  The consistent histories formalism is discussed using path-projected states.
These are used to analyse various criteria for approximate consistency. The
connection between the Dowker-Halliwell criterion and sphere packing problems
is shown and used to prove several new bounds on the violation of probability
sum rules. The quantum Zeno effect is also analysed within the consistent
histories formalism and used to demonstrate some of the difficulties involved
in discussing approximate consistency. The complications associated with null
histories and infinite sets are briefly discussed.
