A theoretical basis is provided for the empirical equations of downstream hydraulic geometry, written as scaling relations, using the equations for momentum, flow resistance, and continuity of gradually varied open channel flow. Width, depth, velocity, and bed slope are expressed as power functions of water discharge and bed sediment size. These theoretical relations apply in alluvial, ice, and bedrock channels and hold for any value of water discharge as opposed to just mean or bank-full values used in empirical equations. A scaling relation for bed load transport rate in alluvial channels is also derived using dimensional analysis. Threshold and regime theories of stable channel design are shown to be consistent with the requirements of hydraulic similitude as defined by the scaling relations. If field measurements confirm that the scaling relations apply in braided alluvial channels, then predictions of a threshold bed slope between states of meandering and braiding will not be valid.