We have made a substantial number of small additions to the text and a small number of substantial ones. The small additions are usually in the form of extra exercises and applications which have occurred to us, or have otherwise come to our attention, since the first edition went to press. For example, in chapter 7 we give a pleasant application of caustics and orthotomics to seismology and a discussion of developable surfaces which generalizes the previous material on the developable of a space curve. Our criterion for the inclusion of these extras has always been that they should further illuminate the theory in a way that is geometrically satisfying.

There are several more substantial additions, some of which are complete expositions of topics, while others are sketches intended to whet the reader's appetite for more advanced treatments. An example of the former is the discussion of the Morse lemma in chapter 4. It seemed to us worth while to include, in the first half of the book, a full treatment of the simplest case of classification of a function of two or more variables, namely the case of nondegenerate or Morse functions. This has a number of striking geometrical applications, and some of these are given as exercises.