The $U(1)$-gauge theory with the Villain action is considered in a cubic lattice approximation of three- and four-dimensional tori. As the lattice spacing tends to zero, the naturally defined correlation functions converge to the correlation functions of the $\mathbf R$-gauge electrodynamics on three- and four-dimensional tori only for a special scaling, which depends on the correlation functions. Another scaling gives degenerate continuum limits. The Wilson criterion for the confinement of charged particles is fulfilled for the $\mathbf R$-gauge electrodynamics on a torus. If the radius of the initial torus tends to infinity, then the correlation functions converge to the correlation functions of the $\mathbf R$-gauge Euclidean electrodynamics.