§1.2## 2.
2.
Scholia Vaticana in Pappum III, p. 1171. α΄.
Scholia Vaticana in Pappum III, p. 1171. 1.
Ὀκτάεδρον ἔχει τρίγωνα δ, ἑξάγωνα δὲ δ, πλευρὰς ιη, γωνίας δὲ στερεὰς ιβ, ἑκάστη δὲ στερεὰ γωνία περιέχεται ὑπὸ γ γωνιῶν ἐπιπέδων, ὧν δύο μὲν ἑξαγωνικαί, μία δὲ τριγωνική, ὥστε λείπειν τῶν δ ὀρθῶν μιᾶς ὀρθῆς γωνίας δύο τριτημορίοις.
The octahedron has 4 triangles and 4 hexagons, 18 edges, and 12 solid angles; and each solid angle is contained by 3 plane angles, of which two are hexagonal and one is triangular, so as to fall short of 4 right angles by two-thirds of one right angle.
Τοῦτο γεννᾶται ἐκ τῆς πρώτης πυραμίδος διαιρουμένων τῶν πλευρῶν αὐτῆς εἰς ἴσα καὶ διὰ τῶν τομῶν ἐπιπέδων ἐκβαλλομένων καὶ τῶν γωνιῶν ἐκπιπτουσῶν. β΄.
This is generated from the first pyramid, its edges being divided into equal parts, and planes being extended through the cuts, and the angles falling away. 2.
Τεσσαρεσκαιδεκάεδρον περιέχεται ὑπὸ μὲν τριγώνων η, ὑπὸ δὲ τετραγώνων Ϛ΄, ἔχει δὲ πλευρὰς κδ, γωνίας δὲ στερεὰς ιβ, ἑκάστη δὲ στερεὰ γωνία περιέχεται ὑπὸ δ γωνιῶν ἐπιπέδων, ὧν δύο μὲν τετραγωνικαί, β δὲ τριγωνικαί, ὥστε λείπειν τῶν δ ὀρθῶν μιᾶς γωνίας ὀρθῆς δύο τριτημορίοις.
The tetradecahedron is contained by 8 triangles and 6 squares, and has 24 edges and 12 solid angles; and each solid angle is contained by 4 plane angles, of which two are square and two are triangular, so as to fall short of 4 right angles by two-thirds of one right angle.
Τοῦτο γεννᾶται ἐκ τοῦ κύβου διαιρουμένων δίχα τῶν πλευρῶν αὐτοῦ καὶ διὰ τῶν τομῶν ἐπιπέδων ἐκβαλλομένων τῶν η γωνιῶν ἐκπιπτουσῶν. γ΄.
This is generated from the cube, its edges being divided in half, and planes being extended through the cuts, and the 8 angles falling away. 3.
Τεσσαρεσκαιδεκάεδρον περιέχεται ὑπὸ μὲν τετραγώνων Ϛ΄, ὑπὸ δὲ ἑξαγώνων η, ἔχει δὲ πλευρὰς λϚ΄, γωνίας δὲ στερεὰς κδ, ἑκάστη δὲ στερεὰ γωνία περιέχεται ὑπὸ γωνιῶν ἐπιπέδων, ὧν δύο μὲν ἑξαγωνικαί, μία δὲ τετραγωνική.
The tetradecahedron is contained by 6 squares and 8 hexagons, and has 36 edges and 24 solid angles; and each solid angle is contained by plane angles, of which two are hexagonal and one is square.
Τοῦτο γεννᾶται ἐκ τοῦ ὀκταέδρου τεμνομένης τρίχα ἑκάστης τῶν αὐτοῦ πλευρῶν καὶ διὰ τῶν τομῶν ἐπιπέδων ἐκβαλλομένων καὶ τῶν Ϛ΄ γωνιῶν ἐκπιπτουσῶν. δ΄.
This is generated from the octahedron, each of its edges being trisected, and planes being extended through the cuts, and the 6 angles falling away. 4.
Τὸ δὲ τρίτον, ἐπεὶ περιέχεται τριγώνοις η καὶ ὀκταγώνοις Ϛ΄, ἕξει στερεὰς μὲν γωνίας κδ·
And the third, since it is contained by 8 triangles and 6 octagons, will have 24 solid angles; and each is contained by 3 plane angles, of which two are octagonal and one is triangular; and it has 36 edges.
ἑκάστη δὲ περιέχεται ὑπὸ γ γωνιῶν ἐπιπέδων, ὧν δύο ὀκταγωνικαί, μία δὲ τριγωνική πλευρὰς δὲ ἔχει λϚ΄, Τοῦτο γεννᾶται ἐκ τοῦ κύβου τεμνομένης ἑκάστης αὐτοῦ πλευρᾶς οὕτως, ὥστε γίνεσθαι τρία τμήματα, ὧν τὸ μέσον ἑκατέρου τῶν ἄκρων διπλάσιόν ἐστιν δυνάμει. ε΄.
This is generated from the cube, each of its edges being cut in such a way that three segments are made, of which the middle one is double in power of each of the extremes. 5.
Ἑκκαιεικοσάεδρον γεννᾶται ἐκ τοῦ τεσσαρεσκαιδεκαέδρου τοῦ περιεχομένου ὑπὸ η τριγώνων καὶ Ϛ΄ τετραγώνων τεμνομένης ἑκάστης αὐτοῦ πλευρᾶς δίχα καὶ διὰ τῶν τομῶν ἐκβαλλομένων ἐπιπέδων καὶ
The icosihexahedron is generated from the tetradecahedron contained by 8 triangles and 6 squares, each of its edges being divided in half, and planes being extended through the cuts, and...
§1.3## 3.
3.
Hero, Definitiones, ed.
Hero, Definitiones, ed.
J. L. Heiberg, p. 66.
J. L. Heiberg, p. 66.
Ὧν εἰδέναι καὶ Πλάτωνα τὸ τεσσαρεσκαιδεκάεδρον, εἶναί τε τοῦτο διπλοῦν, τὸ μὲν ἐξ ὀκτὼ τριγώνων καὶ τετραγώνων ἓξ σύνθετον, ἐκ γῆς καὶ ἀέρος, ὅπερ καὶ τῶν ἀρχαίων τινὲς ᾔδεσαν, τὸ δὲ ἕτερον πάλιν ἐκ τετραγώνων μὲν ὀκτώ, τριγώνων δὲ Ϛ΄, ὃ καὶ χαλεπώτερον εἶναι δοκεῖ.
Of these, Plato too is said to have known the tetradecahedron, and that this is twofold: one composed of eight triangles and six squares, from earth and air, which indeed some of the ancients knew, and the other, again, of eight squares and six triangles, which also seems to be more difficult.