 The differences between the FG and Eu approaches to formalizing Euclid’s constructions can be understood as representing different general conceptions of the role of diagrams in mathematics. FG embodies a conception where diagrams concretely realize a range of mathematical possibilities. They support mathematical inference by furnishing direct access to these possibilities. Eu in contrast embodies a conception where diagrams serve to represent in a single symbol the various components of a complex mathematical situation. They support mathematical inference by allowing the mathematical reasoner to consider all these components in one place, and to focus on those components relevant to a proof.