  A canonical analysis of the Einstein-Hilbert action S_d (d>2) is considered,
using the first order form with the metric and affine connection as independent
fields. We adopt a conservative approach to using the Dirac constraint
formalism; we do not use equations of motion which are independent of time
derivatives and correspond to first class constraints to eliminate fields.
Applying the Dirac procedure, we find that the primary constraints lead to
secondary constraints which are equations of motion not involving time
derivatives, and that those secondary constraints which are first class imply
novel tertiary constraints which are also first class. Once the constraints and
their associated gauge conditions are used to eliminate the non-dynamical
degrees of freedom in S_d, there are d(d-3) degrees of freedom left in phase
space. We also consider the simpler limiting case of the non-interacting
graviton in the first order formalism as well as the effect of adding the
action for a massless scalar field to the Einstein-Hilbert action.
