This is essentially a prolegomenon to the Partial Preface and serves primarily
to place in a proper context the contents of this book and how they relate to the
original six chapters, to which it refers.
Those six chapters, originally by W. F. Eberlein, sought to relate the ideas of
Abel to the later work of Jacobi and concluded with the transformation theory
of the theta functions. The first chapter began with the differential equation
associated with the motion of a simple pendulum, very much in the tradition of
Greenhill’s‘Applications of Elliptic Functions’ (1892), but much influenced by
the spirit of modern analysis. (Greenhill’s obituary reads that ‘his walls (were)
festooned with every variety of pendulum, simple or compound.’1) The version
given here is inspired by those early chapters and, apart from the addition of
illustrative examples and extra exercises, is essentially unchanged