Unit 1: Grounding Geometry & Linear Betweenness

Moritz Pasch — Vorlesungen über neuere Geometrie (1882), Vorwort, Einleitung & § 1 (pp. 1–17)

[Folio Vorwort & Einleitung | p. 1]

Preface & Introduction (Vorwort & Einleitung, Giessen, 1882)

Modern projective geometry—elevated by Karl Georg Christian von Staudt into an autonomous science independent of measurement—still leaves unexamined a multitude of elementary assumptions concerning the relative order of points on a straight line and the separation of regions in a plane. Even in Euclid’s Elements, proofs continually rely upon facts read tacitly off a drawn figure rather than deduced from explicitly stated principles.

To place geometry upon an unassailable deductive foundation, we must separate two tasks strictly:

  1. First, isolate the irreducible primitive concepts (Stammbegriffe: point, straight segment gerade Strecke, planar surface ebene Fläche, and later congruence) and state every empirical observation required of them in a closed list of nuclear propositions (Grundsätze or Stammsätze).

  2. Second, derive every subsequent concept (such as the full straight line, the ray, the entire plane, and ideal projective sheaves) and every theorem (Lehrsatz) by pure logical inference alone, without ever consulting sensible intuition or figures again.

Notice that in immediate physical observation we never encounter an infinite straight line, but only a bounded straight segment joining two endpoints $A$ and $B$, together with points lying interior to the segment (Punkte innerhalb der Strecke, i.e., points lying between $A$ and $B$).

[Folio § 1 | p. 4]

§ 1. On the Straight Line (Von der geraden Linie, pp. 4–17)

Grundsatz I. Between two distinct points one can always draw a straight segment, and indeed only one.

We denote the straight segment joining $A$ and $B$ by $AB$ or $BA$; $A$ and $B$ are its endpoints (Endpunkte). A statement “$C$ lies within segment $AB$” means that $C$ lies between (zwischen) $A$ and $B$.

Grundsatz II. One can always specify a point that lies inside a given straight segment.

Grundsatz III. If the point $C$ lies inside the segment $AB$, then the point $A$ lies outside the segment $BC$.

Grundsatz IV. If the point $C$ lies inside the segment $AB$, then all points of the segment $AC$ belong also to the segment $AB$.

Grundsatz V. If the point $C$ lies inside the segment $AB$, then a point belonging to neither segment $AC$ nor segment $BC$ cannot belong to segment $AB$ (save $C$ itself).

(From Grundsätze I–V follow immediately the partition and transitivity laws of open interior segments: if $C$ lies between $A$ and $B$, and $D$ lies between $A$ and $C$, then $D$ lies between $A$ and $B$ while $C$ lies between $D$ and $B$.)

Grundsatz VI (Segment Prolongation). If $A$ and $B$ are arbitrary distinct points, one can always choose a point $C$ in such a way that $B$ lies inside the segment $AC$.

Grundsatz VII (Uniqueness of Prolongation). If the point $B$ lies inside both the segment $AC$ and the segment $AD$, then either $C$ coincides with $D$, or $C$ lies inside $BD$, or $D$ lies inside $BC$.

Grundsatz VIII (Disjointness of Distinct Lines). If $B$ lies inside $AC$ and $A$ lies inside $B'C$, while $B$ and $B'$ do not belong to the same straight line with $A$, no contradiction arises; on a single straight line, two distinct straight lines cannot share two distinct points.

Moritz Pasch (1882) §§ 1 & 11: Open Linear Betweenness vs. 4-Point Projective Separation Segment interior points C between A and B vs. exterior point D, showing pair separation AB // CD preserved across perspective section. S A C (inside AB) B D (outside AB) A' C' B' D' Pasch (1882) § 1 Grundsätze V–VIII & Lehrsätze 19–23: C lies between A and B while D lies outside AB, yielding invariant pair separation AB // CD
**Pasch's Nuclear Axioms of Linear Betweenness (1882 § 1, pp. 4–16):** Bounded segments $AB$, interior density ($C$ between $A$ and $B$), prolongation ($B$ between $A$ and $D$), and linear partition of four collinear points into a unique ordered quadruple $A\text{--}C\text{--}B\text{--}D$.

Definition of the Full Straight Line (*Gerade*). Given two distinct points $A$ and $B$, the straight line $[AB]$ consists of:

  1. the endpoints $A$ and $B$,

  2. all points $X$ lying inside segment $AB$ ($A\text{--}X\text{--}B$),

  3. all points $Y$ such that $B$ lies inside $AY$ ($A\text{--}B\text{--}Y$), and

  4. all points $Z$ such that $A$ lies inside $ZB$ ($Z\text{--}A\text{--}B$).

Lehrsätze 1–23 (Selected Deductive Consequences on the Line).

  • Lehrsatz 4. Every straight line $[AB]$ is uniquely determined by any two distinct points lying upon it.

  • Lehrsatz 12 (Ray Separation by a Single Point). Any point $O$ on a straight line $g$ divides all remaining points of $g$ into two non-empty classes (Halbstrahlen or opposite rays): two points $P, Q \neq O$ lie on the same side of $O$ when $O$ does not lie inside $PQ$, and on opposite sides of $O$ when $O$ does lie inside $PQ$.

  • Lehrsatz 19 (Four-Point Linear Order Theorem). Given any four distinct collinear points, they can always be lettered in a unique way (up to reversal) as $A, B, C, D$ such that $B$ lies both between $A, C$ and between $A, D$, while $C$ lies both between $A, D$ and between $B, D$.