I set my bow in the cloud, and it shall be a sign of the covenant between me and the earth...Genesis

The rainbow arises when the sun shines upon humid air; or again by a certain peculiar blending of light with air, which will cause either all the distinctive qualities of these colors or else some of them belonging to a single kind, and from the reflection of this light the air all around will be colored as we see it to be, as the sun shines upon its parts. The circular shape which it assumes is due to the fact that the distance of every point is perceived by our sight to be equal; or it may be because, the atoms in the air or in the clouds and deriving from the sun having been thus united, the aggregate of them presents a sort of roundness. Epicurus of Samos, Letter to Pythocles.

Sometimes we can recognize both parts of a rainbow. The first rainbow arc is at 42 degrees and the secondary less visible at 50 degrees. The colours of the main rainbow vary smoothly from red on the outside of the arc to violet on the inside. The secondary rainbow displays colors in the reverse order of the main. Between these two rainbow arcs there is a dark band called Alexander's dark band as Alexander of Aphrodisias was the first who described this phenomenon.

At sea aboard USS Harry S. Truman (CVN 75) Jan. 1, 2003 -- A rainbow welcomes in the new year for Sailors and Marines aboard the Harry S. Truman in Souda Bay, Crete, Greece Source.

Alexander of Aphrodisias was a Peripatetic philosopher and commentator, active in the late second and early third century AD. Aristotle wrote extensively on rainbows in his De Meteorologica, and even speculated that a rainbow is caused by the reflection of sunlight from the drops of water in a cloud.

René Descartes was the first person to give a full explanation of how a rainbow is formed. He showed that if one traces the path through a spherical raindrop of parallel light-rays entering the drop at different points on its surface, each emerges in a different direction, but there is a concentration of emerging rays at an angle of 42 degrees from the reverse direction to the incident rays, in exact agreement with the observed angular size of rainbows. Furthermore, since some colours are refracted more than others in a raindrop, the rainbow angle is slightly different for each colour, so a raindrop disperses the sun's light into a set of nearly overlapping coloured arcs.

Descartes showed that light-rays which are internally reflected twice inside a raindrop emerge concentrated at an angle of 50 degrees from the reverse direction to the incident rays. This rainbow is naturally less intense than the primary rainbow, since a light-ray always loses some of its intensity at every reflection or refraction. The 50 degrees represents the angle of maximum deviation of doubly reflected light from the reverse direction (i.e., doubly reflected light can deviate by more than this angle, but not by less). Thus, we expect the secondary rainbow to have a diffuse outer edge and a sharp inner edge. We expect doubly reflected violet light to be refracted more strongly than doubly reflected red light in a raindrop. Red is concentrated on the inside of the secondary rainbow and violet is concentrated on the outside. Since no reflected light emerges between the primary and secondary rainbows i.e. in the angular range 42 to 50 degrees, relative to the reverse direction, this region of the sky looks slightly less bright than the other surrounding regions of the sky, which explains Alexander's dark band.

References

http://plato.stanford.edu/entries/alexander-aphrodisias/