Alhazen's famous Billiard Table problem

Ibn al-Haytham (965 in Basra - c. 1039 in Cairo), also know as Alhazen, is regarded as the "father of modern optics” and was a great scientist. 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 of a circle meeting at a point on the circumference and making equal angles with the normal at that point.

This is equivalent to finding the point on the edge of a circular Billiard Table at which a cue ball at a given point must be aimed in order to canon off the edge of the table 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 equation of the fourth degree. This eventually led Ibn al-Haytham to derive the earliest formula for the sum of fourth powers; by using an early proof by mathematical induction, he developed a method that can be readily generalized to find the formula for the sum of any integral powers. He applied his result of sums on integral powers to find the volume of a paraboloid through integration.

He was thus able to find the integrals for polynomials up to the fourth degree, and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal and integral calculus. Ibn al-Haytham eventually solved the problem using conic sections and a geometric proof, though many after him attempted to find an algebraic solution to the problem, which was finally found in 1997 by the Oxford mathematician Peter M. Neumann.