Editing Ibn al-Haytham (section)

=== Alhazen's problem ===
{{Main|Alhazen's problem}}
[[File:Theorem of al-Haitham.JPG|thumb|The [[Circle#Theorems|theorem of Ibn Haytham]]]]
His work on [[catoptrics]] in Book V of the Book of Optics contains a discussion of what is now known as Alhazen's problem, first formulated by [[Ptolemy]] in 150 AD. It comprises drawing lines from two points in the [[plane (mathematics)|plane]] of a circle meeting at a point on the [[circumference]] and making equal angles with the [[Normal (geometry)|normal]] at that point. This is equivalent to finding the point on the edge of a circular [[billiard table]] at which a player must aim a cue ball at a given point to make it bounce off the table edge and hit another ball at a second given point. Thus, its main application in optics is to solve the problem, "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer." This leads to an [[Quartic equation|equation of the fourth degree]].<ref>{{harvnb|O'Connor|Robertson|1999}}, {{harvnb|Weisstein|2008}}.</ref> This eventually led Alhazen to derive a formula for the sum of [[fourth power]]s, where previously only the formulas for the sums of squares and cubes had been stated. His method can be readily generalized to find the formula for the sum of any integral powers, although he did not himself do this (perhaps because he only needed the fourth power to calculate the volume of the paraboloid he was interested in). He used his result on sums of integral powers to perform what would now be called an [[Integral|integration]], where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref>{{harvnb|Katz|1995|pp=165–169, 173–174}}.</ref> Alhazen eventually solved the problem using [[conic section]]s and a geometric proof. His solution was extremely long and complicated and may not have been understood by mathematicians reading him in Latin translation.
Later mathematicians used [[Descartes]]' analytical methods to analyse the problem.<ref>{{harvnb|Smith|1992}}.</ref> An algebraic solution to the problem was finally found in 1965 by Jack M. Elkin, an actuarian.<ref>{{Citation|last=Elkin|first=Jack M.|title=A deceptively easy problem|journal=Mathematics Teacher|volume=58|issue=3|pages=194–199|year=1965|doi=10.5951/MT.58.3.0194|jstor=27968003}}</ref> Other solutions were discovered in 1989, by Harald Riede<ref>{{Citation|last=Riede|first=Harald|title=Reflexion am Kugelspiegel. Oder: das Problem des Alhazen|journal=Praxis der Mathematik|volume=31|issue=2|pages=65–70|year=1989|language=de}}</ref> and in 1997 by the [[University of Oxford|Oxford]] mathematician [[Peter M. Neumann]].<ref>{{Citation|last=Neumann|first=Peter M.|author-link=Peter M. Neumann|title=Reflections on Reflection in a Spherical Mirror|journal=[[American Mathematical Monthly]]|volume=105|issue=6|pages=523–528|year=1998|jstor=2589403|mr=1626185|doi=10.1080/00029890.1998.12004920}}</ref><ref>{{Citation|last=Highfield |first=Roger |author-link=Roger Highfield |date=1 April 1997 |title=Don solves the last puzzle left by ancient Greeks |journal=[[Electronic Telegraph]] |volume=676 |url=https://www.telegraph.co.uk/htmlContent.jhtml?html=/archive/1997/04/01/ngre01.html|url-status=dead |archive-url=https://web.archive.org/web/20041123051228/http://www.telegraph.co.uk/htmlContent.jhtml?html=%2Farchive%2F1997%2F04%2F01%2Fngre01.html |archive-date=23 November 2004 }}</ref>
Recently, [[Mitsubishi Electric Research Laboratories]] (MERL) researchers solved the extension of Alhazen's problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors.<ref>{{harvnb|Agrawal|Taguchi|Ramalingam|2011}}.</ref>