Unit 1 Summary: Bounded Segments & Linear Betweenness Axioms

Analytical Synopsis of Vorlesungen über neuere Geometrie (1882), Vorwort, Einleitung & § 1

1. Bounded Segments Before Infinite Lines (*Einleitung*): Pasch begins from finite observable straight segments $AB$ and the ternary relation “$C$ lies inside $AB$” (betweenness) rather than assuming completed infinite lines a priori.

2. Eight Nuclear Linear Axioms (§ 1, *Grundsätze* I–VIII): Axiomatize unique segment connection (I), interior density (II), asymmetry (III), convex inclusion (IV), two-part partition around an interior point (V), arbitrarily repeatable prolongation (VI–VII), and unique two-point line determination (VIII).

3. Derived Line, Opposite Rays & Ordered Quadruples (*Lehrsätze* 1–23): Constructs the full line $[AB]$, proves two-ray division by any single point $O$ (Lehrsatz 12, ruling out closed spherical/elliptic circles!), and proves canonical ordering $A\text{--}B\text{--}C\text{--}D$ of any four collinear points (Lehrsatz 19).