**Al-Haytham** is also known as **Alhazen**. He was an Islamic mathematician who wrote early works on optics as well as geometry and number theory.

- We shall discuss this problem, and ibn al-Haytham's other work, after giving some biographical details.
- In contrast to our lack of knowledge of the lives of many of the Arabic mathematicians, we have quite a number of details of ibn al-Haytham's life.
- It is worth commenting that an autobiography written by ibn al-Haytham in 1027 survives, but it says nothing of the events his life and concentrates on his intellectual development.
- Since the main events that we know of in ibn al-Haytham's life involve his time in Egypt, we should set the scene regarding that country.
- These events were happening while ibn al-Haytham was a young boy growing up in Basra.
- We know little of ibn al-Haytham's years in Basra.
- However, ibn al-Haytham became increasingly unhappy with his deep studies of religion and made a decision to devote himself entirely to a study of science which he found most clearly described in the writings of Aristotle.
- Having made this decision, ibn al-Haytham kept to it for the rest of his life devoting all his energies to mathematics, physics, and other sciences.
- Ibn al-Haytham went to Egypt some considerable time after he made the decision to give up his job as a minister and to devote himself to science, for he had made his reputation as a famous scientist while still in Basra.
- We do know that al-Hakim was Caliph when ibn al-Haytham reached Egypt.
- Al-Hakim, despite being a cruel leader who murdered his enemies, was a patron of the sciences employing top quality scientists such as the astronomer ibn Yunus.
- Our knowledge of ibn al-Haytham's interaction with al-Hakim comes from a number of sources, the most important of which is the writings of al-Qifti.
- We are told that al-Hakim learnt of a proposal by ibn al-Haytham to regulate the flow of water down the Nile.
- He requested that ibn al-Haytham come to Egypt to carry out his proposal and al-Hakim appointed him to head an engineering team which would undertake the task.
- However, as the team travelled further and further up the Nile, ibn al-Haytham realised that his idea to regulate the flow of water with large constructions would not work.
- Ibn al-Haytham returned with his engineering team and reported to al-Hakim that they could not achieve their aim.
- Al-Hakim, disappointed with ibn al-Haytham's scientific abilities, appointed him to an administrative post.
- At first ibn al-Haytham accepted this but soon realised that al-Hakim was a dangerous man whom he could not trust.
- It appears that ibn al-Haytham pretended to be mad and as a result was confined to his house until after al-Hakim's death in 1021.
- According to al-Qifti, ibn al-Haytham lived for the rest of his life near the Azhar Mosque in Cairo writing mathematics texts, teaching and making money by copying texts.
- Since the Fatimids founded the University of Al-Azhar based on this mosque in 970, ibn al-Haytham must have been associated with this centre of learning.
- A different report says that after failing in his mission to regulate the Nile, ibn al-Haytham fled from Egypt to Syria where he spent the rest of his life.
- This however seems unlikely for other reports certainly make it certain that ibn al-Haytham was in Egypt in 1038.
- One further complication is the title of a work ibn al-Haytham wrote in 1027 which is entitled Ibn al-Haytham's answer to a geometrical question addressed to him in Baghdad.
- Yet another version has ibn al-Haytham pretending to be mad while still in Basra.
- Ibn al-Haytham's writings are too extensive for us to be able to cover even a reasonable amount.
- A seven volume work on optics, Kitab al-Manazir, is considered by many to be ibn al-Haytham's most important contribution.
- The previous major work on optics had been Ptolemy's Almagest Ⓣ(The major thesis: from the Arabic 'al-majisti' -- the Arabic translation of the Greek 'Mathematike Syntaxis' later translated into Latin as 'Magna Syntaxis') and although ibn al-Haytham's work did not have an influence to equal that of Ptolemy's, nevertheless it must be regarded as the next major contribution to the field.
- The work begins with an introduction in which ibn al-Haytham says that he will begin "the inquiry into the principles and premises".
- Although we have quoted the problem for spherical mirrors, ibn al-Haytham also considered cylindrical and conical mirrors.
- Ibn al-Haytham's study of refraction led him to propose that the atmosphere had a finite depth of about 15 km.
- Abu al-Qasim ibn Madan was an astronomer who proposed questions to ibn al-Haytham, raising doubts about some of Ptolemy's explanations of physical phenomena.
- Ibn al-Haytham wrote a treatise Solution of doubts in which he gives his answers to these questions.
- There are strange contrasts in ibn al-Haytham's work relating to Ptolemy.
- In Al-Shukuk ala Batlamyus (Doubts concerning Ptolemy), ibn al-Haytham is critical of Ptolemy's ideas yet in a popular work the Configuration, intended for the layman, ibn al-Haytham completely accepts Ptolemy's views without question.
- One of the mathematical problems which ibn al-Haytham attacked was the problem of squaring the circle.
- Whether ibn al-Haytham suspected that the problem was insoluble or whether he only realised that he could not solve it, in an interesting question which will never be answered.
- In Opuscula ibn al-Haytham considers the solution of a system of congruences.
- Here ibn al-Haytham gives a general method of solution which, in the special case, gives the solution (7 - 1)! + 1.
- Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem).
- Another contribution by ibn al-Haytham to number theory was his work on perfect numbers.
- Ibn al-Haytham's main purpose in Analysis and synthesis is to study the methods mathematicians use to solve problems.
- The ancient Greeks used analysis to solve geometric problems but ibn al-Haytham sees it as a more general mathematical method which can be applied to other problems such as those in algebra.
- In this work ibn al-Haytham realises that analysis was not an algorithm which could automatically be applied using given rules but he realises that the method requires intuition.

Born 965, (possibly) Basra, Persia (now Iraq). Died 1040, (possibly) Cairo, Egypt.

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Ancient Arab, Ancient Greek, Astronomy, Geometry, Origin Iraq, Number Theory, Physics, Puzzles And Problems, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive