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

Stereographic Projection. 123
EF. Let IC be perpendicular to AB. Join AE, AF, intersecting IC in the points G, IT. Now, since GAH is a right
angle, and A C is perpendicular to GEI, the angle AHG = GA C
(Euc. VI. viII.) = AEC (Eue. I. v.). Hence the triangles EAF
and HA G are inversely similar, and therefore the section made
by the plane of the primitive with the cone, whose vertex is A,
and which stands on the great circle EF, is an antiparallel
section. Hence it is a circle.
Cor. 1.-The projections of the poles P, Q of the great circle
EFwill be inverse points with respect to its projection.
Cor. 2.-If the plane of a small circle be parallel to the
plane of EF, the projection of P and Q will be inverse points
with respect to its projection.
Cor. 3.-A system of small circles, whose planes are parallel,
will project into a system of coaxal circles.
Cor. 4.-Every circle whose plane passes through the pole
of the primitive is projected into a right line.
118. The angle made by any two circles on the sphere is equal
to the angle made by their projections on the plane of the primitive.
DEM.-Let O be the pole of the primitive, M2 the point in
which the circles intersect; and let MT, MFV the tangents to
the circles at Jf meet the tangent plane to the sphere at O in
the points T, V. Join OT, 0V, TV; then evidently the angle
2T'V = TOV; but since the tangent plane at O is parallel to
the plane of the primitive, the lines OT, 0 V are parallel to
the projections of the lines MT, MV. Hence the angle TO Vis
equal to the angle between the projections.
Cor. 1. -Any circle whose plane is perpendicular to the
plane of the primitive is projected into a circle orthogonal to
the primitive.
Cor. 2.-A system of coaxal circles on the sphere is projected
into a system of coaxal circles on the plane of the primitive.