the principle. In fact Laplace was aware that Fermat's principle follows from Huygens' construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplace's report, printed in 1810. Young's claim was more general than Laplace's, and likewise upheld Fermat's principle even in the case of extraordinary refraction, in which the rays are generally not perpendicular to the wavefronts. Unfortunately, however, the omitted middle sentence of the quoted paragraph by Young began "The motion of every undulation must necessarily be in a direction perpendicular to its surface..." (emphasis added), and was therefore bound to sow confusion rather than clarity. No such confusion subsists in Augustin-Jean Fresnel's "Second Memoir" on double refraction (Fresnel, 1827), which addresses Fermat's principle in several places (without naming Fermat), proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time. (In the following summary, page numbers refer to Alfred W.Hobson's translation.) For refraction of a plane wave at parallel incidence on one face of an anisotropic crystalline wedge (pp.291–2), in order to find the "first ray arrived" at an observation point beyond the other face of the wedge, it suffices to treat the rays outside the crystal as normal to the wavefronts, and within the crystal to consider only the parallel wavefronts (whatever the ray direction). So in this case, Fresnel does not attempt to trace the complete ray path. Next, Fresnel considers a ray refracted from a point-source M inside a crystal, through a point A on the surface, to an observation point B outside (pp.294–6). The surface passing through B and given by the "locus of the disturbances which arrive first" is, according to Huygens' construction, normal to "the ray AB of swiftest arrival". But this construction requires knowledge of the "surface of the wave" (that is, the secondary wavefront) within the crystal. Then he considers a plane wavefront propagating in a medium with non-spherical secondary wavefronts, oriented so that the ray path given by Huygens' construction — from the source of the secondary wavefront to its point of tangency with the subsequent primary wavefront — is not normal to the primary wavefronts (p.296). He shows that this path is nevertheless "the path of quickest arrival of the disturbance" from the earlier primary wavefront to