articles

Exploratory Experimentation: Goethe, Land, and Color Theory

Johann Wolfgang von Goethe's Theory of Colors¹ has continued to fascinate physicists for almost two centuries since its publication in 1810. Hermann von Helmholtz, Werner Heisenberg, Walter Heitler, and Carl Friedrich von Weizsäcker are among those who have written substantial essays on Goethe.² More recently, chaos theorist Mitchell Feigenbaum consulted Goethe's work and was surprised to find that "Goethe had actually performed an extraordinary set of experiments in his investigation of colors." Not only that, Feigenbaum persuaded himself that "Goethe had been right about color!"³

We agree with Feigenbaum that the experiments contained in Theory of Colors are what gives Goethe's work its abiding interest. In this article, we suggest that Goethe was a remarkable representative of a research style that we call exploratory experimentation. Long ignored by historians and philosophers of science, exploratory experimentation has nevertheless played a crucial role in the history of physics. Among others, Michael Faraday's investigations into electromagnetism followed the exploratory approach; they are discussed in box 1 and box 2 on pages 48 and 49. In the main, though, we tell the story of exploratory experimentation by looking at two investigations of color from different historical periods--Goethe's experiments with prismatic colors and Edwin Land's experiments on color vision. To understand properly their work, one must first consider a classic example of a very different approach to understanding color, that of Isaac Newton.

Figure 1

Newton first announced his "New Theory about Light and Colours" in a famous letter to the Royal Society of London.⁴ In it, he described several experiments, including the classic one--illustrated in Figure 1a--in which a beam of sunlight refracted by a prism casts an oblong colored spectrum on a wall. Starting from the observation that the image was not circular like the original sunbeam, Newton inferred the principles of his new theory: that sunlight was a mixture of rays of different "refrangibility," that colors were not "Qualifications of Light, . . . but Original and connate properties," and that colors could be either simple or compound. The crucial element of Newton's theory was his extension of the traditional concept of the ray, which he associated with a definite degree of refrangibility. Newton's reliance on this new concept served both to guide the interpretation of his experiments and to rule out competing explanations. Chief among those explanations were theories that attributed color to some modification of the light during refraction, such as mingling with shadow or (as was proposed by René Descartes) a frictional slowing of the rotation rates of light corpuscles. Newton ascribed to his experiments extraordinary demonstrative power: His theory, he said in his letter, "is not an Hypothesis but most rigid consequence, not conjectured by barely inferring 'tis thus because not otherwise or because it satisfies all phaenomena . . . but evinced by ye mediation of experiments concluding directly & wthout any suspicion of doubt." Indeed, Newton chose to model his later Opticks after Euclid's Elements, with each of the central propositions followed by a "Proof by Experiments."⁵

The role of preexisting hypothesis in Newton's optical work is revealed by an early notebook (1664-65), which gives a more reliable picture of his research practice than the carefully constructed letter of 1672.⁶ Newton reported that when he looked through a prism at a cardboard with its two halves painted in different colors, he observed colored fringes at the border between the two, as illustrated in Figure 1b. Newton varied the colors of the cardboard halves and noted in a table the colors of the resulting fringes. He connected those observations immediately with considerations of how different velocities of light "globuli" cause different color sensations. Evidently a corpuscular theory of light formed the background of Newton's experiments, serving both to guide their design and to conceptualize their results. For example, the corpuscular hypothesis implied that swifter rays would be less refracted by a prism than slower ones because they were exposed for a shorter time to the prism's influence.

In contrast to Goethe and Land, Newton was not primarily concerned with color as such, which he regarded as an indicator of more abstract and mathematizable properties of light rays. Among the few propositions in Opticks that deal with color per se is one (book one, part 2, proposition 6) proposing a geometrical procedure for determining the compound color that results from mixing simple spectral colors. The color wheel Newton used for this purpose is illustrated in Figure 1c. He left unclear, though, how the relative "number of rays" of each of the component colors was to be defined, and Newton admitted that he was unable to generate white from two colors, although his scheme predicted that possibility. But those failures did not worry him: Such color-mixing problems, he remarked,⁷ were "Curiosities of little or no moment to understanding the Phaenomena of Nature."

Goethe's scientific interest in color was inspired by the natural optical phenomena and the coloristic traditions of Renaissance painting that he encountered during his first journey to Italy (1786-88). Goethe's first publication on color theory, Contributions to Optics followed a few years later.¹ The Contributions centered around a series of experiments in which Goethe viewed various painted images on paper through a prism. Like Newton before him, he observed colored fringes along boundaries. Unlike Newton, however, Goethe systematically varied the experimental conditions--the shape, size, color, and orientation of the images viewed; the refracting angle of the prism; and the distance of the prism from the Figure--to determine how they influenced what he saw.

Figure 2

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.

Figure 3

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 as 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.

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. Particularly important are Goethe's experiments on colored shadows, such as one in which the shadow of a pencil cast by a lighted candle and illuminated by the setting sun is observed to be bright blue. Goethe was among the first to recognize the importance of this phenomenon, for which no account is given in Newton's theory.

Figure 4

As an epitome of his research, Goethe proposed a symmetric color circle, illustrated in Figure 4, that applied in all areas he had studied. By contrast, Newton's color circle, with seven colors subtending unequal angles, did not exhibit the symmetry and complementarity that Goethe regarded as essential characteristics of color. For Newton, only spectral colors could count as fundamental. By contrast, Goethe's more empirical approach led him to recognize the essential role of (nonspectral) magenta in a complete color circle, a role that it still has in all modern color systems. Artisans such as painters, dyers, and tanners, who had to deal practically with color, generally felt much more attracted to Goethe's color circle than to Newton's. One painter strongly influenced by Goethe's work was J. M. W. Turner (1775-1851), whose annotated copy of Theory of Colors is extant and whose painting "Light and Colour (Goethe's Theory)" is on display at Tate Britain in London.

Newton's and Goethe's respective approaches to color illustrate two very different approaches to experimental research. We call them theory-oriented and exploratory experimentation. Theory-oriented experimentation is often regarded as the only relevant kind: It corresponds roughly to the "standard" view in the philosophy of science that experiments are designed with previously formulated theories in mind and serve primarily to test or demonstrate them. Such a view was stated forcefully by Karl Popper, who wrote, "The theoretician puts certain definite questions to the experimenter, and the latter, by his experiments, tries to elicit a decisive answer to these questions, and to no others. . . . Theory dominates the experimental work from its initial planning up to the finishing touches in the laboratory."⁸ According to this view, it makes sense to perform an isolated experiment, and in particular an experimentum crucis, designed to judge between competing hypotheses. Newton largely followed such an approach in his experiments on color.

By contrast, exploratory experimentation has been relatively neglected by historians and philosophers of science. Its defining characteristic is the systematic and extensive variation of experimental conditions to discover which of them influence or are necessary to the phenomena under study. The focus is less on the connection between isolated experiments and an overarching theory, and more on the links among related experiments. Exploratory experimentation aims to open up the full variety and complexity of a field, and simultaneously to develop new concepts and categories that allow a basic ordering of that multiplicity. Exploratory experimentation typically comes to the fore in situations in which no well-formed conceptual framework for the phenomena being investigated is yet available; instead, experiments and concepts codevelop, reinforcing or weakening each other in concert.

Exploratory experimentation often results in the establishment of a hierarchy within a realm of phenomena. At the pinnacle are those phenomena--Goethe calls them primordial--that involve only the essential conditions and that are therefore attributed a special status. All other effects can be deduced or explained from those elementary ones by progressively complicating the experimental arrangement and adding new conditions. The connection between a particular effect and an elementary phenomenon is revealed by establishing a chain of intermediate effects. In his methodological essay The Experiment as Mediator Between Object and Subject,⁹ Goethe described the result of such an approach as a "series of experiments that border on one another closely and touch each other directly; and which indeed, if one knows them all exactly and surveys them, constitute as it were a single experiment. . . ." He regarded this care to connect the "closest to the closest" as an experimental analog of mathematical deduction, which "on account of its deliberateness and purity reveals every leap into assertion." In that context, isolated experiments are not very informative, let alone demonstrative, as they well might be in theory-oriented work. The difference is nicely illustrated by the exchange between Newton and an early critic, the Liège Jesuit Anthony Lucas, who brought forward many new experiments (including variations of Newton's own), which he claimed could not be accounted for by Newton's theory. Newton's response was to insist that one "try only the experimentum crucis [Opticks, book 1, part 1, experiment 6]," for "where one will do, what need of many?"¹⁰

The now-classic experiments on color vision begun in the 1950s by Land are not only a fine example of exploratory experimentation at the frontier between physics and biology, they also have a direct bearing on the theoretical content of Goethe's Theory of Colors. Land's research began with a simple experiment using two black-and-white transparencies of the same colored scene. The first transparency, the "long record," was taken through a filter that passed only long-wavelength light. The second, the "short record," was taken through a filter that passed only short wavelengths. The two records differed only in the lightness or darkness of corresponding points; neither had any color. The transparencies were then projected onto a screen, directly on top of one another, using a beam of light from the red part of the spectrum for the long record and a beam of incandescent light for the short record. According to the classical color theory based on the work of Newton, Thomas Young, James Clerk Maxwell, and Hermann von Helmholtz, the image on the screen could only be some shade of pink. What the observer saw, however, was an image brilliantly and diversely colored, almost like the original scene.

Although Land was not the first to observe such two-color projection effects, his observation initiated a program of exploratory experimentation lasting more than two decades. He began with a series of 22 variations on the two-projector experiment. Those experiments demonstrated that the unexpected or "nonclassical" colors appeared essentially instantaneously, and could not be explained by time-dependent adaptations in the eye. The experiments also showed that the colors were not substantially affected by such factors as the intensities of the ambient illumination or of the projecting beams, the angle subtended by the image, or the filters used to produce the short and long records. Land then performed a more precise series of experiments using a dual monochromator that allowed the experimenter to vary at will the wavelengths of the projecting beams, and to study the range of colors observed as a function of those wavelengths.¹¹

From the experiments, Land concluded that classical color theory was valid only for spots of light observed in totally dark surroundings and that it had only limited relevance to color perception in natural situations involving multiple objects and variable illumination. In particular, he concluded that the stimulus for the color seen at a point in an image was not, as usually supposed, the wavelength composition of the radiant energy reaching the eye from that point. His subsequent experiments were aimed at uncovering the nature of the stimulus. Most of these experiments used "Mondrians," collages of paper rectangles with different shapes and colors.

Figure 5

Land began with experiments in which colorless Mondrians in white, gray, and black were viewed through dark goggles that allowed only the eye's rod (night-vision) system to operate. By adjusting the illumination of the Mondrians, Land showed that the patches maintained a constant rank order of perceived lightness, even though a patch that appeared dark might be sending much more light to the eye than one that appeared light. This suggested to Land that the eye was able to discover lightness values independent of the flux of energy it received; the reflectance, the physical correlate of lightness, might be the color stimulus he was seeking. This idea led Land to a series of experiments in which he illuminated colored Mondrians with long-, middle-, and short-wavelength light that could be mixed in any proportion (see Figure 5). In one set of experiments, the illumination was adjusted so that, for example, a white area of one Mondrian sent to the eye exactly the same triplet of radiant energies as a green area of another Mondrian. The two areas continued to appear white and green, a dramatic demonstration that their perceived colors were independent of the flux of energy they emitted as a function of wavelength. In another set of experiments, observers were asked to choose from a standard set of 1150 color chips the one that best matched the color of a given area on an illuminated Mondrian. Land found that when a match was made, it was the reflectances of the two areas that corresponded, and not the triplets of radiant energy being sent to the eye in the three illuminating wave bands.¹²

The "retinex" theory of color vision that Land developed on the basis of his experiments has two essential elements: It recognizes lightness (that is, reflectance) as the fundamental stimulus of color, and it emphasizes the importance of boundaries, which allow the eye to estimate lightness by seeking out singularities in the ratio of energy flux from closely spaced points. The parallel with Goethe's theory, which itself emphasizes the crucial roles of lightness and of boundaries, is striking.

Textbook accounts of the history of physics usually highlight discoveries involving simple systems, that is, those consisting of relatively few interacting elements. Such systems lend themselves to study by means of isolated experiments designed to demonstrate directly an underlying physical principle. Most of the celebrated experiments of physics, from Galileo's with balls on inclined planes to Robert Millikan's with oil drops, are of this type. The physicist studying a simple system deliberately removes complicating influences, like an intensely focused road builder cutting a straight road with little interest in the surrounding landscape.

Newton's investigations into optics were guided by the metaphysical belief that color was merely a subjective correlate of mechanical properties of light rays. He therefore abstracted from the complex world of normal visual perception, working in a dark chamber illuminated only by a single sunbeam. The system he studied was thus a simple one, comprising entities of a single kind--rays with diverse refrangibility--whose mutual interactions, such as color mixing, were purely superpositional. Newton's approach was entirely reasonable given his aim: His mathematization of light and color could best take flight from a few particular effects. But the price paid was that his experiments had only limited relevance to color as usually perceived.

Physicists studying complex systems that consist of numerous interacting elements face a task different in kind from that confronting Newton. They often start with a multitude of empirical findings whose interconnections and underlying principles are unclear. They must use experiments not so much to demonstrate propositions as to develop the concepts needed to make sense of multiplicity. The traditional isolated experiment is of little help here. Instead, the student of complexity must be an explorer, performing numerous laboratory or numerical experiments under different conditions, sufficiently "close" to one another that no important feature of the behavior is missed. Such a physicist is not so much a road builder as a mapmaker, whose principal interest is the physiognomy of a complex landscape.

The role of relative complexity in motivating the choice of experimental strategy is clearly illustrated by the contrast between Newton and the exploratory cases we have discussed. Goethe and Land were interested in color as an irreducible quality, not as an epiphenomenon. Recognizing that the human eye and the external world constitute a complex interactive system, both chose to explore it under diverse aspects, performing literally hundreds of experiments during their careers. The result was a deeper understanding of the complexity of the conditions under which colors appear in the world of everyday experience. Faraday also studied phenomena that exhibited a bewildering diversity and complexity in which many interacting factors played important roles: the shapes of wires; the strength of magnets; the speed and direction of the relative motion between them; and the strength, direction, and time-dependence of currents. Although the laws describing these phenomena may seem simple to us today, this simplicity was not evident to Faraday, who chose to follow an exploratory path.

Theory-oriented and exploratory experimentation are not exclusive categories, but rather members of a spectrum of experimental research strategies. Which is more productive in a given context depends on many factors, including a field's state of development, the sort of knowledge (for example, underlying mechanisms versus phenomenal regularities) sought by the physicist, and the complexity of the system being studied. Our aim in emphasizing the exploratory path has been to bring to light an experimental style that has played an important, but hitherto underrecognized, role in the history of physics.

The authors thank Gérard Bienfait and Catherine Carbonne for assistance with the preparation of Figure 2.

1. The standard edition of Goethe's scientific writings, which contains both Theory of Colors and Contributions to Optics, is G. Schmidt, W. Troll, L. Wolf, D. Kuhn, W. von Engelhardt, eds., Die Schriften zur Naturwissenschaft, Böhlaus, Weimar, Germany (1947- ), in particular, vols. 3-6. Modern English translations of portions of Theory of Colors and related works can be found in J. W. von Goethe, Scientific Studies, D. E. Miller, ed. and trans., Suhrkamp, New York (1988). All Goethe quotes in this article are given in our own translation.
2. H. von Helmholtz, in R. Kahl, ed., Selected Writings of Hermann von Helmholtz, Wesleyan U. Press, Middletown, Conn. (1971), chaps. 2 and 18; W. Heisenberg, Across the Frontiers, Harper & Row, New York (1974), chap. 10; W. Heitler, Man and Science, Basic Books, New York (1963), chap. 2; C. F. von Weizsäcker, in E. Trunz, ed., Goethes Werke, Hamburger Ausgabe, Band 13: Naturwissenschaftliche Schriften, Wegner, Hamburg, Germany (1955), p. 537.
3. J. Gleick, Chaos: Making a New Science, Penguin, New York (1987), pp. 164-165.
4. I. Newton, Philos. Trans. R. Soc. London, 6, 3075 (1672), reprinted in I. B. Cohen, ed., Isaac Newton's Papers & Letters on Natural Philosophy and Related Documents, Harvard U. Press, Cambridge, Mass. (1958).
5. See, for example, I. Newton, Opticks: A Treatise of the Reflections, Refractions, Inflections & Colours of Light, Dover, New York (1952), based on the 4th edition, London (1730).
6. J. E. McGuire, M. Tamny, eds., Certain Philosophical Questions: Newton's Trinity Notebook, Cambridge U. Press, New York (1983), pp. 430-434.
7. I. Newton, ref. 5, p. 157.
8. K. R. Popper, The Logic of Scientific Discovery, Basic Books, New York (1959), p. 107.
9. J. W. von Goethe, in D. E. Miller ed. and trans., ref. 1, p. 16.
10. I. Newton, in I. B. Cohen, ed., ref. 4, p. 174.
11. E. H. Land, Proc. Natl. Acad. Sci. USA 45, 115 (1959); 45, 636 (1959). E. H. Land, Sci. Am., December 1977, p. 108. A recent account of Land's work and its historical context is given by S. Zeki, A Vision of the Brain, Blackwell Scientific, Boston (1993).
12. J. L. Benton, J. Opt. Soc. Am. 59, 103 (1969). E. H. Land, Sci. Am., May 1959, p. 84.
13. M. Faraday, Experimental Researches in Electricity, vol. 2, R. Taylor, London (1844), p. 140.
14. M. Faraday, Experimental Researches in Electricity, vol. 1, R. Taylor, London (1839), pp. 1-41.
15. T. Martin, ed., Faraday's Diary, vol. 1, Bell and Sons, London (1932), p. 367.
16. H. von Helmholtz, in R. Kahl, ed., ref. 2, chap. 15.