Euclid Was Wrong
An earlier contribution from this author was called The Definition of Parallelism. The article stated that the definition of parallelism in Euclidean geometry is wrong. The motivation was that the lack of a point common to two lines was used in the definition. Instead it was stated that only existing concepts should be used in a definition. The discussion following the article gave no important arguments neither for nor against the suggested idea. The intention with this article is therefore to provide more arguments regarding the definition of parallelism in Euclidean geometry.
Euclid’s greatest error
Euclid had a focus on straight lines and defined parallelism with a tacit assumption that this restriction also was valid for the concept of parallelism. Therefore , he never discovered that curved lines also can be parallel. We can see this in an example with two circles. If the two circles are concentric they also are parallel. A bending railroad is also an example of parallel curved lines. An engineer would not define the railroad by stating that the rails should not be crossing each other. From these examples we can conclude that parallelism must be defined by constant separation between two lines. Separation is constant if the two lines are bending around the same center of cuvature. If this center also is fixed we get two concentric and parallel cicles.
Separation can be defined as a distance from (and in a right angle to) one line to the other line.
With the asumptions made here we find that the Euclidean definition is wrong by reasoning in absurdum. We find that according Euclid’s definition one circle and one straight line can be parallel to each other. We can also see that in the case restricted to straight lines we can easily find a second point. With two points a unique straight line is defined. Therefore, one point and one straight line can also define a straight line. We do not have to demand the defining point to be outside the defining line.
Euclid was right regarding his fifth postolate. However, he could not prove that since he used a wrong definition of parallelism.