This work provides a lucid and rigorous account of the foundations of algebraic geometry. The authors have confined themselves to fundamental concepts and geometrical methods, and do not give detailed developments of geometrical properties but geometrical meaning has been emphasised throughout. Here in this volume, the authors have again confined their attention to varieties defined on a ground field without characteristic. In order to familiarize the reader with the different techniques available to algebraic geometers, they have not confined themselves to one method and on occasion have deliberately used more advanced methods where elementary ones would serve, when by so doing it has been possible to illustrate the power of the more advanced techniques, such as valuation theory. The other two volumes of Hodge and Pedoe's classic work are also available. Together, these books give an insight into algebraic geometry that is unique and unsurpassed.