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

Exact Symbolic Reconstruction, Latin Philology & Canon Reception · 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. Centerless Cosmology & Kinematic Relativity (DI II.11–12, 1440)

Over a century before Copernicus’s De revolutionibus (1543), Bruno’s De l’infinito universo et mondi (1584), and Galileo’s Dialogo (1632), DI II.11–12 deduces three revolutionary physical corollaries directly from learned ignorance:

  1. Impossibility of an Immobile Cosmic Center or Fixed Poles (`II.11 §§1–4`): Because physical positions, figures, and motions in the contracted cosmos always admit of more and less, exact equidistance (aequedistantia praecisa) never occurs in nature; hence neither the earth nor the sun nor the eighth sphere possesses an immobile geometric center or fixed celestial poles.

  2. Ship-in-Midstream Principle of Relative Motion (`II.12 §1`): Motion is detectable solely relative to a chosen reference point (Si enim quis ignoraret aquam fluere et ripas non videret existendo in navi in medio aquae, navem quomodo apprehenderet moveri?). Every observer on Earth T, Sun S, Moon L, or Mars constructs his own local horizon and poles, rendering the universe quasi habens undique centrum et nullibi circumferentiam.

  3. Astrophysical Homogeneity (`II.12 §§3–4`): Rejecting the Aristotelian sublunary/superlunary chasm, Cusa declares the moving earth a stella nobilis shining into space through its outer fiery envelope just as the sun shines toward us.

II. Exact Algebraic Audit of the Scripta Mathematica (1445–1459)

  1. *De Geometricis Transmutationibus* (1445, `§§9–11`): In the unit circle \(R = 1\) centered at A at \((0, 0)\), the inscribed equilateral triangle B–C–D has side \(|BC| = \sqrt{3}\) and perimeter \(P_3 = 3\sqrt{3}\). Dividing \(BC\) into four equal segments at E, F, G, the midpoint F satisfies \(Q(A, F) = \frac{1}{4}\) and the quarter-point E satisfies:

    \[Q(A, E) = |AF|^2 + |FE|^2 = \frac{1}{4} + \left(\frac{\sqrt{3}}{4}\right)^2 = \frac{7}{16} \implies |AE| = \frac{\sqrt{7}}{4}.\]

    Extending \(AE\) in sesquiquartal ratio \(5:4\) gives Cusa’s isoperimetric radius \(|AH| = \frac{5\sqrt{7}}{16}\) (\(Q(A, H) = \frac{175}{256}\)), implying:

    \[\pi_{\text{Cusa},1445} = \frac{P_3}{2|AH|} = \frac{3\sqrt{3}}{5\sqrt{7}/8} = \frac{24\sqrt{21}}{35} \approx 3.14233762,\]

    which exceeds true \(\pi \approx 3.14159265\) by only \(+7.45 \times 10^{-4}\) (lying strictly inside Archimedes' 96-gon bounds \(3\frac{10}{71} < \pi < 3\frac{1}{7}\), though Regiomontanus later showed via tighter bounds that \(\frac{24\sqrt{21}}{35} > \pi\); regiomontanus:quadratura:1533).

  2. *De Mathematicis Complementis* (1453, `§§3–10`): Cusa astutely observes that Archimedes’ spiral \(r(\varphi) = a\varphi\) presupposes the synchronized velocity ratio of uniform rectilinear motion along the radius to uniform circular motion along the circumference—the very transcendental ratio \(1 : 2\pi\) sought by quadrature! His dissection proof in §10 establishes \(\operatorname{Area}(P_n) = r_n \cdot \frac{P_n}{2}\) purely by Euclidean rectangle-halving.

  3. *De Mathematica Perfectione* (1458) & *Aurea Propositio in Mathematicis* (1459): Both treatises converge on the Cusa–Snellius–Huygens Law (cusa:perf:1458, cusa:aurea:1459, snellius:huygens:1654). Let rays at angles \(\pm\theta\) (\(0 < \theta \le \frac{\pi}{4}\)) and central ray \(0\) issue from A at \((0, 0)\) across unit circle \(R = 1\), meeting a vertical rectifying line \(x = x_K\) at I, K, L, so \(|AK| = x_K\) and \(|AI| = |AL| = \frac{x_K}{\cos\theta}\). Enforcing Cusa’s Three-Ray Sum Invariance:

    \[|AI| + |AK| + |AL| = 3 \iff x_K\!\left(1 + \frac{2}{\cos\theta}\right) = 3 \implies x_K = \frac{3\cos\theta}{2 + \cos\theta}.\]

    At \(\theta = \frac{\pi}{6}\) (modeled over \(\mathbb{Q}(\sqrt{3})\) in cusa_aurea_propositio_1459.svg), \(x_K = \frac{3\sqrt{3}/2}{2 + \sqrt{3}/2} = \frac{12\sqrt{3} - 9}{13}\), the outer rays have length \(|AI| = |AL| = \frac{24 - 6\sqrt{3}}{13}\), the combined endpoint ascent \(2(|AI| - 1) = \frac{22 - 12\sqrt{3}}{13}\) equals the midpoint descent \(1 - x_K\), and the rectified length is:

    \[L_{\text{Cusa}}(\theta) = 2 x_K \tan\theta = \frac{6\sin\theta}{2 + \cos\theta} = 2\theta - \frac{\theta^5}{90} - \frac{\theta^7}{756} - O(\theta^9),\]

    so for the half-arc \(\theta - \frac{3\sin\theta}{2 + \cos\theta} = \frac{\theta^5}{180} + \frac{\theta^7}{1512} + O(\theta^9)\)! Because \(\cos\theta\) and \(\sin\theta\) cancel through third order in the [1/1] Padé-type rational weight \(\frac{3}{2 + \cos\theta}\), Cusa achieved fifth-order accuracy—yielding at \(\theta = \frac{\pi}{6}\) an implied \(\pi\) estimate of \(\frac{18}{2 + \sqrt{3}/2} = \frac{36(4 - \sqrt{3})}{26} \approx 3.140237\), within \(0.043\%\) of \(\pi\)!

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 V live in src/geometor/cusa/ and the companion walkthrough Demonstrations.