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

Steregraphic Projection. 127
bc, cd; then drawing bt, ct tangents to bc, and cu du, tangents to cd, we
have in the plane hexagon abtcud the sum of the angles
A + + B+C+ D+t+u=47r; but A+ + B -C D-)=2r+22E;. 2E =27r-t-u.
Again, if the circles bc, cd intersect again in c', c' is the stereographic
projection of the antipodes. Hence the right line ca produced will pass
through c. Join bc, bc'; dc, dc', then the angle tbc = bc'c. Hence bc'c is
half the supplement of t, and cc'd half the supplement of u;.'. 2bc'd+t + u=2r;.'. bc'd =E.
Now from the plane triangle bc'd, we have
in 'd = sin2 1 = (b' + bd - c'd)(bd + c'd - bc')
4bc'. dc'
but
sin f cos ~b cos b
bd = osi c d= s
cos I a cos cos da COS, sin c' cos d sin e.sin2 E =
(sin le. sin f + cos a cos c - cos cos csd) (sin e sin f - cos 1a cos c + cos b cos d}
4 cos a cos b cos c cos d
8. If a spherical quadrilateral be cyclic, prove that. in2 1 = -æ)in (,( s - b)a) sin (s-b) sin (s - c)sin (s-d) (4-)
sin" E = 2ff 2,424
a b c d *
cos cos - cos - cos -
2 2 2 2
9. If the cyclic quadrilateral be circumscribed to another circle, prove
sin2 E = tan a. tan b tan c. tan2 d. (425)
10. Being given four circles in a plane, prove that the plane can be inverted into a sphere, so that the four circles on the plane will be the
stereographic projections of four equal circles on the sphere.-(STEINER.)
11. If A', B', C' be the stereographic projections of the angular points of
the spherical triangle ABC; and if the angles of the plane triangle A'B'C'
be respectively equal to those of the spherical triangle, each diminished by
one-third of the spherical excess, prove that the arcs drawn from A, B, C to
one of the poles of the primitive divides the area of ABC into three equal
parts.
Observation.-The applications of Stereographic Projection to Spherical
Trigonometry, contained in ~ 119 and in Exercises xxxii., are taken from
M. PAUL SERRET, Méthodes des Géométrie, pp. 30-44.