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

136 Polyhedra.
Let DA = a, DB = b, DC= c, BC = a', CA = b', AB = O',
DA'= a, DB' = S, DC = y, B' C= a', C'A'= 3', A'B'= y',
we have
a' _3 _y, _ a' a'/3b a'.aa a
a= - = - = =aa.;
a' c b c bc bc abc
/3 = bb*. b Y= cc'.
abc abcC
If O' be the second extremity of the diameter DO' of the
sphere, we have
aa = b/3 = cy = 2R2 = DO. D0'.
aa'. 2R2 bb'. 2R2 cc'. 2R2
Hence a' =, ---, ' y=
abc abc abc
Putting aa' = a, bb'= b,, cc' = ci, and al + bl + cl = 21s, we have
triangle
4R4 4R4
'B' C' = 222 s. (sl - al) (sl - bi) (sl - Ci) = a s2.2
We have also
DAB C abc a2 b'2 a2 b2c2
DA'B'C' afly aa. b,8. cy 8R6 '
then
DABC a2b2c2,S1
4R4 =;.' R = 6. DABC- (441)
a2b2,02
SECOND METHOD (DOSTOR, Nouvelles Annales, 1874, p. 523):Let DABC be the tetrahedron, O its circumcentre; A', B', C'
the middle points of the edges DA, DB, DC, which, as before,
are denoted by a, b, c, respectively. Draw 0f parallel to DA,
meeting the face BDC in I, and M1N parallel to BD. Now,