Cross-Corpus Concordance¶
Linking Cusanus (1440–1459) to Ancient, Scholastic, Renaissance & Modern Geometry
Systematic concordance of 15 canonical loci across De Docta Ignorantia and Cusa’s Scripta Mathematica, cross-referenced to plato, aristotle, euclid/elements, archimedes, proclus, aquinas, hoenen, pacioli, durer, kepler, casey, klein, and wildberger.
De Docta Ignorantia, Dedicatio & Epistola auctoris (Feb 12, 1440)¶
Codex Unit: Unit I — Learned Ignorance, Polygon Exhaustion & the Coincidence of Maximum and Minimum
Central Thesis: Wonder (admiratio) & the Sea-Voyage Illumination of Coincidentia Oppositorum
Symbolic Verification Hook:
geometor.cusa.coincidentia_circle_line.verify_sagitta_and_curvature_limits
Ancient & Medieval Antecedents
plato: Theaetetus 155d (wonder as origin of philosophy)
aristotle: Metaphysics I.2, 982b12 (propter admirari philosophari)
dionysius: Mystica Theologia I.1–3 (apophatic ascent)
Renaissance & Modern Mathematical Reception
kepler: Mysterium Cosmographicum (1596), Cap. II (‘divinus mihi Cusanus’)
hoenen: De Noetica Geometriae (1954), Cap. I (intellectual insight)
De Docta Ignorantia I.1 (Quomodo scire est ignorare)¶
Codex Unit: Unit I — Learned Ignorance, Polygon Exhaustion & the Coincidence of Maximum and Minimum
Central Thesis: Comparative Inquiry via Proportion, Number & Socratic Learned Ignorance
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.compute_polygon_exhaustion_table
Ancient & Medieval Antecedents
plato: Apology 21d & Meno 80d–84c (knowing one’s ignorance as spur to inquiry)
aristotle: Metaphysics II.1, 993b9–11 (night-owl / bat before the sun)
euclid: Elements I, Common Notions & Book V (reduction to first principles via media)
Renaissance & Modern Mathematical Reception
pacioli: De Divina Proportione (1509), Caps. V–VII (proportion as medium of all science)
hoenen: De Noetica Geometriae (1954), Cap. IV (axiomatic nexus)
De Docta Ignorantia I.2 & I.4 (Maximum absolutum & Coincidentia minimi)¶
Codex Unit: Unit I — Learned Ignorance, Polygon Exhaustion & the Coincidence of Maximum and Minimum
Central Thesis: The Absolute Maximum and Coincidence of Maximum & Minimum Beyond Non-Contradiction
Symbolic Verification Hook:
geometor.cusa.coincidentia_circle_line.verify_mobius_inversion_coincidence
Ancient & Medieval Antecedents
plato: Parmenides 137c–142a & Philebus 24a–25d (the ‘more and less’ vs. uncontracted One)
aristotle: Metaphysics IV.3, 1005b19 (principle of non-contradiction in discursive ratio)
aquinas: Summa Theologiae I, q. 7 & In Boethium de Trinitate q. 6 (judgment above imagination)
Renaissance & Modern Mathematical Reception
klein: Elementarmathematik II (projective identification of +oo and -oo on P^1(R))
De Docta Ignorantia I.3 (Quod praecisa veritas sit incomprehensibilis)¶
Codex Unit: Unit I — Learned Ignorance, Polygon Exhaustion & the Coincidence of Maximum and Minimum
Central Thesis: Inscribed Polygon Exhaustion & Incommensurability of Non-Circle with Circle
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_polygon_exhaustion
Ancient & Medieval Antecedents
euclid: Elements IV.2–15 & XII.2 (inscribed/circumscribed polygons & exhaustion)
archimedes: Dimensio Circuli Props. 1–3 (96-gon rational bounds on pi)
aristotle: Physics VII.4, 248a10–b7 (incomparability of circular arc and straight line)
Renaissance & Modern Mathematical Reception
pacioli: De Divina Proportione (1509), Cap. V (irrationality as divine emblem)
durer: Underweysung der Messung (1525), Book II (constructive polygon series)
De Docta Ignorantia I.11 (Quod mathematica nos iuvet plurimum)¶
Codex Unit: Unit II — Incorruptible Certitude & the Three-Step Method of Mathematical Transsumption
Central Thesis: Incorruptible Certainty of Mathematical Signs & Refutation of Indivisible Atoms
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_transsumptio_ascent
Ancient & Medieval Antecedents
plato: Republic VI–VII, 510b–527c (mathematicals drawing soul toward essence)
aristotle: Metaphysics VIII.3, 1043b33 (species compared to numbers) & De Anima II.3, 414b29 (triangle in quadrangle)
proclus: In Euclidem Prologue I, Friedlein pp. 3–32 (intermediate status of mathematicals)
aquinas: In Boethium de Trinitate q. 5 a. 3 & q. 6 a. 2 (materia intelligibilis & certitudo mathematicae)
Renaissance & Modern Mathematical Reception
hoenen: De Noetica Geometriae (1954), Caps. II–III (abstraction of geometric extension)
De Docta Ignorantia I.12 (Quomodo signis mathematicalibus sit utendum)¶
Codex Unit: Unit II — Incorruptible Certitude & the Three-Step Method of Mathematical Transsumption
Central Thesis: The Three-Step Method of Mathematical Transsumption (Finitum -> Infinitum -> Simplex)
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_transsumptio_ascent
Ancient & Medieval Antecedents
plato: Republic VI, 510b–511d (ascent from hypothetical figures to unhypothetical first principle)
proclus: In Euclidem pp. 51–56 (projection of forms in geometric imagination)
Renaissance & Modern Mathematical Reception
kepler: Ad Vitellionem Paralipomena (1604), Cap. IV.4 (principle of continuity from ellipse to parabola to straight line)
De Docta Ignorantia I.13 (De passionibus lineae maximae et infinitae)¶
Codex Unit: Unit III — Coincidence of Infinite Circle, Straight Line, Triangle & Sphere
Central Thesis: Coincidence of the Infinite Circle & Tangent Arc with the Infinite Straight Line
Symbolic Verification Hook:
geometor.cusa.coincidentia_circle_line.construct_tangent_circle_pencil
Ancient & Medieval Antecedents
plato: Parmenides 137e (definitions of straight and round)
euclid: Elements III.16 (horn angle between circle and tangent smaller than any rectilinear angle)
proclus: In Euclidem pp. 103–113 (dialectic of straight and circular lines)
Renaissance & Modern Mathematical Reception
kepler: Mysterium Cosmographicum (1596), Cap. II (curved and straight in Cusanus)
casey: Sequel to Euclid (1886), VI.5 (inversion of coaxial tangent circle pencil into parallel lines)
klein: Elementarmathematik II (Möbius group Equivalence of lines and circles on S^2)
De Docta Ignorantia I.14 (Quod infinita linea sit triangulus)¶
Codex Unit: Unit III — Coincidence of Infinite Circle, Straight Line, Triangle & Sphere
Central Thesis: Collapse of the Flattening Triangle onto the Straight Line as Apex Approaches Two Right Angles
Symbolic Verification Hook:
geometor.cusa.coincidentia_triangle_line.construct_flattening_triangles
Ancient & Medieval Antecedents
euclid: Elements I.20 (triangle inequality BA + AC > BC) & I.32 (angle sum = 2 right angles)
proclus: In Euclidem pp. 321–326, 379–384 (on Elements I.20 and I.32)
Renaissance & Modern Mathematical Reception
wildberger: Divine Proportions (2005), Chs. 3–4 (apex spread s_A = 4n^2/(n^2+1)^2 -> 0 & Triple Quad residual 16/n^2 -> 0)
De Docta Ignorantia I.15 (Quod ille triangulus sit circulus et sphaera)¶
Codex Unit: Unit III — Coincidence of Infinite Circle, Straight Line, Triangle & Sphere
Central Thesis: Generative Circumduction of the Infinite Line into Triangle, Circle & Sphere
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_generative_unfolding
Ancient & Medieval Antecedents
euclid: Elements I, Post. 3 & XI, Def. 14 (sphere defined by semicircle rotation around fixed diameter)
proclus: In Euclidem pp. 185–187 (kinematic generation of circle by revolving line)
Renaissance & Modern Mathematical Reception
durer: Underweysung der Messung (1525), Book I (kinematic generation of curves and solids)
klein: Elementarmathematik II (compactification of R^2 to S^2)
De Docta Ignorantia I.16–18 (Mensura adaequatissima & Participatio rectitudinis)¶
Central Thesis: The Infinite Line as Exact Measure of All Lines & Rectitude as Measure of the Curved
Symbolic Verification Hook:
geometor.cusa.coincidentia_circle_line.verify_sagitta_and_curvature_limits
Ancient & Medieval Antecedents
plato: Phaedo 100b–e (participation in the single Idea, cited via Calcidius in I.17)
aristotle: Metaphysics X.1, 1052b18 (first as measure) & De Anima I.5, 411a5 (rectum est iudex sui et obliqui)
aquinas: Summa Theologiae I, q. 13 & q. 44 (exemplar causality and analogical participation)
Renaissance & Modern Mathematical Reception
hoenen: De Noetica Geometriae (1954), Cap. VII (measure and continuum)
De Docta Ignorantia I.19, I.21–23 (Triangulus, Circulus & Sphaera Infinita)¶
Central Thesis: Coincidence of Center, Diameter & Circumference in the Infinite Circle and Sphere
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_generative_unfolding
Ancient & Medieval Antecedents
plato: Parmenides 137e & Timaeus 33b (spherical equidistance and perfection)
proclus: In Euclidem pp. 146–155 (symbolic theology of center, radii, and circumference)
dionysius: De Divinis Nominibus V.6 & IX (center containing all radii)
Renaissance & Modern Mathematical Reception
pacioli: De Divina Proportione (1509), Cap. V (trinitarian geometry)
kepler: Harmonices Mundi (1619), Book IV, Ch. 1 (spherical archetype of the Trinity)
De Docta Ignorantia II.11–12 (Corollaria de motu & De condicionibus terrae)¶
Central Thesis: Decentered Universe, Perspectival Relativity of Motion & the Noble Moving Earth
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_cosmic_relativity_frame
Ancient & Medieval Antecedents
plato: Timaeus 30b, 34a–b, 40b–c (world as living organism & axial/cosmic motions)
aristotle: De Caelo II.13–14, 293a15–298a20 (geocentric rest refuted by Cusa)
Renaissance & Modern Mathematical Reception
kepler: Astronomia Nova (1609) & Epitome Astronomiae Copernicanae (1618) (optical relativity of motion & celestial physics)
klein: Erlangen Program (invariance under change of observer frame)
De Geometricis Transmutationibus (1445), §§1–5, 9–11¶
Central Thesis: Quarter-Side Sesquiquarta (5:4) Rectification of the Inscribed Equilateral Triangle
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_transmutatio_isoperimetra
Ancient & Medieval Antecedents
euclid: Elements IV.2, IV.4 & I.47 (inscribed triangle, incircle, and Pythagorean quadrance Q(A,E)=7/16)
archimedes: Dimensio Circuli Prop. 3 (pi < 22/7; Cusa obtains 24*sqrt(21)/35 ~= 3.142337 < 22/7)
Renaissance & Modern Mathematical Reception
durer: Underweysung der Messung (1525), Book II (compass-and-straightedge approximate quadratures)
wildberger: Divine Proportions (2005) (exact rational quadrance Q(A,H)=175/256)
De Mathematicis Complementis (1453), §§1–10¶
Central Thesis: Critique of Archimedes’ Spiral & Theory of Prima Linea (Inradius) and Secunda Linea (Circumradius)
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.compute_polygon_exhaustion_table
Ancient & Medieval Antecedents
archimedes: De Lineis Spiralibus Prop. 18 & Dimensio Circuli Prop. 1
euclid: Elements VI.13 (mean proportional squaring rectangle r * (P/2))
Renaissance & Modern Mathematical Reception
regiomontanus: De Quadratura Circuli (1464/1533) (trigonometric audit of Cusa’s polygon interpolations)
De Mathematica Perfectione (1458) §§1–6 & Aurea Propositio in Mathematicis (1459) §§1–7¶
Central Thesis: Three-Ray Chord–Arc–Tangent Invariance & Fifth-Order Snellius–Cusa Rectification Law
Symbolic Verification Hook:
geometor.cusa.polygon_exhaustion_and_complementa.construct_aurea_propositio
Ancient & Medieval Antecedents
euclid: Elements III.16, III.18 & VI.4 (similar right triangles equating chord and tangent three-ray ratios)
archimedes: Dimensio Circuli Prop. 3 (half-angle bisection improved from O(theta^3) to O(theta^5))
Renaissance & Modern Mathematical Reception
snellius & huygens: Cyclometricus (1621) & De Circuli Magnitudine Inventa (1654) (theta - 3 sin theta / (2 + cos theta) = theta^5/180 + O(theta^7))