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

140 Polyhedra.
6~. The medians PMQ, QN, RS are perpendicular to their corresponding edges.
129. The four solid angles G- AB C, G- BD, -, G-BCD,
G -A CD, are equal. Hence the spherical triangles which they
intercept on the sphere are each equal in area to one-fourth of
the spherical surface, and therefore the spherical excess of each
is two right angles.
Again, the angle JlGg = supplement of iXGD = supplement of MGC = PGC = A GB. Then, in the spherical
triangle ABC, the arc which joins a summit to the middle of
the opposite side is equal to the supplement of haZf that side,
and the arcjoining the point of concourse of the medians to the
middle of any side is equal to half that side. HIence the spherical
triangle ABC is divided by the antipodes of the point D into
three diametral triangles.
130. The volume of the tetrahedron is double of the octahedron MINPQRS = 16SMNG = 8 GM. GN2. GS; but
GS2 + GM2 iBC2C2= G2 + GN2 32, GN2+ GS2= 72;
9. F- /(2+7 a2 2 _ )(y2+-2 _ 2) (a2+2 2) (445)
Cor. 1.-The square of the radius of the circumscribed
sphere
a2 + 32 + y2
8= a +, + 2 (446)
For A G2= AfZM + ffG2 = 2+ 22 2+
C8 -- 8
3V
Cor. 2.-The radius of the inscribed sphere = AB (447)
ja.Jj(