CROSS-CORPUS CONCORDANCE¶
Bidirectional Loci Across Klein (1872/1893 & 1895/1897) and Nine GEOMETOR Sister Corpora
This master concordance maps the algebraic constructibility criteria, restricted-instrument theorems, transcendence proofs, and transformation-group hierarchies of Felix Klein across Famous Problems of Elementary Geometry (1895/1897) and The Erlangen Program (1872/1893) directly to nine sister repositories in the GEOMETOR ecosystem: euclid, kepler, hilbert, bertrand-russell, hoenen, lemoine, petersen, pappus, and wildberger.
Summary Matrix¶
Foundational Theme |
Famous Problems (1895/1897) |
Erlangen Program (1872/1893) |
GEOMETOR Sister Corpora |
|---|---|---|---|
Euclidean Straightedge-and-Compass Postulates & Iterated Square-Root Towers |
Introduction & Part I, Ch. I (§§ 1–15, EN pp. 1–12 / DE S. 1–10) |
§ 1 (Principal Group of Similarities) |
|
Delian Cube Duplication, Angle Trisection, and Plane vs. Solid Problems |
Part I, Ch. II (§§ 1–3, EN pp. 13–15 / DE S. 10–12) & Ch. V (§§ 2–5, EN pp. 42–47 / DE S. 32–37) |
§ 10 (Galois Group Analogy for Algebraic Equations) |
|
Cyclotomy, Kepler’s Unknowable Heptagon (n = 7), and Gauss’s Heptadecagon (n = 17) |
Part I, Ch. III (§§ 1–8, EN pp. 16–23 / DE S. 13–19) & Ch. IV (§§ 1–6, EN pp. 24–33 / DE S. 19–25) |
§ 10 (Commutative / Abelian Group Reductions) |
|
Restricted Instruments: Poncelet–Steiner Fixed Circle, Mascheroni Compasses, and Geometrography |
Part I, Ch. IV (§§ 7–9, EN pp. 33–41 / DE S. 25–32) & Ch. V (§ 1, Paper Folding) |
§§ 2–3 (Projective Straightedge Constructions vs. Metric Circle) |
|
Cantor’s Continuum, Transcendence of e and pi, and Quadrature of the Circle |
Part II, Chs. I–V (EN pp. 49–80 / DE S. 38–66) |
§ 8.2 (Analysis Situs & Continuous Manifolds) |
|
The Erlangen Hierarchy: Principal Group, Affine Geometry, and Projective Collineations |
Part I, Ch. IV (§ 7, Rational Projective vs. Quadratic Euclidean Operations) |
Introduction & §§ 1–5 (EN pp. 215–230 / DE S. 3–21) |
|
Cayley–Klein Projective Metrics, the Absolute Quadric, and Rational Trigonometry |
Introduction (Theoretical Exactitude vs. Empirical Approximation) |
§§ 1–2, § 6, § 10.2 & Note V (On the So-Called Non-Euclidean Geometry) |
|
Reciprocal Radii (Möbius Inversion), Plücker Line Geometry, and Lie’s Sphere Geometry |
Part I, Ch. V (§§ 3–4: CircularPoints at Infinity on Cissoid & Conchoid) |
§§ 5–9 & Note VII (EN pp. 226–238, 248 / DE S. 15–32, 42) |
|
Detailed Analytical Concordance Entries¶
1. Euclidean Straightedge-and-Compass Postulates & Iterated Square-Root Towers¶
Famous Problems (1895 / 1897): Introduction & Part I, Ch. I (§§ 1–15, EN pp. 1–12 / DE S. 1–10)
Erlangen Program (1872 / 1893): § 1 (Principal Group of Similarities)
Cross-Corpus Relations:
geometor/euclid — The Thirteen Books of Euclid’s Elements (Heath, 1908),
Book I, Postulates 1–3; Book II, Prop. 14; Book VI, Props. 12–13: Klein’s Fundamental Theorem (Hauptsatz) translates Euclid’s Postulates 1–3 into the analytic closure of rational operations $(+, -, \cdot, /)$ and square roots $\sqrt{R}$.geometor/petersen — Methods and Theories for the Solution of Problems of Geometrical Constructions (1879),
Algebraic Appendix & Theorie der algebraischen Gleichungen (1878): Klein explicitly singles out Julius Petersen in the Introduction (EN p. 2 / DE S. 2) as the sole textbook author presenting elementary algebraic impossibility proofs for geometric constructions.geometor/hilbert — Grundlagen der Geometrie (1899 / 1902),
Chapter VII (§§ 36–39: Geometrical Constructions Based Upon Axioms I–V): Hilbert refines Klein’s quadratic-tower criterion by isolating the Pythagorean subfield (sqrt(1 + omega^2), solvable with straightedge and segment-transferrer Eichmass) inside the full Euclidean square-root field.
2. Delian Cube Duplication, Angle Trisection, and Plane vs. Solid Problems¶
Famous Problems (1895 / 1897): Part I, Ch. II (§§ 1–3, EN pp. 13–15 / DE S. 10–12) & Ch. V (§§ 2–5, EN pp. 42–47 / DE S. 32–37)
Erlangen Program (1872 / 1893): § 10 (Galois Group Analogy for Algebraic Equations)
Cross-Corpus Relations:
geometor/pappus — Synagoge (Collection), Books III, IV & VII,
Book III (Delian Mean Proportionals) & Book IV (Plane, Solid, and Linear Problems; Conchoid Trisection): Klein’s degree-2^h impossibility proofs for x^3 = 2 and x^3 = cos(phi) + i*sin(phi), paired with conic intersections, Diocles’ cissoid, and Nicomedes’ conchoid in Ch. V, supply the exact Galois-algebraic proof of Pappus’s ancient classification of plane, solid, and linear loci.geometor/euclid — The Thirteen Books of Euclid’s Elements (Heath, 1908),
Book I, Prop. 9 (Angle Bisection) & Book XI, Props. 31–33 (Parallelepipeds & Cubes): Contrasts elementary quadratic angle bisection (Euclid I.9) with the irreducible cubic monodromy of general angle trisection.
3. Cyclotomy, Kepler’s Unknowable Heptagon (n = 7), and Gauss’s Heptadecagon (n = 17)¶
Famous Problems (1895 / 1897): Part I, Ch. III (§§ 1–8, EN pp. 16–23 / DE S. 13–19) & Ch. IV (§§ 1–6, EN pp. 24–33 / DE S. 19–25)
Erlangen Program (1872 / 1893): § 10 (Commutative / Abelian Group Reductions)
Cross-Corpus Relations:
geometor/kepler — Harmonices Mundi, Liber I (1619),
Book I, Props. XLV–XLVII (De heptagono et figuris inscibilibus): Kepler argued philosophically and algebraically that the regular heptagon (n = 7) is ‘unknowable’ (inscibilis) because its side ratio obeys a cubic equation irreducible to planar bisections; Klein Ch. III (§§ 5–7) proves rigorouly via Eisenstein’s criterion that z^6 + … + 1 = 0 (real cubic u^3 + u^2 - 2u - 1 = 0) has degree 6 != 2^h, while p = 17 = 2^(2^2) + 1 splits into four quadratics.geometor/euclid — The Thirteen Books of Euclid’s Elements (Heath, 1908),
Book IV, Props. 11, 15, 16 (Regular 5-, 6-, and 15-gons): Klein Ch. III, § 2 explains Euclid IV.16 (regular 15-gon) arithmetically via Bezout partial fractions 1/15 = 2/3 - 3/5.
4. Restricted Instruments: Poncelet–Steiner Fixed Circle, Mascheroni Compasses, and Geometrography¶
Famous Problems (1895 / 1897): Part I, Ch. IV (§§ 7–9, EN pp. 33–41 / DE S. 25–32) & Ch. V (§ 1, Paper Folding)
Erlangen Program (1872 / 1893): §§ 2–3 (Projective Straightedge Constructions vs. Metric Circle)
Cross-Corpus Relations:
geometor/lemoine — Géométrographie, ou Art des constructions géométriques (1902),
Part I (Operational Symbols R_1, R_2, C_1, C_2, C_3) & Poncelet-Steiner / Mascheroni Comparisons: Klein’s sharp distinction between theoretical instrument classes—straightedge alone (rational field), Mascheroni compasses alone, and Poncelet–Steiner–von Staudt–Schröter straightedge + one fixed circle—parallels Lemoine’s quantitative operational measure of ruler vs. compass simplicity.geometor/pappus — Synagoge (Collection), Book VII,
Complete Quadrilateral Harmonic Construction & Perspectivity: Pure straightedge constructions (Lambert, Brianchon, von Staudt) construct precisely projective rational functions via harmonic ranges, requiring one fixed metric circle C_0 to adjoin square roots.
5. Cantor’s Continuum, Transcendence of e and pi, and Quadrature of the Circle¶
Famous Problems (1895 / 1897): Part II, Chs. I–V (EN pp. 49–80 / DE S. 38–66)
Erlangen Program (1872 / 1893): § 8.2 (Analysis Situs & Continuous Manifolds)
Cross-Corpus Relations:
geometor/bertrand-russell — The Principles of Mathematics (1903) & An Essay on the Foundations of Geometry (1897),
PoM Part V (Infinity and Continuity, Cantor's Aleph_0 vs. Continuum) & Part VI: Klein’s exposition of Cantor’s countability of algebraic numbers by height N and diagonal uncountability of the continuum (Part II, Ch. I) anticipates Russell’s Cantorian resolution of spatial continuity in 1903.geometor/hoenen — De Noetica Geometriae (1954),
Caput III (§ 2.6, Continuous Extension vs. Point-Sets) & Caput VII: Contrasts point-set enumerations of algebraic vs. transcendental coordinates with intuitive continuous geometric extension traced by mechanical curve-integrators (Integraph, Ch. V).
6. The Erlangen Hierarchy: Principal Group, Affine Geometry, and Projective Collineations¶
Famous Problems (1895 / 1897): Part I, Ch. IV (§ 7, Rational Projective vs. Quadratic Euclidean Operations)
Erlangen Program (1872 / 1893): Introduction & §§ 1–5 (EN pp. 215–230 / DE S. 3–21)
Cross-Corpus Relations:
geometor/pappus — Synagoge (Collection), Book VII,
Props. 129–145 (Cross-Ratio Invariance Under Central Projection): Elevates Pappus’s cross-ratio invariance from an isolated synthetic lemma to the defining invariant of the projective collineation group encompassing the Principal Group.geometor/hilbert — Grundlagen der Geometrie (1899 / 1902),
Chapters V–VI (Desargues's & Pascal's Theorems) & 1902 Appendix on Transformation Groups: Connects Klein’s continuous group hierarchy directly to Hilbert’s axiomatic Characterization of Desarguesian skew-fields, Pappian commutative fields, and Helmholtz–Lie transformation axioms.
7. Cayley–Klein Projective Metrics, the Absolute Quadric, and Rational Trigonometry¶
Famous Problems (1895 / 1897): Introduction (Theoretical Exactitude vs. Empirical Approximation)
Erlangen Program (1872 / 1893): §§ 1–2, § 6, § 10.2 & Note V (On the So-Called Non-Euclidean Geometry)
Cross-Corpus Relations:
geometor/wildberger — Divine Proportions (2005) & Universal Hyperbolic Geometry,
Quadrance, Spread, Cross-Ratio, and Projective Null Conic Metrics: Wildberger’s quadrance Q and spread s strip away transcendental logarithms from Klein’s Cayley–Klein metric d(A,B) = c*log(A,B;I_1,I_2), expressing Principal, hyperbolic, elliptic, and Möbius invariants as pure rational functions over Q.geometor/bertrand-russell — An Essay on the Foundations of Geometry (1897) & Principles of Mathematics (1903),
EFG Ch. I (§§ 30–43: Cayley & Klein) & PoM Ch. XLVIII (§§ 405–411: Projective Theory of Distance): Russell explicitly adopts Klein’s 1871–1872 projective classification of hyperbolic, elliptic, and parabolic geometries relative to the Absolute while debating whether cross-ratio metrics replace philosophical intuition of distance.geometor/hoenen — De Noetica Geometriae (1954),
Caput II–IV (Critique of Non-Euclidean Projective Interpretations & Structuralism): Hoenen examines Klein’s Principle of Transference (§§ 4–5) and Cayley–Klein models (Note V), arguing that relative consistency inside projective Euclidean models presupposes an intuitive ambient continuum.
8. Reciprocal Radii (Möbius Inversion), Plücker Line Geometry, and Lie’s Sphere Geometry¶
Famous Problems (1895 / 1897): Part I, Ch. V (§§ 3–4: CircularPoints at Infinity on Cissoid & Conchoid)
Erlangen Program (1872 / 1893): §§ 5–9 & Note VII (EN pp. 226–238, 248 / DE S. 15–32, 42)
Cross-Corpus Relations:
geometor/petersen — Methods and Theories for the Solution of Problems of Geometrical Constructions (1879),
Chapter III (Method of Rotation, Similarity, and Inversion / Reciprocal Radii): Petersen’s constructive problem-solving transformations (translation, rotation, homothety, circle inversion) instantiate in planar synthesis the exact subgroups systematized in §§ 1 & 6 of Klein’s Erlangen Program.geometor/lemoine — Géométrographie (1902),
Transformation by Reciprocal Radii & Apollonius Contact Constructions: Links Lie’s contact geometry of tangency among circles and spheres (§ 7) to classical Apollonius contact constructions.