Bilingual Mathematical & Axiomatic Glossary

Technical German & English Terminology of von Staudt (1847, 1857) and Pasch (1882)

Geometrie der Lage — Geometry of Position (Pure Projective Geometry)

Von Staudt’s pure synthetic science of spatial incidence, order, harmonic separation, and polarity (1847 § 1), developed without any reliance on metrical magnitude (length, angle, area, or numerical ratio).

Einförmiges Grundgebilde — Uniform Fundamental Form (One-Dimensional Primitive Form)

Any one of the three projectively dual one-dimensional carriers of 1847 § 5: the rectilinear range of points (gerades Gebilde), the flat pencil of rays (ebenes Strahlbüschel), and the axial pencil of planes (Ebenenbüschel).

Unendlich fernes Element (Scheinpunkt / Scheingerade) — Ideal Element (Point / Line at Infinity)

A direction or parallel sheaf adjoined to ordinary space (1847 § 2, arts. 16–22) so that every two coplanar lines meet in a unique point and every two planes meet in a unique line; reconstructed strictly from finite proper segments via sheaves (Bündel) in Pasch 1882 §§ 8–9.

Gesetz der Reciprocität — Law of Reciprocity (Projective Duality)

The structural symmetry of three-dimensional projective space whereby points and planes are dual primitive elements while straight lines are self-dual (1847 § 3, art. 25), and in a single plane points and lines exchange incidence roles.

Vollständiges ebenes Viereck — Complete Plane Quadrangle

Four coplanar points (no three collinear) joined pairwise by six sides (three pairs of opposite sides) intersecting in three diagonal points (1847 § 6, art. 66); plane-dual to the complete quadrilateral (vollständiges Vierseit).

Harmonisches Gebilde (Harmonische Punkte) — Harmonic Form (Harmonic Tetrad)

Four collinear points $A, C, B, D$ such that two opposite sides of a complete quadrangle pass through $A$, two opposite sides pass through $B$, and the two remaining diagonals pass respectively through $C$ and $D$ (1847 § 6, arts. 67–71). Satisfies $(A, B; C, D) = -1$ without any appeal to distance.

Projektivische Grundgebilde — Projectively Related Fundamental Forms

Two uniform fundamental forms placed in bijection so that every harmonic tetrad of one corresponds to a harmonic tetrad of the other (1847 § 8, art. 93).

Ebenes Polarsystem — Planar Polar System (Polarity)

An involutory reciprocal correlation between the points and lines of a plane (1847 § 18, arts. 235–240) assigning to each point (Pol) a unique line (Polare) such that if $Q$ lies on the polar of $P$, then $P$ lies on the polar of $Q$.

Ordnungscurve (Kurve zweiter Ordnung) — Order Curve (Point-Conic)

Von Staudt’s pure incidence definition of a conic section (1847 § 19, art. 242) as the locus of points in a planar polar system that are incident with their own polars, dual to the class-conic (Hüllcurve) of self-conjugate tangents.

Wurf — Throw

An ordered tetrad $(ABCD)$ of four elements of a uniform fundamental form, considered up to projectivity (1857 § 19, arts. 256–257); the synthetic projective entity whose numerical avatar is the cross-ratio.

Rechnung mit Würfen — Algebra of Throws

Von Staudt’s pure straightedge addition ($X + Y$) and multiplication ($X \cdot Y$) of points or throws relative to three fixed gauge elements $(P_0, P_1, P_\infty)$ on a carrier line (1857 §§ 19–20, arts. 256–269), endowing $\ell \setminus \{P_\infty\}$ with the structure of a commutative field.

Harmonische Reihe (Harmonisches Netz) — Harmonic Scale (Möbius Net)

The sequence of integer-abscissa points $P_{-1}, P_0, P_1, P_2, P_3, \dots, P_\infty$ generated by iterated harmonic reflection $(P_{n-1}, P_{n+1}; P_n, P_\infty) = -1$, together with all rational subdivisions constructible by straightedge alone.

Projektivische Abscisse — Projective Abscissa

The intrinsic coordinate assigned to a point $X \in \ell \setminus \{P_\infty\}$ relative to the frame $(P_0, P_1, P_\infty)$ by the Algebra of Throws (1857 § 29, arts. 403–404), equal algebraically to $(X, P_1; P_0, P_\infty)$.

Zwischenbeziehung (Zwischen) — Linear Betweenness

Pasch’s primitive ternary order relation “$C$ lies between $A$ and $B$ on the straight segment $AB$” (1882 § 1, Grundsätze I–VIII), axiomatizing open segments prior to congruence or measure.

Getrennte Punktepaare — Mutually Separating Point Pairs

Two pairs $(A, B)$ and $(C, D)$ on a closed projective line such that each pair divides the carrier into two segments containing one point of the other pair; satisfied by every harmonic tetrad.

IV. Grundsatz der Ebene (Axiom von Pasch) — Pasch’s Planar Separation Axiom

Pasch’s planar order axiom (1882 § 2, p. 21): if a coplanar straight line meets one side $AB$ of a triangle $\triangle ABC$ internally and passes through no vertex, it must also meet internally one (and by 10. Lehrsatz, p. 25, only one) of the other two sides $AC$ or $BC$.

Stammbegriffe (Kernbegriffe) — Primitive / Nuclear Concepts

In Pasch’s axiomatic methodology (1882 § 12), the irreducible undefined primitives (point, straight segment, planar surface, congruence) out of which every derived geometric concept must be explicitly defined.

Stammsätze (Grundsätze / Kernsätze) — Nuclear Propositions (Axioms)

In Pasch’s 1882 § 12 manifesto (p. 98), the complete system of foundational axioms distilled initially from empirical observation, after which every theorem must follow by pure logical deduction without any appeal to spatial intuition or diagrams.