The manner in which projective space can be derived from Euclidean by adjoining a hyperplane at infinity is familiar. B6cherl has shown that conformal space (i.e. space of the geometry of inversion)2 can be derived by adjoining instead a cone at infinity. In the two-dimensional case, the cone is merely a pair of intersecting lines, and B6cher remarks that the complete space is perfectly representable on an ordinary quadric surface. In this representation, generators of the quadric correspond to lines, and so do conic sections by planes through a definite point 0 of the quadric; the generators through 0 give the two lines at infinity. The object of the present paper is to work out the analogous representation of the conformal spaces of three and four dimensions on the quadric hypersurfaces in projective space of four and five dimensions, the cone at infinity corresponding in each case to the tangent section at a special point 0 of the hypersurface. When we have established this correspondence between conformal geometry and the geometry of a quadric, we find that each may be a help to our understanding of the other. It is our habitual use of Cartesian coordinates that makes us regard a hyperplane as the natural element with which to augment Euclidean space. If instead we use the coordinates that for n = 2 and 3 are called tetracyclic3 and pentaspherical,4 we are as unequivocally led to the ideal cone. The notion of undertaking this work, and the principal facts about conformal space, were given to me by Professor 0. Veblen, to whom I would express my gratitude.