The question of teaching mathematics at high-school and university level has been debated ever since Felix Klein, at the end of the 19th century, presented his famous Erlanger program. Klein wanted to enliven teaching by giving prominence, in all fields of exact science-analysis, geometry and physicsto the concept of a group and to the principles of transformations. The aim of theory is that of uniting and studying the invariant elements which remain unchanged during group transformations. These invariant laws constitute the heart of the theory, what is often referred to as its mathematical structure. Naturally enough, Klein, who contributed greatly to the development of group theory, was inclined to overemphasize to some extent the importance of the group concept. In fact, during the subsequent progress of science, this concept has become a special case of the more general concept of mapping, which, influenced by the epoch-making ideas of Gauss and Riemann, is assuming an increasingly dominant position in science today. With this reservation, it can be agreed that Klein in his program clearly laid down a main line for present and future science. His program has contributed essentially to the development of teaching at the university level. These views of Klein must be given less prominence in the high school curriculum. Here, Klein's most important contribution takes another direction. He considered the time was ripe for an extension of high school mathematics so as to include the calculus, the basis of higher mathematics. The idea has gradually gained ground. Nowadays, the syllabus in almost all countries includes a section on the concept of limit, and the elements of infinitesimal calculus. The extent to which this involves actual progress is a controversial matter. In the experience of many university teachers, the inadequate knowledge of the calculus gained by mathematics students during their high school years seems rather to hamper them at university level, at which students are prepared for more advanced mathematics at a standard of strictness which is quite different. This sceptical attitude is not entirely unjustified. Nevertheless, a return to the former state of affairs is neither possible nor desirable. By virtue of the constantly growing significance of mathematics as a fundamental technical instrument in many different fields where the chief emphasis is laid on the application of the calculus-rather than on complete mastery of the logical structure of the technique-it is impossible to deny the need for certain, if no more than superficial, knowledge of the first principles of the infinitesimal calculus before and on university entrance. Since the beginning of this century, mathematics has expanded enormously in various directions. Modern scientific progress is based essentially upon the