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Ibn alHaytham (Alhazen)  

Born  CE^{[1]} (354 AH)^{[2]} July 1, 965 Basra, Buyid Emirate 
Died  March 6, 1040(430 AH)^{[3]} (aged 74)^{[1]} Cairo, Egypt, Fatimid Caliphate 
Residence  
Fields  
Known for 

Ibn alHaytham (Alhazen)  

Born  CE^{[1]} (354 AH)^{[2]} July 1, 965 Basra, Buyid Emirate 
Died  March 6, 1040(430 AH)^{[3]} (aged 74)^{[1]} Cairo, Egypt, Fatimid Caliphate 
Residence  
Fields  
Known for 

Abū ʿAlī alḤasan ibn alḤasan ibn alHaytham (Arabic: أبو علي، الحسن بن الحسن بن الهيثم), frequently referred to as Ibn alHaytham (Arabic: ابن الهيثم, Latinized as Alhazen^{[Notes 1]} or Alhacen; c. 965 – c. 1040), was an Arab^{[8]} scientist, polymath, mathematician, astronomer and philosopher who made significant contributions to the principles of optics, astronomy, mathematics, meteorology,^{[9]} visual perception and the scientific method.
He has been described as the father of modern optics, ophthalmology,^{[10]} experimental physics and scientific methodology^{[11]}^{[12]}^{[13]} and the first theoretical physicist^{[14]} In medieval Europe, he was nicknamed Ptolemaeus Secundus ("Ptolemy the Second")^{[15]} or simply called "The Physicist".^{[16]} He is also sometimes called alBasri (Arabic: البصري) after Basra, his birthplace.^{[17]}
According to one version of his biography,^{[18]} alHaytham, confident about the practical application of his mathematical knowledge, assumed he could regulate the floods of the Nile. Having been ordered to do so by AlHakim biAmr Allah, the sixth ruler of the Fatimid caliphate, he quickly realised its impossibility. Fearing for his life, he feigned madness^{[1]}^{[19]} and was placed under house arrest. Once AlHakim had died, he was able to prove that he was not mad and for the rest of his life made money copying texts while writing mathematical works and teaching.^{[20]}
Born c. 965 in Basra, which was then part of the Buyid emirate,^{[1]} to an Arab family,^{[21]}^{[22]} he lived mainly in Cairo, Egypt, dying there at age 74.^{[15]} During the Islamic Golden Age, Basra was a "key beginning of learning",^{[23]} and he was educated there and in Baghdad, the capital of the Abbasid Caliphate, and the focus of the "high point of Islamic civilization".^{[23]} During his time in Basra, he trained for government work and became Minister for the area.^{[17]}
One account of his career has him called to Egypt by AlHakim biAmr Allah, ruler of the Fatimid Caliphate, to regulate the flooding of the Nile, a task requiring an early attempt at building a dam at the present site of the Aswan Dam.^{[24]} After deciding the scheme was impractical and fearing the caliph's anger, he feigned madness. He was kept under house arrest from 1011 until alHakim's death in 1021.^{[25]} During this time, he wrote his influential Book of Optics. After his house arrest ended, he wrote scores of other treatises on physics, astronomy and mathematics. He later traveled to Islamic Spain. During this period, he had ample time for his scientific pursuits, which included optics, mathematics, physics, medicine, and practical experiments.
Some biographers have claimed that Alhazen fled to Syria, ventured into Baghdad later in his life, or was in Basra when he pretended to be insane. In any case, he was in Egypt by 1038.^{[17]} Accounts state that during his time in Cairo he lived near and studied at the AlAzhar Mosque.^{[26]}
Among his students were Sorkhab (Sohrab), a Persian from Iran's Semnan who was his student for over 3 years, and Abu alWafa Mubashir ibn Fatek, an Egyptian scientist who learned mathematics from Alhazan.^{[27]}
Alhazen made significant improvements in optics, physical science, and the scientific method. Alhazen's work on optics is credited with contributing a new emphasis on experiment. Alhazen is considered by BritishIraqi physicist, Jim AlKhalili, to be the "first true scientist" based on Alhazen's pioneering work on the scientific method.^{[14]}
His main work, Kitab alManazir (Book of Optics) had little direct influence in the Islamic Middle East, being known mainly through a thirteenthcentury commentary by Kamāl alDīn alFārisī, the Tanqīḥ alManāẓir lidhawī labṣār wa lbaṣā'ir.^{[28]} In Islamic Spain, it was used by the eleventhcentury mathematician, alMu'taman ibn Hūd, and a Latin translation was produced in the thirteenth century.^{[29]}^{[30]} This translation was read by and greatly influenced a number of Western scientists including: Roger Bacon,^{[31]} Robert Grosseteste,^{[32]} Witelo, Giambattista della Porta,^{[33]} Leonardo Da Vinci,^{[34]} Galileo Galilei,^{[35]} Christian Huygens,^{[36]} René Descartes,^{[35]} and Johannes Kepler.^{[37]} His research in catoptrics (the study of optical systems using mirrors) centred on spherical and parabolic mirrors and spherical aberration. He made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the problem known as "Alhazen's problem".^{[38]} Meanwhile in the Islamic world, Alhazen's work influenced Averroes' writings on optics,^{[39]} and his legacy was further advanced through the 'reforming' of his Optics by Persian scientist Kamal alDin alFarisi (died ca. 1320) in the latter's Kitab Tanqih alManazir (The Revision of [Ibn alHaytham's] Optics).^{[40]} Alhazen wrote as many as 200 books, although only 55 have survived, and many of those have not yet been translated from Arabic.^{[citation needed]} Some of his treatises on optics survived only through Latin translation. During the Middle Ages his books on cosmology were translated into Latin, Hebrew and other languages. The crater Alhazen on the Moon is named in his honour,^{[41]} as was the asteroid 59239 Alhazen.^{[42]} In honour of Alhazen, the Aga Khan University (Pakistan) named its Ophthalmology endowed chair as "The IbneHaitham Associate Professor and Chief of Ophthalmology".^{[43]} Alhazen, by the name Ibn alHaytham, is featured on the obverse of the Iraqi 10,000dinar banknote issued in 2003,^{[44]} and on 10dinar notes from 1982. A research facility that UN weapons inspectors suspected of conducting chemical and biological weapons research in Saddam Hussein's Iraq was also named after him.^{[44]}^{[45]}
Alhazen's most famous work is his sevenvolume treatise on optics Kitab alManazir (Book of Optics), written from 1011 to 1021.
Optics was translated into Latin by an unknown scholar at the end of the 12th century or the beginning of the 13th century.^{[46]} It was printed by Friedrich Risner in 1572, with the title Opticae thesaurus: Alhazeni Arabis libri septem, nuncprimum editi; Eiusdem liber De Crepusculis et nubium ascensionibus (English : Optics treasure: Arab Alhazeni seven books, published for the first time: The book of the Twilight of the clouds and ascensions).^{[47]} Risner is also the author of the name variant "Alhazen"; before Risner he was known in the west as Alhacen, which is the correct transcription of the Arabic name.^{[48]} This work enjoyed a great reputation during the Middle Ages. Works by Alhazen on geometric subjects were discovered in the Bibliothèque nationale in Paris in 1834 by E. A. Sedillot. Other manuscripts are preserved in the Bodleian Library at Oxford and in the library of Leiden.
Two major theories on vision prevailed in classical antiquity. The first theory, the emission theory, was supported by such thinkers as Euclid and Ptolemy, who believed that sight worked by the eye emitting rays of light. The second theory, the intromission theory supported by Aristotle and his followers, had physical forms entering the eye from an object. Previous Islamic writers (such as alKindi) had argued essentially on Euclidean, Galenist, or Aristotelian lines; Alhazen's achievement was to come up with a theory which successfully combined parts of the mathematical ray arguments of Euclid, the medical tradition of Galen, and the intromission theories of Aristotle. Alhazen's intromission theory followed alKindi (and broke with Aristotle) in asserting that "from each point of every colored body, illuminated by any light, issue light and color along every straight line that can be drawn from that point".^{[49]} This however left him with the problem of explaining how a coherent image was formed from many independent sources of radiation; in particular, every point of an object would send rays to every point on the eye. What Alhazen needed was for each point on an object to correspond to one point only on the eye.^{[49]} He attempted to resolve this by asserting that only perpendicular rays from the object would be perceived by the eye; for any one point on the eye, only the ray which reached it directly, without being refracted by any other part of the eye, would be perceived. He argued using a physical analogy that perpendicular rays were stronger than oblique rays; in the same way that a ball thrown directly at a board might break the board, whereas a ball thrown obliquely at the board would glance off, perpendicular rays were stronger than refracted rays, and it was only perpendicular rays which were perceived by the eye. As there was only one perpendicular ray that would enter the eye at any one point, and all these rays would converge on the centre of the eye in a cone, this allowed him to resolve the problem of each point on an object sending many rays to the eye; if only the perpendicular ray mattered, then he had a onetoone correspondence and the confusion could be resolved.^{[50]} He later asserted (in book seven of the Optics) that other rays would be refracted through the eye and perceived as if perpendicular.^{[51]}
His arguments regarding perpendicular rays do not clearly explain why only perpendicular rays were perceived; why would the weaker oblique rays not be perceived more weakly?^{[52]} His later argument that refracted rays would be perceived as if perpendicular does not seem persuasive.^{[53]} However, despite its weaknesses, no other theory of the time was so comprehensive, and it was enormously influential, particularly in Western Europe: "Directly or indirectly, his De Aspectibus inspired much of the activity in optics which occurred between the 13th and 17th centuries." ^{[54]} Kepler's later theory of the retinal image (which resolved the problem of the correspondence of points on an object and points in the eye) built directly on the conceptual framework of Alhazen.^{[54]}
Alhazen showed through experiment that light travels in straight lines, and carried out various experiments with lenses, mirrors, refraction, and reflection.^{[38]} He was the first to consider separately the vertical and horizontal components of reflected and refracted light rays, which was an important step in understanding optics geometrically.^{[55]}
The camera obscura was known to the Chinese, and Aristotle had discussed the principle behind it in his Problems, however it is Alhazen's work which contains the first clear description^{[56]} and early analysis^{[57]} of the device.
Alhazen studied the process of sight, the structure of the eye, image formation in the eye, and the visual system. Ian P. Howard argued in a 1996 Perception article that Alhazen should be credited with many discoveries and theories which were previously attributed to Western Europeans writing centuries later. For example, he described what became in the 19th century Hering's law of equal innervation; he had a description of vertical horopters which predates Aguilonius by 600 years and is actually closer to the modern definition than Aguilonius's; and his work on binocular disparity was repeated by Panum in 1858.^{[58]} Craig AaenStockdale, while agreeing that Alhazen should be credited with many advances, has expressed some caution, especially when considering Alhazen in isolation from Ptolemy, who Alhazen was extremely familiar with. Alhazen corrected a significant error of Ptolemy regarding binocular vision, but otherwise his account is very similar; Ptolemy also attempted to explain what is now called Hering's law.^{[59]} In general, Alhazen built on and expanded the optics of Ptolemy.^{[60]}^{[61]} In a more detailed account of Ibn alHaytham's contribution to the study of binocular vision based on Lejeune^{[62]} and Sabra,^{[18]} Raynaud^{[63]} showed that the concepts of correspondence, homonymous and crossed diplopia were in place in Ibn alHaytham's optics. But contrary to Howard, he explained why Ibn alHaytham did not give the circular figure of the horopter and why, by reasoning experimentally, he was in fact closer to the discovery of Panum's fusional area than that of the ViethMüller circle. In this regard, Ibn alHaytham's theory of binocular vision faced two main limits: the lack of recognition of the role of the retina, and obviously the lack of an experimental investigation of ocular tracts.
Alhazen's most original contribution was that after describing how he thought the eye was anatomically constructed, he went on to consider how this anatomy would behave functionally as an optical system.^{[64]} His understanding of pinhole projection from his experiments appears to have influenced his consideration of image inversion in the eye,^{[65]} which he sought to avoid.^{[66]} He maintained that the rays that fell perpendicularly on the lens (or glacial humor as he called it) were further refracted outward as they left the glacial humor and the resulting image thus passed upright into the optic nerve at the back of the eye.^{[67]} He followed Galen in believing that the lens was the receptive organ of sight, although some of his work hints that he thought the retina was also involved.^{[68]}
“  The duty of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and,.. attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency.  ” 
—Alhazen, ^{[18]} 
An aspect associated with Alhazen's optical research is related to systemic and methodological reliance on experimentation (i'tibar)(Arabic: إختبار) and controlled testing in his scientific inquiries. Moreover, his experimental directives rested on combining classical physics (ilm tabi'i) with mathematics (ta'alim; geometry in particular). This mathematicalphysical approach to experimental science supported most of his propositions in Kitab alManazir (The Optics; De aspectibus or Perspectivae) and grounded his theories of vision, light and colour, as well as his research in catoptrics and dioptrics (the study of the refraction of light).^{[40]} Bradley Steffens in his book Ibn AlHaytham: First Scientist has argued that Alhazen's approach to testing and experimentation made an important contribution to the scientific method. According to Matthias Schramm, Alhazen:
was the first to make a systematic use of the method of varying the experimental conditions in a constant and uniform manner, in an experiment showing that the intensity of the lightspot formed by the projection of the moonlight through two small apertures onto a screen diminishes constantly as one of the apertures is gradually blocked up.^{[69]}
G. J. Toomer expressed some skepticism regarding Schramm's view, arguing that caution is needed to avoid reading anachronistically particular passages in Alhazen's very large body of work, and while acknowledging Alhazen's importance in developing experimental techniques, argued that he should not be considered in isolation from other Islamic and ancient thinkers.^{[70]}
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 carom 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.^{[17]}^{[71]} This eventually led Alhazen to derive a formula for the sum of fourth powers, 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 integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.^{[72]} Alhazen eventually solved the problem using conic sections 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,^{[73]} with a new solution being found in 1997 by the Oxford mathematician Peter M. Neumann.^{[74]} Recently, Mitsubishi Electric Research Laboratories (MERL) researchers Amit Agrawal, Yuichi Taguchi and Srikumar Ramalingam solved the extension of Alhazen's problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors.^{[75]} They showed that the mirror reflection point can be computed by solving an eighth degree equation in the most general case. If the camera (eye) is placed on the axis of the mirror, the degree of the equation reduces to six.^{[76]} Alhazen's problem can also be extended to multiple refractions from a spherical ball. Given a light source and a spherical ball of certain refractive index, the closest point on the spherical ball where the light is refracted to the eye of the observer can be obtained by solving a tenth degree equation.^{[76]}
The Kitab alManazir (Book of Optics) describes several experimental observations that Alhazen made and how he used his results to explain certain optical phenomena using mechanical analogies. He conducted experiments with projectiles, and a description of his conclusions is: "it was only the impact of perpendicular projectiles on surfaces which was forceful enough to enable them to penetrate whereas the oblique ones were deflected. For example, to explain refraction from a rare to a dense medium, he used the mechanical analogy of an iron ball thrown at a thin slate covering a wide hole in a metal sheet. A perpendicular throw would break the slate and pass through, whereas an oblique one with equal force and from an equal distance would not."^{[77]} He also used this result to explain how intense, direct light hurts the eye, using a mechanical analogy: "Alhazen associated 'strong' lights with perpendicular rays and 'weak' lights with oblique ones. The obvious answer to the problem of multiple rays and the eye was in the choice of the perpendicular ray since there could only be one such ray from each point on the surface of the object which could penetrate the eye."^{[77]}
Sudanese psychologist Omar Khaleefa has argued that Alhazen should be considered be the "founder of experimental psychology", for his pioneering work on the psychology of visual perception and optical illusions.^{[78]} Khaleefa has also argued that Alhazen should also be considered the "founder of psychophysics", a subdiscipline and precursor to modern psychology.^{[78]} Although Alhazen made many subjective reports regarding vision, there is no evidence that he used quantitative psychophysical techniques and the claim has been rebuffed.^{[59]}
Alhazen offered an explanation of the Moon illusion, an illusion that played an important role in the scientific tradition of medieval Europe.^{[79]} Many authors repeated explanations that attempted to solve the problem of the Moon appearing larger near the horizon than it does when higher up in the sky, a debate that is still unresolved. Alhazen argued against Ptolemy's refraction theory, and defined the problem in terms of perceived, rather than real, enlargement. He said that judging the distance of an object depends on there being an uninterrupted sequence of intervening bodies between the object and the observer. When the Moon is high in the sky there are no intervening objects, so the Moon appears close. The perceived size of an object of constant angular size varies with its perceived distance. Therefore, the Moon appears closer and smaller high in the sky, and further and larger on the horizon. Through works by Roger Bacon, John Pecham and Witelo based on Alhazen's explanation, the Moon illusion gradually came to be accepted as a psychological phenomenon, with the refraction theory being rejected in the 17th century.^{[80]} Although Alhazen is often credited with the perceived distance explanation, he was not the first author to offer it. Cleomedes (c. 2nd century) gave this account (in addition to refraction), and he credited it to Posidonius (c. 13550 BC).^{[81]} Ptolemy may also have offered this explanation in his Optics, but the text is obscure.^{[82]} Alhazen's writings were more widely available in the Middle Ages than those of these earlier authors, and that probably explains why Alhazen received the credit.
Besides the Book of Optics, Alhazen wrote several other treatises on the same subject, including his Risala fi lDaw’ (Treatise on Light). He investigated the properties of luminance, the rainbow, eclipses, twilight, and moonlight. Experiments with mirrors and magnifying lenses provided the foundation for his theories on catoptrics.^{[83]}
In his treatise Mizan alHikmah (Balance of Wisdom), Alhazen discussed the density of the atmosphere and related it to altitude. He also studied atmospheric refraction.^{[38]}
Alhazen discussed the physics of the celestial region in his Epitome of Astronomy, arguing that Ptolemaic models needed to be understood in terms of physical objects rather than abstract hypotheses; in other words that it should be possible to create physical models where (for example) none of the celestial bodies would collide with each other. The suggestion of mechanical models for the Earth centred Ptolemaic model "greatly contributed to the eventual triumph of the Ptolemaic system among the Christians of the West". Alhazen's determination to root astronomy in the realm of physical objects was important however, because it meant astronomical hypotheses "were accountable to the laws of physics", and could be criticised and improved upon in those terms.^{[84]}
In Mizan alHikmah (Balance of Wisdom), Alhazen discussed the theories of attraction between masses.^{[38]} He also wrote Maqala fi daw alqamar (On the Light of the Moon).
In his work, Alhazen discussed theories on the motion of a body.^{[83]} In his Treatise on Place, Alhazen disagreed with Aristotle's view that nature abhors a void, and he used geometry in an attempt to demonstrate that place (almakan) is the imagined threedimensional void between the inner surfaces of a containing body.^{[85]}
In his On the Configuration of the World Alhazen presented a detailed description of the physical structure of the earth:
The earth as a whole is a round sphere whose center is the center of the world. It is stationary in its [the world's] middle, fixed in it and not moving in any direction nor moving with any of the varieties of motion, but always at rest.^{[86]}
The book is a nontechnical explanation of Ptolemy's Almagest, which was eventually translated into Hebrew and Latin in the 13th and 14th centuries and subsequently had an influence on astronomers such as Georg von Peuerbach^{[1]} during the European Middle Ages and Renaissance.^{[87]}^{[88]}
In his AlShukūk ‛alā Batlamyūs, variously translated as Doubts Concerning Ptolemy or Aporias against Ptolemy, published at some time between 1025 and 1028, Alhazen criticized Ptolemy's Almagest, Planetary Hypotheses, and Optics, pointing out various contradictions he found in these works, particularly in astronomy. Ptolemy's Almagest concerned mathematical theories regarding the motion of the planets, whereas the Hypotheses concerned what Ptolemy thought was the actual configuration of the planets. Ptolemy himself acknowledged that his theories and configurations did not always agree with each other, arguing that this was not a problem provided it did not result in noticeable error, but Alhazen was particularly scathing in his criticism of the inherent contradictions in Ptolemy's works.^{[89]} He considered that some of the mathematical devices Ptolemy introduced into astronomy, especially the equant, failed to satisfy the physical requirement of uniform circular motion, and noted the absurdity of relating actual physical motions to imaginary mathematical points, lines and circles:^{[90]}
Ptolemy assumed an arrangement (hay'a) that cannot exist, and the fact that this arrangement produces in his imagination the motions that belong to the planets does not free him from the error he committed in his assumed arrangement, for the existing motions of the planets cannot be the result of an arrangement that is impossible to exist... [F]or a man to imagine a circle in the heavens, and to imagine the planet moving in it does not bring about the planet's motion.^{[91]}^{[92]}
Having pointed out the problems, Alhazen appears to have intended to resolve the contradictions he pointed out in Ptolemy in a later work. Alhazen's belief was that there was a "true configuration" of the planets which Ptolemy had failed to grasp; his intention was to complete and repair Ptolemy's system, not to replace it completely.^{[89]}
In the Doubts Concerning Ptolemy Alhazen set out his views on the difficulty of attaining scientific knowledge and the need to question existing authorities and theories:
Truth is sought for itself [but] the truths, [he warns] are immersed in uncertainties [and the scientific authorities (such as Ptolemy, whom he greatly respected) are] not immune from error...^{[18]}
He held that the criticism of existing theories—which dominated this book—holds a special place in the growth of scientific knowledge.
Alhazen's The Model of the Motions of Each of the Seven Planets was written c. 1038. Only one damaged manuscript has been found, with only the introduction and the first section, on the theory of planetary motion, surviving. (There was also a second section on astronomical calculation, and a third section, on astronomical instruments.) Following on from his Doubts on Ptolemy, Alhazen described a new, geometrybased planetary model, describing the motions of the planets in terms of spherical geometry, infinitesimal geometry and trigonometry. He kept a geocentric universe and assumed that celestial motions are uniformly circular, which required the inclusion of epicycles to explain observed motion, but he managed to eliminate Ptolemy's equant. In general, his model made no attempt to provide a causal explanation of the motions, but concentrated on providing a complete, geometric description which could be used to explain observed motions, without the contradictions inherent in Ptolemy's model.^{[93]}
Alhazen wrote a total of twentyfive astronomical works, some concerning technical issues such as Exact Determination of the Meridian, a second group concerning accurate astronomical observation, a third group concerning various astronomical problems and questions such as the location of the Milky Way; Alhazen argued for a distant location, based on the fact that it does not move in relation to the fixed stars.^{[94]} The fourth group consists of ten works on astronomical theory, including the Doubts and Model of the Motions discussed above.^{[95]}
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."^{[96]}
In geometry, Alhazen developed analytical geometry and the link between algebra and geometry.^{[96]} He developed a formula for adding the first 100 natural numbers, using a geometric proof to prove the formula.^{[97]}
Alhazen explored the Euclidean parallel postulate, the fifth postulate in Euclid's Elements, using a proof by contradiction,^{[98]} and in effect introducing the concept of motion into geometry.^{[99]} He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn alHaytham–Lambert quadrilateral".^{[100]} His theorems on quadrilaterals, including the Lambert quadrilateral, were the first theorems on elliptical geometry and hyperbolic geometry. These theorems, along with his alternative postulates, such as Playfair's axiom, can be seen as marking the beginning of nonEuclidean geometry. His work had a considerable influence on its development among the later Persian geometers Omar Khayyám and Nasīr alDīn alTūsī, and the European geometers Witelo, Gersonides, and Alfonso.^{[101]}
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.^{[17]} 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.^{[102]}
His contributions to number theory include his work on perfect numbers. In his Analysis and Synthesis, Alhazen may have been the first to state that every even perfect number is of the form 2^{n−1}(2^{n} − 1) where 2^{n} − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century).^{[17]}
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.^{[17]}
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 the question of whether animals could react to music, for example whether a camel would increase or decrease its pace.
In engineering, one account of his career as a civil engineer has him summoned to Egypt by the Fatimid Caliph, AlHakim biAmr Allah, to regulate the flooding of the Nile River. He carried out a detailed scientific study of the annual inundation of the Nile River, and he drew plans for building a dam, at the site of the modernday Aswan Dam. His field work, however, later made him aware of the impracticality of this scheme, and he soon feigned madness so he could avoid punishment from the Caliph.^{[103]}
In his Treatise on Place, Alhazen disagreed with Aristotle's view that nature abhors a void, and he used geometry in an attempt to demonstrate that place (almakan) is the imagined threedimensional void between the inner surfaces of a containing body.^{[85]} Abdellatif, a supporter of Aristotle's philosophical view of place, later criticized the work in Fi alRadd ‘ala Ibn alHaytham fi almakan (A refutation of Ibn alHaytham’s place) for its geometrization of place.^{[85]}
Alhazen also discussed space perception and its epistemological implications in his Book of Optics. In "tying the visual perception of space to prior bodily experience, Alhacen unequivocally rejected the intuitiveness of spatial perception and, therefore, the autonomy of vision. Without tangible notions of distance and size for correlation, sight can tell us next to nothing about such things."^{[104]}
Alhazen was a devout Muslim, though it is uncertain which branch of Islam he followed. He may have been either a follower of the Ash'ari school of Sunni Islamic theology according to Ziauddin Sardar^{[105]} and Lawrence Bettany^{[106]} (and opposed to the views of the Mu'tazili school),^{[106]} a follower of the Mu'tazili school of Islamic theology according to Peter Edward Hodgson,^{[107]} or a follower of Shia Islam possibly according to A. I. Sabra.^{[108]}
Alhazen wrote a work on Islamic theology, in which he discussed prophethood and developed a system of philosophical criteria to discern its false claimants in his time.^{[109]} He also wrote a treatise entitled Finding the Direction of Qibla by Calculation, in which he discussed finding the Qibla, where Salat prayers are directed towards, mathematically.^{[110]}
He wrote in his Doubts Concerning Ptolemy:
Truth is sought for its own sake ... Finding the truth is difficult, and the road to it is rough. For the truths are plunged in obscurity. ... God, however, has not preserved the scientist from error and has not safeguarded science from shortcomings and faults. If this had been the case, scientists would not have disagreed upon any point of science...^{[111]}
In The Winding Motion, Alhazen further wrote:
From the statements made by the noble Shaykh, it is clear that he believes in Ptolemy's words in everything he says, without relying on a demonstration or calling on a proof, but by pure imitation (taqlid); that is how experts in the prophetic tradition have faith in Prophets, may the blessing of God be upon them. But it is not the way that mathematicians have faith in specialists in the demonstrative sciences.^{[112]}
Alhazen described his theology:
I constantly sought knowledge and truth, and it became my belief that for gaining access to the effulgence and closeness to God, there is no better way than that of searching for truth and knowledge.^{[113]}
According to medieval biographers, Alhazen wrote more than 200 works on a wide range of subjects, of which at least 96 of his scientific works are known. Most of his works are now lost, but more than 50 of them have survived to some extent. Nearly half of his surviving works are on mathematics, 23 of them are on astronomy, and 14 of them are on optics, with a few on other subjects.^{[114]} Not all his surviving works have yet been studied, but some of the ones that have are given below.^{[95]}^{[110]}
"The first clear description of the device appears in the Book of Optics of Alhazen."
"The principles of the camera obscura first began to be correctly analysed in the eleventh century, when they were outlined by Ibn alHaytham."
In seventeenth century Europe the problems formulated by Ibn alHaytham (965–1041) became known as 'Alhazen's problem'. [...] AlHaytham’s contributions to geometry and number theory went well beyond the Archimedean tradition. AlHaytham 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. AlHaytham was a scientist who made major contributions to the fields of mathematics, physics and astronomy during the latter half of the tenth century.
In effect, this method characterised parallel lines as lines always equidistant from one another and also introduced the concept of motion into geometry.
Three scientists, Ibn alHaytham, Khayyam and alTusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the nineteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the thirteenth century, while revising Alhazen's Book of Optics (Kitab alManazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Gersonides, who lived in southern France, and by the abovementioned Alfonso from Spain directly border on Ibn Alhazen's demonstration.



