Semi-arc and speculum

Meaning

When you stand at a given geographic latitude φ and watch a star or planet rise in the east, culminate on the meridian, and set in the west, the path it traces is a diurnal parallel of declination δ. Half of its time above the horizon is spent rising from the eastern horizon (ASC) to the Midheaven (MC), and the other half descending from the Midheaven to the western horizon (DSC).

• The Diurnal Semi-Arc (DSA) is the equatorial rotation (in degrees of right ascension) required to carry a body from the eastern horizon to the upper meridian (MC), or from MC to the western horizon (DSC).
• The Nocturnal Semi-Arc (NSA) is the equatorial rotation required to carry the same body from the western horizon to the lower meridian (IC), or from IC back to the eastern horizon (ASC).

Because a complete semicircle from upper meridian (MC) to lower meridian (IC) is always exactly 180° of right ascension (12 sidereal hours), every non-circumpolar point on the celestial sphere obeys the Fundamental Semi-Arc Invariant:

DSA + NSA = 180°

Ascensional Difference and Spherical Derivation

On the celestial equator (δ = 0°), a point spends exactly half the day above the horizon and half below, so DSA = NSA = 90°. When a planet has northern or southern declination δ at observer latitude φ, its rising and setting times shift by its Ascensional Difference (AD):

sin(AD) = tan(φ) · tan(δ)

From the ascensional difference follow both semi-arcs:

DSA = 90° + AD, NSA = 90° − AD = 180° − DSA

In the Northern Hemisphere (φ > 0), a body with northern declination (δ > 0, such as a planet in Cancer or Gemini) has AD > 0, so its Diurnal Semi-Arc exceeds 90° (reaching 115°–125° at mid-latitudes) and its Nocturnal Semi-Arc shrinks correspondingly.

The Tabula Speculi and Placidian Mundane Fraction (f)

Before computing any primary direction, traditional astronomers construct a Speculum (Tabula Speculi, "Mirror of the Sky") listing every natal planet's equatorial and horizon-meridian coordinates:

In Moira's Speculum, the Placidian mundane fraction f ∈ [−2, +2] provides an exact, latitude-invariant coordinate for every planet's mundane seat:

• f = 0.000: Culminating on the Midheaven (MC)
• f = −1.000: Rising on the Ascendant (ASC) (eastern horizon; −1/3 is the 11th cusp, −2/3 is the 12th cusp)
• f = +1.000: Setting on the Descendant (DSC) (western horizon; +1/3 is the 9th cusp, +2/3 is the 8th cusp)
• f = ±2.000: Anti-culminating on the Imum Coeli (IC)

Under Ptolemy's and Placidus's semi-arc doctrine, two bodies occupy the same mundane position when their proportional fractions f are equal—even if their declinations and semi-arcs differ [1][2].