Plane Geometry - Bruce E. Shapiro

108 SECTION 22. NEUTRAL GEOMETRYElliptic Parallel Postulate. For every line l and every point P ∉ l thereis no line m such that P ∈ m and m ‖ l.We will see that this postulate is incompatible with Neutral Geometry.In fact, if we accept Neutral geometry then we may extend it with eitherthe Euclidean axiom or the Hyperbolic axiom, but that these two axiomsare the only possibilities. If we reject one, than the other one immediatelybecomes fact!Theorem 22.1 Under the conditions of neutral geometry, either the EuclideanParallel Postulate or the Hyperbolic Parallel Postulate must hold.The proof will be given later.This does not mean that the Elliptic postulate is without use. In factthe subject of Elliptic, and in particular, spherical geometry has importantpractical applications in navigation and measurement on a planetary surface.However, accepting elliptic measurement means throwing out one fothe other postulates, usually replacing the concept of betweenness with oneof separation. A more general formulation of geometry that was formalizedby Riemann, known as differential geometry or Riemannian geometry,describes geometry on curved surfaces (or manifolds).Following Venema’s development we have defined a different set of axiomsthat give the same results as Euclid’s first four axioms. We will call thesethe Axioms of Neutral Geometry:Neutral Geometry Axiom 1 (Existence Postulate) (Axiom 14.1) Thecollection of all points forms a nonempty set with more than one (i.e., atleast two) points.Neutral Geometry Axiom 2 (Incidence Postulate) (Axiom 14.5) Everyline is a set of points. For every pair of distinct points A, B there isexactly one line l = ←→ AB such that A, B ∈ l.Neutral Geometry Axiom 3 (Ruler Postulate) (Axiom 14.11) Forevery pair of points P, Q there is a number P Q called the distance fromP to Q. For each line l there is a one-to-one mapping f : l ↦→ R such thatif x = f(P ) and y = f(Q) then P Q = |x − y| is the value of the distance.Neutral Geometry Axiom 4 (Plane Separation Postulate) (Axiom15.2) For every line l the points that do not lie on l form two disjoint,convex non-empty sets H 1 and H 2 , called half-planes, bounded by l suchthat if P ∈ H 1 and Q ∈ H 2 then P Q intersects l.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012