Unit 2 Summary: Quadrangle Harmonicity & the Fundamental Theorem

Analytical Synopsis of Geometrie der Lage (1847), §§ 6–8

1. Complete Plane Quadrangle (§ 6, art. 66): Four general coplanar points $E, F, G, H$ determine six sides in three opposite pairs meeting at three diagonal points; plane-dual to the complete quadrilateral.

2. Metric-Free Harmonic Fourth (§ 6, art. 67): Four collinear points $(A, B; C, D)$ are harmonic iff two opposite quadrangle sides meet at $A$, two meet at $B$, and the remaining two cut $[AB]$ at $C$ and $D$—requiring 7 straightedge lines and 0 circles.

3. 3D Incidence Proof of Uniqueness (§ 6, art. 70): Lifting to an auxiliary plane $\alpha'$ through $\ell$ proves purely from spatial plane intersections (equivalent in the plane to Desargues’s Two-Triangle Theorem) that $D$ is independent of the quadrangle.

4. Harmonic Definition of Projectivity (§ 8, art. 93): Replaces Steiner’s and Chasles’s cross-ratio definition with harmonic-tetrad preservation.

5. Fundamental Theorem (§ 8, art. 96): Three distinct elements $(A, B, C) \mapsto (A', B', C')$ uniquely determine a projectivity on a uniform fundamental form.