  Let M be a geometrically finite pinched negatively curved Riemannian manifold
with at least one cusp. Inspired by the theory of diophantine approximation of
a real (or complex) number by rational ones, we develop a theory of
approximation of geodesic lines starting from a given cusp by ones returning to
it. We define a new invariant for M, the {\it Hurwitz constant} of M. It
measures how well all geodesic lines starting from the cusp are approximated by
ones returning to it. In the case of constant curvature, we express the Hurwitz
constant in terms of lengths of closed geodesics and their depths outside the
cusp neighborhood. Using the cut locus of the cusp, we define an explicit
approximation sequence for geodesic lines starting from the cusp and explore
its properties. We prove that the modular once-punctured hyperbolic torus has
the minimum Hurwitz constant in its moduli space.
