
Many sources claim that the beginnings of the modern theory of the rainbow originated with Rene Descartes in 1637. While it is certainly true that Descartes' work was the first comprehensive, correct mathematical treatment of the rainbow, it was the culmination of the work of many investigators, dating back to Aristotle. Not to take anything away from Descartes, but he certainly was not the first person to recognize that rainbows form as a result of the interaction of light rays and individual, spherical droplets of water in the air. Such notables as Kepler and Harriot made contributions, as well as many other less well know scientists. In fact, according to Boyer, Leibniz and Newton all but accused (quite unfairly, as it turns out) Descartes of plagiarizing the work of Francisco Maurolio (14941573) and Marco Antonio de Dominis (15661624) in his work on the rainbow.^{1}
Certainly, rainbows are the result of the interaction of light and rain drops. Unraveling some of the details of this interaction is very interesting.
The diagram above shows the path that a light ray might take through a spherical rain drop. The geometry of the situation is very elegant, but, contrary to some explanations (text p. 455 Fig. 2927, for instance) it is not a complete explanation of how rainbows form.
Why? Because parallel rays from the sun strike each rain drop at many angles (labeled "a" in the diagram above), and for each angle in air, the refracted and reflected light ray leaves the drop at a different angle. So where does this 42^{o} rainbow angle come from? Why aren't rainbows smeared all over the sky? Why is light "concentrated at the angles shown"(quoted from the text) in Fig. 2927 of the text? We have to look more closely.
^{1}Carl B. Boyer, "Kepler's Explanation of the Rainbow", American Journal of Physics 18, 360366(1950)