Editing Ibn al-Haytham (section)

== Mathematical works ==
[[File:AlhazenSummation.svg|thumb|upright|Alhazen's geometrically proven summation formula]]
In [[Islamic mathematics|mathematics]], Alhazen built on the mathematical works of [[Euclid]] and [[Thabit ibn Qurra]] and worked on "the beginnings of the link between [[algebra]] and [[geometry]]". Alhazen made developments in [[conic section]]s and number theory.<ref>{{harvnb|Faruqi|2006|pp=395–396}}:
In seventeenth century Europe the problems formulated by Ibn al-Haytham (965–1041) became known as 'Alhazen's problem'. ... Al-Haytham's contributions to geometry and number theory went well beyond the Archimedean tradition. Al-Haytham also worked on analytical geometry and the beginnings of the link between algebra and geometry. Subsequently, this work led in pure mathematics to the harmonious fusion of algebra and geometry that was epitomised by Descartes in geometric analysis and by Newton in the calculus. Al-Haytham was a scientist who made major contributions to the fields of mathematics, physics and astronomy during the latter half of the tenth century.</ref>

He developed a formula for summing the first 100 natural numbers, using a geometric proof to prove the formula.<ref>{{harvnb|Rottman|2000}}, Chapter 1.</ref>

=== Geometry ===
[[File:LunesOfAlhazen.svg|thumb|upright|The lunes of Alhazen. The two blue lunes together have the same area as the green right triangle.]]
Alhazen explored what is now known as the [[Euclidean geometry|Euclidean]] [[parallel postulate]], the fifth [[Axiom|postulate]] in [[Euclid's Elements|Euclid's ''Elements'']], using a [[Reductio ad absurdum|proof by contradiction]],<ref>{{harvnb|Eder|2000}}.</ref> and in effect introducing the concept of motion into geometry.<ref>{{harvnb|Katz|1998|p=269}}: "In effect, this method characterised parallel lines as lines always equidistant from one another and also introduced the concept of motion into geometry."</ref> He formulated the [[Lambert quadrilateral]], which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral".<ref>{{harvnb|Rozenfeld|1988|p=65}}.</ref> He was criticised by Omar Khayyam who pointed that Aristotle had condemned the use of [[Motion (geometry)|motion in geometry]].<ref>{{Cite book |last1=Boyer |first1=Carl B. |url=https://books.google.com/books?id=bR9HAAAAQBAJ&dq=motion+geometry+alhazen&pg=PA220 |title=A History of Mathematics |last2=Merzbach |first2=Uta C. |date=2011 |publisher=John Wiley & Sons |isbn=978-0-470-63056-3 |language=en |access-date=19 March 2023 |archive-date=7 September 2023 |archive-url=https://web.archive.org/web/20230907232753/https://books.google.com/books?id=bR9HAAAAQBAJ&dq=motion+geometry+alhazen&pg=PA220 |url-status=live }}</ref>

In elementary geometry, Alhazen attempted to solve the problem of [[squaring the circle]] using the area of [[Lune (mathematics)|lunes]] (crescent shapes), but later gave up on the impossible task.<ref name="{{harvnb|o'connor|robertson|1999}}.">{{harvnb|O'Connor|Robertson|1999}}.</ref> The two lunes formed from a [[right triangle]] by erecting a semicircle on each of the triangle's sides, inward for the hypotenuse and outward for the other two sides, are known as the [[Lune of Hippocrates|lunes of Alhazen]]; they have the same total area as the triangle itself.<ref>{{Harvnb|Alsina|Nelsen|2010}}.</ref>

=== Number theory ===

Alhazen's contributions to [[number theory]] include his work on [[perfect number]]s. In his ''Analysis and Synthesis'', he may have been the first to state that every even perfect number is of the form 2<sup>''n''−1</sup>(2<sup>''n''</sup>&nbsp;−&nbsp;1) where 2<sup>''n''</sup>&nbsp;−&nbsp;1 is [[Prime number|prime]], but he was not able to prove this result; [[Leonhard Euler|Euler]] later proved it in the 18th century, and it is now called the [[Euclid–Euler theorem]].<ref name="{{harvnb|o'connor|robertson|1999}}." />

Alhazen solved problems involving [[congruence relation|congruences]] using what is now called [[Wilson's theorem]]. In his ''Opuscula'', Alhazen considers the solution of a system of congruences, and gives two general methods of solution. His first method, the canonical method, involved Wilson's theorem, while his second method involved a version of the [[Chinese remainder theorem]].<ref name="{{harvnb|o'connor|robertson|1999}}." />

=== Calculus ===
Alhazen discovered the sum formula for the fourth power, using a method that could be generally used to determine the sum for any integral power. He used this to find the volume of a [[paraboloid]]. He could find the integral formula for any polynomial without having developed a general formula.<ref>{{cite journal |doi=10.2307/2691411 |author=Katz, Victor J. |author-link=Victor J. Katz |title=Ideas of Calculus in Islam and India |jstor=2691411 |journal=Mathematics Magazine |year=1995 |volume=68 |issue=3 |pages=163–174 [165–169, 173–174]year=1995}}</ref>