Summary — Chapter XLVII: Metrical Geometry (§§ 388–404)

Primary Text: Read Verbatim Treatise Track (pp. 404–418) · Commentary: Critical Notes

Executive Summary

Chapter XLVII (Metrical Geometry, §§388–404, pp. 404–418) is the celebrated locus classicus cited by both Fr. Peter Hoenen, S.J. (De Noetica Geometriae, Caput IV) and Sir Thomas L. Heath (Euclid Vol. I, pp. 227, 249). In §§389–391 (pp. 404–407), Russell delivers his incisive critique of Euclid’s tacit appeals to spatial intuition in Prop. I.1 (two-circle intersection) and of the method of superposition in Prop. I.4 (‘strikes every intelligent child as a juggle’), exposing the logical circularity of defining spatial congruence through rigid material bodies (the thermometer by the window and fire) and formulating the exact three-step congruence postulate required to replace motion. Sections 392–404 then build metrical geometry rigorously from stretches, magnitude of divisibility, and angles defined as stretches of rays.

Argumentative Architecture

flowchart TD
            A["§389 (pp. 404–405): Critique of Euclid I.1 Circle Intersection"] --> B["§390 (pp. 405–406): Refutation of Superposition & Rigid Bodies (Euclid I.4)"]
            B --> C["§391 (pp. 406–407): Axiomatic Reconstruction of I.4, I.8, I.16, I.26"]
            C --> D["§§392–399: Distance vs. Stretch & Magnitude of Divisibility"]
            D --> E["§§400–404: Angle as Stretch of Rays, Area, Volume & Chirality"]
        

Key Arguments & Theses

§389 (pp. 404–405): Critique of Euclid I.1 and Implicit Intuition

Russell shows that I.1 assumes without proof that two coplanar circles whose centers lie on each other’s circumference intersect, and I.4, I.7, I.16 rely on unaxiomatized planar order (inside/outside) and betweenness.

§390 (pp. 405–406): Why Superposition Fails Logically

First, spatial points are immutable positions (‘can no more change its position than the leopard can change his spots’). Second, moving material triangles requires rigidity, whose very definition presupposes prior spatial equality—illustrated by setting up $\triangle ABC$ by the window and $\triangle DEF$ by the fire, where a heated thermometer changes equally with the triangles.

§391 (pp. 406–407): The Congruence Postulate Replacing Superposition

Following Hilbert, Russell states the exact triangle-determination and SAS congruence axioms needed to validate I.4, I.8, and I.26 without motion, while noting that I.16 fails in spherical/elliptic space unless open serial order holds.

§§396–402: Magnitude of Divisibility and Angle as Stretch of Rays

Distances are derived from the magnitude of divisibility of generating serial relations rather than point-set cardinality ($2^{\alpha_0}$); angles must be defined as stretches of rays rather than unbounded planar sectors (§402), because in hyperbolic space a proper sub-sector could otherwise exceed the whole.