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Non Euclidean parallelsPosted by DickT on September 25, 2002 at 08:13:16: In Reply to: Axiom of Choice posted by sol on September 25, 2002 at 05:20:12: Sol, Not a line and it's own intersection. Pick a line, any line, in the plane. Draw a line across it (a transversal). Let the angle between the line and the transversal be A. Now move some distance along the transversal away from the intersection to a point p and draw another line through p that makes the same angle A with the transversal (and on the same side as the original angle). Then Euclid's parallel postulate says: But the non Euclidean geometries deny conclusion (2). In "elliptic" non Euclidean geometry there are no parallels. In "hyperbolic" non Euclidean geometry there is an infinite set of different lines through p that are parallel to the original line (in the sense of never meeting it). Both of these non Euclidean geometries are just as internally consistent as Euclid's geometry. So there is no mathematical way to decide which one is "right". It becomes a matter of empirical testing to see which geometry is "the one" in any given region of space. This was the position when Riemann began his study of "the concepts which lie at the base of Geometry. Regards, Follow Ups: (Reload page to see most recent)
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