Introduction. It is a classical problem to construct a complex projective space, using as ‘imaginary points’ objects of some kind from the geometry of real projective space. (The various solutions to this problem have been reviewed in a book by Coolidge ((1)).) The object of the present paper is to solve the corresponding problem for n-dimensional projective spaces over two skew fields, one being a second-rank extension of the other. The method used seems not to have been published before, even in connexion with the classical problem.