  We extend finding geometrically-significant preserved quantities by solving
specific PDEs to 1-$d$ projective transformations and subgroups. This can be
viewed not only as a purely geometrical problem but also as a subcase of
finding physical observables, and furthermore as part of extending the
comparative study of Background Independence level-by-level in mathematical
structure to include projective structure. Full 1-$d$ projective invariants are
well-known to be cross-ratios. We moreover rederive this fact as the unique
solution of 1-$d$ projective geometry's preserved equation PDE system. We also
provide the preserved quantities for the 1-$d$ geometries whose only
transformations are 1) special-projective transformations $Q$, giving
differences of reciprocals. 2) $Q$ alongside dilations $D$, now giving ratios
of difference of reciprocals. This analysis moreover firstly points to a new
interpretation of cross-ratio: those ratios of differences that are
concurrently differences of reciprocals, and secondly motivates 1) and 2) as
corresponding to bona fide and distinctive Geometries.
