HYPSICLES :: Hypsiclis liber sive elementorum liber xiv qui fertur

TLG 0717 001 :: HYPSICLES :: Hypsiclis liber sive elementorum liber xiv qui fertur HYPSICLES Astron. Math. (Alexandrinus: 2 B.C.) Scholia: Cf. SCHOLIA IN EUCLIDEM (5022) Hypsiclis liber sive elementorum liber xiv qui fertur Source: Stamatis, E.S. (post J.L. Heiberg) (ed.), Euclidis elementa, vol. 5.1, 2nd edn. Leipzig: Teubner, 1977: 1–22. Cf. et EUCLIDES Geom. (1799 001) Citation: Section — (line)

Pr	Βασιλείδης ὁ Τύριος, ὦ Πρώταρχε, παραγενηθεὶς εἰς Ἀλεξάνδρειαν καὶ συσταθεὶς τῷ πατρὶ ἡμῶν διὰ τὴν ἀπὸ τοῦ μαθήματος συγγένειαν συνδιέτριψεν αὐτῷ τὸν πλεῖ‐ στον τῆς ἐπιδημίας χρόνον. καί ποτε ζητοῦντες τὸ ὑπὸ
5	Ἀπολλωνίου συγγραφὲν περὶ τῆς συγκρίσεως τοῦ δωδε‐ καέδρου καὶ τοῦ εἰκοσαέδρου τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφομένων, τίνα ἔχει λόγον πρὸς ἄλληλα, ἔδοξαν ταῦτα μὴ ὀρθῶς γεγραφηκέναι τὸν Ἀπολλώνιον, αὐτοὶ δὲ ταῦτα καθάραντες ἔγραψαν, ὡς ἦν ἀκούειν τοῦ πατρός. ἐγὼ δὲ
10	ὕστερον περιέπεσον ἑτέρῳ βιβλίῳ ὑπὸ Ἀπολλωνίου ἐκδε‐ δομένῳ περιέχοντί τινα ἀπόδειξιν περὶ τοῦ προκειμένου, καὶ μεγάλως ἐψυχαγωγήθην ἐπὶ τῇ τοῦ προβλήματος ζητήσει. τὸ μὲν οὖν ὑπὸ Ἀπολλωνίου ἐκδοθὲν ἔοικε κοινῇ σκοπεῖν· καὶ γὰρ περιφέρεται δοκοῦν ὕστερον γεγράφθαι
15	φιλοπόνως· ὅσα δ’ ἐγὼ δοκῶ δεῖν, ὑπομνηματισάμενος ἔκρινα προσφωνῆσαί σοι διὰ μὲν τὴν ἐν ἅπασι τοῖς μαθή‐ μασι, μάλιστα δὲ ἐν γεωμετρίᾳ προκοπὴν ἐμπειρικῶς κρι‐ νοῦντι τὰ ῥηθησόμενα, διὰ δὲ τὴν πρὸς τὸν πατέρα συν‐ ήθειαν καὶ τὴν πρὸς ἡμᾶς εὔνοιαν εὐμενῶς ἀκουσομένῳ τῆς	1
20	πραγματείας. καιρὸς δ’ ἂν εἴη τοῦ μὲν προοιμίου πεπαῦ‐ σθαι, τῆς δὲ συντάξεως ἄρχεσθαι.
1	Ἡ ἀπὸ τοῦ κέντρου κύκλου τινὸς ἐπὶ τὴν τοῦ πενταγώνου πλευρὰν τοῦ εἰς τὸν αὐτὸν κύκλον ἐγγραφομένου κάθετος [Omitted graphic marker]ἀγομένη ἡμίσειά ἐστι συν‐ αμφοτέρου τῆς τε τοῦ ἑξα‐
5	γώνου καὶ τῆς τοῦ δεκαγώ‐ νου πλευρᾶς τῶν εἰς τὸν αὐ‐ τὸν κύκλον ἐγγραφομένων. Ἔστω κύκλος ὁ ΑΒΓ, καὶ ἐν τῷ ΑΒΓ κύκλῳ ἔστω
10	πενταγώνου πλευρὰ ἡ ΒΓ, καὶ εἰλήφθω κέντρον τοῦ κύκλου τὸ Δ, καὶ ἐπὶ τὴν ΒΓ ἀπὸ τοῦ Δ κάθετος ἤχ‐ θω ἡ ΔΕ, καὶ ἐκβεβλήσθω‐
15	σαν ἐπ’ εὐθείας τῇ ΔΕ εὐθεῖαι αἱ ΕΖ, ΔΑ. λέγω, ὅτι ἡ ΔΕ ἡμίσειά ἐστι τῆς τοῦ ἑξαγώνου καὶ τῆς τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγραφομένων.
15	Ἐπεζεύχθωσαν γὰρ αἱ ΔΓ, ΓΖ, καὶ κείσθω τῇ ΕΖ ἴση ἡ ΗΕ, καὶ ἀπὸ τοῦ Η σημείου ἐπὶ τὸ Γ ἐπεζεύχθω ἡ ΗΓ.	2
20	ἐπεὶ οὖν πενταπλασία ἐστὶν ὅλου τοῦ κύκλου ἡ περιφέρεια τῆς ΒΖΓ περιφερείας, καί ἐστι τῆς μὲν ὅλου τοῦ κύκλου περιφερείας ἡμίσεια ἡ ΑΓΖ, τῆς δὲ ΒΖΓ ἡμίσεια ἡ ΖΓ, καὶ ἡ ΑΓΖ ἄρα περιφέρεια πενταπλασία ἐστὶ τῆς ΖΓ περιφερείας. τετραπλῆ ἄρα ἐστὶν ἡ ΑΓ τῆς ΖΓ. ὡς δὲ ἡ
25	ΑΓ πρὸς τὴν ΖΓ, οὕτως ἡ ὑπὸ ΑΔΓ πρὸς τὴν ὑπὸ ΖΔΓ γωνίαν. τετραπλῆ ἄρα ἡ ὑπὸ ΑΔΓ τῆς ὑπὸ ΖΔΓ. διπλῆ δὲ ἡ ὑπὸ ΑΔΓ τῆς ὑπὸ ΕΖΓ· διπλῆ ἄρα καὶ ἡ ὑπὸ ΕΖΓ τῆς ὑπὸ ΗΔΓ. ἔστι δὲ καὶ ἡ ὑπὸ ΕΖΓ ἴση τῇ ὑπὸ ΕΗΓ. διπλῆ ἄρα καὶ ἡ ὑπὸ ΕΗΓ τῆς ὑπὸ ΗΔΓ. ἴση ἄρα ἡ ΔΗ
30	τῇ ΗΓ. ἀλλὰ ἡ ΗΓ τῇ ΖΓ ἐστιν ἴση. ἴση ἄρα καὶ ἡ ΔΗ τῇ ΓΖ. ἴση δὲ καὶ ἡ ΗΕ τῇ ΕΖ. ἴση ἄρα καὶ ἡ ΔΕ συναμ‐ φοτέρῳ τῇ ΕΖΓ. κοινὴ προσκείσθω ἡ ΕΔ. συναμφότερος ἄρα ἡ ΔΖΓ διπλῆ τῆς ΔΕ. καί ἐστιν ἡ μὲν ΔΖ ἴση τῇ τοῦ ἑξαγώνου πλευρᾷ, ἡ δὲ ΖΓ ἴση τῇ τοῦ δεκαγώνου· ἡ ΔΕ
35	ἄρα ἡμίσειά ἐστι τῆς τε τοῦ ἑξαγώνου καὶ τῆς τοῦ δεκαγώ‐ νου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγραφομένων.
35	Φανερὸν δὴ ἐκ τοῦ ἐν τῷ ιγʹ βιβλίῳ θεωρήματος, ὅτι ἡ ἀπὸ τοῦ κέντρου τοῦ κύκλου ἐπὶ τὴν πλευρὰν τοῦ ἰσοπλεύ‐ ρου τριγώνου κάθετος ἀγομένη ἡμίσειά ἐστι τῆς ἐκ τοῦ	3
40	κέντρου τοῦ κύκλου. Ὁ αὐτὸς κύκλος περιλαμβάνει τό τε τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφομένων. τοῦτο δὲ γράφεται ὑπὸ μὲν Ἀρισταίου ἐν τῷ ἐπιγραφομένῳ τῶν ε σχημάτων συγκρίσει,
45	ὑπὸ δὲ Ἀπολλωνίου ἐν τῇ δευτέρᾳ ἐκδόσει τῆς συγκρίσεως τοῦ δωδεκαέδρου πρὸς τὸ εἰκοσάεδρον, ὅτι ἐστὶν ὡς ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπι‐ φάνειαν, οὕτως καὶ αὐτὸ τὸ δωδεκάεδρον πρὸς τὸ εἰκοσά‐ εδρον διὰ τὸ τὴν αὐτὴν εἶναι κάθετον ἀπὸ τοῦ κέντρου τῆς
1 (50)	σφαίρας ἐπὶ τὸ τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον. γραπτέον δὲ καὶ ἡμῖν αὐτοῖς, ὅτι ὁ αὐτὸς κύκλος περιλαμβάνει τό τε τοῦ δωδεκαέδρου πεν‐ τάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον τῶν εἰς τὴν αὐ‐ τὴν σφαῖραν ἐγγραφομένων, προγραφέντος τοῦδε.
2	Ἐὰν εἰς κύκλον πεντάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον ἐγγραφῇ, ἡ ὑπὸ δύο πλευρὰς ὑποτείνουσα καὶ ἡ τοῦ πεν‐ ταγώνου συναμφότερος δυνάμει τῆς ἐκ τοῦ κέντρου πεν‐ ταπλασία ἐστίν.
5	Ἔστω κύκλος ὁ ΑΒΓ, καὶ ἐν τῷ ΑΒΓ κύκλῳ πενταγώ‐
5	νου πλευρὰ ἔστω ἡ ΑΓ, καὶ εἰλήφθω τὸ κέντρον τοῦ κύκ‐ λου τὸ Δ, καὶ ἐπὶ τὴν ΑΓ κάθετος ἤχθω ἡ ΔΖ καὶ ἐκ‐ βεβλήσθω ἐπὶ τὰ Β, Ε, καὶ ἐπεζεύχθω ἡ ΑΒ. λέγω, ὅτι τὰ ἀπὸ τῶν ΒΑ, ΑΓ τετράγωνα[Omitted graphic marker]	4
10	πενταπλάσιά ἐστι τοῦ ἀπὸ τῆς ΔΕ τετραγώνου. Ἐπεζεύχθω ἡ ΑΕ· δεκαγώνου ἄρα ἐστὶν ἡ ΑΕ. καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΒΕ τῆς ΕΔ, τετραπλάσιον
15	ἄρα ἐστὶ τὸ ἀπὸ τῆς ΒΕ τοῦ ἀπὸ τῆς ΕΔ. τῷ δὲ ἀπὸ τῆς ΒΕ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΒΑΕ. τετραπλά‐ σια ἄρα τὰ ἀπὸ τῶν ΒΑΕ τοῦ ἀπὸ τῆς ΔΕ. πενταπλάσια ἄρα ἐστὶ τὰ ἀπὸ τῶν ΒΑΕ, ΕΔ
20	τοῦ ἀπὸ τῆς ΔΕ. τοῖς δὲ ἀπὸ τῶν ΔΕ, ΕΑ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΑΓ. πενταπλάσια ἄρα τὰ ἀπὸ τῶν ΑΒ, ΑΓ τοῦ ἀπὸ τῆς ΔΕ. Τούτου δεδειγμένου δεικτέον, ὅτι ὁ αὐτὸς κύκλος περι‐ λαμβάνει τό τε τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ
25	εἰκοσαέδρου τρίγωνον τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφο‐ μένων.
3	Ἐκκείσθω ἡ τῆς σφαίρας διάμετρος ἡ ΑΒ, καὶ ἐγγεγρά‐
3	φθω εἰς αὐτὴν δωδεκάεδρόν τε καὶ εἰκοσάεδρον, καὶ ἔστω ἓν μὲν τοῦ δωδεκαέδρου πεντάγωνον τὸ ΓΔΕΖΗ, τοῦ εἰκοσαέδρου δὲ τρίγωνον τὸ ΚΛΘ. λέγω, ὅτι αἱ ἐκ τῶν[Omitted graphic marker]	5
5	κέντρων τῶν περὶ αὐτὰ κύκλων ἴσαι εἰσί, τουτέστιν ὅτι ὁ αὐτὸς κύκλος περιλαμβάνει τό τε ΓΔΕΖΗ πεντάγωνον καὶ τὸ ΘΚΛ τρίγωνον. Ἐπεζεύχθω ἡ ΔΗ· κύβου ἄρα ἐστὶν ἡ ΔΗ. ἐκκείσθω δή τις εὐθεῖα ἡ ΜΝ, ὥστε πενταπλάσιον εἶναι τὸ ἀπὸ τῆς
10	ΑΒ τοῦ ἀπὸ τῆς ΜΝ. ἔστι δὲ καὶ ἡ τῆς σφαίρας διάμετρος δυνάμει πενταπλασία τῆς ἐκ τοῦ κέντρου τοῦ κύκλου, ἀφ’ οὗ τὸ εἰκοσάεδρον ἀναγέγραπται. ἡ ΜΝ ἄρα ἴση ἐστὶ τῇ ἀπὸ τοῦ κέντρου τοῦ κύκλου, ἀφ’ οὗ τὸ εἰκοσάεδρον ἀναγέ‐
10	γραπται. τετμήσθω ἡ ΜΝ ἄκρον καὶ μέσον λόγον κατὰ τὸ	6
15	Ξ, καὶ ἔστω μεῖζον τμῆμα ἡ ΜΞ. δεκαγώνου ἄρα ἡ ΜΞ. καὶ ἐπεὶ πενταπλάσιόν ἐστι τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΜΝ, τριπλάσιον δὲ τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΔΗ, τρία ἄρα τὰ ἀπὸ ΔΗ ἴσα εἰσὶ πέντε τοῖς ἀπὸ ΜΝ. ὡς δὲ τρία τὰ ἀπὸ ΔΗ πρὸς τρία τὰ ἀπὸ ΓΗ, οὕτως πέντε τὰ ἀπὸ ΜΝ
20	πρὸς πέντε τὰ ἀπὸ ΜΞ. πέντε δὲ τὰ ἀπὸ ΜΞ καὶ πέντε τὰ ἀπὸ ΜΝ ἴσα εἰσὶ πέντε τοῖς ἀπὸ ΚΛ. πέντε ἄρα τὰ ἀπὸ ΚΛ ἴσα εἰσὶ τρισὶ τοῖς ἀπὸ ΓΗ καὶ τρισὶ τοῖς ἀπὸ ΔΗ. ἀλλὰ πέντε μὲν τὰ ἀπὸ ΚΛ ἴσα εἰσὶ δεκαπέντε τοῖς ἀπὸ τῆς ἐκ τοῦ κέντρου τοῦ περιγραφομένου περὶ
25	τὸ ΘΚΛ τρίγωνον κύκλου, τρία δὲ τὰ ἀπὸ ΔΗ καὶ τρία τὰ ἀπὸ ΓΗ ἴσα εἰσὶ ιε τοῖς ἀπὸ τῆς ἐκ τοῦ κέντρου τοῦ περιγραφομένου κύκλου περὶ τὸ ΓΔΕΖΗ· προεδείχθη γὰρ τὸ ἀπὸ τῆς ΔΗ μετὰ τοῦ ἀπὸ τῆς ΓΗ πενταπλάσια τοῦ ἀπὸ τῆς ἐκ τοῦ κέντρου τοῦ κύκλου τοῦ περιγραφο‐
30	μένου περὶ τὸ πεντάγωνον τὸ ΓΔΕΖΗ. δεκαπέντε ἄρα τὰ ἀπὸ τῆς ἐκ τοῦ κέντρου ἴσα ἐστὶ δεκαπέντε τοῖς ἀπὸ τῆς ἐκ τοῦ κέντρου· ἡ ἄρα διάμετρος ἴση τῇ διαμέτρῳ. Ὁ αὐτὸς ἄρα κύκλος περιλαμβάνει τό τε τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον τῶν εἰς τὴν	7
35	αὐτὴν σφαῖραν ἐγγραφομένων.
4	Ἐὰν ᾖ πεντάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον καὶ περὶ αὐτὸ κύκλος, καὶ ἀπὸ τοῦ κέντρου κάθετος ἐπὶ μίαν πλευ‐ ρὰν ἀχθῇ, τὸ τριακοντάκις ὑπὸ μιᾶς τῶν πλευρῶν καὶ τῆς καθέτου ἴσον ἐστὶ τῇ τοῦ δωδεκαέδρου ἐπιφανείᾳ.
5	Ἔστω πεντάγωνον ἰσόπλευρόν τε καὶ ἰσογώνιον τὸ ΑΒΓΔΕ καὶ περὶ τὸ πεντάγωνον κύκλος ὁ ΑΓΔ, καὶ [Omitted graphic marker]εἰλήφθω τὸ κέντρον τοῦ κύκλου τὸ Ζ, καὶ ἀπὸ τοῦ Ζ ἐπὶ τὴν ΓΔ κάθετος ἤχθω ἡ ΖΗ. λέγω, ὅτι
10	τὸ τριακοντάκις ὑπὸ ΓΔ, ΖΗ ἴσον ἐστὶ δώδεκα πενταγώνοις τοῖς ΑΒΓΔΕ. Ἐπεζεύχθωσαν αἱ ΓΖ, ΖΔ. ἐπεὶ οὖν τὸ ὑπὸ ΓΔ, ΖΗ διπλά‐
15	σιόν ἐστι τοῦ ΓΖΔ τριγώνου, τὸ ἄρα πεντάκις ὑπὸ ΓΔ, ΖΗ δέκα τρίγωνά ἐστι. καὶ πάντα ἑξάκις. τὸ ἄρα τριακοντάκις ὑπὸ ΓΔ, ΖΗ ἴσον ἐστὶ τῇ
15	τοῦ δωδεκαέδρου ἐπιφανείᾳ.	8
5	Ὁμοίως δὴ δείξομεν, ὅτι, ἐὰν ᾖ ἰσόπλευρον τρίγωνον τὸ ΑΒΓ καὶ περὶ αὐτὸ κύκλος καὶ τὸ κέντρον τοῦ κύκλου τὸ Δ, κάθετος δὲ ἐπὶ τὴν ΒΓ ἡ ΔΕ,[Omitted graphic marker] τὸ τριακοντάκις ὑπὸ ΒΓ, ΔΕ ἴσον
5	ἐστὶ τῇ τοῦ εἰκοσαέδρου ἐπιφα‐ νείᾳ. Ἐπεὶ γὰρ πάλιν τὸ ὑπὸ ΔΕ, ΒΓ διπλάσιόν ἐστι τοῦ ΔΒΓ τριγώ‐ νου, δύο ἄρα τρίγωνα τὰ ΔΒΓ ἴσα
10	ἐστὶ τῷ ὑπὸ ΔΕ, ΒΓ. καὶ πάντα τρίς· ἓξ ἄρα τρίγωνα τὰ ΔΒΓ ἴσα τρισὶ τοῖς ὑπὸ ΔΕ, ΒΓ. ἓξ δὲ τρίγωνα τὰ ΔΒΓ δύο ἐστὶ τρίγωνα τὰ ΑΒΓ. τρία ἄρα τὰ ὑπὸ ΔΕ, ΒΓ ἴσα ἐστὶ δυσὶ τοῖς ΑΒΓ. καὶ πάντα δεκάκις. τὸ ἄρα τριακοντάκις
15	ὑπὸ ΔΕ, ΒΓ ἴσον ἐστὶν εἴκοσι τοῖς ΑΒΓ τριγώνοις, τουτ‐ έστι τῇ τοῦ εἰκοσαέδρου ἐπιφανείᾳ. ὥστε καὶ ὡς ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφά‐ νειαν, οὕτως τὸ ὑπὸ τῆς πλευρᾶς αὐτοῦ καὶ τῆς ἀπὸ τοῦ κέντρου τοῦ περὶ τὸ ΑΒΓΔΕ πεντάγωνον κύκλου ἐπ’
20	αὐτὴν καθέτου ἀγομένης πρὸς τὸ ὑπὸ τῆς πλευρᾶς τοῦ εἰκοσαέδρου καὶ τῆς ἀπὸ τοῦ κέντρου τοῦ περὶ τὸ τρίγωνον κύκλου ἐπ’ αὐτὴν καθέτου ἀγομένης τῶν εἰς τὴν αὐτὴν
20	σφαῖραν ἐγγραφομένων εἰκοσαέδρου καὶ δωδεκαέδρου.	9
6	Τούτου δήλου ὄντος δεικτέον, ὅτι ὡς ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν, οὕτως ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου πλευράν. Ἐκκείσθω κύκλος ὁ περιλαμβάνων τό τε τοῦ δωδεκα‐
5	έδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον τῶν εἰς [Omitted graphic marker]τὴν αὐτὴν σφαῖραν ἐγγραφο‐ μένων ὁ ΑΒΓ, καὶ ἐγγεγρά‐ φθω εἰς τὸν ΑΒΓ κύκλον εἰκο‐ σαέδρου μὲν πλευρὰ ἡ ΓΔ,
10	δωδεκαέδρου δὲ ἡ ΑΓ. τριγώ‐ νου μὲν ἄρα ἰσοπλεύρου ἐστὶ πλευρὰ ἡ ΓΔ, πενταγώνου δὲ ἡ ΑΓ. καὶ εἰλήφθω τὸ κέν‐ τρον τοῦ κύκλου τὸ Ε, καὶ
15	ἀπὸ τοῦ Ε ἐπὶ τὰς ΔΓ, ΓΑ κάθετοι ἤχθωσαν αἱ ΕΖ, ΕΗ, καὶ ἐκβεβλήσθω ἐπ’ εὐθείας τῇ ΕΗ εὐθεῖα ἡ ΗΒ, καὶ ἐπεζεύχθω ἡ ΒΓ, καὶ ἐκκείσθω κύβου πλευρὰ ἡ Θ. λέγω, ὅτι ἐστὶν ὡς ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ
20	εἰκοσαέδρου ἐπιφάνειαν, οὕτως ἡ Θ πρὸς τὴν ΓΔ. Ἐπεὶ γὰρ συναμφοτέρου τῆς ΒΕ, ΒΓ ἄκρον καὶ μέσον λόγον τετμημένης τὸ μεῖζον τμῆμά ἐστιν ἡ ΒΕ, καί ἐστι συναμφοτέρου μὲν τῆς ΕΒΓ ἡμίσεια ἡ ΕΗ, τῆς δὲ ΒΕ
20	ἡμίσεια ἡ ΕΖ, τῆς ΕΗ ἄρα ἄκρον καὶ μέσον λόγον τεμνο‐	10
25	μένης τὸ μεῖζον τμῆμά ἐστιν ἡ ΕΖ. ἔστι δὲ καὶ τῆς Θ ἄκρον καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμα ἡ ΑΓ. ὡς ἄρα ἡ Θ πρὸς τὴν ΓΑ, οὕτως ἡ ΗΕ πρὸς τὴν ΕΖ. ἴσον ἄρα τὸ ὑπὸ ΖΕ, Θ τῷ ὑπὸ ΓΑ, ΕΗ. καὶ ἐπεί ἐστιν ὡς ἡ Θ πρὸς τὴν ΓΔ, οὕτως τὸ ὑπὸ ΖΕ, Θ πρὸς τὸ ὑπὸ
30	ΓΔ, ΖΕ, τῷ δὲ ὑπὸ ΖΕ, Θ ἴσον ἐστὶ τὸ ὑπὸ ΓΑ, ΗΕ, ὡς ἄρα ἡ Θ πρὸς τὴν ΓΔ, οὕτως τὸ ὑπὸ τῶν ΓΑ, ΗΕ πρὸς τὸ ὑπὸ τῶν ΓΔ, ΖΕ, τουτέστιν ἡ τοῦ δωδεκαέδρου ἐπι‐ φάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν. ὡς ἄρα ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου
35	ἐπιφάνειαν, οὕτως ἡ Θ πρὸς τὴν ΓΔ.
7	Καὶ ἄλλως δεῖξαι, ὅτι ἐστὶν ὡς ἡ τοῦ δωδεκαέδρου ἐπι‐ φάνεια πρὸς τὴν τοῦ εἰκοσα‐ [Omitted graphic marker] έδρου ἐπιφάνειαν, οὕτως ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ
5	εἰκοσαέδρου πλευράν, προ‐ γραφέντος τοῦδε· Ἔστω κύκλος ὁ ΑΒΓ, καὶ ἐγγεγράφθωσαν εἰς τὸν ΑΒΓ κύκλον πενταγώνου ἰσοπλεύ‐
10	ρου πλευραὶ αἱ ΑΒ, ΑΓ, καὶ ἐπεζεύχθω ἡ ΒΓ, καὶ εἰλή‐
10	φθω τὸ κέντρον τοῦ κύκλου τὸ Δ, καὶ ἀπὸ τοῦ Α ἐπὶ τὸ Δ ἐπεζεύχθω ἡ ΑΔ, καὶ ἐκβεβλή‐ σθω ἐπ’ εὐθείας τῇ ΑΔ εὐθεῖα ἡ ΔΕ, καὶ κείσθω τῆς μὲν	11
15	ΑΔ ἡμίσεια ἡ ΔΖ, ἡ δὲ ΗΓ τῆς ΓΘ ἔστω τριπλῆ. λέγω, ὅτι τὸ ὑπὸ ΑΖ, ΒΘ ἴσον ἐστὶ τῷ πενταγώνῳ. Ἀπὸ γὰρ τοῦ Β ἐπὶ τὸ Δ ἐπεζεύχθω ἡ ΒΔ. ἐπεὶ διπλῆ ἐστιν ἡ ΑΔ τῆς ΔΖ, ἡμιολία ἄρα ἐστὶ τῆς ΑΔ ἡ ΑΖ. πά‐ λιν ἐπεὶ τριπλῆ ἐστιν ἡ ΗΓ τῆς ΓΘ, διπλῆ ἡ ΗΘ τῆς
20	ΘΓ. ἡμιολία ἄρα ἐστὶν ἡ ΗΓ τῆς ΘΗ. ὡς ἄρα ἡ ΖΑ πρὸς τὴν ΑΔ, οὕτως ἡ ΓΗ πρὸς τὴν ΗΘ. ἴσον ἄρα τὸ ὑπὸ ΑΖ, ΗΘ τῷ ὑπὸ ΑΔ, ΓΗ. ἡ δὲ ΓΗ τῇ ΗΒ ἴση. τὸ ἄρα ὑπὸ ΑΔ, ΗΒ τῷ ὑπὸ ΖΑ, ΗΘ ἴσον ἐστίν. ἀλλὰ τὸ ὑπὸ ΑΔ, ΗΒ δύο ἐστὶ τρίγωνα τὰ ΑΒΔ. καὶ τὸ ὑπὸ ΖΑ, ΗΘ ἄρα
25	δύο ἐστὶ τρίγωνα τὰ ΑΒΔ. ὥστε καὶ πέντε ἄρα τὰ ὑπὸ ΑΖ, ΗΘ δέκα ἐστὶ τρίγωνα. δέκα δὲ τρίγωνα δύο ἐστὶ πεντάγωνα. πέντε ἄρα τὰ ὑπὸ ΑΖ, ΗΘ δύο πενταγώνοις ἴσα ἐστίν. ἐπεὶ οὖν διπλῆ ἐστιν ἡ ΗΘ τῆς ΘΓ, τὸ ὑπὸ ΑΖ, ΗΘ διπλοῦν ἐστι τοῦ ὑπὸ ΑΖ, ΘΓ. δύο ἄρα τὰ ὑπὸ ΑΖ,
30	ΘΓ ἴσα ἐστὶ τῷ ὑπὸ ΑΖ, ΗΘ. καὶ δέκα ἄρα τὰ ὑπὸ ΑΖ, ΘΓ ἴσα ἐστὶ πέντε τοῖς ὑπὸ ΑΖ, ΗΘ, τουτέστι δύο πεντα‐
30	γώνοις. ὥστε πέντε τὰ ὑπὸ ΑΖ, ΘΓ ἴσα ἐστὶν ἑνὶ πεντα‐ γώνῳ. πεντάκις δὲ τὰ ὑπὸ ΑΖ, ΘΓ ἴσα ἐστὶ τῷ ὑπὸ ΑΖ, ΒΘ, ἐπειδὴ πενταπλῆ ἐστιν ἡ ΒΘ τῆς ΘΓ, καὶ κοινὸν	12
35	ὕψος ἐστὶν ἡ ΑΖ. τὸ ἄρα ὑπὸ ΑΖ, ΒΘ ἴσον ἐστὶν ἑνὶ πενταγώνῳ.
8	Τούτου δήλου ὄντος νῦν ἐκκείσθω ὁ περιλαμβάνων κύ‐ κλος τό τε τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ εἰκο‐ σαέδρου τρίγωνον τῶν εἰς τὴν[Omitted graphic marker] αὐτὴν σφαῖραν ἐγγραφομέ‐
5	νων ὁ ΑΒΓ, καὶ ἐγγεγρά‐ φθωσαν εἰς τὸν ΑΒΓ κύκλον πενταγώνου ἰσοπλεύρου πλευ‐ ραὶ αἱ ΒΑ, ΑΓ, καὶ ἐπ‐ εζεύχθω ἡ ΒΓ, καὶ εἰλήφθω
10	τὸ κέντρον τοῦ κύκλου τὸ Ε, καὶ ἀπὸ τοῦ Α ἐπὶ τὸ Ε ἐπ‐ εζεύχθω ἡ ΑΕ, καὶ ἐκβεβλή‐ σθω ἐπὶ τὸ Ζ, καὶ ἔστω ἡ ΑΕ τῆς ΕΗ διπλῆ, τριπλῆ δὲ
15	ἡ ΚΓ τῆς ΘΓ, καὶ ἀπὸ τοῦ Η τῇ ΑΖ πρὸς ὀρθὰς ἤχθω ἡ ΗΜ, καὶ ἐκβεβλήσθω ἐπ’ εὐθείας ἡ ΗΔ τῇ ΗΜ. τριγώ‐ νου ἄρα ἰσοπλεύρου ἐστὶν ἡ ΔΜ. ἐπεζεύχθωσαν αἱ ΑΔ, ΑΜ. ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΔΜ τρίγωνον. καὶ ἐπεὶ τὸ μὲν ὑπὸ ΑΗ, ΘΒ ἴσον ἐστὶ τῷ πενταγώνῳ, τὸ δὲ ὑπὸ
20	ΑΗΔ τῷ ΑΔΜ τριγώνῳ, ἔστιν ἄρα ὡς τὸ ὑπὸ ΑΗ, ΘΒ
20	πρὸς τὸ ὑπὸ ΔΗΑ, οὕτως τὸ πεντάγωνον πρὸς τὸ τρίγω‐ νον. ὡς δὲ τὸ ὑπὸ ΒΘ, ΑΗ πρὸς τὸ ὑπὸ ΔΗΑ, οὕτως ἡ ΒΘ πρὸς τὴν ΔΗ. καὶ ὡς ἄρα δώδεκα αἱ ΒΘ πρὸς εἴκοσι τὰς ΔΗ, οὕτως δώδεκα πεντάγωνα πρὸς εἴκοσι τρίγωνα,	13
25	τουτέστιν ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκο‐ σαέδρου. καί εἰσι δώδεκα μὲν αἱ ΒΘ δέκα αἱ ΒΓ· ἡ μὲν γὰρ ΒΘ τῆς ΘΓ ἐστι πενταπλῆ, ἡ δὲ ΒΓ τῆς ΓΘ ἐστιν ἑξαπλῆ. ἓξ ἄρα αἱ ΒΘ ἴσαι εἰσὶ πέντε ταῖς ΒΓ. καὶ τὰ διπλάσια δέ. εἴκοσι δὲ αἱ ΔΗ δέκα εἰσὶν αἱ ΔΜ· διπλῆ γὰρ
30	ἡ ΔΜ τῆς ΔΗ. ὡς ἄρα δέκα αἱ ΒΓ πρὸς δέκα τὰς ΔΜ, οὕτως ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκο‐ σαέδρου ἐπιφάνειαν. καί ἐστιν ἡ μὲν ΒΓ ἡ τοῦ κύβου πλευ‐ ρά, ἡ δὲ ΔΜ ἡ τοῦ εἰκοσαέδρου. καὶ ὡς ἄρα ἡ τοῦ δωδε‐ καέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν,
35	οὕτως ἡ ΒΓ πρὸς τὴν ΔΜ, τουτέστιν ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου πλευράν.
9	Δεικτέον δή, ὅτι καὶ εὐθείας οἱασδηποτοῦν τμηθείσης ἄκρον καὶ μέσον λόγον, ὡς ἔχει ἡ δυναμένη τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ μείζονος τμήματος πρὸς τὴν δυνα‐ μένην τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ ἐλάττονος τμήματος,
5	τοῦτον ἔχει τὸν λόγον ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ
5	εἰκοσαέδρου πλευράν. Ἔστω ὁ περιλαμβάνων κύκλος τό τε τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφομένων ὁ ΑΘΒ, καὶ εἰλήφθω τὸ	14
10	κέντρον τοῦ κύκλου τὸ Γ, καὶ[Omitted graphic marker] προσεκβεβλήσθω τις, ὡς ἔτυχεν, ἀπὸ τοῦ Γ σημείου ἡ ΓΒ καὶ τετμήσθω ἄκρον καὶ μέσον λόγον κατὰ τὸ Δ, καὶ
15	τὸ μεῖζον τμῆμα ἔστω ἡ ΓΔ. δεκαγώνου ἄρα πλευρά ἐστιν ἡ ΓΔ τοῦ εἰς τὸν αὐτὸν κύ‐ κλον ἐγγραφομένου. ἐκκεί‐ σθω δὴ εἰκοσαέδρου πλευρὰ
20	ἡ Ε, δωδεκαέδρου δὲ ἡ Ζ, κύβου δὲ ἡ Η. ἡ μὲν ἄρα Ε τριγώνου ἰσοπλεύρου ἐστὶ πλευρά, ἡ δὲ Ζ πενταγώνου τοῦ εἰς τὸν αὐτὸν κύκλον ἐγ‐
25	γραφομένου, ἡ δὲ Ζ τῆς Η μεῖζόν ἐστι τμῆμα ἄκρον καὶ μέσον λόγον τεμνομένης. ἐπεὶ ἡ Ε ἴση ἐστὶ τῇ τοῦ ἰσοπλεύρου τριγώνου πλευρᾷ, ἡ δὲ τοῦ τριγώνου τοῦ ἰσοπλεύρου πλευρὰ δυνάμει τριπλασία ἐστὶ τῆς ΒΓ [τριπλάσιον ἄρα ἐστὶ τὸ ἀπὸ τῆς Ε τοῦ ἀπὸ τῆς ΒΓ],
30	ἔστι δὲ καὶ τὰ ἀπὸ τῶν ΓΒΔ τριπλάσια τοῦ ἀπὸ ΓΔ, ὡς ἄρα τὸ ἀπὸ τῆς Ε πρὸς τὸ ἀπὸ τῆς ΓΒ, οὕτως τὰ ἀπὸ τῶν ΓΒ, ΒΔ πρὸς τὸ ἀπὸ τῆς ΓΔ. ἐναλλάξ, ὡς τὸ ἀπὸ τῆς Ε πρὸς τὰ ἀπὸ τῶν ΓΒ, ΒΔ, οὕτως τὸ ἀπὸ τῆς ΓΒ πρὸς τὸ ἀπὸ τῆς ΓΔ. ὡς δὲ τὸ ἀπὸ τῆς ΒΓ πρὸς τὸ ἀπὸ τῆς ΓΔ,	15
35	οὕτως τὸ ἀπὸ τῆς Η πρὸς τὸ ἀπὸ τῆς Ζ· μεῖζον γάρ ἐστι τμῆμα ἡ Ζ τῆς Η. καὶ ὡς ἄρα τὸ ἀπὸ τῆς Ε πρὸς τὰ ἀπὸ τῶν ΓΒ, ΒΔ, οὕτως τὸ ἀπὸ Η πρὸς τὸ ἀπὸ Ζ. ἐναλλὰξ καὶ ἀνάπαλιν· ὡς ἄρα τὸ ἀπὸ Η πρὸς τὸ ἀπὸ Ε, οὕτως τὸ ἀπὸ Ζ πρὸς τὰ ἀπὸ τῶν ΓΒΔ. τῷ δὲ ἀπὸ τῆς Ζ ἴσα τὰ ἀπὸ τῶν
40	ΒΓΔ· ἡ γὰρ τοῦ πενταγώνου πλευρὰ δύναται τήν τε τοῦ ἑξαγώνου πλευρὰν καὶ τὴν τοῦ δεκαγώνου τῶν εἰς τὸν αὐτὸν κύκλον ἐγγραφομένων. ὡς ἄρα τὸ ἀπὸ τῆς Η πρὸς τὸ ἀπὸ τῆς Ε, οὕτως τὰ ἀπὸ τῶν ΒΓΔ πρὸς τὰ ἀπὸ τῶν
40	ΓΒΔ. καὶ ὡς ἄρα τὸ ἀπὸ τῆς Η πρὸς τὸ ἀπὸ τῆς Ε, οὕτως	16
45	εὐθείας ἄκρον καὶ μέσον λόγον τεμνομένης τὸ ἀπὸ τῆς δυναμένης τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ μείζονος τμήματος πρὸς τὸ ἀπὸ τῆς δυναμένης τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ ἐλάττονος τμήματος. καί ἐστιν ἡ μὲν Η κύβου πλευρά, ἡ δὲ Ε εἰκοσαέδρου.
9 (50)	Ἐὰν ἄρα εὐθεῖα ἄκρον καὶ μέσον λόγον τμηθῇ, ἔσται ὡς ἡ δυναμένη τὴν ὅλην καὶ τὸ μεῖζον τμῆμα πρὸς τὴν δυνα‐ μένην τὴν ὅλην καὶ τὸ ἔλαττον τμῆμα, οὕτως ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου πλευρὰν τῶν εἰς τὴν αὐ‐ τὴν σφαῖραν ἐγγραφομένων.
10	Καὶ δεικτέον, ὅτι ὡς ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου, οὕτως τὸ στερεὸν τοῦ δωδεκαέδρου πρὸς τὸ στερεὸν τοῦ εἰκοσαέδρου. Ἐπεὶ γὰρ ἴσοι κύκλοι περιλαμβάνουσι τό τε τοῦ δωδε‐
5	καέδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφομένων, ἐν δὲ ταῖς σφαίραις οἱ ἴσοι κύκλοι ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου, αἱ ἄρα ἀπὸ τοῦ κέντρου τῆς σφαίρας ἐπὶ τὰ τῶν κύκλων ἐπίπεδα κάθετοι ἀγόμεναι ἴσαι τέ εἰσι καὶ ἐπὶ τὰ κέντρα τῶν κύκλων πεσοῦν‐
10	ται. ὥστε αἱ ἀπὸ τοῦ κέντρου τῆς σφαίρας ἐπὶ τὸ κέντρον τοῦ περιλαμβάνοντος κύκλου τό τε τοῦ εἰκοσαέδρου τρίγω‐
10	νον καὶ τὸ τοῦ δωδεκαέδρου πεντάγωνον κάθετοι ἀγόμεναι ἴσαι εἰσίν. ἰσοϋψεῖς ἄρα εἰσὶν αἱ πυραμίδες αἱ βάσεις ἔχου‐ σαι τὰ τοῦ δωδεκαέδρου πεντάγωνα καὶ αἱ βάσεις ἔχουσαι	17
15	τὰ τοῦ εἰκοσαέδρου τρίγωνα. αἱ δὲ ἰσοϋψεῖς πυραμίδες πρὸς ἀλλήλας εἰσὶν ὡς αἱ βάσεις. ὡς ἄρα τὸ πεντάγωνον πρὸς τὸ τρίγωνον, οὕτως ἡ πυραμίς, ἧς βάσις μέν ἐστι τὸ πεντάγω‐ νον, κορυφὴ δὲ τὸ κέντρον τῆς σφαίρας, πρὸς τὴν πυρα‐ μίδα τὴν βάσιν μὲν ἔχουσαν τὸ τρίγωνον, κορυφὴν δὲ τὸ
20	κέντρον τῆς σφαίρας. καὶ ὡς ἄρα δώδεκα πεντάγωνα πρὸς εἴκοσι τρίγωνα, οὕτως δώδεκα πυραμίδες πενταγώνους βά‐ σεις ἔχουσαι πρὸς εἴκοσι πυραμίδας τριγώνους βάσεις ἐχούσας. καὶ δώδεκα μὲν πεντάγωνα ἡ τοῦ δωδεκαέδρου ἐστὶν ἐπιφάνεια, εἴκοσι δὲ τρίγωνα ἡ τοῦ εἰκοσαέδρου ἐπι‐
25	φάνεια. ἔστιν ἄρα ὡς ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου, οὕτως ιβ πυραμίδες πενταγώνους ἔχουσαι βάσεις πρὸς εἴκοσι πυραμίδας τριγώνους βάσεις ἐχούσας. καί εἰσι ιβ μὲν πυραμίδες πενταγώνους βάσεις ἔχουσαι τὸ στερεὸν τοῦ δωδεκαέδρου, εἴκοσι δὲ πυραμί‐
30	δες τριγώνους βάσεις ἔχουσαι τὸ στερεὸν τοῦ εἰκοσαέδρου. καὶ ὡς ἄρα ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν, οὕτως τὸ στερεὸν τοῦ δωδεκαέδρου πρὸς τὸ στερεὸν τοῦ εἰκοσαέδρου. ὡς δὲ ἡ ἐπιφάνεια τοῦ
30	δωδεκαέδρου πρὸς τὴν ἐπιφάνειαν τοῦ εἰκοσαέδρου, ἐδεί‐	18
35	χθη ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου πλευ‐ ράν. καὶ ὡς ἄρα ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκο‐ σαέδρου πλευράν, οὕτως τὸ στερεὸν τοῦ δωδεκαέδρου πρὸς τὸ στερεὸν τοῦ εἰκοσαέδρου.
11	Ὅτι δέ, ἐὰν δύο εὐθεῖαι ἄκρον καὶ μέσον λόγον τμη‐ θῶσιν, ἐν ἀναλογίᾳ εἰσὶ τῇ ὑποκειμένῃ, δείξομεν οὕ‐ τως· Τετμήσθω γὰρ ἡ μὲν ΑΒ ἄκρον καὶ μέσον λόγον κατὰ τὸ
5	Γ, καὶ τὸ μεῖζον αὐτῆς τμῆμα ἔστω ἡ ΑΓ. ὁμοίως δὲ καὶ[Omitted graphic marker] ἡ ΔΕ ἄκρον καὶ μέσον λόγον τετμήσθω κατὰ τὸ Ζ, καὶ τὸ μεῖζον αὐτῆς τμῆμα ἔστω ἡ ΔΖ. λέγω, ὅτι ὡς ὅλη ἡ ΑΒ πρὸς τὴν ΑΓ, οὕτως ὅλη ἡ ΔΕ πρὸς τὸ μεῖζον τμῆμα τὴν ΔΖ.
10	Ἐπεὶ γὰρ τὸ μὲν ὑπὸ ΑΒΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ, τὸ δὲ ὑπὸ ΔΕΖ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΔΖ, ἔστιν ἄρα ὡς τὸ ὑπὸ ΑΒΓ πρὸς τὸ ἀπὸ τῆς ΑΓ, οὕτως τὸ ὑπὸ ΔΕΖ πρὸς τὸ ἀπὸ τῆς ΔΖ. καὶ ὡς τὸ τετράκις ἄρα ὑπὸ ΑΒΓ
10	πρὸς τὸ ἀπὸ τῆς ΑΓ, οὕτως τὸ τετράκις ὑπὸ ΔΕΖ πρὸς	19
15	τὸ ἀπὸ ΔΖ. καὶ συνθέντι ὡς τὸ τετράκις ὑπὸ ΑΒΓ μετὰ τοῦ ἀπὸ τῆς ΑΓ πρὸς τὸ ἀπὸ τῆς ΑΓ, οὕτως τὸ τετράκις ὑπὸ ΔΕΖ μετὰ τοῦ ἀπὸ τῆς ΔΖ πρὸς τὸ ἀπὸ τῆς ΔΖ. ὥστε καὶ ὡς τὸ ἀπὸ συναμφοτέρου τῆς ΑΒΓ πρὸς τὸ ἀπὸ τῆς ΑΓ, οὕτως τὸ ἀπὸ συναμφοτέρου τῆς ΔΕΖ πρὸς τὸ
20	ἀπὸ τῆς ΔΖ. καὶ μήκει ὡς συναμφότερος ἡ ΑΒΓ μετὰ τῆς ΑΓ, τουτέστι δύο αἱ ΑΒ, πρὸς τὴν ΑΓ, οὕτως συναμφό‐ τερος ἡ ΔΕΖ μετὰ τῆς ΔΖ, τουτέστι δύο αἱ ΔΕ, πρὸς τὴν ΔΖ. καὶ τὰ ἡμίση, ὡς ἡ ΑΒ πρὸς τὴν ΑΓ, οὕτως ἡ ΔΕ πρὸς τὴν ΔΖ.
12	Καὶ ὅτι εὐθείας οἱασδηποτοῦν ἄκρον καὶ μέσον λόγον τμηθείσης τὸν λόγον, ὃν ἔχει ἡ δυναμένη τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ μείζονος τμήματος πρὸς τὴν δυναμένην τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ ἐλάττονος τμήματος, τοῦτον
5	ἔχει ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου πλευ‐ ράν. δεδειγμένου δὲ καὶ τοῦδε, ὅτι ὡς ἡ τοῦ κύβου πλευρὰ
5	πρὸς τὴν τοῦ εἰκοσαέδρου πλευράν, οὕτως ἡ τοῦ δωδε‐ καέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφομένων, προσενηνεγμέ‐	20
10	νου δὲ καὶ τοῦδε, ὅτι ὡς ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν, καὶ αὐτὸ τὸ δωδεκάεδρον πρὸς τὸ εἰκοσάεδρον διὰ τὸ ὑπὸ τοῦ αὐτοῦ κύκλου περι‐ λαμβάνεσθαι τό τε τοῦ δωδεκαέδρου πεντάγωνον καὶ τὸ τοῦ εἰκοσαέδρου τρίγωνον, δῆλον, ὅτι, ἐὰν εἰς τὴν αὐτὴν
15	σφαῖραν ἐγγραφῇ δωδεκάεδρόν τε καὶ εἰκοσάεδρον, λόγον ἕξει εὐθείας ἡσδηποτοῦν ἄκρον καὶ μέσον λόγον τμηθείσης ὡς ἡ δυναμένη τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ μείζονος τμήματος πρὸς τὴν δυναμένην τὸ ἀπὸ τῆς ὅλης καὶ τὸ ἀπὸ τοῦ ἐλάττονος τμήματος.
20	Τούτων δὴ πάντων γνωρίμων ἡμῖν γενομένων δῆλον, ὅτι, ἐὰν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφῇ δωδεκάεδρόν τε καὶ εἰκοσάεδρον, τὸ δωδεκάεδρον πρὸς τὸ εἰκοσάεδρον λόγον ἕξει εὐθείας ἡσδηποτοῦν ἄκρον καὶ μέσον λόγον τεμνο‐ μένης ὡς ἡ δυναμένη τὴν ὅλην καὶ τὸ μεῖζον τμῆμα πρὸς
25	τὴν δυναμένην τὴν ὅλην καὶ τὸ ἔλαττον τμῆμα. ἐπεὶ γάρ ἐστιν ὡς τὸ δωδεκάεδρον πρὸς τὸ εἰκοσάεδρον, οὕτως ἡ τοῦ δωδεκαέδρου ἐπιφάνεια πρὸς τὴν τοῦ εἰκοσαέδρου ἐπιφάνειαν, τουτέστιν ἡ τοῦ κύβου πλευρὰ πρὸς τὴν τοῦ εἰκοσαέδρου πλευράν, ὡς δὲ ἡ τοῦ κύβου πλευρὰ πρὸς τὴν
30	τοῦ εἰκοσαέδρου πλευράν, οὕτως εὐθείας ἡσδηποτοῦν ἄκρον καὶ μέσον λόγον τετμημένης ἡ δυναμένη τὴν ὅλην καὶ τὸ μεῖζον τμῆμα πρὸς τὴν δυναμένην τὴν ὅλην καὶ τὸ ἔλαττον τμῆμα, ὡς ἄρα τὸ δωδεκάεδρον πρὸς τὸ εἰκοσά‐ εδρον τῶν εἰς τὴν αὐτὴν σφαῖραν ἐγγραφομένων, εὐθείας	21
35	ἡσδηποτοῦν ἄκρον καὶ μέσον λόγον τετμημένης ἡ δυνα‐ μένη τὴν ὅλην καὶ τὸ μεῖζον τμῆμα πρὸς τὴν δυναμένην	21
35	τὴν ὅλην καὶ τὸ ἔλαττον τμῆμα.	22

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