This long-awaited book by two of the foremost researchers and writers in the field is the first part of a treatise that covers the subject in breadth and depth, paying special attention to the historical origins, partly in applications, e. g. from geometrical optics, of parts of the theory. A variety of aids to the reader is provided: besides the very detailed table of contents, an introduction to each chapter, section and subsection, an overview of the relevant literature (in Vol. 2) plus the references in the Scholia to each chapter, in the (historical) footnotes, and in the bibliography, and finally an index of the examples used throughout the book. Both individually and collectively these volumes have already become standard references.