The Rayleigh‐Ritz method of eigenvalue approximation is used to obtain a solution for the free vibrations of an elastic solid. The resonant frequencies of an isotropic cube, calculated by this method, agree with the frequencies of the Lamé modes, for which an exact solution is possible, up to eight significant digits. Experimental measurements of the 13 lowest‐frequency modes of a cube of fused quartz are all within 0.5% of the results computed by using the elastic constants determined by another method. These constants differ from the elastic constants determined directly from the experimental frequencies by less than 0.2% for the shear modulus and by less than 0.003 for the value of Poisson's ratio. The cube‐resonance method of elastic constant determination is also applicable to substances of more general crystallographic symmetry.