Gauss and the definition of the plane concept in Euclidean elementary geometry

Abstract

On July 28, 1797, Gauss wrote in his diary: "Plani possibilitatem demostravi" ["I have proved the possibility of the plane"]. On January 27, 1829, in a letter to Bessel, he was skeptical about a definition of the plane in Euclidean geometry according to which the plane is a surface having the property that the connecting line of two of its points lies on it. He voiced the same criticism of this definition some years later in a letter to Wolfgang Bolyai dated March 6, 1832. In his Nachlaβ, a note was found with the title "Zur Begründung des Planum" ["On the Founding of the Plane"]. All his letters and notes on this topic became known only after Gauss's Nachlaβ had been published, and they have not yet been discussed in their totality. Here, I present and thoroughly discuss all the letters and notes in which Gauss struggled with the concept of the plane, especially with his own definition of the latter within Euclidean elementary geometry. © 1990 Academic Press, Inc.