Variations on a theme : Clifford’s parallelism in elliptic space

In 1873, W. K. Clifford introduced a notion of parallelism in the three-dimensional elliptic space that, quite surprisingly, exhibits almost all properties of Euclidean parallelism in ordinary space. The purpose of this paper is to describe the genesis of this notion in Clifford’s works and to provide a historical analysis of its reception in the investigations of F. Klein, L. Bianchi, G. Fubini, and E. Bortolotti. Special emphasis is placed upon the important role that Clifford’s parallelism played in the development of the theory of connections.