Abū Naṣr’s Alternative Method for Determining the Solar Orbital Elements

In Almagest III.4, Ptolemy explains the Seasons Method for determining the structural parameters of the Sun. During the Islamic period, alternative procedures were proposed to eliminate observational errors arising from observations near solstices: Using the midpoints of the seasons in the early ninth-century Mumtaḥan observations; Abū Ja ̔far al-Khāzin’s recommendation to observe the Sun from longitude 120° through 240°, and from 300° to 60°; Ibrāhīm b. Sinān’s method of observing the Sun at two pairs of ecliptic positions symmetric with respect to the apsidal line; and Abū al-Rayḥān al-Bīrūnī’s use of two diametrically opposed longitudes with a third point near one of them, employed by Taqī al-Dīn Muḥammad b. Ma ̔rūf. A method devised by Abū Naṣr Manṣūr b. ̔Alī b. ̔Irāq (ca. 960-1036) survives in a single leaf of his Shāhī Majisṭī (Royal Majisṭī), preserved in the British Library. It uses three closely spaced points in spring or summer that intercept two arcs forming acute angles. We explain the mathematics underlying Abū Naṣr’s description and estimate its theoretical deviation from the mismatch between eccentric and elliptical models. The deviations are negligible: ±0.00014 in eccentricity and +0.16° to –0.28° in the orientation of the apsidal line. We present a passage from ̔Abd al-Raḥmān al-Khāzinī’s Experimental Astronomy indicating Khāzinī’s use of Abū Naṣr’s method.