Projective geometry and Euclid both
measure intervals by intervals, but
Euclid
Projective Geometry and Euclid part company here, because there is no exclusively geometric way to validate these assumptions, meaning that, for
Projective Geometry, size and equality do not exist, A List of Consequences :-
Points haven't size (not even zero):
Lines haven't length (not even ∞): Planes haven't area (not even ∞^{2}). |
As there are three elements – point, line and plane – there are three fundamental types of interval ...
We note in passing that, for every type, whenever one interval is formed, two are in fact formed.
For example, we may trace out one interval going right from
A
to B on the line l, shown on the left above,
and may also trace out a second interval going left
from A to B. (The latter tracing works
because a line has no ends.) This fact is of significance when
investigating the validity of infinitesimals for non-Euclidean, projective
geometry. Intervals have ends, Elements do not. |