One of the fundamental difficulties in teaching the calculus is that we are obliged to introduce the concepts of the Theory of Functions and yet make no use of the real number theory on which the Theory of Functions is based. We must talk of limits, continuous functions, derivatives, definite integrals and of existence theorems like Rolle’s and Taylor’s, and must endeavour to hide the deficiencies of our reasoning beneath the kindly cloak of geometrical intuition. The Calculus course is usually a compromise, completely rigorous proofs of the basic theorems being deemed too difficult for the type of student for whom the course is planned, but I believe a far more rigorous account than the customary one is possible without greatly increasing the strain on the students’ powers of understanding, this can be achieved, however, only by a new approach to the subject and it is with the first steps in such an approach that we are here concerned.