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

CHAPTER V.
SPHERICAL EXCESS.
77. In the preceding chapters we have made frequent use
of the fumtion of the angles of a triangle, called the spherical
excess. In this chapter we shall enter into further detail, and
give a more systematic account of its theory than could have
been conveniently given in those chapters.
SECTION I.-FORMULAE RELATIVE TO E.
78. Lemma. —If the triangle BCA' be colunar with ABC, it
has two sides equal to r - b, r - c, and their included angle is
equal to A; therefore if 2E be the spherical excess of ABC,
2A - 2E will be the spherical excess of A'BC. Hence we
have the following rule of transformation:RuLE. —In any formula containing the elements b, c, A, E of a
spherical triangle, we may change the sides b, c into their supplement, and E into A - E.
This rule supplies easy proofs of several propositions.
79. Let ABC be a spherical triangle; bisect BC, AC in
C
D
A
Fig. 31.
A', B'. Join A'B', and produce to meet AB in D, E; let fall