Probably also the concept of distance? (Somewhat implicitly, I guess, in the form of |AB|=|CD|?) Boris Tsirelson 19:12, 28 March 2010 (UTC)
The following text (by Paul Wormer), moved from "Plane", could be used later in the history of Euclidean geometry. Boris Tsirelson 07:24, 30 July 2010 (UTC)
Until well into the nineteenth century it was thought that the only geometry possible was Euclidean and consequently this definition of "plane" was considered satisfactory. However, with the birth of non-Euclidean geometry and attention to the logical foundations of mathematics in the second half of the nineteenth century, doubts arose about the exactness and the limitations of the Euclidean definition of a plane.In 1899 David Hilbert published his seminal book Grundlagen der Geometrie [Foundations of Geometry] in which he re-investigated and rephrased Euclid's two-millennia-old axioms and propositions. Hilbert begins with listing undefined concepts, among which are "point", "line", and "plane". In terms of these undefined concepts Hilbert formulates sets of axioms. The first axiom regarding the plane is axiom I4: Three points A, B, C that are not on one and the same line determine always a plane α. He adds that this is expressed as "A, B, and C lie in α", or "A, B, and C are points of α". His axiom I5 is a subtle extension of I4: Any three points in plane α that are not on one line determine plane α.
- D. Hilbert, Grundlagen der Geometrie, B. G. Teubner, Leipzig (1899) 2nd German edition