The elastic constants of a composite material containing periodically arranged elliptic inclusions are numerically analyzed by the body force method. Denoting Young's modulus and Poisson's ratio by (E_M, v_M) and (E_I, v_I), where the suffixes M and I indicate the matrix and the inclusion respectively, the numerical analyses are carried out for both cases of E_M>E_I and E_M<E_I. First, the method of analysis is explained briefly and the accuracy of the method is checked by comparing the present results with the previous solutions which were obtained for a few simple cases by other researchers. And then, many numerical results are obtained for various shape ratios of elliptic inclusions under square and diagonal configurations. Based on the numerical results, it is concluded that the conventional prediction method of Young's modulus by the volume fraction of inclusion does not necessarily give good approximations. A more rational method for the approximate prediction of Young's modulus of a composite material is proposed.