  We have analytically determined the refractive index for the mechanical
refraction of a relativistic particle for its all possible speeds. We have
critically analysed the importance of Descartes' metaphysical theory and
extended it in this regard. We have considered the conservation of the
tangential component of the relativistic momentum and the relativistic energy
of the particle in the process of the mechanical refraction within the
optical-mechanical analogy. Our result for the mechanical refractive index
exactly matches with the forms of both the Fermat's result on Snell's law of
optical refraction at the ultra-relativistic limit and the Descartes'
metaphysical result on the pseudo-Snell law of optical refraction at the
non-relativistic limit.
