Our library does not have von Staudt's "Geometrie der Lage". My private copy
is at home and I will not come to the departement 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
>
> Hello Everyone,
>
> I am exploring projective proofs of Desargues triangle theorem [If
> two triangles are perspective from a point, they are perspective from a
> line]. I would appreciate any information you can provide regarding
> the following questions. It has been a while since I read the
> excellent book by J.V. Field and Jeremy Gray (The Geometrical Work of
> Girard Desargues, Springer Verlag, 1986), but I don't recall these
> questions being addressed there.
>
> 1. Who gave the first purely projective proof? Was it Von Staudt in 1847?
>
> 2. Mario Pieri (1897) suggested the provability of the theorem from the
> equivalent of following axioms (I've condensed his):
>
> 1. There exist a point and a line that are not incident.
> 2. Every line is incident with at least three distinct points.
> 3. Any two distinct points are incident with exactly one line.
> 4. A version of Pasch's axiom.
> 5. If A,B,C are non-collinear points, then there exist at least one
> point that does not belong to the plane ABC. [Pieri defines plane ABC as
> the union of lines joining A to BC, given that A,B,C are non-collinear].
>
> A.N. Whitehead used Pieri's axioms in The Axioms of Projective Geometry
> [1906] to sketch a proof. But Pieri himself did not do the proof in 1897.
>
> The fact that Pieri did not actually prove the theorem on the basis of
> these axioms leads me to believe there may have existed such a proof
> based on so few axioms prior to 1897, but I haven't found one yet. Any
> advice will be appreciated.
>
> Sincerely yours,
>
> Elena Marchisotto
> Professor of Mathematics
> California State University, Northridge
> email: emarchisotto@csun.edu
>