in the case of one or more equations with many variables; the only thing unusual about this naming is, that I keep it when no equations between the variables is given whatsoever. In this case I call the total (set) of solutions the -fold totality; whereas when equations are given, the total of their solutions is called respectively (an) -fold, -fold, -fold, ... Continuum. From the notion of the solutions contained in a totality comes forth that of the independence of their relative positions (of the variables) in the system of variables used, insofar as new variables could take their place by transformation. This independence is expressed in the inalterability of that, which I call the distance between two given solutions (), () and define in the easiest case by: while at the same time I call a system of variables orthogonal [...] We can see how he is still thinking of points in -dimensional space as solutions to linear equations, and how he is considering a system without any equations, thus obtaining all possible points of the , as we would put it now. He disseminated the concept in the articles he published in the 1850s and 1860s, and it matured rapidly. By 1867 he starts an article by saying "We consider the space of -tuples of points. [...]". This indicates not only that he had a firm grip on things, but also that his audience did not need a long explanation of it. Polytopes In Theorie der Vielfachen Kontinuität he goes on to define what he calls polyschemes, nowadays called polytopes, which are the higher-dimensional analogues to polygons and polyhedra. He develops their theory and finds, among other things, the higher-dimensional version of Euler's formula. He determines the regular polytopes, i.e. the -dimensional cousins of regular polygons and platonic solids. It turns out there are six in dimension four and three in all higher dimensions. Although Schläfli was familiar to his colleagues in the second half of the 19th century, especially for his contributions to complex analysis, his early geometrical work failed to attract notice for many years. At the beginning of the twentieth century Pieter Hendrik Schoute started to work on polytopes together with Alicia Boole Stott. She reproved Schläfli's result on regular polytopes for dimension 4 only and afterwards rediscovered his book. Later Willem Abraham Wijthoff studied semi-regular polytopes and this work was continued by H.S.M.