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Elements (2)

Chiefly about Lines

Line is extension, not place.
Hence,

  • a line has no place (no “location”),
A Line is not an interval.
Accordingly,
  • a line has no size,
of any value whatsoever, especially not ∞.

   
 

‘Ends’

A line has no ends, because
  • ends are points
  • lines are not points
  • and lines do not consist of points.

Incidence

  • Two points can have a line in common,
  • –but need not, because elements do not define each other.
  • Two planes cannot avoid having a line in common.
  • Two lines can have a point in common,
  • –but need not, because elements do not define each other.
  • If two lines have either a plane or a point in common, they must have both a point and a plane in common.
  • If two lines DO NOT have either a point or a plane in common, they must have neither in common, and are skew.
   
 

‘Motion’

If a projective line-without-ends is to be moved, it must be moved projectively, namely by projective transformation, and must necessarily move as
  • an unended thing,
and to preserve continuity during the rotation, two successive orientations of the line must always project each into the other, which implies that an appropriate condition of incidence must exist (namely, there must be a plane, and the corresponding point, common to both orientations). In consequence—
  • a line can move around just one point (P →),
  • and in just one plane (π →).
This implies that a line,
  • can move only by rotation, and
  • cannot move by translation.

One immediate consequence is that
there is no linear (point-to-point) distance between lines,
because lines are not points.

It implies, also,that lines cannot, by single projection, move skew to themselves, as there is no condition of incidence that can project a first line into a second line skew to the first, because they have nothing in common: hence, continuity cannot be established.

rotate_line

   
  Thus, also, a line cannot and does not bend, like a wire rod, because a line, bent, would end at the bend, twice. Lines have no ends.

It follows at once that

by which we see that the words, ‘extension’, and, ‘straightness’, are synonyms for the quality that defines a line.

bent_line    
         
         

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