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— |
The 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? 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. 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. |
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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