The following questions are discussed: how can measures be used when lengths are not directly measurable? Is there a way of resolving the differences, between the infinite process of approximation, and the problem of incommensurability? what sort of new « rationality » would allow us to treat the irrational ratios of incommensurable magnitudes as numbers? During this course of history, these questions sometimes made demands on our three protagonists: numbers, magnitudes and measures. This article is aimed at teachers of mathematics for use as a means of introducing a historical perspective into the teaching of mathematics. It also contains exercises to be solved according to ancient and modern methods. The chapter ends with a bibliography which contains, in addition to the historical sources that have been used, a certain number of books recommended for further study of the subject. (ZDM/Mathdi)