previous next

Click on a word to bring up parses, dictionary entries, and frequency statistics

τῇ ἐκ δύο μέσων μήκει σύμμετρος καὶ αὐτὴ ἐκ δύο μέσων ἐστὶ καὶ τῇ τάξει αὐτή.

ἔστω ἐκ δύο μέσων ΑΒ, καὶ τῇ ΑΒ σύμμετρος ἔστω μήκει ΓΔ: λέγω, ὅτι ΓΔ ἐκ δύο μέσων ἐστὶ καὶ τῇ
5τάξει αὐτὴ τῇ ΑΒ.

ἐπεὶ γὰρ ἐκ δύο μέσων ἐστὶν ΑΒ, διῃρήσθω εἰς τὰς μέσας κατὰ τὸ Ε: αἱ ΑΕ, ΕΒ ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι. καὶ γεγονέτω ὡς ΑΒ πρὸς ΓΔ, ΑΕ πρὸς ΓΖ: καὶ λοιπὴ ἄρα ΕΒ πρὸς
10λοιπὴν τὴν ΖΔ ἐστιν, ὡς ΑΒ πρὸς ΓΔ. σύμμετρος δὲ ΑΒ τῇ ΓΔ μήκει: σύμμετρος ἄρα καὶ ἑκατέρα τῶν ΑΕ, ΕΒ ἑκατέρᾳ τῶν ΓΖ, ΖΔ. μέσαι δὲ αἱ ΑΕ, ΕΒ: μέσαι ἄρα καὶ αἱ ΓΖ, ΖΔ. καὶ ἐπεί ἐστιν ὡς ΑΕ πρὸς ΕΒ,
15 ΓΖ πρὸς ΖΔ, αἱ δὲ ΑΕ, ΕΒ δυνάμει μόνον σύμμετροί εἰσιν, καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει μόνον σύμμετροί εἰσιν. ἐδείχθησαν δὲ καὶ μέσαι: ΓΔ ἄρα ἐκ δύο μέσων ἐστίν.

λέγω δή, ὅτι καὶ τῇ τάξει αὐτή ἐστι τῇ ΑΒ.

ἐπεὶ γάρ ἐστιν ὡς ΑΕ πρὸς ΕΒ, ΓΖ πρὸς ΖΔ,
20καὶ ὡς ἄρα τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ὑπὸ τῶν ΑΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ὑπὸ τῶν ΓΖΔ: ἐναλλὰξ ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΓΖ, οὕτως τὸ ὑπὸ τῶν ΑΕΒ πρὸς τὸ ὑπὸ τῶν ΓΖΔ. σύμμετρον δὲ τὸ ἀπὸ τῆς ΑΕ τῷ ἀπὸ τῆς ΓΖ: σύμμετρον ἄρα καὶ τὸ ὑπὸ τῶν
25ΑΕΒ τῷ ὑπὸ τῶν ΓΖΔ. εἴτε οὖν ῥητόν ἐστι τὸ ὑπὸ τῶν ΑΕΒ, καὶ τὸ ὑπὸ τῶν ΓΖΔ ῥητόν ἐστιν καὶ διὰ τοῦτό ἐστιν ἐκ δύο μέσων πρώτη. εἴτε μέσον, μέσον, καί ἐστιν ἑκατέρα δευτέρα.

καὶ διὰ τοῦτο ἔσται ΓΔ τῇ ΑΒ τῇ τάξει αὐτή:
30ὅπερ ἔδει δεῖξαι.

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 United States License.

An XML version of this text is available for download, with the additional restriction that you offer Perseus any modifications you make. Perseus provides credit for all accepted changes, storing new additions in a versioning system.

load focus English (Thomas L. Heath, Sir Thomas Little Heath, 1956)
load Vocabulary Tool
hide Display Preferences
Greek Display:
Arabic Display:
View by Default:
Browse Bar: