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Preamble: the (distinct) lines of a line-pair in a plane are parallel if they form an angle of zero. The vertex of that angle, i.e., the meeting point of the pair, lies at—and exactly specifies the whereabouts of—infinity.

So what is Measurement? To approach an answer, we might ask what we actually do when we measure? We lay down a ruler of a known number of supposedly equal units, with one end matching the start of the thing to be measured. We note where, on the measured thing, the other end of our ruler falls, mark that place, then we lift the ruler away, and re-lay it with its first end matching the mark - and so on, until we have spanned the thing we are measuring - and we count layings as we go.  That done, we multiply the number of units on the ruler by the number of layings. (We'll worry about mis-matches and fractions later) So the procedure is: lift and lay, rhythmically and serially, counting (adding) until done. Let us now cast this procedure in the simplest geometrical terms that we can find.


We measure a table-top along the line m, as on the left. We lift the red ruler, keeping it parallel to itself in the blue plane π (which has the line m in the plane of the table-top), and so that the endpoints of it move in parallel (white) lines toward point U at , and stop in the line i, parallel to m. We re-lay the ruler in like manner, except the direction of motion of the endpoints changes to the parallel (black) lines from point D at , so as to place the left end of the ruler where the right end was. Count, and repeat.

We get a total of something more than 3 ruler's-worth of  length for this table-top.

Note that point X, at , is in plain sight, or rather - because points are invisible - in our field of vision. So are points D and U, also at ∞. If we turn our eyes to put X as far out-of-sight as it can ever be, then m, i and x will seem parallel, and all the intervals will seem equal in the accustomed way. We get the Draughtsman's ‘Side Elevation’.


However, because Synthetic Geometry preserves only elementary incidence,
it has no Axiom of Equality.

Consequently, distance cannot be expressed as a sum of equal units.

So the matter is more subtle.

We are called upon to realise that

distance is not absolute.

It is quite well known that
we cannot find a point at absolute rest.

It is perhaps not as well known that
we cannot find an interval with absolute size.


I wish seriously to suggest that it is because our faculty of sight somehow incorporates and understands this transformational process as a largely non-analytic, "instinctual" part of seeing, that we "know" the intervals are "equal", as immediately as we "know" this text is red, not yellow.

Perhaps this is what distinguishes sight from perception.

There must be such a distinction, since there are people who can see perfectly, but not recognise what it is they see (Oliver Sacks: "The Man who Mistook his Wife for a Hat")


We can now generalise, and say that

any such iterated process obeying only the laws of incidence
can produce a ‘measure’ with geometrically-identical, but not equal, intervals.

Specifically, lines i and x, above, may also pass through distinct points, X and Y, in line m, as below.

The interactive construction below "rings the changes". The construction adapts immediately to movable elements being left-clicked and dragged to new places.

Please enable Java for an interactive construction (with Cinderella).

By all the foregoing, in the diagram above,

all intervals are geometrically identical
wherever they are on the line,

so, as far as this transformation is concerned, blue interval one is geometrically the same as red interval two.

The interactive construction below "rings the changes". The construction adapts immediately to movable elements being left-clicked and dragged to new places.

Please enable Java for an interactive construction (with Cinderella).

If we place the intervals end to end, and add some more of them by continuing the stepping, as in the diagrams above, we arrive at a perfectly respectable linear measure, from which we can obtain counts of sequenced, identical intervals, with which to measure other intervals on just this line,
but not “Euclidean” sums of equal units.

Variations on the Theme of Linear Measure.

In this interactive, you may shift the black lines by dragging white points A, B and C, and rotate these lines around their white points, and so set the positions of invariant points X and Y. They may be sent to infinity, one at a time, or together, or they may be set locally, separately or together, illustrating the variations listed below.

Please enable Java for an interactive construction (with Cinderella).

By the axioms of incidence, two real points may be related in exactly two ways: that is, they may be

  • distinct, or
  • coincident.

If infinity is the meeting place of parallels, then the points of this (real) pair may be

  • at infinity, or
  • "local" (i.e., not at infinity),
so that,
  1. both points may be local and distinct
  2. both points may be local and coincident
  3. both points may be at infinity—and must be coincident, since there is just one place at infinity on a line
  4. one point may be local, while the other is at infinity

These four “states of incidence” correspond to four “styles” of linear measure,
pertaining to translation of points in a line , according to the ‘constraints’ on the invariants:

  1. "Growth measure" the least constrained case
  2. "Perspective measure" the third-most constrained case
  3. "Equal step measure" the most constrained case: it is taken as the Standard Reference Measure (SRM)
  4. "Exponential (or Geometric) measure" the second-most constrained case
When the invariant vertices are imaginary, then the measure is angular, and pertains to rotation of lines in a point.

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