by Ted Huntington

06/10/2008

Ted Huntington

ABSTRACT

With enough precision, a curved line will always be measured as a curved line and never as a straight line for any segment of the line measured. This casts doubt on the claim implied by Lobacevskii in 1829 that, over a small segment, a large enough curve is indistinguishable from a straight line. In addition, there are clear distinctions to be made between curved line geometry, the actual claim of Euclid’s fifth (parallel) postulate, an adapted, more inclusive postulate (an additional postulate which includes curved lines), so-called “Euclidean geometry” viewed as a two-dimensional concept, and the actual theory behind the rise of so-called “non-Euclidean” geometry. The author finds that any surface geometry is a subset of an infinite space of the same number of dimensions, that this principle is true for any number of dimensions, and concludes that as opposed to the implicit assumption of Lobacevskii, a curved line on an infinite circle or sphere will always be theoretically measured as a curve on any curve segment and will not appear to be a straight line provided that the measuring tool is precise enough.