 Repudiating any use of spatial or temporal intuition in an exact determination of the continuum, Cantor undertakes its precise arithmetical definition. Making reference to the definition of real number he has already provided (i.e., in terms of fundamental sequences), he introduces the \(n\)-dimensional arithmetical space \(G_n\) as the set of all \(n\)-tuples of real numbers \(\langle x_1,x_2 ,\ldots ,x_n\rangle\), calling each such an arithmetical point of \(G_{n.}\) The distance between two such points is given by