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

Spherical Triangles. 15
3~. If the arc A'B' be applied to AB, the point C' will be
one of the points of intersection of the arcs of two small circles,
described from A and B as poles, and passing through the
point C: these arcs will intersect in two points C, C", placed
on opposite sides of AB; then, if the elements are disposed in
the same order in both triangles, C' will coincide with C. If
in a different order, the triangle A'B'C' can be superposed on
the antipodal triangle of ABC, and in each case we have the
corresponding angles equal each to each.
19. If two sides AB, AC of a spherical triangle be egual1~. The angles B, C are equal. 2~. The median AD, which bisects
BC, bisects the angle A.
DEM.-The arc AD divides the triangle ABC into two triangles, which are symmetrically equal (18, 3~).
Cor.-If two angles B, C are equal, the opposite sides AB,
AC, are equal. For the polar triangle of AB C has two sides
equal. Ience, &c.
20. Tofind the area of a lune.
F E
B
Fig. 9.
Let A CBBD, AEBF be two lunes having equal angles at A;
then, by superposition, it is evident that these lunes are equal.