Goethe's edge spectra
If the slit of the spectral apparatus is extremely widened or if a broad white strip is observed against black paper through a prism, as described by Goethe in the Didactic Part of his Theory of Color, the edge spectrum shown in Fig. 7a will be perceived. Goethe explains these edge spectra as being the shift of the objects from their real position caused by the effect of the prism. According to Goethe, the image is not shifted completely as if it in fact resisted the shift. As a result, a ”secondary image” is produced which slightly precedes the actual image. If the bright rectangle is viewed through a prism, it is shifted to the left by refraction, and the bright secondary image is superposed on the dark paper. Goethe propounds that bright on dark produces blue which changes into violet if the effect of the dark increases. On the right edge, the image of the dark surface shifts over the remaining bright ”principal image”. Dark on bright produces yellow which, according to Goethe, accounts for the yellow seam. Where the effect of the dark increases, yellow changes into red.
From the physico-optical viewpoint, it is an untenable interpretation that edge spectra should be caused by principal and secondary images and their resistance to displacement. In the case of a wide slit, the edge spectra can be demonstrated to result from the overlap of monochrome slit images, as illustrated in Fig. 7b. For greater clarity, the slit images of the individual colors are shown in a vertical arrangement. On the right, (starting from 1), the red edge spectrum is very obvious because both red and yellow are fully represented here. On the left in the illustration, the blue edge spectrum is visible (at 1’ and 2’). At the position marked with 4, all colors are present and produce white.
Extraordinary observations were made by Goethe on the ”negative slit” (Fig. 8a): Unlike the experiment described above, a broad black strip is viewed against a white background through the prism. An unusual ”reversed spectrum” is observed here, displaying the respective complementary colors of the previously described edge spectrum. The formation of this ”reversed spectrum” can be demonstrated in Fig. 8b. Starting at the top, a dark field should be drawn in the middle between the strips of the same color. The background at 0 and 0’ – previously black – is now white because all colors are present here. The previously white center at 4 is now black due to the lack of any color. On the left, the sequence of colors towards the edge is red (3’), reddish yellow (2’) and yellow (1’), and on the right violet (3), blue (2) and bluish green (1). Goethe lists the following ”elements” between white and white, from right to left: blue, bluish red, black, reddish yellow, yellow (Theory of Colors; Didactic Part § 246), corresponding to the positions marked here with 2, 3, 4, 2’, 1’. If the normal slit or the white strip becomes increasingly narrow, the standard prismatic spectrum is gradually obtained, with green instead of white in the middle. If the ”negative slit” or the black strip becomes increasingly narrow, the red and violet spectral ends overlap at position 4 to form purple, the complementary color of green, as can be seen in the illustration. As a result, the following color sequence is obtained with a thin black strip or negative slit: white, yellow, orange, red, purple, violet, blue, bluish green, white.
This excerpt is from an article written for a magazine published for the famous Carl Zeiss Company. Its authors are: Prof. Lutz Wenke (Dean of the Faculty of Physics and Astronomy at the Friedrich Schiller University in Jena), Dr. Friedrich Zollner, Manfred Tettweiler (both from the Institute of Applied Optics) and Hans-Joachim Teske (Manager of the Astronomical Instruments business unit at Carl Zeiss). There are a few interesting points you must have noticed in the “demonstration” above. Firstly, the orientation of the prism is not mentioned, and the sentence which was probably meant to reveal that orientation (“If the bright rectangle is viewed through a prism, it is shifted to the left by refraction...”) is still not clear enough. In any event, we know the orientation necessary to produce the colours observed: The prism has to be oriented with its refractive angle (vertex) pointing to the observer’s left. Secondly, the spectral colours do not extend for the whole width of the white strip. This is rather odd and, in any case, it’s an ad hoc decision. Thirdly, I’m sure you have noticed how convoluted the ‘explanation’ is (especially for the so-called “negative spectrum”), considering how simply it could have been shown where the observed colours originate. Fourthly, the “demonstrations” illustrated in the figures make no sense, when the orientation of the prism is taken into consideration—for in figures 7b and 8b the spectral colours are depicted to run at a 90 degree angle to the normal way of refraction!
This is truly a very strange “demonstration” and I wonder how many physicists, apart from these authors, are accepting it. There could be, however, a possibility that I may have misunderstood something in the “demonstrations” above, and in that case I would love to hear from those who could clarify the situation. On the other hand, I (and I have reasons to believe that you, too) can explain the colours observed much, much easier.
The second example I want to give you is from a paper written by David Seamon, titled “Goethe’s way of science as a phenomenology of nature”.
To understand Goethe’s style of looking and seeing, I want to focus on the prism experiments in part two of Theory of Color. These easy-to-do exercises are a helpful way to introduce students to phenomenological looking because a phenomenon is present—the appearance of color in a prism—which, on one hand, most people are unfamiliar with yet which, on the other hand, can be readily examined, described, and verified through sustained work with the prisms. Table 1 indicates the kind of questions one should keep in mind in doing these experiments and, for that matter, all Goethean science.
Participants are asked to begin by simply looking through the prism, seeking to become more and more familiar with what is seen. They record their observations in words and colored drawings. Ideally, the experiments are done by a group of four or five, so that participants can report their observations to each other and bring forth descriptive claims—e.g., “I see a halo of color around all objects” or “I notice that there only seem to be colors along edges of objects.” Other participants can then confirm or reject these observations in their own looking and seeing. Gradually, the group moves toward a consensus as to exactly how, where, and in what manner colors appear.
This process of looking is slow and requires continual presentation, corroboration, recognition of error, and correction. Eventually, however, group members can establish a thorough picture of what their experience of color through the prism is and end with a set of descriptive generalizations like those in table 2.
SEEING AND UNDERSTANDING BROADER PATTERNS
The general exercise of looking through the prism just described is excellent for introducing students to the effort, care, and persistence required to produce accurate phenomenological description, but Goethe’s aim is considerably larger: to discover a theory of color that arises from the colors themselves through our growing awareness and understanding of them.
Here, we move into a stage of looking and seeing that explores the wholeness of color by describing in what ways the colors arrange themselves in relationship to each other and to the edge of light and darkness that, as discovered in the experiment just described, seems to be a prerequisite for any color to arise at all.
To identify such patterns and relationships, Goethe presents a series of experiments using a set of cards with black and white patterns that are to be viewed carefully through the prism and results accurately recorded. Examples of these cards are illustrated in table 3 and instructions for the use of three of these cards is provided in tables 4 and 5.
The value of the cards in these experiments is that they provide a simple way to direct the appearance of color and, thereby, provide a more manageable and dependable context for looking and describing. Rather than seeing color along any edge, participants are now all looking at the same edge displaced in the same way so they can be certain that they will see the same appearance of colors.
In regard to card A, for example, we begin with the white area above the black and, through the prism, look at the white-black horizontal edge in the middle of the card. If the image that we see is displaced by the prism below the actual card, then at the edge we see the darker colors of blue above violet (see drawing 1). If we turn the card upside down so that black is above white, we now see something quite different—a set of lighter edge colors that, from top down, are red-orange and yellow (see drawing 2).
As drawings 3 and 4 indicate, the experiments with cards B and C are perhaps the most intriguing because they generate two colors not as regularly seen as in the dominant spectra of yellow-orange-red and blue-indigo-violet. As one moves card B farther away toward arm’s length, there is a point at which the yellow and blue edges merge, and a vivid green appears horizontally so that the original white rectangle is now a band of rainbow (drawing 3). For card C, a similar point is reached where the red and violet edges merge to create a brilliant magenta (drawing 4)
The first thing you might have realised is that from what we have discussed thus far you can explain why the colours in the drawings 1 and 2 are observed. You might have also established—without looking through a prism or reading the instructions—that the colours in those drawings are observed only when the prism is oriented with the vertex pointing down. In fact, you might have realised that, in principle, you could predict what colours would be observed by looking through a prism at all six cards in table 3—even though we haven’t yet discussed the colours observed in experiments like those depicted in drawings 3 and 4. The new colours seen in those drawings (green and magenta) are the products of a mixture of certain spectral colours, as indeed it is explained. But the most important thing you have probably realised is the simplicity with which you can predict and explain the origins of the colours observed in Goethean experiments. You might thing that this explanation is accommodated by common sense and that—therefore—it is rather conspicuous. You might also think that it follows directly from Newton’s reply to Lucas and, on that basis alone, that physicists would have embraced it a long time ago. You might indeed think about all this, but you would be wrong. As you have seen in the first example I gave you earlier, and as it will become evident from the next two examples written by physicists, the explanation that can best predict and describe the observations of Goethean prismatic experiments is not part of the defensive arsenal of the scientific community.
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 [red-yellow] and [violet-blue] and the two aperture-spectra [cyan-magenta-yellow] and [red-green-violet].
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.
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.
The above was written by the physicist Pehr Sällström. Let us compare our (unconventional) explanation of the colours observed in the picture given with Goethe’s explanation (as described by the named physicist). Before doing that, however, notice that the orientation of the prism is not mentioned. Nevertheless, we don’t need to look through a prism in order to establish the orientation of the prism that will result in the specific colours displayed—we can firmly deduce that the only orientation which will result in the depicted colours is an orientation of the prism pointing with its refractive angle (vertex) to the right. I should also mention here that we’ll ignore some of the spectra that will be generated by the picture above, concentrating only on the spectra discussed. To make our task easier I have numbered the spectra of interest in the figure below, and I have also numbered the sources responsible for those spectra (according to our understanding, of course).
On the right picture I have numbered the four spectra observed, in the order in which they were listed in the article. On the left picture I have correspondingly numbered the sources generating the spectra, as I said before. Now, in the article there isn’t an explanation per se of where the four spectra come from—there is only a rather observational (phenomenological) comment that accompanies the picture. Apart from that comment the author says: “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”. These observations are quite useless in understanding the phenomenon—they are similar to learning the multiplication table by heart, where knowing the answer does not mean understanding the principle. (Besides that, Goethe’s enthusiasm for finding a physical basis for “colour-complementarity” is rather mystifying, in my opinion. But that’s another story.)
From our perspective, the spectra observed can be easily explained as being generated by an observation through a prism of four ‘independent’ sources.
Thus, spectrum number 1 in the picture on the right (the red-yellow) is one half of the spectrum generated by the white rectangle 1 in the left picture. The other half of the spectrum generated by the rectangle 1 is formed by a violet-blue combination, which is contributing to the creation of spectrum 3. The orientation of the colours in spectrum 1 points to where the other half is. Spectrum 1 is observed to appear towards the base of the prism, while the other half (the violet-blue combination) is observed to appear towards the vertex of the prism—just like we’ve already established.
Spectrum 2 (the violet-blue) is one half of the full spectrum generated by the white rectangle 2. The other half of that full spectrum is the red-yellow combination, and it can be seen (albeit, less vividly than its counterpart) at the border between the white rectangle 2 and the grey background of the page.
Spectrum 3 (the blue-magenta-yellow) is formed by the violet-blue half of the spectrum generated by the rectangle 1 and the red-yellow half spectrum generated by the white rectangle 3. In effect, the magenta component of spectrum 3 is formed by the mixing of red and violet—the blue and yellow components remaining unaffected. The violet-blue combination of the full spectrum generated by rectangle 3 can be seen at its border with the page itself.
Spectrum 4 (the red-green-violet) is the full spectrum generated by the narrow white rectangle 4. The green component of that spectrum is the result of the mix of its yellow and blue components. Spectrum 4 displays three colours (red-green-violet) only if the observation is conducted from a distance greater than approx. 20cm. If you look at spectrum 4 from a smaller distance you will see the yellow and blue components instead.
This is my explanation for those so-called four spectra. Compare it with Goethe’s, or with the one offered by the physicists from whom I gave you the first example, and judge for yourself. I know that my explanation can account for all possible subjective prismatic experiments, and that it can also predict what colours will be seen in all circumstances. This explanation is so accurate and comprehensive that I will therefore call it, henceforth, the law of colour-display in subjective prismatic experiments. In the last example I want to show you we will apply the law of colour-display to some more complicated shapes. Then in the next chapter I’ll continue to test the law and I will also showwhere the spectral colours originate and how they come into observation.
The final example I want to show you comes from an article titled “Exploratory Experimentation: Goethe, Land, and Color Theory” which appeared in Physics Today in July 2002.
Goethe's experimental procedure comprised two stages: an analytic one that moved from complex appearances through simpler ones to a first principle, and a synthetic stage that moved in reverse order, showing how more complex appearances are related to the first principle. The analytic stage is illustrated by a set of experiments with black-and-white images. Figure 2 shows how a few of the images Goethe used look when viewed through a prism with its refracting angle held downward. The general law determined by Goethe was that colored fringes arose at black-white borders parallel to the prism's axis: yellow and red when the white was below the black, blue and violet when it was above, as shown in the prism view of Figure 2e. For Goethe, these fringes constituted an elementary appearance of prismatic color from which all others could be derived. For example, Goethe's experiments with black and white rectangles showed that the Newtonian and complementary spectra (see the prism views of Figures 2c and d) were generated when the colored fringes from two closely spaced black-white boundaries encountered each other: The yellow and blue fringes mixed to produce green; the red and violet produced magenta. For Goethe, therefore, the Newtonian and complementary spectra were compound phenomena that could be derived from the law of colored fringes.
The synthetic stage of Goethe's investigation is illustrated by his experiments on the colored fringes that appear when gray and colored images on various backgrounds are viewed through a prism. Figure 3 shows how part of one of Goethe's diagrams (see the cover of this issue), from Theory of Colors, looks through a prism with its refracting angle held downward. Experiments with squares in different shades of gray against white and black backgrounds showed that the intensity of the colored fringes increased with the lightness contrast at the boundary. More complex phenomena were seen using colored squares, which exhibited fringes with new colors not seen in the previous experiments. Goethe argued, quite plausibly, that those new colors were due to the mixing of the elementary fringe colors with the colors of the squares themselves. Goethe regarded that mixing the true explanation of Newton's observation that a red square, viewed through a prism against a black background, appears displaced slightly higher than a blue one, as seen in the upper right of Figure 3. Whereas Newton had adduced this observation to prove that different colors of light have different refrangibilities—the first proposition of his Opticks—Goethe saw it as merely a special case of the more general law of colored fringes.
Goethe's analytic investigations proceeded from the complex to the simple. Shown are five black-and-white images selected from a series studied by Goethe, viewed with the naked eye (top, adapted from Contributions to Optics, ref. 1) and through a prism with its refracting angle held downward (bottom). The up-down sequence of all the colors is reversed if the refracting angle is held upward. (a) An irregular arrangement of black and white exhibited colored fringes with no apparent order. (b) The colors generated by a simpler checkerboard pattern were periodic and exhibited regular changes as the checkerboard was rotated, but were still too complicated to be expressed in a law. (c) The colored fringes generated by a white rectangle depended on the width of the rectangle and its distance from the prism. A very narrow rectangle, or one at a great distance, exhibited a spectrum with just three colors. Wider rectangles, such as the one shown, displayed fringes whose colors--red, yellow, green, blue, and violet--were consistent with those of the Newtonian spectrum. (d) A black rectangle on a white background exhibited a spectrum—blue, violet, magenta, red, and yellow—complementary to that of (c). The complementary spectrum's central magenta, called "pure red" by Goethe, is not in the Newtonian spectrum. (e) The boundaries of wider rectangles acted as isolated black-white contrasts, displaying red and yellow fringes when the black was above, blue and violet when it was below. No colors appeared at vertical black-white borders.
The experiments just described are only a small fraction of those that Goethe performed during his career. Others included novel experiments with refracted sunlight that displayed at a glance the evolution of both the Newtonian and complementary spectra as a function of distance from the prism, and careful replications and variations of many of the experiments in book 1 of Newton's Opticks.
We shall pay close attention to this description of Goethe’s work, for it is a good summary and it mentions the most important aspects of Goethe’s theory of colours. In the first paragraph cited we encounter again Goethe’s explanation of the colours observed in subjective prismatic experiments like those depicted in figure 2. Notice that Goethe’s mechanistic observation is called “general law”, although it falls well short of accounting for all subjective experiments—as it will become evident soon. The stipulations of GMore complex phenomena were seen using colored squares, which exhibited fringes with new colors not seen in the previous experiments. Goethe argued, quite plausibly, that those new colors were due to the mixing of the elementary fringe colors with the colors of the squares themselves. Goethe regarded that mixing the true explanation of Newton's observation that a red square, viewed through a prism against a black background, appears displaced slightly higher than a blue one, as seen in the upper right of Figure 3. Whereas Newton had adduced this observation to prove that different colors of light have different refrangibilities—the first proposition of his Opticks—Goethe saw it as merely a special case of the more general law of colored fringes. Goethe’s “general law” we already discussed. A more interesting observation is mentioned in the second paragraph.
More complex phenomena were seen using colored squares, which exhibited fringes with new colors not seen in the previous experiments. Goethe argued, quite plausibly, that those new colors were due to the mixing of the elementary fringe colors with the colors of the squares themselves. Goethe regarded that mixing the true explanation of Newton's observation that a red square, viewed through a prism against a black background, appears displaced slightly higher than a blue one, as seen in the upper right of Figure 3. Whereas Newton had adduced this observation to prove that different colors of light have different refrangibilities—the first proposition of his Opticks—Goethe saw it as merely a special case of the more general law of colored fringes.
This is the most important contribution Goethe made to the research into the nature of colour. It is also the only observation that truly shows deficiency in Newton’s theory—although, alas, it failed to attract the attention it genuinely deserves. In fact, as you will see, Goethe’s argument on this issue is not only quite plausible—it is undoubtedly true. We shall analyse that argument in detail, and then you can assess my claim.