Ibn al-Haytham (Alhazen) was born c. 965 in Basra, which was then part of the Buyid emirate,^{[1]} to an Arab family.^{[16]}^{[17]}
Alhazen arrived in Cairo under the reign of Fatimid Caliph al-Hakim, a patron of the sciences who was particularly interested in astronomy.^{[18]} He proposed to the Caliph a hydraulic project to improve regulation of the flooding of the Nile, a task requiring an early attempt at building a dam at the present site of the Aswan Dam,^{[18]} but later his field work convinced him of the technical impracticality of this scheme.^{[19]} Alhazen continued to live in Cairo, in the neighborhood of the famous University of al-Azhar, until his death in 1040.^{[12]} Legend has it that after deciding the scheme was impractical and fearing the caliph‘s anger, Alhazen feigned madness and was kept under house arrest from 1011 until al-Hakim’s death in 1021.^{[20]} During this time, he wrote his influential Book of Optics and continued to write further treatises on astronomy, geometry, number theory, optics and natural philosophy.
Among his students were Sorkhab (Sohrab), a Persian from Semnan who was his student for over 3 years, and Abu al-Wafa Mubashir ibn Fatek, an Egyptian prince who learned mathematics from Alhazen.^{[21]}
Legacy[edit]
Front page of the Opticae Thesaurus, which included the first printed Latin translation of Alhazen’sBook of Optics. The illustration incorporates many examples of optical phenomena including perspective effects, the rainbow, mirrors, and refraction.
Alhazen made significant contributions to optics, number theory, geometry, astronomy and natural philosophy. Alhazen’s work on optics is credited with contributing a new emphasis on experiment.
His main work, Kitab al-Manazir (Book of Optics) was known in Islamicate societies mainly, but not exclusively, through the thirteenth-century commentary by Kamāl al-Dīn al-Fārisī, the Tanqīḥ al-Manāẓir li-dhawī l-abṣār wa l-baṣā’ir.^{[22]} In al-Andalus, it was used by the eleventh-century prince of the Banu Hud dynasty of Zaragossa and author of an important mathematical text, al-Mu’taman ibn Hūd. A Latin translation of the Kitab al-Manazir was made probably in the late twelfth or early thirteenth century.^{[23]}^{[24]} This translation was read by and greatly influenced a number of scholars in Catholic Europe including: Roger Bacon,^{[25]} Robert Grosseteste,^{[26]}Witelo, Giambattista della Porta,^{[27]} Leonardo Da Vinci,^{[28]} Galileo Galilei,^{[29]}Christiaan Huygens,^{[30]} René Descartes,^{[29]} and Johannes Kepler.^{[31]} His research incatoptrics (the study of optical systems using mirrors) centred on spherical andparabolic 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“.^{[32]} Meanwhile in the Islamic world, Alhazen’s work influenced Averroes‘ writings on optics,^{[33]} and his legacy was further advanced through the ‘reforming’ of his Optics by Persian scientist Kamal al-Din al-Farisi (died ca. 1320) in the latter’s Kitab Tanqih al-Manazir (The Revision of [Ibn al-Haytham’s] Optics).^{[34]} Alhazen wrote as many as 200 books, although only 55 have survived. Some of his treatises on optics survived only through Latin translation. During the Middle Ages his books oncosmology were translated into Latin, Hebrew and other languages. The crater Alhazen on the Moon is named in his honour,^{[35]} as was the asteroid 59239 Alhazen.^{[36]} In honour of Alhazen, the Aga Khan University (Pakistan) named its Ophthalmology endowed chair as “The Ibn-e-Haitham Associate Professor and Chief of Ophthalmology”.^{[37]} Alhazen, by the name Ibn al-Haytham, is featured on the obverse of the Iraqi 10,000-dinar banknote issued in 2003,^{[38]} and on 10-dinar notes from 1982.
One of the major scientific anniversaries that will be celebrated during the 2015 International Year of Light is: the works on optics by Ibn Al-Haytham (1015).
Book of Optics[edit]
Main article: Book of Optics
Alhazen’s most famous work^{[39]} is his seven-volume treatise on optics Kitab al-Manazir (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.^{[40]} It was printed by Friedrich Risnerin 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).^{[41]} 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.^{[42]} 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. In all, A. Mark Smith has accounted for 18 full or near-complete manuscripts, and five fragments, which are preserved in 14 locations, including one in the Bodleian Library at Oxford, and one in the library of Bruges.^{[43]}
Theory of vision[edit]
Alhazen on Iraqi 10 dinars
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 oflight. 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 al-Kindi) had argued essentially on Euclidean, Galenist, or Aristotelian lines. The strongest influence on the Book of Optics was from Ptolemy’sOptics, while the description of the anatomy and physiology of the eye was based on Galen’s account.^{[44]} 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 al-Kindi (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”.^{[45]} 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.^{[45]} 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 one-to-one correspondence and the confusion could be resolved.^{[46]} He later asserted (in book seven of theOptics) that other rays would be refracted through the eye and perceived as if perpendicular.^{[47]}
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?^{[48]} His later argument that refracted rays would be perceived as if perpendicular does not seem persuasive.^{[49]} 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 Aspectibusinspired much of the activity in optics which occurred between the 13th and 17th centuries.” ^{[50]} Kepler‘s later theory of theretinal 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.^{[50]}
Alhazen showed through experiment that light travels in straight lines, and carried out various experiments with lenses,mirrors, refraction, and reflection.^{[32]} 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.^{[51]}
The camera obscura was known to the ancient Chinese and was described by the Han Chinese polymathic genius Shen Kuoin his scientific book Dream Pool Essays which was printed and published in the year 1088 C.E.. Aristotle had discussed the basic principle behind it in his Problems, however Alhazen’s work also contained the first clear description, outside of China, ofcamera obscura in the areas of the middle east, Europe, Africa and India.^{[52]} and early analysis^{[53]} 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.^{[54]} Craig Aaen-Stockdale, 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.^{[55]} In general, Alhazen built on and expanded the optics of Ptolemy.^{[56]}^{[57]} In a more detailed account of Ibn al-Haytham’s contribution to the study of binocular vision based on Lejeune^{[58]} and Sabra,^{[59]} Raynaud^{[60]} showed that the concepts of correspondence, homonymous and crossed diplopia were in place in Ibn al-Haytham’s optics. But contrary to Howard, he explained why Ibn al-Haytham 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 Vieth-Müller circle. In this regard, Ibn al-Haytham’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.^{[61]} His understanding of pinhole projection from his experiments appears to have influenced his consideration of image inversion in the eye,^{[62]} which he sought to avoid.^{[63]} 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.^{[64]} 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.^{[65]}
Alhazen’s synthesis of light and vision adhered to the Aristotelian scheme, exhaustively describing the process of vision in a logical, complete fashion.^{[66]}
Scientific method[edit]
Hevelius‘s Selenographia, showing Alhasen [sic] representing reason, and Galileo representing the senses.
“ |
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, ^{[59]} |
An aspect associated with Alhazen’s optical research is related to systemic and methodological reliance on experimentation (i’tibar)(Arabic: إعتبار) 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 mathematical-physical approach to experimental science supported most of his propositions in Kitab al-Manazir (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).^{[34]} 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 light-spot 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.^{[67]}
- 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.^{[68]}
- Mark Smith’s critical editions (2001, 2006, 2008, 2010) ofDe Aspectibuscontain a Latin glossary with page numbers of each occurrence of the words, to illustrate Alhacen’s experimental viewpoint. Smith shows that Alhacen was received well in the West because he reinforced the importance of the Hellenic tradition to them.^{[69]}
Alhazen’s problem[edit]
Main article: Alhazen’s problem
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 thecircumference 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 anequation of the fourth degree.^{[14]}^{[70]} 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.^{[71]} 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,^{[72]} with a new solution being found in 1997 by the Oxford mathematician Peter M. Neumann.^{[73]} 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.^{[74]} 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.^{[75]} 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.^{[75]}
Refraction[edit]
Smith 2010 has noted that Alhazen’s treatment of refraction describes an experimental setup without publication of data.^{[76]}Ptolemy published his experimental results for refraction, in contrast. One generation before Alhazen, Ibn Sahl discovered his statement of the lengths of the hypotenuse for each incident and refracted right triangle, respectively. This is equivalent to Descartes’ formulation for refraction. Alhazen’s convention for describing the incident and refracted angles is still in use. His failure to publish his data is an open question.
Other contributions[edit]
The Kitab al-Manazir (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 sub-discipline 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.^{[55]}
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 toPosidonius (c. 135-50 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.
Other works on physics[edit]
Optical treatises[edit]
Besides the Book of Optics, Alhazen wrote several other treatises on the same subject, including his Risala fi l-Daw’ (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]}
Celestial physics[edit]
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]}
He also wrote Maqala fi daw al-qamar (On the Light of the Moon).
Mechanics[edit]
In his work, Alhazen discussed theories on the motion of a body.^{[83]} In his Treatise on Place, Alhazen disagreed withAristotle‘s view that nature abhors a void, and he used geometry in an attempt to demonstrate that place (al-makan) is the imagined three-dimensional void between the inner surfaces of a containing body.^{[85]}
Astronomical works[edit]
On the Configuration of the World[edit]
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 non-technical 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]}
Doubts Concerning Ptolemy[edit]
In his Al-Shukūk ‛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…^{[59]}
He held that the criticism of existing theories—which dominated this book—holds a special place in the growth of scientific knowledge.
Model of the Motions of Each of the Seven Planets[edit]
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, geometry-based 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]}
Other astronomical works[edit]
Alhazen wrote a total of twenty-five 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 theDoubts and Model of the Motions discussed above.^{[95]}
Mathematical works[edit]
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]}
He developed a formula for summing the first 100 natural numbers, using a geometric proof to prove the formula.^{[97]}
Geometry[edit]
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 parallel postulate, the fifth postulate inEuclid’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 al-Haytham–Lambert quadrilateral”.^{[100]} His theorems onquadrilaterals, including the Lambert quadrilateral, were the first theorems on elliptical geometryand hyperbolic geometry. These theorems, along with his alternative postulates, such as Playfair’s axiom, can be seen as marking the beginning of non-Euclidean geometry. His work had a considerable influence on its development among the later Persian geometers Omar Khayyámand Nasīr al-Dīn al-Tūsī, 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.^{[14]} 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]}
Number theory[edit]
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 2^{n}^{−1}(2^{n} − 1) where 2^{n} − 1 is prime, but he was not able to prove this result; Euler later proved it in the 18th century.^{[14]}
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.^{[14]}
Other works[edit]
Influence of Melodies on the Souls of Animals[edit]
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.