Geometry

Although the main achievements of eighteenth-century mathematics were connected with the development of mathematical analysis, important discoveries were made in the course of the century in geometry also. The development of analysis was linked in the first instance with the development of analytic geometry. Plane analytic geometry, which had appeared in the work of Descartes and Fermat, was significantly advanced in the late seventeenth century and the first half of the eighteenth century in the work of Newton, Hermann, Stirling, Maupertuis, and Cramer, and assumed a form very close to its modern form in the second volume of Leonhard Euler’s Introductio in analysin infinitorum (1748), and in Clairaut’s book on curves of double curvature (1731). Analytic geometry was developed in three dimensions in the appendix to the second volume of Euler’s Inlroductio; it was further developed in papers of Monge (1794–1805). In connection with the development of the concept of a function geometers made ever more extensive use of geometric transformations: Clairaut and Euler laid the foundations of the subject of affine transformations, d’Alembert and Euler founded the subject of conformal mappings, and Waring and Monge studied projective transformations from various points of view. Johann Bernoulli. Clairaut. and Euler solved a number of problems in the differential geometry of curves in space, including in particular the theory of geodesies on a surface. In his Recherches sur la courbure des surfaces (1767) Euler laid the foundations of the differential geometry of surfaces, which was further developed in the work of Monge.