7.1. In the last six chapters we have analysed Cauchy's work on complex function theory from 1814 to 1831, and we have indicated some of the background to the development of his ideas. In the present chapter we shall try to draw the threads together. There are several themes that recur throughout the story; we shall look at these in turn, and try to trace how Cauchy's approach to each of them changed during the period.

To begin with, we shall have to say something about the kind of functions that Cauchy regarded as admissible, and then about the development of his concept of the integral. After a brief sketch of some earlier ideas about complex functions and about techniques for the evaluation of definite integrals, we shall show how, by combining some of these ideas, he was able to obtain results equivalent to special cases of ‘Cauchy's theorem’ and the residue theorem. We shall then follow the development of his techniques, which enabled him to establish more general forms of these results, and we shall indicate the role played here by his tentative introduction and gradually increasing use of a geometrical picture of the complex plane.