 It seems nevertheless true that the origins of geometric thinking have, in different cultures, always been associated with visuo-spatial reasoning, and in particular with figures and diagrams. This is a key reason for the celebrated beauty and appeal of geometry. Kant came close to capturing the spirit of Euclid’s style of work when he insisted that mathematical proof is based on the “construction of concepts” in intuition. The geometer works with carefully defined concepts, for which Euclid always shows how to construct instances (thus, e.g., proposition 1 shows how to construct an equilateral triangle given a line segment, and prop. 31 how to construct a parallel to a given line). But, as Kant emphasized in a famous passage of the Critique, logical analysis of the concepts could never provide the means of proof – an appeal to the construction (say, of a triangle) is required, offering a concrete instance either in empirical intuition (an external representation) or in pure intuition. Next, on the basis of the postulates, new elements are incorporated into the construction, which make it possible to interconnect concepts and obtain new results. So it is that one can prove diagrammatically, e.g., that the angles of a triangle add up to 180º (which, by the way, requires an appeal to the parallel postulate).