Commentary on the first book of _"Elements"_ — Definitions

This commentary examines the definitions of Book I of Euclid’s Elements as a structured system for determining the nature and reference of geometric entities. Rather than treating the definitions as isolated descriptions, it investigates the conceptual order established between fundamental structures, such as points, lines, and surfaces, and the progressively more complex entities formed from them. Drawing on the ancient and late-antique commentarial traditions, the commentary distinguishes between entities that function as principles of geometric determination and those whose applicability presupposes an already individuated object. This distinction is formulated through the concepts of referential constituents and referential derivatives: a referential constituent is that without which the reference of a geometric entity cannot be fixed according to its definition, whereas a referential derivative characterizes an entity whose reference has already been established. The analysis is applied particularly to Euclid’s definitions of the point, line, surface, angle, figure, and circle, with special attention to the status of relations and events in the constitution of geometric objects. The resulting framework proposes that Euclidean definitions encode not merely properties of figures, but an ordered process of geometric individuation and classification. In this sense, the definitions provide the semantic foundation upon which the deductive structure of the Elements is subsequently developed.