On the role of mass in the mathematical structure of Newtonian and special relativistic mechanics

We consider five-dimensional real linear spaces with a (otherwise well-known) linear action of the Galilei and the Poincare group on them, describe the geometry of these two spaces, and show, that these geometries comprise the notions of space-time, mass, momentum, force and physical dimensions in a natural way. In this way we geometrize the quantity of mass and integrate it together with space-time into two geometries in a natural way, so that these geometries are perfectly suitable for underlying the Newtonian and special relativistic mechanics of pointlike bodies.