Does a curve violate the first postulate of Euclidean Geometry?

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Do the definitions and properties of curve secant and tangent lead to the violation of the first postulate of Euclidean Geometry?

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To my knowledge, the first postulate is that given any two points, there is a line which has them as endpoints; this has little to do with curves, so how could the definition violate that? Actually, a definition in itself cannot violate any statement at all. Even

Definition. A pair of two points is called a Jabberwocky if there does not exist a line which has them as endpoint.

does not contradict the first postulate. However, using the first postulate we quickly arrive at

Theorem. There is no Jabberwocky.