Article ID: CBB892484458

An Analysis of Medieval Solar Theories (2018)


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.

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Authors & Contributors
Kunitzsch, Paul
Ihsanoglu, Ekmeleddin
Sidoli, Nathan Camillo
Berggren, John Lennart
Kapel, Martin
Janssens, Jules L
Tarikh-e Elm (The Iranian Journal for the History of Science)
Archive for History of Exact Sciences
Bollettino di Storia delle Scienze Matematiche
Early Science and Medicine: A Journal for the Study of Science, Technology and Medicine in the Pre-modern Period
Suhayl: Journal for the History of the Exact and Natural Sciences in Islamic Civilisation
Journal for the History of Astronomy
Edwin Mellen Press
Centre de recherche en histoire des sciences
Johns Hopkins University Press
Arab/Islamic world, civilization and culture
Western world, civilization and culture
Cross-cultural interaction; cultural influence
Science and religion
Aristarchos of Samos
al-Ghazālī, Abū Ḥāmid Muḥammad ibn Muḥammad
Henry of Ghent
Suhrawardī, Yaḥyá ibn Ḥabash
Quṭb al-Dīn Shīrāzī
Time Periods
Early modern
14th century
Ottoman Empire
Mediterranean region
Florence (Italy)

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