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Ibn Sahl (mathematician)

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Mathematician (c. 940-1000)

Reproduction of Millī MS 867 fol. 7r, showing hisdiscovery of the law of refraction (from Rashed, 1990). The lower part of the figure shows a representation of aplano-convex lens (at the right) and itsprincipal 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 afocal point on the axis at the left.

Interpretation of Ibn Sahl's construction. If the ratio of lengths $L_{1}/L_{2}$ is kept equal to $n_{1}/n_{2}$ then the rays satisfy the law of sines, or Snell's law. The innerhypotenuse of the right-angled triangle shows the path of anincident ray and the outer hypotenuse shows an extension of the path of therefracted 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]

{\displaystyle L_{1}/L_{2}} — Interpretation of Ibn Sahl's construction. If the ratio of lengths $L_{1}/L_{2}$ is kept equal to $n_{1}/n_{2}$ then the rays satisfy the law of sines, or Snell's law. The innerhypotenuse of the right-angled triangle shows the path of anincident ray and the outer hypotenuse shows an extension of the path of therefracted 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]

Ibn Sahl (full name:Abū Saʿd al-ʿAlāʾ ibn SahlPersian:ابوسعدالعلاءِبن سعل (ابن سهل); c. 940–1000) was aPersian^[2]^[3]^[4]^[5] orArab^[6] mathematician andphysicist of theIslamic Golden Age,^[7] associated with theBuyid court ofBaghdad. Nothing in his name allows us to glimpse his country of origin.^[8]

He is known to have written an optical treatise around 984. The text of this treatise was reconstructed byRoshdi 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 titleFī al-'āla al-muḥriqa "On the burning instruments", the Tehran manuscript has a title added in a later handKitāb al-harrāqāt "The book of burners".

Ibn Sahl is the first Muslim scholar known to have studiedPtolemy'sOptics, and as such an important precursor to theBook of Optics byIbn Al-Haytham (Alhazen), written some thirty years later.^[9]Ibn Sahl dealt with the optical properties ofcurved mirrors andlenses and has been described as the discoverer of the law ofrefraction (Snell's law).^[10]^[11]^[12]Ibn Sahl uses this law to derive lens shapes that focus light with no geometric aberrations, known asanaclastic lenses.In the remaining parts of the treatise, Ibn Sahl dealt withparabolic mirrors,ellipsoidal mirrors,biconvex lenses, and techniques for drawinghyperbolic arcs.

Ibn Sahl designed convex lenses that focus light rays that are parallel, which can cause an object to burn at a specific distance.^{[citation needed]}

References

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^Kurt Bernardo Wolf,Geometric Optics on Phase Space, p. 9, Springer, 2004,ISBN 3-540-22039-9 online
^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"
^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)."
^Shoja, Mohammadali M.; Agutter, Paul S.; Tubbs, R. Shane (2015)."Rhazes Doubting Galen: Ancient and Medieval Theories of Vision".International Journal of History and Philosophy of Medicine.5: 10510.doi:10.18550/ijhpm.011515.0510 (inactive 29 July 2025). Archived fromthe original on 16 November 2019.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.{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link)
^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)."
^"Ibn Saul Discovers the Law of Refraction : History of Information".www.historyofinformation.com. Retrieved2025-11-11.
^Hogendijk, Jan P.; Sabra, Abdelhamid I., eds. (2003).The Enterprise of Science in Islam: New Perspectives. Cambridge, Massachusetts; London, England: MIT Press. p. 89.ISBN 0-262-19482-1.
^"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 withanother 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.
^Rashed (1990:"Ibn al-Haytham was not the first to have effectively used Ptolemy'sOptics, [...] al-Kindi was not the only significant figure in the history of Arabic optics before Ibn al-Haytham"
^Papadopoulos, Athanase (2017)."Roshdi Rashed, Historian of Greek and Arabic Mathematics". p. 12.
^Zghal, Mourad; Bouali, Hamid-Eddine; Ben Lakhdar, Zohra; Hamam, Habib (2015)."The first steps for learning optics: Ibn Sahl's, Al-Haytham's and Young's works on refraction as typical examples"(PDF). In Nantel, Marc (ed.).Tenth International Topical Meeting on Education and Training in Optics and Photonics. Proceedings of the SPIE. Vol. 9665.doi:10.1117/12.2207465.R. Rashed credited Ibn Sahl with discovering the law of refraction [23], usually called Snell's law and also Snell and Descartes' law."
^Smith, A. Mark (2015).From Sight to Light: The Passage from Ancient to Modern Optics. University of Chicago Press. p. 178.ISBN 9780226174761.

Sources

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Rashed, Roshdi (1990). "A Pioneer in Anaclastics: Ibn Sahl on Burning Mirrors and Lenses".Isis.81 (3):464–491.doi:10.1086/355456.JSTOR 233423.
Rashed, R.,Géométrie et dioptrique au Xe siècle: Ibn Sahl, al-Quhi et Ibn al-Haytham. Paris: Les Belles Lettres, 1993
Zghal, Mourad; et al. (2007)."The first steps for learning optics: Ibn Sahl's, Al-Haytham's and Young's works on refraction as typical examples"(PDF). In Nantel, Marc (ed.).Tenth International Topical Meeting on Education and Training in Optics and Photonics. Proceedings of the SPIE. Vol. 9665. p. 3.Bibcode:2007SPIE.9665E..09Z.doi:10.1117/12.2207465.S2CID 13875045. Retrieved20 June 2011.
Berggren, Len (2007)."Ibn Sahl: Abū Saʿd al-ʿAlāʾ ibn Sahl". In Thomas Hockey; et al. (eds.).The Biographical Encyclopedia of Astronomers. New York: Springer. p. 567.ISBN 978-0-387-31022-0. (PDF version)
Selin, Helaine (1997).Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures.Springer Science & Business Media. pp. 432–433.ISBN 978-07-92-34066-9.
Sarmiş, İbrahım (1999).İBN SEHL - An article published in Turkish Encyclopedia of Islam. Vol. 20. Istanbul:TDV İslâm Ansiklopedisi. p. 312.ISBN 978-97-53-89447-0.

Mathematics in the medieval Islamic world

Mathematicians

9th century	'Abd al-Hamīd ibn Turk Sanad ibn Ali al-Jawharī Al-Ḥajjāj ibn Yūsuf Al-Kindi Qusta ibn Luqa Al-Mahani al-Dinawari Banū Mūsā brothers Hunayn ibn Ishaq Al-Khwarizmi Yusuf al-Khuri Ishaq ibn Hunayn Na'im ibn Musa Thābit ibn Qurra al-Marwazi Abu Said Gorgani
10th century	Abu al-Wafa al-Khazin Al-Qabisi Abu Kamil Ahmad ibn Yusuf Aṣ-Ṣaidanānī Sinān ibn al-Fatḥ al-Khojandi Al-Nayrizi Al-Saghani Brethren of Purity Ibn Sahl Ibn Yunus al-Uqlidisi Al-Battani Sinan ibn Thabit Ibrahim ibn Sinan Al-Isfahani Nazif ibn Yumn al-Qūhī Abu al-Jud Al-Sijzi Al-Karaji al-Majriti al-Jabali
11th century	Abu Nasr Mansur Alhazen Kushyar Gilani Al-Biruni Ibn al-Samh Abu Mansur al-Baghdadi Avicenna al-Jayyānī al-Nasawī al-Zarqālī ibn Hud Al-Isfizari Omar Khayyam Muhammad al-Baghdadi
12th century	Jabir ibn Aflah Al-Kharaqī Al-Khazini Al-Samawal al-Maghribi al-Hassar Sharaf al-Din al-Tusi Ibn al-Yasamin
13th century	Ibn al‐Ha'im al‐Ishbili Ahmad al-Buni Ibn Munim Alam al-Din al-Hanafi Ibn Adlan al-Urdi Nasir al-Din al-Tusi al-Abhari Muhyi al-Din al-Maghribi al-Hasan al-Marrakushi Qutb al-Din al-Shirazi Shams al-Din al-Samarqandi Ibn al-Banna' Kamāl al-Dīn al-Fārisī
14th century	Nizam al-Din al-Nisapuri Ibn al-Shatir Ibn al-Durayhim Al-Khalili al-Umawi
15th century	Ibn al-Majdi al-Rūmī al-Kāshī Ulugh Beg Ali Qushji al-Wafa'i al-Qalaṣādī Sibt al-Maridini Ibn Ghazi al-Miknasi
16th century	Al-Birjandi Muhammad Baqir Yazdi Taqi ad-Din Ibn Hamza al-Maghribi