  Riemann extends to n dimensions the methods employed by Gauss (1828) in his study of the intrinsic geometry of curved surfaces embedded in Euclidean space (called ‘intrinsic’ because it describes the metric properties that the surfaces display by themselves, independently of the way they lie in space). Looking back at Gauss's work one gets a better intuitive feel for Riemann's concepts (see Torretti 1978, pp. 68–82). However, for the sake of conciseness and perspicuity, it is advisable to look forward and to avail oneself of certain concepts introduced by later mathematicians as they tried to make sense of Riemann's proposal. Consider the modern formulation of Riemann's theory in the supplement  A Modern Formulation of Riemann's Theory.