the point of tangency. In a later heading (p.305) he declares that "The construction of Huygens, which determines the path of swiftest arrival," is applicable to secondary wavefronts of any shape. He then notes that when we apply Huygens' construction to refraction into a crystal with a two-sheeted secondary wavefront, and draw the lines from the two points of tangency to the center of the secondary wavefront, "we shall have the directions of the two paths of swiftest arrival, and consequently of the ordinary and of the extraordinary ray." Under the heading "Definition of the word Ray" (p.309), he concludes that this term must be applied to the line which joins the center of the secondary wave to a point on its surface, whatever the inclination of this line to the surface. As a "new consideration" (pp.310–11), he notes that if a plane wavefront is passed through a small hole centered on point E, then the direction ED of maximum intensity of the resulting beam will be that in which the secondary wave starting from E will "arrive there the first", and the secondary wavefronts from opposite sides of the hole (equidistant from E) will "arrive at D in the same time" as each other. This direction is not assumed to be normal to any wavefront. Thus Fresnel showed, even for anisotropic media, that the ray path given by Huygens' construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary "wave surface" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam. However, establishing the general equivalence between Huygens' construction and Fermat's principle would have required further consideration of Fermat's principle in point-to-point terms. Hendrik Lorentz, in a paper written in 1886 and republished in 1907, deduced the principle of least time in point-to-point form from Huygens' construction. But the essence of his argument was somewhat obscured by an apparent dependence on aether and aether drag. Lorentz's work was cited in 1959 by Adriaan J. de Witte, who then offered his own argument, which "although in essence the same, is believed to be more cogent and more general." De Witte's treatment is more original than that description might suggest, although limited to two dimensions; it uses calculus of variations to show that Huygens' construction and