A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Regular Polyhedra. 129
Since the denominator in these expressions must be positive,
— + - must be greater than;but n cannot be iess than 3,
mn n 2i
since a solid angle cannot be formed by less than three plane
angles. Hence m cannot be greater than 5. The following
will be found to be the only admissible system of values for
m and n, viz.,
3, 3; 4, 3; 3,4; 5, 3; 3,5;
and the corresponding polyhedra are the Tetrahedron, Cube,
Octahedron, Dodecahedron, and Icosahedron, or solids of 4, 6, 8,
12, 20 faces, which have respectively 4, 8, 6, 20, 12 vertices.
122. If I denote the inclination of two adjacent faces of a
regular polyhedron,
7r 7r
sin I = cosec -. cos -. (428)
m n
B
D
Fig. 50.
DEM.-Let AB be the side common to the two faces, C and
D their centres, from which let CE, DE be drawn perpendicular to AB; then the angle between CE and ED will be
equal to I. In the plane of the lines CE, DE, let CO and DO
be drawn at right angles to them, and meeting in O. Join
O0A, OE, OB, and from O as centre suppose a sphere to be
E