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)