The normal road to geometry: Euclid's Elements and the mathematical competence of his audience

Euclid famously stated that there is no royal road to geometry, but his use of δ® does give an indication of the minimum level of knowledge and understanding which he required from his audience. The aim of this article is to gain insight into his interaction with his audience through a characterization of the use of δ® in the Elements. I will argue that the primary use of δ® indicates a lively interaction between Euclid and his audience. Furthermore, the specific contexts in which δ® occurs reveal the considerable mathematical competence that Euclid expected from his audience.