n of the
crystal, the proportion of which is as 5 to 3, and which always raises
the letters equally, and higher than the irregular refraction does.
For one sees the letters and the paper on which they are written, as
on two different stages at the same time; and in the first position of
the eyes, namely, when they are in the plane through AH these two
stages are four times more distant from one another than when the eyes
are in the plane through EF.

We will show that this effect follows from the refractions; and it
will enable us at the same time to ascertain the apparent place of a
point of an object placed immediately under the crystal, according to
the different situation of the eyes.

40. Let us see first by how much the irregular refraction of the plane
through AH ought to lift the bottom of the crystal. Let the plane of
this figure represent separately the section through Q_q_ and CL, in
which section there is also the ray RC, and let the semi-elliptic
plane through Q_q_ and CM be inclined to the former, as previously, by
an angle of 6 degrees 40 minutes; and in this plane CI is then the
refraction of the ray RC.

[Illustration]

If now one considers the point I as at the bottom of the crystal, and
that it is viewed by the rays ICR, _Icr_, refracted equally at the
points C_c_, which should be equally distant from D, and that these
rays meet the two eyes at R_r_; it is certain that the point I will
appear raised to S where the straight lines RC, _rc_, meet; which
point S is in DP, perpendicular to Q_q_. And if upon DP there is drawn
the perpendicular IP, which will lie at the bottom of the crystal, the
length SP will be the apparent elevation of the point I above the
bottom.

Let there be described on Q_q_ a semicircle cutting the ray CR at B,
from which BV is drawn perpendicular to Q_q_; and let the proportion
of the refraction for this section be, as before, that of the line N
to the semi-diameter CQ.

Then as N is to CQ so is VC to CD, as appears by the method of finding
the refraction which we have shown above, Article 31; but as VC is to
CD, so is VB to DS. Then as N is to CQ, so is VB to DS. Let ML be
perpendicular to CL. And because I suppose the eyes R_r_ to be distant
about a foot or so from the crystal, and consequently the angle RS_r_
very small, VB may be considered as equal to the semi-diameter CQ, and
DP as equal to CL; then as N is to CQ so is CQ to DS. But N is valued
at 156,962 parts, of