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 20. These are the only surfaces that reflect light accurately in the manner we have described, which the simplest of all curved surfaces, the sphere, does not do; as to other surfaces, if we had occasion to treat of their reflexion, we should proceed in a manner similar to that used for the sphere, but in most instances if it were required to investigate the simple case of a small pencil of rays incident perpendicularly, it would be easiest to substitute for the surface its osculating sphere, that is, provided there were one, which is always the case at the vertex of a surface of revolution.

All this will, however, be much better understood after going through the next Chapter, which places the subject in a more general point of view. 21. have hitherto considered only the places where reflected rays intersect the axis of a spherical mirror; we will now examine their intersections with each other, and treat the question in a more general manner, that is, we will suppose a cone of rays to be reflected at any surface, any how disposed with respect to the radiant point. We will, however, simplify the question a little by taking only a plane section of the surface through the radiant point.

In Fig. 15, represents a point from which proceed rays, ,  … which are reflected at the curve …  into the directions , ….

The question being merely one of Plane Geometry, it will immediately occur to the reader that these reflected rays must be very