The use of the differential geometry of a Riemannian space in the
mathematical formulation of physical theories led to important
developments in the geometry of such spaces. The concept of
parallelism of vectors, as introduced by Levi-Civita, gave rise to a
theory of the affine properties of a Riemannian space. Covariant
differentiation, as developed by Christoffel and Ricci, is a
fundamental process in this theory. Various writers, notably
Eddington, Einstein and Weyl, in their efforts to formulate a combined
theory of gravitation and electromagnetism, proposed a simultaneous
generalization of this process and of the definition of parallelism.
This generalization consisted in using general functions of the
coordinates in the formulas of covariant differentiation in place of
the Christoffel symbols formed with respect to the fundamental tensor
of a Riemannian space. This has been the line of approach adopted also
by Cartan, Schouten and others. When such a set of functions is
assigned to a space it is said to be affinely connected.

From the affine point of view the geodesics of a Riemannian space are
the straight lines, in the sense that the tangents to a geodesic are
parallel with respect to the curve. In any affinely connected space
there are straight lines, which we call the paths. A path is uniquely
determined by a point and a direction or by two points within a
sufficiently restricted region. Conversely, a system of curves
possessing this property may be taken as the straight lines of a space
and an affine connection deduced therefrom. This method of departure
was adopted by Veblen and Eisenhart in their papers dealing with the
geometry of paths, the equations of the paths being a generalization
of those of geodesics by the process described in the first paragraph.

In presenting the development of these ideas Eisenhart begins with a
definition of covariant differentiation which involves functions
$L^i_{jk}$ of the coordinates, the law connecting the corresponding
functions in any two coordinate systems being fundamental. Upon this
foundation a general tensor calculus was built and a theory of
parallelism.