What Lobachevski meant by parallel lines

[nomadreid]

I am not sure that this is the right rubric for this question, as it is historical, but as it is part of the history of Model Theory, I am putting it here. I will not be offended if the moderators decide that it doesn't belong here.

In https://arxiv.org/pdf/1008.2667.pdf, the author states that Lobachevsky
"calls 'parallels' (not just non-intersecting straight lines but) the two boundary lines which separate secants from non-secants (i.e. parallels in the usual terminology) passing through a given point."
whereby he earlier defines "secant" as follows
"For a terminological convenience I shall call a given straight line secant of another given straight line when the two lines intersect (in a single point)."

I do not understand what "boundary lines" here mean. Can someone clarify? Thanks.

 

[jedishrfu]

Perhaps this writeup on hyperbolic geometry explains it:

https://en.wikipedia.org/wiki/Hyperbolic_geometry

I think in this case given a line and a point not on the line in a hyperbolic geometric plane then there are many lines going through the point that don't intersect with the given line and there are many lines that do intersect with the given line. Hence there is a boundary between those lines that don't and those that do and that is termed the parallel line.

Here's the article exerpt on it:

These non-intersecting lines are divided into two classes:

• Two of the lines (x and y in the diagram) are limiting parallels (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it.
• All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting.

Some geometers simply use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting.

 

[fresh_42]

Moved to General Mathematics, as we don't have a specifically geometry forum.

 

[nomadreid]

Thanks, Jedishrfu. That explains it well; my question is thereby answered. I just have not figured out why he needs to refer to limiting parallel lines, if the purpose is merely to exhibit a geometry which violates Playfair's axiom: the "ultraparallel" lines do that sufficiently, don't they?

 

[jedishrfu]

I think the edge case is more interesting from a geometric point of view.

 

[nomadreid]

Three questions: first, why is it more interesting, and two, do you think that is why Lobachevsky did that, to make it more interesting, and three, was it really necessary in order to make a consistent non-Euclidean (in this case, hyperbolic) geometry?