The rheology of suspensions of Brownian, or colloidal, particles (diameter~$d \lesssim 1~\mu$m) differs markedly from that of larger grains ($d \gtrsim 50~\mu$m). Each of these two regimes has been separately studied, but the flow of suspensions with intermediate particle sizes ($1~\mu\textrm{m} \lesssim d \lesssim 50~\mu$m), which occur ubiquitously in applications, remains poorly understood. By measuring the rheology of suspensions of hard spheres with a wide range of sizes, we show experimentally that shear thickening drives the transition from colloidal to granular flow across the intermediate size regime. This insight makes possible a unified description of the (non-inertial) rheology of hard spheres over the full size spectrum. Moreover, we are able to test a new theory of friction-induced shear thickening, showing that our data can be well fitted using expressions derived from it.