on the relationships between colors

CLASSICAL ARTICLES IN COLOR

On the Relationships betweenColors

Tobias Mayerde Affinitate Colorum, inOpera Inedita Tobiae Mayeri,Ed. G. C. Lichtenberg, Go¨ttingen, 1775

Translated by Adriana Fiorentini

Commentary by Barry B. LeeMax Planck Institute for Biophysical Chemistry, 37077 Go¨ttingen, Germany

Received 19 February 1999; accepted 17 April 1999

As the bicentennial of Thomas Young’s Bakerian lecture1

approaches, we reproduce here the first formal attempt toconstruct a trichromatic color mixture space, as put for-ward in Gottingen in 1758 by Tobias Mayer. His formula-tion is of historical interest, since during his stay in Go¨t-tingen in the last decade of the eighteenth century Youngbecame acquainted with Mayer’s work .

Tobias Mayer was born in 1723 in Marbach in southernGermany. Despite little formal education, in 1750 he wasoffered a professorship at the George–August University inGottingen, and soon took over the newly built astronomicalobservatory there. Prior to taking up the appointment inGottingen, he had been employed in the HohmannschenKartenverlag in Nuremberg, where he was active in devel-oping and using astronomical measurement techniques forcartography. In this capacity, he came into contact with theleading astronomers of Europe and soon became an inter-nationally recognized figure in astronomical measurement.

Barely fifty years after a framework had been provided bythe Newtonian revolution, these techniques were of greatpractical importance for cartography and navigation. Sev-eral other universities attempted to recruit Mayer, but heremained in Go¨ttingen until his early death in 1762. In1753, he was persuaded to submit his lunar navigationaltables to the British Admiralty in an application for theprize for a method of longitude estimation, and in 1765 heposthumously shared the £10,000 prize with John Harrison,the English inventor of the marine chronometer.2

Mayer was interested in psychophysical performance,and published a report on visual acuity as a function ofretinal illuminance.3,4 Apparently his interest was stimu-lated by differences in his ability to read graduation markson his telescope at night. He noted that the deterioration inacuity at low light levels was small in comparison with achange of illuminance of many log units. There is no certainorigin for his interest in color, but perhaps through hisexperience in map production he became familiar with thepractice of color mixture and printing; a few years earlierthe first color prints had been produced.5 On the other hand,he was a talented artist and may have developed an interest

Correspondence to: Dr. B. B. Lee, Max Planck Institute for BiophysicalChemistry, 37077 Go¨ttingen, Germany (e-mail: [email protected])© 2000 John Wiley & Sons, Inc.

66 CCC 0361-2317/00/010066-09 COLOR research and application

in color mixture from this source. In any event, in 1758 hegave a public lecture in which he gave trichromatic mixturea mathematical format; he constructed the first color trian-gle. This lecture was extensively reported in the Go¨t-tingische Anzeigen von gelehrten Sachen,6 an internationalscientific news periodical. Lambert read this report and wasprompted to extend Mayer’s ideas.7 A full manuscript wasnot published in Mayer’s lifetime, and finally appeared in acollection of Mayer’s unpublished work (Opera Inedita)edited by Georg Christof Lichtenberg.8 Figure 1 shows thefrontispiece of this article.

Lichtenberg was a remarkable figure, famous as an es-sayist and for his aphorisms, as well as a scientist; he wasProfessor of Physics in Go¨ttingen from 1775 to his death in1799. His main interest was electricity and magnetism, andmagnetic lines of force as revealed by iron filings still bearhis name for German students. He published an addendumto Mayer’s article in which he normalized each primary totheir sum, as in current CIE practice, and discussed colormixture methods. He maintained an interest in vision; in1794 he contributed to a textbook of optometry edited by hisfriend Soemmering, Professor of Medicine in Frankfurt andthe first to describe the human fovea.9 He was certainlyaware of the trichromatic theory of George Palmer, sinceLichtenberg’s less famous brother edited Lichtenberg’smagazine,10 where Palmer’s theory is discussed. WhenThomas Young wrote his doctoral thesis in Go¨ttingen in1795–1796, he noted that Lichtenberg was one of the fewprofessors with whom he became personally acquainted.11

Young’s visits are noted in Lichtenberg’s diaries, but un-fortunately the subjects of their discussions are not de-scribed.

Mayer used red (rot), yellow (gelb), and blue (blau) ashis primaries, as Thomas Young suggested in his Bakerianlecture, and the addition of black and white produceddarker and lighter colors. It is unclear from Mayer’s article

what status he assigns to these primaries. From hints in theGottingische Anzeigen, he believed trichromatic color mix-ture to be primarily a perceptual phenomenon but wasuncertain as to whether light was a continuous spectrum ordiscrete, with three component rays. This confusion betweenperceptual trichromacy and a physically continuous spec-trum brought forth a vigorous reaction from Euler,12 andwas not resolved until Thomas Young’s formulation of thetrichromatic theory.1

At that time, the distinction between additive and sub-tractive color mixture had not been recognized. The meth-ods used by Mayer to test mixtures are uncertain. BothMayer and Lichtenberg stress the use of mixing pigments,but in a dry, powdered form. With a grain size coarserelative to the wavelength of light, this approximates anadditive mixture, but with a reduction in luminance. Bothauthors imply that the darker, subtractive colors obtainedwhen mixing pigments in solution were due to pH changes.Lichtenberg13 was also familiar with the use of a rotatingdisc with segments of different color as a means of colormixture. Mayer appears to have used his mixture principlesto construct a wax representation for use in color printing(first reported in the Go¨ttingsche Anzeigen14), but the de-tails reported by Lichtenberg make it difficult to recon-struct. However, Lichtenberg clearly states that the equa-tions described refer to mixing “colors” rather than“pigments,” and he remarks that the problem of mixingpigments might have been solved had Mayer the opportunityto continue his work. Nevertheless, the failure to distinguishadequately between additive and subtractive mixtures led toconsiderable confusion. Despite the fact that Mayer’s andLichtenberg’s theoretical formulations are clearly additive,much of their practical discussion seem to refer to subtrac-tive mixture. The failure to distinguish between the physicalmixture of light per se and its relation to psychophysicalperformance (e.g., in chromatic discrimination) also caused

FIG. 1. The title page of Mayer’s article, put together with a selection of other writings, into the Opera Inedita by Lichtenberg.

Volume 25, Number 1, February 2000 67

difficulties, although Lichtenberg was apparently aware ofthe distinction as the footnote in which he compares his andMayer’s formulation indicates.

In Lichtenberg’s publication of the Opera Inedita, ahand-painted color triangle found in Mayer’s papers isreproduced. This, of course, would have been a subtractivemixture, so that producing the triangle must have involveda deviation from Mayer’s principles. The problems thatarose from this failure to distinguish between additive andsubtractive color mixture are apparent in Lichtenberg’scommentary, where he devotes a long section to the diffi-culties he had in getting Mayer’s color plate correct; even-tually this seems to have been achieved on an empiricalbasis.

Both Mayer’s and Lichtenberg’s articles are written inelegant Latin. We reproduce here a translation of Mayer’stext, and that segment of Lichtenberg’s commentary thathas theoretical significance; the segments omitted have todo with a number of topics of a practical nature. A partialtranslation (into German) of these two manuscripts togetherwith a discussion of the relations between Mayer’s, Lam-bert’s and Lichtenberg’s theories may be found elsewhere.15

Today, the inadequacies of Mayer’s approach, especiallythe failure to adequately distinguish between additive andsubtractive color mixture, are easy to pick out. Neverthe-less, these early steps toward a color science provide aninteresting insight into the workings of visual science in theeighteenth century. Mayer’s and Lichtenberg’s primary in-terests lay in physics rather than perception. However,especially in the transcripts of Lichtenberg’s lectures,13

their concern with the nature of sensation cannot but befamiliar to present day psychophysicists.

Acknowedgments: We would like to thank John Mollon,who many years ago drew our attention to the involvementof Gottingen in early days of color science, We would alsolike to thank Sergio Tavano for significant help with thetranslation and Jan Kremers, Joel Pokorny, and VivianneSmith, who commented on the text.

ADRIANA FIORENTINI

Istituto de Neurofisiologia del CNRPisaItaly

BARRY LEE

Max Planck Institute forBiophysical Chemistry37077 Go¨ttingenGermany

1. Young T. On the theory of light and colours. Phil Trans London1802;92:12–48.

2. Forbes EG. The birth of navigational science. Nat Maritime Museum,London, 1980.

3. Mayer T. Erfahrungen u¨ber die Scha¨rfe des Gesichtsinn. Go¨ttingischeAnzeige von gelehrten Sachen 1754;401–402.

4. Grusser OJ. Quantitative visual psychophysics during the period ofEuropean enlightenment. The studies of the astronomer and mathema-tician Tobias Meyer (1723–1762) on visual acuity and colour percep-tion. Doc Ophthamol 1989;71:93–111.

5. Blon II JCL. Colorito: On the harmony of colouring in painting.London, 1721.

6. Mayer T. von Messung der Farben. Go¨ttingische Anzeige von gele-hrten Sachen. 1758;1385–1389.

7. Lambert JH. Beschreibung einer mit dem Calauschen Wachse ausge-malten Farbenpyramide, wo die Mischung jede Farbe aus Weiß unddrey Grundfarben angeordnet, dargelegt und derselben Berechnungund veilfacher Gebrauch gewiesen wird. Berlin, 1772.

8. Mayer T. Opera Inedita. Lichtenberg GC, editor. Go¨ttingen, 1775.9. Soemmering ST. Adams, Busch und Lichtenberg u¨ber einige Pflichten

gegen die Augen. Frankfurt am Main, 1794.10. Walls GL. The G. Palmer story. J History Med Allied Sci 1956;11:

66–96.11. Peacock G. The Life of Thomas Young. London, 1855.12. Euler M. Lettres a une Princesse d’Allemagne sur differentes Ques-

tions de Physique et de Philosophie. Paris: Royez, 1787.13. Lichtenberg GC. Lichtenberg u¨ber Luft und Licht. Vienna and Trieste:

Geistinger’s Buchhandlung; 1811.14. Mayer T. Neue Kunst Gema¨lde mit naturlichen Farben zu Drucken.

Gottigische Anzeige von gelehrten Sachen. 1759;402.15. Lang H. Drei Farbsysteme des 18. Jahrhuderts von Mayer, Lambert

und Lichtenberg. Farbe Design 1980;15:52–59.

On the Relationships betweenColours§1—Concerning the number, measurement and mixture ofcolors it appears to me that, despite their great variety, thefirst principle is as follows:

Three, and no more, are the simple or primary colors,from which all others are produced by mixture, but whichcannot be obtained in any way by mixing others in whateverproportion: they are red, yellow and blue. These are seen inthe rainbow, and even more vividly in a ray of sunlightpassing through a glass prism, although in the latter casethey appear flanked by other secondary colors. Some takethose secondary colors also as simple colors, so that theycite seven (and why not more, since in fact their number isinfinite): red, orange, yellow, green, blue, purple and violet.But since they do not explain clearly what they mean by asimple color, nothing follows from their opinion that con-tradicts my claim. If we define as primary those colors bymixture of which all others can derive, and which cannot beproduced in any way by the others, then even those whohave a different opinion on this question will easily agreewith us; and indeed, on the base of experience, they cannotlist other colors, beside the three quoted above, which sharethese properties.

§2—I shall not include white and black among colors thatcan be properly so called, although they are commonlygiven this name. Indeed, the former is more correctly com-pared with light, the latter with shadow and darkness, andphysicists do not deny that white is a perfect mixture of allcolors, so that they even consider it as a light source; black,on the contrary, is equivalent to the absence of color, and islike darkness. But I shall say more extensively below whatI think about these two.

§3—As regards the secondary colors, some result fromthe mixture of only two primary colors, others from all three

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of them, and these mixtures differ from each other depend-ing upon whether one primary is contained in a higher orlower proportion with respect to the others. And since thesimple colors can be mixed in an infinite variety of differentproportions, there is obviously an infinite number of differ-ent colors. Let us cite as examples only the most important.

§4—Mixtures of red and yellow give rise to saffronyellow, golden yellow, orange, flame-scarlet and deep red.The three former colors are obtained when the mixturecontains more yellow, the three latter, more red. If they aremixed in equal amounts, the resulting color could properlybe called red-yellow, because it is difficult to give a name toevery single color.

§5—Mixtures of yellow and blue yield all kinds of green.Properly speaking, green is the color that contains equalamounts of yellow and blue, and changes to yellowish-green if it contains more yellow; mixing more blue insteadof yellow yields aquamarine.

§6—Mixtures of red and blue exhibit a purple color, andproperly this name pertains to colors that contain equalamounts of red and blue. And indeed, rose-red has more red,violet has more blue.

§7—These are thus the colors that can be obtained bymixing two simple colors. All these can be seen in therainbow and in a sunbeam passed through a prism. Thereappear between red and yellow all kinds of reddish colors,from red to more and more yellowish; and between yellowand blue, all kinds of greens. From red on the one side toblue on the other, various shades result in purple or violet.

§8—Now the colors which originate from mixture of thethree primary ones do not appear in the rainbow or in therefracted sunbeam. And indeed that light beam itself, if webelieve the physicists, may be a mixture of three simplecolors. What color it has is not easy to say. White occurswhen it falls on a white surface, as does red when it falls ona red one, and yellow on a yellow one, and perhaps it ismore appropriate to consider it not a color, but more cor-rectly to be the source of color. In fact, this is not the placeto get involved in difficulties which, even if we couldresolve them, would not contribute at all to the presentquestion. What is certain, contrary to Newton’s claim, is thefact that mixtures of red, yellow and blue neither yield lightnor white, but all kinds of dark shades, browns, reddishbrowns, blue-grays, and ash greys, depending on whetherthey contain more of one or the other simple color.

§9—White mixed with either simple or complex colorsdoes not produce any new color, but merely makes thempaler, and that is why it can hardly be included amongcolors. However, if somebody prefers to consider these palecolors as new ones, I would not object if he includes whiteas a color; it is a matter of terminology. Black behaves in asimilar way so that, when mixed with other colors it doesnot generate new hues, but merely darker colors. And somecolors can be called black, if they are very dark. Commonlyblack is considered only the color that, if mixed with whiteor attenuated in any other way, becomes ash gray. Thenblack has to be included among the complex colors, because

ash gray originates from red, yellow and blue mixed insuitable proportions.*

§10—One can easily see that, when I deal with colormixtures, this has nothing to do with those artificial mix-tures or chemical paints like that, for example, which areused to produce the black color of the ink of our clerk. It isknown that by adding some acids, it is possible to removethe natural colors of many objects, or rather, to change theircolors into different ones. In some objects this occurs as aconsequence of the action of sunlight or of air. I do notmean here any such violent mixing and modification. There-fore, if one wants to do experiments in order to convinceoneself that what is said about the production of complexcolors is true, one has to use pigments not containing anyacid which may act upon the color of the pigments aftermixing. It will be safest to use pigments that, reduced to apowder, can be mixed dry.

§11—Although the fractions of simple colors in the mix-tures can be varied in infinite proportion, and therefore theremust be innumerable different colors, not all these colors aredistinguishable. If for example only 1/20th or 1/30th of blueis added to one quantity of yellow, the color obtained willsimulate some aspect of green, but this will differ so slightlyfrom yellow itself, that just by eye it could scarcely, or notat all, be distinguished. Thus, the colors that can be discrim-inated only with great difficulty from their neighbors, wewill call indistinguishable from them.

§12—To develop a reliable method, if colors distinctfrom others have to be produced, it is convenient to followthe rule that architects and musicians prescribe for their ownuse. That is, one has to establish proportions of colors to bemixed that are restricted in number. And indeed, the fewerthe number, the more distinct are the colors mixed from theprimaries according to these proportions. Neither in archi-tecture nor in music are proportions easily accepted greaterthan twelve, because greater ratios are hardly perceivable bythe senses. Similarly, in this quasi-music of colors (as wemay not improperly call the whole of the visual arts) it isbest to be content with the number twelve. In this way,between two simple colors we shall obtain eleven interme-diate mixture colors that can be distinguished by the eye,while we shall rightly consider indistinguishable the colorsthat fall in between these.

§13—This rule should not only be followed for the colorsthat can be obtained from two primaries, but also for thosehaving all three primaries as ingredients. Having establishedthis once, we may do something that at first might appearimpossible, i.e., count up exactly all and every single colorso that for any given color it will be possible to say at oncefrom which proportions of primaries it is composed. To dothis conveniently, we shall indicate red by the letterr ,yellow by g and blue byb; the complex colors we shallindicate by letters related to the simple colors of which theyare composed. Thus, for instance colors resulting from amixture of red and yellow will be indicated byrg or gr,

* Its symbol, as explained below, is r3 g2b7.


colors composed by the three simple ones byrgb, etc. Andthe relative amounts of the three simple colors composing acertain color will be indicated by numbers above the letters,like exponents, so that a color consisting of 3 parts of yellowand 2 of blue will be indicated byg3b2. I have to stress thatthese numbers are not to be considered as exponents, as inalgebra. To distinguish them by name, let us call themcoefficients.

§14—It is obvious, however, that if the coefficients of acertain color have a common factor, once divided by thatfactor they represent the same color. And indeed, only therelative values of the coefficients, not their absolute values,are considered in color mixtures. For instance,r8g4 indi-cates the same color asr2g1. In turn, if all the coefficientsare multiplied by the same number, the color does notchange. In general, a coefficient should be considered as thenumerator of a fraction where the denominator is the sum ofall the coefficients present in the color symbol. For instance,r8g4 is the same asr8/12g4/12, but for brevity the denomina-tor is omitted.

§15—In order to list the single colors that differ at leastby 1/12 of the total from any other color, either simple orcomplex, the problem is reduced to finding out which arethe possible ways to decompose 12 into the sum of two orthree whole numbers and how many times the order of thesenumbers attributed to the lettersrg, gb, rb , rgb can bevaried. Solving this problem is not difficult. From the com-bination, these colors come out to be:

1) Primary colors

r 12 g12 b12

2) Colors composed of two primaries

r 11 g1 g11 b1 r 11 b1

r 10 g2 g10 b2 r 10 b2

r 9 g3 g9 b3 r 9 b3

. . . . . . . . .r 1 g11 g1 b11 r 1 b11

3) Colors composed of three primaries

r 1 g1 b10 r 2 g1 b9 . . . r 8 g1 b3 r 9 g1 b2 r 10 g1 b1

r 1 g2 b9 r 2 g2 b8 . . . r 8 g2 b2 r 9 g2 b1

r 1 g3 b8 r 2 g3 b7 . . . r 8 g3 b1

. . . . . . . . . . . .r 1 g7 b4 r 2 g7 b3 . . . r 4 g7 b1

r 1 g8 b3 r 2 g8 b2 r 3 g8 b1

r 1 g9 b2 r 2 g9 b1

r 1 g10 b1

§16—In this list are to be found all the colors which differfrom their neighbors by 1/12; that is the amount which canbe reliably distinguished visually. And indeed no color canbe shown that is not so similar to one of the colors listed thattheir difference does not exceed 1/24, namely by an imper-

ceptible amount: therefore, we can state that all and everysingle color is contained in this matrix. Thus, the followingnumbers may be calculated:

Simple colors 3Colors composed of two primaries 33Colors composed of three primaries 55

—–And the total number of different colors 91

Simple colors 3 Colors composed of two primaries 33Colors composed of three primaries 55 And the total num-ber of different colors 91§17—Since this is a triangular number with side 13, allcolors can be represented in a triangle, subdivided into 91small areas in the following way: the three simple colors atthe vertices, colors with two components along the sides andthose with three components within the triangle, beingcloser to the simple colors from which they are morestrongly derived. This is the arrangement shown in Figure 1(Fig. 2 in this translation).

§18—If each of the areas is filled with the color corre-sponding to the symbol inscribed in that area, as can beeasily done by those who know how to use pigments, a colorscale is obtained. By means of this scale (as for an alloy ofgold and silver on a touchstone) every color can be identi-fied and named, and the ratio of the three simple colorswhich compose that color can be defined .

§19—Such a color scale will be particularly useful in thevisual arts. And indeed, since there is not a pigment forevery single color, in most cases it would be difficult toproduce a certain color by mixing pigments. Apparently,painters have succeeded in doing so only by trial and error.If the relation of each color to the simple ones has beenchosen once and for all by this scale, it will be easy to obtainany color that does not have its own pigment, by mixingother pigments. In turn, it will be possible to find out whichcolor would result from the mixture of pigments in a givenproportion.

§20—Some examples will make this clearer. Suppose itwas found that the following values are appropriate, accord-ing to the color scale:

r12 for a pigment made ofcinnabar,r8g4 for a pigment made ofred lead,r4g6b2 for a pigment made ofocher,r4b11 for a pigment made ofPrussian blue;

and similarly for the symbols of other pigments. Now weask which color will result from the mixture of three parts ofred lead, two of Prussian blue and four of ocher. To answerthis question, multiply the coefficient of the symbol as-signed to each pigment by the number of parts used in themixture, as follows:

red lead ocher Prussian bluer 8g4 r 4g6b2 r 2b11

3 4 2________ __________ _________r 24g12 r 16g24b8 r 4b22


then add all the coefficients of the same letter, to obtainr44g36b30 and divide by the number of parts: 31 4 1 2 59.The color to be found will be:r41(2/3)g4b31(1/3), which,because of the fractional coefficients, has to be assigned tothe indistinguishable colors; but the closest color among thedistinguishable ones will ber5g4b3 ;that is the color con-sisting of 5 parts of red, 4 of yellow and 3 of blue.

§ 21—Another example. Given a color with symbolr7g2b3, one wants to know what parts of cinnabar, ocher andPrussian blue have to be mixed to obtain that color. Let uscall the parts of

cinnabar 5 x, ocher 5 y, Prussian blue5 z,

and therefore, according to the previous example, the re-sulting color will be:

12x1 4y 1 z

rx1y1z

6y

gx1y1z

2y 1 11z

bx1y1z

and, if this is alsor7g2b3, we have:

12x1 4y 1 z

x 1 y 1 z5 7 or 5x2 3y 2 6z5 0

6y

x 1 y 1 z5 2 or 2x2 4y 1 2z5 0

2y 1 11z

x 1 y 1 z5 3 or 3x2 2y 2 8z5 0†

Since the sum of the second and third equations is equalto the first one, the problem is clearly indeterminate andtherefore has infinite solutions. And indeed one has x5(7/4) z; y5 (11/8) z. But in color mixing one is concernedwith only the proportions of the components and thereforethere is only one solution, whatever number is taken for z.Taking the smallest whole numbers one has x5 14, y5 11,z 5 8. And the required color r7g2b3 will obviously be

† Editor’s Note: The equation is reprinted as published, but the correctequation is 3x1 y 2 8z 5 0.

FIG. 2. Tobias Mayer’s color triangle from the Opera Inedita, showing the 13 steps to a side.


obtained by mixing 14 parts ofcinnabar, 11 of ocher and8 of Prussian blue.

§22—For the same reason the problem will be indeter-minate if more than three colors were mixed to produce acertain color. Sometimes the problem is insoluble, whenparts obtained by solving the equations are negative. Butthis and similar questions, that may derive from the rules formixing the colors, I do not pursue here.

§23—So far we have consideredperfectcolors, namelythose that, because of their brightness, are most vivid. Thereremain those which fall short of this, in a sense, ideal stateand tend towards either light or darkness, or, if you will,towards paleness or shadow. Every color is made lighter ifmixed with white, and this is because white is the lightest ofall colors. Therefore, since every perfect color can be indi-cated byrmgnbt, if we indicate white with the letterw, thecolor made lighter by mixing it with white, will be:wp(rmgnbt)q, or wp(rmqgnqbtq), where p stands for the pro-portion of white, and q for the proportion of the color in themixture.

§24—Here too one must point out what was noted aboveabout perfect colors: mixing color and white in differentproportions does not always result in distinct colors. Thus,even if the two perfect colorsr12 and r11b1 differ only byone degree from each other, when mixed with equalamounts of white they will no longer differ by one degree,but only half of that, and it is much more difficult todistinguish them if they contain even more white. There-fore, if we assume that the difference between a givensimple color and white consists of twelve degrees and wemix one quantity of white with eleven quantities of red,yellow or blue, the number of intermediate colors (namelythose obtained by mixing these simple colors attenuatedwith white) will be smaller than the number of those ob-tained by mixing the same perfect colors. And since thenumber of the latter is triangular with side 13, it is easy toshow that the number of those that contain one part of whiteis triangular with side 12. Similarly, the number of thosethat contain two parts of white is triangular with side 11, andso on. This becomes straightforward if in the generalexpression given above for colors mixed with white,wprmqgnqbtq, the coefficients are whole numbers and theirsum is 12. Since it would be tiresome to list the symbols ofall these colors, which anyone can easily derive, I show onlytheir total. There will be therefore the following distinctcolors:

containing one part of white w1 r m gn b11-m-n 78

containing 2 parts of white w2 r m gn b10-m-n 66

" 3 " " " w3 r m gn b9-m-n 55

" 4 " " " w4 r m gn b8-m-n 45

" 5 " " " w5 r m gn b7-m-n 36

" 6 " " " w6 r m gn b6-m-n 28

" 7 " " " w7 r m gn b5-m-n 21

" 8 " " " w8 r m gn b4-m-n 15

" 9 " " " w9 r m gn b3-m-n 10

" 10 " " " w10 r m gn b2-m-n 6

" 11 " " " w11 r m gn b1-m-n 3

" 12 or perfect white 1

_____________________

thus the total number of colors mixed with white: 364

Note that this number is pyramidal, with side 12.

§25—There remain the dark colors, which tend towardblack or darkness. To better understand their nature, the firstthing to consider is that every perfect color is endowed witha degree of brightness that makes it most distinct. If this iscompletely removed from that color, were it a mixture or aprimary, what remains is perfect black. If only a part isremoved, the result is a dark color, albeit of the same name.But to distinguish such a color from others of lower bright-ness will be the more difficult, the darker the colors. Andindeed, if all light were removed, all colors would becomeblack and no difference would remain between them.

§26—Next it must be noted that a dark color may bemade lighter by mixing it with white, and that it is possibleto add to it so much that, from being dark, it first becomesperfect, and then a lighter color. There is no perfect colorthat cannot be produced in this way from a darker color ofthe same name. Therefore, in every perfect color one has toimagine a certain amount of lightness that, if separated fromit either totally or in part, would leave black or a darkercolor.

§27—From this it is clear, I believe, that the same crite-rion holds for producing dark colors as white and lightcolors . Either removing lightness from a color, or separat-ing white from it, makes the color dark or black. Butlightness differs from whiteness in that the former, spread ina dark or black color, does not make it brighter or closer toa perfect color, while the latter, mixed with black or a darkcolor, can produce any perfect color. The nature of perfectcolors must be such that they occupy a place halfwaybetween white and black, and that they are produced whenequal amounts of black and white are mixed together. Forthis reason, since we numbered 12 intermediate degreesfrom a perfect color to white, we have to state that everyperfect color must be separated by the same amount fromblack.

§28—From this it is possible to conclude that dark col-ors, which are seen as mixtures of perfect colors and black,have rather to be considered as mixtures of perfect colorsand with ”negative“ white. In fact black is a color somewhatvariegated and indeterminate, since it can originate from


any perfect color and, therefore, when mixed with white oranother color, produces sometimes one color, sometimesanother. This ambiguity is eliminated if black is thought ofas absence of white.

§29—Therefore, if as above we indicated a color mixedwith white with the symbolwp(rmgnbt)q, now a color mixedwith a negative white, i.e., tending to black or dark, will beindicated with the same symbol, but with the coefficient p ofwhite changed into a negative number, and sow-p(rmgnbt)q.Thus, as many dark colors are produced as there are palecolors, i.e., 364. In this way it will finally be possible toassign the total number of all distinct colors, the perfectones and both the light and the dark ones. And so:

Perfect colors - - - 91

Light colors - - - 364

Dark colors - - - 364

_______________

The sum will be 819

§30—Anyone who considers seriously the enormous vari-ety of colors must admire the wide range of visual arts thatcan make use of all these colors. If all languages arecomposed of the combination of 20 letters of the alphabet,or a few more, and if the musical arts can derive so manydifferent harmonic compositions from a variable order ofmusical tones, the number of which rarely exceeds 50, howmuch richer and prolific must be pictorial art, that has thechoice of over eight hundred colors, from which variableorder and composition produces its effects? But this appliesalso to the wisdom which has created this world; it hasprovided the greatest purpose with the richest variety ofobjects.

Lichtenberg’s 1775 Commentary to Mayer’s Triangle

This study, now appearing for the first time, seems tohave lost its originality. The eminent Lambert, having beenmade aware of Mayer’s discovery through the Go¨ttingscheAnzeigen of 1758, and impressed by its novelty and ele-gance, illustrated and extended it to such an extent that itbecame his own. By referring the reader to that work, Icould easily refrain from dealing any longer with this sub-ject, but this is nevertheless required by the need to describeboth the triangle painted by Mayer and my contribution tothis issue.

Before doing so, it is perhaps useful to deal with somemore general questions about Mayer’s theory in order tobetter appreciate the link between all these triangles, andtheir purpose and scope.

Let us imagine, therefore, subdividing a triangle, whichwe may assume to be equilateral, into infinitely small areas.And that the small areas at the three vertices of the triangleeach contain its own primary color,r , b, g, the weighting of

which, at the vertices, is made equal to 1, while the strengthsof colors in mixtures is taken to be proportional to theirweighting, so that for instance mr 1 mb gives a colorexactly intermediate betweenr andb.

Moreover, let us assume that in other small areas of thetriangle, the coloursr , b, g are distributed according to therule that, in any given area, their weights are proportional tothe lengths r, b, g of the perpendiculars drawn from that areato the sides of the triangle. Thus, with the height of thetriangle 5 1, the amount of red contained in a given areawill be 5 r r , the amount of blue5 b b, the amount ofyellow 5 g g and the expression for the mixture, that is thecolour of that area, can be conveniently written as A5 rr 1 b b 1 g g, wherer, b, g are positive variables and theirsum, if the triangle is equilateral, is constant and equal to thetriangle height.

If one of these quantities becomes zero, the other twoproduce a color on one side of the triangle, a mixture of twoprimaries, and if one of them is unity, or alternatively thesum of two of them is zero, the equation represents aprimary color.

Since, given three numbers x, y, z either rational orirrational, it is always possible to express their ratios by thethree perpendiculars from a given point to the sides of thetriangle, it is evident that this triangle contains all thecolours that can be obtained by mixing the three primariesat the vertices; and it is impossible that, for instance, acombination of a mixture color with either another mixtureor a primary will result in a color not already contained inthe triangle. Therefore, whatever mixtures are combined,the result of the mixture can always be reduced to theexpression Rr 1 Bb 1 Gg. Thus, by drawing straight linesparallel to the sides of the triangle opposing vertexr andvertexb, respectively, the point where the two lines inter-sect, located at distance R / (R1 B 1 G) from the formerside and at distance B / (R 1 B 1 G) from the latter, willbe the one given by Rr 1 Bb 1 Gg. And indeed, the colorof this area will be

Rr 1 Bb 1 Gg

R 1 B 1 G, where

r 5R

R 1 B 1 G, b 5

B

R 1 B 1 G, g 5

G

R 1 B 1 G

But even if this triangle contains an infinite number ofcolours, it does not contain all. The lightness and darknessof all mixtures contained within it are limited. Thus, forinstance, in our triangle the colour (r 1 b 1 g) / 3, whichrefers to that at the midpoint, will belong to the family ofachromatic colours, that is of those colours in which neitherred nor blue nor yellow predominate, not even slightly; butunlessr , b, g are extremely dark, it will not be actuallyblack, but rather gray, which will become closer to black ifthe vertex colors are all darker, or, other things being equal,if just one or two of them are darker.

What I said about gray can be applied to the othercolours, because all end up as their own black, and arerevealed again when mixed with white, as gray can be


produced from a black such as burnt ivory or from otherburnt materials. This can be illustrated by placing a coloredglass between the eye and, for instance, a white surface wellilluminated by sunlight. If thick, these glasses prevent anylight from reaching the eye and appear black; if thin, theyappear coloured and the thinner they are, the lighter theyappear.

To our triangle belong colours, for example greens, someof which would be called light, and others that would becalled dark. They are all indicated by the same formula,bb 1 g g, whereb andg are variables. They may differ notonly in brightness, as would a glass 1/10 of a line thickdiffer from a slide of the same glass 1 line thick, but also inthe ratio of the mixture. However, if you found in ourtriangle the color of one of these two slides of the sameglass, it would be useless to look for the other, since for allslides of the same coloured glass, whatever their thickness,the ratiob : g is the same. Therefore, in order to describewith the equation given above all colour triangles contain-ing all possible colours, one is forced to assume that thecoloursr , b, g are variable and can vary continuously fromextremely light or equal to white to extremely dark or equalto black.

Since we consider the vertex colours to be simple, it is notimproper to indicate this variability by means of exponents,so that rn, bn, gn will indicate colours closer to perfectdarkness than to perfect lightness, andr -n, b-n, g-n, colourscloser to perfect lightness than to perfect darkness, and thetermsr`,b`,g` andr -`, b-`, g-` indicate colors equivalent toblack and white, respectively.

Having established this, it would be easy to provide ageneral equation for an infinite number of chromatic trian-gles, but it is convenient to retain our proven method, theequilateral triangle that we assumed previously, for whichwe can write as the most general formula : A5 r rn 1 b bn

1 g gn.It is thus clear that the system of all colors can be

represented by a prism* having as bases equilateral trian-gles, one representing, according to the equation, infinitelylight or white colours and the other, infinitely dark colours,while the sections of the prism parallel to its bases satisfy allpositive and negative values of n.

The triangle obtained from the general equation for n50 is Mayer’s color triangle, and is obtained by cutting the

prism on a plane parallel to its base, through the midpoint ofits axis. This triangle contains all colours that lie exactlymidway between white and black and shows the greatestvariety of distinct colours of all the equilateral chromatictriangles. Therefore it is more worthy of attention than theother triangles in a general treatment such as the presentcommentary. Lambert’s triangle, that has as vertex colors avery dark Prussian blue, scarlet and gum yellow, and cantherefore be indicated byr rm 1 b bm1p1g g-n, yields avery oblique section through our prism. The field of colourswith positive and negative exponents, each of which can befurther manipulated by mixing with white, is of the widestand thus most common use in the visual arts.

In Mayer’s triangle, any painter can produce all colourswith negative exponents, by adding ”white pigment,“ butone cannot produce good colours with positive exponentstending to black, since there is no black pigment that is nota combination ofr, b, g, and that, once mixed withr , b, g,will not elevate them differentially to higher exponents. Theblack that can be obtained fromr , b, g, mixtures and that isdescribed by the formular` 1 b` 1 g` will produce colorsof higher exponent which are muddy, matt and dull, beinglocated near the axis of the prism. Nevertheless, Mayer’striangle is quite consistent with this ingenious author’spurpose, to give a general method to produce colours and todescribe all those that can be perceived. This was certainlythe aim that this most skilled author wished to pursue, as isevident from his paper and also from the triangle he made,as we shall see.

As to the pigments, when he talked about them he did soonly for purposes of illustration, as geometrists illustratetheorems with lines of chalk on a blackboard. No doubt,were the distinguished author still alive, he would apply thegeneral propositions of his commentary on colors to pig-ments, as a practical aid for artists. And who could do thisbetter than Mayer, an ingenious innovator in the theory ofart, and who already practiced it with great success as ayouth?

There follows an extensive discussion of the color trian-gle, which Lichtenberg received with Mayer’s papers, andthe methods Mayer may have used to prepare it. Lichten-berg also discusses the problems he had producing 600copies of the triangle for the printer of the Opera Inedita.After these difficulties, he apparently carried out a series ofcolor mixture experiments himself with dry pigments, whichhe describes in some detail. To determine the ”coloringpower“ of three primaries, he mixed pairs of them indifferent proportions to find mixtures the colors of whichobservers perceived as midway between the primary pairs;in the case of blue and yellow, he asked observers toproduce a unique green. He also describes the wax model ofMayer and concludes with a discussion of color printing.

* This result can be reached more elegantly with two identical pyramidshaving a common base, if we assume that the number of distinguishablecolour mixtures in the triangle diminish in inverse proportion to the squareof the distance from the extremes, i.e., their number decreases with n2. Thisis similar to what we have already assumed for the triangle, and is clearfrom Mayer’s commentary and can be easily derived from our equation.But these and other questions which derive either from considering obliquesections of these solids or from computation, and are not without use incolor science, I leave to be discussed elsewhere in the future.


on the relationships between colors

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