This book has grown out of a set of lecture notes for a graduate course in numerical analysis which I gave at the Computation Laboratory of Harvard University during the years 1947 to 1949. Professor Howard Aiken of that laboratory foresaw the great need that would be felt throughout the country for men trained in the design and use of electronic digital computers and initiated at that early date a set of courses to meet this need. I was privileged during that time to devote full time to the preparation of the course on numerical methods. It was felt that such a course should cover those topics most directly needed for an understanding of the methods used in the numerical solution of differential equations, both ordinary and partial, and in the solution of integral equations. Thus in this book, as in the lectures, considerable time is devoted to finite-difference tables and notation, to numerical differentiation and integration, but peripheral subjects such as the smoothing of experimental data, least-squares approximation, and harmonic analysis are omitted. These omissions made it possible to include in one book all the background material needed for obtaining numerical solutions to advanced problems in applied mathematics. While the main reason for writing this book has been to acquaint the student with the best procedures available for obtaining numerical solutions to problems arising in applied mathematics, methods of doubtful practicality have often been included to broaden the student's understanding of the essential unity of all finite difference methods. Since the goal sought is largely a pragmatic one, only a modest attempt has been made to preserve mathematical rigor and free use has been made of symbolic methods and heuristic arguments.