The proof of Parr and Chattaraj for the principle of maximum chemical hardness is analyzed in some detail, It is shown that it is very general and can be applied to many more observables. The case of the physical, or mechanical, hardness of a solid is taken as an example. It is shown that this also should be a maximum in an equilibrium system. Assuming the validity of the argument leads to new information about the compressibility of solids. A simple expression is also given for the Gruneisen constant.