Euclid's Parallel Postulate

Date: 14 Feb 1995 11:36:20 -0500
From: Cerreta, Pat
Subject: parallel postulate

I am studying Euclid's Parallel Postulate.  Why did mathematicians 
disagree with him?  What other geometries resulted from this 
disagreement?  What postulate replaced the Parallel Postulate?

Thank You,
Byram Hills Freshman Geometry Class
Armonk,New York

Date: 14 Feb 1995 14:59:02 -0500
From: Dr. Ethan
Subject: Re: parallel postulate

Howdy folks,

Well, to fully explain you need to make sure you know the difference 
between a theorem and a postulate.  A postulate is a statement about a
geometrical world that is accepted without justification and makes the 
frame work for the geometrical world.  A theorem is a conclusion that 
follows logically from the postulates using rational argument and does 
require justification.  It is a result of the framework of the 
geometrical world.

So now to Euclid's fourth and most troublesome postulate:

For every line, l, and point, P, which is not on that line, there exists a 
unique line, m, through P that is parallel to l.  (Remember parallel means 
that they do not intersect.  It does not necessarily mean that they are
always the same distance apart.)

This is a postulate, so when you say that you disagree with it, that 
doesn't mean it is wrong it just means that you don't want to assume that 
it is true in your geometrical world.

The disagreement came when mathematicians tried to figure out whether
or not the parallel postulate had to be taken as a postulate, or whether it
followed from the other four.  This was hotly debated but eventually it was
determined that no, it did not follow from the other three.  It had to be
assumed in order to get Euclidean geometry (Which for a long time was 
the only geometry that anybody thought mattered).  

From our perspective, the most important of these attempts was that
made by Girolamo Saccheri.  He hoped to assume that the postulate 
wasn't true, and show that that led to something that made no sense; 
thereby proving the postulate true.  He was unable to find a contradiction 
but did unknowingly discover the first non-Euclidean geometry (i.e. one in 
which the fifth postulate doesn't hold)

Well I guess I'm going to skip a lot of history here and skip to the results.  
It was eventually decided that the fifth postulate was indeed a postulate 
so we didn't have to assume it.  The two assumptions that have been used 
instead are these:

   1. Given a line, l, and a point, P, not on that line there are infinitely 
      many lines passing through P that are parallel to l.
      (This produces hyperbolic geometry)

   2. Given a line, l, and a point not on that line, P, there are no lines
      passing through P parallel to l.
           (This produces projective geometry)

If this is unclear or you would like more information, please write back 
and I will tell you more.

                Ethan Doctor On Call

Date: 14 Feb 1995 21:53:39 -0500
From: Dr. Ken
Subject: Re: parallel postulate

Hello there!

I just thought I'd add to the wonderful answer of my colleague and buddy
Ethan.  The two kinds of Non-Euclidean geometry are what we get by 
assuming the _negation_ of the parallel postulate.  Since the parallel 
postulate says, "for any line, l, and any point, p, not on that line, there 
is exactly one line through p which is parallel to l," the negation of it is, 
"there exists a line, l, and a point, p, not on that line such that there is 
either (a) no line through p which is parallel to l, or (b) more than one 
line through p which is parallel to l."

It turns out that if there is one place in our geometry where we're in case
(a), we can prove that we're in case (a) _everywhere_, i.e. that there are
no parallel lines whatsoever in our whole system of geometry.  We call this
geometry (as Ethan said) projective geometry or spherical geometry.  The
definition is that "there exists a line, l, and a point, p, not on that line
such that there is no line through p which is parallel to l," but we can
prove that "given any line, l, and any point, p, there is no line through p
which is parallel to l."

Also, if there's any place where we're in case (b), then we're in case (b)
everywhere.  This is known as Hyperbolic geometry.  The definition is 
"there exists a line, l, and a point, p, not on that line such that there is 
more than one line through p parallel to l," but we can actually prove that 
the following holds: "given any line, l, and any point, p, not on that line, 
there are infinitely many lines through p which are parallel to l."  

For what that's worth.

-Ken "Dr." Math