Subject: Re: [HM] When is Playfair's postulate important?
From: AJ Franco de Oliveira (francoli@dmat.uevora.pt)
Date: Wed Jan 12 2000 - 12:50:28 EST
At 10:28 10-01-2000 -0500, Daina Taimina wrote:
> ... I came up with the following questions: Where do we ever
> care that parallel (non-intersecting) lines are unique?
> Is there any construction that needs this result?
> ...
> Is uniqueness of non-intersecting lines (by itself) useful except
> as an historically important distinction between the plane and
> the non-Euclidean geometries?"
> ...
You speak of construction, and not of theorem or proposition, but of
course constructions have to be justified by theorems. There are many
theorems of plane Euclidean theorems that depend essentially on the
uniqueness of the parallel: angle sum = 180º, converse of alternate
interior angles, r//s and s//t implies r=t or r//t, most theorems on
similarity of triangles, for instance, to construct a triangle with
one side given similar to given triangle, to construct a parallelogram!,
etc. AJFO
This archive was generated by hypermail 2b28 : Wed Jan 12 2000 - 13:05:52 EST