Thread: All Hail the Parallel Postulate

I was reading that John Playfair in 1795 was able to reduce the parralell postulate to more acceptable terms:
"Given a line and a point not on the line, then one can draw one and only one exactly one absolutly one parallel line through the point parallel to the given line." This is more elegant, but it did not give his reasoning. May someone explain if you know this?

It makes me want to laugh, people were working of simplifing the parralel postulate for close to 2000 years, and the result they got was that you can draw parrallel lines. If a non-mathematician would hear of this he will think we are insane to try to simplify something so obvious like this!

Originally Posted by ThePerfectHacker

I was reading that John Playfair in 1795 was able to reduce the parralell postulate to more acceptable terms:
"Given a line and a point not on the line, then one can draw one and only one exactly one absolutly one parallel line through the point parallel to the given line." This is more elegant, but it did not give his reasoning. May someone explain if you know this?

It makes me want to laugh, people were working of simplifing the parralel postulate for close to 2000 years, and the result they got was that you can draw parrallel lines. If a non-mathematician would hear of this he will think we are insane to try to simplify something so obvious like this!

The reason for seeking alternate formulations of the parallel postulate is
that for more than 2000 years Mathematicians have not found the existing
formulations obvious.

This even lead Gauss to go out and attempt to measure the angle sum
of a large triangle to see if it was measurably different from 180 degrees.

The resolution of this problem is that the parallel postulate is what
differentiates Euclidean geometry from other similar geometries which
differ only in terms of how many parallels exist through a given point to a
given line.

We have:
Elliptic geometry - no parallels
Hyperbolic geometry - more than one parallel.

These are homogeneous geometries, if we lift the homogeneity constraint
we allow inhomogeneous geometries, which include our best current
models of the geometry of space time.

RonL

1)Have you ever studied non-euclidean geometries?
2)Is it heavily dependent on concepts of analysis, or does it use its own math, just like Euclidean geometry?
3)Is it part of topology?

You can study non-euclidean geometries, either in the classic sense (replace the 5th postulate and proceed to prove theorems) or in the -much more rewarding- manifold theory.

Only that the latter demands some analytic machinery (as it should, if it is to be more rewarding ). And yes, you can get amazing results, using the underlying topology of the manifold.

If this sounds not clear, wait till you hear about this. In manifold theory, a fundamental notion is the metric: the distance between two points. You can visualize the three geometries, as follows:

-Euclidean Geometry, as the plane, with all its straight lines.

-Riemmanian Geometry, as the sphere. "Straight lines" are all the great circles. Here, given a point p not on a "straight line" s, every great circle passing from p will intersect s; So there are no parallel "straight lines" to s from p.

-Hyperbolic Geometry, as the barrel of a trumpet, where "straight lines" are semicircular arcs, with endpoints on the lip of the barrel. (not only, but these will do.) Here, given a point p not on a "straight line" s, there are infinitely many "straight lines" passing through p, and not intersecting s.

To prove these results -manifoldwise- you need some knowledge of ...differential equations. These "straight lines" are called geodesics for ellegance, and their defining formulae are systems of dif. equations, for us to cry over.

Hope this has been helpful...

Accept this as the Postulate:
"Given a line and a point not on the line, then it is possible to draw one and only one parrallel line to the point".
Prove, the Parrallel Postulate.

This page is excellent. :O

It shows that, the intermediate between the two forms of the postulate, was the -equivalent- Pythagorean Theorem!

Originally Posted by Rebesques

This page is excellent. :O

It shows that, the intermediate between the two forms of the postulate, was the -equivalent- Pythagorean Theorem!

Thank you, it really is a good link.