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© Copyright 1998, Jim Loy
Are these
two lines parallel? It is hard to tell. The line segments are on the same
plane, and they do not meet. But, they are just a small part of the entire
infinite lines. If you extend the segments to infinity, do they ever intersect?
There can be pairs of lines which obviously will intersect, if extended. But,
there are also pairs of lines that you can not be sure about.
The situation is this: "What happens at infinity is not obvious." There are some things, that happen at infinity, which ARE obvious (more or less). An example is that a straight line just keeps being straight forever.
One version of Euclid's Fifth Postulate says:
Through a point (P in the diagram) not on a line (l1), one and only one line (l2) can be drawn parallel to the given line.
This version is called Playfair's Axiom. While you may believe that postulate to be true, it is not obvious. Other postulates, having nothing to do with parallels, are very obvious.
Throughout history, mathematicians were dissatisfied with this overly-complex postulate. They sometimes came up with equivalent postulates. For example, assuming that the sum of the angles of a triangle is 180 degrees (two right angles) is equivalent to the Fifth Postulate. But, no equivalent postulate was much simpler.
Then Lobachevsky and Riemann (and a few other people) tried alternative postulates which contradicted the Fifth Postulate. Lobachevsky's was that there are at least two lines parallel to l1, through P. Riemann's postulate was that there are no lines parallel to l1, through P. It turns out that Riemann's postulate violates the postulate that a line has an infinite length. So, that postulate must be reconsidered, too.
Lobachevsky's and Riemann's postulates may seem to be good attempts to prove Euclid's Fifth Postulate, by contradiction. That is what Saccheri attempted to do, much earlier. But, that does not work. It has been shown that, of the three alternative postulates (Euclid's, Lobachevsky's, or Riemann's), none of them contradicts any of the other postulates. They are all consistent with the rest of geometry.
So, which of the three is true? If Euclid's Postulate is true, then Lobachevsky's and Riemann's can never be disproved. Cosmologists seem to think that Riemann's is true, except that the universe (and the length of a line) is currently expanding.
Addendum:
This is Euclid's actual Fifth Postulate:
If a straight line (the diagonal line in the diagram) intersecting two straight lines make the interior angles (a and b in the diagram) on the same side less than two right angles (180 degrees), the two straight lines, if extended indefinitely, meet on that side on which the angles are less than the two right angles.
This may seem obvious, from the diagram. But, as the sum of the two angles gets closer and closer to 180 degrees, the situation becomes not so obvious.
This may seem to be a valid alternative to the above
statement:
If a straight line intersecting two straight lines (diagram on the right) make the alternate (b and c) angles equal to one another, the straight lines will be parallel (do not intersect) to one another.
That is Euclid's Proposition (Theorem) 27. It was proved without using the Fifth Postulate. One problem is that in Lobachevsky's geometry, there are other lines, through the same points, which do not intersect. And in Riemann's geometry, the proof is not valid as it requires extending the lines infinitely. The same problems arise from this statement (using the same diagram):
If a straight line intersecting two straight lines make the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another.
That is part of Euclid's Proposition 28, which follows from Proposition 27, above. As I said above, the following (Playfair's Axiom) is equivalent to the Fifth Postulate:
Through a point not on a line , one and only one line can be drawn parallel to the given line.
Another equivalent statement is this, which Proclus thought he could prove:
If any straight line intersects one of two parallel lines, it intersects the other.
He used this interesting idea as an assumption (postulate), apparently from Aristotle:
Two intersecting lines, if extended indefinitely, get farther and farther apart without limit.
That is true in Euclid's and Lobachevsky's geometries, but is not true in Riemann's where lines cannot be extended indefinitely. He also made the unstated assumption that parallel lines do not get farther and farther apart without limit. In Lobachevsky's geometry, they do. Other statements which are equivalent to the Fifth Postulate are these:
Note: I have used subscripts (l1) above. If your browser does not support subscripts, hopefully their use did not interfere too much.