Unit IV · Planisphaerium: Stereographic Projection & Astrolabe Geometry

Planisphaerium, Sections 1–4 (Heiberg 1907, Vol. II, pp. 227–234)

Section 1 · Stereographic Projection of Equator, Tropics & Ecliptic (Heiberg II, pp. 227–229)

^{p. 227} Since it is possible, O Syrus (Jesure), and frequently necessary, that the circles occurring upon the solid sphere be represented in a flat plane just as if it actually were flat, in harmony with those things demonstrated upon the solid sphere, I judged it essential to investigate this method of construction as well.

Its principle rests upon placing one single circle, abgd, in place of the celestial equator (qui rectam speram determinat), described around center e, which takes the place of the visible [North] Pole (manifesti poli). Through the center we draw two diameters cutting one another at right angles, ag and bd, representing two great circles drawn through the poles perpendicular both to the equator and to one another (the solstitial and equinoctial colures).

Next, every circle parallel to the equator—because all points along its circumference maintain an equal distance from the poles—will likewise be described in the plane around the same center e, at distances cut off by visual rays drawn from the hidden [South] Pole (South Pole):

• Circles on the sphere lying toward the visible [North] Pole (such as the Tropic of Cancer) fall inside the equatorial circle abgd, because straight lines drawn from the South Pole through northern parallels meet the equatorial plane closer to e (tl).

• Circles on the sphere lying toward the hidden [South] Pole (such as the Tropic of Capricorn) fall outside the equatorial circle abgd, because rays from the South Pole through southern parallels diverge outward before piercing the extended equatorial plane (km).

^{p. 228} Theorem (The Projected Oblique Zodiacal Circle Bisects the Equator). Now let tl be the parallel nearest the visible pole (Cancer) and km the parallel nearest the hidden pole (Capricorn), cut by meridian km at t, l, k, and m. Bisect segment tm at center r, and with radius rm = rt describe the oblique zodiacal circle mbtd, tangent to the summer tropic at t and to the winter tropic at m. I say that this circle passes directly through points b and d, thereby bisecting the equatorial circle abgd in the plane just as the ecliptic bisects the equator on the solid sphere.

Demonstration. Fold the meridian circle onto the plane so that d serves as the South Pole of projection while b serves as the North Pole, and draw straight rays from d to m (dm) and to t (dt):

1. On the meridian circle, the summer solstice and winter solstice are diametrically opposite on the sphere; hence the arc intercepted between the two solstitial points across the circumference is a semicircle ($180^\circ$).

2. Therefore, by Eucl. III.31, the inscribed angle ∠mdt subtending that diameter at pole d is a right angle (angulum mdt rectum esse)!

3. Consequently, in the right-angled triangle mdt with hypotenuse tm bisected at r, the circle described on diameter tm around center r must pass through the right-angled vertex d—and by symmetry across diameter km, through the opposite equinoctial point b!

Sections 2–3 · The Horizon & Inclined Great Circles (Heiberg II, pp. 229–232)

^{p. 229–232} By the identical right-angle principle, Ptolemy demonstrates in Sections 2 and 3 that every great circle inclined to the equator—in particular the local Horizon (circulus finitor) for any terrestrial latitude $\varphi$—projects stereographically onto an eccentric circle whose center lies along the meridian line and which bisects both the equatorial circle (at a and g) and the oblique zodiacal circle mbtd at two diametrically opposite ecliptic degrees (six signs always rising above the horizon and six below).

Section 4 · Circular Inversion & Numerical Radii from the Almagest (Heiberg II, pp. 233–234)

^{p. 233} Theorem (Inversion of Opposite Parallels Across the Equator). Take any two parallel circles equally distant on opposite sides of the celestial equator—specifically the winter tropic radius ez ($= r_{\text{Cap}}$) and the summer tropic radius ek ($= r_{\text{Can}}$) along meridian eg. Because the stereographic rays from the South Pole form right-angled similar triangles sharing the equatorial radius ed ($= R_{\text{eq}}$) as geometric mean:

$$\frac{ez}{ed} = \frac{ed}{ek} \quad\Longleftrightarrow\quad r_{\text{Cap}} \cdot r_{\text{Can}} = R_{\text{eq}}^2.$$

^{p. 234} Explicit Sexagesimal Evaluation Citing the *Almagest*. Ptolemy now inserts his own Almagest I.12 obliquity and I.11 chord values (”quemadmodum in Almagesti constitutum est”) to compute the exact dimensions of an astrolabe plate of equatorial radius $R_{\text{eq}} = ed = 60^p$:

• Tropical distance from the equator: $\varepsilon = 23^\circ 51' 20''$ (partibus XXIII punctis LI secundis XX).

• Supplementary colure arcs: $90^\circ + \varepsilon = 113^\circ 51' 20''$, with supplement $90^\circ - \varepsilon = 66^\circ 8' 40''$.

• Corresponding Almagest chords ($D = 120^p$):

  $$\operatorname{crd}(113^\circ 51' 20'') = 100;33,28^p, \qquad \operatorname{crd}(66^\circ 8' 40'') = 65;29,0^p.$$

• Since $ez : ed = ed : ek = 100;33,28 : 65;29,0$ and $ed = 60^p$, Ptolemy obtains:

  1. Winter Tropic (Capricorn) Radius:

     $$r_{\text{Cap}} = ez = 92;8,15^p \qquad (\text{Latin: }\textit{partibus XCII puncta VIII secunde XV}).$$

  2. Summer Tropic (Cancer) Radius:

     $$r_{\text{Can}} = ek = 39;4,19^p \qquad (\text{Latin: }\textit{partes XXXIX puncta IIII secunde XVIIII}).$$

  3. Total Diameter of the Projected Ecliptic (`tm = ez + ek`):

     $$D_{\text{ecl}} = 92;8,15^p + 39;4,19^p = 131;12,34^p, \qquad R_{\text{ecl}} = 65;36,17^p,$$

     with inversion check $92;8,15 \times 39;4,19 = 3599.991 \approx 3600 = 60^2$!