We have calculated for the first time the contribution to the shear viscosity η of a dilute suspension of hard spheres from the full hydrodynamic interactions involving three spheres. Using the multiple scattering theory, we find the third order coefficient k3 in η=η0(1+k1Φ+k2Φ2+k3Φ3 +⋅⋅⋅) to be 6.40 for a suspension of hard spheres and stick boundary conditions. η0 is the viscosity of the solvent and Φ is the volume fraction of particles in the suspension. We conclude that the empirical equations used extensively in the literature are in reasonable agreement with our precise calculations of the hydrodynamic correlations of three spheres.