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During their first years at the university the students mainly encounter that part of analysis that was developed in the 17th and 18th centuries. The focus is on computations and very little time is spent on theoretical questions and proofs. The results are rather concrete and most students may feel that they are intuitively correct,or even self evident. So the need for proofs do not seem so urgent. But in the 19th century there was a revolutionary development in analysis, algebra and logic. Old concepts where abstracted and generalized and there was a greater need for rigorous proofs of their new highly non trivial results. It is this remarkable evolution of mathematics that this course reflects.