Commentarii in libros de sphaera et cylindro

Ἔστωσαν τέσσαρες ὅροι οἱ Α, Γ, △, Β. Λέγω ὅτι ὁ συγκείμενος λόχος ἐκ τοῦ τοῦ ὑπὸ τῶν Α, Β πρὸς τὸ ἀπὸ Γ μετὰ τοῦ τῆς Β πρὸς λόγου ὁ αὐτός ἐστι τῷ τοῦ ὑπὸ Α, Β ἐπὶ τὴν Β πρὸς τὸ ἀπὸ Γ ἐπὶ τὴν △.

Ἔστω γὰρ τῷ μὲν ὑπὸ Α, Β ἴσος ὁ Κ, τῷ δὲ ἀπὸ Γ ἴσος ὁ Λ, καὶ γεχονέτω ὡς ὁ Β πρὸς △, οὕτως ὁ Λ πρὸς Μ· ὁ ἄρα τοῦ Κ πρὸς Μ λόχος σύγκειται ἐκ τοῦ τοῦ πρὸς Λ, τουτέστι τοῦ ὑπὸ Α, Β πρὸς τὸ ἀπὸ Γ, καὶ τοῦ Λ πρὸς Μ, τουτέστι τοῦ Β πρὸς △. Ὁ δὴ Κ τὸν Β πολλαπλασιάσας τὸν Ν ποιείτω, ὁ δὲ Λ τὸν Β πολλαπλασιάσας τὸν Ξ ποιείτω, τὸν δὲ △ πολλαπλασιάσας τὸν Ο. Ἐπεὶ οὖν τὸ ὑπὸ τῶν A, Β ὁ Κ ἐστίν, ὁ δὲ Κ τὸν Β πολλαπλασιάσας τὸν Ν πεποίηκεν, ὁ ἄρα Ν ἐστὶν τὸ ὑπὸ Α, Β ἐπὶ τὸν Β, Πάλιν, ἐπεὶ τὸ ἀπὸ Γ ὁ Λ ἐστίν, ὁ δὲ Λ τὸν △ πολλαπλασιάσας τὸν O πεποίηκεν, ὁ O ἄρα ἐστὶν τὸ ἀπὸ τοῦ Γ ἐπὶ τὸν △ ὥστε ὁ τοῦ ὑπὸ Α, Β ἐπὶ τὸν Β πρὸς τὸ ἀπὸ Γ ἐπὶ τὸν △ λόφος ὁ αὐτός ἐστι

τῷ τοῦ Ν πρὸς Ο. Δεῖ ἄρα δεῖξαι ὅτι ὁ τοῦ πρὸς Μ λόγος ὁ αὐτός ἐστι τῷ τοῦ Ν πρὸς Ο.

Ἐπεὶ οὖν ἑκάτερος τῶν Κ, Λ τὸν Β πολλαπλασιάσας ἑκάτερον τῶν Ν, Ξ πεποίηκεν, ἔστιν ἄρα ὡς ὁ Κ πρὸς τὸν Λ, οὕτως ὁ Ν πρὸς Ξ. Πάλιν, ἐπεὶ ὁ Λ ἑκάτερον τῶν Β, △ πολλαπλασιάσας ἑκάτερον τῶν Ξ, Ο πεποίηκεν, ἔστιν ἄρα ὡς ὁ Β πρὸς △, ὁ Ξ πρὸς Ο. Ἀλλ᾿ ὡς ὁ Β πρὸς △, ὁ Λ πρὸς τὸν Μ· καὶ ὡς ἄρα ὁ πρὸς Μ, ὁ Ξ πρὸς Ο. Οἱ ἄρα Κ, Λ, Μ τοῖς Ν, Ξ, Ο ἐν τῷ αὐτῷ λόγῳ εἰσὶν σύνδυο λαμβανόμενοι· καὶ διʼ ἴσου ἄρα ἐστὶν ὡς ὁ Κ πρὸς Μ, οὕτως ὁ Ν πρὸς Ο. Καὶ ἔστιν ὁ τοῦ Κ πρὸς Μ λόγος ὁ αὐτὸς τῷ συγκειμένῳ ἐκ τοῦ τοῦ ὑπὸ A, Β πρὸς τὸ ἀπὸ Γ καὶ τοῦ ὃν ἔχει ὁ Β πρὸς △, ὁ δὲ τοῦ Ν πρὸς Ο λόγος ὁ αὐτὸς ἐστι τῷ τοῦ ὑπὸ Α, Β ἐπὶ τὸν Β πρὸς τὸ ἀπὸ Γ ἐπὶ τὸν ὁ ἄρα σγκείμενος λόγος ἐκ τοῦ τοῦ ὑπὸ Α, Β πρὸς τὸ ἀπὸ Γ καὶ τοῦ ὃν ἔχει ὁ Β πρὸς △ ὁ αὐτός ἐστι τῷ τοῦ ὑπὸ Α, Β ἐπὶ τὸν Β πρὸς τὸ ἀπὸ Γ ἐπὶ τὸν △.

Φανερὸν δὲ καὶ ὅτι τὸ ὑπὸ Α, Β ἐπὶ τὸν Β ἴσον ἐστὶ τῷ ἀπὸ τοῦ Β ἐπὶ τὸν Α. Ἐπεὶ γάρ ἐστιν ὡς ὁ Α πρὸς τὸν Β, οὕτως τὸ ὑπὸ Α, Β πρὸς τὸ ἀπὸ τοῦ Β τοῦ Β κοινοῦ ὕψους λαμβανομένου, ἐὰν δὲ τέσσαρες ὅροι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν ἄκρων ἴσον ἐστὶ τῷ ὑπὸ τῶν μέσων, τὸ ἄρα ὑπὸ Α, Β ἐπὶ τὸν Β ἴσον ἐστὶ τῷ ἀπὸ τοῦ Β ἐπὶ τὸν Α.