The first edition of this book gave a systematic exposition of the Weinstein method of intermediate problems for calculating lower bounds of eigenvalues. From the reviews of this edition and from subsequent shorter expositions it has become clear that the method is of considerable interest to the mathematical world. The method of intermediate problems consists of finding a base problem, constructing intermediate problems, and solving the intermediate problems, theoretically and numerically, in terms of the base problem. In the last thirty years fundamental progress has been made in all three steps, but the logical scheme remains the same as given originally by Weinstein. The method was investigated and extended in a profound way by Aronszajn and in recent years the interest in it has been greatly increased by the success of Bazley, Bazley and Fox, and others (see, for example, Bazley (1, 2) and Bazley and Fox (1-10)) in simplifying and extending the numerical applications, particularly in quantum mechanics.