عنوان مجله
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ARCHIVE FOR HISTORY OF EXACT SCIENCES
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چکیده
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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 ¯λ0. 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 Mumtah. an (830): − 0.1 × 10−4, B¯ır¯un¯ı (1016): + 0.4 × 10−4,
Ulugh Beg (1437): − 0.9 × 10−4, and Taq¯ı al-D¯ın (1579): − 1.1 × 10−4. Except
for al-Kh¯azin¯ı (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-Sh¯at.ir, 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° (Taq¯ı al-D¯ın) to 1.9° (al-Kh¯azin¯ı) with themean absolute errorMAE
0.87°, μ−0.71° and σ 0.65°. The errors in λ for the 13,000-day ephemerides
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