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

124 Inversions.
For a system of coaxal circles on the sphere is intersected
orthogonally by a system of circles passing through the two
limiting points (~ 94). Hlence the projections are intersected
orthogonally by a system of circles passing through two points.
119. Applications to Spherical Trigonometry.
Let ABC be a spherical triangle, AB'C a colunar triangle;
then, if 0, the pole of the primitive, be the antipodes of A, the
sides AB, A C, AB' will project into right lines ab, ac, ab', and
the circle BCB' into the circle bcb'. Join bc, cb'; then the
angles of the figure formed by the lines ab, ac, and the arc bc,
are respectively equal to the angles of the spherical triangle
ABC (Art. 117); but the sum of the angles of the rectilineal
triangle abc is two right angles, hence the sum of the angles
formed by the arc bc with its chord is equal to the spherical
excess 2E; therefore one of them is equal to E.
A
OFig. 4
Fig. 48.
Or thus:-If a circle be described about the triangle abc, the
angle made by this circle with the arc bo is (~ 118) equal to
the angle made by the circumcircle of the triangle ABC with
the side BC, and this is equal to (A - E) (Exercises xxIIi. 15);