The principle of equivalence says that at any given space–time point, there exists a local coordinate system with respect to which the three-acceleration of a freely falling test body vanishes regardless of its three-velocity. In this article, a more intrinsic and geometric criterion for free fall motion is provided. The criterion is much simpler than a recently proposed criterion that is based on the Desargues property. For space–times of dimension greater than 2, the Desargues property is a theorem. It is shown that it suffices to require that a version of Pasch’s axiom is satisfied; that is, two paths in a plane intersect up to corrections of order ε3, where ε is the scale parameter of a shrinking process.