_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.
