In her recent doctoral dissertation, Konstantina Zormbala provides a
thorough investigation on the historical development of the concept and
definition of the plane in axiomatic elementary geometry [1]. She
discusses many 19th century attempts carried out to find a satisfactory
*definition of the plane*.
It has been suggested that Gauss's first idea concerning a good
*definition of the straight line* goes back to 1796, and that in the
following year he goes on to discuss the *definition of the plane*.
Gauss's notes on these issues were not published at the time, and
we are acquainted with them through his _Nachlass_.
While in his teens, Gauss used the so-called "Simson's definition"
of the plane; viz. a surface on which lies the straight line connecting
any two of its points, so named after Robert Simson (1687-1768) who
stated it in his own editions of "The Elements of Euclides".
Gauss criticized this traditional definition, but his own attempts to
solve this conceptual issue involved intuition anyway, and it does seem
that he was not aware of the need of Pasch-like properties -- despite
the fact that one might claim that he implicitly uses such ideas in his
demonstrations.
The issue of the concept and definition of the plane comes up in Gauss's
note of Euclid's Theorem I.7 (1829), as well as in his note "Begru"ndung
des Planum" (1832) -- "a surface in which every straight line AD passing
through a given point A and forming a right angle with a given straight
line AB lies", for which he proves Simson's property. [2]
An 'epistola' from Gauss to his student Christian L. Gerling (1788-1864)
seems to have been the starting point for Gerling's own misgivings on
this subject. Perhaps Gerling is remembered as an astronomer, and also
for his achievements in geodesy, but his maths books, and his role in
founding Marburg's Institute of Mathematics and Physics are well-known.
It happens that Gerling later motivated Heinrich Wilhelm Feodor Deahna
(1815-1844), whose thesis treated the same subject [3].
And the story continues...
[1] Zormbala, Konstantina:
"Die historische Entwicklung des Begriffs und der Definition der Ebene
in der Axiomatik der Elementargeometrie", 392 pages, Bielefeld: Univ.
Dissertation, 1995.
[2] Zormbala, Konstantina:
"Gauss and the Definition of the Plane Concept in Euclidean Elementary
Geometry", _Historia Mathematica_, vol. 23, no. 4, pp. 418-436, 1996.
[3] Deahna, Feodorus:
"Demonstratio theorematis geometrici fundamentalis, atque huiusque pro
axiomate sumis esse superficium planam", Marburg: University of Marburg,
Phil. Diss., 1837.
Greetings from Montevideo,
Julio GC
On Tue, 15 Sep 1998, Jeremy J. Gray <j.j.gray@open.ac.uk> wrote:
| Are there any good sources for critiques of Euclidean geometry in
| the 19th Century other than those on the parallel postulate? I have
| in mind discussions of what counts as a plane, missing axioms (such
| as Pasch's), deliberate errors (such as the famous purported proof
| that all triangles are isosceles) and I suspect there were a number
| of other enquiries, all cumulatively undermining confidence that
| Euclid's Elements were indeed the epitome of reasoning. I'm in a
| position to recognise the tips of some icebergs, but I doubt if I'm
| looking at every ocean.