Le baiser de Leibniz : et la courbure devint une quantité (1680-1686)

In the early 1680s, Gottfried Wilhelm Leibniz conducted research to make sense of the concept of the ‘angle between two curves’ and, in so doing, questioned the ancient and controversial concept of the angle of contact, which appeared in Euclid but had not been thematised. His research led to conceptual results, which he published in 1686. He generalised the concept of the contact angle in an original way by introducing the concept of the osculation angles, which enabled him to provide unprecedented mathematical clarifications concerning the curvature of a curve connecting the latter to the angle between two curves. However, Leibniz did not accompany his conceptual inventions with techniques that would allow them to be represented by means of analytical expressions. On the other hand, in attempting to give mathematical meaning to this new type of angle, he was led to formulate and reformulate concepts fundamental to Euclidean geometry, such as similarity and homogeneity. Only some of these results have been published. Our contribution will attempt to show how the analysis of several of Leibniz's mathematical drafts, most of which remain unpublished, can shed light on the genesis of the concept of osculation in the context of research into the foundations of geometry.