Bilingual Technical Glossary

French (1902), German, Latin, and English Terminology of Conventionalism & Non-Euclidean Geometry

Raisonnement par récurrence — Reasoning by Recurrence (Complete Mathematical Induction) [fr]

Poincaré’s term in Chapter I for mathematical induction—proving a property holds for n = 1, proving that if it holds for n - 1 it holds for n, and concluding it holds for all integers. Poincaré identifies it as the sole instrument capable of leaping from the finite to the infinite, constituting a genuine synthetic a priori intuition.

Continu physique — Physical (Perceptual) Continuum [fr]

A sensory domain characterized by Poincaré’s contradictory triad of threshold observations: A = B, B = C, yet A < C (drawn from Fechner’s psychophysics). To resolve this logical contradiction against transitivity, the mind invents the mathematical continuum.

Coupure (Schnitt) — Dedekind Cut [fr/de]

Richard Dedekind’s partition of the rational numbers Q into two non-empty classes between which an incommensurable symbol (such as √2) is created to guarantee geometric intersection—forming what Poincaré calls the mathematical continuum of the second order.

Continu du troisième ordre — Continuum of the Third Order (Non-Archimedean Continuum) [fr]

An enriched ordered continuum constructed by inserting actual infinitesimals (Veronese 1891; Hilbert 1899) between Dedekind real numbers, violating Archimedes’ axiom.

Analysis Situs — Topology (Qualitative Geometry of Position) [la/fr]

Leibniz’s and Poincaré’s name for topology—the study of qualitative properties of multidimensional continua invariant under arbitrary continuous homeomorphisms prior to any metric or projective group.

Postulatum d’Euclide — Euclid’s Fifth (Parallel) Postulate [fr]

Playfairs/Euclid’s parallel axiom asserting that through a given point passes at most one parallel to a given straight line. Denied toward multiplicity by Bolyai and Lobachevsky (hyperbolic geometry) and toward non-existence by Riemann (spherical/elliptic geometry).

Dictionnaire de Poincaré — Poincaré’s Non-Euclidean Translation Dictionary [fr]

Poincaré’s two-column semantic mapping in Chapter III translating every primitive term of Lobachevsky’s geometry (‘space’, ‘plane’, ‘line’, ‘sphere’, ‘distance’, ‘displacement’) into ordinary Euclidean objects orthogonal to a fundamental horizon circle/sphere Ω, proving that any contradiction in Lobachevsky geometry would immediately induce a contradiction in Euclid.

Plan fondamental / Cercle fondamental (Ω) — Fundamental Horizon Plane or Circle [fr]

The ideal boundary ∂D (in 2D, the unit circle Ω: x² + y² = 1; in 3D upper half-space, z = 0) representing infinity in Poincaré’s conformal model. All hyperbolic geodesics meet Ω orthogonally at ideal points.

Inversion par rapport à Ω — Horizon Circle Inversion [fr]

The involutive Euclidean transformation P ↦ P* = P / ‖P‖² satisfying OP · OP* = 1. Any Euclidean circle passing through P and P* cuts the unit horizon Ω orthogonally (‖OC‖² − r² = 1), providing the exact straightedge-and-compass generator of hyperbolic geodesics.

Rapport anharmonique — Cross-Ratio (Anharmonic Ratio) [fr]

The projective/Möbius four-point invariant χ(P, Q; U, V) = (‖PV‖·‖QU‖) / (‖PU‖·‖QV‖) along a geodesic meeting the horizon Ω at ideal points U, V. Its natural logarithm ln(χ) equals the hyperbolic distance d_H(P, Q).

Quadrance hyperbolique rationnelle — Wildberger’s Rational Hyperbolic Quadrance [fr/en]

Transcendental-free algebraic measure of separation in hyperbolic geometry: conformal quadrance τ(P, Q) = ‖P - Q‖² / ‖1 - P̄Q‖² = tanh²(d_H/2) and Universal Hyperbolic Quadrance q_UHG(P, Q) = -4τ/(1 - τ)² = -sinh²(d_H), taking pure rational values in Q whenever P, Q ∈ Q² ∩ D.

Parallèle limite (Horoparallèle) — Limiting Parallel (Horoparallel) [fr]

One of the two boundary Lobachevskian parallels through P ∉ L_hyp that share an ideal endpoint U or V on ∂Ω with L_hyp, separating secant geodesics from ultraparallels.

Ultraparallèle (Droite non-sécante divergente) — Ultraparallel (Hyperparallel) [fr]

A hyperbolic geodesic through P ∉ L_hyp disjoint from L_hyp both inside the open disk D and on the ideal boundary ∂Ω, admitting a unique common perpendicular geodesic.

Défaut angulaire — Hyperbolic Angular Defect [fr]

The positive deficit Δ(ABC) = π - (α + β + γ) > 0 by which the interior angle sum of a hyperbolic triangle falls short of two right angles, proportional to its hyperbolic area and reaching its supremum π for an ideal triangle.

Quatrième géométrie — Poincaré’s Fourth (Indefinite / Lorentzian) Geometry [fr]

The non-Riemannian constant-curvature geometry introduced by Poincaré in Chapter III § 8 (by taking the region outside the fundamental sphere/circle), in which rotations become hyperbolic boosts and a real null line can be perpendicular to itself.

Conventionnalisme géométrique (Définitions déguisées) — Geometric Conventionalism (Disguised Definitions) [fr]

Poincaré’s central epistemological thesis (Ch. III § 11): geometric axioms are neither synthetic a priori intuitions nor empirical laws, but conventions guided (not imposed) by experience with approximately rigid bodies; Euclidean geometry is retained not as ‘truer’ than Lobachevsky’s, but as the simplest and most convenient convention.

Monde sphérique / Disque chauffé de Poincaré — Poincaré’s Heated-Disk World [fr]

Poincaré’s celebrated thought experiment in Chapter IV: an enclosure of radius R kept at temperature T ∝ R² - r² where thermal expansion makes every mobile rod shrink as R² - r² and refractive index varies as 1/(R² - r²). Sentient inhabitants adopting solid-body and optical conventions inevitably construct Lobachevskian hyperbolic geometry.

Groupe des déplacements — Group of Displacements (Lie Transformation Group) [fr]

The six-parameter (in 3D) or three-parameter (in 2D) continuous Lie group of motions without deformation. For Poincaré, although particular spatial sensations are empirical, the general notion of a mathematical group pre-exists in the mind as a form of understanding.