The Unified Modeling Language (UML) is likely to be-come an important industry standard language for modelling object-oriented systems. However, its use as a precise analysis tool is limited due to a lack of precise semantics and practical analysis techniques. This paper proposes a rigorous analysis technique for UML based on the use of diagrammatical transformations. A precise description of a subset of UML class diagrams is presented. This is then used to identify a number of deductive transformations on class diagrams. Conditions for checking the soundness of the rules are also given. Because the reasoning system is based on the manipulation of diagrams, it is proposed that they can be successfully used by practitioners without recourse to complex linguistic proof techniques.