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Metrics
Pure Projective Geometry does not define absolute size,
either linear or angular.
There is no inch or metre, no radian or degree.
there are no square units of area,
or cubic units of volume,
or the steradians of solid angle.

But it does define intervals which are exactly constant
with regard to how they are devised
.

Such intervals are geometrically indistinguishable,
meaning that they cannot be told apart,
and if that is so,
then they must simply be the same.
They are in every geometric respect equal to each other.

We adopt this notion as
our
definition of equality, and say that
two, distinct intervals on the same element,
devised in identically the same, projective manner,
are equal in size.

Thus an interval may be identical to another, but not unique, simply because there can be more than one.

If two geometrically identical, and geometrically indistinguishable intervals are nevertheless to be distinguished, then that can only be by whatever it is that allows them to be non-unique, and countable as two (in short: by their countability), and this will probably be mostly a matter of their provenance—of their history.

 

An analogy:

Bricks may be indistinguishable, one from another, but may nonetheless be distinguishable

  • by their positions in the building of which they are parts,
  • by the routes they took to these positions and
  • by the times of their arrivals.

The Standard Reference Measure(SRM)

derive SRM

We may also compare intervals devised in different ways on the same element,
and adopt a convention by which
one way is taken as the standard.
In Illustration—

geometric seriesThe sketch on the left depicts two linear measures on the same (red) line, l. Let us call them the "black" and the blue".

One of these, the blue, is the Standard Measure, derived as in the animation above right.

The black is new. It has one of the vertices, X, of its (non-degenerate) triangle XYZ placed locally on line l - on the page, so to speak - and Y on l at the infinite. Accordingly, the intermediate line i through Y and Z lies parallel to l. Line m, the bearer of directing points U and D, passes through X and Z, as always.

The (semi-degenerate) blue triangle X'Y'Z', on the other hand, has all three of its vertices on l and all at the same place, namely infinity, so lines l, m' and i' (made distinct) are parallel. We already know that the blue measure has "equal" intervals, even as "equal" is ordinarily understood.

How is that with the black measure?

Well, as we have seen, it too, within its own context, has equal intervals. Yet,with the exception of the intervals between 0 and 1 (here deliberately made to match), correspondingly-numbered black and blue intervals clearly differ.

Both measures were produced by stepping in identical ways:
the only difference between them concerns the locations of vertices X and Z.

The black versions of these are local. All others, black or blue, lie together in the same place on l, namely, at the infinite.

However, the black interval between 1 and 2 has been chosen to match the blue interval between 1 and 3, so by the blue measure, that black interval is two blue intervals in length. If you look to the first black interval to the left of interval (0,1), you will see that it has a length of half a blue interval, and then you will see that the black interval to the left of that black interval has a length of one quarter of a blue interval - that is, half of a half of a blue interval. It appears that if we step left with the black intervals, we will find that each is half the size of the one before, as guaged against the blue measure. Conversely, if we instead step right, successive black intervals will be found to double in length, again as guaged against the blue measure.

We have a constant common ratio, or characteristic multiplier, which is 2.0 going right, and 0.5 going left. By the blue measure, the black measure is a geometric series, or geometric progression. Now, this common ratio is a "nice" one, achieved by conscious construction: it is either an integer, or that integer's reciprocal, perforce a nicely rational number. It will not always be so nice with other, less deliberate constructions, but it will always be constant.

As we step left, the black intervals diminish with respect to to the blue, and it is clear that no number of steps will take the black measure past vertex X - but it is equally clear that an exactly infinite number of steps will bring the measure exactly to X. X is a limit. By the same reasoning, so is Y.

The black series, going right, must end precisely at infinity, at Y (this may come as a surprise: the German word for infinity is Unendlichkeit, a 'state of unending'. All the same, it is an ending-place for this measure).

On the other hand, the line l, bearing these measures, does not stop at Y, as lines have no ends. Neither does line l halt at X, the other limit of this measure. This fetches us up sharply against the realisation that some measures do not span the line in which they sit. The black measure is one. The blue is not.

The general lesson here is that, although no one measure can be judged absolute, or superior to any other, any one may nevertheless be taken as standard, and used to "measure" the others, just as we have done above. It is a matter of convention. It is natural and convenient for us to use the blue measure, and we conventionally do—and, though it is not, we are probably predisposed to regard it as absolute.

Now let us try our technique of "looking askance" (to bring points at infinity into our field of view) on the black measure, just identified as a geometric progression above.

We station ourselves near X, and glance along l towards Y. This is to place the line with its measure into perspective, with Y as the"vanishing point". What do we "see"?

We see a measure constructed on an all-distinct, non-degenerate triangle—in other words, a "growth" measure—from which we may say that a growth measure is a geometric progression in perspective!

Having got this far—that is, as far as seeing that we may standardise on one style of measure, and then compare others to it—we can "algebraicise" our various measures.

We can 'algebraicise' because now the idea of "Countable Units" can be correctly applied.

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