Summary — Chapter XLVI: Descriptive Geometry (§§ 374–387)

Primary Text: Read Verbatim Treatise Track (pp. 393–403) · Commentary: Critical Notes

Executive Summary

Chapter XLVI (Descriptive Geometry, §§374–387, pp. 393–403) examines spaces in which two points determine an open serial line rather than a closed projective circuit—the common ordinal substrate of Euclidean and hyperbolic geometry prior to congruence. Synthesizing Moritz Pasch (1882) and Giuseppe Peano (1889/1894), Russell axiomatizes betweenness, proves planar intersection properties via Pasch’s transversal axiom (§380), and demonstrates how adjoining ideal points, lines, and planes defined as logical classes (sheaves of lines) embeds every descriptive three-space into a projective space (§§383–387).

Argumentative Architecture

flowchart LR
            A["§§374–378 Pasch–Peano Betweenness & Serial Rays"] --> B["§§379–382 Segments, Pasch Planar Axiom & Solid Space"]
            B --> C["§§383–386 Ideal Points, Lines & Planes as Logical Sheaves"]
            C --> D["§387 Equivalence with Punctured Projective Space"]
        

Key Arguments & Theses

§§375–378: Pasch–Peano Betweenness vs. Serial Relations

Lines may be generated either from Peano’s primitive open segment $ab$ ($c \in ab$) or from a class $K$ of transitive asymmetrical ray-relations $R$ with converse $\breve{R}$.

§380: Pasch’s Axiom and the Descriptive Plane

A plane cannot be defined merely as lines joining $A$ to points of $BC$, because parallels or hyperbolic ultra-parallels leave gaps; Pasch’s triangle-transversal postulates close the plane.

§§383–387: Logical Construction of Ideal Elements

Instead of feigning non-existent ‘points at infinity’, Pasch and Russell define an ideal point as the class (sheaf) of coplanar ray-pairs satisfying three-plane concurrence—securing the existence-theorem purely logically.