There have been two earlier Cambridge Tracts that have touched upon the Hardy–Littlewood method, namely those of Landau, 1937, and Estermann, 1952. However there has been no general account of the method published in the United Kingdom despite the not inconsiderable contribution of English scholars in inventing and developing the method and the numerous monographs that have appeared abroad.

The purpose of this tract is to give an account of the classical forms of the method together with an outline of some of the more recent developments. It has been deemed more desirable to have this particular emphasis as many of the later applications make important use of the classical material.

It would have been useful to devote some space to the work of Davenport on cubic forms, to the joint work of Davenport and Lewis on simultaneous equations, to the work of Rademacher and Siegel that extends the method to algebraic numbers, and to the work of various authors, culminating in the recent work of Schmidt, on bounds for solutions of homogeneous equations and inequalities. However this would have made the tract unwieldy. The interested reader is referred to the Bibliography.

It is assumed that the reader has a familiarity with the elements of number theory, such as is contained in the treatise of Hardy and Wright. Also, in dealing with one or two subjects it is expected that the reader has a working acquaintance with more advanced topics in number theory.