spheroid QC_qg_M. Then having drawn CO perpendicular to the ray CR and
situate in the plane through CR and AH, let there be adjusted, across
the angle ACO, the straight line OK equal to N and perpendicular to
CO, and let it meet the straight line AH at K. Supposing consequently
that CL is perpendicular to the surface of the crystal AEHF, and that
CM is the refraction of the ray which falls perpendicularly on this
same surface, let there be drawn a plane through the line CM and
through KCH, making in the spheroid the semi-ellipse QM_q_, which will
be given, since the angle MCL is given of value 6 degrees 40 minutes.
And it is certain, according to what has been explained above, Article
27, that a plane which would touch the spheroid at the point M, where
I suppose the straight line CM to meet the surface, would be parallel
to the plane QG_q_. If then through the point K one now draws KS
parallel to G_g_, which will be parallel also to QX, the tangent to
the Ellipse QG_q_ at Q; and if one conceives a plane passing through
KS and touching the spheroid, the point of contact will necessarily be
in the Ellipse QM_q_, because this plane through KS, as well as the
plane which touches the spheroid at the point M, are parallel to QX,
the tangent of the spheroid: for this consequence will be demonstrated
at the end of this Treatise. Let this point of contact be at I, then
making KC, QC, DC proportionals, draw DI parallel to CM; also join CI.
I say that CI will be the required refraction of the ray RC. This will
be manifest if, in considering CO, which is perpendicular to the ray
RC, as a portion of the wave of light, we can demonstrate that the
continuation of its piece C will be found in the crystal at I, when O
has arrived at K.
38. Now as in the Chapter on Reflexion, in demonstrating that the
incident and reflected rays are always in the same plane perpendicular
to the reflecting surface, we considered the breadth of the wave of
light, so, similarly, we must here consider the breadth of the wave CO
in the diameter G_g_. Taking then the breadth C_c_ on the side toward
the angle E, let the parallelogram CO_oc_ be taken as a portion of a
wave, and let us complete the parallelograms CK_kc_, CI_ic_, Kl_ik_,
OK_ko_. In the time then that the line O_o_ arrives at the surface of
the crystal at K_k_, all the points of the wave CO_oc_ will have
arrived at the rectangle K_c_ along lines parallel to OK; and from the
points of their incidences there will originate, beyond that, in the
crystal partial hemi-spheroids, similar to the hemi-spheroid QM_q_,
and similarly disposed. These hemi-spheroids will necessarily all
61