Snell's law (also known as the Snell–Descartes law, and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air. In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. The law is also satisfied in meta-materials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

The law states that, for a given pair of media, the ratio of the sines of angle of incidence ${\displaystyle \left(\theta _{1}\right)}$ and angle of refraction ${\displaystyle \left(\theta _{2}\right)}$ is equal to the refractive index of the second medium with regard to the first (${\displaystyle n_{2,1}}$) which is equal to the ratio of the refractive indices ${\displaystyle \left({\tfrac {n_{2}}{n_{1}}}\right)}$ of the two media, or equivalently, to the ratio of the phase velocities ${\displaystyle \left({\tfrac {v_{1}}{v_{2}}}\right)}$ in the two media.

The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

Ptolemy, in Alexandria, Egypt, had found a relationship regarding refraction angles, but it was inaccurate for angles that were not small. Ptolemy was confident he had found an accurate empirical law, partially as a result of slightly altering his data to fit theory (see: confirmation bias).

The law was eventually named after Snell, although it was first discovered by the Persian scientist Ibn Sahl, at Baghdad court in 984. In the manuscript On Burning Mirrors and Lenses, Sahl used the law to derive lens shapes that focus light with no geometric aberration.

Alhazen, in his Book of Optics (1021), came close to rediscovering the law of refraction, but he did not take this step.

The law was rediscovered by Thomas Harriot in 1602, who however did not publish his results although he had corresponded with Kepler on this very subject. In 1621, the Dutch astronomer Willebrord Snellius (1580–1626)—Snell—derived a mathematically equivalent form, that remained unpublished during his lifetime. René Descartes independently derived the law using heuristic momentum conservation arguments in terms of sines in his 1637 essay La Dioptrique, and used it to solve a range of optical problems. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time. Descartes assumed the speed of light was infinite, yet in his derivation of Snell's law he also assumed the denser the medium, the greater the speed of light. Fermat supported the opposing assumptions, i.e., the speed of light is finite, and his derivation depended upon the speed of light being slower in a denser medium. Fermat's derivation also utilized his invention of adequality, a mathematical procedure equivalent to differential calculus, for finding maxima, minima, and tangents.

In his influential mathematics book Geometry, Descartes solves a problem that was worked on by Apollonius of Perga and Pappus of Alexandria. Given n lines L and a point P(L) on each line, find the locus of points Q such that the lengths of the line segments QP(L) satisfy certain conditions. For example, when n = 4, given the lines a, b, c, and d and a point A on a, B on b, and so on, find the locus of points Q such that the product QA*QB equals the product QC*QD. When the lines are not all parallel, Pappus showed that the loci are conics, but when Descartes considered larger n, he obtained cubic and higher degree curves. To show that the cubic curves were interesting, he showed that they arose naturally in optics from Snell's law.