ESHS/HSS Annual Meeting

Dotted Lines and the Reconceptualization of Quantity in Early Modern Editions of Euclid’s Elements

• In Event: Pluralizing the Heritage: Ancient Authors Between Transmission, Contestation, and Reinvention

English Abstract

This paper examines how the evolving visual language of dotted lines in early modern editions of Euclid’s Elements participated in reshaping the concept of quantity. Though the Elements remained a canonical mathematical work since antiquity, its content and presentation underwent continuous adaptations. Scholarship mapped many textual revisions, but its diagrams, nowadays recognized as integral to mathematical reasoning, remained understudied. This paper demonstrates that the various uses of dotted lines across the Elements’ editions offer a lens through which to trace how early modern editions worked at the boundary between arithmetic and geometry notions of quantity toward a more abstract and unified conception. It further argues that the visual strategies filtering in from practical and didactic oriented editions, as well as pragmatic considerations of print, were part of the early modern reconceptualisation of quantity.
The study draws on dozens of printed editions of the Elements published before 1650 together with a secondary group of mathematical treatises. Its methodology combines digital humanities in a two-step process: automatic extraction and classification of diagrams from hundreds of pages per edition followed by close reading that situates each diagram within its context.
The findings show that dotted segments appear in the arithmetic sections of the earliest printed editions. There, each dot stood for a unit and supplied readers with concrete numerical examples that the text itself lacked. As editors confronted challenges such as representing large numbers or reductio ad absurdum proofs, the vocabulary expanded, Hindu-Arabic numerals were used with increasing flexibility and visual devices previously constrained to the geometric diagrams appeared in the arithmetic ones. Toward the end of the sixteenth century dotted lines entered the geometric sections as pedagogical tools and in relation to concrete geometrical applications. Furthermore, textbooks relying on moveable type and woodcuts further constrained the available visual vocabulary, shaping readers’ horizons of expectations and editors’ strategies.
By the early seventeenth century the two visual representations of quantity increasingly intersected. The changing uses of dotted lines make visible how arithmetic and geometric conceptions of quantity were negotiated through pedagogy, print constraints and diverse readerships rather than through a single coherent mathematical programme.

Author

• Mia Joskowicz, Tel-Aviv University - CNRS-SPHère Paris