Visual Representations of Euclidean Geometry: Diagrammatic Reasoning in Oliver Byrne’s Work

Around the half of the nineteenth century the Irish mathematician and civil engineer Oliver Byrne published a peculiar version of Euclid’s Elements where he displays Euclid’s proofs in form of pictures with little to no explanatory text (or the usual letter symbols) associated. That is, Byrne uses graphical representations (encoded using standardized colors and shapes), corresponding with diagrammatic parts of geometrical entities, which inhere in propositions. In this way, Byrne proves by picturing Euclid’s propositions. A recent, renewed interest in the importance of mathematical practice had led to a reconsideration of the role and scope of diagrams and diagrammatical reasoning in mathematics. In this context, Byrne’s peculiar use of diagrammatical representations can be seen as both closer to the original Greek and Euclidean ideals of rigor and justification in mathematics, as well as an effective example of how visualizations in mathematics may serve as efficient cognitive-representational tools.