I 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
these divergences in the direction parallel to the surface of the
crystal.

30. To find the length of the line N, in proportion to CP, CS, CG, it
must be determined by observations of the irregular refraction which
occurs in this section of the crystal; and I find thus that the ratio
of N to GC is just a little less than 8 to 5. And having regard to
some other observations and phenomena of which I shall speak
afterwards, I put N at 156,962 parts, of which the semi-diameter CG is
found to contain 98,779, making this ratio 8 to 5-1/29. Now this
proportion, which there is between the line N and CG, may be called
the Proportion of the Refraction; similarly as in glass that of 3 to
2, as will be manifest when I shall have explained a short process in
the preceding way to find the irregular refractions.

31. Supposing then, in the next figure, as previously, the surface of
the crystal _g_G, the Ellipse GP_g_, and the line N; and CM the
refraction of the perpendicular ray FC, from which it diverges by 6
degrees 40 minutes. Now let there be some other ray RC, the refraction
of which must be found.

About the centre C, with semi-diameter CG, let the circumference _g_RG
be described, cutting the ray RC at R; and let RV be the perpendicular
on CG. Then as the line N is to CG let CV be to CD, and let DI be
drawn parallel to CM, cutting