Mission & Axiomatic Synthesis

Why von Staudt (1847, 1857) and Pasch (1882) Stand Together in GEOMETOR

Across the nineteenth century, synthetic geometry confronted two foundational crises:

1. The Circularity of Metrical Projective Geometry: Jean-Victor Poncelet, Jakob Steiner, and Michel Chasles discovered that incidence, harmonic division, and conic polarity remain invariant under central projection—the most general spatial transformations—yet defined those invariants using Euclidean segment lengths and signed arithmetic cross-ratios $\frac{AC}{BC} : \frac{AD}{BD}$. How could the science of pure position depend upon compass-based congruence?

2. The Unstated Order Assumptions of Classical & Projective Deduction: Both Euclid (in Elements I.16 and I.21) and von Staudt himself (in assuming that the rational harmonic scale exhausts continuous projective order) relied silently on visual inspection of betweenness and planar separation.

geometor/staudt unites the two treatises that resolved these twin crises and forged modern axiomatic geometry:

• Karl Georg Christian von Staudt (Geometrie der Lage, 1847; Beiträge, Heft 2, 1857): Constructs the harmonic fourth $D$ of three collinear points $(A, B, C)$ purely by unmarked straightedge via the complete plane quadrangle, defines conics as self-conjugate loci of planar polarities, and erects the Algebra of Throws ($X + Y$, $X \cdot Y$), generating intrinsic coordinates over any commutative field before a single distance is measured.

• Moritz Pasch (Vorlesungen über neuere Geometrie, 1882): Isolates the eight nuclear axioms of open linear betweenness (§ 1) and Pasch’s Planar Separation Axiom (IV. Grundsatz, § 2, p. 21), grounds projective ideal elements strictly on finite sheaves (Bündel, §§ 8–11), and states the modern hypothetico-deductive criterion (§ 12, p. 98) directly inherited by David Hilbert, Bertrand Russell, Oswald Veblen, Fr. Petrus Hoenen, and Alfred Tarski.

Exact Symbolic Realization in geometor.staudt

In keeping with GEOMETOR’s mandate of 100% exact symbolic algebra and strict content/code separation:

• Every straightedge construction in geometor.staudt executes on Model(..., auto_intersect=False) over $\mathbb{Q}$ and $\mathbb{Q}(\phi)$ and enforces assert_straightedge_only(model) (zero circles, len(model.circles) == 0, Lemoine class S = R).

• All historical and translated codex tracks in Codex I and Codex II remain pure scholarly text accompanied by standalone .euclid-graphic SVG figures, while executable Python constructions reside cleanly in src/geometor/staudt/ and demos/.