  By measuring the density fluctuations in a highly elongated weakly
interacting Bose gas, we observe and quantify the transition from the ideal gas
to a quasi-condensate regime throughout the dimensional crossover from a purely
1D to an almost 3D gas. We show that that the entire transition region and the
dimensional crossover are described surprisingly well by the modified Yang-Yang
model. Furthermore, we find that at low temperatures the linear density at the
quasi-condensate transition scales according to an interaction-driven scenario
of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales
according to the degeneracy-driven critical scenario of transverse condensation
of a 3D ideal gas.
