Natural Geometry Based on the Rectilinear Plane

Abstract

The aim of this note is to introduce natural geometry, in which the rectilinear plane is a primitive. This plane is admitted together with its rectitude, and its rectitude states that a Lambert quadrilateral has its fourth angle right. The fourth angle of a Lambert quadrilateral may be right, acute or obtuse, and this trichotomy distinguishes three planes. Starting from the rectitude, we prove the uniqueness of the parallel by a chain of neutral steps: the rectangle, the triangle whose angles sum to $π$, and Legendre's theorem \cite{legendre}, all borrowed from the classical literature. That uniqueness is equivalent to the fifth postulate, which we thus reach in Euclid's own wording: the equivalence is read in the natural direction, and it is not a derivation of the fifth postulate from the other four, since the independence of that postulate is not in question. In the rectilinear plane the parallels are characterised as exactly the images of a line under a point-reflection, and equivalently as the pairs of equidistant lines. The other two planes are then placed beside the rectilinear one, and the same definitions serve for all three.