Pathfinders

by Jim Al-Khalili

  To appreciate what Ibn Sahl did, we must again return to the Greeks and the writings of Ptolemy, who had described the refraction of light in his Optics. He presented tables of angles of incidence and refraction from different materials, and is credited with proposing an approximate version of the law of refraction by suggesting that the ratio of the angles made by a beam of light on either side of a surface boundary, as it travels between two transparent media, is a constant. We know today that the constancy of this ratio of angles only holds when the rays of light travel between the two media at almost right angles to their surface boundary. This is because what is really constant is not the ratio of the angles themselves but that of their trigonometric sines, and this correct ratio only approximates to that of the angles when they are small: when the rays enter almost vertically. This correct trigonometric ratio was stated by Snell and confirmed independently by Descartes. It is these two men who are therefore usually credited with the law of refraction. However, the now incontrovertible fact is that Ibn Sahl arrived at the same result 650 years earlier. He correctly stated the law geometrically, as the ratio of the sides of triangles of light rays, which is exactly equivalent to the ratio of the sines.5 Thus, while Europeans have argued over whether it should be referred to as Snell’s law or Descartes’ law, there is no doubt that the real credit should go to Ibn Sahl. And while much of the work on refraction that was used by the Europeans in the seventeenth century came from Ibn al-Haytham’s Optics, it is to Ibn Sahl that equal debt is owed.

  Despite this, Ibn Sahl was not a scientist in the way that Ibn al-Haytham was. Unlike Ibn al-Haytham, he did not carry out experiments or attempt to understand the physical cause of refraction as a slowing down of light when it enters a denser medium. Ibn Sahl was only interested in understanding enough to construct lenses for the purpose of burning. Nevertheless, his work was hugely influential on Ibn al-Haytham, as were the Arabic translations of a number of Greek texts.

  The revolution brought about by Ibn al-Haytham’s Book of Optics can be understood at many levels. First and foremost, it was a real science textbook, with detailed descriptions of experiments, including the apparatus and the way it was set up, the measurements taken and the results. These were then used to justify his theories, which he developed using mathematical (geometrical) models. The work can be roughly divided into two parts: Books I, II and III are devoted to the theory of vision and the associated physiology of the eye and the psychology of perception, while Books IV to VII cover traditional physical optics. It became a far more important text than Ptolemy’s Optics and certainly the most influential work in the field until Kepler.

  The first Latin translation of Ibn al-Haytham’s Book of Optics was made during the late twelfth or early thirteenth century as De aspectibus.6 In England, it was to have a great influence on Roger Bacon (c. 1214–c. 1292), who wrote a summary of it, as well as on his Polish contemporary Witelo (b. c. 1230), and it was soon being widely cited across Europe – and would continue to be for several hundred years, far more so than any of the books on optics by Greeks like Euclid, Aristotle and Ptolemy.7 Equally importantly, later Islamic scholars would make great use of his work and extend it further, such as the Persians al-Shirāzi and al-Fārisi in the thirteenth century, the latter using it for the very first correct mathematical explanation of the rainbow (at the same time as, but independently of, the German Theodoric of Freiberg).

  The only Latin printed edition of the Book of Optics was published by Friedrich Risner in 1572 under the name Opticae Thesaurus, which contained, along with Ibn al-Haytham’s Optics, Witelo’s Perspectiva and the work of a lesser-known scholar by the name of Ibn Mu’ādh, which had been translated into Latin even before any of Ibn al-Haytham’s work, as Liber de crepusculis.8 This short treatise on the nature of dawn and twilight is fascinating, and for years it was wrongly attributed to Ibn al-Haytham.

  When it comes to vision, you might have thought that the way we see things is straightforward. I do not mean by this how the light entering our eyes forms an image on the retina, which is then sent as an electric signal through the optic nerve to the brain to interpret. I mean the far more basic idea that we see things because light from them enters our eyes. Surely this is obvious and always has been. It is remarkable therefore that, until Ibn al-Haytham, scholars’ understanding of how vision works was a confused mess. The Greeks had several theories of vision. Euclid and Ptolemy believed in what is called the emission theory, in which we see objects because rays of light are emitted from our eyes to illuminate them; the rays of light leave the eye in straight lines emanating like a cone. An opposing, more sensible view was held by Aristotle, who argued for an intromission theory of vision whereby light enters our eyes from the object we are looking at. However, he did not follow Euclid’s geometrical model of rays of light simply reversed in direction; for Aristotle, the space itself between the object and the eye lights up so that the image of the object enters the eye instantaneously.

  This is how Euclid would have defended his emission theory: we have difficulty seeing a small object like a needle even if it is in front of our eyes; we would have to look directly at it. This experience would be hard to understand if the needle is sending rays to the eye all the time; we should be able to see it if our eyes are open regardless of where we ‘look’. The only sensible solution, he claimed, is that the eye must be actively sending rays at the needle in order to see, rather than just waiting passively for the needle to send its rays.9

  To confuse matters further, Plato and Galen had a combined emission/intromission theory, whereby the eye sends out rays of light to the object being looked at and this then reflects the light back into the eye. This idea was favoured by Islamic scholars such as al-Kindi and Hunayn ibn Ishāq. Ibn al-Haytham changed all this and resolved an issue that so many great minds could not sort out. He begins his Book of Optics thus: ‘We find that when the eye looks into exceedingly bright lights, it suffers greatly because of them and is injured. For when an observer looks at the body of the sun, he cannot see it well, since his eye suffers pain because of the light.’10 This implies that it must be the sun’s action on the eye that inflicts the injury rather than anything to do with the eye emitting rays. He also uses arguments associated with the phenomenon of afterimage (when we look for a long time at a bright object then look away). He also reiterates an argument of Aristotle’s as to the ludicrous notion that we see the stars in the night sky by sending rays out to reach them. Ibn al-Haytham concludes that ‘all these things indicate that light produces some effect in the eye’.11

  He next uses faultless logic to show the absurdity of the emission theory. He argues that if we see because rays are emitted from the eye onto an object, then either the object sends back some sort of signal to the eye or it does not. If it does not, then how can the eye perceive what its rays have fallen on? Therefore there must be light coming back to the eye and this is how we see. But if so, what use is there for the original rays emitted by the eye? The light could come directly from the object if it is luminous, or be reflected from it if it is not. Thus, the rays from the eye are an unnecessary complication and should be dropped. In this way, he used a form of Occam’s Razor, the dictum that a phenomenon should be explained using as few assumptions as necessary, attributed to the fourteenth-century English philosopher William of Occam.

  But Ibn al-Haytham went further than philosophical arguments, for he did something quite astounding. He used Euclid’s geometrical model of the emission theory and applied it to the intromission theory. Now, it is rays emanating from the object that spread out radially in straight lines. In this way, he ‘mathematized’ his theory of vision.

  Interestingly, what he did not do, despite giving the first optical description of the camera obscura,12 was to connect this to the way the eye works (by projecting an inverted image of the objects we see onto the retina). Thus, while making huge advances on everything that had come before, even to the extent of understanding that vision
works via refraction of light through the eye’s lens, when it came to how the rays entered the eye, he did not make that final step of explaining how the real image of the object being perceived forms on the retina. Instead, he switched from physics to psychology as soon as the light entered the eye in order to explain how we ‘perceive’. This was partly due to the incomplete understanding of the workings of the eye that he had learnt from Galen. It would not be until the turn of the seventeenth century that Kepler would provide the correct explanation by describing the eye as like a camera.

  A famous optical effect that Ibn al-Haytham helped clarify is known today as the moon illusion. But until he explained it, no one had realized it was an illusion at all. It is the phenomenon in which the moon appears larger near the horizon than it does when higher up in the sky – and the same occurs with the sun and the star constellations. The earliest recorded mention of the moon illusion goes back to the seventh century BCE and a clay tablet found in the Assyrian King Ashurbanipal’s great library of Nineveh. A popular belief among the ancient Greeks, and described by Ptolemy, held that the moon appears larger near the horizon due to a real magnification effect caused by refraction of light by the earth’s atmosphere. But on the contrary, we now know that atmospheric refraction actually causes the moon to appear about 1.5 per cent smaller when it is near the horizon than when it is high in the sky. In fact, if we ignore this small effect, it can be shown by direct measurement that the angle the full moon subtends at an observer’s eye remains constant as it rises or sinks in the sky. Photographs of the moon at different elevations also show that its size remains the same. A simple way of demonstrating that the effect is an illusion is to hold up a small coin at arm’s length with one eye closed, positioning it next to the moon when it is at different positions in the sky, revealing that there is no change in size.

  Ibn al-Haytham was the first person to explain this phenomenon as a psychological rather than a physical effect. The first three volumes of his Book of Optics contain ideas on the psychology of perception, and it is here that he dismisses the Greeks’ idea that the moon appears larger when it is low in the sky because of the refraction of its light through the atmosphere. He shows that this is due to the subjective nature of perspective and is nothing more than an optical illusion. When the moon is high in the sky, there is no other reference body nearby to compare its size to and therefore no way of mentally determining how far away it is. But when it is just above the horizon, it appears closer to us and so we imagine it to be larger. This explanation became accepted throughout Europe after Ibn al-Haytham’s translated work, De aspectibus, was studied by Roger Bacon and Witelo in the thirteenth century.

  After dealing with the nature of vision, Ibn al-Haytham went on to tackle and extend the geometrical optics of Ptolemy and Ibn Sahl – although he never stated the law of refraction in its correct form as Ibn Sahl did, but rather followed the approximate version of Ptolemy. The difference between the two approaches can best be described in the following way. It is the difference between measuring the distance between two points on the perimeter of a circle as being the length of the arc (Ptolemy and Ibn al-Haytham: wrong) and the length of the straight-line chord between them (Ibn Sahl: correct). It is only when the points are close together that the two versions roughly agree, for this is when there is so little curvature that the arc approximates to a straight line.

  The difference between the Greeks’ version of the law of refraction and the correct one described by Ibn Sahl. If you consider a beam of light entering water from above, its path will be bent towards the vertical once it enters the water. According to Ptolemy, it is the ratio of the lengths of the two arcs of the circle (shown as solid curves) that is constant. Ibn Sahl corrected this by stating that it is in fact the ratio of the two straight-line chords just inside the arcs that is constant. This is today described as the ratio of trigonometric sines of the angles made by the path of the light beam in the air and the water.

  Where Ibn al-Haytham did go further than Ibn Sahl was in trying to understand the underlying physics of refraction by making use of the idea of resolving vectors describing separately the vertical and horizontal components of velocities, and he understood the notion of light travelling at different speeds in different media. Like Ibn Sahl, he carried out all his work geometrically rather than algebraically, and did not use trigonometric relations, even though others before him, such as the Syrian astronomer al-Battāni, had written extensively on trigonometry. On the whole, although Ibn al-Haytham added much to the science of refraction which he inherited from the Greeks, it is probably fair to say that his contribution to this field was more preservative than creative.13

  One of the new ideas he did introduce was in the study of atmospheric refraction (the bending of light received on the surface of the earth from celestial bodies). In common with other contemporaries of his, such as Ibn Sīna, Ibn al-Haytham believed that the speed of light was finite. Where they differed was in their notions about the underlying nature of light. Ibn al-Haytham believed light to be a continuous ray, whereas Ibn Sīna believed it to be composed of particles (a remarkable insight, given that Newton much later also attributed just such a ‘corpuscular’ nature to light and Einstein proved it in his work on the photoelectric effect that won him the Nobel Prize in 1921 – nine hundred years after Ibn Sīna and Ibn al-Haytham).14

  Ibn al-Haytham also carried out some of the first experiments on the dispersion of light into its constituent colours and studied shadows, rainbows and eclipses, and his work decisively influenced the theory of perspective that developed in Renaissance Europe, in both science and art. In the fourteenth century his Book of Optics was translated from Latin into common Italian, making it accessible to a much wider number of people, including several Renaissance artists, such as the Italians Leon Battista Alberti and Lorenzo Ghiberti, and, indirectly, the Dutchman Jan Vermeer. All made use of his discussions on perspective to create the illusion of three-dimensional depth on canvas and in carvings.15

  As a brief aside, I mentioned earlier Ibn Mu’ādh’s eleventh-century work on twilight, translated into Latin as Liber de crepusculis, which has been wrongly attributed to Ibn al-Haytham.16 The reason this is of interest is because, in it, Ibn Mu’ādh gives a pretty good estimate of the height of the atmosphere. He recognized that the twilight following a sunset must be due to illuminated water vapour high in the upper reaches of the atmosphere reflecting sunlight long after the sun has set. He gave a value for the angle of the sun below the horizon at the end of the evening twilight of 19 degrees. This, he argued, is the lowest elevation at which the sun’s rays can still meet the upper vestiges of the atmosphere. Through the use of simple geometric ideas and a value for the size of the earth provided by al-Ma’mūn’s astronomers, Ibn Mu’ādh calculated the height of the atmosphere to be around 52 miles. His work found wide interest in the Latin Middle Ages and in the Renaissance. His method and understanding of atmospheric optics was improved upon only when the Danish astronomer Tycho Brahe raised the issue of atmospheric refraction at the end of the sixteenth century and the subsequent optical work of the great Johannes Kepler was published in 1604. But Ibn Mu’ādh’s value for the height of the atmosphere is still pretty good. Indeed, the boundary between the earth’s atmosphere and outer space, known as the Kármán line, is at an altitude of 62 miles.

  Ibn Mu’ādh’s method for calculating the height of the atmosphere. If an observer at A catches the last glimpse of twilight on the horizon along his line of sight, at point B, this means there should be matter at B that is still illuminated by the sun. This is when the sun is 19 degrees below the horizon with its rays coming in along the tangent CB. A little Euclidean geometry tells us that the angle AOC must also be 19 degrees, or that AOB is 9.5 degrees. It is then easy to establish using trigonometry and a knowledge of the size of the earth that the height of B above the surface (Ibn Mu’ādh’s edge of the atmosphere) is 52 miles.

  In mathematics, Ibn al-Haytham�
��s name is probably best known in association with a famous problem in geometry that came about from his study of the reflection of light from curved mirrors. First described by Ptolemy, it became known in Europe as Alhazen’s Problem since it was discussed extensively in his Book of Optics. It can be stated as the problem of finding the point of reflection on a concave mirror of a light source in order to reach a given point. It is also often described as the billiard-ball problem: by considering a circular billiard table, find the point on the cushion that you need to bounce the ball off in order for it to hit another. Of course, we are familiar with finding the angles on a traditional ‘straight’ cushion, and this can easily be solved to make sure the angle of incidence is equal to the angle of reflection. While the same law of reflection also holds for a curved reflector, the problem can no longer be solved using geometry (with a ruler and compass). Instead, Ibn al-Haytham showed his readers that one must use and solve a difficult algebraic equation called a ‘quartic’ (i.e. involving x4).