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On Measurement

Ceva's Theorem

Allowing Euclidean Absolutes, we have —

The Cross Ratio
(On the Algebra of Measure)

The proof that cross ratio is preserved under central projection requires acceptance and deployment of the Notion of Area.

When a range of points on one line is projected from a point into another range on another line, metric quantities, such as length, or angle, are not preserved from range to range, simply because, for pure projective geometry, which takes cognisance only of straightness and elementary incidence, there are no such quantities to preserve!

However, if it is insisted, with Euclid, that intervals can be defined as a sum of equal units, then under the terms of that definition, it is possible to assign numerical sizes to the intervals between points, and then to use standard Euclidean/Analytic ‘geometry’. It is found that –

a ratio of ratios, or "Cross Ratio", of any three such intervals is preserved (i.e. is constant) under projection from range to range.

As this (also in its dual form) is the only metric quality to be preserved under projection, Arthur Cayley and Felix Klein were moved to use a form of it actually to define distance, and with it a metric, for projective geometry.

It will be appreciated that this was a curious and possibly redundant thing to do, because they already had distance - from Euclid - and had indeed used it (albeit embedded in the notion of "area") to establish the invariance of cross ratio in the first place—that is, they used distance to get distance. Did they blunder, or were they privy to a deeper truth?

For intervals a, b and c on one line projecting into A, B and C respectively on another line, we may, for example, write

[a/(b+c)]/[(a+b)/c] = [A/(B+C)]/[(A+B)/C]

A Euclidean/analytic proof, using area and trigonometry, of this result is given in the panels to the left and right.

There is of course noelementary geometric proof.

Area entails Eucliean Absolutes:

• ‘equal’ distances and ‘equal’ angles
• special, ‘right’ angles.
They are not native to Geometry, and represent artificial conflation of Number with the Elements of Geometry. They are grafts.

Allowing Euclidean Absolutes, we have —

This demonstrates that
Cross Ratio is preserved under central projection.

We stress again that this is true only under the assumption, tacit for Euclidean Geometry, that ratios of lengths can altogether be formed.

However,
Lengths are not available to Projective Geometry,
so neither are ratios of them.