258 Prof. Grassmann on the Theory of Compound Colours. 



with a gives the tint a, whilst the colour which has the tint x 

 and the intensity y furnishes quite a different tint ; but these two 

 colours mixed with a, with tlie same intensity y, have two in- 

 finitely close tints ; that is to say, these two colours mixed with 

 a pass continuously into one another, so that (according to the 

 second proposition) the mixture must continuously change, as 

 also its tint ; but this should be quite different. Thus the sup- 

 position that the transition from a to x may be negative leads to 

 a contradiction, that is, it must necessarily be positive. For 

 the same reason, if x lies at an infinitely small distance from a 

 towards the negative side, a negative transition from a to x will 

 take jjlace. If now the tint x be supposed to change continu- 

 ously from a towards the positive side, so as to pass through the 

 entire series of colours back to a, the corresponding transition 

 of the mixture, which in each case is eft'ected by the elevation 

 of //, as it is at first positive and afterwards negative, must 

 necessai'ily change its sign somewhere. Let a! be a tint in 

 which this change occurs, so that before x reaches it the trans- 

 ition is positive, but as soon as it has passed it becomes negative. 

 If the tint x passes continuously through the tint a', the tint of the 

 mixture must continuously vary with every value of the intensity 

 y, hence the whole of the tints which result from the increase of 

 the intensity y lie extremely close together in both cases (when 

 X lies at an infinitesimal distance from a' on the right or left 

 side) . This, however, is impossible, as some of them lie on the 

 positive and the others on the negative transition from a to a!. 

 Thus the supposition that there is no homogeneous colour, which, 

 when mixed with a, furnishes white, leads to a contradiction ; 

 i. e. every colour has another homogeneous colour, which, when 

 mixed with it, furnishes white. Q. E. D. 



I have chosen the indirect form of proof, because in this 

 manner the greatest possible exactness is mo.:t readily obtained 

 without digression. Moreover, it is evident that this indirect 

 form of proof carries with it the direct assertion that the colour «', 

 at which the character of the transition changes, is the same 

 which, when mixed with a, in any degree of intensity must give 

 coloui'less light. 



If now we test Helmholtz's experiments, we obtain from 

 them, at least approximately, the colour which is capable of 

 furnishing colourless light with any other given colour. For 

 yellow, according to Helmholtz, this is indigo, a result which is 

 bv no means so divergent from the Newtonian theory of com- 

 pound colours as it appears to be at first sight. Helmholtz has 

 more exactly determined the two colours, which, according to 

 him, furnish white ; for the yellow lies between the lines D and 

 E of Fraunhofer, and about three times as far from E as from D, 



