for the claim. The fourth group consists of ten works on astronomical theory, including the Doubts and Model of the Motions discussed above. Mathematical works In 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". He developed a formula for summing the first 100 natural numbers, using a geometric proof to prove the formula. Geometry Alhazen explored what is now known as the Euclidean parallel postulate, the fifth postulate in Euclid's Elements, using a proof by contradiction, and in effect introducing the concept of motion into geometry. He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral". In elementary geometry, Alhazen attempted to solve the problem of squaring the circle using the area of lunes (crescent shapes), but later gave up on the impossible task. 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 lunes of Alhazen; they have the same total area as the triangle itself. Number theory Alhazen's contributions to number theory include his work on perfect numbers. In his Analysis and Synthesis, he may have been the first to state that every even perfect number is of the form 2n−1(2n − 1) where 2n − 1 is prime, but he was not able to prove this result; Euler later proved it in the 18th century, and it is now called the Euclid–Euler theorem. Alhazen solved problems involving 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. 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. Other works Influence of Melodies on the Souls of Animals Alhazen also wrote a Treatise on the Influence of Melodies on the Souls of Animals, although no copies have survived. It appears to have been concerned with