Differential Geometry of Surfaces - History of Surfaces

History of Surfaces

Isolated properties of surfaces of revolution were known already to Archimedes. The development of calculus in the seventeenth century provided a more systematic way of proving them. Curvature of general surfaces was first studied by Euler. In 1760 he proved a formula for the curvature of a plane section of a surface and in 1771 he considered surfaces represented in a parametric form. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in 1795. The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 1825 and 1827. This marked a new departure from tradition because for the first time Gauss considered the intrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space. The crowning result, the Theorema Egregium of Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e. invariant under local isometries. This point of view was extended to higher-dimensional spaces by Riemann and led to what is known today as Riemannian geometry. The nineteenth century was the golden age for the theory of surfaces, from both the topological and the differential-geometric point of view, with most leading geometers devoting themselves to their study. Darboux collected many results in his four-volume treatise Théorie des surfaces (1887–1896).

The presentation below largely follows Gauss, but with important later contributions from other geometers. For a time Gauss was Cartographer to George III of Great Britain and Hannover; this royal patronage could explain why these papers contain practical calculations of the curvature of the earth based purely on measurements on the surface of the planet.