Snel's re-discovery of the law came in 1621; but he did not publish it before his death in 1626. Descartes certainly was the first person to publish it, in his famous “Discourse on Method” (1637).
Consequently, the law is usually known in France as “Descartes's Law.” The great French physicist François Arago, who was addicted to questions of priority and national pride, asserted that Descartes was the discoverer; after referring to Ptolemy's work on the errors caused by atmospheric refraction in astronomical observations, he says:
As for the mathematical law of these deviations, which the Arab Alhasen, the Pole Vitellio, Kepler and other physicists have sought in vain, it is due to Descartes. I say Descartes, and Descartes alone; since, if the belated claims of Huyghens in favor of his compatriot Snellius were accepted, one would have to give up forever the writing of the history of science.
It is not clear whether Descartes discovered the law independently or not. Nevertheless, it is certain that both Harriot and Snel knew the law well before Descartes.
In 1990, Roshdi Rashed published a paper on a mathematical work from the year 984 that uses the true law of refraction to show that lenses should have hyperboloidal surfaces. (Such lenses are, in fact, used in the condensers of slide projectors today.)
In Rashed's summary of Ibn Sahl's monograph, the sine law seems to appear from nowhere. It is stated (in geometrical rather than trigonometrical terms: the ratio of two line segments has a fixed ratio, which corresponds to the reciprocal of the refractive index, in modern terminology), and then used to derive the hyperboloidal surface. No numerical values or experimental measurements are offered.
The 10th-Century paper, of which only portions survive, is obviously a masterpiece of geometric virtuosity. The full force of solid geometry, including conic sections, is displayed. The question is, should we regard this as a discovery of the law of refraction?
I am inclined to say no. Although Ibn Sahl had a practical application in mind (burning glases), there's no evidence that he himself was anything but a pure theoretician. Geometers often work problems backwards and then present the argument in the order opposite to that which was actually used; I suspect he began with conic surfaces of revolution, and discovered that what we would call the sine law turned out to be a simple property that would allow such surfaces to be useful for making burning glasses.
I'd be interested to hear from anyone who has more definite evidence on this matter.
© 2002, 2005, 2006, 2012 Andrew T. Young