Measurement Preamble: the (distinct) lines of a linepair 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 relay 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 mismatches 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 tabletop 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 tabletop), 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 relay 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'sworth of length for this tabletop. 

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 outofsight 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, So the matter is more subtle. We are called upon to realise that distance is not absolute. It is quite well known that It is perhaps not as well known that 

I wish seriously to suggest that it is because our faculty of sight somehow incorporates and understands this transformational process as a largely nonanalytic, "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") 

any such iterated process
obeying only the laws of incidence 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 leftclicked and dragged to new places.
By all the foregoing, in the diagram above, all intervals are geometrically identical 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 leftclicked and dragged to new places. 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, 

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.
By the axioms of incidence, two real points may be related in exactly two ways: that is, they may be
If infinity is the meeting place of parallels, then the points of this (real) pair may be


These four “states of incidence” correspond to four “styles” of linear measure,

