# Ibn Sahl (mathematician)

Reproduction of Millī MS 867 fol. 7r, showing his discovery of the law of refraction (from Rashed, 1990).
Interpretation of Ibn Sahl's construction. If the ratio of lengths ${\displaystyle L_{1}/L_{2}}$ is kept equal to ${\displaystyle n_{1}/n_{2}}$ then the rays satisfy the law of sines, or Snell's law. The inner hypotenuse of the right-angled triangle shows the path of an incident ray and the outer hypotenuse shows an extension of the path of the refracted ray if the incident ray met a crystal whose face is vertical at the point where the two hypotenuses intersect. The ratio of the length of the smaller hypotenuse to the larger is the reciprocal of the refractive index of the crystal.[1] The lower part of the figure shows a representation of a plano-convex lens (at the right) and its principal axis (the intersecting horizontal line). The curvature of the convex part of the lens brings all rays parallel to the horizontal axis (and approaching the lens from the right) to a focal point on the axis at the left.

Ibn Sahl (full name Abū Saʿd al-ʿAlāʾ ibn Sahl أبو سعد العلاء ابن سهل; c. 940–1000) was a Persian[2][3][4][5] mathematician and physicist of the Islamic Golden Age,[6] associated with the Buwayhid court of Baghdad. Nothing in his name allows us to glimpse his country of origin.[7]

He is known to have written an optical treatise around 984. The text of this treatise was reconstructed by Roshdi Rashed from two manuscripts (edited 1993).: Damascus, al-Ẓāhirīya MS 4871, 3 fols., and Tehran, Millī MS 867, 51 fols. The Tehran manuscript is much longer, but it is badly damaged, and the Damascus ms. contains a section missing entirely from the Tehran ms. The Damascus ms. has the title Fī al-'āla al-muḥriqa "On the burning instruments", the Tehran ms. has a title added in a later hand Kitāb al-harrāqāt "The book of burners".

Ibn Sahl is the first Muslim scholar known to have studied Ptolemy's Optics, and as such an important precursor to the Book of Optics by Ibn Al-Haytham (Alhazen), written some thirty years later.[8] Ibn Sahl dealt with the optical properties of curved mirrors and lenses and has been described as the discoverer of the law of refraction (Snell's law).[9][10] Ibn Sahl uses this law to derive lens shapes that focus light with no geometric aberrations, known as anaclastic lenses. In the remaining parts of the treatise, Ibn Sahl dealt with parabolic mirrors, ellipsoidal mirrors, biconvex lenses, and techniques for drawing hyperbolic arcs.