The theory of manifolds goes back to Riemann’s lecture “On the hypotheses which lie at the foundations of geometry,” delivered in 1854 at the University of Gottingen. In fact, it was precisely the efforts to clarify and deepen Riemann’s ideas (as understood by his successors) that led to manifolds and Riemannian spaces as we understand them today. Nevertheless, Riemann himself gave examples of manifolds in his sense (e.g., “the possibilities for a function in a given region”) which are not (finite dimensional) manifolds in the modern sense. Here we touch upon one of the limitations of the category ~2 of P-manifolds and P-maps: A? is not Cartesian closed; in particular the space of P-maps between two P-manifolds is not a P-manifold. A limitation of a different nature is the absence of a convenient language to describe things in the “infinitely small.” In particular infinitesimals, which had played such an important role in analysis and geometry until the beginning of this century, have been exorcized by the modern theory of manifolds, although they are still mentioned as a heuristic help in understanding. In this way, they have been forced to play, literally, the role of “ghosts of departed quantities” that Bishop Berkeley had assigned them. If we look at the works of geometers like Darboux, Lie, and Cartan, as well as those of contemporary engineers and physicists, we find (at least) two kinds of infinitesimals; the nilpotent injhitesimals (e.g., “first-order infinitesimals”) which are used to deal with notions like forms and parallel transport, and the invertible infinitesimals, employed for instance in the theory of improper functions of which the S function of Dirac is the best known example. Furthermore, these invertible infinitesimals come together with infinitely large natural numbers, used already by Leibniz and Euler to deal with series, infinite products, and the like. Several attempts have been made to remove (some of) the limitations of 229 OOOl-8708/87 $7.50