The normal road to geometry: \(\delta\acute{\eta}\) in Euclid's \textit{Elements} and the mathematical competence of his audience (Q2834388)

A pragmatic analysis of the use of \(\delta\acute{\eta}\) in Euclid's \textit{Elements}, limited to Books 1, 7, and 9 (given that these ``contain relatively simple and short propositions, which allows for an analysis that is not unnecessarily complex on account of mathematical difficulties''). There are 151 cases of \(\delta\acute{\eta}\) within those books, from a total of 703 in all of the \textit{Elements}. In 142 of those 151 occurrences, one encounters its use in one of the following five uses: (1) ``proceeding to the next step''; (2) ``transferring from enunciation to proofsetting''; (3) ``analogous cases''; (4) ``exclusive disjunctions''; (5) ``rephrasing''. The function of each of these in their respective proofs, as well as that of the isolated cases, belonging to none of the five uses, are analyzed in detail.NEWLINENEWLINEThe overall conclusion is that ``In its most abstract sense, the use of \(\delta\acute{\eta}\) in Euclid's \textit{Elements} can be tentatively characterized as signalling that Euclid expects the statement to be obvious in some sense to those audience members who are familiar with mathematical proof procedures.'' This also indicates that a ``standardization [of proof structures] had developed to such an extent that Euclid could confidently express the expectation of his audience's familiarity with the unwritten rules of mathematical proof procedures''.