ABSTRACT
This chapter explores an important and yet neglected episode in the history of the foundations of mathematics, and its relations with the transmission and transformation of the text of Euclid’s Elements: the early modern debate on the continuity of geometrical figures and the theory of intersections in elementary geometry. The introduction of new axioms was generally seen as a way of making explicit the consequences of the Euclidean definitions of the geometrical terms, and offering some further statements that might be useful in the proofs. This seems to have been the case with the axioms on intersections as well, and the latter were seen by several early modern mathematicians as grounded on the notions of continuity and magnitude. The grounding of the continuity of space itself, therefore, had to wait for a roundabout historical path that passed through the foundations of analysis and, later on, the definition of completeness in the domain of real numbers.
