Wallis’s Use of Innovative Diagrams

Mathematical research is best characterized by problem- solving activities which make use of a variety of modes of representation. Against this background, my aim is to discuss the epistemic value of diagrammatic representation in problem-solving. To make my point, I consider a case study selected from Wallis’s work on the quadrature of conic sections. Wallis’s definition of conic sections is given in terms of algebraic equations setting them free from ‘the embrangling of the cone’. This suggests the aim to eliminate figures and other iconic elements with a view to attaining higher level of abstraction but, in Wallis’s work, geometric diagrams display relations that can be fruitfully used to calculate arithmetically the area of a figure. The use of displayed relations leads to the formulation of algebraic equations defining curves and it is also what makes room for arithmetical calculations. Accordingly, the notion of a general method of resolution is grounded on properties read off the diagram so that despite Wallis’s insistence on algebraic representation -I argue- diagrams remain essential working tools.