Chapter I: The Five Groups of Axioms — Summary¶
Structural & Mathematical Overview — Kapitel I: Die fünf Axiomgruppen
Architectural Summary of Chapter I (§§ 1–8)¶
Chapter I constructs the complete deductive foundation of three-dimensional Euclidean space across eight sections and twenty foundational theorems (Theorems 1–20):
§ 1 — Primitive Systems: Replaces Euclid’s descriptive definitions of point, line, and plane (I.Def.1, 4, 7) with three unspecified systems of things bound solely by mutual relations across five axiom groups.
§ 2 — Group I: Connection (I 1–7, Theorems 1–2): Separates planar incidence (I 1–2: existence and uniqueness of the straight line joining two distinct points) from spatial incidence (I 3–7: plane determination by three non-collinear points, line-in-plane containment, two-plane intersection dimension, and existence of four non-coplanar points). Deduces unique intersection properties (Theorem 1) and plane determination by a line and external point or two intersecting lines (Theorem 2).
§ 3 — Group II: Order (II 1–5): Axiomatizes the ternary collinear relation between (II 1–4), defines the segment $AB$, and introduces Moritz Pasch’s planar triangle-piercing postulate (II 5, Axiom von Pasch).
§ 4 — Consequences of Order (Theorems 3–7): Proves unlimited linear density (Theorem 3), total ordering of any finite collinear point set $A, B, C, D, E, \ldots, K$ (Theorem 4), planar half-plane separation (Theorem 5), half-rays, Camille Jordan’s polygonal separation theorem (Theorem 6), and spatial half-space separation (Theorem 7) without needing any spatial order axiom.
§ 5 — Group III: Parallels (Axiom III, Theorem 8): Adopts Playfair’s uniqueness form of Euclid’s Fifth Postulate in a plane $\alpha$ and proves transitivity of non-intersection (Theorem 8). (Note: renumbered Group IV in later editions.)
§ 6 — Group IV: Congruence (IV 1–6): Eliminates physical motion and superposition (ἐφαρμόζειν) via six static axioms: segment transfer (IV 1), congruence transitivity (IV 2), segment additivity (IV 3), angle transfer (IV 4), angle transitivity (IV 5), and the triangle bridge axiom (IV 6) deriving base-angle congruence from two congruent sides and the included angle. (Note: renumbered Group III in later editions.)
§ 7 — Consequences of Congruence (Theorems 9–20): Proves congruent point-series order preservation (Theorem 9), SAS triangle congruence (Theorem 10, Euclid I.4), ASA congruence (Theorem 11), supplementary angle congruence (Theorem 12, Euclid I.13), interior/exterior angle ray decomposition (Theorems 13–14), congruence of all right angles (Theorem 15, proving Euclid Postulate 4!), SSS congruence (Theorem 16, Euclid I.8), rigid figure extension in the plane and 3-space (Theorems 17–18), alternate/interior angle parallelism (Theorem 19), and the $180^\circ$ triangle angle-sum theorem (Theorem 20).
§ 8 — Group V: Continuity: Formulates the linear Axiom of Archimedes (V) and—in the 1900/1902 Remark—the metamathematical Axiom of Completeness (Vollständigkeit, later V 2).