COLOR METRICS.

Theory and Techniques of Colorimetry.

N. T. Fedorov, Moscow.

§ 1. The foundations of the entire well-ordered edifice of the modern measuring doctrine of colors were laid by Newton¹, strengthened and expanded by Maxwell² in his classic memoirs on color mixing, the first of which appeared in 1855.

The further development of this science through the classical investigations of Helmholtz³, König and Dieterici⁴, Fr. Exner⁵, Ives⁶ and others leads to the four remarkable memoirs of E. Schrödinger⁷, who for the first time showed with extraordinary clarity that the laws of color mixing can be established quite independently of the question of the brightness of different colors, forming a completely self-contained complex of facts, which he named

¹ Newton. Optics (1704). Russian trans. by Prof. S. I. Vavilov, p. 107 ff. GIZ, 1927.

² Maxwell. Scient. Papers, 1890.

³ Helmholtz. Hand. d. Physiol. Opt. (2nd ed.), 1896.

⁴ König und Dieterici. ZS. f. Psych. und Physiol. d. Sinnesorgane 4, S. 241, 1892.

⁵ F. Exner. A series of communications in “Berichte d. Wiener Akad.” from 1902 to 1922.

⁶ Ives. Jour. of the Frankl. Inst. 195, 25, 1923.

⁷ E. Schrödinger. Ann. d. Phys. 63, 397, 427, 481 and 62, 603, 1920. See also Müller-Pouillet, Lehrb. d. Ph., vol. II.

lower metric of color, as opposed to the higher metric of color, which embraces questions of heterochromatic photometry, of the eye’s sensitivity to changes in chromaticity, etc.

Closely adjoining Schrödinger’s works are the investigations of Guild and Rösch, who constructed very simple and ingenious instruments for the colorimetric analysis of the colors of transparent and opaque bodies.

The study of colors may be presented either as a chapter of physiology and biological physics1, or as a chapter of measurement physics. In our survey we shall present it precisely from this latter point of view, without in any way touching upon the question of the essence of our color vision or the physiological processes underlying the facts set forth by us.

Unlike other branches of measurement physics, where we always deal only with spatio-temporal coincidences (of a pointer, a spot of light, the top of a mercury column with one or another division of a scale, of the image of a crosshair with the image of one or another star, etc.) or, finally, with the temporal coincidence of two such spatial coincidences, etc., in every measurement of color we have, in addition to arrangements of this kind, at least one arrangement of an entirely special sort. In the simplest case this arrangement consists in the fact that two adjacent color fields, at some definite value of one or another variable parameter of the apparatus (the angle of rotation of a Nicol prism, the size of a diaphragm, the position of a collimator, etc.), become for us indistinguishable with respect to their color—in the fact, consequently, that we ascertain a coincidence in color together with one of the coincidences of the first kind. In other cases, as, for example, in the study—

our ability to distinguish chromaticity, we, on the contrary, proceed from a state of complete equality, changing one of the physical parameters until we obtain a barely noticeable difference of the fields. Further, in some methods of heterochromatic photometry, for example, we are dealing with a setting not for equality of the fields, but for their greatest similarity, and, finally, there is also a possible setting for the greatest difference, the greatest contrast. These four kinds of settings exhaust all possible cases.

The results of measurements of the first kind obey extremely simple regularities, established formally already by Grassmann; moreover, the system of these regularities—the usual laws of color mixing—Schrödinger calls the lower metric of color, in contrast to the much more complicated and much less studied laws of the higher metric of color, which deals with the results of measurements of the remaining kinds.

To the lower metric of color, with which we are constantly concerned in colorimetric practice, our first article will be devoted.

§ 2. We know that visible light consists of all possible mixtures of rays whose wavelengths lie within the limits of approximately from 400 to 750 \(m\mu\). The diversity of possible mixtures made up of these simple rays is infinitely great, since each kind of ray, taken at any intensity, may be mixed with all the others, each of which in turn may be taken at any intensity. On the other hand, experience teaches us that the series of color tones known to us is exhausted by the series of spectral colors, if to it we add also mixtures composed of the extreme colors of the spectrum (red and violet), forming the series of so-called purple tones. All colors known to us, however complicated from the physical point of view they may be in their composition, we can obtain by mixing, in one or another proportion, some spectral (or purple) color with white and varying the intensity of the resulting mixture (“setting for equality”).

This spectral (or purple) color is called the hue of our color; moreover, depending on the amount of white added, one speaks of a greater or lesser “saturation” or “purity” of the color.

For the quantitative determination of the color hue, purity, and relative brightness of any color, the American physicist Nutting1 constructed a special instrument, which he called a monochromatic colorimeter. The scheme of this instrument is extremely simple. It is (Fig. 1) a spectroscope with collimator \(A\), prism \(P\)

Fig. 1.

and telescope \(D\), the eyepiece of which has been removed and replaced by a slit. If the Lummer–Brodhun cube is removed, then the eye, placed at the slit, will see the prism \(P\), illuminated by monochromatic light.

A beam of rays of white light is introduced by means of collimator \(B\), and these rays are reflected from the surface of the prism and joined to the spectral ones. The analyzed light passes through a third collimator onto the Lummer–Brodhun cube and is reflected by it into the observer’s eye. The observer thus sees the middle part of the cube, illuminated by a mixture of spectral rays \(+\) white, and the periphery, illuminated by the light under investigation. By changing the wavelength, the colli-

the amount of white added to the spectral rays and, by increasing or decreasing the brightness of the mixture, one can obtain in the field of view a color identical with the sample being analyzed and from this determine the corresponding hue \((\lambda)\), purity, and relative brightness.\(^{1}\)

In Fig. 2 the actual scheme of this instrument is given in the form in which it has been put on sale by the firm A. Hilger.\(^{2}\)

It is more obvious that, since the spectral colors together with the purples form a certain closed one-dimensional sequence, they, together with their unsaturated, pale shades, form a certain two-dimensional manifold, which may be compared with the points of a certain plane, at least of a circle, at the center of which lies white. If, moreover, we add to this that we can vary arbitrarily the intensity of each (both saturated and pale) color, then it is evident that the totality of all colors forms a certain three-dimensional manifold, comparable with the points of some solid, for example a cone, whose vertex corresponds

Fig. 2.

\(^{1}\) Details on methods for determining relative brightness will be given in the second article, devoted to the higher metrology of color.

\(^{2}\) A similar instrument was recently constructed by Prof. V. V. Shuleikin.

objective darkness, whereas each generator represents some spectral (or purple) color in all possible degrees of its intensity, and each half-line going from the vertex into the cone corresponds to various degrees of intensity of some more or less pale shade of one or another saturated color (the axis of such a cone would represent the white achromatic color, whose intensity would increase with distance from the vertex).

Fig. 3.

After all that has been said, the following definition of color will also be understandable to us (according to Schrödinger): radiant energy entering our eye and producing one or another sensation of color is some function of the wavelength. All these functions \(f(\lambda)\) can be divided into large groups in such a way that radiant energy belonging to one of such groups, when falling on the retina of our eye (in fovea centralis, ee), produces one and the same color sensation.

Fig. 4.

Examples:

1) White sunlight and white obtained by mixing two complementary colors (Fig. 3).

2) Light of the green thallium line + light of the red lithium line and light of the yellow sodium line + a little white (Fig. 4). Each such group \(f(\lambda)\) we shall call a color. The word “mixing” or “addition,” as applied to radiant energy, denotes simply the superposition of several beams of rays. Denoting the wave function of one of such beams by \(f(\lambda)\), and of another by \(\varphi(\lambda)\), we can

denote their sum or mixture as follows: \(f(\lambda)+\varphi(\lambda)\). The fundamental empirical fact underlying everything that follows is the possibility of speaking likewise of the mixing or addition of colors. A priori it would seem that in this case the operation of addition cannot be unambiguous, since, as was indicated above, one and the same color may be composed in the most diverse ways. Experience shows, however, that this is not in fact the case, and that for determining the color of a mixture only the color of the components matters; this was expressed by Grassmann in the form of the following proposition: “Colors that look the same give, when mixed, mixtures that look the same” (Grassmann’s third law). This experimental fact—and this alone—allows us to construct colorimetry while paying no attention at all to the physical composition of the light that gives rise to the given color sensation. As symbols for denoting color we shall use capital Latin letters, and as the symbol for denoting the mixing of colors the sign \(+\) \((A+B;\ A+B+C+D\), etc.).

To denote that two colors look the same, we shall use the sign \(=\). If some color \(L\) can also be obtained by mixing the colors \(A\) and \(B\), we may write symbolically:

\[ L=A+B \]

(for example, Grassmann’s third law can be written as follows: if \(A=A_1\) and \(B=B_1\), then also \(A+B=A_1+B_1\)).

Similarly we can define “subtraction”1, and “multiplication” by some number, and “division” of a color; moreover, the possibility of multiplication by any positive number is already contained in Grassmann’s second law, which Schrödinger formulates in the following way: “If the light (radiant energy acting on our eye) changes continuously, then its color changes likewise.” Further experience teaches us that for a person possessing normal color—

COLOR METRIC

... vision, the manifold of all colors is a manifold of three dimensions, i.e., three linearly independent colors can be found, but four colors will always be related to one another. If \(A\), \(B\), and \(C\) are three such linearly independent colors, and \(F\) any fourth one, then we may write:

\[ \alpha A+\beta B+\gamma C+\xi F=0, \]

whence

\[ F=x_1A+x_2B+x_3C. \]

All the properties of color listed here formally coincide with those axioms which underlie the affine geometry of a bundle of vectors issuing from some point. The manifold of colors, or in other words the color space, is therefore a certain three-dimensional image having—from the standpoint of settings of the first kind—an affine structure (in the higher metric of color, which deals with settings for greatest similarity or barely noticeable difference, the color space already has its own peculiar metric structure).

Among the experimental works in this field we shall mention first of all Maxwell’s investigations1, who, taking as primary colors spectral red with wavelength \(630\,\mu\mu\), green—\(528\,\mu\mu\), and blue—\(457\,\mu\mu\), showed that indeed all the remaining colors are linear functions of these primaries. It turned out, however, that the possibility of obtaining all colors by mixing three primaries is valid only with certain restrictions. Thus, for example, although from the three colors taken by Maxwell one can obtain all spectral tones, their saturation will be less than in the spectrum. Yellow, for example, obtained by mixing red and green, always proves to be less saturated than in the spectrum. Adding to the mixture a third color—blue—would only make matters worse, reducing the saturation still further. The same applies also to mixtures intermediate between green and blue: they are obtained as whitish in comparison with the spectral ones, and approx—

are drawn to white still more, when red is added. In order to obtain complete identity in the color of a mixture of two primary colors (for example red and green)—to set up the so-called color equation—with some intermediate spectral color (yellow in our example), it is necessary to add to this intermediate color a certain quantity of the third primary color.

Symbolically this may be written as follows:

\[ F + x_{3}B = x_{1}R + x_{2}G \]

where \(R\), \(G\), and \(B\) are the three primary colors, and \(F\) is some fourth color to be determined; the equality sign indicates the identity of the color of both mixtures. In this “color equation” \(x_{1}\), \(x_{2}\), \(-x_{3}\) are called the coordinates of the color \(F\) with respect to the primary colors \(R\), \(G\), and \(B\). It should be emphasized that only saturated spectral colors can have a negative coordinate, whereas their mixtures with white (the great majority of the colors of bodies) are characterized by three positive numbers.

Further, from Grassmann’s second law it follows that, when any two beams of light are mixed, their corresponding coordinates are added—red with red, green with green, blue with blue. These coordinates, therefore, characterize not only the color of a given light taken separately, but also its influence as a constituent part of some mixture. Therefore, having “graduated” the spectrum once and for all, i.e., having determined the coordinates of all spectral colors, we shall be able in advance to predict the color of any, however complex, mixture of light rays, knowing only its physical composition and determining its color coordinates by summation or by graphical integration, as will be shown below.

After Maxwell, the “graduation” of the spectrum with considerably greater accuracy was carried out by König and Dieterici, who above all took into account the inexpediency of стрем-

lations obtain any spectral color by mixing the three primaries in various proportions, since experiment shows that when two colors lying far apart in the spectrum are mixed (for example, red and green), the result always contains a considerable amount of white, which makes difficult and inaccurate—owing to the low sensitivity of our eye to differences in the color of such pale colors—the establishment of equality of the visual fields. Their method is therefore reduced to the following.

Let us suppose that we are given \(n\) points of the spectrum, for which it is necessary to find the coordinates \(R'\), \(G'\), and \(B'\) (as in Maxwell), or \(R'\), \(G'\), and \(V'\) (as König and Dieterici took them). For this it is by no means necessary to set up equations connecting \((n-3)\) of the spectral colors given to us with the three colors we have adopted as primaries. It is quite sufficient to set up \((n-3)\) equations connecting the various spectral colors with one another in groups of 4; moreover, these 4 colors should be chosen so that the accuracy of the measurements is greatest. Thus, for example, one chooses three spectral colors lying not far from one another and, in order to obtain an identity of color between a mixture of the two extreme ones and the middle color, a small amount of the color complementary to it, or close to such a color, is added to the latter, as a result of which this third color becomes somewhat less saturated and can be equated to the mixture of the first two. From these \((n-3)\) equations one can then, by calculation, find the required coordinates \(R'\), \(G'\), and \(V'\) of all \((n-3)\) colors of the spectrum. If the extreme red, the extreme violet, and spectral green \((\lambda = 505\,\mu\mu)\) are chosen as the primary colors, then the results of König and Dieterici’s measurements can be represented graphically by plotting wavelengths along the axis of abscissas, and \(x_1\), \(x_2\), and \(x_3\) along the axis of ordinates (Fig. 5). Starting from these curves, we can already construct the color body more accurately. For this purpose we take any three, proce—

Fig. 5.

...vectors \(R'\), \(G'\), and \(V'\) issuing from a certain point, and we assume quite arbitrarily that they correspond to certain “unit quanta” of the three colors we have chosen as fundamental: red, green, and violet (Fig. 6). Any other color will then correspond to a vector that is the sum

\[ x_1 R' + x_2 G' + x_3 V'', \]

where one of the coefficients may be negative. Each color will, consequently, be uniquely determined by the vector corresponding to it, or simply by the point of space corresponding to the tip of this vector. An approximate representation of the form of such a color body can be obtained from Fig. 6, in which, for greater clarity, there is also given a section of this body by a plane cutting off equal segments of the three fundamental vectors.

Fig. 6.

The generators of the cone passing through the curve \(R'G'V'\) correspond to spectral colors; the vectors lying in the plane \(OR'V'\) correspond to purple mixtures, and the vector \(OW\) to white. From this same drawing we see that the vector corresponding to the mixture of any two colors is coplanar with the vectors of these colors. Its intersection with the plane \(R'G'V'\) therefore lies on the line joining, in this plane, the points of intersection with it of the vectors of the component colors, and always between them. Between white and one of the...

saturated colors of the surface of the cone therefore include the pale, unsaturated shades of the latter. When mixing the primary colors pairwise (\(R'\) with \(G'\) or \(G'\) with \(V'\)), owing to the form of the curve \(R'G'V'\), we shall obtain not the spectral colors in their full saturation, but only these unsaturated shades. The complementary colors lie in one plane with the vector \(OW\), which in this case must pass between them.

In order to get rid of negative coefficients in the color equations, instead of the vector \(OG^1\) one may, as was done by König and Dieterici, take some other vector \(G''\), lying outside the color cone and obtained by intersecting the planes tangent to the extreme rectilinear segments of the curve \(R'G'V'\). This new vector, lying outside the color cone, will evidently correspond to a certain unreal green color, more saturated than is found in the spectrum, an idea of which we can obtain by fatiguing our eye with red and then looking at the green part of the spectrum. The transition from one system of “primary” vectors to another is carried out purely formally, without any new measurements. In this new system of “primary” vectors all the coefficients \(x_1\), \(x_2\), and \(x_3\) will always be positive, and the curves \(R''\), \(G''\), and \(V''\) will take the following form (Fig. 7). (In constructing these curves, König and Dieterici quite arbitrarily assumed that for white \(R'' = G'' = V''\),

and therefore the areas bounded by these curves and by the axis of abscissas have been made equal by them).

In this choice of the fundamental colors there is, of course, a certain arbitrariness. Instead of the vectors \(R'\), \(G''\), and \(V''\) one may take any three that would not give negative values of \(x_1\), \(x_2\), and \(x_3\), and for this reason König and Dieterici, on the basis of a study of the phenomena of color blindness, made an attempt to find three such fundamental vectors as would correspond to certain real, physiological features of our eye.

It is known that among the color-blind the most frequently encountered are the so-called red- and green-blind, or, in Kries’s terminology, protanopes and deuteranopes. For both the former and the latter it is characteristic that any spectral color given to them, say, on one field of a Helmholtz apparatus for mixing colors can be reproduced for them with perfect accuracy on the other field of this instrument by means of a mixture of the extreme rays of the spectrum: red and blue-violet. Their world of colors has, consequently, only two dimensions. Further, both for the protanope and for the deuteranope there is characteristically present in the spectrum a certain narrow neutral zone, which appears to them achromatic, gray; for protanopes this point lies at about \(495\,\mu\mu\), and for deuteranopes at about \(505\,\mu\mu\). By studying quantitatively their mixing of spectral rays, it was possible to determine the relative amount of red and blue-violet rays required for reproducing the remaining colors of the spectrum. To each spectral color, say (Fig. 6) “\(b''\),” there will correspond for them a certain color “\(l''\)” on the line \(R'V'\). Drawing the planes \(Oa_1\), \(Ob_1\), and so on, we shall see that, for the protanope, they all intersect along a certain line \(OR\). This vector will evidently correspond to the red color missing in the protanope, since adding it to the vectors characterizing the spectral colors does not change them. Having made the same construction for the deuteranope, we obtain the vector \(G\), corresponding to the green color missing in the deuteranope. The position of the vector \(B\), corresponding to the fundamental blue-violet color, which

would be possible to determine from the study of cases of the corresponding color blindness, is still determined insufficiently accurately, and therefore many (for example, the American Optical Society) prefer to use the primary colors \(R'\), \(G'\), \(V'\), and not \(R\), \(G\), and \(B\).

Formally, however, all these systems are entirely equivalent, since they are obtained from one another by simple mathematical transformations. The curves of König and Dieterici (with Ives’ corrections\(^1\)) for white daylight\(^2\) in these new coordinates are shown in Fig. 8; the numerical value of the ordinates at intervals of \(10\,m\mu\) is given in Table I (see p. 106).

Despite the fact that the spatial representation of colors is the most natural, in practice a simpler method is used, localizing all colors in a single plane and introducing the third measurement of color—its relative brightness—by means of a certain empirical formula connecting this coordinate of color with the quantities \(R\), \(G\), and \(B\).

Fig. 8.

Such a two-dimensional representation of the manifold of colors we obtain by taking some arbitrary plane section of the color solid, usually called a color

\(^1\) Ives. J. Frankl. Inst. Vol. 195, 1923. These curves themselves, still not very accurate, agree with experiment better than all the remaining variants. A new determination of these curves for a whole series of persons appears to be highly desirable.

\(^2\) The curves \(R\), \(G\), and \(B\) for any other light source are obtained by dividing the ordinates of these curves by the ordinates of the energy-distribution curve in the spectrum of daylight and multiplying the quotient by the corresponding ordinates of the energy-distribution curve in the spectrum of the other source.

TABLE I.

triangle (the triangle $RGB$ in Fig. 6). All colors which, in the spatial representation, lie on one half-line passing through the point $O$, i.e. all possible degrees of intensity of a certain definite color, we localize at one point of the color triangle, through which this half-line passes. All colors possessing the same color tone and the same saturation will, consequently, be characterized by a certain, quite definite ratio of their coordinates. These

coordinates can therefore be considered geometrically as projective (trilinear) homogeneous coordinates on the plane. As is known, a plane trilinear system of coordinates consists of a certain coordinate triangle and a “unit point,” with both the former and the latter chosen arbitrarily. In the theory of colors, ever since Maxwell, the center of gravity of the triangle has usually been chosen as the “unit point,” and, moreover, the triangle itself is taken to be equilateral. This special choice of the coordinate triangle entails that the point corresponding to white (the unit point) is at the same time the center of gravity and the geometric center of the coordinate triangle. In consequence of choosing the center of gravity as the unit point, the position in the triangle of some color with coordinates \(x_1, x_2\), and \(x_3\) coincides with the center of gravity of the coordinate triangle at whose vertices the weights \(x_1, x_2\), and \(x_3\) are placed. Hence there follows a very simple method for finding the place of a given color in the color triangle.

Fig. 9.

One makes, for example, the point \(R\) coincide with the origin of a rectangular coordinate system \(\xi\eta\), directing the \(\xi\)-axis along \(RB\). Then, in order to determine the position of the color \((x_1, x_2, x_3)\), we must find the coordinates \(\xi\) and \(\eta\) of the center of gravity of the system of masses \(x_1, x_2, x_3\), applied at the points \(R, G\), and \(B\). If the side of the triangle is equal to \(l\), then these coordinates are determined by the formulas:

\[ \xi = \frac{1}{2}l \cdot \frac{x_2 + 2x_3}{x_1 + x_2 + x_3} \quad \text{and} \quad \eta = \frac{1}{2}l \cdot \frac{x_1\sqrt{3}}{x_1 + x_2 + x_3}. \]

The position of the spectral colors in such a color triangle is shown in Fig. 9, the values of \(x_1, x_2\), and \(x_3\)

taken from the König–Aïssa curves. On this same triangle a grid of trilinear coordinates has been drawn, allowing \(x_1, x_2\), and \(x_3\) to be determined even without the mediation of the Descartes coordinates \(\xi\) and \(\eta\) (for this it is only necessary to express \(x_1, x_2\), and \(x_3\) in such units that \(x_1+x_2+x_3=l\). Then, laying off \(R\) along the side \(GR\) in the direction indicated by the arrow, \(G\) along the side \(BG\), and \(B\) along the side \(RB\), and drawing lines parallel respectively to the sides of the triangle \(GB\), \(RB\), and \(GR\), we shall find the required color at the intersection of these lines).

The point determined in the figure by the intersection of the three heavy lines has the coordinates: \(R=7.67\), \(G=7.29\), and \(B=5.03\).

§ 3. The problem of practical colorimetry is to find the position of any color given to us within the manifold of other colors. In practice we usually deal with the colors of various bodies (fabrics, paper, glass, solutions, etc.), whose spectra contain rays of every possible wavelength, but in proportions—compared with white light—such as is seen, for example, from Fig. 10, where reflection spectra are given for a number of standardized colors of Ostwald’s atlas, measured by Kohlrausch in Vienna; along the ordinate axis the values of \(\frac{I}{I_0}\) are plotted. If we know the spectrum of some body and do not have at hand any of the colorimetric instruments that will be described below, then we can find the coordinates \(x_1, x_2, x_3\) in the following way. Denoting \(\frac{I}{I_0}\) by \(\rho(\lambda)\), we obtain the coordinates of our color by calculating three such integrals:

Fig. 10.

\[ x_f'=\int x_1(\lambda)\rho(\lambda)\,d\lambda;\qquad x_f''=\int x_2(\lambda)\rho(\lambda)\,d\lambda \quad \text{and} \]

\[ x_f'''=\int x_3(\lambda)\rho(\lambda)\,d\lambda. \]

Finding these integrals, for which one must form the products \(x_1(\lambda)\rho(\lambda)\), \(x_2(\lambda)\rho(\lambda)\), and \(x_3(\lambda)\rho(\lambda)\) over the entire spectrum and integrate them (by weighting, with the aid of a planimeter, according to Simpson’s formula, or simply by summing the ordinates at every \(10\,\mu\mu\)), takes a fair amount of time and can be considerably simplified by means of the following method, proposed by Prof. R. Luther.1

Having once and for all compiled tables of the values of the integrals:

\[ \int_{700}^{\lambda} x_1(\lambda)\,d\lambda;\qquad \int_{700}^{\lambda} x_2(\lambda)\,d\lambda \quad\text{and}\quad \int_{700}^{\lambda} x_3(\lambda)\,d\lambda \]

for the entire visible spectrum at every \(10\,\mu\mu\), we deform the axis of abscissas, plotting the wavelengths not uniformly, but at distances from the origin of the coordinates (\(700\,\mu\mu\)) proportional to the written integrals, as is shown in Fig. 11. If in these coordinate grids we plot the curve \(\rho(\lambda)\), then the areas of the resulting curves will be proportional respectively to \(x_1\), \(x_2\), and \(x_3\) (at the top of Fig. 11 is given the spectrum of emerald according to F. Exner, and the same spectrum deformed according to Luther. The hatched areas are equal to the areas bounded by the axis of abscissas and the continuous curves).

Fig. 11.

However, even this method proves in practice still too complicated, requiring for each sample knowledge of its spectrum. Much more quickly do colorimetric methods lead to the goal, based on matching the color of a given sample by a mixture of several colors once and for all preca—

calibrated standards. The simplest of these methods is a modification of Maxwell’s method1 and requires only a small motor and three disks: red, green, and blue-violet (Maxwell colored the red disk with cinnabar, the green with emerald green, and the blue with ultramarine. One may, of course, use any paints of the corresponding colors).

These disks must be carefully calibrated, i.e., for each of them the reflectance must be measured, and then the following computed:

\[ x_{f_1}',\ x_{f_1}'',\ x_{f_1}''';\quad x_{f_2}',\ x_{f_2}'',\ x_{f_2}'''\ \text{and}\ x_{f_3}',\ x_{f_3}'',\ x_{f_3}'''. \]

The sample whose color we wish to analyze is pasted onto a small cardboard disk, which is likewise cut along a radius. These disks are mounted on the common axis of the motor, as shown in Fig. 12; moreover, in order to obtain, during rotation, complete identity of the colors of the large and small disks, it is sometimes necessary to add, by sector, white and black, either both to the small disk or black to the large one.

Fig. 12.

We vary the relative magnitude of the sectors of the three standards, of the disk being tested, and of the white and black disks until, during rotation, the color on the outside and on the inside becomes completely identical. Let this be so if we take \(a^\circ\) of red, \(b^\circ\) of green, \(c^\circ\) of blue, \(d^\circ\) of the color being analyzed, \(y^\circ\) of white, and \(z^\circ\) of black, in the small disk. We may then write that our analyzed color plus a certain amount of achromatic color will contain:

\[ \begin{aligned} d x_f' + y x_w &= a x_{f_1}' + b x_{f_1}'' + c x_{f_1}''';\\ d x_f'' + y x_w &= a x_{f_2}' + b x_{f_2}'' + c x_{f_2}''';\\ d x_f''' + y x_w &= a x_{f_3}' + b x_{f_3}'' + c x_{f_3}'''. \end{aligned} \]

after which we can also determine the position of the color in the color triangle. Instead of adding white we can, on the basis of what was said above, make the large circle from two standards, and add the third to the small circle being analyzed.

Much more accurate results can be obtained with one of the colorimeters of the English physicist Guild, or with the aid of Z. Révész’s very simple and ingenious instrument.

In the colorimeter, called by Guild the trichromatic one, we equate the color given to us to an optical mixture of three primary colors (red, green, and blue), to which one may also add, if necessary, a certain amount of white. The diagram of this Guild instrument is given in Fig. 13.

Fig. 13.

A parallel bundle of rays from the lamp shown on the left falls on the left wall of a box containing three sector-shaped apertures, each of about 59°. These apertures can be covered with red, green, and blue gelatin light filters. On the inner side of this wall is a prism \(CD\), which, by means of a small motor, can rotate rapidly in front of the wall about an axis passing through \(D\). During rotation the reflecting surface \(C\) moves before the apertures; the light passing through them is reflected at \(C\) and \(D\), and travels in the direction \(D\). This light, after passing through a lens, falls on a Lummer–Brodhun cube and, after total internal reflection, into the eye of the observer looking through a telescope set at infinity. With sufficiently rapid rotation of the prism \(CD\), the photometric field appears colored in a homogeneous color,

corresponding mixture of the colors of the three sectors. By selecting, on the one hand, suitable light filters and changing, by means of movable shutters, the magnitudes of the sectors, one can obtain any color. The rays from the colored object being analyzed, after passing through lens \(L\) and falling on the Lummer–Brodhun cube, illuminate another part of the field of vision.

The above-mentioned reduction in the saturation of the color under investigation can be effected in the following way. With the aid of prisms with total internal reflection \(L\) and \(M\), and of a plate of mirror glass placed at an angle of \(45^\circ\), white light from the lamp is mixed with the light beam under investigation. The intensity of this added light can be varied in a measurable way by means of a circular gray wedge \(N\). Instead of adding white light, it is also possible to add to the color being analyzed a color corresponding to one of the three primary light filters, the corresponding light filter then being placed behind \(M\). As for the choice of light filters, in this respect there is wide freedom. It is only necessary that the color of one of them cannot be composed from the colors of the other two. For practical purposes the light filters of the firm Wratten are convenient, their colors corresponding to wavelengths of \(630\,\mu\mu\), \(537\,\mu\mu\), and \(450\,\mu\mu\). It goes without saying that for the light filters the quantities \(x_{f1}'\), \(x_{f1}''\), \(x_{f1}'''\), \(x_{f2}'\), \(x_{f2}''\), \(x_{f2}'''\), and \(x_{f3}'\), \(x_{f3}''\), \(x_{f3}'''\) must be determined in advance. (At present the three-color colorimeter of Guild is being marketed by the firm A. Hilger.)

Another colorimeter, constructed by the same Guild and called “vectorial,” is based on a completely different principle. We saw above (p. 106) that all spectral colors are arranged in the color triangle along a certain curved line \(RGV\). If we are given some color \(C\), lying inside the color triangle (Fig. 14), then we can connect it by straight lines with two points lying near the ends of the curve of spectral colors, say with the points corresponding to wavelengths \(\lambda = 670\,\mu\mu\) and \(\lambda = 470\,\mu\mu\). Extending these straight lines beyond the point \(C\), we find their two other intersections with the curve of spectral colors, in two other,

corresponding, let us assume, to wavelengths at the 496 and 586 mµ points.

If we did not know in advance the position of the color in the color triangle, then it will be found when we determine two such spectral rays (586 mµ and 496 mµ) which, being mixed respectively with 470 and 670 mµ, will give us our color \(C\). This sought color will then lie in the color triangle at the intersection of the lines joining the points \(470\,m\mu\)—586 and \(670\,m\mu\)—496 mµ. For practical application of this method, Guild constructed an apparatus, the scheme of which reduces to the following (Fig. 15). The spectroscope has two collimators \(C_1\) and \(C_2\); in front of their slits a gray wedge \(W_1\) and \(W_2\) is placed. The light source is a point

Fig. 14.

Fig. 15.

lamp illuminating both slits. In front of \(C_2\) there is placed either a red or a blue light filter \(F\). First, let it be, say, red. The red light is reflected by a plane-parallel

Proceedings of the Physical Sciences, vol. IX, issue 1.

with the plate \(G\) into the viewing tube of the spectral apparatus and, after a twofold reflection in the prism \(P\), enters the photometric part of the instrument proper, shown in the figure on the right. The Lummer-Brodhun cube is illuminated on one side by light coming from \(P\), and on the other—through \(L_5\)—by light whose color we wish to analyze. \(C_4\) is the viewing tube through which the observations are made. As long as the collimator \(C_1\) is closed, the two fields of the Lummer-Brodhun prism will in general have different coloration. By opening it, however, and rotating the prism in the proper manner, we can find such a wavelength at which both fields will have the same color. After this the red light filter is replaced by a blue one and the required wavelength is again found. Having determined in advance the position of the colors of both light filters in the color triangle, from these two experiments we find the position in the triangle of the tested color given to us, as was shown above.

Fig. 16.

Quite recently (in 1928) another very simple and ingenious instrument was described, constructed by S. Rösch1. This instrument is based on the idea of the so-called optimal pigment colors, expressed by E. Schrödinger in 19202. These optimal colors possess the property that, for a given hue and saturation, they are the brightest, and for a given hue and lightness—the most saturated. The optimal colors may have a spectrum of one of four types (Fig. 16), i.e., in other words:

1) their reflecting or transmitting capacity may have values of 0 or 1,

2) it can have, over the entire extent of the spectrum, only two jumps (from 1 to 0 and from 0 to 1).

If we are given some color sample, we can always find two such wavelengths \(\lambda_1\) and \(\lambda_2\) such that the segment of the spectrum bounded by them, when mixed, gives a color identical with this sample in hue and saturation. Since, however, optimal colors are limiting, practically unattainable ideal cases, in order to obtain identity we must, to one degree or another, weaken the intensity of such an optimal color, making the ordinates of its spectrum equal not to 0 and 1, but to 0 and \(a < 1\). The triad \((\lambda_1,\lambda_2\) and \(a)\) exhaustively characterizes any color, and \((1-a)\) essentially corresponds to what W. Ostwald calls the “content of black in the color.” Fig. 17 shows the actual spectrum of a certain green paint and its metameric deformation according to E. Schrödinger.

Fig. 17.

Reesch’s apparatus (now manufactured by the firm C. Zeiss) makes it possible, extremely simply and quickly, to find \(\lambda_1\) and \(\lambda_2\), and \(a\), and to pass from this coordinate system to the three-color system.

Fig. 18.

The diagram of this apparatus is shown in Fig. 18 and, in its essential features, reduces to the following.

The analyzed opaque sample, placed at \(P\) and illuminated by some light source \(L\), is viewed through lens \(O\) and Hüfner prism, filling with its color one half of the field of view. At \(P\) there is placed a matte white ...

plate which directs the rays of the same light source \(L'\) into the spectroscope, the grating of which is denoted by the letter \(G\), and the slit by \(Sp\). The actual spectrum of the lamp is obtained in the plane \(Sch\), where there is a movable template that makes it possible to cut out any part of the spectrum and has the form shown in Fig. 19 (\(a\) or \(b\)). Template \(a\) can be moved in two mutually perpendicular directions (from right to left and from top to bottom); template \(b\) moves only in the horizontal direction, while the vertical displacement is here replaced by rotational motion about an axis passing through the center of the disk. In \(Z\) there is a cylindrical lens, which connects the rays cut out by the template in such a way that the corresponding half of Roffner’s prism appears colored in a certain homogeneous color of their mixture. It is easy to see that with the aid of these templates we shall be able to reproduce all four types of spectra of optimal pigment colors shown in Fig. 16. The relative intensity of the spectral rays (the quantity \(a\)) is varied by means of the achromatic wedge \(n\) in front of the slit \(Sp\). The procedure of measurements with this instrument is as follows. First the zero position of the achromatic wedge is found by placing identical white surfaces in \(P\) and \(P'\); then, by moving \(Sch\) and \(n\), both fields of view are set to equality, after which the color coordinates are found immediately: \(\lambda_1\), \(\lambda_2\), and \(a\), or, equivalently, the width of the cutout, \(sp\), the wavelength \(\lambda_m\) corresponding to its middle, and \(a\).

Fig. 19.

In order to translate these color designations into the language of the three-component theory, it is necessary only once and for all to deter-

divide and plot in the color triangle a series of curves of “equal notch width \(sp\),” and curves of “one and the same \(\lambda_m\)” (Fig. 20; the length of the entire spectrum is taken as 300; spectra of types \(c\) and \(d\) correspond to the lines \(k_0\) and \(k_b\); above these lines lie the colors corresponding to spectra of type “\(a\),” below them “\(b\)”; curves of equal \(\lambda\) are shown by dashed lines). From Fig. 21

Fig. 20.

Fig. 21.

we see that these systems of curves uniquely and exhaustively determine the quality of a color from the point of view of the three-component theory, and therefore, having constructed these curves on a large scale and in sufficient number, we can pass from one system of color coordinates to another without any calculations. In addition, Rösch constructed a diagram which makes it possible, from the magnitude \(a\) (the relative intensity of a color in comparison with the optimal one), to find the lightness of the color as well. In this diagram (Fig. 21) he draws, within the triangle, lines of equal lightness of the optimal pigment colors (“isochrons”), from which it is easy, having found \(a\) experimentally, to pass to the lightness of the pigment. But this will be discussed in greater detail in the second article, devoted to higher color metrics and to various color systems.

1. S. Rösch. Phys. ZS. 29, S. 83—91, 1928. ↩↩↩↩↩↩↩

2. E. Schrödinger. Ann. d. Phys. 62, 603—622, 1920. ↩