any wavelike disturbance and explain Fermat's principle in purely mechanistic terms, without any imputation of knowledge or purpose. The principle applies to waves in general, including (e.g.) sound waves in fluids and elastic waves in solids. In a modified form, it even works for matter waves: in quantum mechanics, the classical path of a particle is obtainable by applying Fermat's principle to the associated wave — except that, because the frequency may vary with the path, the stationarity is in the phase shift (or number of cycles) and not necessarily in the time. Fermat's principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves. Equivalence to Huygens' construction In this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below. Let the surface be a wavefront at time , and let the surface be the same wavefront at the later time (Fig.4). Let be a general point on . Then, according to Huygens' construction, is the envelope (common tangent surface), on the forward side of , of all the secondary wavefronts each of which would expand in time from a point on , and if the secondary wavefront expanding from point in time touches the surface at point , then and lie on a ray. The construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray. The ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront (cf. Fig.2), and therefore from the normal of the primary wavefront at the point of tangency. Hence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction. Now let be a point on close to , and let be a point on close to . Then, by the construction, the time taken for a secondary wavefront from to reach has at most a second-order dependence on the displacement , and the time taken for a secondary wavefront to