The stereographic projection of the sphere is a perspective projection from a point on the surface onto the opposite diametral plane. Analytical geometry provides a unified method of treating the stereographic projection and problems in spherical trigonometry. The basic result is the equation for the projection of a general circle (eq. 1). This is applied to the polar, equatorial and oblique stereographic projections, and a simple proof of Cayley’s theorem is obtained. Next, the cosine formula for a right-angled spherical triangle and the relation between angles measured along different great circles are derived. Finally, the construction used on astrolabes for determining planetary time is shown to be the stereographic projection of a problem on the sphere, which is solved.