the Ellipse _g_MG at I; then joining CI,
this will be the required refraction of the ray RC. Which is
demonstrated thus.

[Illustration]

Let CO be perpendicular to CR, and across the angle OCG let OK be
adjusted, equal to N and perpendicular to CO, and let there be drawn
the straight line KI, which if it is demonstrated to be a tangent to
the Ellipse at I, it will be evident by the things heretofore
explained that CI is the refraction of the ray RC. Now since the angle
RCO is a right angle, it is easy to see that the right-angled
triangles RCV, KCO, are similar. As then, CK is to KO, so also is RC
to CV. But KO is equal to N, and RC to CG: then as CK is to N so will
CG be to CV. But as N is to CG, so, by construction, is CV to CD. Then
as CK is to CG so is CG to CD. And because DI is parallel to CM, the
conjugate diameter to CG, it follows that KI touches the Ellipse at I;
which remained to be shown.

32. One sees then that as there is in the refraction of ordinary
media a certain constant proportion between the sines of the angles
which the incident ray and the refracted ray make with the
perpendicular, so here there is such a proportion between CV and CD or
IE; that is to say between the Sine of the angle which the incident
ray makes with the perpendicular, and the horizontal intercept, in the
Ellipse, between the refraction of this ray and the diameter CM. For
the ratio of CV to CD is, as has been said, the same as that of N to
the semi-diameter CG.

33. I will add here, before passing away, that in comparing together
the regular and irregular refraction of this crystal, there is this
remarkable fact, that if ABPS be the spheroid by which light spreads
in the Crystal in a certain space of time (which spreading, as has
been said, serves for the irregular refraction), then the inscribed
sphere BVST is the extension in the same space of time of the light
which serves for the regular refraction.

[Illustration]

For we have stated before this, that the line N being the radius of a
spherical wave of light in air, while in the crystal it spread through
the spheroid ABPS, the ratio of N to CS will be 156,962 to 93,410. But
it has also been stated that the proportion of the regular refraction
was 5 to 3; that is to say, that N being the radius of a spherical
wave of light in air, its extension in the crystal would, in the same
space of time, form a sphere the radius of which would be to N as 3 to
5. Now 156,962 is