  A generalisation of Riemannian geometry is considered, based exclusively on
the minimal assumptions that the line element $ds$ is a regular function of
position and direction and that the distance of every point from itself is
equal to zero. Besides the Riemannian line element, also Riemann's residual
hypotheses of homogeneity and symmetry are dropped. Surprisingly, the
infinitesimal Pythagorean distance formula reemerges, without the need of being
postulated, as a first approximation to $almost$ $every$ geometry that is
invariant with respect to direction reversal. More in general, the first
approximation to $almost$ $every$ geometry is a one-parameter family of
homogeneous Riemann-Randers line elements, naturally providing the geometrical
framework for a unified theory of the classical electromagnetic and
gravitational fileds. Geometry naturally accounts for the hierarchy between
electromagnetic and gravitational interactions, their different
attractive/repulsive nature, electric charge, CPT symmetry, Maxwell and
Einstein equations. Within this framework higher order terms could describe new
spacetime degrees of freedom of possible large-scale relevance.
