The pure mathematician makes no attempt to imagine a space of four dimensions; he lays no claim to visualising a world that is inconceivable to other men. Only he finds that certain notions in algebra are discussed most readily in terms adopted from geometry and given a meaning entirely algebraic, and since it is to the mathematician alone that algebraic problems are of concern in themselves, fear lest the man in the street should mistake the very subject of a mathematical conversation he might overhear has not prevented the mathematician from using the vocabulary he finds best suited to his own needs.