page_title: Snell's law
section_title: History
hierarchical_section_title: Snell's law / History
caption_attribution_description: English: Milli MS 867, fol. 7r, Milli Library, Tehran (Ibn Sahl's treatiser on optics). reproduced from Rashed, R. "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses", Isis 81, p. 464–491, 1990. [p. 465].
context_page_description: Snell's law 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 metamaterials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.
  Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalent to the reciprocal of the ratio of the indices of refraction:
  with each as the angle measured from the normal of the boundary, as the velocity of light in the respective medium, and as the refractive index of the respective medium.
  The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.
context_section_description: 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 fudging his data to fit theory (see: confirmation bias). Alhazen, in his Book of Optics (1021), came closer to discovering the law of refraction, though he did not take this step.
  The law eventually named after Snell was first accurately described by the Persian scientist Ibn Sahl at the 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 aberrations.
  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the Dutch astronomer Willebrord Snellius (1580–1626)—Snell—derived a mathematically equivalent form, that remained unpublished during his lifetime.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 Dioptrics, 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.
  According to Dijksterhuis, "In De natura lucis et proprietate (1662) Isaac Vossius said that Descartes had seen Snell's paper and concocted his own proof.  We now know this charge to be undeserved but it has been adopted many times since."  Both Fermat and Huygens repeated this accusation that Descartes had copied Snell. In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes."
  In his 1678 Traité de la Lumière, Christiaan Huygens showed how Snell's law of sines could be explained by, or derived from, the wave nature of light, using what we have come to call the Huygens–Fresnel principle.
  With the development of modern optical and electromagnetic theory, the ancient Snell's law was brought into a new stage.  In 1962, Bloembergen showed that at the boundary of nonlinear medium, the Snell's law should be written in a general form. In 2008 and 2011, plasmonic metasurfaces were also demonstrated to change the reflection and refraction directions of light beam.
caption: Reproduction of a page of Ibn Sahl's manuscript showing his discovery of the law of refraction.
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