  We define the {\it Wirtinger number} of a link, an invariant closely related
to the meridional rank. The Wirtinger number is the minimum number of
generators of the fundamental group of the link complement over all meridional
presentations in which every relation is an iterated Wirtinger relation arising
in a diagram. We prove that the Wirtinger number of a link equals its bridge
number. This equality can be viewed as establishing a weak version of Cappell
and Shaneson's Meridional Rank Conjecture, and suggests a new approach to this
conjecture. Our result also leads to a combinatorial technique for obtaining
strong upper bounds on bridge numbers. This technique has so far allowed us to
add the bridge numbers of approximately 50,000 prime knots of up to 14
crossings to the knot table. As another application, we use the Wirtinger
number to show there exists a universal constant $C$ with the property that the
hyperbolic volume of a prime alternating link $L$ is bounded below by $C$ times
the bridge number of $L$.
