Chapter 5 focus on the essay on conic sections that Blaise Pascal published in 1640, at the age of sixteen, and his, unfortunately lost, main geometrical work, the Conicorum opus completum, the content of which Pascal presented to the Parisian academy in 1654. His 1640 essay, contains his most famous theorem, today known as Pascal’s hexagon theorem, that generalizes Pappus’s theorem which regards the degenerate case when the conic splits in a pair of straight lines. Although much praised by the contemporaries Pascal’ treatise was never printed and the manuscript became lost forever a few decades after Pascal’s death. However, the description he left of this work, and the notes which G. Wilhelm Leibniz took during his reading of the manuscript, and left behind him, have allowed to understand the real nature of what Pascal’s himself called “the mystic hexagram” which goes beyond his hexagon theorem, and a conjectural reconstruction of Pascal’s treatise in the light of results first by Newton and then by Brianchon almost two centuries later. All this is described at some length in this chapter.