## ReplicationCan the same ruler be constructed from other elements? On this page, we see a ruler created
from two, contiguous intervals and one “terminator" — which is to say, from
four, given points in the line. It is reproduced below.If the ruler is to be and stay useful,
we need to be sure that every possible construction on those first four, given
points will lead to the same, fifth point, and to the same reference interval.In fact, it is guaranteed, as we
now see, by the
Theorem of Desargues^{[1] [2]}. |
The red and blue constructions are on different planes,
incident in the black line, and each
is the other's “shadow”, from “source” P.
On the black line (the “perspectrix”), the shadows are merged (degenerate)._{0}So, we have several perspectivities in P,
comparing several Desargues triangles, and
finding them equivalent._{0} |
Merely by
inspection, we see that the points on the perspectrix must stay exactly
where they are, wherever the source goes,
as these points and their shadows are the same things (try dragging P!) – so their _{0}distribution must remain constant,
too. Formally, they are
invariant under central
projection, and Desargues provides the formal proof of it. |