  The problem of determining whether or not a non-elementary subgroup of
$PSL(2,\CC)$ is discrete is a long standing one. The importance of two
generator subgroups comes from J{\o}rgensen's inequality which has as a
corollary the fact that a non-elementary subgroup of $PSL(2,\CC)$ is discrete
if and only if every non-elementary two generator subgroup is. A solution even
in the two-generator $PSL(2,\RR)$ case appears to require an algorithm that
relies on a the concept of {\sl trace minimizing} that was initiated by
Rosenberger and Purzitsky in the 1970's Their work has lead to many
discreteness results and algorithms. Here we show how their concept of trace
minimizing leads to a theorem that gives bounds on the hyperbolic lengths of
curves on the quotient surface that are the images of primitive generators in
the case where the group is discrete and the quotient is a pair of pants. The
result follows as a consequence of the Non-Euclidean Euclidean algorithm.
