Ray coming most obliquely in the Line MC [in _Fig._
1.] be refracted at C by the Plane RS into the Line CN, and if it be
required to find the Line CE, into which any other Ray AC shall be
refracted; let MC, AD, be the Sines of Incidence of the two Rays, and
NG, EF, their Sines of Refraction, and let the equal Motions of the
incident Rays be represented by the equal Lines MC and AC, and the
Motion MC being considered as parallel to the refracting Plane, let the
other Motion AC be distinguished into two Motions AD and DC, one of
which AD is parallel, and the other DC perpendicular to the refracting
Surface. In like manner, let the Motions of the emerging Rays be
distinguish'd into two, whereof the perpendicular ones are MC/NG x CG
and AD/EF x CF. And if the force of the refracting Plane begins to act
upon the Rays either in that Plane or at a certain distance from it on
the one side, and ends at a certain distance from it on the other side,
and in all places between those two limits acts upon the Rays in Lines
perpendicular to that refracting Plane, and the Actions upon the Rays at
equal distances from the refracting Plane be equal, and at unequal ones
either equal or unequal according to any rate whatever; that Motion of
the Ray which is parallel to the refracting Plane, will suffer no
Alteration by that Force; and that Motion which is perpendicular to it
will be altered according to the rule of the foregoing Proposition. If
therefore for the perpendicular velocity of the emerging Ray CN you
write MC/NG x CG as above, then the perpendicular velocity of any other
emerging Ray CE which was AD/EF x CF, will be equal to the square Root
of CD_q_ + (_MCq/NGq_ x CG_q_). And by squaring these Equals, and adding
to them the Equals AD_q_ and MC_q_ - CD_q_, and dividing the Sums by the
Equals CF_q_ + EF_q_ and CG_q_ + NG_q_, you will have _MCq/NGq_ equal to
_ADq/EFq_. Whence AD, the Sine of Incidence, is to EF the Sine of
Refraction, as MC to NG, that is, in a given _ratio_. And this
Demonstration being general, without determining what Light is, or by
what kind of Force it is refracted, or assuming any thing farther than
that the refracting Body acts upon the Rays in Lines perpendicular to
its Surface; I take it to be a very convincing Argument of the full
truth of this Proposition.

So then, if the _ratio_ of the Sines of Incidence and Refraction of any
sort of Rays be found in any one case, 'tis given in all cases; and this
may