Goethe, Newton and the physics of colour
Johann Wolfgang von Goethe, the well-known German author and poet was born in 1749 and died in 1832, which is the same as to say that he lived during a period of intense development of the foundations of chemistry and electricity. Two relatively young sciences that have since had thoroughgoing influence as theoretical basis for the technological civilization we live in today. Although a man of the arts and humanities, Goethe took great interest in scientific matters. This interest was rooted in his fascination with the various phenomena of nature that met his eye, such as the rainbow and other colourful events, as well as the shape of clouds that come and go on the sky, and the wonderful metamorphosis of plants and other organisms during their life cycle. He made careful observations and wrote a number of articles dealing with issues belonging to geology, botany, zoology as well as physics and general morphology.
One of the great puzzles he devoted himself to, during more than half his life, was human colour vision and the wonders of light and colour, in the open air as well as in connection with various arts and crafts. In 1791, after his famous "Italian journey", he published a small book titled "Beiträge zur Optik", dealing with the colour phenomena observable by help of a glass prism. His main work in the field of colour was however printed twenty years later, in 1810, and called simply "Zur Farbenlehre". It consisted of three books, the first of which, "Entwurf einer Farbenlehre", presented a design for a theory of colour. The second volume presented excerpts and summaries of documents pertaining to the history of colour science from antiquity up to Goethes own time. The third volume was a critical appraisal of the theory of light and colour presented in "Opticks", the famous book written by Isaac Newton and published in 1704, in other words about a hundred years before Goethe.
Goethe's prismatic demonstrations
That Goethe, in his writings on light and colour, argued so strongly and uncompromisingly against Newton's ideas, was what once caught my own interest and aroused my curiosity. Would it be possible for me, as a physicist, to understand the essential contents of this critical appraisal? Was there any sense in it? Or is it true, as many a commentator on Goethe has concluded, that he was wrong; incapable of understanding the merits of a scientific way of reasoning? I made up my mind to find some kind of answer to these questions. So I took a close look at "Zur Farbenlehre".
The first volume of it, dealing with an outline of a theory of colour, has three main chapters: Physiological colours, Physical colours, and Chemical colours, and is rounded up with a famous chapter, dealing with the aesthetics, psychology and practical use of colour, titled "Sinnlich-sittliche Wirkung der Farbe". His work could thus be characterized as an inter-disciplinary, encyclopaedic inventory of colour.
Let us dwell for a while on Goethes treatment of "Physical colours". What he aimed at by using this term was those colour phenomena which are conjured up as if out of nowhere, at the interaction of light with colourless material bodies, and being due essentially to the geometrical constraints, characterizing the situation. To this class of colour phenomena belongs above all the rainbow, the lustrous colours of mother-of-pearl and the like, but also the colourful spectra created when the rays of the sun come to play with the crystal glass pendants of a chandelier.
In "Beiträge zur Optik" Goethe advises us to look through a glass prism and observe the colour phenomena that appear. It soon becomes evident to the observer that colours appear at distinct borders between dark and bright areas in the field of view. If you vary the geometrical conditions you find that all of the various configurations can be boiled down to four principal spectra: The two border-spectra and and the two aperture-spectra and .
An essential feature of the world of prismatic colours is a basic symmetry: whenever white and black are interchanged in the pattern, the other colours are interchanged specifically, i.e. yellow is interchanged with violet, purple with green, and cyan with red.
black-and-white picture to the left is viewed through a
glass prism. It then looks as shown to the right. Colours
appear at the borders between white and black, i.e. where
light and darkness meet. When the borders are close
together, the simple colour spectra meet and mix, either
additively to make purple, or subtractively to make
( Digital photo taken through a flint glass prism of 30 deg. diffracting angle. )
Thus, if the upper half of the picture (in the illustration above) should be additively superimposed upon the lower half, the result would ideally be a full white rectangle. If they were instead superimposed subtractively, i.e. as colour slides, laid upon each other, then the result would be a wholly black rectangle. The two halves are perfectly complementary: They have not a single wavelength in common and together cover the whole range.
Goethe was enthusiastic over the discovery he had made, namely that the complementary relationship among colours, since long well known to the painters, had such an evident foundation in the physics of colour. For that reason he was anxious to stress that all four spectra had to be considered as basis for a true theory of colour -- not only the particular one, obtained in case of a narrow aperture, studied by Newton. The physicists of Goethe's time told him that all these phenomena could very well be explained by help of Newton's concept of rays of light, differently refrangible. But Goethe stubbornly maintained that it was not just a question of explanation but of basic principles.
Pondering things over during the years, I think I have come to an understanding of what Goethe was after. He was pointing out a lack, or shall we say imperfection, in Newton's theory, especially as this theory was propagated by Newton's followers and late disciples.
Newton's colour theory
To make this incompleteness evident to you, let me refresh your mind with some of the main traits of Newton's colour theory. The experimentum crucis of this theory seemed to prove unobjectionably that any beam of light can be described as a simple sum of independent elementary constituents (what Newton called Rays of homogenous light), each one characterized by its particular angle of deviation when passing through a glass prism. Newton explicitly states that
"By the Rays of Light I understand its least Parts, and those as well Successive in the same Lines, as Contemporary in several Lines."
What he had in mind was probably some kind of atomic constituents of the hypothetical enigmatic entity light. Accordingly, we nowadays speak of photons, a photon having its characteristic frequency (or corresponding wavelength, if you prefer). Any light beam is a flux of a great many independent photons, and is characterized by the photon density as a function of wavelength -- the spectral distribution, for short.
Now, what is "colour" according to Newton? Well, he is a bit vague on this point. In "Opticks", Part II, Proposition II, he states that:
"All homogeneal Light has its proper Colour answering to its Degree of Refrangibility"
which is what we see, when looking at a spectrum. It does not tell us what colour non-homogeneal light, which is the normal case, should have. The Proposition also seems to leave open the possibility that homogeneal Light, under certain circumstances, could have another colour than the "proper" one. By way of explication, Newton adds:
"The Rays, to speak properly, are not coloured. In them there is nothing else than a certain Power and Disposition to stir up a Sensation of this or that Colour. (...) Colours in the Object are nothing but a Disposition to reflect this or that sort of Rays more copiously than the rest; in the Rays they are nothing but their Disposition to propagate this or that Motion into the Sensorium."
This statement is compatible with a modern point of view, describing the light entering the eye of an observer, looking at an object, as a flux of photons, emitted from the surface of the illuminated object. It seems reasonable to associate the perceived colour of the object with its selective reflectance. This is determined by the fact, that the probability for an incident photon to be absorbed by the material of the surface, instead of being reflected from it (or being transmitted, in case of a transparent object), depends specifically on the wavelength of the photon. Consequently, the spectral distribution of the reflected, or transmitted, light is in general different from that of the incident light. Hence, the light beam, entering the observers eye, through its spectral distribution brings with it information about the reflectance of the surface from which it emerges. How the "decoding" of this information is accomplished concerns the physiological and psychophysical aspects of colour vision.
Newton says, "The Rays are not coloured". In our words: photons do not have any colour. In fact, they do not even have a disposition to stir up the sensation of this or that particular colour. Each individual photon has its characteristic wavelength, determining among other things its deflection when passing a prism or a ruling, but it does not have any proper colour. (There is a probability that it should be absorbed by any of the three types of receptor pigments in the retina.) Only a flux of great many photons, i.e. a light beam, has a disposition to stir up a specific excitation of the visual organ. The excitation is normally the statistical effect of millions of photon-catches. It is spectral distribution over wavelength, not wavelength per se, which is the physical entity of relevance, when modelling colour vision.
This is self-evident, in a way. The reflectance of macroscopic bodies is a property of their surfaces, not of the molecules of which the body is a conglomerate. (A ball has a spherical shape, mathematically definable and physically measurable, but it would be nonsense to say that this implies that the molecules building it necessarily should be spherical or to maintain that the ball cannot be spherical because the molecules actually do not have any shape.) Shape and colour are seeable entities, properties of the furniture of earth, as Susan Stebbing called it in her classical essay on philosophy of science. They have no relevance at the atomic or molecular level, but are of course nevertheless real.
An apparent contradiction
The rays are not coloured, but what we usually call light, a manifold of rays, i.e. light as illumination, may be coloured. If sunlight passes through a pane of coloured glass, it transforms into coloured illumination, the hue of which may be observable if the light falls on a white surface. And we are all familiar with the golden light from the setting sun. Evidently, we have reason to regard colour as a property of (macroscopic) light as well as a property of (macroscopic) material objects.
But isnt there a contradiction involved here? If we choose to define colour (physically) as the spectral reflectance of a surface, or transmittance of a clear object, then the colour should be independent of variations in the illumination. If, on the other hand, we regard the spectral distribution of the light, reaching our eyes from the objects surface, as the measurable correlate of colour, then we should expect to perceive continually shifting colours of the objects, since the spectral distribution of the illumination varies appreciably during the day, and as soon as the object is turned or moved around.
Colour cannot be a property of light and a property of material bodies at the same time. Well then, maybe at different times? The perception psychologist David Katz discusses this in a clever way. He gives us the following example. Say that we have a yellow sample of paper illuminated by bluish light. Do we see the sample as green? Imagine the equivalent situation: a blue sample, illuminated by yellowish light. Do we see the sample as blue, or does it look green? The optical stimulus, the light rays coming from the sample to the eye, might very well be the same in the two situations and moreover identical with the reflection from a green sample, illuminated with ordinary, neutral, white light.
So the visual system is confronted with an ambiguity, which can be solved only if it gets some information about the prevailing illumination. In other words, if you look at the sample through a narrow tube, so that you only see a part of it and nothing else, then it will look green, in all three cases. If you place the sample on a white background and look at it without restrictions, the background will furnish information concerning the character of the illumination whether it is cool, warm or neutral and the sample will look different in the three cases. When we perceive the colours of objects, we also perceive the objects as so and so illuminated. That is, the perception in the three cases of our example will be: a yellow sample in bluish light; a blue sample in yellowish light; a green sample in neutral light.
Evidently, the same light may be perceived either as coloured or as neutral, according to circumstances. I think this is a possibility that Newton never pondered about, namely that colour vision could involve a sophisticated information processing activity in the brain! In other words, that we should have to treat a ray of light as a carrier of information, rather than as a stimulus, capable of stirring up a colour sensation.
This way of reasoning solves the apparent contradiction between the alternatives colour as reflectance and colour as spectral distribution.
The importance of white
It is a popular misunderstanding of Newton's theory that what we, with our eyes, perceive as colour, corresponds to what the physicists measure as wavelengths. What is wrong with this? To begin with, the perceived colour of a patch of light depends on the over all shape of the spectral composition of the flux of light, not on the absence or presence of any particular wavelengths in it. Furthermore, usually perceived colour is related to the reflectance of object surfaces, rather than to the spectral state of the reflected light, reaching the eye. As is evident from the quotation above, Newton was in favour of this idea, but he didnt elaborate that aspect of his colour theory any further. Being obsessed by his main invention -- namely to describe the action of a light flux as a sum of the actions of elementary constituents of it -- he did not take sufficient care to avoid speaking carelessly of light as a mixture of colours.
If you insist on defining colour in terms of the spectral state of the flux of light, reaching the eye, you will face a problem, which turns out to be a serious shortcoming of the resulting theory: There is no place for the achromatic colours. There is no "black light", nor is there "grey light". Moreover, the often-used expression "white light" is misleading. It designates essentially colourless light. So called "white light" is not white in the same sense as a white paper, or snow or the clouds in the sky. This ambiguity in the use of the term "white" is dangerous, since it can lead the thoughts astray. From the point of view of physics, light is energy, radiation, and darkness is nothing but the absence of light. Nevertheless, in our perceptual experience both light and darkness are qualities. White and black deserve to be counted as colours among the other colours.
Furthermore, light as radiation can be arbitrarily intense; there is no limit, no point, which could be appointed as the white pole of the system. (The "black-pole" could be chosen as the case of zero intensity.)
Phenomenologically colour is always colour of something. As elaborated above, it seems reasonable to try to correlate colour with the reflectance property of material surfaces. Then it is straightforward to define white, black and greys, namely as special cases, on a par with other cases, within the totality of physically possible colours.
In a colour system based on reflectance, white and black are the two extremes in between which the greys and the chromatic colours can be positioned. In this case, there is a white pole, namely 100 % reflectance, as well as a black pole, corresponding to zero reflectance. (They would suitably be placed as opposite poles in a geometrical representation of the colour system.)
And, what is more: We can define the concept of complementary colours the important symmetry within the world of physical colours that Goethe was so proud of having established in his series of observations with a glass prism. In other words, the existence of complementary pairs of colours need not be regarded as a psychological, not even physiological oddity, but has a solid foundation already within the physics of colour (i.e. the spatio-material aspect of colour phenomena).
Let us return to the four prismatic spectra, illustrated above. The particular spectral distributions of the light fluxes forming these spectra, together with all intermediate stages between them, are called "ideal". The Newtonian rays of homogenous light, mixed with various proportions of white light (e.g. full-spectrum radiation) does not cover the totality of all possible colour stimuli. The purples are missing, since the spectrum of pure wavelengths is open-ended. As was later on shown mathematically by Erwin Schrödinger, the system of ideal colours does however represent the totality of possible optical stimuli to the human (trichromatic) eye.
A simple model
To clarify the said, let me sketch a simple model of a physical colour system. Say that we have some source of full-spectrum light, for instance thermal radiation at 6500 K. Out of this unit light a manifold of colours can be generated by means of colour filters with various selective absorption. (Say that you project light patches on a white screen. There you have a representation of a model colour world).
Each colour filter is characterized by its transmittance, i.e. the percentage of the incident light it lets through, at each wavelength. Thus, the transmittance is represented by a function, defined over a range of wavelengths between 400 nm and 700 nm, and taking on values between 0 and 1. White (i.e. complete transparency) is the special case f=1, independent of wavelength, and black (complete opacity) the special case f=0, independent of wavelength. Now, in general, to any colour, given by the transmittance f, there exist (in principle) a colour given by g =1-f, and this is the colour complementary to the first mentioned one. In the example below, a purple colour is the complement of a certain green.
Later on in the history of colour science, Wilhelm Ostwald took up this approach, in constructing his Colour System and Colour Atlas, where each specific colour is characterized by its content of white, black and pure hue (Vollfarbe).
Let me carry this idealized physical colour theory still a step further. You can even make a consistent description of the world of colour without introducing light as a physical entity having properties. What is light? The nature of light has always puzzled the philosophers of science. In the final analysis, light cannot be wholly understood within classical physics. It is essentially a quantum mechanical phenomenon. (For instance, as concerns the relationship between circular and linear polarization of light.)
Let the term light refer to the state of an optical system. All properties are assigned to material objects that modify the light state. A set of linear operators are introduced, to represent sources, absorbers, translators and detectors (in our case retinal receptors). You have an algebra in terms of these operators: fg represents two absorbers modifying the state in succession; f+g two absorbers acting in parallel (e.g. two projectors with one filter in each, illuminating the same place on the white screen).
The states are of course not quantum mechanically pure states, but mixed states (what already Newton saw) and the modifications introduced by selective absorbers essentially change the coefficients of the density matrix, describing the mixture. In this model, there is no need for interpreting light as some kind of quasi-substance. Light just denotes the fact that the system is an interacting system and not only a conglomerate. As long as it is dark, the system (arrangements of material objects together with an observer) is only potentially active. The light source is switched on and the system is actualized, i.e. turned into optical reality.
Effective causes and formal causes
Why was Goethe so emphatic in his refusal to accept the Newtonian description of light and its role in mediating colour vision? And why were the physicists of his time so obsessed by the idea that the spectral composition of light should furnish the ultimate explanation (as far as physics is concerned) of colour phenomena?
Because the physicists were not primarily interested in understanding the phenomena (as such) but were looking for the forces acting behind the curtain. Physics deals primarily with effective causes, causa effectus, the immediate, local means through which the configurations of the world steadily changes. Of course, we physicists are conscious of the need for boundary conditions for a full explanation of the phenomena, that is, in order to specify the particular situation. However, in a sense, boundary conditions are accidental and momentary. The laws of nature are general and permanent.
Goethe says that colours are the Taten und Leiden des Lichts. They are the result of its action, as well as of how it is modified, restrained by matter. In other words, there are effective as well as formal causes of colour - causa formalis. Goethe seems to have been primarily interested in the what rather than the how of colour. What are the phenomena we associate with the term colour? What is the structure of this world of colour? Compare for instance the way he explains the blue colour of the sky! =>
Accordingly, Goethe was sceptical against people showing interest in the mechanisms behind the phenomena. As soon as you ask for these, you risk deceiving yourself. (As e.g. if you throw a stone and then ask: What makes it move through space? From hence comes that force continually pushing it forward? Galileis answer was: Nothing makes it move; movement with constant speed and direction is just a possible state of existence. We need to think of forces acting on it only to change its state of movement. To be stationary is only a special case of movement.)
It is amusing that Goethe was explicitly careful to avoid making any hypothesis about light. He states already from the beginning of his Farbenlehre that he dares not try to say what light is. Newton was famous for his declaration hypotheses non fingo I make no hypothesis. Still he maintained that light should consist of rays, differently refrangible, the rays being conceived as its least parts. However, to be fair against Newton, maybe he would have been ready to change his expression to experiments with a beam of light turns out as if it were a mixture of different elementary sorts of light. It was the application of the tool of linear algebra (component analysis) as an effective mathematical description of the observed phenomena that was his essential contribution.
It is Goethes contribution to have pointed out that the essential aspect of colour phenomena are their formal aspect : What they tell us about the state of the world and our own participation in this world, through sight.
© Pehr Sällström. febr. 2006