  Weyl wanted a metric geometry which would not permit distance  comparison of length between two vectors located at finitely  different points. In a pure infinitesimal geometry, Weyl  argued, if attention is restricted to a single point of the manifold, then some standard of length or gauge must be chosen  arbitrarily before the lengths of vectors can be determined.  Therefore, all that is intrinsic to the notion of a pure  infinitesimal metric differential geometry is the ability to  determine the ratios of the lengths of any two vectors and  the angle between any two vectors, at a point. Such a pure  infinitesimal metric manifold must have at least a  conformal structure \(C\). 