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

16 Spherical Geometry.
Hence, by a process similar to that employed in Esc. VI.,
I. and xxxiii., it may be proved that lunes are proportional
to their angles. Therefore a lune: the whole spherical surface:: angle of lune: 27r. Now if r denote the radius of the
sphere, its surface is 47rr2 (Eue., App. 7). Hence, if A denotes
the angle of the lune, its area is 2Ar2. (7)
21. Girard's Theorem.Thie area of a spherical triangle = 2-Er2 (Def. xv.).
DEM.-Produce the base AB round the sphere, and produce
B C, A C to meet it in E and D; also produce CB, CA through
B and A to meet again in F. Then the spherical triangle
BAF is antipodal to the triangle ED C, and therefore (Art. 17)
equal in area to it. Hence the lune C is equal to the sum of
Fig.
Fig. 0.
the two triangles ABC, CED; also the lune A4 = to the sum
of the triangles ABC, BCD), and the lune B = to the sum of
ABC, CEA. Hence the sum of the three lunes is equal to
twice the area of the spherical triangle AB C, together with the
area of the hemisphere C = ABGDEIT. Silence, if A denote
the area of the triangle AB C, we have
2Ar2 + 2Br2 + 2 Cr2 = 27rr2 + 2A;.. = (A + B +C- C r) r2= 2r. (8)