The treatment of finite systems of equations, as reflected in the 1952 course from which the first edition of this book developed, has been completely superseded by the genera] theory of equations in Banach spaces. Thus it was felt that the old title was no longer adequate. My intent with this new edition has been to bring the reader up to date with the developments in this theory, correct what I felt were some deficiencies in both previous editions, and to retain my original hope, as expressed in the preface to the first edition, that the book would "help to a certain degree to bridge the gap that still exists between 'pure' and 'practical' mathematics."