Unit I · Executive Summary¶
Deductive Structure of Almagest I.1–I.9 & I.10a¶
Philosophical Grounding (*Almagest* I.1): Following Aristotle (Metaphysics E.1), Ptolemy locates mathematics as the unique theoretical science capable of unshakable demonstration (βεβαίαν καὶ ἀμετάπειστον τὴν εἴδησιν), mediating between invisible theology and mutable sublunary physics through arithmetic and geometry.
Spherical Cosmology (*Almagest* I.2–I.9): Establishes the geometric frame of spherical astronomy—a central point-like Earth $O$ inside the rotating celestial sphere, organized around two great circles inclined at obliquity $\varepsilon$: the celestial equator and the ecliptic.
The Sexagesimal Chord Circle (*Almagest* I.10a): Sets circumference $= 360^\circ$ and diameter $D = 120^p$ ($R = 60^p$), replacing Egyptian/Greek unit fractions (μοναριασμοί) with uniform base-60 positional fractions ($p; a, b = p + \frac{a}{60} + \frac{b}{3600}$).
Single-Circle Golden Pentagon & Decagon Construction: On semicircle $AB\Gamma$ with $\Delta B \perp A\Gamma$ and $E$ bisecting $\Delta\Gamma$, swinging $EB = 30\sqrt{5}$ onto $\Delta A$ at $Z$ yields in a single right triangle $\triangle B\Delta Z$:
$$\Delta Z = s_{10} = \frac{R}{\phi} = 30(\sqrt{5} - 1), \qquad BZ = s_5 = \sqrt{R^2 + s_{10}^2} = 30\sqrt{10 - 2\sqrt{5}}.$$
Primary Regular Polygon Chords ($R = 60^p, D = 120^p$)¶
Inscribed Polygon |
Arc $\theta$ |
Exact Symbolic Radical ($\mathbb{Q}(\sqrt{2}, \sqrt{3}, \phi)$) |
Computed Base-60 |
Heiberg (1898) |
|---|---|---|---|---|
Regular Decagon ($s_{10}$) |
$36^\circ$ |
$30(\sqrt{5} - 1) = 60\phi^{-1}$ |
|
|
Regular Hexagon ($s_6$) |
$60^\circ$ |
$60$ |
|
|
Regular Pentagon ($s_5$) |
$72^\circ$ |
$30\sqrt{10 - 2\sqrt{5}}$ |
|
|
Inscribed Square ($s_4$) |
$90^\circ$ |
$60\sqrt{2}$ |
|
|
Equilateral Triangle ($s_3$) |
$120^\circ$ |
$60\sqrt{3}$ |
|
|
Decagon Supplement |
$144^\circ$ |
$30\sqrt{10 + 2\sqrt{5}}$ |
|
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