Everybody can sympathise with Cauchy's students who just wanted to pass their exams and with his colleagues who just wanted the standard material taught in the standard way. Most people neither need nor want to know about rigorous analysis. But there remains a small group for whom the ideas and methods of rigorous analysis represent one of the most splendid triumphs of the human intellect. We echo Cauchy's defiant preface to his printed lecture notes.
As to the methods [used here], I have sought to endow them with all the rigour that is required in geometry and in such a way that I have not had to have recourse to formal manipulations. Such arguments, although commonly accepted ... cannot be considered, it seems to me, as anything other than [suggestive] to be used sometimes in guessing the truth. Such reasons [moreover] ill agree with the mathematical sciences' much vaunted claims of exactitude. It should also be observed that they tend to attribute an indefinite extent to algebraic formulas when, in fact, these formulas hold under certain conditions and for only certain values of the variables involved. In determining these conditions and these values and in settling in a precise manner the sense of the notation and the symbols I use, I eliminate all uncertainty. ... It is true that in order to remain faithful to these principles, I sometimes find myself forced to depend on several propositions that perhaps seem a little hard on first encounter .... But, those who will read them will find, I hope, that such propositions, implying the pleasant necessity of endowing the theorems with a greater degree of precision and restricting
statements which have become too broadly extended, will actually benefit analysis and will also provide a number of topics for research, which are surely not without importance.