Unit I — Learned Ignorance, Polygon Exhaustion & the Coincidence of Maximum and Minimum — Critical Commentary & Notes

Exact Symbolic Reconstruction, Latin Philology & Canon Reception · DI Ded., I.1–4, Ep. (1440)

De Docta Ignorantia I.3 — Polygon Exhaustion & Incommensurability of the Circle ( O V0 ): (x - (0))^2 + (y - (0))^2 - (1) = 0 [ W1 W2 ]: (0)*x + (1)*y + (-1) = 0 [ W2 W3 ]: (1)*x + (0)*y + (1) = 0 [ W3 W4 ]: (0)*x + (1)*y + (1) = 0 [ W4 W1 ]: (1)*x + (0)*y + (-1) = 0 [ O S4 ]: (0)*x + (1)*y + (0) = 0 [ T1 T2 ]: (3/2)*x + (-sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ T2 T3 ]: (0)*x + (1)*y + (1/2) = 0 [ T3 T1 ]: (-3/2)*x + (-sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ S1 S2 ]: (1)*x + (-1)*y + (1) = 0 [ S2 S3 ]: (1)*x + (1)*y + (1) = 0 [ S3 S4 ]: (-1)*x + (1)*y + (1) = 0 [ S4 S1 ]: (-1)*x + (-1)*y + (1) = 0 [ H1 H2 ]: (1/2)*x + (-sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ H2 H3 ]: (1)*x + (0)*y + (sqrt(3)/2) = 0 [ H3 H4 ]: (1/2)*x + (sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ H4 H5 ]: (-1/2)*x + (sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ H5 H6 ]: (1)*x + (0)*y + (-sqrt(3)/2) = 0 [ H6 H1 ]: (-1/2)*x + (-sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ O F ]: (1)*x + (0)*y + (0) = 0 W1 = (1, 1) W2 = (-1, 1) W3 = (-1, -1) W4 = (1, -1) O = (0, 0) S4 = (1, 0) H1 = (0, 1) H3 = (-sqrt(3)/2, -1/2) H5 = (sqrt(3)/2, -1/2) S2 = (-1, 0) H4 = (0, -1) H2 = (-sqrt(3)/2, 1/2) H6 = (sqrt(3)/2, 1/2) F = (0, -1/2) O S4 H1 H3 H5 S2 H4 H2 H6 F
Book I, Chapter 3 — Inscribed Regular Polygons P_3, P_4, P_6, P_8, P_12 in Unit Circle C: Monotonically Increasing Apothem r_n = cos(pi/n) -> 1 Without Finite Identity Between Polygon and Circle.

I. Exact Symbolic Analysis of Polygon Exhaustion (DI I.3)

In DI I.3 §2, Nicholas of Cusa elevates Archimedes' method of exhaustion (archimedes:dimensio) into the foundational epistemological theorem of learned ignorance. For a regular \(n\)-gon inscribed in the unit circle \(x^2 + y^2 = 1\):

\[r_n = \cos\!\left(\frac{\pi}{n}\right), \qquad s_n = 1 - r_n = 2\sin^2\!\left(\frac{\pi}{2n}\right), \qquad P_n = 2n\sin\!\left(\frac{\pi}{n}\right).\]

Across the constructible tower \(n \in \{3, 4, 6, 8, 12\}\) modeled in geometor.cusa.polygon_exhaustion_and_complementa, every vertex, apothem, and sagitta lies in an exact nested quadratic radical field over \(\mathbb{Q}\):

  • \(n = 3\): \(r_3 = \frac{1}{2}\), \(s_3 = \frac{1}{2}\), \(Q_{\text{side}} = 3\).

  • \(n = 4\): \(r_4 = \frac{\sqrt{2}}{2}\), \(s_4 = 1 - \frac{\sqrt{2}}{2}\), \(Q_{\text{side}} = 2\).

  • \(n = 6\): \(r_6 = \frac{\sqrt{3}}{2}\), \(s_6 = 1 - \frac{\sqrt{3}}{2}\), \(Q_{\text{side}} = 1\).

  • \(n = 8\): \(r_8 = \frac{\sqrt{2 + \sqrt{2}}}{2}\), \(Q_{\text{side}} = 2 - \sqrt{2}\).

  • \(n = 12\): \(r_{12} = \frac{\sqrt{6} + \sqrt{2}}{4}\), \(Q_{\text{side}} = 2 - \sqrt{3}\).

Crucially, Cusa distinguishes metric convergence (\(s_n \to 0\), \(P_n \to 2\pi\)) from ontological-definitional identity: for every finite \(n \in \mathbb{N}\), the polygon remains piecewise-rectilinear with \(n\) discrete vertices where curvature is concentrated, whereas the circle’s being consists in uniform, indivisible equidistance from its center. Finite reason moves endlessly along the sequence \(n \mapsto 2n\); only intellectus grasps the limit where polygon resolves into circle.

II. Latin Philology & Textual Witnesses

  • Title & Socratic-Aristotelian Synthesis (`DI I.1 §2`): Cusa weaves together Socrates' Apology (21d), Ecclesiastes 1:8, Job 28:20–21, and Aristotle's night-owl simile (Metaphysics II.1, 993b9–11, Latin nocticorax). Our archival inspection of Paolo Rotta's 1913 edition (cusa:rotta:1913, p. 4) confirms the spelling nocticoraci alongside Rotta’s apparatus notes to Bonaventure and Dante.

  • Anselmian Formula Transposed (`DI I.2 §1` & `I.4 §§1–2`): Cusa adopts Anselm's Proslogion definition (quo nihil maius cogitari / esse potest) but immediately pairs it with its symmetric superlative twin, quo minus esse non potest (minimum). Subtracting quantitative contraction (subtrahendo intellectualiter magnum et parvum), both superlatives coincide in absolute maximality.

  • The Sea-Voyage Colophon (`DI Ep.`): Returning from Constantinople with the Byzantine delegation to the Council of Ferrara–Florence between November 27, 1437 and February 8, 1438, Cusa received the governing intuition of incomprehensibilia incomprehensibiliter amplecti and signed the completed codex in his native Kues (Complevi in Cusa 1440, XII. februarii).

III. Reception in the GEOMETOR Canon

Cusa’s contrast between comparative ratio and transcendent intellectus transforms Plato's Divided Line (Republic VI, 509d–511e; plato:opera), Proclus's theory of mathematical projection in the imagination (proclus:in-euclidem), and Thomas Aquinas's doctrine of materia intelligibilis and judicium (aquinas:boethius), anticipating Petrus Hoenen's De Noetica Geometriae (hoenen:noetica:1954).

IV. Executable Symbolic Verification

In accordance with GEOMETOR’s strict content/code separation for historical editions, executable geometor.model SymPy constructions, Rational Trigonometry proofs, and series-error audits for Unit I live in src/geometor/cusa/ and the companion walkthrough Demonstrations.