Prof. Grassmann on the Theory of Compound Colours. 261 



Let « be a homogeneous colour, and a' that homogeneous 

 colour which furnishes white when mixed with a. For the sake 

 of clearness, let us suppose a and a' to be represented by two 

 lines of equal length running from the same point in opposite 

 directions (fig. 5). Let b also be a colour, which, when 

 mixed with a, furnishes an equal quantity of white to that pro- 

 duced by its mixture with a' ; and in order to express this 

 equal relation of b to a and «', let b be represented by an- 

 other line perpendicular to a and a'. The intensity of the 

 colour b is to be so chosen, that if V be the colour which 

 gives white with b, the intensity of the light resulting from 

 this mixture may be equal to that of the light produced by 

 the mixture of a and a!. This may be represented in the 

 figure by making the line which expresses the colour b as long- 

 as those repi'esenting a and «', whilst the complementary colour 

 of b may be repi'esented by the line b', of equal length with b, 

 but running in the opposite direction. We will suppose that, 

 of the two colours b and b', b is that which lies upon the side of 

 positive transition from a. It is evident that if the colour a 

 be given, a', b, and b* may be found by observation. For 

 instance, if a be yellow, a' is indigo ; between a and a' on the 

 side of positive transition lie the various tints of green and 

 blue ; greenish-yellow mixed with yellow («) gives a very small, 

 but with indigo (a') a very considerable intermixture of white. 

 Proceeding from greenish-yellow on the positive side, the inter- 

 mixture of white will increase by mixture with yellow, and 

 decrease by mixture with indigo. A tint will consequently 

 lie on the course of the transition, which will furnish an equal 

 quantity of white when mixed with yellow, as with indigo. 

 This must be about green, so that b will be green and b' purple. 

 Now it is evident, that, by the mixture of two of these four 

 colours, all tints must be obtainable. These tints may be 

 ascertained by observation for all degrees of intensity of the 

 homogeneous colours a and b, b and a' , a' and U, b' and a. 

 Suppose the intensities of the two colours to be mixed to be 

 signified by the length of the lines representing them, so that 

 if a colour has the tone a, and its intensity is in the same pro- 

 portion to that of a as m to 1, then that colour may be repre- 

 sented by a line having the same direction as a, but 7)i times its 

 length. Having represented in this manner the two colours 

 geometrically, let us construct from these lines the geometrical 

 sum, that is, the diagonal of the parallelogram which has the two 

 lines for its sides*, and assume that this sum or diagonal shall 



* The idea of this geometrical sum was first developed by Mobius in his 

 Mechanik des Himmels (1843), and by myself in my Ausdehmmgslehre 

 (1844). 



