The surfaces capable of division into Infinitesimal Squares by their Curves of Curvature

Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory better, it is described how to give the classical characterization a rigorous meaning in the sense of modern differential geometry.