Edit: It was pointed out in the comments, sheers are transformations which are volume preserving, and not orthogonal. That was sloppy of me. Consider that part of my question answered.
I believe that the terms length and distance are synonymous in this context.
I often read that in the most general affine geometry there is 'no metric', 'no concept of distance', etc. Often the same authors will soon after speak of things such as 'the rate of change of a dependent variable along a path', or 'the value represented by the segment between two points of intersection on a line', etc.
For examples:
Misner, Thorner and Wheeler call proper time the "affine parameter" of a world-line in their chapter on affine geometry.
In listing subgroups of the group of all affine transformations, Schouten informs the reader that under the group of all orthogonal homogeneous transformations concepts of length and angle exist. Which implies they do not exists under the group of all affine transformations.
I understand that a metric (scalar product) provides a distance function between arbitrary points, and also provides a definition of angle. But, it appears that we have some notion of distance along a path (or line) under the most general affine group.
To add to my confusion, Schouten defines the group of equivoluminar linear homogeneous transformations, that is, with $\Delta =\pm 1$, in which "volumes may be compared". He states that the associated space is a centered affine space with given unit volume. This is defined separately from the group of all orthogonal homogeneous transformations.
But, I thought that having a unit determinant, and being an orthogonal transformation were the same thing. Edit:wrong!
Schouten later states:
The components [of the contravariant vector, (AKA differential 1-form)] with respect to [the coordinate system] are the inverses of the segments cut out on the axes, measured by the corresponding units on these axes.
But he never says what such a measurement means.
Edwards takes a different approach to volume, and geometry in general. He begins with the standard basis spanning $\mathbb{R}^{n}$, and defines volume as the real number product of the components of an interval in $\mathbb{R}^{n}$.
So I ask: In the most general affine geometry is it meaningful to speak of distance along a coordinate curve? What is the most general subgroup of the group of all affine transformations under which there is a measure of volume? What is an example of a centered affine transformation with $\Delta =\pm 1$ which is not orthogonal?