English translation of A. Fresnel, "Extrait d'un Mémoire sur la double réfraction", Annales de Chimie et de Physique, Ser. 2, vol. 28, pp. 263–79 (1825), as reprinted in Oeuvres complètes d'Augustin Fresnel, vol. 2 (1868), pp. 465–78, with the corresponding extract from the "Table Analytique" in Oeuvres complètes..., vol. 3 (1870), at pp. 659–62. Translator's abstract: The author's earlier explanation of the double refraction of uniaxial crystals (such as calcite), based on the hypotheses that light consists of transverse waves and that a birefringent medium has different elasticities in different directions, led to the prediction that in a biaxial crystal (such as topaz), there should be no true ordinary ray—that is, no ray having a speed independent of direction. This has been confirmed by two experiments, one using interference and the other using refraction. The optical properties of a uniaxial or biaxial crystal are determined by the lengths of three perpendicular axes of elasticity, whose directions are those in which the restoring force is parallel to the displacement, and whose lengths are the principal axes of the surface of elasticity, of which each radius vector gives, by its length, the speed of propagation of vibrations parallel thereto. In any plane wavefront, the permitted directions of vibration, into which any other vibration is resolved, are the directions of the longest and shortest radius vectors, these being the directions in which the reaction has a component parallel to the vibration and (at most) another component normal to the wavefront. If the surface of elasticity has three unequal principal semi-axes, two planes through the center cut the surface in circles. The directions perpendicular to these planes, being the directions of a single wave speed, could be called the optical axes. The angle between these axes varies due to dispersion, if the principal semi-axes vary in different proportions. For an object point so far away that the incident wavefront can be assumed plane, the ordinary and extraordinary images can be located if we know the deviations of the respective wavefronts. For a closer object point, however, we need to know the equation of the wave surface, which takes the place of the secondary wavefronts in Huygens' construction. This surface must be tangential to every plane wavefront that travels from the origin in the same time; the equation of the surface that satisfies this condition is given. If two semi-axes of elasticity are equal, the equation of the wave surface can be factored into the equation of a sphere and that of a spheroid; this is the case of uniaxial crystals. The general wave surface can be constructed from the diametral sections of an ellipsoid having the same principal semi-axes as the surface of elasticity. The directions perpendicular to the two circular sections of this ellipsoid, being different from the corresponding directions for the surface of elasticity, are alternative candidates for the term optical axes. The ellipsoid construction leads to the confirmation of Biot's sine-product law (with Biot's ray speeds replaced by their reciprocals), and the approximate confirmation of Biot's dihedral law for the planes of polarization. The full memoir explains why the refraction of a homogeneous medium is never more than double, and why there cannot be more than two optical axes. However, the theory does not cover the rotation of the plane of polarization in quartz, which seems to imply that the homologous faces of its molecules are not all parallel.