Unit IV · Executive Summary¶
Mathematical Architecture of Ptolemy’s Planisphaerium¶
South-Pole Stereographic Map: Projects every point $(X, Y, Z) \in S^2 \setminus \{(0,0,-R)\}$ from the South Celestial Pole $(0, 0, -R)$ onto the equatorial plane $Z = 0$:
$$(x, y) = \left(\frac{R\,X}{R + Z},\; \frac{R\,Y}{R + Z}\right), \qquad (X, Y, Z) = R\left(\frac{2Rx}{R^2 + r^2},\; \frac{2Ry}{R^2 + r^2},\; \frac{R^2 - r^2}{R^2 + r^2}\right).$$
Circle Preservation: Every plane section $aX + bY + cZ + d = 0$ of $S^2$ maps to an exact circle $(c - d)(x^2 + y^2) + 2Rax + 2Rby + R^2(c + d) = 0$ (or a straight line through $e$ when $c = d$, i.e. meridians through the South Pole).
Conformal Pullback: $dX^2 + dY^2 + dZ^2 = \frac{4R^4}{(R^2 + x^2 + y^2)^2}(dx^2 + dy^2)$, preserving all angles of intersection between celestial circles!
Circular Inversion of Opposite Parallels (§4): Parallel circles at declinations $+\delta$ and $-\delta$ project to concentric circles of radii $r(\delta) = R\tan\left(\frac{90^\circ - \delta}{2}\right)$ satisfying:
$$r(+\delta)\cdot r(-\delta) = R^2.$$
Comparison of Exact Rational (u = 1/5) & Historical (ε = 23°51’20’’) Astrolabe Radii¶
Astrolabe Element |
Rational Model ($\tan\frac{\varepsilon}{2} = \frac{1}{5}, \varepsilon \approx 22^\circ 37'$) |
Ptolemy’s Planisphaerium §4 ($\varepsilon = 23^\circ 51'20''$) |
|---|---|---|
Equatorial Radius $R_{\text{eq}} = ed$ |
$60$ |
|
Tropic of Cancer Radius $r_{\text{Can}}$ |
$40$ |
|
Tropic of Capricorn Radius $r_{\text{Cap}}$ |
$90$ |
|
Inversion Product $r_{\text{Can}} \cdot r_{\text{Cap}}$ |
$40 \times 90 = 3600 = 60^2$ |
$39;4,19 \times 92;8,15 \approx 3600 = 60^2$ |
Oblique Ecliptic Center & Radius |
Center $(0, 25)$, Radius $65$ |
Center $(0,\; 26;31,58)$, Radius |