When nonequilibrium molecular dynamics is used to impose isothermal shear on a two-body periodic system of hard disks or spheres, the equations of motion reduce to those describing a Lorentz gas under shear. In this shearing Lorentz gas a single particle moves, isothermally, through a spatially periodic shearing crystal of infinitely massive scatterers. The curvilinear trajectories are calculated analytically and used to measure the dilute Lorentz gas viscosity at several strain rates. Simulations and solutions of Boltzmann's equation exhibit shear thinning resembling that found in N-body nonequilibrium simulations. For the three-dimensional Lorentz gas we obtained an exact expression for the viscosity which is valid at all strain rates. In two dimensions this is not possible due to the anisotropy of the scattering.