fall upon the Circumference at L. Parallel to the incident Ray AN draw
the Diameter BQ, and let the Sine of Incidence out of Air into Water be
to the Sine of Refraction as I to R. Now, if you suppose the Point of
Incidence N to move from the Point B, continually till it come to L, the
Arch QF will first increase and then decrease, and so will the Angle AXR
which the Rays AN and GR contain; and the Arch QF and Angle AXR will be
biggest when ND is to CN as sqrt(II - RR) to sqrt(3)RR, in which
case NE will be to ND as 2R to I. Also the Angle AYS, which the Rays AN
and HS contain will first decrease, and then increase and grow least
when ND is to CN as sqrt(II - RR) to sqrt(8)RR, in which case NE
will be to ND, as 3R to I. And so the Angle which the next emergent Ray
(that is, the emergent Ray after three Reflexions) contains with the
incident Ray AN will come to its Limit when ND is to CN as sqrt(II -
RR) to sqrt(15)RR, in which case NE will be to ND as 4R to I. And the
Angle which the Ray next after that Emergent, that is, the Ray emergent
after four Reflexions, contains with the Incident, will come to its
Limit, when ND is to CN as sqrt(II - RR) to sqrt(24)RR, in which
case NE will be to ND as 5R to I; and so on infinitely, the Numbers 3,
8, 15, 24, &c. being gather'd by continual Addition of the Terms of the
arithmetical Progression 3, 5, 7, 9, &c. The Truth of all this
Mathematicians will easily examine.[M]
Now it is to be observed, that as when the Sun comes to his Tropicks,
Days increase and decrease but a very little for a great while together;
so when by increasing the distance CD, these Angles come to their
Limits, they vary their quantity but very little for some time together,
and therefore a far greater number of the Rays which fall upon all the
Points N in the Quadrant BL, shall emerge in the Limits of these Angles,
than in any other Inclinations. And farther it is to be observed, that
the Rays which differ in Refrangibility will have different Limits of
their Angles of Emergence, and by consequence according to their
different Degrees of Refrangibility emerge most copiously in different
Angles, and being separated from one another appear each in their proper
Colours. And what those Angles are may be easily gather'd from the
foregoing Theorem by Computation.
For in the least refrangible Rays the Sines I and R (as was found above)
are 108 and 81, and thence by Computation the greatest Angle AXR will be
found 42 Degrees and 2 Minutes, and the least Angle AYS, 50 Degrees and
57 Minutes. And in the most refrangible Rays the Sines I and R are 109
95