which CM contains 100,000 and CQ 105,032. Then DS
will have 70,283. But CL is 99,324, being the sine of the complement
of the angle MCL which is 6 degrees 40 minutes; CM being supposed as
radius. Then DP, considered as equal to CL, will be to DS as 99,324 to
70,283. And so the elevation of the point I by the refraction of this
section is known.

[Illustration]

41. Now let there be represented the other section through EF in the
figure before the preceding one; and let CM_g_ be the semi-ellipse,
considered in Articles 27 and 28, which is made by cutting a
spheroidal wave having centre C. Let the point I, taken in this
ellipse, be imagined again at the bottom of the Crystal; and let it be
viewed by the refracted rays ICR, I_cr_, which go to the two eyes; CR
and _cr_ being equally inclined to the surface of the crystal G_g_.
This being so, if one draws ID parallel to CM, which I suppose to be
the refraction of the perpendicular ray incident at the point C, the
distances DC, D_c_, will be equal, as is easy to see by that which has
been demonstrated in Article 28. Now it is certain that the point I
should appear at S where the straight lines RC, _rc_, meet when
prolonged; and that this point will fall in the line DP perpendicular
to G_g_. If one draws IP perpendicular to this DP, it will be the
distance PS which will mark the apparent elevation of the point I. Let
there be described on G_g_ a semicircle cutting CR at B, from which
let BV be drawn perpendicular to G_g_; and let N to GC be the
proportion of the refraction in this section, as in Article 28. Since
then CI is the refraction of the radius BC, and DI is parallel to CM,
VC must be to CD as N to GC, according to what has been demonstrated
in Article 31. But as VC is to CD so is BV to DS. Let ML be drawn
perpendicular to CL. And because I consider, again, the eyes to be
distant above the crystal, BV is deemed equal to the semi-diameter CG;
and hence DS will be a third proportional to the lines N and CG: also
DP will be deemed equal to CL. Now CG consisting of 98,778 parts, of
which CM contains 100,000, N is taken as 156,962. Then DS will be
62,163. But CL is also determined, and contains 99,324 parts, as has
been said in Articles 34 and 40. Then the ratio of PD to DS will be as
99,324 to 62,163. And thus one knows the elevation of the point at the
bottom I by the refraction of this section; and it appears that this
elevation is greater than that by the refraction o