Abstract
In this article, drawing on the works of E. Schrödinger, Kohlrausch, and my own, I shall attempt to show in the most concise form that the so-called Ostwald theory of colors is incorrect in its very foundations and that its apparent simplicity is purchased at the cost of truth.
Full Text
FOUNDATIONS AND CRITIQUE OF OSTWALD’S THEORY OF COLORS1
Klemens Schaefer, Breslau.
§ 1. In this article, relying on the works of E. Schrödinger2, Kohlrausch3, and my own4, I shall try to show, in the most concise form, that the so-called Ostwald theory of colors is incorrect in its very foundations, and that its apparent simplicity is bought at the price of truth.
For greater clarity of exposition, I shall first give several definitions and explanations, to which I shall refer in what follows.
a. We shall consider here only body colors, i.e., the colors of completely matte (not glossy) pigments (for example, colored fabrics, dyed paper, etc.), illuminated by white sunlight. This last condition is necessary, since the color of pigments depends, of course, on the kind of illumination; the choice of white sunlight as “normal light” is in itself arbitrary, although it can be justified phylogenetically.
b. Such pigments generally “reflect” sunlight with unequal strength in different parts of the spectrum. In the special case when the “reflecting power” does not depend on wavelength, i.e., the body reflects light of all wavelengths with the very same relative intensities as in the light source, we call the pigment “white,” “gray,” or “black.” It is white in the case when all the light is completely reflected (“reflecting power” \(R = 1\)), gray if \(0 < R < 1\), and, finally, black when \(R = 0\). Such pigments we shall call “achromatic.”
c. In the general case, when \(R = \psi(\lambda)\), the mixture of rays entering our eye after reflection by the body will, generally speaking, be colored; we then call the pigment “chromatic.”
d. The color of any chromatic pigment arises, consequently, as a result of the mixing of all the light waves reflected by it; if the brightness of sunlight in the interval of wavelengths from \(\lambda\) to \(\lambda+d\lambda\) is denoted by \(f(\lambda)\,d\lambda\), then the brightness of the reflected light with wavelength \(\lambda\) will be equal to \(f(\lambda)\psi(\lambda)\,d\lambda\), while the brightness of the entire mixture of reflected rays will be determined by the sum (integral)
\[ \int_{\lambda_0}^{\lambda_1} f(\lambda)\psi(\lambda)\,d\lambda, \]
where \(\lambda_0\) and \(\lambda_1\) are the limits of the visible spectrum.
e. Consequently, in order to determine the color of any pigment, it is necessary in every case to proceed from its “reflection curve” \(R(\lambda)\); without exact knowledge of it we shall be able to say nothing about the color of the pigment.
f. If a pigment color (or a mixture of spectral rays) is considered foveally, i.e. in such a way that the light stimulus of the eye is limited to the Fovea centralis, and if, in addition, the surroundings of the color are completely dark (for example, a sufficiently small colored spot lying on black paper or viewed through a narrow tube, or, finally, light in the slit of a spectral apparatus), then, as Helmholtz and Maxwell showed, the color of a pigment for a “normal” eye can be characterized by three independent parameters or coordinates. As such, one may, for example, following the example of the named scholars, choose the hue (i.e. an indication of the wavelength of the same or of the complementary color in the spectrum), the saturation (which decreases with an increase in the content of white light), and brightness. The choice precisely of these coordinates is not necessary: instead of them one could introduce any three other mutually independent characteristics. Only one thing is essential: one must always take three mutually independent coordinates of color; hence the expression: the space of colors is three-dimensional.
Pigment colors (or spectral color) considered under such conditions Ostwald calls “absolute” (unbezogenen); we shall adhere here to this terminology, although it is self-evident that these pigment colors too are not “absolute,” but relate to known normal conditions (foveal viewing, dark colorless surroundings). This is, if one likes, an arbitrary restriction, though of exactly the same kind as the condition mentioned above of illumination by white sunlight. These restrictions have, of course, their quite definite basis, which will become clear from what follows.
Having adopted this terminology, we shall say more precisely: the space of “absolute” pigment colors is three-dimensional.
g. The colors of pigments considered not under these “normal conditions” (for example, large pigments whose image on the retina extends beyond the fovea; pigments in an arbitrarily colored illuminated environment, for example against a white, yellow, or green background, and so forth) are called by Ostwald “relative” (bezogenen). Owing to the special properties of our eye, especially owing to the so-called phenomena of contrast, such “relative” colors appear to the eye in certain cases as entirely different from the very same pigment under normal conditions. For example—and this is one of the most important cases—a pigment which under “normal” conditions appears yellow, when taken under other conditions, may, depending on the nature of the illumination of the background, seem brown or olive, and may assume all possible intermediate shades between them.
From this it follows that the “relative” colors of pigments depend on a large number of coordinates, in the general case even on an infinitely large number, since we can vary the external conditions in any way. The space of “relative” pigment colors is therefore not three-dimensional, but in the general case of an infinitely large number of dimensions.
Practically speaking, the theory of colors of bodies is possible only for the so-called “absolute” colors, since only these colors can be determined by a finite, small number (3) of coordinates. This circumstance is, of course, also the reason why only “absolute” colors have always been taken as the basis of the so-called theory of colors.
§ 2. After these preliminary remarks let us turn to Ostwald’s theory of colors; in doing so we shall rely on his own survey article1.
Leaving aside certain inessential details, we may first of all state that Ostwald too defines his colors by three data which, to be sure, differ from the Helmholtz–Maxwell ones indicated above (hue, brightness, and saturation). In Ostwald these coordinates are called: hue, white content, and black content. As was already emphasized above, what is essential is only the number of coordinates (three); their particular choice, on the contrary, is immaterial. Therefore there would in essence be nothing improbable if the Ostwald definitions of color turned out to be not only admissible, but even far more suitable than the Helmholtz–Maxwell ones. In that case, of course, the Ostwald coordinates would have to be single-valued functions of the Helmholtz coordinates and conversely. Before going more deeply into this fundamental question, it is necessary to emphasize that already from the number of coordinates it immediately follows that Ostwald too bases his theory on
and can place only “absolute” pigments (for the justification see § 1, g). Ostwald himself, to be sure, asserts the opposite; in Ostwald’s opinion, his merit lies precisely in having created a theory of “relative” colors, into which, for example, brown and olive-green colors fit. This is, however, an obvious error; we shall return later to the particular case of the color brown.
As a first result, therefore, one may write: Ostwald’s theory of pigment colors likewise deals only with absolute colors.
§ 3. Let us now see how Ostwald defines, in principle, his new coordinates—the content of black and of white. Ostwald arranges his pigment colors of equal saturation (the fact that his saturation is nothing other than the corresponding concept in Helmholtz we may, as incidental, leave without attention here; a more precise definition of Ostwaldian saturation is given below) in “color circles,” which in the general case are divided into 100 parts, the separate color tones being designated by numbers from 00 to 100. By means of the added purple tones, which do not occur in the spectrum, they form a closed series: from lemon-yellow (00) through green, green-blue, blue, indigo, violet, purple, red, orange back to lemon-yellow (100 = 00). Complementary colors are situated at the ends of one and the same diameter and therefore lie at diametrically opposite points of the color circle. 00 and 50, 25 and 75, and so on, consequently designate complementary pairs of colors.
Ostwald further asserts that, on the basis of experimental investigations, he came to the conclusion that the “most saturated” pigment—let us say, lemon-yellow 00—reflects the rays of all colors within a symmetrically situated “color semicircle”—in our example, consequently, all colors from 75 through 00 to 25; from this he creates the concept of the “full color,” i.e. of an ideally saturated pigment, which in the “color semicircle” situated symmetrically with respect to the color number reflects everything, and in the other, opposite semicircle absorbs everything. In our example, for the full color 00 we would have (see Fig. 1):
\[ R = 1 \text{ from } 00 \text{ to } 25;\quad R = 0 \text{ from } 25 \text{ to } 75;\quad R = 1 \text{ from } 75 \text{ to } 0). \]
The color semicircle from 25 through 50 to 75 corresponds, as is easy to see, to the “region of absorption,” and the color semicircle from 75 through 00 to 25 to the “region of reflection.”
“Full colors” do not exist in nature, and therefore Ostwald considers pigments which differ from full-colored ones in that in the region of reflection the reflection is not complete, but \(R\) is somewhat less than 1, and, correspondingly, in the region of absorption \(R\) is not equal to 0, but somewhat greater
FOUNDATIONS AND CRITIQUE OF THE THEORY OF COLORS
0, while, however, both the law concerning the color semicircle and the assumption that in both regions \(R\) is constant are preserved. (See Fig. 2, where these relations are presented for lemon-yellow pigment 00.) Consequently, neither in the “region of reflection” do we have complete reflection, nor in the “region of absorption” complete absorption. Ostwald explains both as follows: when particles of black paint are added to the pigment, nothing changes in the region of absorption, since there the reflection is in any case equal to zero; in the region of reflection, however, the reflection becomes smaller.
Fig. 1.
Fig. 2.
The distance \(S\) of the reflection curve in the region of reflection from 1 (see Fig. 2) is, therefore, the so-called “black content.” Conversely, when white particles are mixed with a “full-color” pigment, nothing changes in the region of reflection, since there the reflection is in any case equal to 1; in the region of absorption, however, the curve rises above 0; the distance from the axis of abscissas therefore measures the “white content” in the pigment. The difference \(1-(S+W)\) Ostwald calls the “color content” \(F\), and thus obtains the equation: \(F+S+W=1\), which he considers fundamental. Colors of equal saturation according to Ostwald are, consequently, those for which either \(F=\mathrm{const}\), or \(S+W=\mathrm{const}\). However, there are no pigments in nature that would have a reflection curve like that shown in Fig. 2. Pigment 00 of lemon-yellow color in reality has approximately the following reflection curve (the solid curve in Fig. 3), and the situation is entirely analogous in all cases. Ostwald therefore replaces the reflection curve actually observed by a curve like that shown in Fig. 2, constructing this rectangular curve, shown by dashes in Fig. 3, so that the lowest point of the actual reflection curve
Fig. 3.
determines approximately the content of white, and the highest—the content of black; moreover, the “law of the color semicircle” remains in force. This means that Ostwald asserts: two colored plates, one of which has the reflection curve shown by the solid line in Fig. 3, and the other the one shown in the same figure by hatching, will appear to the eye identical, indistinguishable.
§ 4. We shall now undertake a critique of this attempt by Ostwald to characterize pigment colors, apart from hue, by the content of white $W$ and black $S$.
First, one further remark. From all that has been set forth above it follows that Ostwald, in essence, also proceeds from the reflection curve of the pigment, which, according to § 1, c, is unconditionally necessary. But he schematizes it in the way shown in Figs. 2 and 3 only in order to make it possible to apply convenient methods, requiring little time, for determining $W$ and $S$.
But it is immediately evident that the replacement of the actual reflection curve of Fig. 3 by a rectangular curve is in the general case—why only in the general case will be stated below—impermissible, since in doing so the relative quantities of the colors mixed are also changed, and consequently, in the general case, the color of the mixture as well. If we thus disregard the special cases, possible with the qualification “in the general case,” which we shall discuss below, we may say: even a schematic representation of the actual reflection curve by means of a rectangular curve is impermissible. To this is added the further fact that the content of white and black is regarded by Ostwald as something completely equivalent and synonymous, which plainly follows from his considerations on white and black particles (§ 3), according to which, indeed, the mechanical admixture of white and black particles to colored ones is completely equivalent. In reality, however, the admixture of black to a chromatic pigment (by either the direct mixing in of black particles, or the introduction of a black sector on a disk for mixing colors) is something entirely different from the corresponding admixture of white. For, as experiment shows1, the addition of black produces only a darkening of the pigment, which can always be completely compensated by stronger illumination (of course, by white sunlight). In other words, a given pigment and that same pigment with an admixture of black may be made indistinguishable to the eye if the latter is illuminated more strongly. The situation is quite otherwise with the addition of white.
It changes the quality of the color and can never be compensated by a change in the intensity of illumination. Thus, two entirely different things are regarded in Ostwald’s “theory” as equivalent.
For Ostwald, the doctrine of the content of black and white has exceptional significance. For, as indicated above, he thinks and asserts that his systematics also embraces “relative” colors, which, as we have already explained above, is incorrect. Ostwald believes that an admixture of black to yellow makes it brown and that, consequently, he can also characterize brown pigments, which no other theory is capable of doing. But this is an error for two reasons. First, the content of black in a yellow pigment in itself has nothing in common with the brown coloration of pigments considered under “normal” conditions; with an admixture of black there never arises a brown color, which is a typical example of a “relative” color. And, secondly, Ostwald’s theory also applies to “absolute” colors.
§ 5. Although even these objections are already sufficiently weighty, the question is not yet exhausted.
Let us consider further pigments that have one and the same hue of color, but different reflection curves, whose maxima and minima coincide, as is shown in Fig. 4. All these pigments—the number of which can be increased at will—have, according to Ostwald, the same content of black and white and, consequently, in his system occupy one and the same place, are identical. And yet to the eye they will in general appear different, since each of them represents a mixture of colors different from the others. According to Ostwald’s measurements, all these different pigments would not differ from one another, having one and the same numerical designation. In a certain sense the converse is also true. In Ostwald’s system it is possible that one and the same pigment color may be represented by infinitely varied combinations of numerical characteristics. To understand this, let us recall that color space is only three-dimensional, whereas the totality of spectra is of infinitely many dimensions. And this means that there are infinitely many different mixtures of colors which are indistinguishable to the eye. For example, any two complementary colors give the very same white: in Ostwald’s color circle, for instance, the pairs 00 and 50; 01 and 51; ... 25 and 75... 49 and 99. The same is true for all colors. Each of them, in the general case, can be obtained with the aid of an infinitely large
Fig. 4
of different mixtures of spectral rays, which in pigment colors corresponds to an infinitely large number of pigments having different reflection curves. According to Ostwald, each of these reflection curves would in general also have a different content of white and black, as a result of which one and the same pigment colors would figure in infinitely many places of his system and, being indistinguishable to the eye, would have different Ostwald designations.
Both of these circumstances are fatal for the Ostwald system.
§ 6. The latter of them—namely, that one and the same complex color can in general be obtained in an infinitely large number of different ways—is the alpha and omega of the doctrine of color mixture; it is, so to speak, the “normal phenomenon” of this doctrine. And all the more surprising is the fact that Ostwald apparently did not know of its existence, since he discovers it anew and gives it a name which alone is new here: “metamerism.” According to Ostwald, “metameric pigments” are those which, despite different reflection curves, are indistinguishable to the eye. As was already indicated above, this circumstance is fatal for Ostwald’s systematics, destroying the uniqueness of the arrangement of colors. Ostwald and his pupils themselves feel this perfectly well, but they remove from their path the solution of this difficulty by referring to the fact that—even with respect to a normal phenomenon!—there is not yet sufficient experimental material. Meanwhile, the phenomenon of metamerism could probably have served to give Ostwald’s systematics such a direction as would have made it acceptable. For in § 4, when criticizing the replacement of the reflection curve by a certain rectangular curve, we had to add the restriction “in the general case.” “In the general case,” as we said, the replacement of the continuous actual reflection curve in Fig. 3 by the hatched curve of the same figure is inadmissible, since “in the general case” different colors correspond to these reflection curves. Only in the case where both reflection curves belong to metameric pigments—and this is the possible exception mentioned above—would the Ostwald method be admissible.
But then the question immediately arises: is it not possible, for each given pigment, to deform its actual reflection curve in such a “metameric” way that it becomes an Ostwald rectangle? If this question could be answered affirmatively, then all difficulties would thereby be eliminated, and the content of white and black together with the color tone would indeed determine the pigment uniquely. In that case one could still argue about whether the name is expedient: content of white and black, but in essence everything would be in order, and then it would also be necessary
it may turn out that \(W\) and \(S\) are in a simple relation to the Helmholtz coordinates: brightness and saturation. Unfortunately, investigation1 shows that this question must be answered in the negative. In order for the actual and the Ostwald schematized reflection curves to belong to metameric pigments, some definite condition2 must evidently be fulfilled, which is clear even without calculation. This condition is fulfilled if one rejects or modifies Ostwald’s law of the color semicircle, Ostwald’s most brilliant creation, in the opinion of his adherents. But then everything is again destroyed, since only in the case of preserving this law will the Ostwald schematization of the reflection curve be established in a genuinely unambiguous way.
§ 7. To avoid misunderstandings, one more remark should be added. We do not at all wish to deny (but also do not assert) that the Ostwald systematics may in other cases prove suitable with a certain approximation. This would occur when the condition indicated above (§ 6) is accidentally and approximately fulfilled. Our critique is above all a critique in substance. The exact foundations of the theory of color mixture, and exact knowledge of reflection spectra, are replaced by the Ostwald systematics with a crude scheme and, moreover, to a certain degree simply for the sake of convenience. It rejects, in favor of an apparent simplicity of exposition and of methods of measurement, the necessary exact formulation. Its claim that it is the first to encompass the quantitative side of the colors of bodies is inadmissible for two reasons. First, because in fact it does not do this, and, secondly, because this was already done long ago in the method that takes its origin from Newton and was developed by Helmholtz and Maxwell, which characterizes colors by their hue, brightness, and saturation.
Our conviction that Ostwald’s theory represents a significant step backward in relation to what had been done before it, and that, because of the vagueness of its fundamental concepts, it may cause quite incredible harm, has determined the choice of the points of this theory considered here. Even if all the other points of the theory were irreproachable, what has been set forth here is, in our opinion, sufficient to reject a theory which, owing to its apparent simplicity (and to the energetic propaganda of its creator), may penetrate into still wider circles. In his autobiography3 Ostwald, it is true, holds a different opinion: “I emphasize already here that I regard the creation of a doctrine of color measurement as the highest achievement which I have been able to accomplish.” But this is—fortunately—a mistake.