5. Through a point outside a given line, one and only one line may be drawn parallel to that line
can also be interpreted in analytic terms as the tangent line by saying that the original line describes the diameter of a circle and the parallel
5. Through a point outside a given line, one and only one line may be drawn parallel to that line
can also be interpreted in analytic terms as the tangent line by saying that the original line describes the diameter of a circle and the parallel line describes the limit point of the tangent.
Now, I'm not a mathematician by any stretch of the imagination but I don't think that there is any way of going around Euclid's fifth postulate even in "non-Euclidean" geometries. It seems to be a general property (consequence?) of abstract space in describing continuous processes/curves; even in the chaos theory the center may vary continuously but the limit point that is the tangent remains faithful to its center.