# Ibn Sahl (mathematician)

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

Reproduction of Millī MS 867 fol. 7r, showing his discovery of the law of refraction (from Rashed, 1990). 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.
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 change of medium 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 ratio of the refractive indices of the media.[1]

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 manuscript contains a section missing entirely from the Tehran manuscript. The Damascus manuscript has the title Fī al-'āla al-muḥriqa "On the burning instruments", the Tehran manuscript 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][11] 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.

Ibn Sahl designed convex lenses that focus lights rays that are parallel, which can cause an object to burn at a specific distance. A biconvex lens has the ability to focus at a specific point at an infinite distance. Ibn Sahl has made many contributions to optics, he also wrote an article about the celestial sphere. A constant ratio is the main focus point of his study, and it allows for a better understanding of refraction lenses. Ibn Sahl did an experiment where a piece of crystal was used to propagate a ray of light through the crystal which then refracts in the air.

## References

1. ^ Kurt Bernardo Wolf, Geometric Optics on Phase Space, p. 9, Springer, 2004, ISBN 3-540-22039-9 online
2. ^ Enterprise of Science in Islam: New Perspectives - J. P. Hogendijk,A. I. Sabra "The first clear evidence we have of a correct understanding of Ptolemy's theory of refraction does not appear in the Arabic sources available to us until the second half of the tenth century, when the Persian mathematician al-Ala ibn Sahl was able to put Ptolemy's ideas to use in formulating entirely original geometrical arguments for the construction of burning instruments by means of refraction"
3. ^ http://www.iranicaonline.org/articles/optics,"There are a number of optical texts by authors with a Persian ethnicity or association. The earliest is Abu Saʿd al-ʿAlāʾ Ebn Sahl at the Persian Buyid court (945–1055), better known for his early conception of the “sine law of refraction” and burning mirrors (Rashed, 1990, pp. 464-68; 1993; 2005) than his work on optics proper (Sabra, 1989, pp. lix-lx; 1994)."
4. ^ https://ijhpm.org/index.php/IJHPM/article/download/111/62,"Exploiting the 10th-century Persian mathematician Ibn Sahl’s development on Ptolemy’s studies of refraction,48 he generalized the relationship between incident and refracted rays in a form that presaged Snell’s law."
5. ^ https://www.sciencelearn.org.nz/resources/1867-light-ideas-and-technology-timeline,"Persian scientist Ibn Sahl writes On burning mirrors and lenses, which sets out his understanding of how curved mirrors and lenses bend and focus light. He discovers a law of refraction mathematically equivalent to Snell’s law (1615)."
6. ^ Hogendijk, edited by Jan P.; Sabra, Abdelhamid I. (2003). The enterprise of science in Islam : new perspectives. Cambridge, Mass. ; London: MIT. p. 89. ISBN 0-262-19482-1. `{{cite book}}`: `|first1=` has generic name (help)
7. ^ "Nothing in his surname and given names, however, allows us to glimpse either his country of origin or his social and religious allegiance — unless a link may be established with another Ibn Sahl of the same period, who was an astrologer concerned with mathematics; for the time being, however, this connection has no historical value." Roshdi Rashed, Geometry and Dioptrics in Classical Islam, London (2005), p. 3.
8. ^ Rashed (1990:"Ibn al-Haytham was not the first to have effectively used Ptolemy's Optics, [...] al-Kindi was not the only significant figure in the history of Arabic optics before Ibn al-Haytham"
9. ^ "Roshdi Rashed, Historian of Greek and Arabic Mathematics". 2017: 12. `{{cite journal}}`: Cite journal requires `|journal=` (help)
10. ^ http://spie.org/etop/2007/etop07fundamentalsII.pdf," R. Rashed credited Ibn Sahl with discovering the law of refraction [23], usually called Snell’s law and also Snell and Descartes’ law."
11. ^ Smith, A. Mark (2015). From Sight to Light: The Passage from Ancient to Modern Optics. University of Chicago Press. p. 178. ISBN 9780226174761.