certain rational varieties (spaces of straight lines, of conics, etc. ), whereas we shall emphasize the geometry on an arbitrary variety, or at least on a variety without multiple points. The theory of intersection-multiplicities, however, occupies such a centrat position among the topics which constitute the founda­ tions of algebraic geometry, that a complete treatment of it necessarily supplies the tools by which many other such topics can be dealt with. In deciding be­ tween alternative methods of proof for the theorems in this book, consistency, and the possibility of applying these methods to further problems, have been the main considerations; for instance, one will find here all that is needed for the proof of Bertini's theorems, for a detailed ideal-theoretic study (by geometric means) of the quotient-ring of a simple point, for the elementary part of the theory of linear series, and for a rigorous definition of the various concepts of equivalence. In consequence, the author has deliberately avoided a few short cuts; this is not to say that there may not be many more which he did not notice, and which our readers, it is hoped, may yet discover. Our method of exposition will be dogmatic and unhistorical throughout, formal proofs, without references, being given at every step.