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

CHAPTER VIII.
POLYHEDRA.
SECTION I.-REGULAR POLYHEDRA.
120. If S be the number of solid angles, F the number of
faces, E the number of edges of any polyhedron (see Appendix to
EUCLID, 6th edition, p. 283), 8 + F=E + 2.-(EULER.)
DEM.-With any point in the interior of the polyhedron as
centre, describe a sphere of radius r, and draw lines from the
centre to each solid angle; let the points in which these meet
the surface of the sphere be joined by arcs of great circles.
These arcs will divide the surface into F spherical polygons.
Now if s denote the sum of the angles of any of these polygons,
and m the number of its sides, its area is r2(s - (m - 2)7r), but
the sum of the areas of all the polygons is equal to the surface of
the sphere or 47rr2. Hence, since there are F polygons, we have
47r = ss - 7rm + 2F7r; but Ss is evidently equal to 27rS, and
2m is the number of the sides of all the polygons, and therefore
equal to 2E. Ience 47r = 27rS - 2Er + 2F7r;.-. S+_F=E+2. (426)
121. There can be onlyfive regular polyhedra.
DDEM.-Let m be the number of sides in each face, and n the
number of plane angles in each solid angle, then the entire
number of plane angles is equal to mF or nS or 2E. Hence
we have the equations
mF=nS=2E and S+F=E+2.
Therefore solving for S, F, and L, we get
4m _4n 2mn
B -.F li=. E=
2(m+n)-mn' 2(m+ n)-mn' 2(m+n)-mn
(427)