§1.1#1## 1.
1.
Pappus V, 34, ed.
Pappus V, 34, ed.
Hultsch, p. 352.
Hultsch, p. 352.
Ταῦτα δ᾿ ἐστὶν οὐ μόνον τὰ παρὰ τῷ θειοτάτῳ Πλάτωνι πέντε σχήματα, τουτέστιν τετράεδρόν τε καὶ ἑξάεδρον, ὀκτάεδρόν τε καὶ δωδεκάεδρον, πέμπτον δ᾿ εἰκοσάεδρον, ἀλλὰ καὶ τὰ ὑπὸ Ἀρχιμήδους εὑρεθέντα τρισκαίδεκα τὸν ἀριθμὸν ὑπὸ ἰσοπλεύρων μὲν καὶ ἰσογωνίων, οὐχ ὁμοίων δὲ πολυγώνων περιεχόμενα.
These are not only the five figures in the most divine Plato, namely, the tetrahedron, hexahedron, octahedron, dodecahedron, and fifthly the icosahedron, but also the thirteen in number discovered by Archimedes, which are contained by equilateral and equiangular, but not similar, polygons.
Τὸ μὲν γὰρ πρῶτον ὀκτάεδρόν ἐστιν περιεχόμενον ὑπὸ τριγώνων △Ζ καὶ ἑξαγώνων δ.
For the first is an octahedron contained by 6 triangles (△) and 4 hexagons.
Τρία δὲ μετὰ τοῦτο τεσσαρεσκαιδεκάεδρα, ὧν τὸ μὲν πρῶτον περιέχεται τριγώνοις η καὶ τετραγώνοις ς΄, τὸ δὲ δεύτερον τετραγώνοις ς΄ καὶ ἑξαγώνοις η, τὸ δὲ τρίτον τριγώνοις η καὶ ὀκταγώνοις ς΄.
And after this, three tetradecahedra, of which the first is contained by 8 triangles and 6 squares, the second by 6 squares and 8 hexagons, and the third by 8 triangles and 6 octagons.
Μετὰ δὲ ταῦτα ἑκκαιεικοσάεδρά ἐστιν δύο, ὧν τὸ μὲν πρῶτον περιέχεται τριγώνοις η καὶ τετραγώνοις ιη, τὸ δὲ δεύτερον τετραγώνοις ιβ, ἑξαγώνοις η καὶ ὀκταγώνοις ς΄.
And after these, two icosihexahedra, of which the first is contained by 8 triangles and 18 squares, and the second by 12 squares, 8 hexagons, and 6 octagons.
Μετὰ δὲ ταῦτα δυοκαιτριακοντάεδρά ἐστιν τρία, ὧν τὸ μὲν πρῶτον περιέχεται τριγώνοις κ καὶ πενταγώνοις ιβ, τὸ δὲ δεύτερον πενταγώνοις ιβ καὶ ἑξαγώνοις κ, τὸ δὲ τρίτον τριγώνοις κ καὶ δεκαγώνοις ιβ.
And after these, three triacontadihedra, of which the first is contained by 20 triangles and 12 pentagons, the second by 12 pentagons and 20 hexagons, and the third by 20 triangles and 12 decagons.
Μετὰ δὲ ταῦτα ἔν ἐστιν ὁκτωκαιτριακοντάεδρον περιεχόμενον ὑπὸ τριγώνων λβ καὶ τετραγώνων ς΄.
And after these, one triacontaoctahedron contained by 32 triangles and 6 squares.
Μετὰ δὲ τοῦτο δυοκαιεξηκοντάεδρά ἐστι δύο, ὧν τὸ μὲν πρῶτον περιέχεται τργώνοις κ καὶ τετραγώνοις λ καὶ πενταγώνοις ιβ, τὸ δὲ δεύτερον τετραγώνοις λ καὶ ἑξαγώνοις κ καὶ δεκαγώνοις ιβ.
And after this, two hexacontadihedra, of which the first is contained by 20 triangles, 30 squares, and 12 pentagons, and the second by 30 squares, 20 hexagons, and 12 decagons.
Μετὰ δὲ ταῦτα τελευταῖόν ἐστιν δυοκαιενενηκοντάεδρον, ὃ περιέχεται τριγώνοις π καὶ πενταγώνοις ιβ.
And after these, the last is a nonacontadihedron, which is contained by 80 triangles and 12 pentagons.
Ὅσας δὲ γωνίας ἕκαστον ἔχει στερεὰς τῶν ιγ τούτων σχημάτων πολυέδρων καὶ ὅσας πλευράς, διὰ τοῦδε τοῦ τρόπου θεωρεῖται ὅσων μὲν γὰρ ἁπλῶς πολυέδρων αἱ στερεαὶ γωνίαι τρισὶν ἐπιπέδοις περιέχονται γωνίαις, ἐξαριθμηθεισῶν τῶν ἐπιπέδων γωνιῶν, ἃς ἔχουσιν πᾶσαι αἱ ἕδραι τοῦ πολυέδρου, δῆλον ὡς ὁ τῶν στερεῶν γωνιῶν ἀριθμὸς τρίτον μέρος ἐστὶ τοῦ γενομένου ἀριθμοῦ, ὅσων δὲ πολυέδρων ἡ στερεὰ γωνία περιέχεται τέσσαρσιν ἐπιπέδοις, ἐξαριθμηθεισῶν πασῶν τῶν ἐπιπέδων γωνιῶν, ἃς ἔχουσιν αἱ ἕδραι τοῦ πολυέδρου, τοῦ γενομένου ἀριθμοῦ τὸ τέταρτον μέρος ἐστὶν ὁ ἀριθμὸς ὁ τῶν στερεῶν γωνιῶν τοῦ πολυέδρου ὁμοίως δὲ καὶ ὅσων πολυέδρων ἡ στερεὰ γωνία περιέχεται ὑπὸ ε γωνιῶν ἐπιπέδων, τὸ πέμπτον τοῦ πλήθους τῶν ἐπιπέδων γωνιῶν ἐστιν ὁ ἀριθμὸς τοῦ πλήθους τῶν στερεῶν γωνιῶν.
How many solid angles and how many edges each of these 13 polyhedral figures has is observed in the following manner. For as many polyhedra whose solid angles are simply contained by three plane angles, when all the plane angles which all the faces of the polyhedron have are counted, it is clear that the number of solid angles is one-third of the resulting number; and for as many polyhedra whose solid angle is contained by four plane angles, when all the plane angles which the faces of the polyhedron have are counted, the number of the solid angles of the polyhedron is one-fourth of the resulting number; likewise also for as many polyhedra whose solid angle is contained by five plane angles, one-fifth of the multitude of the plane angles is the number of the multitude of the solid angles.
Τῶν δὲ πλευρῶν τὸ πλῆθος, ἃς ἕκαστον ἔχει τῶν πολυέδρων, τόνδε τὸν τρόπον εὑρήσομεν.
And we shall find the multitude of the edges which each of the polyhedra has in this manner.
Ἐξαριθμηθεισῶν γὰρ πασῶν τῶν πλευρῶν, ἂς ἔχει τὰ ἐπίπεδα τὰ περιέχοντα τὸ πολύεδρον, ὁ ἀριθμὸς αὐτῶν δῆλον ὡς ἴσος ἐστὶν τῷ πλήθει τὸν ἐπιπέδων γωνιῶν.
For when all the edges which the planes containing the polyhedron have are counted, it is clear that their number is equal to the multitude of the plane angles.
Ἀλλ᾿ ἐπειδὴ δύο ἐπιπέδων ἑκὰστη τῶν πλευρῶν αὐτοῦ κοινή ἐστιν, δῆλον ὅτι τοῦ πλήθους τὸ ἥμισυ αἱ πλευραί εἰσι τοῦ πολυέδρου.
But since each of its edges is common to two planes, it is clear that half of the multitude is the edges of the polyhedron.