Edit to add: This clearly deals with projective geometry. If I figure it out well enough to post a satisfactory answer, I will do so. If someone else posts a decent answer before then, I will be happy to give you the brownie points.
This question pertains to Tensor Analysis for Physicists, Second Edition, By: J. A. Schouten.
My understanding of what has been said leading to the quoted statement is as follows:
A space of all allowable coordinate systems under the affine group of $\mathbb{R}^n$ is called an affine space, or an $E_{n}$. The space of all allowable coordinate system under the centered affine group of $\mathbb{R}^n$ is called a centered $E_{n}$.
The space of all allowable coordinate systems resulting from centered affine transformations of the solution set of the system of $n-p$ linearly independent equations
$$C_{\nu}^{i}x^{\nu}+C^{i}=0,$$
is called a flat sub-manifold of $E_{n}$, and is an $E_{p}$. Two $E_{p}$'s are said to be parallel if they can be transformed into each other by a translation. Two parallel $E_{p}$'s are said to have the same $p-\text{direction}.$
After introducing the concept of the intersection of $E_{p}$'s and $E_{q}$'s, Schouten states:
[W]e consider the points of $E_{p}$ 'at infinity' as an improper $E_{p-1}$ (or $E_{p-1}$ at infinity. Then two parallel $E_{p-1}$'s intersect in an improper $E_{p-1},$ and from this we see that an improper $E_{p-1}$ may be considered as a $p-\text{direction}.$
That statement makes very little sense to me. Can this be stated in terms of a traditional Euclidean 2-plane in 3-space, in a way that clarifies Schouten's statement? Am I to envision translating the plane parallel to itself to "positive infinity"? Am I to consider the "boundary" of the plane at the infinite extremes of its coordinate axes?
The statement appears following equation 3.9 on page 5. It is available in the preview.