§13.prop.18#2μείζων ἄρα ἐστὶν ἡ ΓΚ τῆς ΓΔ. κείσθω τῇ ΓΚ ἴση ἡ ΓΛ, καὶ ἀπὸ τοῦ Λ τῇ ΑΒ πρὸς ὀρθὰς ἤχθω ἡ ΛΜ, καὶ ἐπεζεύχθω ἡ ΜΒ. καὶ ἐπεὶ πενταπλάσιόν ἐστι τὸ ἀπὸ τῆς ΒΓ τοῦ ἀπὸ τῆς ΓΚ, καί ἐστι τῆς μὲν ΒΓ διπλῆ ἡ ΑΒ, τῆς δὲ ΓΚ διπλῆ ἡ ΚΛ, πενταπλάσιον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΚΛ. ἔστι δὲ καὶ ἡ τῆς σφαίρας διάμετρος δυνάμει πενταπλασίων τῆς ἐκ τοῦ κέντρου τοῦ κύκλου, ἀφʼ οὗ τὸ εἰκοσάεδρον ἀναγέγραπται.
Therefore GK is greater than GD. Let GL be made equal to GK, and from L let LM be drawn at right angles to AB, and let MB be joined. And since the square on BG is quintuple of the square on GK, and AB is double of BG, and KL is double of GK, therefore the square on AB is quintuple of the square on KL. But the diameter of the sphere is also in power quintuple of the radius of the circle from which the icosahedron has been described.
καί ἐστιν ἡ ΑΒ ἡ τῆς σφαίρας διάμετρος· ἡ ΚΛ ἄρα ἐκ τοῦ κέντρου ἐστὶ τοῦ κύκλου, ἀφʼ οὗ τὸ εἰκοσάεδρον ἀναγέγραπται·
And AB is the diameter of the sphere; therefore KL is the radius of the circle from which the icosahedron has been described.
ἡ ΚΛ ἄρα ἑξαγώνου ἐστὶ πλευρὰ τοῦ εἰρημένου κύκλου.
Therefore KL is the side of the hexagon of the said circle.
καὶ ἐπεὶ ἡ τῆς σφαίρας διάμετρος σύγκειται ἔκ τε τῆς τοῦ ἑξαγώνου καὶ δύο τῶν τοῦ δεκαγώνου τῶν εἰς τὸν εἰρημένον κύκλον ἐγγραφομένων, καί ἐστιν ἡ μὲν ΑΒ ἡ τῆς σφαίρας διάμετρος, ἡ δὲ ΚΛ ἑξαγώνου πλευρά, καὶ ἴση ἡ ΑΚ τῇ ΛΒ, ἑκατέρα ἄρα τῶν ΑΚ, ΛΒ δεκαγώνου ἐστὶ πλευρὰ τοῦ ἐγγραφομένου εἰς τὸν κύκλον, ἀφʼ οὗ τὸ εἰκοσάεδρον ἀναγέγραπται.
And since the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the said circle, and AB is the diameter of the sphere, while KL is the side of the hexagon, and AK is equal to LB, therefore each of AK, LB is the side of the decagon inscribed in the circle from which the icosahedron has been described.
καὶ ἐπεὶ δεκαγώνου μὲν ἡ ΛΒ, ἑξαγώνου δὲ ἡ ΜΛ· ἴση γάρ ἐστι τῇ ΚΛ, ἐπεὶ καὶ τῇ ΘΚ· ἴσον γὰρ ἀπέχουσιν ἀπὸ τοῦ κέντρου· καί ἐστιν ἑκατέρα τῶν ΘΚ, ΚΛ διπλασίων τῆς ΚΓ·
And since LB is the side of the decagon, and ML is the side of the hexagon—for it is equal to KL, since it is also equal to ThetaK, because they are at an equal distance from the center; and each of ThetaK, KL is double of KG—therefore MB is the side of the pentagon.
πενταγώνου ἄρα ἐστὶν ἡ ΜΒ. ἡ δὲ τοῦ πενταγώνου ἐστὶν ἡ τοῦ εἰκοσαέδρου· εἰκοσαέδρου ἄρα ἐστὶν ἡ ΜΒ.
καὶ ἐπεὶ ἡ ΖΒ κύβου ἐστὶ πλευρά, τετμήσθω ἄκρον καὶ μέσον λόγον κατὰ τὸ Ν, καὶ ἔστω μεῖζον τμῆμα τὸ ΝΒ· ἡ ΝΒ ἄρα δωδεκαέδρου ἐστὶ πλευρά.
But the side of the pentagon is the side of the icosahedron; therefore MB is the side of the icosahedron. And since ZB is the side of the cube, let it be cut in extreme and mean ratio at N, and let NB be the greater segment; therefore NB is the side of the dodecahedron.
καὶ ἐπεὶ ἡ τῆς σφαίρας διάμετρος ἐδείχθη τῆς μὲν ΑΖ πλευρᾶς τῆς πυραμίδος δυνάμει ἡμιολία, τῆς δὲ τοῦ ὀκταέδρου τῆς ΒΕ δυνάμει διπλασίων, τῆς δὲ τοῦ κύβου τῆς ΖΒ δυνάμει τριπλασίων, οἵων ἄρα ἡ τῆς σφαίρας διάμετρος δυνάμει ἕξ, τοιούτων ἡ μὲν τῆς πυραμίδος τεσσάρων, ἡ δὲ τοῦ ὀκταέδρου τριῶν, ἡ δὲ τοῦ κύβου δύο.
And since the diameter of the sphere was proved to be in power one and a half times the side AZ of the pyramid, and double of the side BE of the octahedron, and triple of the side ZB of the cube, therefore, of such parts as the square on the diameter of the sphere contains six, the square on the side of the pyramid contains four, that on the octahedron three, and that on the cube two.
ἡ μὲν ἄρα τῆς πυραμίδος πλευρὰ τῆς μὲν τοῦ ὀκταέδρου πλευρᾶς δυνάμει ἐστὶν ἐπίτριτος, τῆς δὲ τοῦ κύβου δυνάμει διπλῆ, ἡ δὲ τοῦ ὀκταέδρου τῆς τοῦ κύβου δυνάμει ἡμιολία.
Therefore the side of the pyramid is in power four-thirds of the side of the octahedron, and double of that of the cube, and the side of the octahedron is in power one and a half times that of the cube.
αἱ μὲν οὖν εἰρημέναι τῶν τριῶν σχημάτων πλευραί, λέγω δὴ πυραμίδος καὶ ὀκταέδρου καὶ κύβου, πρὸς ἀλλήλας εἰσὶν ἐν λόγοις ῥητοῖς.
Therefore the said sides of the three figures, I mean those of the pyramid, the octahedron, and the cube, are in rational ratios to one another.
αἱ δὲ λοιπαὶ δύο, λέγω δὴ ἥ τε τοῦ εἰκοσαέδρου καὶ ἡ τοῦ δωδεκαέδρου, οὔτε πρὸς ἀλλήλας οὔτε πρὸς τὰς προειρημένας εἰσὶν ἐν λόγοις ῥητοῖς· ἄλογοι γάρ εἰσιν, ἡ μὲν ἐλάττων, ἡ δὲ ἀποτομή.
But the remaining two, I mean the side of the icosahedron and that of the dodecahedron, are neither in rational ratios to one another nor to the aforesaid sides; for they are irrational, the one being a minor straight line, and the other an apotome.
ὅτι μείζων ἐστὶν ἡ τοῦ εἰκοσαέδρου πλευρὰ ἡ ΜΒ τῆς τοῦ δωδεκαέδρου τῆς ΝΒ, δείξομεν οὕτως.
That the side MB of the icosahedron is greater than the side NB of the dodecahedron, we shall show as follows.