JHA, xliv (2013) 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 S. MOHAMMAD MOZAFFARI, Research Institute for Astronomy and Astrophysics of Maragha 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. Introduction Ptolemy, in Almagest III, 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 ω = 360/TY. 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 λap of the apogee: 0021-8286/13/4403-0313/$10.00 © 2013 Science History Publications Ltd 314 S. Mohammad Mozaffari Fig. 1. The eccentric solar model with constant eccentricity. (1) Therefore, the solar eccentric model has three parameters (TY or ω, e, and λap) but only one anomaly due to its eccentricity. Ptolemy assumes that all three parameters are constant.1 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 Banū Mūsā or Thābit 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 Bīrūnī 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 4.2 × 10–5 per century. TY is also subjected to a slight Limitations of Methods 315 time-dependent change whose rate and sign depend on which of the four cardinal points (solstices and equinoxes) is adopted as the zero-point for measuring the length of TY.3 It is worth noting that the solar apogee also progresses with a rate of 61.9″ per year or, approximately, one degree in 58 years, i.e., faster than the rate of precession, which is one degree in 71.6 years.4 The Earth in reality travels on an elliptical orbit with eccentricity e′ (expressed in terms of the length of the semi-major axis a = 1) around the Sun, where the Sun is located at one of the foci of the orbit. By exchanging the (positions of the) Sun and the Earth, the system is simply transferred to the geocentric mode and therefore the Sun apparently orbits around the Earth in an elliptical orbit. The difference in the solar geocentric longitude between the eccentric and elliptical models and the relation between the two eccentricities, e and e′, are as follows:5 ∆λ ≈ (e − 2e′)sin M + ( 1 e2 − 5 e′2 )sin 2M 2 4 (2) ′ 3 ′2 e ≈ 2e − 2 e cos M , where M is the mean anomaly (M = c). The formula for Δλ in (2) is accurate up to arc seconds, which is more than sufficient for medieval astronomical studies. If the eccentricity of the eccentric model is equated with the distance between the foci of the elliptical orbit, i.e., e = 2e′ (for either R = a = 1 or R = a = 60),6 then it is evident that the difference in the solar longitude given by the two models will be maximum when the Sun reaches its apparent orbital octant, i.e., for M = 45°. In practice the difference will be of the order of one second at most. This implies that the elliptical orbit does not match completely with the eccentric orbit and, under the condition e = 2e′, a deviation of about one arc-second may occur. (Note that we have assumed that also other parameters, i.e., the longitude of the orbit’s apogee and the rate of its progressive motion as well as the mean longitude of the Sun and its mean motion, are taken equal in the two models. This is not the case when we consider the historical values.) In the case of historical values for the eccentricity of the eccentre model, we expect in an ideal manner that the values measured are twice as long as the elliptical eccentricity. However, because of the small incongruence of the two models, this relation e = 2e′ may not be expected to hold exactly. We shall later note and take into account the effect of the mismatch of the two models on the accuracy of the historical values for e obtained from different methods. Nevertheless, the relation e = 2e′ turns out useful as a unified scale in order to give a general perspective of the accuracy and congruency of the historical values. This will be illustrated later in Figure 6(a), where the historical values are shown along with the descending curve of the Earth’s elliptical eccentricity. The solar eccentricity is responsible for the alternative deviation of the solar angular velocity from its mean value as seen from the Earth. For instance, the dotted curves in Figure 2 depict the rate of change in the solar velocity v with respect to its mean value ω as a variable of the mean anomaly (counted from the apogee) for the Ptolemaic eccentricity 2;30 and Muhyī al-Dīn al-Maghribī (d. 1283)’s value 2;6 (com- puted in the late thirteenth century based on the direct observations in the Maragha 316 S. Mohammad Mozaffari Fig. 2. The instantaneous velocity of the Sun according to the eccentric model based on Ptolemaic eccentricity e = 2;30 (dotted curve) and on al-Maghribī’s value e = 2;6 (R = 60) (dotted-dashed curve), and to the Keplerian elliptical orbit based on the modern eccentricity e′ = 0.017 (a = 1) (continuous curve). Observatory, see below). The continuous curve shows the true rate of change in the elliptical orbit with eccentricity e′ = 0.017. The correction for this anomaly, i.e., the equation of centre q (Formula (1)), and the true longitude of the Sun thus depend on the value adopted for its eccentricity. In general, any inaccuracy or error in one of the solar parameters can cause a systematic error in the true longitude of the Sun. In all systems for modelling the planetary motions (geocentric or heliocentric; based on either circular orbits or ellipti- cal ones), the solar model occupies a central position. In the Ptolemaic models, the longitude of the Moon, planets, and stars depends on the longitude of the Sun; for example, the mean motion of the inferior planets is equal to that of the Sun. Further- more, the superior planets’ motion in anomaly directly depends on the solar mean motion (the planet’s mean anomaly is the difference between the mean longitude of the Sun and of the planet).