_The Sine of Incidence is either accurately or very nearly in a given
Ratio to the Sine of Refraction._
Whence if that Proportion be known in any one Inclination of the
incident Ray, 'tis known in all the Inclinations, and thereby the
Refraction in all cases of Incidence on the same refracting Body may be
determined. Thus if the Refraction be made out of Air into Water, the
Sine of Incidence of the red Light is to the Sine of its Refraction as 4
to 3. If out of Air into Glass, the Sines are as 17 to 11. In Light of
other Colours the Sines have other Proportions: but the difference is so
little that it need seldom be considered.
[Illustration: FIG. 1]
Suppose therefore, that RS [in _Fig._ 1.] represents the Surface of
stagnating Water, and that C is the point of Incidence in which any Ray
coming in the Air from A in the Line AC is reflected or refracted, and I
would know whither this Ray shall go after Reflexion or Refraction: I
erect upon the Surface of the Water from the point of Incidence the
Perpendicular CP and produce it downwards to Q, and conclude by the
first Axiom, that the Ray after Reflexion and Refraction, shall be
found somewhere in the Plane of the Angle of Incidence ACP produced. I
let fall therefore upon the Perpendicular CP the Sine of Incidence AD;
and if the reflected Ray be desired, I produce AD to B so that DB be
equal to AD, and draw CB. For this Line CB shall be the reflected Ray;
the Angle of Reflexion BCP and its Sine BD being equal to the Angle and
Sine of Incidence, as they ought to be by the second Axiom, But if the
refracted Ray be desired, I produce AD to H, so that DH may be to AD as
the Sine of Refraction to the Sine of Incidence, that is, (if the Light
be red) as 3 to 4; and about the Center C and in the Plane ACP with the
Radius CA describing a Circle ABE, I draw a parallel to the
Perpendicular CPQ, the Line HE cutting the Circumference in E, and
joining CE, this Line CE shall be the Line of the refracted Ray. For if
EF be let fall perpendicularly on the Line PQ, this Line EF shall be the
Sine of Refraction of the Ray CE, the Angle of Refraction being ECQ; and
this Sine EF is equal to DH, and consequently in Proportion to the Sine
of Incidence AD as 3 to 4.
In like manner, if there be a Prism of Glass (that is, a Glass bounded
with two Equal and Parallel Triangular ends, and three plain and well
polished Sides, which meet in three Parallel Lines running from the
three Angles of one end to the three Angles of the other end) and if the
Refraction of the Light in passing cross this Prism be desired: Let ACB
[in _Fig._ 2.] represent a Plane cutting this Prism transversly to its
three Parallel lines or edges there where the Light passeth through it,
and let DE be the Ray incident upon the first side of the Prism AC where
the Light goes into the Glass; and by putting the Proportion of the Sine
of Incidence to the Sine of Refraction as 17 to 11 find EF the first
refracted Ray. Then taking this Ray for the Incident Ray upon the second
side of the Glass BC where the Light goes out, find the next refracted
Ray FG by putting the Proportion of the Sine of Incidence to the Sine of
Refraction as 11 to 17. For if the Sine of Incidence out of Air into
Glass be to the Sine of Refraction as 17 to 11, the Sine of Incidence
out of Glass into Air must on the contrary be to the Sine of Refraction
as 11 to 17, by the third Axiom.
[Illustration: FIG. 2.]
Much after the same manner, if ACBD [in _Fig._ 3.] represent a Glass
spherically convex on both sides (usually called a _Lens_, such as is a
Burning-glass, or Spectacle-glass, or an Object-glass of a Telescope)
and it be required to know how Light falling upon it from any lucid
point Q shall be refracted, let QM represent a Ray falling upon any
point M of its first spherical Surface ACB, and by erecting a
Perpendicular to the Glass at the point M, find the first refracted Ray
MN by the Proportion of the Sines 17 to 11. Let that Ray in going out of
the Glass be incident upon N, and then find the second refracted Ray
N_q_ by the Proportion of the Sines 11 to 17. And after the same manner
may the Refraction be found when the Lens is convex on one side and
plane or concave on the other, or concave on both sides.
[Illustration: FIG. 3.]