Measurements of the engineering constants E, G, and v are routinely made for polymeric materials. If these materials are isotropic, these measurements should satisfy the relationship G = E/2(1 + v). However, many past measurements have indicated that this relationship is not satisfied. This raises questions about the assumptions of material isotropy and the applicability of Hooke's law. The methods used to measure these engineering constants for a number of different polymers are first described. Then, new results obtained in the current investigation are presented, indicating that the elastic constants do in fact satisfy the isotropic relationship for strains up to 0.5%. However, it is shown that at strain levels above this level, the relationship between stress and strain is nonlinear.