Limitations of Methods: The Accuracy of the Values Measured for the Earth's/Sun's Orbital Elements in the Middle East, A.D. 800–1500, Part 1

Abstract

(ProQuest: ... denotes formulae omitted.)(ProQuest: ... denotes non-USASCII text omitted.)The present paper deals with the methods proposed and the values achieved for the eccentricity and the longitude of apogee of the (apparent) orbit of the Sun in the Ptolemaic context in the Middle East during the medieval period. The main goals of this research are as follows: first, to determine the accuracy of the historical values in relation to the theoretical accuracy and/or the intrinsic limitations of the methods used; second, to investigate whether medieval astronomers were aware of the limita- tions, and if so, which alternative methods (assumed to have a higher accuracy) were then proposed; and finally, to see what was the fruit of the substitution in the sense of improving the accuracy of the values achieved.In Section 1, the Ptolemaic eccentric orbit of the Sun and its parameters are introduced. Then, its relation to the Keplerian elliptical orbit of the Earth, which will be used as a criterion for comparing the historical values, is briefly explained. In Section 2, three standard methods of measurement of the solar orbital elements in the medieval period found in the primary sources are reviewed. In Section 3, more than twenty values for the solar eccentricity and longitude of apogee from the medieval period will be classified, provided with historical comments. Discus- sion and conclusions will appear in Section 4 (in Part 2), followed there by two discussions of the medieval astronomers' considerations of the motion of the solar apogee and their diverse interpretations of the variation in the values achieved for the solar eccentricity.1. IntroductionPtolemy, in Almagest ΠΙ, presented a solar model consisting of an eccentric circle whose centre is displaced from the centre of the Earth by an amount of eccentricity e = TO expressed in terms of an arbitrary length R-60 for the radius of the eccentre (Figure 1). The Sun revolves on the eccentre around the Earth but its motion is uniform with respect to the centre of eccentre, and completes one revolution in a tropical year TY (counted in days), i.e., with a mean motion of ? = 360/7T. Due to the eccentricity of its orbit, the true longitude of the Sun (i.e., its longitude as seen from the Earth T) does not match its mean longitude (its longitude as it appears from the centre O of uniform motion) except at apogee (A) and perigee (Π). The difference between the two, called the 'equation of centre' q, is defined as a one-variable function of the solar mean anomaly c, which is the difference between its mean longitude and the longitude Aap of the apogee:Therefore, the solar eccentric model has three parameters (TY or ?, e, and Aap) but only one anomaly due to its eccentricity.Ptolemy assumes that all three parameters are constant.' Since, at least, the latter part of the ninth century, the Middle Eastern astronomers found (cf. Tables 1 and 2) that the longitude of the solar apogee increases with the passage of time and they (seemingly, first of all, the Banu Musa or Thabit b. Qurra and Habash) assumed its rate of change to be the same as the rate of precession, as is the case with the motion of the apogee of the planets in the Ptolemaic context (the relevant discussion will be presented in Section 4). Also, different values for e and TY were observed during the medieval period. The variety in the values obtained for the two was so large that, for example, Copernicus in the prologue of his De revolutionibus complains that [the astronomers] are uncertain of the motion of the Sun and Moon to such an extent that they cannot demonstrate or observe a constant magnitude for the tropical year,2 while BirunT was forced to resolve his readers' worries about the huge differences in the values measured for the solar eccentricity over a short period (see Section 4).In fact both e and TY are changing with the passage of time, e decreases slowly by the average amount of 4. …