  The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem
that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work
of R. Baer was carried out on closed, orientable surfaces and extended by D. B.
A. Epstein to arbitrary surfaces, compact or not, with or without boundary and
orientable or not. We give a new method of deducing the theorem about homotopic
homeomorphisms from the results about homotopic curves via the hyperbolic
geometry of surfaces. This works on all but 13 surfaces where ad hoc proofs are
needed.
