The classical theory of affine connections.

The theory of affine connections is, roughly speaking, a generalization of certain concepts of parallelism and differentiation defined in plane differential geometry, to the differential geometry of surfaces, and, more generally, to the geometry of differentiable manifolds. It is the purpose of this essay to relate the various stages of this generalization, and to present the essentials of the classical theory of affine connections on a differentiable manifold. [...