The revolution in surface theory had been initiated by Gauss in the first half of the nineteenth century. Before Gauss, surface theory meant the study of surfaces in ordinary space ℝ3. Gauss’s Theorema Egregium (Theorem 17.5, page 536), stating that the curvature K of a surface in ℝ3 does not depend on how it is embedded in ℝ3, led mathematicians such as Riemann to question the necessity of a surface being embedded in ℝ3 in the first place. In this way, the idea of an abstract surface was born. Whilst regular surfaces in ℝ3 are abstract surfaces (see page 858), there exist abstract surfaces that cannot be viewed inside ℝ3.