Re: is sci.math slow? correction to my post to the AP thread

In article <rubrum-B2002C.20115818122007@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> Michael Press <rubrum@xxxxxxxxxxx> writes:
In article <Jt64yv.8wM@xxxxxx>,
"*** T. Winter" <***.Winter@xxxxxx> wrote:
....
Indeed. If there are two lines through a point parallel to a given line,
there are infinitely many lines parallel through the point parallel to
the given line. Consider a simple model of a hyperbolic geometry: the
inside of a circle. Straight lines are the same as the Euclidean
straight lines. Now given a straight line and a point not on that line,
we can draw from that point lines connecting the point to the two
points were the given straight line crosses the circle, and there are
obviously two of them. Clearly, within the hyperbolic geometry they do
not cross the given line (the crossing points are on the circle, and that
is outside the geometry). But also all lines through that point and
between those two lines do not cross the given line.

Depends on how parallel is defined.

Indeed.

Euclid's Postulate 5
That, if a line falling on two lines makes the interior
angles on the same side less thatn two right angles,
the two lines, if produced indefinitely, meet on that
side on which are the angles less than two right angles.

There are two ways Postulate 5 can be reformulated.

That two lines always meet. (Spherical geometry)

That two lines may never meet. (Hyperbolic geometry).

Now let's define parallel.

Given a line AB and a point P not on AB, let AP be the
perpendicular from P to AB. A line PQ is _parallel_ to
AB _through_ P if for any line PS with S lying in BAPQ
and 0 < angle{APS} < angle{APQ}, it follow that PS
meets AP.

That is of course a possible definition, but I use the definition from
Euclid:
Definition 23:
"Parallel straight lines are straight lines which, being in the same
plane and being produced indefinitely in both directions, do not
meet one another in either direction."
And I think that is the common definition.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.