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οἱ ἐπὶ ἴσων βάσεων ὄντες κῶνοι καὶ κύλινδροι πρὸς ἀλλήλους εἰσὶν ὡς τὰ ὕψη.

ἔστωσαν γὰρ ἐπὶ ἴσων βάσεων τῶν ΑΒ, ΓΔ κύκλων κύλινδροι οἱ ΕΒ, ΖΔ: λέγω, ὅτι ἐστὶν ὡς ΕΒ κύλινδρος
5πρὸς τὸν ΖΔ κύλινδρον, οὕτως ΗΘ ἄξων πρὸς τὸν ΚΛ ἄξονα.

Ἐκβεβλήσθω γὰρ ΚΛ ἄξων ἐπὶ τὸ Ν σημεῖον, καὶ κείσθω τῷ ΗΘ ἄξονι ἴσος ΛΝ, καὶ περὶ ἄξονα τὸν ΛΝ κύλινδρος νενοήσθω ΓΜ. ἐπεὶ οὖν οἱ ΕΒ, ΓΜ κύλινδροι
10ὑπὸ τὸ αὐτὸ ὕψος εἰσίν, πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις. ἴσαι δέ εἰσιν αἱ βάσεις ἀλλήλαις: ἴσοι ἄρα εἰσὶ καὶ οἱ ΕΒ, ΓΜ κύλινδροι. καὶ ἐπεὶ κύλινδρος ΖΜ ἐπιπέδῳ τέτμηται τῷ ΓΔ παραλλήλῳ ὄντι τοῖς ἀπεναντίον
15ἐπιπέδοις, ἔστιν ἄρα ὡς ΓΜ κύλινδρος πρὸς τὸν ΖΔ κύλινδρον, οὕτως ΛΝ ἄξων πρὸς τὸν ΚΛ ἄξονα. ἴσος δέ ἐστιν μὲν ΓΜ κύλινδρος τῷ ΕΒ κυλίνδρῳ, δὲ ΛΝ ἄξων τῷ ΗΘ
20ἄξονι: ἔστιν ἄρα ὡς ΕΒ κύλινδρος πρὸς τὸν ΖΔ κύλινδρον, οὕτως ΗΘ ἄξων πρὸς τὸν ΚΛ ἄξονα. ὡς δὲ ΕΒ κύλινδρος πρὸς τὸν ΖΔ κύλινδρον, οὕτως ΑΒΗ κῶνος πρὸς τὸν ΓΔΚ κῶνον. καὶ ὡς ἄρα ΗΘ ἄξων πρὸς τὸν ΚΛ ἄξονα, οὕτως ΑΒΗ κῶνος πρὸς τὸν ΓΔΚ
25κῶνον καὶ ΕΒ κύλινδρος πρὸς τὸν ΖΔ κύλινδρον: ὅπερ ἔδει δεῖξαι.

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