of points such as RH_h_C come to meet the surface ab at the points AK_k_B.  Then instead of the hemispherical partial waves which in a body of ordinary refraction would spread from each of these last points, as we have above explained in treating of refraction, these must here be hemi-spheroids.  The axes (or rather the major diameters) of these I supposed to be oblique to the plane ab, as is AV the semi-axis or semi-major diameter of the spheroid SVT, which represents the partial wave coming from the point A, after the wave rc has reached ab.  I say axis or major diameter, because the same ellipse SVT may be considered as the section of a spheroid of which the axis is AZ perpendicular to AV.  But, for the present, without yet deciding one or other, we will consider these spheroids only in those sections of them which make ellipses in the plane of this figure.  Now taking a certain space of time during which the wave SVT has spread from A, it would needs be that from all the other points K_k_B there should proceed, in the same time, waves similar to SVT and similarly situated.  And the common tangent NQ of all these semi-ellipses would be the propagation of the wave rc which fell on ab, and would be the place where this movement occurs in much greater amount than anywhere else, being made up of arcs of an infinity of ellipses, the centres of which are along the line ab.

24.  Now it appeared that this common tangent NQ was parallel to ab, and of the same length, but that it was not directly opposite to it, since it was comprised between the lines an, BQ, which are diameters of ellipses having A and B for centres, conjugate with respect to diameters which are not in the straight line ab.  And in this way I comprehended, a matter which had seemed to me very difficult, how a ray perpendicular to a surface could suffer refraction on entering a transparent body; seeing that the wave rc, having come to the aperture ab, went on forward thence, spreading between the parallel lines an, BQ, yet itself remaining always parallel to ab, so that here the light does not spread along lines perpendicular to its waves, as in ordinary refraction, but along lines cutting the waves obliquely.

[Illustration]

25.  Inquiring subsequently what might be the position and form of these spheroids in the crystal, I considered that all the six faces produced precisely the same refractions.  Taking, then, the parallelopiped AFB, of which the obtuse solid angle C is contained between the three equal plane angles, and imagining in it the three principal sections, one of which is perpendicular to the face DC and passes through the edge cf, another perpendicular to the face BF passing through the edge CA, and the third perpendicular to the face AF passing through the edge bc; I knew that the refractions of the incident rays belonging to these three planes were all similar.  But there