Ibn al-Haytham

How to use this course

Ibn al-Haytham wrote the most important work on light between antiquity and the seventeenth century. His Book of Optics argued that we see because light enters the eye, gave the law of reflection a complete form, posed and solved a hard geometric problem about mirrors, and treated refraction by measurement. It did all this through controlled tests with purpose-built apparatus, which he called iʿtibār. In Latin translation the book shaped optics in Europe from Roger Bacon to Johannes Kepler. He also wrote a sharp critique of Ptolemy’s astronomy and some fine pieces of mathematics.

His life is known from a short autobiography, three lists of his works, and stories told by biographers who wrote a century or two after his death. The stories disagree. Part I sets out what each source says. Part II follows the optics book by book, then turns to astronomy and mathematics. The exercises use modern notation, but most of them reconstruct arguments he made.

The course on Claudius Ptolemy gives the astronomy and optics he criticized. The course on Nasir al-Din al-Tusi follows the next stage of the critique of Ptolemy, at the Maragha observatory, and the course on Nicolaus Copernicus shows where it led.

Timeline

Dates in Ibn al-Haytham’s life come from the Islamic (hijri) calendar. Where a hijri year spans two Western years, the course gives the approximate one. Birth and death are known only roughly.

Basra and the search for truth

Al-Ḥasan ibn al-Ḥasan ibn al-Haytham was born around 965 in Basra, the port city at the head of the Persian Gulf. The year comes from his own remarks and from the dates in his lists of works, which imply that he was in his sixties in the 1020s. Arabic writers called him al-Baṣrī, “of Basra”, and later al-Miṣrī, “the Egyptian”. Medieval Latin readers knew him as Alhacen; the form Alhazen became common after the printed edition of 1572. Whether his family was Arab or Persian is not recorded.

Basra was then part of the Buyid emirate, under the nominal authority of the Abbasid caliph in Baghdad. It was a commercial city with a long tradition of scholarship in grammar, theology and the sciences translated from Greek. By the tenth century Arabic readers had Euclid, Apollonius, Archimedes, Ptolemy, Aristotle and Galen, and a century of Arabic work that extended them.

The autobiography

In 1027 he wrote an account of his intellectual life, modelled on Galen’s book about his own writings. It says little about events. He describes how, as a young man, he studied the claims of the religious sects and found that each held its beliefs on authority. He resolved to accept only what could be shown by demonstration about things perceived by the senses, and he found that method in the writings of Aristotle and in mathematics. The same commitment opens the Doubts concerning Ptolemy, where he says that the seeker after truth must treat what he reads as an opponent, testing it from every side, and must suspect his own judgement as well.

A career in administration

Ibn Abī Uṣaybiʿa, writing in the thirteenth century, says that Ibn al-Haytham held an administrative post in Basra, sometimes described as a ministry, and that he wished to leave it for study. Al-Bayhaqī and Ibn al-Qifṭī add that he became known for his skill in applied mathematics and engineering. These details come from later writers and cannot be checked against his own words.

Cairo, the Nile and al-Ḥākim

The Fatimid dynasty, a rival caliphate, took Egypt in 969 and built Cairo beside the older city of Fustat. In 996 the eleven-year-old al-Ḥākim became caliph. He founded a library and academy, the Dār al-ʿIlm, in 1005 and supported the astronomer Ibn Yūnus, whose great tables, the Ḥākimī Zīj, bear the caliph’s name. He was also notorious for unpredictable decrees and executions.

The Nile project

Ibn al-Qifṭī (d. 1248), the fullest source, says that Ibn al-Haytham, while still in Iraq, claimed he could build works to regulate the Nile so that Egypt would have water in every season. Al-Ḥākim invited him to Egypt and sent him south with a team of engineers. When he reached the cataracts near Aswan he saw that the ancient Egyptians, whose monuments he passed, would have done it if it were possible. He concluded that the scheme could not work and admitted the failure to the caliph. Al-Ḥākim gave him a government post instead.

The pretended madness

Ibn al-Haytham feared the caliph. In Ibn al-Qifṭī’s account he pretended to be insane, his property was taken into custody, and he was confined to his house until al-Ḥākim’s disappearance in 1021. He then showed that he was sane, recovered his property, and lived near the Azhar mosque, earning his living by copying scientific books. Ibn al-Qifṭī’s informant, Yūsuf al-Fāsī, says he copied Euclid’s Elements, the “middle books” of mathematical astronomy and Ptolemy’s Almagest, and earned about 150 dinars a year.

The story is often told as the origin of the Optics: a scholar under house arrest, studying the light that entered his darkened room. No source says so. The date range 1011–1021 sometimes given for the book rests on that reading of the story, and it should be held loosely.

The sources and their stories

Three kinds of evidence survive, and they are of very different quality.

• His own writings. The autobiography of 1027 and three lists of works, ending on 10 February 1027, 25 July 1028 and 2 October 1038. The last list has 92 titles. Ibn al-Qifṭī gives 69 titles, and all but two appear in the 1038 list. At least 55 works survive. Two manuscripts in his own hand are dated, a copy of Apollonius’s Conics (1024) and a geometrical volume (1040–41).
• Ibn al-Qifṭī, in his History of the Learned: the Nile expedition, the feigned madness in Cairo, the copying of books, and death around 430 AH (1038–39). The volume of 1040–41 shows that this date is too early.
• Other biographers. Al-Bayhaqī (d. 1169–70) tells of a failed meeting with al-Ḥākim and a flight to Syria; Sabra judged this unsupported. Ibn Abī Uṣaybiʿa (d. 1270) places the feigned madness in Basra, as a way to escape his post there. A title in the lists implies a stay in Baghdad around 1027.

One man or two?

Ibn Abī Uṣaybiʿa gives the autobiography and the first two lists under the name Muḥammad ibn al-Ḥasan ibn al-Haytham, and the third under al-Ḥasan ibn al-Ḥasan. Roshdi Rashed argued that these were two people: a philosopher and physician named Muḥammad, and the mathematician al-Ḥasan who wrote the Optics. A. I. Sabra replied in two papers titled “One Ibn al-Haytham or two?” (1998 and 2003) that the evidence points to a single author whose name was garbled in transmission. Sabra’s view is widely accepted, and this course treats all the works as those of one person.

Reading the legends

The madness story has the shape of a moral tale: a learned man outwits a tyrant. It may still be true in outline. What the historian can say is limited: he was in Egypt in the reign of al-Ḥākim; the Nile scheme is reported by more than one source; he was in Cairo in later life; and he made his living, at least in part, by his pen.

Light, rays and the method of test

For the general reader. Greek optics was mostly a geometry of sight lines, drawn from the eye to the object. Ibn al-Haytham asked what light is and how it travels, and he answered by building apparatus: tubes, screens with holes, darkened rooms. He wanted every claim to be checked by an arrangement that anyone could repeat.

The plan of the book

• Book I. Light and colour; the anatomy of the eye; the mode of vision.
• Book II. Perception: how the mind recognizes shape, size, distance and the other visible properties.
• Book III. Errors of direct vision, including binocular vision.
• Book IV. Reflection: its laws and the apparatus for testing them.
• Book V. Images in plane, spherical, cylindrical and conical mirrors, including Alhazen’s problem.
• Book VI. Errors of vision due to reflection.
• Book VII. Refraction, with a refraction apparatus, and errors due to refraction.

The Arabic word iʿtibār, translated into Latin as experimentum, means testing or verification. The Optics uses it for a deliberate check, with an instrument, of a claim stated beforehand. The work cites no authority by name.

The tests

For straight-line propagation he looked through straight tubes, and placed straight rulers along beams entering a darkened room through a hole. For the third principle he described a test with several candles in a row outside a dark chamber. Each candle sends a separate spot of light through a single aperture onto the opposite wall, on the straight line through the hole. Cover one candle, and only its spot disappears. The lights have crossed at the aperture and passed through each other unchanged.

The dark chamber

He called it al-bayt al-muẓlim, the dark room. In a short treatise On the Shape of the Eclipse he discussed why, during a partial solar eclipse, sunlight through a small hole forms a crescent, and under what conditions the image through an aperture is distinct. His answer is incomplete, but it shows that he understood the principle, which can be put as follows.

How we see

For the general reader. Euclid and Ptolemy explained vision by rays going out from the eye. Aristotle and the atomists had the object send something to the eye, but without a geometry. Ibn al-Haytham joined the two: light from every point of the object enters the eye, and a geometrical rule picks out one ray from each point, so that the eye receives an ordered picture.

Against rays from the eye

Strong light hurts the eye, and bright objects leave afterimages: both show that something acts on the eye from outside. When we open our eyes at night we see the stars at once; it is implausible that something issuing from the eye fills the space up to the stars in an instant. And if rays did go out, they would still need to bring something back for the eye to perceive. He concluded that the rays from the eye are superfluous.

The problem of confusion

If every point of an object sends light in all directions, every point of the eye’s surface receives light from every point of the object. Why then do we not see a blur? His answer used the geometry of the eye as Galen described it, a set of nested spherical surfaces sharing a centre.

Refracted oblique rays were real in his account and contributed weakly; Book VII uses them to explain why objects slightly outside the cone are seen. The theory gave Euclid’s visual cone a physical basis with the direction of travel reversed. Its weakness is that the image is formed in front of the centre and is never inverted; Kepler would locate the real image on the retina in 1604, inverted.

Perception and judgement

Book II argues that we perceive size by combining the angle an object subtends with a judgement of its distance, made quickly and without awareness. Historians compare this to Hermann von Helmholtz’s “unconscious inference”. Book III, and a separate letter, explain the Moon illusion by this route. The Moon near the horizon subtends the same angle as at the zenith, but the sky near the horizon, with intervening land, is judged farther away, so the Moon is judged larger. Cleomedes had given a related account in antiquity.

Binocular vision

Book III studies seeing with two eyes. When both eyes fix on a point, objects whose images fall on corresponding places in the two eyes are seen single; others are seen double. He tested this with a board marked with lines and small objects at different distances.

Reflection and mirrors

For the general reader. That a mirror returns light at the angle it receives it was known in antiquity. Ibn al-Haytham stated the law in full, for plane and curved mirrors, and built a brass instrument to test it. He also studied burning mirrors, which concentrate sunlight, and found that spherical ones cannot bring rays to a single point.

For curved mirrors the normal is the normal to the tangent plane: the radius for a sphere, the perpendicular to the axis through the surface point for a cylinder. His instrument was a graduated brass ring set vertically, with sights and interchangeable mirrors at the centre, so that a beam admitted through small holes could be traced after reflection.

Images

Book V adopts the rule, already used by Ptolemy, that the image of a point lies on the perpendicular from the point to the mirror (the cathetus, or for a sphere, the line through the centre), where the reflected ray, extended backward, meets it.

Proof. Reflect \(A\) in the mirror to \(A'\). Then \(AP=A'P\), so \(AP+PB=A'P+PB\ge A'B\), with equality when \(P\) is on segment \(A'B\). At that point the angles are equal by vertical angles.

Burning mirrors

In On Paraboloidal Burning Mirrors he proved what earlier writers had asserted: a mirror shaped as a paraboloid of revolution reflects all rays parallel to its axis through one point, the focus. In On Spherical Burning Mirrors he showed that a concave spherical mirror sends parallel rays at different heights to different points on the axis, the effect now called spherical aberration. The Latin translation of the paraboloid treatise, probably by Gerard of Cremona, was read in medieval Europe.

Alhazen’s problem

For the general reader. In a curved mirror, where must you look to see the reflection of a given object? For a plane mirror the answer takes one line. For a spherical mirror it leads to an equation of the fourth degree. Ibn al-Haytham solved it with conic sections, and the problem kept mathematicians busy for the next nine centuries.

By the law of reflection, \(A\), \(B\), \(P\) and the normal at \(P\) lie in one plane, and the normal passes through the centre \(O\). So \(P\) lies in the plane \(OAB\), and the problem is planar: given a circle and two points, find the points \(P\) of the circle where \(OP\) bisects the angle \(APB\). Ibn al-Haytham showed that there is one such point for a convex mirror and up to four for a concave one. He also treated cylindrical and conical mirrors.

His solution

He reduced the problem to a sequence of six lemmas. The key lemma asks for a line through a given point meeting a circle and a line so that the intercepted segment has a given length, a type of problem the Greeks called neusis. He solved it by intersecting the circle with a hyperbola, following Apollonius. The text has scribal errors and was long thought obscure; modern analysis, notably by Muṣṭafā Naẓīf and A. Mark Smith, shows the argument complete.

The modern equation

Put the circle as the unit circle in \(\mathbb C\), with \(A=a\), \(B=b\), and \(P=z\), \(|z|=1\). The condition that the radius bisects the angle between \(PA\) and \(PB\) is that \((a-z)/z\) and \((b-z)/z\) have arguments symmetric about 0, that is,

Derivation. A complex number \(w\) is real iff \(w=\bar w\). On the unit circle \(\bar z=1/z\), so \(\overline{(a-z)(b-z)\bar z^2}=(\bar a-\bar z)(\bar b-\bar z)z^2=\bar a\bar bz^2-(\bar a+\bar b)z+1\). Setting \((a-z)(b-z)\bar z^2\) equal to this and multiplying by \(z^2\) gives the quartic. For the hyperbola, the imaginary part of \((a-z)(b-z)\bar z^2=ab\,\bar z^2-(a+b)\bar z+1\) must vanish (using \(z\bar z=1\)); expanding \(\bar z=x-iy\) gives the equation. Its quadratic part \(q(x^2-y^2)-2pxy\) has trace zero, so the hyperbola is rectangular. Huygens, between 1669 and 1673, also solved the problem with a single hyperbola valid in every case; the form above is a convenient modern version of such a solution.

Impossibility

In 1965 Jack M. Elkin showed, using the resolvent cubic of the quartic, that for some positions of \(A\) and \(B\) the reflection point is not constructible with straightedge and compass. Peter M. Neumann gave an independent proof in 1998. Ibn al-Haytham’s use of conics was therefore necessary, not a lack of ingenuity.

Refraction

For the general reader. Light bends when it passes from air into water or glass. Ptolemy had measured the bending. Ibn al-Haytham measured it again with a better instrument, stated rules that the measurements obey, and offered a mechanical picture: light moves more slowly in a denser medium, and its motion splits into parts along and across the surface. He did not find the exact law, but his picture became the starting point for René Descartes.

Apparatus and rules

The instrument of Book VII was a graduated brass disc, set vertically and half immersed in water (or with glass blocks at its centre), with sights to admit a narrow beam at a measured angle. Let \(i\) be the angle of incidence, \(r\) the angle of refraction, both from the normal, and \(d=i-r\) the deviation. He preferred to state results in terms of \(d\).

He also stated rules about how the ratio of deviation to incidence changes with \(i\). Muṣṭafā Naẓīf showed that some of these hold only for the media and ranges of angle that were tested.

The mechanical picture

He compared light to a ball thrown at a thin board. A ball striking perpendicularly breaks through; a ball striking obliquely is deflected. He decomposed the motion of light into a component along the normal and a component parallel to the surface. A denser medium resists the motion more, so light is slower there. Motion along the normal, he argued, is the strongest and passes most easily, so a ray entering a denser medium turns toward the normal. He also remarked that refracted light takes a path that is “easier and quicker”.

Proof. Let the light go from \(A=(0,h_1)\) to \(B=(L,-h_2)\), crossing the surface \(y=0\) at \((x,0)\). The time is \(T(x)=\sqrt{x^2+h_1^2}/v_1+\sqrt{(L-x)^2+h_2^2}/v_2\). Setting \(T'(x)=0\) gives \(x/(v_1\sqrt{x^2+h_1^2})=(L-x)/(v_2\sqrt{(L-x)^2+h_2^2})\), which is \(\sin i/v_1=\sin r/v_2\). So light bends toward the normal exactly when it is slower in the second medium, as Ibn al-Haytham held.

The sine law itself had been found around 984 by Ibn Sahl, in a treatise on burning lenses, though it did not enter Ibn al-Haytham’s optics. In Europe Willebrord Snell found it in 1621, and Descartes published it in 1637, deriving it from a decomposition of motion like Ibn al-Haytham’s.

Doubts concerning Ptolemy

For the general reader. Ptolemy’s planetary models predicted well, but some of their circles moved in ways no real body could. Ibn al-Haytham listed these contradictions in a book of doubts. He did not reject Ptolemy’s astronomy; he demanded that its models describe physical bodies consistent with the principles Ptolemy himself accepted.

The configuration of the world

An early treatise, On the Configuration of the World, presents Ptolemy’s models as solid, nested spheres in the manner of the Planetary Hypotheses, with the Earth stationary at the centre. It was translated into Spanish, Latin and Hebrew, and it influenced Georg Peurbach’s Theoricae novae planetarum in the fifteenth century.

The doubts

Al-Shukūk ʿalā Baṭlamyūs, written around 1025–28, examines the Almagest, the Planetary Hypotheses and the Optics. Its main objections concern the astronomy:

• The equant. In the models for the planets the epicycle centre moves uniformly in angle about the equant point, not about the centre of the circle that carries it. A solid sphere cannot rotate uniformly about a point other than its centre.
• The Moon. In Ptolemy’s lunar model the epicycle’s apogee is aligned toward a point other than the centre of its carrier, again a motion no sphere can produce.
• Latitudes. The models for planetary latitudes require circles whose planes oscillate, and the Almagest and Planetary Hypotheses do not agree.

He argued that motions assigned to imagined lines and points cannot cause the motion of real bodies, and that Ptolemy had admitted departing from his own principles. The correct arrangement of bodies, he concluded, must exist and had not yet been found.

Proof. In Ptolemy’s arrangement the equant lies on the apogee side of \(C\). Put \(C=0\), \(Q=e\) on the real axis, apogee at angle 0, and the moving point at \(e^{i\theta}\). Since \(e^{i\theta}-e=e^{i\theta}(1-e\,e^{-i\theta})\), the condition \(\arg(e^{i\theta}-e)=M\) gives \(M=\theta+e\sin\theta+O(e^2)\), so \(\theta=M-e\sin M+O(e^2)\) and \(\dot\theta=\omega(1-e\cos M)+O(e^2)\). As a check, at apogee the point is at distance \(1-e\) from \(Q\), so uniform turning about \(Q\) needs linear speed \((1-e)\omega\) there, and \((1+e)\omega\) at perigee.

The model of the motions

Around 1038 he wrote The Model of the Motions of Each of the Seven Planets, which survives incomplete in a single damaged manuscript. Roshdi Rashed’s edition shows a geocentric kinematics without the equant, using geometry and trigonometry to describe the apparent motions. The Maragha astronomers of the thirteenth century, above all Nasir al-Din al-Tusi, took up the same programme and solved it with new devices.

The Milky Way

In a short work he used parallax to locate the Milky Way. A body near the Earth shifts against the stars as the observer’s position changes; the Milky Way shows no such shift, so it lies far beyond the Moon and is not a phenomenon of the upper air, as Aristotle had held.

Power sums, parallels, lunes, numbers

For the general reader. Ibn al-Haytham wrote about twenty-five mathematical works. Four topics stand out. He found the sum of fourth powers and used it to compute a volume that Archimedes had not. He tried to prove Euclid’s parallel postulate. He generalized the ancient theorem on lunes. And he solved a problem in congruences with what is now called Wilson’s theorem.

Sums of powers

He justified the formulas by a geometric dissection of rectangles, which amounts to the identity below, valid for every \(k\ge1\).

Proof. \(\sum_{p=1}^{n}\sum_{j=1}^{p}j^k=\sum_{j=1}^{n}(n-j+1)\,j^k\). Adding \(\sum j^{k+1}\) gives \(\sum (n+1)j^k\).

The first form is close to the one he stated in words. The lemma with \(k=3\) and the known sum of cubes gives it.

The paraboloid

Archimedes had found the volume of a paraboloid formed by turning a parabolic segment about its axis. Ibn al-Haytham also turned it about the chord perpendicular to the axis, which gives a lemon-shaped solid. Slicing perpendicular to the chord gives discs whose radii are \(k(b^2-y^2)\), and the sum of their volumes needs fourth powers.

Proof by sums. Divide \([0,b]\) into \(n\) parts of width \(b/n\). The slice at \(y_j=jb/n\) has radius \(kb^2(1-j^2/n^2)\). Half the volume is approximated by \(\sum_j \pi k^2b^4(1-j^2/n^2)^2\,b/n=\frac{\pi k^2b^5}{n}\sum_j\big(1-\frac{2j^2}{n^2}+\frac{j^4}{n^4}\big)\). Using \(\sum j^2\sim n^3/3\) and \(\sum j^4\sim n^5/5\), the limit is \(\pi k^2b^5(1-\tfrac23+\tfrac15)=\tfrac{8}{15}\pi k^2b^5\), which is \(\tfrac8{15}\) of the half-cylinder \(\pi(kb^2)^2b\). He bounded the sums above and below and concluded by a double reductio, in the manner of Archimedes.

Parallels

In his commentary on the premises of Euclid’s Elements and a treatise on the parallel postulate, he defined parallel lines as equidistant ones, and argued that a perpendicular moving along a line, keeping at right angles, traces a straight line with its other end. The argument turns on quadrilaterals with right angles, of the kinds later studied by Giovanni Girolamo Saccheri and Johann Heinrich Lambert; for a quadrilateral with three right angles he tried to show that the fourth is also right. In hyperbolic geometry the fourth angle is acute, so the attempt needs an assumption equivalent to the postulate, here the motion argument. Omar Khayyam objected to the use of motion in geometry, and al-Tusi criticized the argument in turn. The three-right-angle figure is sometimes called the Ibn al-Haytham–Lambert quadrilateral.

Lunes and congruences

Hippocrates of Chios had squared certain lunes. Ibn al-Haytham proved a general version: for a right triangle inscribed in a semicircle, the two lunes on the legs together equal the triangle. In number theory he asked for a number that leaves remainder 1 when divided by 2, 3, 4, 5 and 6 and is divisible by 7, and he observed that for a prime \(p\), the number \((p-1)!+1\) is divisible by \(p\). The statement is Wilson’s theorem, named after an eighteenth-century rediscovery.

The Cairo years and the last works

From the 1020s to his death Ibn al-Haytham seems to have lived in Cairo as a private scholar, copying, teaching and writing. The lists of works show a steady output. Between February 1027 and July 1028 alone he added 21 titles. By October 1038 the total was 92. Many works answer questions sent by others, such as the replies to the scholar Ibn Maʿdān on Ptolemy’s account of the apparent enlargement of stars near the horizon.

Among the shorter optical works of these years are On the Light of the Moon, which argues that the Moon shines with borrowed light emitted from every point of its surface, as a self-luminous body would; On the Halo and the Rainbow (1028), whose explanation fails but which Kamāl al-Dīn al-Fārisī later revised; On the Light of the Stars, which holds that the stars and planets other than the Moon shine by their own light; On Shadows, which distinguishes full shadow and penumbra; and the Discourse on Light, a summary of his doctrine. In mathematics he wrote Analysis and Synthesis, on the method of discovery in geometry, arithmetic and astronomy, and On Known Things, on what it means for a quantity or figure to be given.

Biographers mention students, including a certain Surkhāb of Semnan and the Egyptian amīr Abū al-Wafāʾ al-Mubashshir ibn Fātik; the details are thin. He died in Cairo around 1040, or soon after. No grave is known.

From Alhacen to Kepler

In the Islamic world

The Optics was little read in Arabic after the eleventh century, until Kamāl al-Dīn al-Fārisī wrote his commentary, the Tanqīḥ al-Manāẓir (Revision of the Optics), early in the fourteenth century, at the suggestion of his teacher Quṭb al-Dīn al-Shīrāzī. Using a glass sphere filled with water as a model raindrop, al-Fārisī explained the primary and secondary rainbows by two refractions and one or two reflections, at about the same time as Theodoric of Freiberg in Europe. The Doubts set the agenda for the astronomers of Maragha. Al-Tusi criticized Ibn al-Haytham’s treatment of parallels and wrote his own.

In Latin Europe

The Latin De aspectibus, made around 1200 by an unknown translator, survives in about nineteen manuscripts, all missing the opening chapters of Book I. Roger Bacon, John Pecham and Witelo built on it the medieval science of perspectiva, which became a university subject. Lorenzo Ghiberti drew on it in his notes on art. Risner’s edition of 1572 put it in print next to Witelo. Johannes Kepler’s Ad Vitellionem paralipomena (1604) started from this tradition and replaced the rule of perpendicular rays with the retinal image. Snell, Descartes, Pierre de Fermat and Huygens all read him.

Assessment

• Optics: intromission given a geometry; light and vision separated into physics, physiology and psychology; the law of reflection in full; the first systematic use of the dark chamber.
• Method: repeated tests with instruments, stated beforehand, and results reported against claims. Historians differ over how far to call this “the scientific method”; Gerald Toomer warned against isolating him from his predecessors. The Optics is, in any case, an early sustained example of physics argued from controlled experiment.
• Mathematics: Alhazen’s problem; power sums and the paraboloid; the quadrilateral of the parallel theory.
• Astronomy: the demand that models be physically realizable, which led to the equant-free models of Maragha and, through channels still debated, to Copernicus.
• Honours: the lunar crater Alhazen and asteroid 59239 Alhazen; his portrait on the Iraqi 10,000-dinar note (2003); the theme of UNESCO’s International Year of Light in 2015.

Sources and further reading

Biography and context

1. A. I. Sabra, Ibn al-Haytham, Dictionary of Scientific Biography, vol. 6, Scribner, 1972.
2. J. J. O’Connor and E. F. Robertson, Abu Ali al-Hasan ibn al-Haytham, MacTutor History of Mathematics.
3. Ibn al-Haytham, Wikipedia (with extensive references).
4. A. I. Sabra, One Ibn al-Haytham or two? An exercise in reading the bio-bibliographical sources, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 12 (1998); and the sequel in vol. 15 (2002–03).
5. D. C. Lindberg, Theories of Vision from al-Kindi to Kepler, University of Chicago Press, 1976.

Works and mathematics

1. A. I. Sabra, The Optics of Ibn al-Haytham, Books I–III: On Direct Vision, translation and commentary, Warburg Institute, 1989.
2. A. M. Smith, Alhacen’s Theory of Visual Perception (2001), Alhacen on the Principles of Reflection (2006), Alhacen on Image-Formation and Distortion in Mirrors (2008), Alhacen on Refraction (2010), American Philosophical Society.
3. Book of Optics and Alhazen’s problem, Wikipedia.
4. Ibn al-Haytham, The Doubts concerning Ptolemy, ed. A. I. Sabra and N. Shehaby, Cairo, 1971.
5. R. Rashed, Les mathématiques infinitésimales du IXe au XIe siècle, vols. 2–4, Al-Furqan, 1993–2002.
6. J. M. Elkin, A deceptively easy problem, Mathematics Teacher 58 (1965).
7. P. M. Neumann, Reflections on reflection in a spherical mirror, American Mathematical Monthly 105 (1998).
8. V. J. Katz, Ideas of calculus in Islam and India, Mathematics Magazine 68 (1995).
9. R. Rashed, The celestial kinematics of Ibn al-Haytham, Arabic Sciences and Philosophy 17 (2007).