Plane Geometry - Bruce E. Shapiro

Section 22Neutral GeometryNeutral Geometry is the study of geometry without any parallel postulate.Traditionally, it is the geometry based on Euclid’s first four axioms whichare summarized here.Euclid’s Axiom 1 A line may be drawn between any two points.Euclid’s Axiom 2 A line segment may be extended indefinitely.Euclid’s Axiom 3 A circle may be drawn with any give point as centerand any given radius.Euclid’s Axiom 4 All right angles are congruent.Neutral geometry is based entirely on these four postulates, or some equivalentset.To extend neutral geometry we can attempt to add one of the followingparallel postulates giving us either Euclidean Geometry or Hyperbolic Geometry.Euclidean Parallel Postulate. For every line l and every point P ∉ lthere is exactly one line m such that P ∈ m and m ‖ l.Hyperbolic Parallel Postulate. For every line l and every point P ∉ lthere are at least two distinct lines m and n (m ≠ n) such that P ∈ m andP ∈ n and m ‖ l and n ‖ l. (Note that m ̸‖ n because they intersect at P !)If one can imagine a single unique parallel line through a given point ormultiple lines through a given point, one might also ask the question ofwhether there are geometries without any parallel lines. This leads to thefollowing postulate:107