Taking a pen in hand, and calculating by means of Snell's law the
track of every ray through a raindrop, Descartes found that, at one
particular angle, the rays, reflected at its back, emerged from the
drop _almost parallel to each other_. They were thus enabled to
preserve their intensity through long atmospheric distances. At all
other angles the rays quitted the drop _divergent_, and through this
divergence became so enfeebled as to be practically lost to the eye.
The angle of parallelism here referred to was that of forty-one
degrees, which observation had proved to be invariably associated with
the rainbow.

From what has been said, it is clear that two observers standing
beside each other, or one above the other, nay, that even the two eyes
of the same observer, do not see exactly the same bow. The position of
the base of the cone changes with that of its apex. And here we have
no difficulty in answering a question often asked--namely, whether a
rainbow is ever seen reflected in water. Seeing two bows, the one in
the heavens, the other in the water, you might be disposed to infer
that the one bears the same relation to the other that a tree upon the
water's edge bears to its reflected image. The rays, however, which
reach an observer's eye after reflection from the water, and which
form a bow in the water, would, were their course from the shower
uninterrupted, converge to a point vertically under the observer, and
as far below the level of the water as his eye is above it. But under
no circumstances could an eye above the water-level and one below it
see the same bow--in other words, the self-same drops of rain cannot
form the reflected bow and the bow seen directly in the heavens. The
reflected bow, therefore, is not, in the usual optical sense of the
term, the _image_ of the bow seen in the sky.