The geometry of an Archimedes screw is governed by certain external parameters (its outer radius, length, and slope) and certain internal parameters (its inner radius, number of blades, and the pitch of the blades). The external parameters are usually determined by the location of the screw and how much water is to be lifted. The internal parameters, however, are free to be chosen to optimize the performance of the screw. In this paper the inner radius and pitch that maximize the volume of water lifted in one turn of the screw are found. The optimal parameter values found are compared with the values used in a screw described by the Roman architect and engineer Vitruvius in the first century B.C., and with values used in the design of modern Archimedes screws.