Unit III · Executive Summary¶
From Planar Chords to Spherical Astronomy (Almagest I.11–I.14)¶
Table of Chords (I.11): Tabulates $\operatorname{crd}(\theta)$ for $\theta = \frac{1}{2}^\circ, 1^\circ, \dots, 180^\circ$ ($360$ entries) in base-60 parts ($2R = 120^p$) together with sexagesimal sixtieths ($\Delta\operatorname{crd}/30$) per arc-minute and built-in algebraic checksums against scribal corruption.
Obliquity of the Ecliptic (I.12): Confirms Eratosthenes’ & Hipparchus’s solstice-to-solstice ratio $2\varepsilon = \frac{11}{83} \times 360^\circ \approx 47^\circ 42'40''$, fixing $\varepsilon = 23^\circ 51'20''$ ($\operatorname{crd} 2\varepsilon = 48;31,55^p$).
Planar & Spherical Menelaus Calculus (I.13): * Planar Sector Lemma: Drawing $EH \parallel \Gamma\Delta$ proves both $\frac{\Gamma A}{AE} = \frac{\Gamma\Delta}{\Delta Z}\cdot\frac{ZB}{BE}$ (kata synthesin) and $\frac{\Gamma E}{EA} = \frac{\Gamma Z}{Z\Delta}\cdot\frac{\Delta B}{BA}$ (kata diairesin). * 3D Central Projection: Projecting great-circle intersections from sphere center $H$ onto the plane of $\triangle A\Gamma\Delta$ maps great circle $BZE$ to a straight line $\Theta K\Lambda$ (Eucl. XI.3), lifting the planar compound ratios directly onto double-arc chords on the sphere.
Solar Declination Law (I.14): Specializing Spherical Menelaus to the right spherical quadrant $Z\text{–}E\text{–}B$ gives the exact Greek equivalent of $\sin\delta = \sin\varepsilon\sin\lambda$:
$$\operatorname{crd}(2\delta) = \frac{\operatorname{crd}(2\varepsilon)\cdot\operatorname{crd}(2\lambda)}{120}.$$
Exact Symbolic vs. Heiberg (1898) Declination Check¶
Solar Longitude $\lambda$ |
$\operatorname{crd}(2\lambda)$ |
$\operatorname{crd}(2\delta)$ |
Computed $\delta(\lambda)$ |
Heiberg (1898, p. 78) |
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$30^\circ$ (1 sign) |
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$60^\circ$ (2 signs) |
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$90^\circ$ (Solstice) |
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