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

Exact Symbolic Reconstruction, Latin Philology & Canon Reception · 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. Exact Symbolic Demonstrations of DI I.13–15

  1. 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:

    \[h(x, R) = R - \sqrt{R^2 - x^2} = \frac{x^2}{R + \sqrt{R^2 - x^2}}.\]

    Along the secant \(x = 1\) for \(R \in \{1, 2, 4, 8\}\) (modeled in geometor.cusa.coincidentia_circle_line), the exact symbolic heights are \(1\), \(2 - \sqrt{3} \approx 0.26795\), \(4 - \sqrt{15} \approx 0.12702\), and \(8 - 3\sqrt{7} \approx 0.06275\), with \(\lim_{R \to \infty} h(x, R) = 0\), \(\lim_{R \to \infty} R\,h(x, R) = \frac{x^2}{2}\), and curvature \(\kappa(R) = \frac{1}{R} \to 0\).

  2. Projective-Conformal Proof via Möbius Circle Inversion: Under inversion in the unit circle \(\Omega\) centered at \(S = (0, -1)\) (which fixes the point of tangency \(O = (0, 0)\)), every circle \(C_R\) of radius \(R\) maps to a bounded rational circle \(C'_R\) with exact center and radius:

    \[M_R = \left(0,\; -\frac{R}{2R + 1}\right), \qquad \rho_R = \frac{R}{2R + 1}.\]

    Meanwhile, the straight horizon line \(y = 0\) maps to the circle \(C'_\infty\) with center \((0, -\frac{1}{2})\) and radius \(\frac{1}{2}\). Taking \(R \to \infty\) on the Riemann sphere \(\widehat{\mathbb{C}}\), \(M_R \to (0, -\frac{1}{2})\) and \(\rho_R \to \frac{1}{2}\) smoothly in Hausdorff metric: on the conformal sphere, the straight line literally IS the circle through the point at infinity (casey:sequel:1886, klein:elementary:1908).

  3. Rational Trigonometry Proof of the Flattening Triangle (`DI I.14 §§2–3`): For the isosceles triangle \(T_n\) with fixed base \(B = (-1, 0)\), \(C = (1, 0)\) (\(Q_{\text{base}} = 4\)) and apex \(A_n = (0, \frac{1}{n})\) over \(\mathbb{Q}(n)\) (modeled in geometor.cusa.coincidentia_triangle_line using Wildberger's invariants, wildberger:divine:2005):

    \[Q(A_n, B) = Q(A_n, C) = 1 + \frac{1}{n^2} = \frac{n^2 + 1}{n^2}, \qquad s_B = s_C = \frac{1}{n^2 + 1}, \qquad s_{A_n} = \frac{4n^2}{(n^2 + 1)^2}.\]

    Every \(T_n\) satisfies the Triple Spread Formula \((s_{A_n} + 2s_B)^2 - 2(s_{A_n}^2 + 2s_B^2) - 4 s_{A_n} s_B^2 = 0\) identically over \(\mathbb{Q}(n)\), while the Triple Quad Formula collinearity residual equals \((4 + 2Q_{\text{leg}})^2 - 2(16 + 2Q_{\text{leg}}^2) = \frac{16}{n^2} \to 0\) as \(n \to \infty\)!

II. Archival Scan Collation: Rotta (1913) vs. Klibansky & Paris (1514)

Direct multimodal inspection of McGill University Library’s digitized scan of Raymond Klibansky's <klibansky> annotated copy of Paolo Rotta's 1913 Laterza edition (cusa:rotta:1913, pp. 28 and 31; cf. cusa:heidelberg:1932) revealed two classic scribal/typesetting eye-skip omissions (*homeoteleuton*) in Rotta’s printed text, both restored in pencil in Klibansky’s own hand from the 1514 Paris and manuscript witnesses and incorporated in our Latin text:

  • `DI I.13 §2` (Rotta p. 28): Rotta’s typesetter jumped from the first minoris circuli to the second, omitting et ille plus a curvitate recedit quam arcus gh adhuc minoris circuli—even though Rotta’s own marginal woodcut right beside the paragraph explicitly labels all four lines a b, c d, e f, and g h!

  • `DI I.14 §3` (Rotta p. 31): Rotta printed ut b d c, et superficies minor, accidentally dropping tanto minus vincunt lineae bd et dc lineam bc between ut bdc and et superficies minor—again restored in Klibansky’s pencil margin at the foot of page 31.

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