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

144 Applications of Spherical Trigonometry.
2~. Solution of Legendre.-This solution;s applicable only
when the angles of elevation of the objects M1, N are very small;
that is, when b and c are each near 90~. It depends on the
following lemma: —Being given a spherical triangle ABC, the
angle A1 of the rectilineal triangle formed by its chords (called the
chordal triangle) is given by the equation
cos A, = sin b sin - c + cos b cos c cos A. (458)
A
B
o
Fig. 58.
DEiM.-Let A'BC be the colunar triangle, M, N the middle
points of the arcs A'B, A'C; then thc chords AB, C are
parallel to the radii 0f, ON of the sphere. Hence
cos A1 = cos MON= sin b sin - c + cos 1 b cos ] c cos 4.
Cor.-If A1 = A - 0, then cos A4 = cos A + 0 sin A approximately; and substituting in (458) for cos 2b cos 1 c, sin i-b sin ac,
the values cos2 (b+c) - sin2 (b - c), sin24 (b +)- sin2 1 (b - c),
we get, after an easy reduction,
0 = tan - sin2 (b+ c) - cot 2 4 sin.2 (b - c). (459)
Given the oblique angle contained between two objects above the
horizon, tofind the corresponding horizontal angle.