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