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

154 Applications of Spherical Trigonometry.
latitude, and x the longitude of S, denoting the angle SrD by 0, from the
right-angled triangles Sr-D,.SrL, we get
tan A cos ( -c ) sin 1 sin(0 - Co)
cot = sin a cot, = - --- (481)
tan a cos 0 sinm sm O
The first of these equations determines o, and the others x and 1.
11. Being given the latitudes and longitudes of two places on the earth
considered as a perfect sphere, to find the distance between them.
This is evidently a case of ~ 66, viz., when two sides and the contained
angle are given, to find the third side.
12. Find the latitude, being given the declination, and the interval between the time the sun is west and sunset.
13. If the latitudes and longitudes of two places on the earth be given,
show how to find the highest latitude attained by a ship in sailing along
a great circle from one place to the other.
14. Being given the latitudes and longitudes of two places, find the sun's
declination when he is on the horizon of both at the same instant.
15. If the difference between the lengths of the longest and the shortest
day at a given place be six hours, find the latitude.
16. If two stars rise together at two places, prove that the places will
have the same latitude; and if they rise together at one place, and set
together at the other, the places will have equal latitudes of opposite
names.
17. If pi, p2 be the radii vectors of two planets which revolve in circular
orbits, prove, if when they appear stationary to one another, the cotangent
of P2's elongation, seen from Pi, be 2 tan 0, that
2pi = p2 tan O. tan O. (482)
18. If 8 be the declination of a heavenly body, which in its diurnal
motion passes in the minimum time from one to another of two parallels of
altitude, whose zenith distances are Z, Z', prove that
= cos (Z+ Z')
sin ~ (c Z = Z ) -sin lat. (483)
cos (Z - Z') v
19. If I be the latitude, w the obliquity of the ecliptic, prove that if the
lengths of the shadow of an upright rod at noon on the longest and the
shortest days be as 1: n,
sin 21: sin 2w:: n + 1: n - 1. (484)