We study excitons bound to quantum wires of InxGa1-xAs embedded in an InP matrix, where the wires vary from 2.93 to 11.72 angstrom (one to four monolayers) thick and from 25 to 250 angstrom wide. We combine spectroscopic data from measurements of photoluminescence with variational calculations of the binding energies of excitons to the wires to deduce the wire widths and thicknesses. The widths are then related to the growth times to deduce lateral growth velocities in the varpor-levitation-epitaxial technique. Monolayer growth rates, at approximately 80 angstrom/sec, are significantly faster than growth rates for the multilayer wires.