An analysis of medieval solar theories

Abstract

From Antiquity through the early modern period, the apparent motion of the Sun in longitude was simulated by the eccentric model set forth in Ptolemy’s Almagest III, with the fundamental parameters including the two orbital elements, the eccentricity e and the longitude of the apogee λA, the mean motion ω, and the radix of the mean longitude $$ \bar{\lambda }_{0} $$ . In this article we investigate the accuracy of 11 solar theories established across the Middle East from 800 to 1600 as well as Ptolemy’s and Tycho Brahe’s, with respect to the precision of the parameter values and of the solar longitudes λ that they produce. The theoretical deviation due to the mismatch between the eccentric model with uniform motion and the elliptical model with Keplerian motion is taken into account in order to determine the precision of e and λA in the theories whose observational basis is available. The smallest errors in the eccentricity are found in these theories: the Mumtaḥan (830): − 0.1 × 10−4, Bīrūnī (1016): + 0.4 × 10−4, Ulugh Beg (1437): − 0.9 × 10−4, and Taqī al-Dīn (1579): − 1.1 × 10−4. Except for al-Khāzinī (1100, error of ~ + 21.9 × 10−4, comparable to Ptolemy’s error of ~ + 33.8 × 10−4), the errors in the medieval determinations of the solar eccentricity do not exceed 7.7 × 10−4 in absolute value (Ibn al-Shāṭir, 1331), with a mean error μ = + 2.57 × 10−4 and standard deviation σ = 3.02 × 10−4. Their precision is remarkable not only in comparison with the errors of Copernicus (− 7.8 × 10−4) and Tycho (+ 10.2 × 10−4), but also with the seventeenth-century measurements by Cassini–Flamsteed (− 2.4 × 10−4) and Riccioli (+ 5.5 × 10−4). The absolute error in λA varies from 0.1° (Taqī al-Dīn) to 1.9° (al-Khāzinī) with the mean absolute error MAE = 0.87°, μ = −0.71° and σ = 0.65°. The errors in λ for the 13,000-day ephemerides show MAE < 6′ and the periodic variations mostly remaining within ± 10′ (except for al-Khāzinī), closely correlated with the accuracy of e and λA.