Measurement
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 l, as on the left. We lift the ruler, keeping it parallel to itself in the plane π (which has the line l in common with the plane of the table-top), and so that the endpoints of it move in straight lines (u), and stop in the line i, parallel to l. We re-lay the ruler in like manner, except the direction of motion (d) of the endpoints changes so as to place the left end of the ruler where the right end was. Count, and repeat. We get a total of rather less than 4 ruler's-worth of length for this table-top.
It is clear that if the up-motions marked u are all parallel to each other, they must all meet in the same point, say U, at infinity, and likewise, if the down-motions marked d are also parallel, they must all meet in the same point, say D, distinct from U, but also at infinity. There must therefore be a line, say m, wholly at infinity between U and D. Now, by design, we have made i parallel to l, so they meet in a point, say X, which must be at infinity, and because all of these lines are in the same plane, X at infinity must be a point in m at infinity, so all three lines, l, i, and m, meet in that point X—at infinity!
Let us draw this again, twice,
- first in such a way as to bring all these infinite points into sight (as per a casual glance!),
- then again as the architect or engineering draughtsman does it (by having the points-at-infinity away 'to the side', as much out of sight as it is possible for them to be).
Finally, let us do away with the table!
Note that the draughtsman does not need to place line m, or points D and U on it, entirely in the infinite; all he or she need do is make it parallel to i and l. It still works fine. All the intervals on l will still be equal.
So now we see what we must do to get parallel lines to meet, and to have a line carry equal intervals which remain equal, no matter what. We must simply adjust our point of view, and the intervals take care of themselves.The matter is slightly more subtle than this, however.
We are called upon to realise that
there is nothing absolute about distance.
What we get are intervals which are "equal" by virtue of being determined by identically the same process each time an interval is made. That is to say, intervals so made are geometrically indistinguishable one from another, so must be the same. Thus, intervals constructed by the same transformation are geometrically equal, and it is crucially significant that the geometry is unable to specify any other kind of equality.
This means that, if the sizes of the intervals engraved on, say, a school ruler, are in any sense absolute, they cannot be so for geometric reasons.
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")
any such iterated process obeying only the laws of incidence must produce a valid measure.
For example, there is no law of incidence that insists that lines i, m and l must pass through a single point common to all of them. If there were, then we would be able to form only one kind of triangle, the kind with all three vertices collapsed into one (this is a perfectly good triangle, by the way, having the requisite three lines and three points in a plane; it is a degenerate triangle). But it is obviously not the only style possible.
In the diagrams on the left we see lines l, i, and m forming a non-degenerate triangle; that is, one with distinct vertices, X, Y and Z.
Otherwise, everything is the same: the ruler is being picked up via U from position 1, and put down via D in position 2 after reversing, or "bouncing", in i.
By all the foregoing, by the definition of measurement, the ruler must retain its length, appearances notwithstanding, so interval 1 is the same as interval 2 as far as this transformation is concerned
If we place the intervals end to end, and add some more of them by continuing the stepping, as in the diagrams on the right, we arrive at a perfectly respectable linear measure.
The interactive construction below "rings the changes" on the diagram above. The construction adapts immediately to movable elements being left-clicked and dragged to new places.
And now we see something rather significant
When forming a linear measure, we always have had reference to a triangle of some description, and the kind of measure we have obtained has depended on the relative disposition of this triangle's vertices.
In point of fact, it depends on how just two of them (here designated X and Y), both on the measure-bearing line, are related to each other.
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 defined as the single meeting place of parallels, then the points of this (real) pair may be
- at infinity, or
- "local" (i.e., not at infinity),
both points may be local and distinct
both points may be local and coincident
both points may be at infinity—and must be coincident, since there is just one place at infinity on a line
one point may be local, while the other is at infinity
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 above.
These four "states of incidence" correspond to four "styles" of measure:
- "Growth measure" the least constrained case
- "Perspective measure" the third-most constrained case
- "Equal step measure" the most constrained case: it is taken as the Standard Reference Measure (SRM)
- "Exponential (or Geometric) measure" the second-most constrained case