"Gauss (1828) pointed out that there are two distinct but complimentary ways of thinking about the of surfaces. ...this point ...is absolutely fundamental to a clear understanding of the subject. Gauss's dual view of curvature fits precisely the 'two surface' description, and it provides succinctly the geometrical conditions which are necessary if the deformation of the two surfaces is to match. The key variable in this connection turns out to be a scalar quantity; and paradoxically... conventional treatment... in terms of general is... too elaborate to reveal this crucial... quantity."