Yes. Exactly. Thank you for pointing that out. In the previous
email I called Pieri's axiom a "version" of Pasch's axiom, but you were
correct to question that. To be more precise, a "version" of the
Veblen-Young-Pasch axiom can be deduced from Pieri's axiom.
What Pieri postulates is this:
*Pieri's Axiom:
Let A,B,C be non-collinear points. If A' is a point on the line BC,
but different from B and C, and B' is a point on the line AC but different
from A and C, then the lines AA' and BB' meet in a point belonging to the
plane ABC.
So Pieri's axiom is purely projective.
Let's rephrase the Veblen-Young-Pasch axiom to mimic the above:
Let A,B,C be three non-collinear points. Let the line l
intersect the line BC at A' and the line AC at B', then l = A'B', and
so the line A'B' will meet the line AB.
Now we can see that a version of this axiom (excluding the cases where l
intersects the vertices) can deduced from Pieri's axiom* simply by using
Pieri's axiom again on those same points as follows:
We know A,A'C are non-collinear. (assume not. Then since A' is on BC,
A'C = BC, and if A is on A'C, then A is on BC which contradicts our
original assumption of the non-collinearity of A,B,C).
We know A' is on BC by construction above, so B is on A'C.
We constructed B' on AC.
So we have the following
A,A',C are non-collinear. B is on A'C and B' is on AC, so AB and A'B'
meet.
So with respect to Desargues theorem, Pieri actually needs a weaker axiom
than do Veblen and Young. Using this axiom Pieri is also able to deduce
that two coplanar lines intersect,...instead of postulating it as some
other projective geometers do.
Veblen knew of Pieri's 1897 work. I wonder why he did not use Pieri's
axiom in his development.
Thank you so much for your email and the opportunity to discuss this.
Sincerely yours,
Elena Marchisotto
On Fri, 22 Jan 1999, Prof. Lueneburg wrote:
> Dear Elena Marchisotto,
>
> Our library does not have von Staudt's "Geometrie der Lage". My
> private copy is at home and I will not come to the department
> for the next three weeks because I have to undergo a bypass
> operation next week. Nevertheless, I want to make a comment on
> Pasch's axiom. Pasch's axiom says that a line meeting the sides
> AB and BC of the triangle ABC BETWEEN A und B and BETWEEN B and C,
> respectively, meets the side CA OUTSIDE the segment CA. Thus
> Pasch's axiom is an axiom concerning projective geometries with
> a betweenness relation. (I know that I am speaking very sloppily,
> confounding projective and affine geometry.) What you have in mind
> is obviously not this situation. Therefore, you should call the
> axiom you consider by the name of Veblen-Young-axiom, because
> Veblen and Young have the axiom one needs here in their book on
> projective geometry.
>
> The Veblen-Young axiom reads: If a line meets two sides of a
> triangle in two different points then it meets the third side, too,
> side meaning the whole line through two points of the triangle.
>
> I do know that it is common usage in America to call the Veblen-Young
> axiom Pasch axiom. One should try to change this.
>
> Heinz Lueneburg
>