The general Lorentz transformations are derived on the basis of a set of postulates which are fully consistent with the underlying ideas of Newtonian mechanics, but which of course do not include the independent postulate of absolute simultaneity. Thus, the assumption of the constancy of the vacuum speed of lightcis not introduced as a starting point. Instead, the given development leads to a universal critical velocity, and the two alternatives of its value being infinite or finite, correspond to the Newtonian and the relativistic kinematics, respectively, with the critical velocity identified with the experimental valuecin the latter case.