here represented by the line GG,
the refraction of the surfaces which are produced by it will be
governed by the hemi-spheroids GNG, according to what has been
explained in the preceding Theory. Similarly, cutting the Crystal
through NN, by a plane perpendicular to the parallelogram ABKF, the
refraction of the surfaces will be governed by the hemi-spheroids NGN.
And if one cuts it through PP, perpendicularly to the said
parallelogram, the refraction of the surfaces ought to be governed by
the hemi-spheroids PSP, and so for others. But I saw that if the plane
NN was almost perpendicular to the plane GG, making the angle NCG,
which is on the side A, an angle of 90 degrees 40 minutes, the
hemi-spheroids NGN would become similar to the hemi-spheroids GNG,
since the planes NN and GG were equally inclined by an angle of 45
degrees 20 minutes to the axis SS. In consequence it must needs be, if
our theory is true, that the surfaces which the section through NN
produces should effect the same refractions as the surfaces of the
section through GG. And not only the surfaces of the section NN but
all other sections produced by planes which might be inclined to the
axis at an angle equal to 45 degrees 20 minutes. So that there are an
infinitude of planes which ought to produce precisely the same
refractions as the natural surfaces of the crystal, or as the section
parallel to any one of those surfaces which are made by cleavage.

I saw also that by cutting it by a plane taken through PP, and
perpendicular to the axis SS, the refraction of the surfaces ought to
be such that the perpendicular ray should suffer thereby no deviation;
and that for oblique rays there would always be an irregular
refraction, differing from the regular, and by which objects placed
beneath the crystal would be less elevated than by that other
refraction.

That, similarly, by cutting the crystal by any plane through the axis
SS, such as the plane of the figure is, the perpendicular ray ought to
suffer no refraction; and that for oblique rays there were different
measures for the irregular refraction according to the situation of
the plane in which the incident ray was.

Now these things were found in fact so; and, after that, I could not
doubt that a similar success could be met with everywhere. Whence I
concluded that one might form from this crystal solids similar to
those which are its natural forms, which should produce, at all their
surfaces, the sa