Axioms of Color Matching

Based on the observations of many color matching experiments, Grassman (1854) summarized a set of eight axioms:

1. Any color can be matched by a mixture of no more than three colors.
2. A color match at one radiance level holds over a wide range of levels.
3. Components of a mixture of colored lights cannot be resolved by the human eye.
4. The luminance of a color mixture is equal to the sum of the luminance of its components.
5. Law of addition -- if color M matches color N and color P matches color Q, then color M mixed with color P matches color N mixed with color Q:
   
   $$[M]\equiv [N],\;\;[P] \equiv [Q]\;\;\Rightarrow \;\; [M]+[P] \equiv [N]+[Q]$$
   
   

6. Law of subtraction -- if the mixture of M and P matches the mixture of N and Q, and if P matches Q, then M matches N:
   
   $$[M]+[P] \equiv [N]+[Q],\;\;\;\; [P] \equiv [Q],\;\; \Rightarrow \;\;[M] \equiv [N]$$
   
   

7. Transitive law -- if M matches N and N matches P, then M matches P:
   
   $$[M]\equiv [N],\;\;\;[N] \equiv [P] \;\;\;\Rightarrow\;\; [M] \equiv [P]$$
   
   

8. Color matching -- a given color $L$ can be matched in one of three ways:
   1. 
      
      $$[L]\equiv A_M [M] + A_N [N] + A_P [P]$$
      
      

   2. 
      
      $$[L]+ A_M [M] \equiv A_N [N] + A_P [P]$$
      
      

   3. 
      
      $$[L]+ A_M [M] + A_N [N] \equiv A_P [P]$$
      
      

The 8th axiom is a summary of the color matching experiments discussed previously. Because the principle of superposition holds in color mixing, we can re-write a color match

$$[L]\equiv A_1(L) [P_1] + A_2(L) [P_2] + A_3(L) [P_3]$$

as

$$L(\lambda) \equiv A_1(L) P_1(\lambda)+A_2(L) P_2(\lambda)+A_3(L) P_3(\lambda) =\sum_{j=1}^3 A_j P_j(\lambda)$$

Again, the symbol $\equiv$ represents that the energy spectral distributions on both sides are perceived as the same by the human eye, but as functions of wavelength $\lambda$, they are not identical in general.