The chapter is dedicated to the reaction to the algebraic methods in geometry which marked the last years of the eighteenth century, mostly coming from the field of applied geometry. The attention goes to the work of Monge, which invented a new and very ingenious graphical method to describe solid figures in the plane, and in 1799 published his Géométrie descriptive based on his lectures at the École polytechnique. Monge set up a great School of descriptive geometry, and the work of some of his pupils (among the most prominent French geometers) is described here: Carnot, Servois and Brianchon. The role of Carnot’s principle of correlation in renewing synthetic geometry and his theory of transversals, as expounded in his Géométrie de position published in 1803, are analysed and discussed. At the same time, it is commented upon the applications that the younger Servois and Brianchon made of Carnot’s theory of transversals. The work of these three geometers led to many new results of projective character, and to the actual re-discovery of Desargues’s theorem on triangles in perspective, Pascal’s hexagon theorem, and Brianchon to the proof of the “dual” of this last theorem, which today bears his name. Beside Carnot’s effort for giving to synthetic geometry the same generality than the analytic one, in this chapter particular attention is put on the work of Brianchon—both for its intrinsic projective value, and for the great influence it undoubtedly had on the first researches of Poncelet–, and it is discussed quite in depth.