; (to prove) that, as the product of ΘΕ, ΗΖ is to the product of ΘΗ, ΖΕ, so is the product of ΘΒ, ∠Γ to the product of Θ∠, ΒΓ. Let ΚΛ be drawn through Θ parallel to ΖΓΑ, and let ∠Α, ΑΒ meet it at the points Κ, Λ; and let ΛΜ be drawn through Λ parallel to ∠Α, and let it meet ΕΘ at Μ. Since therefore, as ΕΖ is to ΖΑ, so is ΕΘ to ΘΛ, and as Α is to ΖΗ, so is ΘΛ to ΘΜ (for indeed, in the parallel, so is ΘΚ to ΘΗ); therefore by equality, as ΕΖ is to ΖΗ, so is ΕΘ to ΘΜ. Therefore the product of ΘΕ, ΗΖ is equal to the product of ΕΖ, ΘΜ. And there being another arbitrary product, that of ΕΖ, ΘΗ, therefore, as the product of ΕΘ, ΗΖ is to the product of ΕΖ, ΗΘ, so is the product of ΕΖ, ΘΜ to the product of ΕΖ, ΗΘ, that is, ΘΜ to ΘΗ, which is ΛΘ to ΘΚ. By the same reasoning also, as ΚΘ is to ΘΛ, so is the product of Θ∠, ΒΓ to the product of ΘΒ, Γ∠; therefore by inversion it becomes, as ΛΘ is to ΘΚ, so is the product of ΘΒ, Γ∠ to the product of Θ∠, ΒΓ. But as ΛΘ is to ΘΚ, so was the product of ΕΘ, Η to the product of ΕΖ, ΗΘ shown to be; and therefore as the product of ΕΘ, ΗΖ is to the product of ΕΖ, ΗΘ, so is the product of ΘΒ, Γ∠ to the product of Θ∠, ΒΓ.
ἐπεὶ ὁ τοῦ ὑπὸ ΘΕ, ΗΖ πρὸς τὸ ὑπὸ ΘΗ, ΖΕ συνῆπται λόγος ἔκ τε τοῦ, ὃν ἔχει ἡ ΘΕ πρὸς τὴν ΕΖ, καὶ τοῦ, ὃν ἔχει ἡ ΖΗ πρὸς τὴν ΗΘ, καί ἐστιν, ὡς μὲν ἡ ΘΕ πρὸς τὴν ΕΖ, οὕτως ἡ Θ∠ πρὸς τὴν ΖΑ, ὡς δὲ ἡ ΖΗ πρὸς τὴν ΗΘ, οὕτως ἡ ΖΑ πρὸς τὴν ΘΚ, το ἄρα ὑπὸ ΘΕ, ΗΖ προς το ὑπὸ ΘΗ, ΕΖ συνῆπται ἔκ τε τοῦ, ὃν ἔχει ἡ ΘΛ πρὸς τὴν ΖΑ, καὶ τοῦ ὃν ἔχει ἡ ΖΑ πρὸς τὴν ΘΚ. ὁ δὲ συνημμένος ἔκ τε τοῦ τῆς ΘΛ πρὸς τὴν ΖΑ καὶ τοῦ τῆς ΖΑ πρὸς τὴν ΘΚ ὁ αὐτός ἐστιν τῷ τῆς ΘΛ πρὸς τὴν ΘΚ· ἔστιν ἄρα, ὡς τὸ ὑπὸ ΘΕ, ΗΖ πρὸς τὸ ὑπὸ ΘΗ, ΖΕ, οὕτως ἡ ΘΛ πρὸς τὴν ΘΚ. διὰ ταὐτὰ καί, ὡς τὸ ὑπὸ Θ∠, ΒΓ πρὸς τὸ ὑπὸ ΘΒ, Γ∠, οὕτως ἐστὶν ἡ ΘΚ πρὸς τὴν ΘΛ. καὶ ἀνάπαλίν ἐστιν, ὡς τὸ ὑπὸ ΘΒ, Γ∠ πρὸς τὸ ὑπὸ Θ∠, ΒΓ, οὕτως ἡ ΘΛ πρὸς τὴν ΘΚ. ἦν δὲ καί, ὡς τὸ ὑπὸ τῶν ΘΕ, ΖΗ πρὸς τὸ ὑπὸ ΘΗ, ΖΕ, οὕτως ἡ ΘΛ πρὸς τὴν ΘΚ· καὶ ὡς ἄρα τὸ ὑπὸ τῶν ΘΕ, ΖΗ πρὸς τὸ ὑπὸ ΘΗ, ΖΕ, οὕτως τὸ ὑπὸ ΘΒ, Γ∠ πρὸς τὸ ὑπὸ Θ∠, ΒΓ.
But by the compounded ratio thus: Since the ratio of the product of ΘΕ, ΗΖ to the product of ΘΗ, ΖΕ is compounded of that which ΘΕ has to ΕΖ and that which ΖΗ has to ΗΘ, and as ΘΕ is to ΕΖ, so is Θ∠ to ΖΑ, and as ΖΗ is to ΗΘ, so is ΖΑ to ΘΚ, therefore the ratio of the product of ΘΕ, ΗΖ to the product of ΘΗ, ΕΖ is compounded of that which ΘΛ has to ΖΑ and that which ΖΑ has to ΘΚ. And the ratio compounded of that of ΘΛ to ΖΑ and that of ΖΑ to ΘΚ is the same as that of ΘΛ to ΘΚ; therefore as the product of ΘΕ, ΗΖ is to the product of ΘΗ, ΖΕ, so is ΘΛ to ΘΚ. For the same reasons also, as the product of Θ∠, ΒΓ is to the product of ΘΒ, Γ∠, so is ΘΚ to ΘΛ. And by inversion, as the product of ΘΒ, Γ∠ is to the product of Θ∠, ΒΓ, so is ΘΛ to ΘΚ. But also, as the product of ΘΕ, ΖΗ was to the product of ΘΗ, ΖΕ, so was ΘΛ to ΘΚ; and therefore as the product of ΘΕ, ΖΗ is to the product of ΘΗ, ΖΕ, so is the product of ΘΒ, Γ∠ to the product of Θ∠, ΒΓ.