But it must be shown how one can determine the point of contact I. Let
there be drawn parallel to the line KT a line HF which touches the
Ellipse HDE, and let this point of contact be at H. And having drawn a
straight line along CH to meet KT at T, let there be imagined a plane
passing through the same CH and through CM (which I suppose to be the
refraction of the perpendicular ray), which makes in the spheroid the
elliptical section HME. It is certain that the plane which will pass
through the straight line KT, and which will touch the spheroid, will
touch it at a point in the Ellipse HME, according to the Lemma which
will be demonstrated at the end of the Chapter. Now this point is
necessarily the point I which is sought, since the plane drawn through
TK can touch the spheroid at one point only. And this point I is easy
to determine, since it is needful only to draw from the point T, which
is in the plane of this Ellipse, the tangent TI, in the way shown
previously. For the Ellipse HME is given, and its conjugate
semi-diameters are CH and CM; because a straight line drawn through M,
parallel to HE, touches the Ellipse HME, as follows from the fact that
a plane taken through M, and parallel to the plane HDE, touches the
spheroid at that point M, as is seen from Articles 27 and 23. For the
rest, the position of this ellipse, with respect to the plane through
the ray RC and through CK, is also given; from which it will be easy
to find the position of CI, the refraction corresponding to the ray
RC.
Now it must be noted that the same ellipse HME serves to find the
refractions of any other ray which may be in the plane through RC and
CK. Because every plane, parallel to the straight line HF, or TK,
which will touch the spheroid, will touch it in this ellipse,
according to the Lemma quoted a little before.
I have investigated thus, in minute detail, the properties of the
irregular refraction of this Crystal, in order to see whether each
phenomenon that is deduced from our hypothesis accords with that which
is observed in fact. And this being so it affords no slight proof of
the truth of our suppositions and principles. But what I am going to
add here confirms them again marvellously. It is this: that there are
different sections of this Crystal, the surfaces of which, thereby
produced, give rise to refractions precisely such as they ought to be,
and as I had foreseen them, according to the preceding Theory.
67