  We discuss the use of histories labelled by a continuous time in the approach
to consistent-histories quantum theory in which propositions about the history
of the system are represented by projection operators on a Hilbert space. This
extends earlier work by two of us \cite{IL95} where we showed how a continuous
time parameter leads to a history algebra that is isomorphic to the canonical
algebra of a quantum field theory. We describe how the appropriate
representation of the history algebra may be chosen by requiring the existence
of projection operators that represent propositions about time average of the
energy. We also show that the history description of quantum mechanics contains
an operator corresponding to velocity that is quite distinct from the momentum
operator. Finally, the discussion is extended to give a preliminary account of
quantum field theory in this approach to the consistent histories formalism.
