"If our geometry is to resemble differential geometry we must adjoin some uniqueness properties. Now in those geometries the geodesics, and more generally the externals in the calculus of variations, are given by differential equations of the second order, and under the hypotheses usually made in those fields, there is just one solution through a given line element. Thus a geodesics has a unique prolongation, though the shortest geodesic are joining two points even on simple surfaces such as the sphere, need not be unique."