With the triangles in this diagram, Ibn Sahl illustrated the law that is currently known as Snell’s law. ©Iranian National Library, Tehran. Manuscript MS 687
Abu Sa’d al-’Ala’ ibn Sahl, ابن سهل, (c. 940–1000 AD) lived and worked as a geometer at the Abbasid court in Baghdad. The exact details of his biography and ancestry are unknown. He wrote important works on geometric optics, mathematics, and astronomy [1].
While investigating the transparency of the heavenly spheres that occur in Aristotelian cosmology, Ibn Sahl decided to study Ptolemy’s classical work Optics, written in the second century AD. Thus, Ibn Sahl was the first of the Arabic sources to have read and correctly understood Ptolemy’s theory of refraction [1]. Ibn Sahl utilized this theory in an entirely original way for constructing burning instruments such as lenses and glass spheres by means of refraction.
6.1 On Burning Mirrors and Lenses
In the year 984 AD, Ibn Sahl wrote the treatise On Burning mirrors and lenses. In this work, he investigated the optimum (non-spherical) shape of lenses and mirrors to focus light at a given distance. Ibn Sahl “appears to be the first in history to engage in research on burning lenses” [2]. He subsequently treated the parabolic mirror, the ellipsoidal mirror, the plano-convex lens, and the biconvex lens. His calculations on geometric aberration predate similar calculations done by Descartes in the 1620s, as discussed in Part IV. Ibn Sahl carried out these calculations both for light sources at an (almost) infinite distance such as the sun, but also for light sources at finite distances [3]. In these calculations, Ibn Sahl needed a law of refraction. Intriguingly, for this he used a law that is geometrically equivalent to Snell’s law that would be re-discovered in 1602 by Thomas Harriot and in 1621 by Willebrord Snellius. The sine law of refraction was thus discovered by Ibn Sahl [4, 5].
The illustration shown at the beginning of this entry is taken from a page of Ibn Sahl’s manuscript. In the top part of the figure, two overlapping triangles are shown. The hypotenuse of the external triangle represents the direction of incident light, while the hypotenuse of the internal triangle represents the direction of refracted light inside a transparent medium. By demanding that these two direction vectors intersect in one point, constructing the direction of refracted light with this figure is geometrically equivalent to keeping the ratio of sines of incident and refracted angles constant. This is known as Snell’s law of refraction.
Ibn Sahl used this law of refraction several times in the treatise, but without explicitly stating it as a law [4, 5]. Indeed, the concept of natural law did not exist at the time. Ibn Sahl used his law as if it was a mathematical relation that was well known. He repeatedly applied this relation, utilizing the fact that the ratio between the sinuses of incoming and refracted angles is constant. He made no reference to the fact that this constant depends on material dependent properties, i.e., what is now known as their refractive index.
Ibn Sahl’s treatise was later used by Ibn al-Haytham in his investigations of refraction. Interestingly, Ibn al-Haytham apparently did not recognize the law of refraction as used by Ibn Sahl. Instead, Ibn al-Haytham started his own experimental investigation into finding a law of refraction. While Ibn Sahl examined the refraction of light, his writing does not include a reference to vision [1].