Dotted lines and the reconceptualization of quantity in early modern editions of Euclid’s Elements

This paper examines the use of dotted lines in the diagrams of early modern printed editions of Euclid’s Elements and argues that their changing roles are closely tied to shifting conceptions of quantity. Based on a corpus of sixty extant editions published before 1650, the study traces the distribution, functions, and transformations of dotted lines across different editions and editorial contexts. In the arithmetical sections of the Elements , dotted lines represent discrete units and instantiate specific numerical examples. However, their use encounters practical and conceptual constraints in cases involving large values or hypothetical constructions, leading editors to adopt alternative visual strategies. In later editions, dotted lines are increasingly supplemented or replaced by Hindu–Arabic numerals, supporting more abstract representations of numbers and the relations between them. In the geometrical books, dotted lines appear later and mark continuous quantities, serving a range of visual functions and becoming ubiquitous in the seventeenth century. Close analysis of Book X shows how different representational conventions are integrated within a single diagram. Methodologically, the paper combines close visual reading with a digital workflow for large-scale diagram extraction and classification, allowing a systematic analysis. The findings show that dotted lines function as a flexible visual resource whose changing uses reflect and support broader shifts in the conceptualization of quantity in early modern mathematics.