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
and 81, and thence by Computation the greatest Angle AXR will be found
40 Degrees and 17 Minutes, and the least Angle AYS 54 Degrees and 7
Minutes.

Suppose now that O [in _Fig._ 15.] is the Spectator's Eye, and OP a Line
drawn parallel to the Sun's Rays and let POE, POF, POG, POH, be Angles
of 40 Degr. 17 Min. 42 Degr. 2 Min. 50 Degr. 57 Min. and 54 Degr. 7 Min.
respectively, and these Angles turned about their common Side OP, shall
with their other Sides OE, OF; OG, OH, describe the Verges of two
Rain-bows AF, BE and CHDG. For if E, F, G, H, be drops placed any where
in the conical Superficies described by OE, OF, OG, OH, and be
illuminated by the Sun's Rays SE, SF, SG, SH; the Angle SEO being equal
to the Angle POE, or 40 Degr. 17 Min. shall be the greatest Angle in
which the most refrangible Rays can after one Reflexion be refracted to
the Eye, and therefore all the Drops in the Line OE shall send the most
refrangible Rays most copiously to the Eye, and thereby strike the
Senses with the deepest violet Colour in that Region. And in like
manner the Angle SFO being equal to the Angle POF, or 42 Degr. 2 Min.
shall be the greatest in