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

Argumentative Architecture & Interactive Diagram Guide · DI I.13–15 (1440)

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
Book I, Chapter 13 — Tangent Circle Pencil C_1, C_2, C_4, C_8 at Origin O = (0, 0) Above Horizon Line AB (y = 0): Exact Radical Secant Sagittae at x = 1 Shrink as 1, 2 - sqrt(3), 4 - sqrt(15), 8 - 3 sqrt(7) -> 0 as Curvature Resolves into Rectitude.
De Docta Ignorantia I.14–15 — Flattening Triangles & Vanishing Rational Spread [ M A1 ]: (1)*x + (0)*y + (0) = 0 [ B C ]: (0)*x + (1)*y + (0) = 0 [ B A1 ]: (-1)*x + (1)*y + (-1) = 0 [ A1 C ]: (1)*x + (1)*y + (-1) = 0 [ B A2 ]: (-1/2)*x + (1)*y + (-1/2) = 0 [ A2 C ]: (1/2)*x + (1)*y + (-1/2) = 0 [ B A4 ]: (-1/4)*x + (1)*y + (-1/4) = 0 [ A4 C ]: (1/4)*x + (1)*y + (-1/4) = 0 [ B A8 ]: (-1/8)*x + (1)*y + (-1/8) = 0 [ A8 C ]: (1/8)*x + (1)*y + (-1/8) = 0 B = (-1, 0) C = (1, 0) M = (0, 0) A1 = (0, 1) A2 = (0, 1/2) A4 = (0, 1/4) A8 = (0, 1/8) B C M A1 A2 A4 A8
Book I, Chapter 14 — Flattening Isosceles Triangles T_n with Fixed Base BC = 2 and Apex A_n = (0, 1/n): Exact Rational Trigonometry Spreads s_A = 4n^2/(n^2+1)^2 -> 0 and Triple Quad Residual 16/n^2 -> 0 as T_n Resolves into Line BC.
De Docta Ignorantia I.13 — Mobius Inversion Unifying Tangent Circles & Straight Line ( S O ): (x - (0))^2 + (y - (-1))^2 - (1) = 0 ( C1 O ): (x - (0))^2 + (y - (1))^2 - (1) = 0 ( M1 O ): (x - (0))^2 + (y - (-1/3))^2 - (1/9) = 0 ( M2 O ): (x - (0))^2 + (y - (-2/5))^2 - (4/25) = 0 ( M4 O ): (x - (0))^2 + (y - (-4/9))^2 - (16/81) = 0 ( Minf O ): (x - (0))^2 + (y - (-1/2))^2 - (1/4) = 0 [ S Z ]: (1)*x + (0)*y + (0) = 0 [ A B ]: (0)*x + (1)*y + (0) = 0 [ S X ]: (-1)*x + (1)*y + (1) = 0 A = (-3/2, 0) B = (3/2, 0) Z = (0, 3/2) C1 = (0, 1) N1 = (0, -2/3) N2 = (0, -4/5) N4 = (0, -8/9) X = (1, 0) O = (0, 0) M1 = (0, -1/3) M2 = (0, -2/5) M4 = (0, -4/9) Minf = (0, -1/2) Xp = (1/2, -1/2) S = (0, -1) A B Z C1 N1 N2 N4 O M1 M2 M4 M∞ S
Book I, Chapters 13 & 15 — Projective-Conformal Realization via Unit-Circle Inversion at Pole S = (0, -1): Every Tangent Circle C_R Maps to a Finite Rational Circle C'_R Converging Smoothly to C'_inf at (0, -1/2), Radius 1/2—Identical to the Inverted Tangent Line AB.

I. Argumentative Synopsis

  1. Coincidence of Infinite Circle and Straight Line (`DI I.13 §§1–2`): A circle’s circumference is less curved in proportion as its diameter is larger. In the tangent family along straight line A–B, arc cd of a larger circle recedes further from curvature than arc ef of a smaller circle, and ef further than arc gh of a still smaller circle. Hence the circumference of the maximal circle, which cannot be greater, is minimally curved and therefore maximally straight: curvature and rectitude coincide.

  2. Potency of the Finite Line & Actual Infinity (`DI I.13 §§3–5`): Swinging finite segment ab about fixed endpoint a to c generates a triangle; completing the revolution generates a circle; swinging to opposite endpoint d forms diameter bd and a semicircle; revolving that semicircle about bd generates a sphere—the ultimate figure in the potency of the line. Because the infinite line is in act everything that the finite line is in potency, it is simultaneously straight line, triangle, circle, and sphere.

  3. Coincidence of Infinite Triangle and Straight Line (`DI I.14`): Two independent proofs demonstrate that the infinite line is the maximal triangle: (a) Metric-cardinal proof: By the triangle inequality (Euclid Elements I.20), if one side is infinite, the other two cannot be smaller; since every part of the infinite is infinite and there cannot be multiple distinct infinites, the three infinite sides are one simple line. (b) Quantitative limit proof: By Elements I.32, the three angles sum to two right angles; as the apex angle (bac -> bdc) approaches two right angles while base bc stays fixed, the side excess \((|BD| + |DC|) - |BC|\) and enclosed area vanish until the triangle resolves into a single line.

  4. Coincidence with Infinite Circle and Sphere (`DI I.15`): Swinging infinite radius ab generates infinite arc bc, which is both straight and—since a part of the infinite equals the whole—the complete circumference; its return upon itself completes the revolution into the infinite sphere.

II. Interactive Diagram Guide

  • `cusa_circle_line_pencil.svg`: Tangent circles centered at C1, C2, and higher radii share common tangent A–B at O; along vertical secants through X and W, exact radical sagittae at P1, P2, P4, P8 (and symmetric Q1–Q8) converge monotonically toward X.

  • `cusa_flattening_triangles.svg`: Fixed base B–C with flattening apices A1, A2, A4, A8 descending to midpoint M.

  • `cusa_mobius_inversion.svg`: Conformal inversion in unit circle centered at pole S maps the tangent pencil to finite rational circles centered at M1, M2, M4 converging to the inverted horizon circle centered at Minf.