The Planetary Latitude Theory of Qizheng Tuibu (1477)

The models with Northern and Southern limits for the superior planets, inclination and slant for the inferior planets, and the calculation formula underlying in the five planetary latitude tables in the Qizheng Tuibu (1477), have been completely outlined. Ptolemaic planetary latitude theory has been applied in the compilation of these tables with slight revision of the parameters and variables, which are implicitly given in the Sanjufīnī Zīj. The planetary latitude tables for five planets in the Qizheng Tuibu are double-argument tables in which the true centrum κ and the true anomaly av are selected as variables. For the superior planets: the Northern latitude β = c5 (κ) × c3 (av)/60, if κ∈[0°, 90°], [270°, 360°]; the Southern latitude - β = c5 (κ) × c4 (av)/60, if κ∈[90°, 270°]. For the inferior planets: Inclination > 0°, if av∈[90°, 270°]; Inclination ˂ 0°, if κ∈[0°, 90°], [270°, 360°]; Slant > 0°, if av∈[0°, 180°]; Slant ˂ 0°, if av∈[180°, 360°]. The latitude of Venus, β = β1 + β2 + β3 = Inclination × cos (κ + 90°) + Slant × cos κ + 0;13° × cos2κ. The latitude of Mercury, if κ∈[0°, 90°] or [270°, 360°], β = β1 + β2 + β3 = Inclination × cos (κ + 270°) + 0.9 × Slant × cos (κ + 180°) - 0;45° × cos2κ; if κ∈[90°, 270°], β = β1 + β2 + β3= Inclination × cos (κ + 270°) + 1.1 × Slant × cos (κ + 180°) - 0;45° × cos2κ. The possible scribal and calculation errors in the above five tables could be revised and corrected.