We investigate the dynamics of soft sphere liquids through computer simulations for spatial dimensions from d = 3 to 8, over a wide range of temperatures and densities. Employing a scaling of density-temperature dependent relaxation times, we precisely identify the density φ0 which marks the ideal glass transition in the hard sphere limit, and a crossover from sub-to super-Arrhenius temperature dependence. The difference between φ0 and the athermal jamming density φJ , small in 3 and 4 dimensions, increases with dimension, with φ0 > φJ for d > 4. We compare our results with recent theoretical calculations.