Modes of diagrammatic reasoning in Euclid’s Elements
Piotr BÅ‚aszczyk and Anna Petiurenko
Abstract.
The standard attitude to Euclid’s diagrams is focused on assumptions hidden behind intersecting lines. We adopt an alternative perspective and study the diagrams in terms of a balance between the visual and theoretical components involved in a proposition. We consider theoretical components to consist of definitions, Postulates, Common Notions, and references to previous propositions. The residuum makes the visual part of the proof. Through analysis of propositions I.6, I.13, and II.1-4, we show that such residuum actually exists. We argue that it is related to a primitive lesser-greater relation between figures, or an undefined relation of the concatenation of figures. We also identify a tendency in the Elements to eliminate visual aspects in order to achieve generality founded on theoretical grounds alone. Our analysis spans between two versions of the Pythagorean theorem, i.e., I.47 and VI.31. We study the diagrams in Books I through VI in terms of how visual elements are being replaced in favor of theoretical components. That process is crowned by proposition VI.31. None of its parts build on the accompanying diagram. Moreover, it concerns objects that are not represented on the diagram at all. In fact, this pattern applies to most propositions of Book VI. Therefore, we treat VI.31 as a model example of Euclidean methodology, not as an exception.