ween them, and the exterior by two Refractions, and
two sorts of Reflexions between them in each Drop of Water, and proves
his Explications by Experiments made with a Phial full of Water, and
with Globes of Glass filled with Water, and placed in the Sun to make
the Colours of the two Bows appear in them. The same Explication
_Des-Cartes_ hath pursued in his Meteors, and mended that of the
exterior Bow. But whilst they understood not the true Origin of Colours,
it's necessary to pursue it here a little farther. For understanding
therefore how the Bow is made, let a Drop of Rain, or any other
spherical transparent Body be represented by the Sphere BNFG, [in _Fig._
14.] described with the Center C, and Semi-diameter CN. And let AN be
one of the Sun's Rays incident upon it at N, and thence refracted to F,
where let it either go out of the Sphere by Refraction towards V, or be
reflected to G; and at G let it either go out by Refraction to R, or be
reflected to H; and at H let it go out by Refraction towards S, cutting
the incident Ray in Y. Produce AN and RG, till they meet in X, and upon
AX and NF, let fall the Perpendiculars CD and CE, and produce CD till it
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