Content of type 'notes'

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

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

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 \(x^2 + y^2 = 1\):

De Docta Ignorantia I.12 — Three-Step Method of Mathematical Transsumption ( O1 D ): (x - (0))^2 + (y - (-1))^2 - (1) = 0 ( O2 D ): (x - (0))^2 + (y - (-2))^2 - (4) = 0 [ O2 D ]: (1)*x + (0)*y + (0) = 0 [ E G ]: (0)*x + (1)*y + (0) = 0 [ B C ]: (0)*x + (1)*y + (1/2) = 0 [ P Q ]: (0)*x + (1)*y + (1) = 0 E = (-2, 0) G = (2, 0) O1 = (0, -1) O2 = (0, -2) D = (0, 0) B = (-sqrt(3)/2, -1/2) C = (sqrt(3)/2, -1/2) H = (0, -1/2) P = (-sqrt(3), -1) Q = (sqrt(3), -1) E G O1 O2 D B C H P Q

Unit II — Incorruptible Certitude & the Three-Step Method of Mathematical Transsumption — Notes

Why does Nicholas of Cusa privilege geometry over every sensible or verbal analogy for divine science? In DI I.11 §2, Cusa locates mathematicalia at the exact epistemological fulcrum between material instability and pure intellectual simplicity:

De Docta Ignorantia I.13 — Coaxial Tangent Circle Pencil & Vanishing Sagitta ( C1 O ): (x - (0))^2 + (y - (1))^2 - (1) = 0 ( C2 O ): (x - (0))^2 + (y - (2))^2 - (4) = 0 ( C4 O ): (x - (0))^2 + (y - (4))^2 - (16) = 0 ( C8 O ): (x - (0))^2 + (y - (8))^2 - (64) = 0 [ O C8 ]: (1)*x + (0)*y + (0) = 0 [ X P1 ]: (1)*x + (0)*y + (-1) = 0 [ W Q1 ]: (1)*x + (0)*y + (1) = 0 [ A B ]: (0)*x + (1)*y + (0) = 0 [ Q1 P1 ]: (0)*x + (1)*y + (-1) = 0 [ Q2 P2 ]: (0)*x + (1)*y + (-2 + sqrt(3)) = 0 [ Q4 P4 ]: (0)*x + (1)*y + (-4 + sqrt(15)) = 0 [ Q8 P8 ]: (0)*x + (1)*y + (-8 + 3*sqrt(7)) = 0 A = (-2, 0) B = (2, 0) W = (-1, 0) X = (1, 0) C1 = (0, 1) C2 = (0, 2) O = (0, 0) P1 = (1, 1) Q1 = (-1, 1) P2 = (1, 2 - sqrt(3)) Q2 = (-1, 2 - sqrt(3)) P4 = (1, 4 - sqrt(15)) Q4 = (-1, 4 - sqrt(15)) P8 = (1, 8 - 3*sqrt(7)) Q8 = (-1, 8 - 3*sqrt(7)) A B C1 C2 O P1 Q1 P2 Q2 P4 Q4 P8 Q8

Unit III — Coincidence of Infinite Circle, Straight Line, Triangle & Sphere — Notes

Sagitta Asymptotics & Radical Secants (`DI I.13 §2`): Let \(C_R\) be the circle of radius \(R > 0\) with center \(O_R = (0, R)\) tangent to the straight line \(AB\) (\(y = 0\)) at \(O = (0, 0)\), so \(x^2 + (y - R)^2 = R^2\). At fixed horizontal abscissa \(x \neq 0\) (with \(|x| \le R\)), the exact sagitta is:

De Docta Ignorantia I.13, I.15, I.23 — Generative Unfolding of Line, Triangle, Circle & Sphere ( A B ): (x - (0))^2 + (y - (0))^2 - (1) = 0 [ S N ]: (1)*x + (0)*y + (0) = 0 [ L R ]: (0)*x + (1)*y + (-1/2) = 0 [ D B ]: (0)*x + (1)*y + (0) = 0 [ A C ]: (-sqrt(3)/2)*x + (1/2)*y + (0) = 0 [ B C ]: (-sqrt(3)/2)*x + (-1/2)*y + (sqrt(3)/2) = 0 B = (1, 0) D = (-1, 0) R = (sqrt(3)/2, 1/2) L = (-sqrt(3)/2, 1/2) A = (0, 0) N = (0, 1) S = (0, -1) E = (0, 1/2) C = (1/2, sqrt(3)/2) B D R L A N S E C

Unit IV — The Infinite Line as Universal Measure & the Infinite Sphere Where Center, Diameter and Circumference Coincide — Notes

Dedekind-Style Infinite Bijection (`DI I.16 §5`): Four centuries before Bolzano, Dedekind, and Cantor defined infinite sets by equipollence with proper subsets, Nicholas of Cusa argues that an infinite line composed of \(\infty\) one-foot segments and one composed of \(\infty\) two-foot segments are necessarily equal (cum infinitum non sit maius infinito), because every part of the infinite is convertible with the whole infinite.

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

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

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: