Conference rusure::math

    I don't think your problem is really well posed, since a geometry
    is really defined in terms of transformations among lines, points
    and so on and your question thus seems circular.

    You could assume the line to be a geodesic, given the group of
    transformations that make up the geometry under consideration...

    - Jim

re. .1
    
If one body is abraded against another, turning and sliding in all 
possible ways, and then the first against a third, and the second 
against a third, until each pair fit perfectly, the three surfaces 
will be planes. If again the three bodies are abraded so as to produce 
a second plane intersecting the first, the edge where they meet will be 
a straight line. This is a tedious and not very practical way to make 
sure of a straight line.

We generally make use of light rays to test the straightness of a line.
We sight along the edge of a ruler, and correct its edge until the whole
edge can be seen as a point. But this is an optical method, not a geometrical
method: and it may not produce a Euclidean straight line if the light path is
curved as in a space described by Einstein.

Until the year 1864 no striclty geometric method was known by which a
straight line could be generated. Then Peaucellier invented a linkage.
Since then other linkages have been devised that accomplish the same purpose.

-kgg

    seems that the optical method is the most "geometric". The "abrasion"
    method you describe is how those rulers are made in the
    first place, and it is not all that tedious.
    
    If the objection to using light is the curvature of space, well
    then, the ruler will also be equally curved. Relativity showed that
    a ruler that was "straight" in flat space, would still be "straight"
    in curved spce. That is, in both cases it would produce the shortest
    distance between two points.
    
    One common definition of a straight-line is the "taut string".
    
    Light is really very very good at making straight lines.
    
    What is a "linkage"? (I assume you do not mean a mechanical linkage)
                          
                                                   
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    There are lots of spaces where you can NOT draw a straight line
    between two points, because the space is "not connected".
    
    An obvious example is the set of points {(x,y) : x<0 or x>1} with
    two points that you can't draw ANY line between (-1,0) and (2,0),
    yet alone a "straight" one.
    
    Thats why the phrase "Euclidean space" exists - there are LOTS of
    other spaces.
    
    /Bevin

If I recall, Euclid based his geometry on basic tools such as a 
straightedge and compass. The straightedge was postulated, not constructed;
it was just assumed that such a beast existed. So is the question how to 
construct a straightedge, or how to use one? I, too, am confused by the 
question as stated.

    A "linkage" is a mechanical device that can be used to construct
    a straight line.  It consists of about 6 rods connected together
    at their ends (in some arrangement that escapes me), so that the
    rods may rotate w.r.t. each other at these connections.  Two ends
    are 'fixed', and when the linkage is 'moved', the end of one of
    the rods moves in a straight line.

    Actually, the linkage I've just described is constrained to lie
    in a plane -- two such linkages can be combined to trace a line
    if you don't happen to have a flat plane available.

    re .6:
    
    Mechanical linkage, eh? I have serious doubts that that will produce
    a line that is "straighter" than a beam of light. Also, put that
    linkage in a curved space and it'll draw curved lines.
    
    re earlier:
    
    okay I assumed that we would be referring to physical spaces and
    not mathematical spaces.  In which case there is Euclidean or "flat"
    space, Reimann or hyperbolic space, and spherical space.
                                                   
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I regret for not beeing able to peply earlier due to system 
availability.

Until the year 1864 no striclty geometric method was known by which a 
straight line could be generated. Then Peaucellier invented the 
linkage described here (is this mechanical? see for your self).
Since then other linkages have been devised that accomplish the same 
purpose.

A  and  B  are fixed points, separated by a distance  r. At  B  is pivoted
a link whose length is  BC = r. At  C  is pivoted one corner of a rhombus,
formed by four links, and of length  j. Opposite corners of the rhombus
are connected to  A,  by links of length  AM = h.  

Figure 1. To be supplied some other time. (long distance communications
          do not allow slow drawing)

Taking  O  as the center of the rhombus, we have

      AC = AO - CO
      AP = AO + CO 

              =>     AP * AC = (AO)^2 - (CO)^2
                             = (h^2 - x^2) - (j^2 - x^2)
                     taking  MO = x
                             = h^2 - j^2

Observe that  

      h^2 - j^2  is a constant, and we may call it  2rk.

Now  AC = 2r cos (BAP), and  AP = 2rk/AC, hence
                             AP = k / cos (BAP)

This equation shows that P is located on the perpendicular to AB  
whose foot is at distance  k  from  A.

                     P
                     .
                    /|
                  /  |
                /    |
              /      |
            /        |
          /   B      |
      A .____._______|
                k
Figure 2.


Accordingly the path of  P  is a straight line.


Enjoy,

Kostas G.

p.s. Figure 1 to be supplied soon.

    Re .8:
    
    Your demonstration that the linkage makes a straight line uses
    Euclidean geometry.
    
    
    				-- edp