  We are concerned with mapping class groups of surfaces with nonempty
boundary. We present a very natural method, due to Thurston, of finding many
different left orderings of such groups. The construction involves equipping
the surface with a hyperbolic structure, embedding the universal cover in the
hyperbolic plane, and extending the action of the mapping class group on it to
its limit points on the circle at infinity. We classify all orderings of braid
groups which arise in this way. Moreover, restricting to a certain class of
``nonpathological'' orderings, we prove that there are only finitely many
conjugacy classes of such orderings.
