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Topic: Euclids postulates and non-Euclidean geometry
Replies: 18   Last Post: Apr 22, 2006 6:52 PM

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 RCA Posts: 11 Registered: 11/26/05
Re: Euclids postulates and non-Euclidean geometry
Posted: Mar 8, 2006 1:13 PM

When I say plane, I always refer to a flat plane. I can intuitively
perceive the difference between a flat plane and a curved surface. The
fact that a small region on a large curved surface seems flat is just
an illusion and should not enter into a rigorous scientific discussion.

A line always refers to a straight line (except where I explicitly call
it a curved line in quotes) and a triangle always refers to a polygon
made by three straight lines (except where I call it a curved triangle,
in which case the sides are assumed to be curved lines).

If I happen to be dealing with a curved surface, then I know that it is
a curved surface from my frame of reference. I can still perceive a
straight line on a flat plane, living on a curved surface. If I draw a
straight line on a flat piece of paper held perfectly horizontal then
that line is parallel to a tangent to the surface of the earth.

However, if I choose to redefine a line on a spherical surface as
different from a straight line on a plane, then I am again making a
conscious decision to do so, and the redefinition changes the earlier
definition of the line, and hence, it is to be expected that some of
the rules change with that.