  We present quantitative measurements of the spatial density profile of
Bose-Einstein condensates of sodium atoms confined in a new '4D' magnetic
bottle. The condensates are imaged in transmission with near resonant laser
light. We demonstrate that the Thomas-Fermi surface of a condensate can be
determined to better than 1%. More generally, we obtain excellent agreement
with mean-field theory. We conclude that precision measurements of atomic
scattering lengths and interactions between phase separated cold atoms in a
harmonic trap can be measured with high precision using this method.
