From a Particular Diagram to a Universal Result: Euclid's Elements, Book I

Abstract

I argue that Euclid's highly abstract definitions in Book I of the Elements play a key role, enabling general conclusions from particular diagrams. I argue four points. First, definitions limit focus to the most generalized features of elementary geometrical objects. Second, an explicit reciprocal dependency relation exists between figure and boundary. Third, Euclid's derivation style employs a combination of diagrams in tandem with the highly abstract definitions. Fourth, this combination renders all non-universal features of a particular diagram irrelevant. The result limits the role of diagrams in a way that is reminiscent of Aristotle's characterization of the geometer's use of diagrams.