Unit II · Executive Summary

The Five Pillars of Ptolemy’s Chord Calculus (Almagest I.10b)

  1. Ptolemy’s Theorem (pp. 36–37): In any cyclic quadrilateral $AB\Gamma\Delta$, drawing $BE$ with $\angle ABE = \angle \Delta B\Gamma$ splits diagonal $A\Gamma$ at $E$ into two pairs of similar triangles ($\triangle ABE \sim \triangle \Delta B\Gamma$ and $\triangle E B\Gamma \sim \triangle AB\Delta$), proving:

    $$A\Gamma \cdot B\Delta = AB \cdot \Delta\Gamma + A\Delta \cdot B\Gamma.$$

  2. Difference & Sum Laws (pp. 38, 41–42): Setting one diagonal equal to the diameter $D = 120^p$ turns Ptolemy’s Theorem directly into the trigonometric addition/subtraction theorems:

    $$\operatorname{crd}(\alpha \pm \beta) = \frac{\operatorname{crd}(\alpha)\operatorname{crd}(180^\circ - \beta) \pm \operatorname{crd}(\beta)\operatorname{crd}(180^\circ - \alpha)}{120}.$$

  3. Half-Arc Chord Law (pp. 39–40): Bisecting arc $B\Gamma$ at $\Delta$ on semicircle $AB\Gamma$ yields:

    $$\operatorname{crd}^2(\theta / 2) = \frac{D}{2}\bigl(D - \operatorname{crd}(180^\circ - \theta)\bigr) = 60\bigl(120 - \operatorname{crd}(180^\circ - \theta)\bigr).$$

  4. Irreducibility of Angle Trisection (p. 42): Ptolemy explicitly recognizes that $\operatorname{crd}(\frac{1}{2}^\circ)$ cannot be constructed διὰ τῶν γραμμῶν (“by means of lines”) from $\operatorname{crd}(1\frac{1}{2}^\circ)$—anticipating the cubic irreducibility of $4x^3 - 3x = \cos(3\psi)$ over $\mathbb{Q}(\cos 3\psi)$.

  5. Ptolemy’s Inequality Lemma & $\pi \approx 3;8,30$ (pp. 43–46): Proving $\frac{\operatorname{crd}\alpha}{\operatorname{crd}\beta} < \frac{\alpha}{\beta}$ for $\alpha > \beta$ traps $\operatorname{crd}(1^\circ)$ strictly between $\frac{2}{3}\operatorname{crd}(1\frac{1}{2}^\circ) = 1;2,50,0^p$ and $\frac{4}{3}\operatorname{crd}(\frac{3}{4}^\circ) \approx 1;2,50,40^p$, fixing $\operatorname{crd}(1^\circ) = 1;2,50^p = \frac{377}{360}$ and implying $\pi \approx 3 \cdot (1;2,50) = 3;8,30 = \frac{377}{120} \approx 3.1416667$.

Complete Radical & Sexagesimal Verification Cascade

Arc $\theta$

Constructive Derivation

Exact Radical Computed Base-60

Heiberg (1898) Printed Text

$12^\circ$

$\operatorname{crd}(72^\circ - 60^\circ)$ (Diff. Law)

12;32,36

12;32,36 (p. 38)

$6^\circ$

Half-arc of $12^\circ$

6;16,49

6;16,49 (p. 40)

$3^\circ$

Half-arc of $6^\circ$

3;8,28

3;8,28 (p. 40)

$1\frac{1}{2}^\circ$

Half-arc of $3^\circ$

1;34,15

1;34,15 (p. 40)

$\frac{3}{4}^\circ$

Half-arc of $1\frac{1}{2}^\circ$

0;47,7 ($0;47,7,25\dots$)

0;47,8 (p. 40)

$1^\circ$

Inequality sandwich ($1\frac{1}{2}^\circ$ & $\frac{3}{4}^\circ$)

1;2,50 ($= 377/360$)

1;2,50 (p. 46)

$\frac{1}{2}^\circ$

Half-arc of $1^\circ$

0;31,25

0;31,25 (p. 46)