The study of light has been a major topic in the study of mathematics and physics from ancient Greek times up to the present day. This study has on occasion been highly mathematical in nature while at other times it has more relevance to other scientific disciplines. In this article we take a broad look at the topic, although we will emphasis its more mathematical aspects.
The early Greek ideas on natural philosophy, and in particular on the nature of light, would influence the world for two thousand years. Empedocles, in the fifth century BC, postulated that everything was composed of four elements; fire, air, earth and water. He believed that Aphrodite made the human eye out of the four elements and that she lit the fire in the eye which shone out from the eye making sight possible. Now of course if this were true one could see at night, so Empedocles knew that things were somewhat more complicated than this and postulated an interaction between rays from the eyes and rays from a source such as the sun.
Not everyone believed that sight was explained by a beam coming from the eye. Lucretius wrote in On the nature of the Universe (55 BC):-
The light and heat of the sun; these are composed of minute atoms which, when they are shoved off, lose no time in shooting right across the interspace of air in the direction imparted by the shove.Despite this remarkably accurate view, Lucretius's views were not generally accepted and sight was still seen as emanating from the eye.
Long before Lucretius, Euclid had made a mathematical study of light. He wrote Optica in about 300 BC in which he studied the properties of light which he postulated travelled in straight lines. He described the laws of reflection and studied them mathematically. He did question sight being the result of a beam from the eye, for he asks how if one closes ones eyes, then opens them at night one sees the stars immediately. Of course if the beam from the eye travels infinitely fast this is not a problem. In about 60 AD Heron made the interesting observation that when light is reflected by a mirror it travels along the path of least length. Ptolemy, about 80 years after Heron, studied light in his astronomical work. Through accurate measurements of positions of stars, he realised that light is refracted by the atmosphere.
The biggest breakthrough in ancient times was made by al-Haytham around 1000 AD. He argued that sight is due only to light entering the eye from an outside source and there is no beam from the eye itself. He gave a number of arguments to support this claim, the most persuasive being the camera obscura, or pinhole camera. Here light passes through a pinhole shining on a screen where an inverted image is observed. Anyone visiting Edinburgh in Scotland should go to see the camera obscura there near the top of the Royal Mile and marvel at just how effective the camera obscura is in this enjoyable tourist attraction.
Now al-Haytham argued quite correctly that we see objects because the sun's rays of light, which he believed to be streams of tiny particles travelling in straight lines, are reflected from objects into our eyes. He understood that light must travel at a large but finite velocity, and that refraction is caused by the velocity being different in different substances. He also studied spherical and parabolic mirrors, and understood how refraction by a lens will allow images to be focused and magnification to take place. He understood mathematically why a spherical mirror produces aberration.
European scholars who followed al-Haytham did not know of his work. It was not widely available in Europe until the final quarter of the 16th century. However, without making major advances on the Greeks, some Europeans did make some improvements. Grosseteste, in about 1220, stressed the significance of the properties of light to natural philosophy and in turn advocated using geometry to study light. He put forward theories of colour, however, which have little merit. Roger Bacon, about 50 years later, continued to follow his teacher Grosseteste in believing in the importance of the study of light and he did come up with some correct conclusions deduced from experiments carried out in a very scientific way. He believed that the velocity of light is finite, studied convex lenses and advocated their use to correct defective eyesight. About the same time at Roger Bacon was working on optics in England, Witelo was studying mirrors and refraction of light and wrote up his findings in Perspectiva which was a standard text on optics for several centuries.
Following this there was improved understanding of using a lens, and by 1590 Zacharius Jensen even used compound lenses in a microscope. The first person to make a significant step forward after the time of al-Haytham, however, was Kepler at the beginning of the 17th century. Kepler worked on optics, and came up with the first correct mathematical theory of the camera obscura. He also gave the first correct explanation of how the human eye works, with an upside-down image formed on the retina. He correctly explained shortsight and longsight. He gave the important result that the intensity of light observed from a source varies inversely with the square of the distance of the observer from the source. He was wrong, however, in arguing that the velocity of light is infinite. He published his results were published in Supplements to Witelo, on the optical part of astronomy (1604). In fact an important discovery had been made earlier by Thomas Harriot when he discovered the sine law of refraction of light in 1601, but he did not publish the result.
Kepler's work was a nice piece of mathematics, but people did not believe that the eye created an upside-down image on the retina. The argument that we do not observe the world upside-down seemed convincing. Only about five years after the publication of Kepler's work, Galileo constructed a telescope, following ideas of Hans Lippershey from the Netherlands who had constructed one in the previous year. Galileo turned his telescope on Jupiter in 1610 and observed its four major moons. Thomas Harriot in England observed the moons of Jupiter in the same year. In 1611 Kepler published Dioptrice which was another important work on optics. It described how one could put lenses together to give what today is called a telephoto lens. It also described total internal reflection but failed to give the correct law of refraction of light, Harriot's result being unknown to Kepler (or anyone else) although the two had corresponded.
Willebrord Snell discovered the sine law of refraction of light in 1621 but, like Harriot, he did not publish the result. The first to publish the law was Descartes in 1637. In was contained in La Dioptrique published as a supplement to Discours de la méthod pour bien conduire sa raison et chercher la vérité dans les sciences. Descartes and Fermat carried on a discussion after this publication (see  for details) and Fermat initially assumed that they had reached a different law since they had started from different assumptions. Fermat proposed that light follows the path which takes the shortest time, enabling Snell's law of refraction to be deduced mathematically. Other contributions around this time by Descartes was his belief in the mathematical argument by Kepler which showed that the image formed on the retina of the eye should be upside-down. He conducted an experiment with the eye of a dead ox, scraping away the retina and seeing that indeed the image was upside-down. Some of Descartes' claims were fallacious such as his belief that the velocity of light is infinite. He stated, rather foolishly, that he staked his philosophy on that fact. See  for details of why Descartes was so strongly convinced.
In 1647 Cavalieri published an important contribution to optics when he gave the relationship between the curvature of a thin lens and its focal length. Inspired by Kepler's discoveries on light, James Gregory had begun to work on lenses and in Optica Promota (1663) he described the first practical reflecting telescope now called the Gregorian telescope. In fact Gregory made a fundamental discovery about light a few years later while in St Andrews. He discovered diffraction by letting light pass through a feather but he was not the first to investigate this phenomenon as Grimaldi had studied it a few years earlier. Here is Gregory's description:-
Let in the sun's rays by a small hole to a darkened house, and at the hole place a feather (the more delicate and white the better for this purpose), and it shall direct to a white wall or paper opposite to it a number of small circles and ovals (if I mistake them not) whereof one is somewhat white (to wit, the middle which is opposite the sun) and all the rest severally coloured. I would gladly hear Mr Newton's thoughts of it.The reference to Newton brings us to the person who revolutionised thinking on light. We now looking at the last third of the 17th century, a period where major theories on light would be put forward. These resulted from the contributions of Huygens, Hooke and Newton and two opposing theories were supported. In the 1660s Gassendi had put forward the particle theory, suggesting that light was composed of a stream of tiny particles, while Descartes suggested that space was filled with 'plenum' which transmitted pressure from a light source onto the eye. The wave theory by Huygens and Hooke was a development of Descartes' ideas where now they proposed that light be a wave through the plenum, while Newton supported the theory that light rays were composed of tiny particles. Let us first examine Newton's major contribution.
When Newton experimented with passing light through a triangular glass prism in around 1666 it was well known that a spectrum of colours was produced. There was a standard explanation of this, namely that the pure while light was somehow corrupted in passing through the glass. The further it had to travel in the glass the more it was corrupted, hence the different colours that emerged. Newton carried out a very simple experiment. He placed a second triangular prism in the path of the coloured beams of light emerging from the first triangular prism, but he put the second prism the other way up that is standing on its point. The coloured rays of light entered this second prism and a single ray of white light emerged.
Now having passed through the two prisms the light had passed through a longer distance of glass than if it had just passed through one, and it should have been further corrupted, but it was not. The true explanation was clear to Newton. The white light was not pure as believed, it was composed of light of the different colours which combined to give white light. Now Newton used his understanding of colours to design telescopes which had as little chromatic aberration as possible. He now understood what caused chromatic aberration, the coloured fringes seen round objects viewed through a telescope. It was an almost necessary consequence of using lenses. To avoid the problem Newton designed a reflecting telescope.
In 1672 Newton published his theory of colour in the Philosophical Transactions of the Royal Society and in it he gave experimental evidence that light is composed of minute particles. A few years earlier another member of the Royal Society, Robert Hooke, had published a wave theory of light and his own theory of colours. He reacted to Newton's paper by claiming that what was original in the paper was wrong and what was correct in the paper was stolen from him. In  Nakajima discusses the Newton-Hooke controversy of 1672:-
Now Hooke was not the only person to argue against Newton's theory of light. Huygens was developing his wave theory of light at this time and by 1678 he had it worked out in all its mathematical details although he did not publish his Treatise on light until 1690. Like Newton, an interest in telescopes had prompted Huygens to try to understand the nature of light. He proposed a wave theory, but of course a wave has to travel through a medium so Huygens' model included an all pervading aether which carries the wave. This is similar to the way that sound waves travel. Sound waves travel in air and if a bell is placed in a vacuum then nothing is heard. Similarly, it was believed, light waves needed the aether through which to travel. It was a beautifully worked out theory and explained most of the observed phenomena of light such as reflection, refraction, diffraction etc.
After Hooke's death, Newton published Opticks in 1704. It discussed the theory of light and colour and dealt with investigations of the colours of thin sheets, 'Newton's rings', and the diffraction of light. To explain some of his observations Newton had to argue that the corpuscles of light created waves in the aether. However, the work strongly argued for a corpuscular theory of light, with the most telling argument being that light travels in straight lines yet waves are seen to bend into a region of shadow. There was a possible way to distinguish between Newton's corpuscular theory and Huygens' wave theory. In the former theory it was necessary for light to travel faster in a more dense medium, while in the latter theory light needed to travel more slowly.
In fact the speed of light had been calculated by Römer a couple of years before Huygens had completed working out of his wave theory. In 1676 Römer used data from the eclipses of Jupiter's moons to get the first reasonable value for the speed of light. He realised that the reason the time between eclipses of Jupiter's moons by the planet was shorter when the Earth on the same side of the sun as Jupiter and became longer when Earth and Jupiter moved towards opposite sides of the sun was due to the time taken for light to cross the increased distance. He calculated the speed as 225,000 km per second, rather than the correct value of 299,792 km per second, but it was a remarkable achievement and a definite proof that the velocity of light is finite. However distinguishing between the wave theory and the corpuscular theory with experiments on the velocity were quite impossible at this time. It would not be until 1850 that Foucault showed that light travelled more slowly in water than in air showing that Newton was wrong.
During the 18th century most opinion sided with Newton. He had been right on so many things that it was generally assumed that he must be right about light being corpuscular. Not everyone in the 18th century agreed, however, and when Euler published his work on optics Nova theoria lucis et colorum in 1746 it argued strongly for a wave theory of light. Diffraction was the hardest phenomenon to explain with a corpuscular theory, and Euler used it to support his wave theory. He argued strongly for an analogy between light and sound and consequently for the aether which carried light waves as air carries sound waves. The sun, said Euler, is "a bell ringing out light". Euler's theory was in fact the second version of his wave theory of light and details of both theories are considered in .
Little progress had been made between Newton's Opticks of 1704 and Euler's optical work. Perhaps the most significant was James Bradley's calculation of the velocity of light in 1727. This was still an astronomical method, but Bradley used observations of the aberration of light from stars. This is the apparent slight change in the positions of stars caused by the yearly motion of the Earth. It is worth noting that Bradley's work provided first direct evidence that the Earth revolves around the sun.
Euler's support of the wave theory did little to change the general belief in the corpuscular theory. In  Hakfoort studies the work of Nicolas Béguelin of 1772:-
Beguelin compared the Newtonian emission theory of light and the wave theory of Leonhard Euler. Whereas others opted for one of the two theories by invoking arguments or authorities, Beguelin made a systematic search for experiments which he hoped would settle the dispute. Two of these experiments were most original. The first, which Beguelin himself performed, concerned light rays grazing a glass surface. For several reasons it did not have the impact it deserved. The second one was a thought experiment which was meant to illustrate a major tenet of the wave theory, that is, the analogy between light and sound. ... neither of them brought the debate to an end.However Thomas Young produced a major piece of evidence in favour of the wave theory when he carried out experiments on the interference of light between 1797 and 1799 in Cambridge. Young placed a screen with two pin holes in it in front of a point source of light. He published his results in 1801, describing the pattern of dark and light bands seen on the screen behind the holes. He produced these interference patterns also using two slits and he explained the results using a wave theory. In fact he went further than this, explaining Newton's results in terms of his wave theory. The different colours of light, said Young, are different wavelengths of light. He related the amount of refraction of light, or diffraction of light, to its wavelength. According to Young, diffraction fringes occur as a result of interference between the incident wave and a wave arising from the edge of a diffracting aperture or body. He even calculated the wavelengths of the different colours using Newton's own experimental data. His explanation of interference, from his own words of 1807, is as follows :-
The middle of the pattern is always light, and the bright stripes on each side are at such distances that the light coming to them from one of the apertures must have passed through a longer space than that which comes from the other by an interval which is equal to the breadth of one, two, three, or more of the supposed wavelengths, while the intervening dark spaces correspond to a difference of half a supposed wavelength, of one and a half, of two and a half, or more.This description is absolutely correct but it was difficult for people to accept. There is something very definitely counterintuitive in claiming that two rays of light could, under certain conditions, add to give darkness. We should note that Young made other notable discoveries about light, in particular he realised that colour vision was due to the eye having receptors each of which was sensitive to one of the three colours red, green, or blue.
Two scientists who contributed to an understanding that light can be polarised when reflected from a surface were Malus and Brewster. Malus's discovery of the polarisation of light by reflection was published in 1809 and his theory of double refraction of light in crystals was published in the following year. Malus transformed geometrical optics into the study of straight lines and their reflection and refraction at surfaces, see  for details. Brewster's publication in 1811 gave what is now known as Brewster's law, namely that the maximum polarisation of a beam of light occurs when it strikes the surface of a transparent medium so that the refracted ray makes an angle of 90° with the reflected ray.
Dark lines in the spectrum of light had first been observed in 1802 by William Wollaston but the correct explanation of them had to wait a few years until a more thorough investigation by Joseph von Fraunhofer who measured the exact positions of over 500 such lines.
A major triumph of the wave theory of light came through the work of Fresnel. He appears not be have been aware of the wave theories of Huygens, Euler or Young but worked out his own wave theory. The experiment he carried out which convinced him of that his wave theory was correct was to place a small obstruction in a path of light and examine the diffraction patterns formed in the shadow. In 1817 the French Académie des Sciences proposed as their prize topic for the 1819 Grand Prix a mathematical theory to explain diffraction. Fresnel wrote a paper giving the mathematical basis for his wave theory of light and in 1819 the committee, with Arago as chairman, and including Poisson, Biot and Laplace met to consider his work.
It was a committee which was not well disposed to the wave theory of light, most believing in the corpuscular model. However Poisson was fascinated by the mathematical model which Fresnel proposed and succeeded in computing some of the integrals to find other consequences. He wrote :-
Let parallel light impinge on an opaque disk, the surrounding being perfectly transparent. The disk casts a shadow - of course - but the very centre of the shadow will be bright. Succinctly, there is no darkness anywhere along the central perpendicular behind an opaque disk (except immediately behind the disk).This was a remarkable prediction, but Arago asked that Poisson's predictions based on Fresnel's mathematical model be tested. Indeed the bright spot was seen to be there exactly as the theory predicted. Arago stated in his report on Fresnel's entry for the prize to the Académie des Sciences :-
One of your commissioners, M Poisson, had deduced from the integrals reported by [Fresnel] the singular result that the centre of the shadow of an opaque circular screen must, when the rays penetrate there at incidences which are only a little more oblique, be just as illuminated as if the screen did not exist. The consequence has been submitted to the test of direct experiment, and observation has perfectly confirmed the calculation.Fresnel was awarded the Grand Prix and his work was a strong argument for a transverse wave theory of light. Fresnel and Arago subsequently undertook further work, explaining polarisation of light with their theory. In the 1820s and 1830s diffraction was studied by a number of scientists; Fraunhofer published his theory in 1823 while twelve years later Airy mathematically calculated the diffraction pattern produced by a circular aperture. The next major advances were due to Faraday and Maxwell and in some sense these completed the 'classical' understanding of light. By 'classical' here we meant pre-relativity and pre-quantum theory. We will study the developments in relativity-quantum theory era in a separate article; see Light through the ages: Relativity and quantum era. Before we move on to look at Faraday and Maxwell's major contributions let us look briefly at some other contributions from the middle of the 19th century.
Fizeau, in 1849, was the first person to calculate the speed of light without using an astronomical method. He used a rotating wheel with 720 teeth to break up a light beam into a series of pulses. A partially reflecting mirror sent some light through the wheel while some passed through. The light which passed through the wheel was sent on a journey of 17.3 kilometres before being reflected back to interfere with light which had passed through the partially reflecting mirror. He found that it took 0.00056 seconds to make the 17.3 km journey and he calculated a speed of 300,000 kilometres per second with an error of 1000 km per sec. In the following year Foucault used the rotating mirror method to calculate the speed of light in air and in water, finding that the speed was slower in water. The wave theory of light was by now completely established. In 1860 Bunsen and Kirchhoff observed dark lines in the spectrum of a light source passed though burning substances. These were absorption lines as had been observed in the solar spectrum by Wollaston and Fraunhofer earlier.
Faraday did not himself have the necessary mathematical skills but his work was crucial in allowing Maxwell to develop a sophisticated mathematical theory based on the understanding which Faraday had brought to the study of electricity, magnetism, gravity and light. In 1845 Faraday studied the effect of a magnetic field on plane-polarised light. He discovered what is now called the Faraday effect, namely that if a beam of light is passed through a substance which polarises it, then the plane of polarisation is rotated by a magnetic field parallel to the ray of light. In 1846 Faraday gave a lecture at the Royal Institution in which he put forward his view that there is a unity in the forces of nature. He proposed that the lines of electric and magnetic force associated with atoms could provide the medium by which light waves were propagated:-
The view which I am so bold to put forth considers radiation as a high species of vibration in the lines of force which are known to connect particles, and also masses of matter together. It endeavours to dismiss the aether but not the vibrations.Faraday's ideas provided the basis on which Maxwell built his mathematical electromagnetic theory. One of Maxwell's first contributions to light was the creation of the first colour photograph in 1861. He based his idea on Thomas Young's understanding of colour vision. Young had shown that colour vision was due to the eye having three types of receptors each type sensitive to one of the three primary colours red, green, or blue. Maxwell took three black and white photographs of a tartan ribbon, one through a red filter, one through a green filter and one through a blue filter. At a meeting of the Royal Institution, with Faraday in the audience, Maxwell projected the three images, the image made with the red filter being projected with red light and similarly the others. The three images were projected on top of each other to create a colour image of the tartan ribbon on the screen.
In 1862 Maxwell realised that electromagnetic phenomena are related to light when he discovered that they travelled at the same speed. He wrote in On Physical Lines of Force:-
We can scarcely avoid the inference that light consists in the traverse undulations of the same medium which is the cause of electric and magnetic phenomena.In 1864 Maxwell wrote a paper in which he stated (see ):-
This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself (including radiant heat and other radiations) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.The four partial differential equations, now known as Maxwell's equations, which completely describe the classical electromagnetic theory appeared in fully developed form in Maxwell's paper Electricity and Magnetism (1873). Planck, who made one of the next major breakthoughts described in Light through the ages: Relativity and quantum era, said on the occasion of the centenary of Maxwell's birth in 1931, that this theory:-
... remains for all time one of the greatest triumphs of human intellectual endeavor.
Article by: J J O'Connor and E F Robertson
MacTutor History of Mathematics