On “contour apparent”, “courbe de contact” and ramification curves: Duality between a principle and a tool

As has already been described in the literature, it was Julius Plucker who solved the “paradox” of the duality principle in the 1830s – that is, the mystery to the problem of the degree change between a curve and its dual: given a curve, the points on its dual correspond to tangent lines of the given curve. The “paradox”, or, more correctly, the puzzle, began with the expectation that the dual of the dual would be the original curve.