This paper provides a critical analysis of the geometric and physical assumptions underlying the derivation of the Lorentz transformations. It is demonstrated that the classical derivation, based on the geometry of the Michelson-Morley interferometer arms, contains a fundamental kinematic error [?] caused by a misconception of the nature of the luminiferous medium (the World Ether). Applying the Pythagorean equation to the spatial displacements of a moving system, rather than utilizing a vector analysis of a spherical wave front of an elastic wave in the medium, led to a false hypothesis regarding the deformation of space-time. An alternative quaternion description of a wave packet in absolute Euclidean space is presented. The invalidity of the relativistic postulate of the constancy of the speed of light is proven by the results of long-term high-altitude experiments conducted by Dayton Miller, which recorded a real ether drift and the dynamics of a partial drag of the medium within a hydrodynamic boundary layer. The long-standing paradox of stellar aberration is rigorously resolved without invoking the four-dimensional Minkowski metric.