HIGHER METRIC OF COLOR.

N. T. Fedorov, Moscow.

§ 1. In our first article¹ the foundations of the lower metric of color were set forth, dealing with settings for equality. In the present article we shall first consider the experimental material underlying the higher metric of color, dealing primarily with settings for the greatest similarity (heterochromatic photometry) and for a barely perceptible difference (the eye’s sensitivity to changes in hue and saturation), and then formulate the most fundamental propositions of the higher metric of color in the form in which they were given by E. Schrödinger.² In the following paragraph we shall indicate by what route one should approach the construction of the rational body of pigment colors.

If we have two beams of light of unequal brightness but of identical spectral composition, then we can always equalize them in brightness by changing the intensity of one of them by a factor of \(n\); moreover, the one whose intensity must be decreased by a factor of \(n\) for this purpose we shall call \(n\) times brighter. In this definition we assume, therefore, that the brightness of a light beam is proportional to its objective intensity and can therefore be expressed in energy units. In the case, however, when we have two equally bright beams of light of differ—

¹ Uspekhi fiz. nauk, No. 1, 92–118, 1929. Unfortunately, that article contains a number of misprints, often distorting the meaning. They should be corrected according to the list appended to this issue of the journal.

² E. Schrödinger, Ann. d. Physik, 63, 427 and 481, 1920.

not of spectral composition, but identical in color—for example, white sunlight and a mixture of any two complementary spectral rays—their energies in the general case will be different. But even here, if two such beams seem to us unequal in brightness, we can always, by changing the objective intensity of one of them by a factor of \(n\), make them indistinguishable in brightness; moreover, the ratio of their brightnesses, as in the first case, is determined from the relative change in the objective intensity of one of them. The matter is more complicated in the case where we have beams of light of unequal color, red, for example, and green, and when by changing the intensity we will obviously not be able to bring them to identity. In this case, according to Helmholtz,1 we can perform the adjustment only to the greatest similarity. In fact, we are not able qualitatively to assess the degree of difference existing between some definite red and some definite yellow, saying that they differ from one another more or less than some gray color from black; but we can always judge whether this difference decreases or increases under some definite change made with this pair of colors (rotation of a Nicol, rearrangement of a sector on a rotating disk, etc.). If, however, we compare some given color with some continuously varying one-dimensional series of colors of another color tone—for example, with a series of intensity steps of some other color—then it may happen that the difference between the colors being compared will decrease up to some definite member of the series and then increase again. We can always find such a “most similar” member; moreover, it is especially easy to do this in the case when the color tones taken are not very different from one another, just as dropping a perpendicular from some point to a given straight line—finding the nearest point of this straight line—is easier when this perpendicular is small. It is self-evident that

1 H. v. Helmholtz. Zts. für Psych. und Phys. 2, 1, 1891.

in the concept of the “most similar color” there is nothing absolute, and that this color is determined by the series that is given for selection. After what has been said, Helmholtz’s definition becomes clear to us, according to which the setting of two colors \(A\) and \(B\) to equal brightness is reduced merely to finding, in the series of colors that can be obtained from \(B\) by changing its intensity, the color most similar to \(A\).

The well-known practical difficulties of heterochromatic photometry have given rise to an enormous number (eleven!) of different methods, all of which may be divided into two groups: direct and indirect methods. The following three methods belong to the first group.

1. Direct comparison of colors by brightness, even if they differ greatly in color. This includes the old method of Fraunhofer,^1 then the method of König,^2 who determined the distribution of relative brightness (luminosity) in the spectrum by comparing the brightness of various rays of the spectrum with the brightness of certain definite green rays \((\lambda = 535\,m\mu)\), Abney’s method,^3 which determined the brightness of spectral rays by comparison with the brightness of white light, and also the determination of the luminosity of colors by means of the gray scale of Ostwald or Munsell.

2. The method of small intermediate steps, which consists in the fact that intermediate colors are introduced between the colors being compared, and in this way one sharply heterochromatic photometric measurement is decomposed into a series of less heterochromatic operations. (Cf. Exner,^4 Gibson and Tyndall,^5 Kohlrausch,^6 Clemens Schaefer,^7 et al.) and, finally,

^1 Fraunhofer. Ges. Schriften. 1, 1888.
^2 A. König. Gesam. Abh. 144, 1903.
^3 W. Abney. Res. in Colour Vision, 88, 1913.
^4 F. Exner. Wiener. Ber., 129, 1920.
^5 K. S. Gibson and E. P. Tyndall. Sc. Pap. of the Bureau of Stand. No. 475, 1923.
^6 K. W. F. Kohlrausch. Phys. Zts. 21, 396, 1920.
^7 Cl. Schäfer. Phys. Zts. 26, 58 u. 908. 1925.

1. F. Exner’s method for determining the lightness of pigments, which in essence reduces to weakly heterochromatic photometry. The analogous methods of Brücke and Hering also belong here.

All these methods, in their essence, amount to arrangements for obtaining the greatest similarity. The indirect methods are more numerous, and, as Schrödinger rightly points out, each of them brings with it a new definition of heterochromatic “equal brightness.”

Referring those interested to a detailed survey of these methods in the article by the recently deceased Prof. Kries,² we shall in what follows consider in greater detail only one of the most perfected of them—the “flicker method.”

§ 2. Passing to a more detailed analysis of the “direct” methods, we shall first of all establish one necessary condition underlying them, a condition indicated by Schrödinger³ and tested experimentally by Kohlrausch⁴ and Cl. Schäfer⁵ for pigment colors, and by Reichenberg⁶ and ourselves⁷ for spectral colors.

In the photometric comparison of two pigments whose colors lie far apart in the spectrum, we always encounter the difficulty that, for an observer with normal color vision, the impression of the quality of the color is so predominant that often he is not at all able to say which of the pigments appears lighter to him—except, of course, in extreme cases. But it is precisely on these extreme cases, evidently, that our instinctive “feeling” is based that one can in general speak of the lightness of colored pigments. As was already indicated above, in the second of the “direct” methods, in order to—

¹ Fr. Exner. Wien. Ber., 127, 1829, 1918.
² J. v. Kries. Zts. für techn. Phys., 6, 327, 1924.
³ F. Schrödinger, loc. cit.
⁴ Kohlrausch, loc. cit.
⁵ Cl. Schäfer, loc. cit.
⁶ Reichenberg. Dissertation in manuscript (Vienna, 1925).
⁷ N. Fedorow und V. Fedorowa. Zts. f. Physik, 67, 855, 1929.

To avoid the difficulties of heterochromatic photometry, one may insert between the two pigments being compared in their lightness a series of intermediate steps such that the difference in color between two adjacent steps is insignificant. Then it is possible, starting from the first pigment, through a known number of intermediate steps, i.e. almost isochromatically, to reach the last pigment, obtaining as a result a certain number characterizing its lightness in relation to the first pigment.

Fig. 1.

In this connection it is self-evidently assumed that we always arrive at one and the same result, regardless of how many and what intermediate steps we include: the result obtained by us must not depend on the “path.” This is precisely the “Schrödinger condition,” obtained by him theoretically. Both Kohlrausch and Schäfer used

…were used to verify this condition by V. Ostwald’s excellent color atlases. As is known, Ostwald arranges all colors in the form of a double cone (Fig. 1), with the most saturated, “full” colors lying along the circumference of their joined bases; upward they pass into white, and downward into black. The letters denote the content of white (the first) and black (the second) in the color. In Fig. 2 is given the numbering of the color tones contained in any circle of the color body perpendicular to its white–black axis, with an indication of the wavelengths of the spectral rays corresponding to these colors. Each color, consequently, is denoted by one numeral and two letters. We shall denote the luminosity of the lightest pigment 00 by 100. Then, if the Schrödinger condition is fulfilled, then, taking this pigment as the initial one and photometring through equal steps, which must be established experimentally by rational magnitude, we must, on reaching pigment 00 from the other side, obtain for its luminosity the same value—100. Kohlrausch, after measuring 24 colors from the circle “ia,” obtained instead of 100 the number 102.4; for the circle “nc,” 92, which undoubtedly lies within the limits of possible error, which is summed from the errors of the individual (24!) settings.

Fig. 3.

The results obtained by Schäfer¹ for the circle “ia” are shown in Fig. 3, where the results of measurements are given for a different number of intermediate steps—one (00 compared—

¹ He used an ordinary Glan spectrophotometer, the prism of which had been removed, and at the slits real images were obtained of the brightly illuminated pigments being compared. The size of the field of view excluded the possibility of the Purkinje phenomenon appearing.

called with 50), three (00—25; 25—50; 50—75; 75—00), 4, 6, 12, 24, and 48. First of all we see here how well the lightness of color No. 50, obtained by direct comparison of it with yellow No. 00, agrees with what is obtained with a different number of intermediate steps. Next we see that, to within a few percent, Schrödinger’s condition is fulfilled here; the magnitude of the step itself proves immaterial, but with too small a number of intermediate steps the difficulty of separate settings increases; with an excessively large number of steps the total error increases as a result of the accumulation of small errors in the separate intermediate measurements. Schaefer considers it most expedient to take 12 steps. The summed error with 12 steps for the circles “ba,” “ea,” “pa,” and “pm” in Schaefer was found to be respectively 4.1%; 2%; 3.1% and 1.2%. An idea of what is obtained by different observers is given by Fig. 4, where the solid curve was obtained for the circle “la” by observer F. I., and the dashed curve by observer G.

Relative lightness

Fig. 4.

Let us further suppose that we wish to compare the red Ostwald standard 25 pa with the green 75 pa. Then, if Schrödinger’s condition is fulfilled, we must obtain one and the same result both in the case where we pass from red to green through a series of intermediate colors of the same circle, i.e., either through purple, violet, blue, sky-blue, aquamarine, or through orange, yellow, yellow-green (29,33......67,71 or 21,17....83,79), and in the case where

when, without changing the color tone (25), we pass from it to white (aa) through a series of ever paler tones of the same color (25 pa, 25 na, 25 la.....25 ca), and then from white through the series 75 ca.....75 la, 75 na to 75 pa. This task was carried out by K. Schäfer as follows. Having measured the relative lightness of 12 colors of the circle ia, he measured a series of transitional colors from 00 ia to white and thus obtained for the lightness of the color 00 ia (which he had previously assumed to be equal to 100) the number 73.2. He obtained the lightnesses of the other colors by multiplying their relative lightnesses by 0.732. After this he determined the lightness of each of these colors also by another method—through a series of intermediates from each of them to white, i.e., as he had earlier done for the yellow 00 ia. The results he obtained are shown in Fig. 5, where the solid curve was obtained by the first method, and the dashed curve by the second. We see that in this case as well the result does not depend on the nature of the “path.” For spectral colors, Schrödinger’s condition was first verified by Reichenberg, who, in the spectroscopic laboratory of Prof. Haschek at the Second Physical Institute of the University of Vienna, measured by F. Exner’s step (“cascade”) method the distribution of relative brightness in the diffraction spectrum of a voltaic

Fig. 5.

Fig. 6.

arc, the magnitude of the steps being 8 mμ, 4 mμ, and 2 mμ. The visible spectrum had a length of 108 cm (from 700 to 400 mμ). The change in the magnitude of the steps was carried out with the aid of a special apparatus of Prof. Haschek, made for the institute by the firm of Hertz. This apparatus (Fig. 6) consists of 4 prisms, two of which lie below ($P_1'$ and $P_2'$), while the other two are above them ($P_1$ and $P_2$), with

Fig. 7.

the air gap between the pairs $P_1P_1'$ and $P_2P_2'$ being variable by means of a micrometer screw. If we place this apparatus at one or another point of the spectrum, bringing both pairs of prisms into close contact, we shall see both the upper and the lower parts of the field of view illuminated by light of one and the same wavelength $\lambda$. If, however, the prisms are separated, then, as is easy to see, the wavelengths will already be different—there will appear between them a certain difference $\Delta\lambda$, the magnitude of which, for different distances between the prisms, can be calculated or found experimentally. $M$ is a ground-glass plate, and $T$ is a viewing tube containing the Rochon prism $R$ and the Nicol $N$; by rotating the latter

…we shall be able to set the two halves of the field of view to “maximum similarity.” Having measured the ratio \(K=\dfrac{H_{\lambda+\Delta\lambda}}{H_\lambda}\), where \(H_\lambda\) is the relative brightness, for a whole series of wavelengths, we shall be able to construct the curve \(K(\lambda)\), which will lie above 1 on one side of the brightness maximum in the spectrum and below it on the other side.

From this curve, by successive division or multiplication,¹ we can also obtain the curve of the distribution of relative brightness in the spectrum.

Fig. 8.

The results obtained by Reichenberg are shown in Fig. 7.² By the author of the present article, together with V. I. Fedorova, using a simple attachment to Glan’s spectrophotometer constructed by him,³ consisting of two slits (Fig. 8), one of which can be shifted relative to the other; by exactly the same method it was shown that, even when \(\Delta\lambda\) is increased to \(12\,m\mu\), the brightness-distribution curve does not change, as is seen from Fig. 9, where the dashed line (b) gives the curve \(K=\dfrac{H_{\lambda+\Delta\lambda}'}{H_\lambda'}\) for \(\Delta\lambda=12\,m\mu\), constructed from the curve \(H'\), obtained at \(\Delta\lambda=8\,m\mu\) for the prismatic spectrum by the gas-filled milky “Osram” lamp, while crosses indicate the points obtained experimentally for \(\Delta\lambda=12\,m\mu\).⁴

We see, therefore, that Schrödinger’s condition is fulfilled quite exactly when \(\Delta\lambda\) is varied from 2 to

¹ Or from the curve \(\log K(\lambda)\) by successive graphical integration.

² The independence of the curve \(H'\) from \(\Delta\lambda\) for \(2\,m\mu\) and \(4\,m\mu\) was also confirmed by us on the same spectral installation of the University of Vienna, but for a somewhat larger interval (from \(718\,m\mu\) to \(426\,m\mu\)). See the Fifth Congress of Russian Physicists, 1926 (p. 65).

³ N. Fedorow. Reports of the Academy of Sciences of the USSR, 413, 1928.

⁴ N. Fedorow und W. Fedorowa. Die Naturwissenschaften. 16, 757–758, 1928 und Zts. für Physik, 57, 855, 1929.

12 mµ, and in the first case the difference in color of the compared fields over the entire extent of the visible spectrum (about 300 mµ) is noticeable in all only over an interval of 22 mµ, whereas in the latter case the difference in color is noticeable over almost the entire extent of the spectrum. By an analogous cascade method, in the Bureau of Standards in America in 1925, Gibson and Tyndall1 carried out an investigation of the distribution of relative brightness in the spectrum for 52 persons, and the results obtained were recalculated for a spectrum with a uniform distribution of energy. The distribution

Fig. 9.

of luminosity in a spectrum with a uniform distribution of energy obviously characterizes the sensitivity of our eye to radiant energy of different wavelengths—that which is usually called the “visibility” of radiant energy. Having this curve (Fig. 10), we can find the luminosity of colored bodies simply from their reflection spectrum. In fact, we can write the following obvious equality determining luminosity:

\[ H=\frac{\int \rho(\lambda)\,V(\lambda)\,E(\lambda)\,d\lambda} {\int V(\lambda)\,E(\lambda)\,d\lambda}, \tag{1} \]

where \(\rho(\lambda)\) is the reflecting power of the body, \(V(\lambda)\) is the visibility of radiant energy with wavelength \(\lambda\), \(E(\lambda)\) is the ordi-

of the energy-distribution curve in the spectrum of the light source illuminating the colored body taken.1

In the same work cited above, Kl. Schäfer, using the old data of König for the distribution of relative brightness in the spectrum of white sunlight and the curves \(\rho(\lambda)\) for 12 Ostwald colors of the circle “ia,” calculated their luminosity. The results are shown in Fig. 11, and the agreement between the calculated and observed figures must be considered very good, especially if we take into account that König’s data are not very accurate and differ noticeably in places from the corresponding figures that can be obtained, for example, from the data of Tyndall and Gibson. Luminosity can also be measured with thermoelements, if a light filter is placed before them whose curve corresponds to the sensitivity curve of our eye.

Fig. 10.

The light filter of Coblentz and Emerson2 most accurately satisfies this condition; it is a modification of the corresponding Ives light filter and has the following composition:

(For a layer thickness of 1 cm, for the absorption of infrared rays an additional vessel with water, 3 to 4 cm thick, is used).

It should further be pointed out that the data obtained by Gibson and Tyndall come close to what had earlier, in 1918, been obtained for 125 observers by an entirely different, “indirect” method by Coblenz and Emerson,¹ also at the Bureau of Standards in Washington. They used the so-called “flicker method,” which is based on the fact, first established by Rood, that when light beams of different colors enter the eye alternately, at a low rate of alternation we clearly see the change of one color into another; when, however, the rate of flicker is increased, at first the color flickers disappear, and we see a single mixed, average color, while, if the relative brightnesses of the compared beams are unequal, this mixed color appears flickering. Flickers of this kind disappear when the relative brightness of the compared beams is the same, and their disappearance may serve as a criterion that such equality has been attained.

Fig. 11.

In the experiments of Coblenz and Emerson the brightness of spectral rays was compared by this method with the brightness of the white light of a certain standard lamp, not decomposed into a spectrum, and in Fig. 12 are shown the results they obtained: all 125 curves.

¹ Loc. cit.

plotted on a single drawing, from which it is evident that, despite the presence of strong individual deviations, one may speak of a certain average curve of “visibility” of radiant energy. In Fig. 13 a comparison is given of all the data available in the literature, obtained, on the one hand, by the method of small steps and, on the other, by the flicker method.

K. L. Schaefer made, in exactly the same way, a determination of the relative brightness of a whole series of Ostwald standards with the aid of the well-known flicker photometer of Bechstein, a description of which may be found in any textbook of photometry. In order to determine with this photometer the relative luminosity of the Ostwald colors, K. L. Schaefer pasted them in pairs onto the edges of the gypsum prism of the photometer.

Fig. 12.

In Fig. 14, where the solid curve represents the data obtained by him for the relative luminosity of the colors of the circle “ns” by the flicker method, and the dotted curve by the method of matching for greatest similarity, it is evident that both these methods lead practically to one and the same result.

Method of small steps.

• Gibson and Tyndall (52 observations).
• Hyde, Forsythe and Cady (29 observations).

Flicker method.

• Coblenz and Emerson (125 observations).
• Nutting (21 observations).
• Reeves (13 observations).
• Ives (18 observations).
• So (20 observations—Japanese).

Fig. 13.

§ 3. The next question which had to be resolved in order to construct a higher metric of color is the question of whether brightness possesses the property of additivity; in other words, can we, denoting brightness by \(H\), write the following equality:

\[ H(x_1,\ x_2,\ x_3) + H(x_1^{1},\ x_2^{1},\ x_3^{1}) = H(x_1+x_1^{1},\ x_2+x_2^{1},\ x_3+x_3^{1}) \tag{2} \]

where \(x_1, x_2, x_3, x_1^{1}, x_2^{1}, x_3^{1}\) are the trilinear coordinates of our colors?

Fig. 14.

If this equality were valid, then, as a simple mathematical consequence, the following equality would follow from it:

\[ H = ax_1 + bx_2 + cx_3, \tag{3} \]

where \(a\), \(b\), and \(c\) are certain constants.1

The investigations of E. Schrödinger,2 Kohlrausch,3 and F. Exner4 make it possible to answer this question in the affirmative.

Exner, for example, determining the lightnesses of a large number of different colored disks, then made on tops all possible mixtures of them with one another and found the lightness of such mixtures in two ways—by direct measurement and by calculation from the previously found lightnesses of the individual disks; in all cases the additivity was confirmed. The coefficients \(a\), \(b\), and \(c\) in formula (3) are most simply determined from the curve of the distribution of brightness in the spectrum of white light, which can be obtained from the “visibility” curve by multiplying its ordinates by the ordinates of the curve of distribution

of energy in the spectrum of white sunlight, closely coinciding with the corresponding curve for an absolutely black body having a temperature of about \(5000^\circ\) abs. (\(5200^\circ\) abs. according to Priest). Having compiled, for a large number of wavelengths (for example, every \(10\,m\mu\)), equations of type (3), one can determine from them, by the method of least squares, the coefficients “\(a\),” “\(b\),” and “\(c\).”

Taking the values \(x_1\), \(x_2\), and \(x_3\) from the Koenig–Ayves curves cited by us in the first article, we obtain for these coefficients the following numbers: \(a=0.568\); \(b=0.426\); \(c=0.006\).

F. Exner, for his curves, which differ somewhat from Ayves’s, and by an entirely different method, obtained coefficients very close to these: \(a=0.562\); \(b=0.425\), and \(c=0.013\). B. I. Fedorova and I succeeded in approaching the determination of the relative magnitude of the coefficients “\(a\)” and “\(b\)” by yet a third method.1 Having produced temporary color blindness for red, and then for green, by exposing the eye to extremely bright red and specially selected green spectral rays, we measured the curves of the distribution of relative brightness in the spectrum both for the first and for the second case (see Fig. 9). Then from these curves \(H_{r-bl}\) and \(H_{g-bl}\), and from the previously measured curve \(H\) for the unwearied eye, it was possible, by the method of least squares, to find the coefficients “\(a\)” and “\(b\).” (The values \(H\), \(H_{r-bl}\), and \(H_{g-bl}\) were taken within the limits from \(670\,m\mu\) to \(510\,m\mu\), in which we can, with considerable approximation, calculate \(H\) from the formula \(H=ax_1+bx_2\); the influence of the third, very small, term \(cx_3\) appears only for shorter wavelengths.) These coefficients proved to be \(a=0.82\) and \(b'=0.24\). When, however, we took the ratio of the maximum ordinates of our curves \(H_{r-bl}\) and \(H_{g-bl}\) to be not unity—which is, of course, arbitrary—but to the ratio of the maximum ordinates of the curves \(x_1\) and \(x_2\), recalculated for the energy distribution in the prismatic spectrum of our light source, we obtained for “\(a\)” and “\(b\)” the numbers 0.82 and 0.62, whence the ratio \(b:a\) proved to be 0.76 instead of 0.75 in Ayves.

It is possible, of course, to determine “\(a\),” “\(b\),” and “\(c\)” by the method of least squares from previously determined luminosities of pigment colors, for example those of Ostwald, as Kohlrausch did; in this case it is necessary to express \(x_F'\), \(x_F''\), and \(x_F'''\) (see my first article) in relative units, taking the value of the integrals \(\int x_1(\lambda)d\lambda = \int x_2(\lambda)d\lambda = \int x_3(\lambda)d\lambda\), over the entire visible spectrum, as unity. Having determined “\(a\),” “\(b\),” and “\(c\)” once and for all, we shall be able, without any heterochromatic photometry, to determine also the relative brightnesses of any colored surface from its spectrum. In those cases where we have already determined for it the values \(x_F'\), \(x_F''\), and \(x_F'''\), this method leads to the goal faster than any other. Kohlrausch showed that the figures for luminosity calculated in this way agree well with those found experimentally by the method of small steps.

§ 4. Up to now we have been dealing with settings for the greatest similarity, and now we must also acquaint ourselves with the experimental material available in the field of settings for a barely noticeable difference, with which we are concerned in studying the sensitivity of the eye to changes in chromaticity and saturation.

Without dwelling at all on the history of this question, we shall merely indicate that the sensitivity of the eye to changes in chromaticity has been carefully studied in a number of experimental investigations by Olga Steindler (Vienna),1 Jones (America),2 Hamilton and Laurens (America),3 E. Kühnert (Vienna),4 and others; and that, if along the axis of ordinates we plot the change in wavelength \(\Delta \lambda\) that is necessary in order to obtain a barely noticeable change in the chromaticity of the compared fields, we obtain a curve of the following form (Fig. 15 after Steindler). This curve has 4 minima.

(the corresponding maxima of the eye’s sensitivity to a change in chromaticity) at \(\lambda = 435\,m\mu, 497\,m\mu, 585\,m\mu\) and \(636\,m\mu\). Between them lie 3 maxima (minima of the eye’s sensitivity!) at \(\lambda = 454\,m\mu, 535\,m\mu\) and \(624\,m\mu\). The position and magnitudes of these minima and maxima vary somewhat from observer to observer, but their number always remains unchanged and the same. There exists a whole series of special apparatuses intended for studying the sensitivity of the eye to a change in chromaticity, but one may, of course, also use the above-described attachment to the Glan spectrophotometer. Our measurements have shown that the number of maxima and minima obtained in this way is correct, while their position lies within the limits of individual variations.

Fig. 15.

Undoubtedly, furthermore, the sensitivity of the eye to a change in chromaticity must be in some way closely connected with the three basic curves \(R\), \(G\) and \(B\), and, indeed, V. I. Fedorova\(^1\) has recently succeeded in showing that both the number and the position of the maxima and minima can be obtained by assuming that the sensitivity of the eye is determined by the following function

\[ \theta_1 = - \frac{1}{f'(\lambda)} \cdot \frac{d f'(\lambda)}{d\lambda}, \tag{4} \]

where \(f(\lambda)=\dfrac{x_1-x_3}{x_2-x_3}\) in the case when \(x_1 > x_3\) and \(x_2 > x_3\), and by an analogous expression for the other possible cases.

\(^1\) V. I. Fedorova. Zhurn. prikl. fiziki, Vol. 5, pp. 201—207, 1928, and Journal für Psychologie und Neurologie, Bd. 40, H. 3—4, 65, 1929.

There are considerably fewer studies in the field of the sensitivity of our eye to a change in color by saturation. The most thorough and purely executed such study was carried out by Jones and Lowry,^1 who experimentally established the relation between the saturation of a color, determined by the number of barely noticeable transitions of it into white of the same luminosity, and its purity, which they varied by adding to a saturated spectral color one or another amount of white, and determined by the formula

$$ P=\frac{H_\lambda}{H_\lambda+H_w}; \tag{5} $$

where \(H_\lambda\) is the brightness of the spectral component, and \(H_w\) is the brightness of the admixed white. In Fig. 16, where purity is plotted along the abscissa and the number of saturation steps from the color of the given purity to the spectral color is plotted along the ordinate, the curve \(S\) represents the relation thus obtained between saturation and purity for \(\lambda = 540\,\mu\mu\), and we see that for neither very small nor very large purity this relation is rectilinear, i.e., in this middle interval saturation is proportional to purity, which is very important for practice, where it is usually not saturation that is measured, but purity. On this same figure a number of other curves are plotted, obtained by differentiating curves of the first kind and representing the sensitivity of the eye to a change in saturation.

Fig. 16.

^1 L. Jones and Lowry. Journ. Opt. Soc. Amer., 13, 25, 1926.

Higher Metric of Color

§ 5. The experimental material with which we became acquainted in the preceding §§ enabled Schrödinger to approach the construction of the higher metric of color, which, however, as he himself repeatedly points out, is far less developed in comparison with the lower metric. The first problem arising here is to find a quantitative measure of the similarity or difference of any two colors, a measure of their distance in color space, which must pass through a minimum at equal brightness of both colors. If on some surface of equal brightness

\(a x_1 + b x_2 + c x_3 = \mathrm{const.}\),

we fix two points \(A\) and \(B\), then, according to what was said above, the point \(B\) must lie closer to \(A\) than any other point of the ray \(OB\), and conversely, the point \(A\) is closer to \(B\) than any other point of the ray \(OA\). It is easy to see that ordinary Euclidean geometry does not satisfy this condition. Indeed, let us take two rays \(OA\) and \(OB\) (Fig. 17), representing all possible degrees of intensity of any two colors, two vectors of the color body depicted in Fig. 6 of the first of our articles, and choose on one of them some color \(A'\). Then the color “most similar” to \(A'\) (lying closest to it) will obviously be the color \(B'\); but the most similar to \(B'\) is not \(A'\), but \(A''\). Schrödinger therefore, for the linear element of color space measuring the degree of difference between two colors, takes, as does Helmholtz, the following non-Euclidean expression:

Fig. 17.

\[ ds^2=\sum_{i=1}^{3}\sum_{k=1}^{3} a_{ik}(x_1,x_2,x_3)\,dx_i\,dx_k \quad (a_{ik}=a_{ki}) \tag{6} \]

(1st proposition of Schrödinger).

He further assumes that for any two barely distinguishable colors \(dx_i\) (and consequently also \(ds\)) must be considered-

be regarded as differentials, and that \(ds\), for each such pair of colors, has one and the same magnitude (Schrödinger’s 2nd proposition). His third proposition reduces to the fact that the distance between colors more remote from one another is determined by the length \((\int ds)\) of the shortest line connecting them (the geodesic). Let us show more clearly what these three propositions contain, taking the concepts “equidistant” and “different to the same degree” as synonymous.

We regard as equally distant from some point \(F\) of color space all “color points” only just perceptibly different from it (2nd proposition). They all lie on a certain small ellipsoid with center at \(F\) (1st proposition). We regard all axes of such small ellipsoids as “of equal length” (2nd proposition). Equidistant surfaces around \(F\), for small distances, are obtained by increasing all axes of the first ellipsoid in one and the same ratio; all these surfaces are, consequently, ellipsoids similar to it and similarly situated (1st proposition). This Euclidean construction is valid only so long as we are dealing with infinitely small quantities. In order to construct a surface around \(F\), removed from \(F\) by a finite distance, we cannot increase the axes of our small ellipsoid in their directions by the corresponding number of times, but must proceed along the shortest path in the Riemannian sense, i.e. along geodesic lines, by equal “segments” of the integral \(\int ds\), which in this case measures the distance (3rd proposition).

In order to satisfy the relation of F. Exner-Abney \((H = ax_1 + bx_2 + cx_3)\) and the Weber-Fechner law, which requires that \(ds\) should not change under a proportional change of all \(x_i\) and \(dx_i\), Schrödinger chooses the function \(a_{ik}\) so that the expression for \(ds^2\) takes the following form:

\[ ds^2 = \frac{1}{a x_1 + b x_2 + c x_3} \cdot \left( \frac{a\,dx_1^2}{x_1} + \frac{b\,dx_2^2}{x_2} + \frac{c\,dx_3^2}{x_3} \right) \tag{7} \]

Having then established a sufficient and necessary condition for integrability, and thereby also the possibility of determining

equalities in brightness of heterochromatic colors (paragraph 2 of this article), he gives the following expression for \(\int ds\):

\[ \int ds = \sqrt{ \left(\log \frac{H}{H_1}\right)^2 + 4\left( \arccos \frac{ a\sqrt{x_1x_1^{1}}+b\sqrt{x_2x_2^{1}}+c\sqrt{x_3x_3^{1}} }{ \sqrt{HH^{1}} } \right)^2 }, \tag{8} \]

where \(H=ax_1+bx_2+cx_3\), and \(H^{1}=ax_1^{1}+bx_2^{1}+cx_3^{1}\).

For the same color tone and the same saturation we obtain from this that \(s=\log \dfrac{H}{H_1}\), respectively, according to Weber–Fechner’s law. The second term under the radical in formula (8) gives us a measure of the difference of colors in tone and saturation for the same \(H\), since \(\int ds\) in this case will be equal to

\[ 2\arccos \frac{ a\sqrt{x_1x_1^{1}}+b\sqrt{x_2x_2^{1}}+c\sqrt{x_3x_3^{1}} }{H}. \]

As a natural measure for saturation, Schrödinger proposes to take the geodesic distance of a color from the equally bright white, for which \(x_1^{1}=x_2^{1}=x_3^{1}=H\). We may therefore, denoting saturation by \(S\), write that:

\[ S=2\arccos \frac{a\sqrt{x_1}+b\sqrt{x_2}+c\sqrt{x_3}}{\sqrt{H}}. \tag{9} \]

The curves of equal saturation will then be the so-called “geodesic circles” around the white point of the color triangle of equal-brightness tones, i.e. curves whose points lie at a certain constant geodesic distance from the white point. With these facts from the higher metric we shall limit ourselves here, emphasizing once again that the author himself does not yet consider his results definitive. They are, however, of great interest as the first attempt (after Helmholtz, who admitted in his constructions a single error that destroyed them)¹ to create a distinctive metric geometry of color.

¹ See on this in E. Schrödinger, loc. cit.

§ 6. Let us consider, in conclusion, the question of the construction of a rational body of pigment colors, which must lie entirely within the cone of spectral colors \(OR'G'V'\) (Fig. 6 of the first article). We shall not dwell here on Ostwald’s color system, since an article by Kl. Schäfer1 has already been devoted to it in this journal, and we shall briefly consider only one of the schemes for constructing a color body according to P. Luther.2 In Fig. 18 a photograph is given of a model of such a color body, and in Fig. 19 one of its sections, passing through the green color \((500\,m\mu)\) and its complementary purple.

Fig. 18.

The surface of this body is formed by the optimal Schrödinger colors; at the bottom lies black, at the top—white. In the vertical direction the luminosity changes, in the horizontal direction the saturation changes. The dotted lines indicate the vectors of the body \(OR'G'V'\). The point “\(X\)” shows the position in this body of the color emerald, whose spectrum was given in the first article (p. 109). Saturation (more correctly, purity) of this color is determined by the ratio \(\dfrac{UX}{US}\), which,

Fig. 19.

HIGHER METRIC OF COLOR

as is easy to see from the similarity of the triangles \(OUS\) and \(OKP\), is equal to \(\dfrac{PK}{PX}\), i.e., equal to the ratio of the luminosity of the pure spectral component to the total luminosity of the color. The optimal color,

Fig. 20.

corresponding in color and saturation to the color of emerald, but having greater luminosity, lies on the surface of the body, at the point \(O_x\), and we can, obviously, obtain our color by darkening the corresponding optimal one. The construction of such a color body for practice—for example, for the textile industry—is at the present time, when

we can use for this purpose the apparatus of Resch,^1 and this, for example, is quite a real and feasible task, provided physicists and practical colorists work jointly.

In America, as early as 1915, Munsell^2 developed a practical system of colors arranged according to their hue, lightness, and saturation (purity). He also issued an atlas of colors containing, unfortunately, only 10 basic colors. In 1929 the “Munsell Color Company” issued a new edition of this atlas, containing already 20 basic colors.

An idea of this system may be obtained from Fig. 20, where in the center a model of the Munsell color solid is shown, and in the corners four sections of this solid by planes passing through the axis. The height of the color here determines its lightness, and the distance from the achromatic axis—its saturation (purity).

It should be pointed out that, for some of its colors, the Bureau of Standards in Washington measured the spectra and calculated their lightnesses by formula (8); moreover, it turned out that colors having practically identical lightness do indeed lie on one horizontal level.^3 It follows likewise from this that, along a radius from the periphery toward the center, we are dealing with a change in color only with respect to saturation.

Unfortunately, in this system the concept of optimal pigment colors could not be used, without which the construction of a rational color solid is impossible.

^1 See our first article.

^2 Munsell. A Color Notation. 7th ed., 1926. Atlas of the Munsell Color System, 1915. New edition of the atlas: “Book of Color,” 1929 (“Munsell Color Company”).

^3 I. Priest, K. Gibson and H. Nicolas, Techn. Pap. B. of Stand. No. 167, 1920.

1. “Usp. fiz. nauk,” Vol. VII, issue 5, 375, 1927. ↩↩↩↩↩↩

2. R. Luther. Zts. f. techn. Phys. No. 12, 540, 1927. ↩↩↩↩

3. Laurens and Hamilton. Amer. Journ. of Physiol. 115, 3, 1923. ↩↩

4. Ed. Kühnert. Dissertation in manuscript (Vienna, 1924). ↩↩