Technical Glossary

Projective, Descriptive, Metrical, Logicist, and Scholastic Terminology

This lexicon defines 27 technical concepts central to Russell’s 1897 and 1903 geometric treatises and their reception in Hoenen and Heath. Every inline :term: role links directly to its entry below.

anharmonic ratio (cross-ratio)

  • Tradition / Domain: English / French (rapport anharmonique)

  • Concise Formulation: Projective invariant of four collinear points or concurrent rays

The double ratio (AB/CB) / (AD/CD) preserved under perspective projection; in pure projective geometry, assigned numerically via Möbius nets and Dedekind continuity without appeal to metrical distance (EFG §§111–112; PoM §370).

axiom of Free Mobility (Congruence)

  • Tradition / Domain: English

  • Concise Formulation: Invariance of spatial magnitudes under continuous change of position

In EFG (§§146–154), the a priori metrical postulate that spatial magnitudes are independent of place, equivalent to constant Riemannian curvature and homogeneity of space.

betweenness

  • Tradition / Domain: English (Descriptive Geometry)

  • Concise Formulation: Triadic open-order relation of collinear points

The primitive order relation axiomatized by Pasch, Peano, and Hilbert, analyzed by Russell in PoM Part IV and Ch. XLVI as generated by transitive asymmetrical serial relations along a line.

descriptive geometry

  • Tradition / Domain: English (Geometry)

  • Concise Formulation: Geometry of open serial order and betweenness (Pasch–Peano)

In Russell’s 1903 classification (PoM Ch. XLVI, §§374–387), the study of spaces where two points determine an open serial segment (betweenness) rather than a closed projective line—applicable to Euclidean and hyperbolic spaces prior to introducing metrical congruence, and completed into projective space via ideal elements.

form of externality

  • Tradition / Domain: English (1897 Epistemology)

  • Concise Formulation: Relational condition of experiencing simultaneous substantival diversity

Central philosophical concept of Russell’s 1897 Essay (Chs. III–IV): any intuitive relational medium rendering experience of diverse interrelated things possible—from which Russell deduced projective geometry and free mobility as a priori before abandoning transcendental arguments in 1903.

formal implication

  • Tradition / Domain: English (Logicism)

  • Concise Formulation: Universal conditional over variables: φ(x) implies ψ(x) for all x

Russell’s foundational relation in PoM (1903, §§1, 5–6): an implication holding for every value of one or more variables, by which pure mathematics asserts hypothetical deductions (‘if S is a Euclidean space, then P holds of S’) without categorical existential commitments.

harmonic range

  • Tradition / Domain: English (Projective Geometry)

  • Concise Formulation: Four collinear points (A, B, C, D) in harmonic conjugation

Four collinear points such that A and C are harmonic conjugates with respect to B and D: metrically having cross-ratio -1, and projectively defined without metrics by von Staudt’s complete quadrilateral construction (EFG §113; PoM §365).

ideal elements (points, lines, and planes at infinity)

  • Tradition / Domain: English (Descriptive Geometry)

  • Concise Formulation: Logical classes (sheaves/pencils) completing a descriptive space into a projective space

Following Pasch, an ‘ideal point’ is defined logically—without postulating fictitious entities—as an entire bundle (sheaf) of lines meeting pairwise in planes, allowing parallel or ultra-parallel lines to share an ideal intersection (PoM §§383–386).

involution

  • Tradition / Domain: English (Projective Geometry)

  • Concise Formulation: Symmetric projective pairing of collinear points with double points

A projective correspondence on a line that interchanges conjugate pairs; hyperbolic involutions possess two real double points harmonic with every conjugate pair, whereas elliptic involutions have none (PoM §366).

logical constant

  • Tradition / Domain: English (Logicism)

  • Concise Formulation: Indefinable or purely logical primitive notion

In PoM §1 & §10, the sole non-variable constituents permitted in propositions of pure mathematics: implication, class-membership (ε), ‘such that’, relation, and truth.

magnitude of divisibility

  • Tradition / Domain: English (PoM Metrical Geometry)

  • Concise Formulation: Quantitative measure of a stretch, area, or volume as a property of serial relations

Russell’s relational interpretation of extensive spatial quantity (PoM §§162, 396–397): since all continuous stretches have the same Cantor cardinality 2^α₀, length belongs not to the point-set itself but to equivalence classes of generating serial relations.

manifold (Mannigfaltigkeit)

  • Tradition / Domain: English / German

  • Concise Formulation: Continuous or discrete n-tuply extended system of determinations

Riemann’s general class-concept of multiply extended magnitudes (1854), critiqued in EFG Ch. II (§§60–65) for neglecting qualitative relational order and reformulated logically in PoM Ch. XLIV as a multi-dimensional series.

materia intelligibilis

  • Tradition / Domain: Latin (Scholastic / Aristotelian)

  • Concise Formulation: Intelligible matter (pure continuous extension)

Aristotle’s and Aquinas’s designation (ὕλη νοητή) for continuous extension abstracted from sensible qualities, invoked by Hoenen (Caput IV & VII) as the principle of individuation and congruence answering Russell’s attack on Euclid I.4.

measure of curvature (Krümmungsmass)

  • Tradition / Domain: English / German

  • Concise Formulation: Intrinsic Gaussian / Riemannian curvature constant k

Intrinsic parameter characterizing homogeneous spaces admitting free mobility: k = 0 for Euclidean space, k < 0 for Lobachevskian (hyperbolic) space, and k > 0 for spherical and elliptic spaces (EFG §§20–23).

Metageometry

  • Tradition / Domain: English (19th-c. Geometry)

  • Concise Formulation: General / non-Euclidean geometry investigating axiom independence

Term used throughout Russell’s 1897 Essay for the synthetic (Gauss, Lobachevsky, Bolyai), metrical-differential (Riemann, Helmholtz, Beltrami), and projective (Cayley, Klein, Lie) generalizations of Euclidean geometry.

metrical geometry

  • Tradition / Domain: English (Geometry)

  • Concise Formulation: Quantitative geometry of distance, congruence, angle, area, and volume

The branch of geometry introducing quantitative equality (congruence) of stretches, distances, and angles, presupposing projective or descriptive order (EFG Ch. III Sec. B; PoM Ch. XLVII).

motion (geometrical)

  • Tradition / Domain: English (PoM Metrical Geometry)

  • Concise Formulation: Class of metric-preserving one-one point transformations

In PoM §§390 & 395, Russell’s static relational replacement for kinematic transport: a group of bijective point-correlations leaving all distances and metrical properties invariant (axiomatized directly as an indefinable by Pieri, 1899).

Möbius net

  • Tradition / Domain: English (Projective Geometry)

  • Concise Formulation: Countable dense harmonic net on a projective line or plane

The denumerable progression of points generated from three initial collinear points (0, 1, ∞) by repeated harmonic quadrilateral constructions, establishing rational projective coordinates and projective order (PoM §368).

point (spatial element)

  • Tradition / Domain: English (Foundations of Geometry)

  • Concise Formulation: Simple unextended term of spatial relations

Treated in EFG (§§117, 189–204) as infected with a dialectical contradiction between unextended simplicity and spatiality, then vindicated in PoM (Ch. LI, §§423–431) as a logically consistent substantive term of multi-dimensional serial relations.

principle of duality

  • Tradition / Domain: English (Projective Geometry)

  • Concise Formulation: Reciprocity of points and planes in 3-space (or points and lines in 2-space)

The formal symmetry of projective axioms under uniform interchange of ‘point’ and ‘plane’ (leaving ‘straight line’ invariant in three dimensions), doubling every projective theorem (EFG §107; PoM §373).

projective geometry

  • Tradition / Domain: English (Geometry)

  • Concise Formulation: Qualitative geometry of incidence, collineation, and harmonic separation

The geometry of properties invariant under central projection and section, requiring neither betweenness on open lines nor quantitative distance; axiomatized qualitatively via incidence and von Staudt’s quadrilateral construction (EFG Ch. III Sec. A; PoM Ch. XLV).

propositional function

  • Tradition / Domain: English (Logicism)

  • Concise Formulation: Open expression φ(x) yielding a proposition for each value of x

An expression containing one or more real variables which becomes a determinate true or false proposition whenever definite terms are substituted for its variables.

quadrilateral construction

  • Tradition / Domain: English (Projective Geometry)

  • Concise Formulation: Von Staudt’s pure-incidence construction of the harmonic fourth point

Given collinear points A, B, D, constructing the unique harmonic conjugate C of A with respect to B, D by projecting from an external point O and intersecting diagonals of a complete quadrilateral—uniqueness in a plane following from three-dimensional incidence via Desargues’s theorem (EFG §113; PoM §365).

separation of couples

  • Tradition / Domain: English (Order Theory / Projective Geometry)

  • Concise Formulation: Tetradic cyclic order relation on a closed series (ab // cd)

Vailati’s five-axiom symmetrical transitive relation between point-pairs (a, c) and (b, d) on a closed projective or elliptic line where simple triadic betweenness fails because any two points bound two complementary segments (PoM §§194, 367, 398).

stretch

  • Tradition / Domain: English (PoM Theory of Quantity)

  • Concise Formulation: Open or closed segment of points between two endpoints on a line

The class of intermediate terms (or the generating serial relation restricted to them) between two collinear points A and B, contrasted by Russell with ‘distance’, which is an indivisible two-term relation (PoM §§169–170, 393–398).

superposition

  • Tradition / Domain: English (Classical Geometry)

  • Concise Formulation: Geometric proof of congruence by transporting one figure onto another

Euclid’s method in I.4, I.8, and C.N. 4; rejected by Russell in PoM §390 as logically circular because spatial points are immobile positions while moving material bodies presupposes an antecedent metrical definition of rigidity; rehabilitated epistemologically by Hoenen via extension as materia intelligibilis.

the Absolute (Grundgebild)

  • Tradition / Domain: English / German

  • Concise Formulation: Invariant conic or quadric locus defining a Cayley–Klein projective metric

Cayley and Klein’s fixed second-degree locus in projective space relative to which distance and angle are logarithmic functions of cross-ratios: an imaginary quadric yields elliptic geometry, a real non-degenerate quadric yields hyperbolic geometry, and a degenerate imaginary conic in a plane yields Euclidean geometry (EFG §31; PoM §§407–409).