Unit V — Centerless Cosmology & the Scripta Mathematica (1445–1459): Isoperimetric Transmutations and the Golden Proposition — Structural Synopsis

Argumentative Architecture & Interactive Diagram Guide · DI II.11–12 · DGT · DMC · DMP · APM (1440–1459)

De Docta Ignorantia II.11–12 — Perspectival Relativity (Undique Centrum, Nullibi Circumferentia) ( T H1 ): (x - (0))^2 + (y - (0))^2 - (4) = 0 ( S H2 ): (x - (3/2))^2 + (y - (1/2))^2 - (4) = 0 ( P H3 ): (x - (-1))^2 + (y - (3/2))^2 - (4) = 0 [ T L ]: (1/2)*x + (-3/4)*y + (0) = 0 [ S M ]: (7/4)*x + (-1)*y + (-17/8) = 0 [ T S ]: (-1/2)*x + (3/2)*y + (0) = 0 [ T P ]: (-3/2)*x + (-1)*y + (0) = 0 [ S P ]: (-1)*x + (-5/2)*y + (11/4) = 0 H1 = (2, 0) H2 = (7/2, 1/2) H3 = (1, 3/2) T = (0, 0) P = (-1, 3/2) L = (-3/4, -1/2) M = (1/2, -5/4) S = (3/2, 1/2) T P L M S
Book II, Chapters 11–12 — Centerless Cosmic Relativity Frame: Observers on Terra (T), Sol (S), Luna (L), and Polus (P) Construct Symmetric Horizon Circles in a Universe Having Its Center Everywhere and Its Circumference Nowhere.
De Geometricis Transmutationibus (1445) — Quarter-Side Sesquiquarta Rectification ( A D ): (x - (0))^2 + (y - (0))^2 - (1) = 0 ( A F ): (x - (0))^2 + (y - (0))^2 - (1/4) = 0 ( A H ): (x - (0))^2 + (y - (0))^2 - (175/256) = 0 [ A F ]: (1)*x + (0)*y + (0) = 0 [ B C ]: (0)*x + (1)*y + (1/2) = 0 [ C D ]: (-3/2)*x + (-sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ D B ]: (3/2)*x + (-sqrt(3)/2)*y + (sqrt(3)/2) = 0 [ A H ]: (5/8)*x + (-5*sqrt(3)/16)*y + (0) = 0 D = (0, 1) B = (-sqrt(3)/2, -1/2) C = (sqrt(3)/2, -1/2) A = (0, 0) E = (-sqrt(3)/4, -1/2) F = (0, -1/2) G = (sqrt(3)/4, -1/2) H = (-5*sqrt(3)/16, -5/8) D B C A E F G H
De Geometricis Transmutationibus (1445, §§9–11) — Inscribed Equilateral Triangle BCD in Unit Circle A: Quarter-Point E on Side BC Extended in Sesquiquartal Ratio 5:4 Yields Isoperimetric Radius AH = (5/16) sqrt(7), Implying pi ~= (24/35) sqrt(21) ~= 3.142337.
De Mathematica Perfectione (1458) & Aurea Propositio (1459) — Three-Ray Arc Rectification ( A D ): (x - (0))^2 + (y - (0))^2 - (1) = 0 [ E G ]: (1)*x + (0)*y + (-1) = 0 [ A E ]: (-sqrt(3)/3)*x + (1)*y + (0) = 0 [ A D ]: (0)*x + (1)*y + (0) = 0 [ A G ]: (sqrt(3)/3)*x + (1)*y + (0) = 0 [ B C ]: (1)*x + (0)*y + (-sqrt(3)/2) = 0 [ I L ]: (1)*x + (0)*y + (9/13 - 12*sqrt(3)/13) = 0 D = (1, 0) B = (sqrt(3)/2, 1/2) C = (sqrt(3)/2, -1/2) A = (0, 0) H = (sqrt(3)/2, 0) E = (1, sqrt(3)/3) G = (1, -sqrt(3)/3) K = (-9/13 + 12*sqrt(3)/13, 0) I = (-9/13 + 12*sqrt(3)/13, 12/13 - 3*sqrt(3)/13) L = (-9/13 + 12*sqrt(3)/13, -12/13 + 3*sqrt(3)/13) D B C A H E G K I L
Aurea Propositio in Mathematicis (1459, §§1–2) — Exact Q(sqrt(3)) Construction at theta = pi/6: Three-Ray Sum |AI| + |AK| + |AL| = 3 Equals Three Unit Radii, with Endpoint Ascent |AI| - 1 = (11 - 6 sqrt(3))/13 Matching Midpoint Descent 1 - |AK| and Fifth-Order Residual theta^5/180.

I. Argumentative Synopsis

  1. Relativity of Motion & the Noble Moving Earth (`DI II.11–12`, 1440): Because no unqualified minimum of motion or exact physical equidistance can exist in the contracted universe, neither the earth nor any celestial body can occupy a fixed cosmic center, nor do fixed celestial poles exist (II.11). Like a passenger on a ship in midstream unable to see the riverbanks (II.12 §1), every observer—whether on earth, the sun, the moon, or Mars—judges himself to stand at a motionless center. Thus the fabric of the world exists quasi habens undique centrum et nullibi circumferentiam, and the moving earth is rehabilitated from medieval baseness as a stella nobilis.

  2. Isoperimetric Extrema & Sesquiquartal Radius (`De Geometricis Transmutationibus`, 1445): Dedicated to Paolo dal Pozzo Toscanelli, Cusa frames curve-straight rectification via isoperimetric regular polygons: the equilateral triangle has minimal capacity and maximal gap between inradius and circumradius, whereas the circle has maximal capacity where both radii coincide. Dividing side \(BC\) of an inscribed equilateral triangle into quarters at \(E, F, G\) and extending \(AE\) in sesquiquartal ratio \(5:4\) yields \(AH = \frac{5\sqrt{7}}{16}\), equivalent to \(\pi \approx \frac{24\sqrt{21}}{35} \approx 3.142337\).

  3. Critique of Archimedes’ Spiral & Prima/Secunda Linea (`De Mathematicis Complementis`, 1453): Thanking Pope Nicholas V for Jacobus Cremonensis’s Latin translation of Archimedes, Cusa diagnoses why On Spirals is kinematically circular for constructing \(\pi\), defines the first line (inradius) and second line (circumradius), and proves geometrically (§10) that polygon area equals inradius times semiperimeter.

  4. Visio Intellectualis & the Cusa–Snellius Law (`De Mathematica Perfectione`, 1458): Writing to Cardinal Antonio Cerdà, Cusa invokes intellectual vision of the vanishing sagitta where minimal chord and minimal arc coincide, deriving for right triangle \(abc\) the celebrated law \(\theta / \sin\theta \approx 3R / (2R + R\cos\theta)\).

  5. Three-Ray Sum Invariance (`Aurea Propositio in Mathematicis`, August 8, 1459): Cusa’s final mathematical gem reformulates his 1458 law as a symmetric three-ray conservation principle: when circular arc bdc is rectified into perpendicular segment ikl, the outward ascent of endpoints i, l balances the inward descent of midpoint k so that \(|AI| + |AK| + |AL| = 3R\)—yielding \(L_{\text{Cusa}}(\theta) = \frac{6R\sin\theta}{2 + \cos\theta}\) with fifth-order Maclaurin accuracy \(\frac{\theta^5}{180}\).

II. Interactive Diagram Guide

  • `cusa_cosmic_relativity.svg`: Symmetric observer horizons around T (Terra), S (Sol), L (Luna), and P (Polus).

  • `cusa_transmutatio_1445.svg`: Unit circle about A, equilateral triangle B–C–D, quarter-point E, and sesquiquartal radius H.

  • `cusa_aurea_propositio_1459.svg`: Exact \(\mathbb{Q}(\sqrt{3})\) sector (\(\theta = \pi/6\)) showing chord B–H–C, arc midpoint D, tangent E–D–G, and Cusa’s balanced rectifying segment I–K–L.