Ellipse GSP, and from the point of contact I join IC, which will be
the required refraction of the ray RC. The demonstration of this is,
it will be seen, entirely similar to that of which we made use in
explaining ordinary refraction. For the refraction of the ray RC is
nothing else than the progression of the portion C of the wave CO,
continued in the crystal. Now the portions H of this wave, during the
time that O came to K, will have arrived at the surface CK along the
straight lines H_x_, and will moreover have produced in the crystal
around the centres _x_ some hemi-spheroidal partial waves similar to
the hemi-spheroidal GSP_g_, and similarly disposed, and of which the
major and minor diameters will bear the same proportions to the lines
_xv_ (the continuations of the lines H_x_ up to KB parallel to CO)
that the diameters of the spheroid GSP_g_ bear to the line CB, or N.
And it is quite easy to see that the common tangent of all these
spheroids, which are here represented by Ellipses, will be the
straight line IK, which consequently will be the propagation of the
wave CO; and the point I will be that of the point C, conformably with
that which has been demonstrated in ordinary refraction.
Now as to finding the point of contact I, it is known that one must
find CD a third proportional to the lines CK, CG, and draw DI parallel
to CM, previously determined, which is the conjugate diameter to CG;
for then, by drawing KI it touches the Ellipse at I.
29. Now as we have found CI the refraction of the ray RC, similarly
one will find C_i_ the refraction of the ray _r_C, which comes from
the opposite side, by making C_o_ perpendicular to _r_C and following
out the rest of the construction as before. Whence one sees that if
the ray _r_C is inclined equally with RC, the line C_d_ will
necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to
CG. And in consequence I_i_ will be cut at E into equal parts by the
line CM, to which DI and _di_ are parallel. And because CM is the
conjugate diameter to CG, it follows that _i_I will be parallel to
_g_G. Therefore if one prolongs the refracted rays CI, C_i_, until
they meet the tangent ML at T and _t_, the distances MT, M_t_, will
also be equal. And so, by our hypothesis, we explain perfectly the
phenomenon mentioned above; to wit, that when there are two rays
equally inclined, but coming from opposite sides, as here the rays RC,
_rc_, their refractions diverge equally from the line followed by the
refraction of the ray perpendicular to the surface, by considering
55