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

6 Spherical Geometry.
DEP. IX.- When two arcs of circles intersect, the angle of the tangents at their points of intersection is called the angle of the arcs.
8. 27he angle of intersection of two great circles is equal to the
inclination of their planes.
DEM.-Let CG, Cff be tangents to the semicircles ABC,
AEC; then (Euc. XI., Def. ix.), since each is perpendicular
to OC, the angle between them is equal to the angle of inclination of the planes of the great circles; but (Def. ix.) the
angle between CG, CI is the angle of intersection of the
great circles. Therefore the angle between two great circles is
equal to the inclination of their planes.
Cor. 1.-If CB, CE be quadrants, OB, OE are at right
angles to OC, and the angle BOE is equal to the angle of
inclination of the planes of the great circles. HIence the
spherical angle BCE is equal to the angle BOE; but BOE
is measured by the arc BE. Hence the spherical angle contained by any two great circles AB C, AEC is measured by the
arc of a great circle intercepted between them, and having the
point Cfor its pole.
Cor. 2.-The spherical angle BAE is equal to the angle
BCE.
Cor. 3.-The angle of intersection of two great circles is
measured by the arc between their poles.
For, if BEbe produced, since the plane BOEis perpendicular
to OC, BE will pass through the poles of ABC, ÂAEC. Let
these be I, K, respectively; then, evidently, the arcs IB, KE
are quadrants. Hence IX = BE; but BE (Cor. 1) is the
measure of the spherical angle B CE. Hence IK is equal to
the measure of the spherical angle.
9. To find the radius of a solid sphere.
Sol.-From any point of the spherical surface as pole, with
any arbitrary opening of the compass describe a circle ABC;