the Sine of Incidence to the Sine of Refraction, the
Point _q_ shall be the Focus of the refracted Rays.

[Illustration: FIG. 4.]

_Cas._ 2. Let ACB [in _Fig._ 5.] be the reflecting Surface of any Sphere
whose Centre is E. Bisect any Radius thereof, (suppose EC) in T, and if
in that Radius on the same side the Point T you take the Points Q and
_q_, so that TQ, TE, and T_q_, be continual Proportionals, and the Point
Q be the Focus of the incident Rays, the Point _q_ shall be the Focus of
the reflected ones.

[Illustration: FIG. 5.]

_Cas._ 3. Let ACB [in _Fig._ 6.] be the refracting Surface of any Sphere
whose Centre is E. In any Radius thereof EC produced both ways take ET
and C_t_ equal to one another and severally in such Proportion to that
Radius as the lesser of the Sines of Incidence and Refraction hath to
the difference of those Sines. And then if in the same Line you find any
two Points Q and _q_, so that TQ be to ET as E_t_ to _tq_, taking _tq_
the contrary way from _t_ which TQ lieth from T, and if the Point Q be
the Focus of any incident Rays, the Point _q_ shall be the Focus of the
refracted ones.

[Illustration: FIG. 6.]

And by the same means the Focus of the Rays after two or more Reflexions
or Refractions may be found.

[Illustration: FIG. 7.]

_Cas._ 4. Let ACBD [in _Fig._ 7.] be any refracting Lens, spherically
Convex or Concave or Plane on either side, and let CD be its Axis (that
is, the Line which cuts both its Surfaces perpendicularly, and passes
through the Centres of the Spheres,) and in this Axis produced let F and
_f_ be the Foci of the refracted Rays found as above, when the incident
Rays on both sides the Lens are parallel to the same Axis; and upon the
Diameter F_f_ bisected in E, describe a Circle. Suppose now that any
Point Q be the Focus of any incident Rays. Draw QE cutting the said
Circle in T and _t_, and therein take _tq_ in such proportion to _t_E as
_t_E or TE hath to TQ. Let _tq_ lie the contrary way from _t_ which TQ
doth from T, and _q_ shall be the Focus of the refracted Rays without
any sensible Error, provided the Point Q be not so remote from the Axis,
nor the Lens so broad as to make any of the Rays fall too obliquely on
the refracting Surfaces.[A]

And by the like Operations may the reflecting or refracting Surfaces be
found when the two Foci are given, and thereby a Lens be formed, which
shall make the Rays flow towards or from what Place you please.[B]

So th