_Homogeneal Rays which flow from several Points of any Object, and fall
perpendicularly or almost perpendicularly on any reflecting or
refracting Plane or spherical Surface, shall afterwards diverge from so
many other Points, or be parallel to so many other Lines, or converge to
so many other Points, either accurately or without any sensible Error.
And the same thing will happen, if the Rays be reflected or refracted
successively by two or three or more Plane or Spherical Surfaces._
The Point from which Rays diverge or to which they converge may be
called their _Focus_. And the Focus of the incident Rays being given,
that of the reflected or refracted ones may be found by finding the
Refraction of any two Rays, as above; or more readily thus.
_Cas._ 1. Let ACB [in _Fig._ 4.] be a reflecting or refracting Plane,
and Q the Focus of the incident Rays, and Q_q_C a Perpendicular to that
Plane. And if this Perpendicular be produced to _q_, so that _q_C be
equal to QC, the Point _q_ shall be the Focus of the reflected Rays: Or
if _q_C be taken on the same side of the Plane with QC, and in
proportion to QC as 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 then the Meaning of this Axiom is, that if Rays fall upon any Plane
or Spherical Surface or Lens, and before their Incidence flow from or
towards any Point Q, they shall after Reflexion or Refraction flow from
or towards the Point _q_ found by the foregoing Rules. And if the
incident Rays flow from or towards several points Q, the reflected or
refracted Rays shall flow from or towards so many other Points _q_
found by the same Rules. Whether the reflected and refracted Rays flow
from or towards the Point _q_ is easily known by the situation of that
Point. For if that Point be on the same side of the reflecting or
refracting Surface or Lens with the Point Q, and the incident Rays flow
from the Point Q, the reflected flow towards the Point _q_ and the
refracted from it; and if the incident Rays flow towards Q, the
reflected flow from _q_, and the refracted towards it. And the contrary
happens when _q_ is on the other side of the Surface.