  This paper develops some of the methods of the "Italian School" of algebraic
geometry in the context of infinitesimals. The results of this paper have no
claim to originality, they can be found in Severi, we have only made the
arguments acceptable by modern standards. However, as the question of rigor was
the main criticism of their approach, this is still a useful project. The
results are limited to algebraic curves. As well as being interesting in their
own right, it is hoped that these may also help the reader to appreciate their
sophisticated approach to algebraic surfaces and an understanding of
singularities. The constructions are also relevant to current research in
Zariski structures, which have played a major role both in model theoretic
applications to diophantine geometry and in recent work on non-commutative
geometry.
