We apply a simple linearization, used standardly in solid state physiscs, to describe the evolution under its self-gravity of an infinite perfect lattice perturbed from its equilibrium. In the limit that the initial perturbations are restricted to wavelengths much larger than the lattice spacing, the evolution corresponds exactly to that usually derived by cosmologists from an analogous linearization of the Lagrangian formulation of the dynamics of a pressureless self-gravitating fluid, including the so called Zeldovich approximation. Our less restricted approximation, however, allows one to trace the evolution of the fully discrete distribution until the time when the particles approach one another, with modifications of the fluid limit explicitely depending on the lattice spacing.