LIMITATIONS of METHODS: the ACCURACY of the VALUES MEASURED for the EARTH’S/SUN’S ORBITAL ELEMENTS in the MIDDLE EAST, A.D

Home , Octant (instrument), Tropical year

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 limitations, 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:

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 (computed 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 elliptical 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). Moreover, in order to determine the parameters of the Ptolemaic orbit of the Moon (eccentricity and radius of epicycle), the longitude of the Moon is directly derived from the longitude of the Sun at the instant of the synodic phenomenon involved (i.e., the maximum phase of a lunar eclipse for determining the radius of the lunar epicycle and half moon for determining the eccentricity of the lunar deferent). Therefore, a simple error in any solar parameter may ipso facto produce a chain of systematic errors in determining the longitude of other celestial objects. An historical example is Ptolemy’s error in determining the moment of the vernal equinox, by +1 day, which caused the well-known systematic error of around one degree in the longitude of the Sun; this error affected the longitude of the stars and, as a consequence, these were listed in Almagest VII and VIII with a systematic error of about –1°.7

2. The Historical Methods for the Measurement of the Solar Eccentricity The three main historical methods for measuring the solar orbital elements are the Seasons Method, the Mid-Seasons (or, Mid-Signs) Method, and the Three-Point Method, as will be described below. All three methods either are explained in the Limitations of Methods 317

Almagest (III 4) or can be extracted from the mathematical methods by which Ptolemy determined the parameters of his own planetary models (IV 6, IV 11, X, and XI).8 Nevertheless, Bīrūnī attributes the Three-Points Method to his master, Abū Nasr Mansūr b. ‘Irāq, and also mentions that in his (now lost) treatise named al-Istishhād bi-ikhtilāf al-arsād (Producing witness for the difference in the observations), he had proved the superiority of this method over the other two.9 Muhyī al-Dīn al-Maghribī’s account of his measurement of the solar eccentricity in the Maragha Observatory is one of the excellent surviving accounts of the utilization of this method in the medieval period (see below). His older colleague at the Observatory, Mu’ayyad al-Dīn al-‘Urdī (d. 1266), wrote a preliminary treatise to sketch the main outline of the method.10 The Mid-Sign method seems to have been somewhat innovative to medieval astronomers. Although Yahyā b. Abī Mansūr knew and made practical use of this method in the early ninth century, it is not known if he proposed it. In what follows, the methods are explained and some historical and technical considerations concerning each are outlined.

The Seasons Method In this method, first the length of the tropical year (TY), and therewith the mean angular velocity ω (= 360°/TY) of the Sun, and the length of two neighbouring seasons (e.g., the interval between an equinox and its successive solstice and the time from that solstice to the next equinox), are measured by observation. Then, the magnitude of the mean angular motion of the Sun on its geocentric orbit in these two intervals (i.e., the lengths of the arcs a and b in Figure 3) are calculated. (The Sun travels 90° on the ecliptic in each of these intervals.) The eccentricity e = TO in Figure 1 or EC in Figure 3 and the longitude of apogee λap = the angle ϒTA in Figure 1 or the angle ϒEA in Figure 3 are then obtained by the following formulae:

(3) . In order to determine the moments of equinoxes and solstices, medieval astronomers measured the meridian altitude of the Sun by means of meridian instruments, e.g., quadrants, sextants, or the Ptolemaic parallactic instrument. These instruments were installed in the line of meridian and were utilized for measuring the noon altitudes of the Sun, the latitude φ of the place of observation, the inclination ε of the ecliptic from the celestial equator, and so on. We know that various such instruments in different sizes were constructed in the early Islamic period.11

In the equinoxes the solar declination δeq = 0 and in the solstices, δsl = ±ε. Thus, the solar noon altitude h in the days around the equinoxes and solstices are measured in a given place with the latitude φ. If h = 90 – φ or h = 90 – φ ± ε, then the instant of the true noon will be simply the time of, respectively, the equinox or solstice. If not, in order to determine the time of an equinox or of a solstice, the Middle Eastern astronomers utilized some techniques adopting the linear interpolation as follows.

The equinox occurs between the true noon of two successive days when h1 < 90 – φ 318 S. Mohammad Mozaffari

Fig. 3. The measurement of the parameters of the eccentric solar model based on the Seasons Method.

and h2 > 90 – φ (in the case of vernal equinox) and h1 > 90 – φ and h2 < 90 – φ (in the case of autumnal equinox). Then, since the rate of variation in the solar declination

(and so, noon altitude), |Δδeq| = |Δheq| ≈ 0;24°, is known, the instant of the equinox is determined as |(h – 90 + φ)/Δδeq|-th of a day after the true noon of first day or before that of second one. Another variant is that after the measurement of h, one derives δ º/d and next λ; then |λ – λeq/ω| (where ω = 0;59,8 is the solar daily mean motion) will be the time elapsed since or remaining before the equinox. Al-Maghribī,12 for example, employed it in order to determine the instant of the autumnal equinox of 1264 and that of the vernal equinox of 1265; on 16 September 1264 (JD = 2182993): h = 52;21º (true modern: 52;17º); with φ = 37;20,30º for Maragha, δ = –0;18,30º; then with ε = 23;30º,13 λ = 180;46,25º [recomputed: ...,24]. Thus, the autumnal equinox of the year occurred 18;50,20 [text: 18;24,46; recomputed: 18;49,55] hours before the true noon of 16 Sept or on 15 Sept, ~17:02 MLT (true modern: ~15:25). On 14 March 1265 (JD = 2183172): h = 53;8,30º (true modern: 53;5º); then λ = 1;12,44º. Thus, the vernal equinox of the year occurred 1 day and 5;31 hours before the true noon of 14 March or on 13 March, ~6:37 MLT (true modern: ~7:37). In the case of solstices, since the variation in δ/h is very small, ~ 0;0,14°, such a simple method of interpolation is of no practical application. In order to determine the instant of a solstice, Bīrūnī proposed14 two techniques for the solstice interpolation (although he himself declared he was not satisfied with either).

(1) Observing the solar meridian (noon) altitudes on some days prior and posterior to the solstice. Find, first, the meridian altitudeh 1, next, the declination δ1, and then Limitations of Methods 319

the longitude λ1 of the Sun on a day t1 prior to the occurrence of a solstice by a few days; also find the meridian altitudeh 2, δ2, and λ2 for a day t2 after that solstice by a

few days. The time interval between t1 and t2 is then known. Consider the third point on the ecliptic symmetric to λ2 with respect to that solstice, through which the Sun passed at an unknown time t3; it is evident that δ3 = δ2 and λ3 = 180 – λ2. Thus, the time interval between t and t is (180 – – )/v . Therefore, the time of the occurrence 1 3 λ2 λ1 C of the solstice is t + (t – t )/2 + (180 – – )/2v . B illustrates the method as 1 2 1 λ2 λ1 C īrūnī follows. In Jurjāniyya with φ = 42;17° (today, Kunya-Urgench, 42;18°), on 7 June 15 1016 (JD = 2092310): h1 = 70;58° (modern: 70;59°), so δ1 = 23;13° [recomputed:

23;15°], and, since Bīrūnī has ε = 23;35° (modern value for Bīrūnī’s time: ε0 =

23;34°), λ1 = 80;11° [80;38°]. On 23 June 1016 (JD = 2092326): h2 = 71;4° (modern: 16 71;5°), so δ2 = 23;21°, and then λ2 = 98;6° [97;50°]; then λ3=81;54° [82;10°]. Since in the days sufficiently near the solstice the solar velocity has its minimum value (v C °/d = 0;57 ), the time interval between t1 and t3 is (81;54 – 80;11)/0;57 = 1;48 [(82;10 – 80;38)/0;57 = 1;37] days. The time interval between the two observations (i.e., t2 – t1) is 16 days. Therefore, the moment of solstice is 8;54 [8;49] days after the first observation, i.e., 21;36 hours after the true noon on 15 June or 16 June, 9:36 [7:36] (true modern: 10;48, i.e., an error of –1;12 [–3;12] hours). In the end, Bīrūnī remarks about the method that it is theoretically sound but practically unreliable, and that it should be employed in a gradual manner (‘alā sabīl al-tadarruj; an iterative process with the aid of other pairs of the observational data?) until one arrives at the most exact value of the instant of solstice.

(2) Measuring the shadows of a gnomon erected perpendicularly on the ground or on a wall, respectively, for the winter and summer solstice observations, during some days around solstice. Assume A, B, C, D, and E (Figure 4) to be the apex of gnomon shadow in noon on five successive days (t1 to t5) before the solstice and T

and K to be the apex of the gnomon noon shadow on two successive days (t′1 and t′2) — of course, not essentially and necessarily immediately — after that solstice. If the instant of solstice is denoted by S, the point C corresponds to the time difference

S – t3 before the solstice and the point T corresponds to the time difference t′1 – S after that solstice. The difference Δt between the two above time intervals is taken to

be (CT/CD)∙(t4 – t3) where t4 – t3 = 1 day. Then S = t3 + (t′1 + Δt – t3)/2. Or, similarly, if Δt′ is the difference between the time interval S – t3 and t′2 – S, then Δt′ = (CK/

CB)∙(t3 – t2) where t3 – t2 = 1 day. Therefore, S = t3 + (t′2 – Δt′ – t3)/2. No example was given for the method.

Fig. 4. The solstice interpolation: the noon shadow technique. 320 S. Mohammad Mozaffari

The method is evidently based on the linear interpolation in the length of gnomon shadows. Nevertheless, the solar declination (and so its meridian altitude) and thus the length of its shadow does not vary in linear proportions between two successive days around the solstice. Bīrūnī seems to notice the problem17 and, consequently, remarks that the method is not guaranteed. (Some centuries later, Edmond Halley developed a method of the solstice observation based on the hyperbolic interpolation.18) Bīrūnī, however, suggests that in order to obtain a desired value, the gnomon should be erected in the highest places from the ground (i.e., where the length of shadow is greater) and/or the shadow scale should have a long length to be graduated to the seconds (i.e., the whole length of the scale to be divided into 3600 equal parts). In both of the methods, it is assumed that the time intervals between any of two points symmetric with respect to a solstice and that solstice are equal while, because the solar apogee does not coincide with the summer solstice, this may not be the case. A comparison with the altitude observations by the other Islamic astronomers for determining the time of solstices19 shows that only Bīrūnī and Abu al-Hasan al-Hirawī have the observed altitudes for the days sufficiently far from the solstices, which might be applied as the input data in these (or other similar interpolation) methods in order to determine the time of solstice, while the others have solely the observed altitudes at the successive days close to solstices.

The formulae (3) are highly sensitive to changes in the lengths of the two time intervals: for a = 93° and b = 92°, we have e = 2.67 and λap = 78.7º; but if a is increased by only one degree (as previously mentioned, this is the size of the systematic error in Ptolemy’s solar theory, corresponding to the motion in approximately one day), we obtain the value e = 3.31 (i.e., an error of around 24%) and λap = 71.6º (an error of around 9%). The sensitivity to a small change in the input data caused drastically different values to be determined for the solar eccentricity and the longitude of the solar apogee from the observations performed even within a short period. This was a main focus of medieval observational astronomers; for example, Bīrūnī noticed the sensitivity of the method as “a major amount of change in the result caused by/due to a minor difference [from the true value] entered into the observation[-al data]”.20 He listed all values of the solar eccentricity obtained from the Seasons Method by his predecessors as the evidence for his statement (see Table 1). Moreover, as seen above, there are technical and empirical difficulties in the exact determination of the moment of occurrence of solstices. This may make a sound application of the method problematic and may drastically decrease the accuracy of the value achieved. The other two methods that are found in the primary sources try to remedy this problem.

The Mid-Seasons (or Mid-Signs) Method21 This method is a substitute for the Seasons Method. In order to avoid the probable large error in the determination of the solstices, the medieval astronomers remedied the Seasons Method. The equinoxes and solstices were replaced by the middles of seasons, where the daily rate of difference in the solar meridian altitude is only around Limitations of Methods 321

0.3°. In the other words, instead of measuring the intervals between the equinoxes and solstices, the two intervals are observed between the middle of three successive seasons. The astronomers thus had to determine, first, the interval between the times when, for example, the Sun reached a longitude of λ = 315° (the middle of the sign Aquarius) and a longitude of λ = 45° (the middle of the sign Taurus). Then they would determine the time interval during which the Sun travelled from λ = 45° to λ = 135° (the middle of the sign Leo), and so on. In this way four quadrants on the ecliptic are marked: Eastern: mid-winter to mid-spring; Northern: mid-spring to mid-summer; Western: mid-summer to mid-autumn; and Southern: mid-autumn to mid-winter. In order to determine the times of mid-seasons, the Islamic medieval astronomers were again using the interpolation techniques like those described above. The declination of the Sun at the mid-seasons when the Sun reaches the longitudes 45, 135, –1 225, and 315°: δmid = δ45 = δ135 = –δ225 = –δ315 = sin (sin 45° ∙ sin ε). The solar noon

altitude in a given place is hmid = 90 – φ + δmid in mid-spring and mid-summer and hmid = 90 – φ – δmid in mid-autumn and mid-winter. Now, if in a day, the solar noon altitude equals hmid, then the moment of the true noon is the time of mid-season.

Example by Bīrūnī: from the latitude φ = 42;17° for Jurjāniyya and ε = 23;35°, δmid

= 16;26°; for mid-summer (and also mid-spring): hmid = 64;9°; Bīrūnī reports that he measured the Sun’s noon altitude on 2 August 1016 and obtained h = 64;9° = hmid. Thus, the mid-summer of the year of 1016 in Jurjāniyya occurred on the true noon of 2 August. Modern values for the noon: h = 64;7° and λ = 135;1°; mid-season, λ = 135°, occurred at 11:35, i.e., ~ +29 min before true noon at 12:4 MLT..

If the solar noon altitude does not equal hmid, a simple linear interpolation is applied to determine the time of mid-season. Note that since we know the rate of variation in the Sun’s declination (and thus in its noon altitude) between two days around a mid-season, only the measurement of the solar altitude in a day around mid-season is needed in order to compute the time of mid-season. The daily variation in the solar

declination (noon altitude) around the mid-seasons is |Δδmid| = |Δhmid| = 0;18° and the

measured noon altitude is h so that |h – hmid| < 0;18°; thus, the mid-season occurs |(h –

hmid)/Δδmid|-th of a day prior or posterior to the true noon of the day of measurement. Example by Bīrūnī: in order to determine the time of mid-autumn of the year 1016 in Jurjāniyya, on 1 November 1016, Bīrūnī found the Sun’s noon altitude to be more than 31;18°, he thought by 0;0,20°22 (this gives impression that the instrument used

was of the precession/calibration of 1/3′). hmid = (90 – 42;17) – 16;26 = 31;17°. Thus, the mid-autumn occurred (31;18,20 – 31;17)/0;18 = 0;4,27 day (Bīrūnī rounded it as 0;4,30) or around 1;48 hours after the true noon. Modern values for the noon: h ≈ 31;19,40° and λ = 224;55°; the mid-autumn occurs at 13:47 MLT, or 2;2 hours after the true noon at 11:45 MLT, i.e., an error of ~ –14 min. The Mid-Signs Method passed into the Western branch of the medieval Islamic astronomy (Spain and north of Africa) and then into the European literature. A possible way of the transmission appears to have been Ibn al-Zarqālluh’s works: in a Latin text of the late thirteenth century, Bernardus de Virduno’s Tractatus super totam astrologiam, we are told that “he made four observations while the Sun was 322 S. Mohammad Mozaffari 322 S. Mohammad Mozaffari 81;18° 81;18° 81;18 81;18 83;57 83;57 81; 9 9 81; 9 81; 8 81; 84;32 84;32 85;13 85;13 ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ≈ ap λ 80;21,43 80;21,43 2 80;22, 81;18,14 81;18,14 81;18,24 81;17,46 81; 8,44 8,44 81; 8,54 81; 8,15 81; e e 2;19,29 2;19,29 2;19,33 2; 7,45 7,45 2; 7,47 2; 7,38 2; 2; 0,30 0,30 2; 83;56,34 2; 8, 0 8,0 2; 8,2 2; 7,53 2; 2; 4,50 4,50 2; 8 82; 4,26 2; 2 84;32, 2; 3,37 3,37 2; 85;12,38 Recomputed ī n ū r ī ap λ 82; 9,55 9,55 82; 83;51, 1, 1 1 1, 83;51, 81;23,10,10 81;23,10,10 81;38,22,28° 82;7,38,23 82;7,38,23 84;34,45,50 49 49 e e 2;19,17, 6 6 2;19,17, 2; 0,28,15 0,28,15 2; 2; 7,40, 2; 2;10,14,19 2;10,14,19 –––– –––– –––– –––– –––– 4,47 2; 84;47 4,35 2; 87;48,14 2; 4,29,19 4,29,19 2; 4,10,49 2; –––– –––– –––– ap λ ~ 80 80 ~ –––– –––– 80;44,19° 80;44,19° 80;45 85;15 85;15 87;48,54 82;14 82;14 –––– 85;13, 5,24 5,24 85;13, Official values values Official B by Recomputed e e 2;5,32 2;5,32 –––– –––– 2;6,31 2;6,31 2;4,45 2;4,45 2;4,35 2;4,45 2;4,45 –––– 2;3,36,34 2;3,36,34 m m m m d 28 57 13 51 h h h h 4 5 3 5 1. d d d d 93; 9,20 9,20 93; 2,47 93;

93;2,25,25 93;2,25,25 93;4,23,31 93;2,30 93;1,54 93;1,54 93;2,40 93 93 93 93 93;1,52 93;1,52 93;4,18 93;7,10 93;7,33 93;8 93;8 93;9 able m T m m m 24 h h 55 33 d h h 9 7 12 9 d d d d 93 93;54,35 93;54,35 93;32,24 Spring Summer 93;27, 3,45 3,45 93;27, 93;30,34,12 93;40 93;40 93;32,50 93 93 93;40 93;35 93;30,55 8 93;30, 93;27,46 93 93;28 93;28 93;26 539 Y Y 539 1171 February 1170–1 2 February 212 Y Y 212 844 April 843–23 25 April Year 13 April 888–12 April 889 889 April 888–12 13 April 201 Y Y 201 833 April 832–26 27 April Y 251 883 April 882–14 15 April Y 343 975 March 974–22 23 March 447 447 1079 February 1078–24 25 February 12 March 1016–11 March 1017 1017 March 1016–11 12 March Y 385 ī n

n n ī ī ū ā r ī n n dh ā ū Ṭ abar mat of Samarqand of Samarqand ṣ mat Y 257 n al-B I c of of Buzj ā ī ā

of Harr of Al s ā c n b. s ā ū Ī ā

c ī amamd b. amamd M n lid al-Marwar lid ā ḥ ū b. ī ā Al c – –Sanad b. b. –Sanad Astronomers 1 1 –Ban 2 –Kh 3 –Batt 4 –Sulaym 5 –Abu al-Waf 6 –Abu al-Rayh 7 – Mu 8 Anonymous – For notes to Table 1, see p. 327. Table notes to For Limitations of Methods 323 in the middle of the quadrants bounded by the solsticial and equinoctial points”. In it, the values e = 1;58, λap = 85;49º, and the date of observations as 193 years after Battānī (see below, Table 1, no. 3), i.e., a.d. 1075, are associated with him.23 With respect to difficulties that his medieval predecessors confronted in order to determine the time of the solstices, Copernicus used a mixed version of the two methods of Seasons and Mid-Signs as already described by Regiomontanus in his Epitome of the Almagest, III, 14:24 in 1515, he measured the interval from the autumnal equinox to mid-autumn (45;16 days) and then to the consecutive vernal equinox (autumn + 25 winter = 178;53,30) and finally determinede = 1;56 and λap = 96;40º. Later, around 1584–88, Tycho Brahe applied the same method to intervals from the vernal equinox to mid-spring and to mid-summer and obtained qmax = 2;3,15°, which corresponds 26 to e = 2;9, and λap = 95;30º.

Three-Point Method According to the surviving account by al-Maghribī,27 after determining TY (and thus ω), the Sun is observed at three different times other than those near equinoxes and solstices during one tropical year. The meridian (noon) altitude of the Sun and thus its declination is measured. Then, the solar longitude in the three moments is computed by λ = sin −1 (sinδ / sinε ). The differences in the Sun’s true longitude between two pairs of two successive observations are thus known. The differences in its mean longitude between two pairs of two successive observations can also be calculated by multiplying the mean velocity ω with the corresponding time intervals. Now, having the magnitudes of the four arcs BD, DE, FG, and GH, the value of the eccentricity e = CE is simply calculated with the aid of plane trigonometry (Figure 5). The Seasons and Mid-Seasons Methods are both obviously special cases of the Three-Point Method. The basic formula in the first two methods, (3), is sensitive to the input data; similarly, the necessary trigonometric steps in the Three-Point Method are also somewhat sensitive to the input data and thus are not guaranteed to produce results more exact than the other two methods.28 Nevertheless, consider- ing the observational essentials of each method, it becomes evident that in order to determine the length of seasons, we need, at least, to determine the times of two equinoxes as well as to cope with the measurement of the instant of a solstice; for this one must observe the solar meridian altitude/shadow during a good number of days and do some interpolations. In order to determine two pairs of the time intervals between two successive mid-seasons, we need to measure, at least, three noon altitudes and carry out, at most, three interpolations. In the general Three-Point Method, we have to measure only three noon altitudes without needing to do any interpolation. The most important feature of the method is that all the observations are well-timed (from noon to noon), and it is only required to take into account the correction due to the equation of time. No special event needs to be observed and thus no difficulty is expected to arise, and also the mean motion of the Sun between 324 S. Mohammad Mozaffari

Fig. 5. The measurement of the parameters of the eccentric solar model based on the Three-Point Method. any two observations can be easily computed. Thus, the input data needed may be obtained more conveniently and accurately. Therefore, it seems that the Three-Point Method has a more secure observational basis which may appreciably increase the accuracy of the results in comparison with the other two methods. Astronomers of the Islamic period generally did not explain their observations and the input data that led to the specific parameters underlying the tables in theirz ījes.29 Nonetheless, it is somewhat probable that in those zījes of the late Islamic period that were based on direct observations, the Three-Point Method was most likely used for determining the desired values for the structural parameters of the solar orbit. Because the accuracy of the method had been known at least since the latter part of the tenth century, it seemingly had become a ‘standard’ method which was explained in zījes and astronomical treatises as the method for determining the longitude of the Sun from its declination in order to rectify and correct its mean velocity and eccentricity (e.g., Al-Maghribī, Talkhīs al-majistī, MS Leiden, no. Or. 110, IV, 5: fols. 58v–61v).

3. Classification of the Historical Values for the Solar Orbital Elements The most important historical sources used in the present paper are Bīrūnī’s al-Qānūn al-mas‘ūdī, Anonymous Sultānī zīj, and Kamālī’s Ashrafī Zīj.30 Bīrūnī’s al-Qānūn, VI, 7,31 provides invaluable comparative studies on the values measured by medieval astronomers in the period a.d. 800–1000 for the length of the tropical year, the length Limitations of Methods 325 of the seasons, the solar eccentricity, and the longitude of its apogee. Therefore, our discussion about the awareness of medieval astronomers as to limitations of the methods (sensitivity to input data and difficulties in obtaining the required correct observational data) concentrates on the analysis of the contents of this source. In the unpublished sources Sultānī zīj, Ashrafī zīj, and al-Khāzinī’s al-Zīj al-mu‘tabar al-sanjarī, the authors quote their predecessors’ values for the solar eccentricity and/ or preserve the tables for the solar equation of centre from earlier works.32 The values adopted for the solar eccentricity e in other works can be deduced from their tabulated

values of the solar equation of centre qmax, according to Formula (1). Some values for the solar orbital elements based on the Seasons Method measured by the Middle Eastern astronomers between a.d. 800 and 1200 are summarized in Table 1. Bīrūnī presents the results of the measurements of the solar orbital elements based on the Mid-Seasons Method performed by his Islamic predecessors as summarized in Table 2. As can be clearly seen, the application of the method may be traced back at the very least to the observational program conducted by Yahyā b. Abī Mansūr in the reign of al-Ma’mūn (early ninth century). Col. 1 contains the numbers by which I refer to each observation or astronomer. Col. 2 gives the name of astronomer(s). Col. 3 represents the period of observation within a year reckoned according to the Yazdigird era and the corresponding Julian year. (The Yazdigird era, 16 June 632, is used with the Egyptian/Persian year consisting of 12 months of 30 days plus five epagomenal days which, in the early Islamic period, were put after the eighth month. In the late Islamic period, they were transferred to the end of the year.) Cols. 4 and 5 give the length of the seasons (spring and summer) measured in each observation. Cols. 6 and 7 represent, respectively, the values for the solar eccentricity e and longitude of apogee λap quoted from or confidently attributed to the corresponding astronomers. Cols. 8 and 9 and cols. 10 and 11 give the recomputed values, respectively, by Bīrūnī and by the present author based on the input data given in cols. 4 and 5. The modern values for the lengths of spring and summer in the periods given in col. 3 are indicated in bold type.33 It is interesting that the values recomputed by Bīrūnī based on the input observational data obtained by the early Islamic astronomers (Table 1, nos. 1–3; Table 2, nos. 1 and 2) are (although, often of good accuracy) different from the values reported from, or ascribed to, those astronomers. Nonetheless, in our comparative study that follows (Section 4), the formal values are used in order to avoid making any possible anachronistic conclusion. Since there are numerous printing errors in the Hyderabad edition of Bīrūnī’s al-Qānūn (two examples are given below Table 2), all values for e have been rechecked. This was done both using the corresponding amounts given by Bīrūnī

for the maximum solar equation of centre qmax, according to Formula (1), and 326326 S.S. Mohammad Mohammad Mozaffari Mozaffari ap λ 81;12,27º 81;12,27º 81;12,12 84;42,34 84;42,34 80;42,32 80;42,32 85;14,52 85;14,52 See See notes Recomputed e e 2;7,33 2;7,33 2;7,29 2;4,58 2;4,58 2;6,44 2;6,44 2;4,24 2;4,24 ī n ū r ī ap λ 81; 1,50,32 1,50,32 81; 4,22,40 81; 85; 0,15,32 0,15,32 85; 61;23,22,40º 61;23,22,40º 81; 2,29,45 2,29,45 81; –––– –––– . 1059 1059 . pril 851 851 pril March 1003 1003 March e e 5 Dec. 1331 1331 Dec. 5 21 Dec. 1342 1342 21 Dec. 2; 7,18,40 7,18,40 2; 7,19,27 2; 2;14,28,21 2;14,28,21 –––– –––– = 2 Feb. 1172 1172 Feb. 2 = = 5 March 1265 1265 March 5 = ap λ 84;59,11,9 84;59,11,9 ) )  ning of 841 Y = 19 Nov. 1471 1471 19 Nov. = Y of841 ning → e e Official values values Official B by Recomputed  –––– –––– –––– 5,11,17 2; 2; 4,35 4,35 2; 82;39° –––– –––– –––– 6,33,17 2; 4,43,25 2; Time Time ? 2) ( d

88;35,50

88;32, 2 2 88;32,

ap ) λ (1 d 2. 3.

able able 90;55,50 90;55,50

94;48,20

91; 3,16 3,16 91; 91;10,40 4,30 91; 5,30 91; T T e e d nus nus 86;24,21 1232 18 Jan. = Y of601 Beginning

ū ); (2) Text: 18;35,50 (scribal ); error: Text: (2) 18;35,50  94;11,20 94;11,15 94;11,15 8,23 94; 94; 9, 7,30 7,30 9, 94; 8,52 94; 94;10 94;10 8 94; 93;33 93;33 8,30 94; 8,25 94; → max d q 2; 0, 8 0,8 2; 5,47 2; 84;33 23 A = Y of220 Beginning 2;14º 2;14º 2;20 77;55° Fixed 2;13 2;13 2;19 0,30 2; 82;39 6,10 2; 86;10 Fixed 2,0 2; 16 = Y of372 Beginning 7,44 2; 79;12 2 = Y of701 Beginning 2; 0,20 0,20 2; 0,10 2; 5,59 2; 5 84; 5,49 2; ? ? Y = Ibn 2; 0,21 0,21 2; 5,59 2; 88;50,21 Y 26–2–634 1;58,59 1;58,59 4,35 2; 2,36 87;50, Y of541 Beginning 0,29,19 2; 6,9 2; 1;55,53,12 0 90; 1,20 2; 4,48 90;30, Begin = Y of712 Beginning 2;12,25 2;12,25 2;18,38 0 86; 16 Feb = H of451 Beginning  

kir East Q. Q. East Q. North Q. West Q. South 91;45,20 91;45,20 91;38,10 91;45,20 91;34,25 91;30,55

91;46,40 91;46,40 91;30,10 93;16,40 ā mad b. Sh b. mad ḥ

ī n n ī ir ir r r

sim A sim

ī ū ī ā j j āṭ j d d rizm

z ī ā ī ā ā z ī nus nus zin al-D ī ī ū ā sh n y ā al-Q ā ḥ ū Year Y 200–201 833 April 831–26 28 April 345 Y Y 345 977 March 976–21 22 March 199 Y Y 199 831 April 830–27 28 April 355 Y Y 355 987 March 986–19 20 March 385 Y Y 385 1017 March 1016–11 12 March Ilkh ī

n ī n n 1 1 al-Khw 2 Ab 3 3 al-Nayr 6 6 Y Ibn 4 4 Am Ibn 7 7 al-Kh 9 9 Mu ū 5 5 Ibn al-A‘lam 8 8 al-Fahh n ā r r r For notes to Table 2, see p. 329; for Table 3, p. 330. Table 2, see p. 329; for Table notes to For ī 11 Ibn al-Sh 12 al-K 10 13 Beg Ulugh ā ṣū agh Ṣ n al-B Man

of of Buzj ā ī ī ā ī Al c b. Ab b. rizm ā mid al- Ḥā mid ā al-Rayh y ū ū ḥ –Sanad b. b. –Sanad –Khw –Ya Astronomers 1 2 3 –Abu al-Waf 4 –Ab 5 –Ab of scribal Examples 94;98,20 errors: (1)(scribal error: Text: Limitations of Methods 327

re-computing based on the given lengths of seasons. Note that each 0;0,1 difference in e when R = a = 60 corresponds to a difference of ~2.3∙10–6 in e while R = a = 1. Accordingly, the deviations up to 4 seconds in e (R = a = 60) may be of much less effect when we wish to express the corresponding value for it by the factor of 10–5 when R = a = 1 (Section 4). We do not know the value adopted by some astronomers for TY (e.g., Table 1, nos. 2, 4 and 5; Table 2, nos. 3 and 4). Nonetheless, re-computation by taking three different values for TY in the case of the Banū Mūsā’s two sets of the observational data and by taking two values for TY in the case of al-Marwarūdhī’s, for example, ensure that adopting a mean historical value TY = 365;14,30d may lead us to the secure values; i.e., those whose differences from the values resulted from adopting the other historical values for TY (which are all close to 365;14,30d) are within the desired accuracy of the present study. Although the values recomputed for e by the present author for those five cases (i.e., Table 1, nos. 2, 4 and 5; Table 2, nos. 3 and 4) do not deviate from Bīrūnī’s by more than ¼′, preference is given to the first, because of the very possible printing errors. However, as we shall see later in Section 4, adopting each of them does not cause any difference in the conclusion. Although in the case of Bīrūnī’s second measurement (Table 2, no. 4), the value 2;4,43… may be assumed the printing error of the recomputed value 2;4,23… ( →  ), nevertheless preference is given to the printed value because Bīrūnī appears to have adopted the value e = 2;4,39 in order to construct his own table of the solar equation of centre,34 which, roughly speaking, seems to be the average of his measured values by the Seasons Method, 2;3,36…, and the Mid-Seasons Method, 2;4,43…. Of course, choosing either 2;4,23 or 2;4,43 does not lead to any change in the conclusion (Section 4). The values adopted or reported for the solar eccentricity in the Islamic zījes can be deduced from the tabulated values of qmax, according to Formula (1). Also, in some sources like Sultānī zīj, Ashrafī zīj, and al-Zīj al-mu‘tabar, as previously referred to, some values used by the astronomers of different periods are listed.35 In the Ashrafī zīj also the complete tables for the solar equation from now lost early Islamic zījes, e.g., those of Ibn al-A‘lam (d. 985) and al-Nayrīzī (9th cent.), are presented. These known values for the solar eccentricity that have not yet been mentioned before are summarized in Table 3. Here, other than Muhyī al-Dīn’s value, there is no verified evidence to show that the values listed are the results of the application of the Three- Point Method. The value used by al-Khwārizmī belongs to the Pre-Islamic Persian astronomy (see below), and the values nos. 2 and 3 were likely measured either by the Seasons Method or by the Mid-Signs one.

Notes to Table 1 [1] Seemingly, the results of the measurements done simultaneously with (or, maybe, accompanied by) the first observational program in the period of al-Ma’mūn (Table 2, nos. 1 and 2). Bīrūnī’s source is (the now lost) On the solar year (Fī sanat al-shams) of the Banū Mūsā (the eldest brother, Abū Ja‘far Muham- mad, died in Jan.–Feb. 873). Bīrūnī says36 that this treatise is attributed to Thābit b. Qurra (a.d. 836–901). 328 S. Mohammad Mozaffari

(Banū Mūsā were the supporters and supervisors of Thābit in his first steps of scholarly career.) However, Ibn al-Nadīm37 and Ibn Yūnus38 ascribed Fī sanat al-shams to Thābit. It is probable that this treatise is the same De anno solis preserved in Latin and generally attributed to Thābit b. Qurra.39 Ibn Yūnus40 gives the values measured by Banū Mūsā as the solar mean motion in a Persian year = 359;45,39,58,2d (correspond- d ing to TY ≈ 365;14,32,33 ), qmax= 2;0,50, λap = 80;44,19° (the year of measurement is missing from the manuscript), which is close enough to the value 80;45° found in De anno solis,41 and the rate of the solar apogee’s motion = 1°/66y. Bīrūnī first gives the results calculated on the basis of a solar year of 365;15d length (= Julian year) and then based on the tropical year of 365;14,24 (= 365.25 – 1/100) days, attributed

to Banū Mūsā / Thābit. Therefore, the two sets of values for e and λap are presented. It seems that in the second calculation Bīrūnī made a mistake. The values recomputed by Bīrūnī are slightly different from (although more exact than) the value formally associated with Banū Mūsā / Thābit. Three sets of values in the last two columns are recomputed based upon the three values for the length of year, respectively (up to down), Banū Mūsā’s TY = 365;14,33d, Thābit’s TY = 365;14,24d, and the Julian year of 365;15d. [2] Bīrūnī quotes the result of this measurement by al-Ma’mūn’s astronomers from (the now lost) Epitome of the Almagest (Tafsīr al-majistī) of Abū Ja‘far al-Khāzin (d. c. 970–980).42 For this measurement, Bīrūnī says, the Ptolemaic solar year length of 365;14,48d was used. The values recomputed in the last two columns are the results of applying, respectively, Ptolemy’s value and the mean historical value 365;14,30d for TY. This was the second observational program performed during al-Ma’mūn’s reign, which was carried out on Mt Qāsiyūn in the vicinity of Damascus after the caliph became displeased with the results of the observational program supervised by Yahya b. Abī Mansūr in the Shammāsiyya quarter of Baghdad (see Table 2, nos. 1 and 2).43 Ibn Yūnus44 associates the value e = 2;5,32 with the observations in Damascus 45 (corresponding to qmax = 1;59,54° which is close to the value 1;59,56° attributed to Yahyā in some sources),

and gives e = 2;4,35 (corresponding to qmax = 1;59°) as the value found in the al-Ma’mūnic observations by Yahyā in Baghdad (see below, Table 2, nos. 1 and 2). This may explain the reason why the two values are found in connection with Yahyā whose zīj was taken to represent the Mumtahan tradition. Still, there is the other (fourth) value measured in this period, by Abū al-Qāsim Ahmad b. Mūsā b. Shākir in 220 Yazdigird (= a.d. 851–2); see Table 3, no. 2. Ibn Yūnus gives the difference between the longitude of apogee computed in al-Ma’mūnīc observations in Baghdad and in Damascus as 3°. From the value λap = 82;39° for the Baghdad observations (see Table 2, nos. 1 and 2), the value measured in the Damascus

observations for λap appears to have been about 80°.

[3] The value Battānī observed is e = 2;4,45, on the basis of which he derived qmax = 1;59,10. He applied it to his table of the solar equation of centre.46 Battānī’s value for TY is 365;14,26d. He performed the solar observations after the year 1194 Alexander (after a.d. 882). His value for e is repeatedly referred to in Islamic astronomical treatises47 and also in the Western literature, e.g., Copernicus’s De revolutionibus.48 [4] None of the writings of Ibn ‘Ismat of Samarqand has been preserved. Nevertheless, Bīrūnī in his works reported some of his astronomical activities and observations as well as the parameters determined by him.49 Calling him “diligent in research by extreme ability”, Bīrūnī shows a considerable reliance on his observations.50 [5] From Abu al-Wafā no independent work discussing the solar parameters is known (they are reported in Bīrūnī’s al-Qānūn).51 His astronomical tables are lost.52 [6] In the same year, 1016, Bīrūnī re-determined the solar eccentricity by means of the Mid-Signs Method (cf. Table 2, no. 5). [7] Abū Ja‘far Muhamamd b. Tabarī observed TY = 365d 5h 46m and the time interval from the vernal equinox to the autumnal equinox as 186d 15h 13m (true: 186d 13h 52m). He then computed e = 2;4,45 (–2″ in error), the same as Battānī’s value.53 [8] Besides the anonymous Sultānī zīj, MS Iran, Parliament, no. 184 contains (on folios. 243v–244r) part of another anonymous treatise named On the proof of the solar equation (of centre), its (longitude of) apogee, its mean motion, and its eccentricity by the observations. In it, the author gives the result of his observations performed in 539 y (a.d. 1170–71) as follows: a rounded value for TY as 365d 5h 48m = 365;14,30d (which is identical to the value observed by al-Maghribī at the Maragha Observatory around one century later; see below, Table 3, no. 9) and another value for it as 365d 5h 46m 52s 27′′′, the time interval from the vernal equinox to the autumnal equinox as 186d 15h 24m (near to the value given by Tabarī; true: 186d 13h 52m), and the length of spring as mentioned in the Table (given in the text as 93d 9h 13m in Limitations of Methods 329

Abjad numerals and 93d 9h 39m in written-out numbers, both of which are scribal errors, as shown by the value given for the corresponding angle a in what follows). From them, he computes the angles a + b = 183;57,50° (+1″ in error) and a = 92;3,27°, and computes e = 2;4,35, which is, interestingly, the same

value obtained by Yahyā b. Abī Mansūr (see Table 2, no. 2), and λap = 87;48,54° (recomputed: …,14; at first sight, it seems to be a simple scribal error; however, this is not the case because the value given by our author agrees with the magnitude he gives later for the daily motion of the solar apogee). Our author compares his measured value with λap = 82;39° obtained by Yahyā in the first al-Ma’mūnic observations and, giving the time interval between the two as 124511 days, computes the daily motion of the apogee as 0;0,0,8,57,36,…° (≈ 1°/66y). Our author appears to achieve success, as the title of the treatise indicates, in proving that the values obtained during al-Ma’mūn’s observational program are correct.

Notes to Table 2

[1] and [2] The results of the first observational program in al-Ma’mūn’s day. Ibn Yūnus54 gives Yahyā’s value for the solar mean motion in a Persian year as 359;45,45,14° which corresponds to the value TY ≈ 365;14,27d. Another value attributed to the Mumtahan tradition is 359;45,39,54º in a Persian year, corresponding to TY ≈ 365;14,32,38d, which, as Ibn Yūnus notices,55 is very close to Banū Mūsā’s (Table 1, no. 1). The results of [1] are amazing, for λap is less than the ancient determinations, e.g., the Hipparchan and Ptolemaic value (65;30°) and the Indian values. Note that if the values given are the lengths of the western and southern quadrants, as Bīrūnī says, then from the corresponding arcs 93.445° and 87.326°,

it follows that the vector of eccentricity is located in the western quadrant, e = 3;13,40, and λap = 142;11°.

Rather, it seems they should be the lengths of the northern and western quadrants; if so, λap = 52;11°, i.e., prior in longitude to the orientation determined by Ptolemy. The source of error in this determination, as Bīrūnī noticed,56 was in the solar observation in the middle of the summer. Bīrūnī quotes the results of [2] from the (now lost) On the solar year (Fī sanat al-shams) of the Banū Mūsā / Thābit. The times of the observations, counted from the solar meridian transit, are given in fractions of a day (the magnitudes of error based on Baghdad local time within parentheses): mid-winter 30 Jan 832 A.D., 0;35,30d (–1;15h) mid-spring 1 May 832 A.D., 0;20,50 (+1;20 ) mid-summer 3 Aug. 832 A.D., 0;32, 5 (+2;39 ) Bīrūnī notes that the time intervals include the corrections due to the equation of time. For this change from the mean time interval (zamān al-mutlaq, lit. “absolute time”) to the true time interval (zamān al-mu‘addal, lit. “adjusted time”), he subtracts 0;0,5d (= 2 min) from the time interval of the Northern Quadrant (second line in row 2). (The true amount of the equation of time in the three above-mentioned periods of time is, respectively, –16, +5 and –4 minutes. This implies that one should add nine minutes to the time interval of the Northern Quadrant and subtract 21 minutes from the time interval of the Eastern Quadrant in order to obtain the true time intervals.) He then points out that this minor shift causes a significant change in the 57 values of e and λap, and so he again draws attention to the sensitivity of these methods. Based on the account given by Ibn Yūnus (see above, Table 1, no. 2), the value e = 2;4,35 was achieved

by Yahyā b. Abī Mansūr which is in agreement with the value qmax = 1;59° found in the manuscripts of the Mumtahan zīj58 and also in a, presumably, thirteenth-century recension of habash’s zīj.59 As we said 60 earlier, another source ascribes the value qmax = 1;59,56° to Yahyā, which, based on the account given by Ibn Yūnus (see Table 1, no. 2), was the achievement of the observations in Damascus and might have been entered into the latter editions of the Mumtahan zīj (Ibn al-Nadīm notes the two editions of this zīj).61 It is not known why Khwārizmī did not apply this new value to his zīj and preferred to stay with the

Old Persian value 2;20 (corresponding to qmax ≈ 2;14°); see Table 3, no. 1. The value given by Yahyā 62 (frequently mentioned by Ibn Yūnus) for λap is 82;39°. [3] and [4] The values Bīrūnī quotes from Abu al-Wafā and al-Saghānī are mentioned nowhere else. [5] Bīrūnī mentions that he repeated three times the measurement of the time interval of the Northern Quadrant and that of the Western Quadrant in Jurjāniyya and obtained two set of values (see the two first lines in row [5]). He attributes the great deviation in these values to a defect in the instrument used as well as to his inability to determine a single amount. He then mentions that since it was impossible for 330 S. Mohammad Mozaffari him to repeat his measurements in Ghazni, he decided to determine the eccentricity using the results of the Mid-Signs Method found in Khwārizmī. Bīrūnī only reports in detail the measurement of the solar meridian altitude on 30 April 1016 for determining the moment of mid-spring.63

Notes to Table 3

64 [1] His values for e and λap both belong to the pre-Islamic Persian astronomy (see Part 2). [2] The solar mean motion in a Persian year is given as 359;45,40d (seemingly, the rounded Banū Mūsa’s value: Table 1, no. 1). The value for eccentricity is very close to the value obtained by Ibn al-A‘lam around one century later.65 [3] Two zījes presumably written by al-Nayrīzī have not been preserved.66 This value is also quoted in the Ashrafī zīj67 and in the prologue of al-Khāzinī’s Wajīz.68 Al-Khāzinī says that he has quoted the value from al-Nayrizī’s Majistī, which is also lost now. Bīrūnī reports some confusion in al-Nayrīzī’s works concerning the motion of the solar apogee. The latter’s value for e belongs to the pre-Islamic Persian astronomy (see Part 2) and for λap is the same as Yahyā’s (Table 2, nos. 1 and 2). 69 70 [4] Ibn Yūnus says that in the Badī‘ zīj of Ibn Amajūr, the value qmax = 2;0,50 (the same value found by Banā Mūsā; see Table 1, no. 1) and the magnitude of the solar mean motion in a Persian year as 359;45,39,45d (corresponding to TY ≈ 365;14,32,47d) are to be found while Muflih b. Yūsuf (an assist- ant to Ibn Amājūr) ascribes qmax = 2;0,20 and λap = 84;5° to Ibn Amajūr. The year of measurement is not indicated. The value of eccentricity is equivalent to that al-Maghribī found at Maragha around three-and- a-half centuries later (below, no. 9). [5] Ibn al-A‘lam’s Zīj al-‘Adūdī is now lost. His value is quoted by Shīrāzī, who calls it “the value that the recent predecessors/moderns have obtained by means of observations”.71 Shīrāzī gives the value e = 2;5,51 for the corresponding eccentricity. It is not known whether this value was explicitly given by Ibn al-A‘lam or whether Shīrāzī computed it from the maximum equation of centre. Ibn al-A‘lam’s table for the solar equation of centre is preserved in the Ashrafī zīj.72 [6] and [10] The table of the solar equation of centre may be found in Ibn Yūnus’s zīj.73 This value was used in the Īlkhānī zīj.74 [7] Khāzinī’s table for the Sun’s equation of centre is also preserved in the Ashrafī zīj.75 The solar apogee in al-Khāzinī’s Zīj al-Mu‘tabar.76 [8] The table of the solar equation of centre in the Ashrafī zīj 77 and the epoch value for the solar apogee 78 79 in al-Fahhād’s Zīj al-‘Alā’ī. In the Greek sources, λap = 87;50,43º. Kennedy lists six zījes attributed to al-Fahhād, five of which are lost.80 Only his Zīj al-‘Alā’ī seems to have attracted much attention from later astronomers and has been preserved in a Greek translation made by Gregory Chioniades in Tabriz around the 1290s.81 Recently a partial Persian manuscript of the ‘Alā’ī zīj has been identified in the Salar Jung Library in Hyderabad (no. H17). Parts of it are contained in some later zījes, e.g. the Nāsirī zīj written by a certain Muhammad b. ‘Umar (India, c. 1250) and the Muzaffarī zīj by al-Fārisī (Yemen, c. 1270).82 It may be worth while to note here that one of the six zījes of al-Fahhād, the so-called Zīj al-Mughnī,83 is mentioned by Kamālī as a zīj by a certain Muntakhab al-Dīn Sakkāk (or Hakkāk), an astronomer of Yazd.84 [9] This value was measured on the basis of the direct observations made in the Maragha Observatory (northwestern Iran) in the 1260s. Al-Maghribī explains his observations and calculations in his Talkhīs al-Majistī IV, 5.85 Note that there exists a systematic error of around +4′ in his measurements of the solar altitude which, as seen earlier (Section 2), was the case with his vernal equinox observations, too.86 Since his value for the latitude of Maragha is in error by around –3′, the error in the solar declination is reduced, and a mean error of –1′ occurs in the solar longitude. In Talkhīs III, 1, he compared his equinox observations (Section 2) with those by Yahyā, al-Marwarudhī, …, and Ptolemy to determine the length of tropical year about the rounded value 365;14,30d.87

h ob- h δ δ λ λ time of solar meridian JDN served true from h true from h true transit (MLT) 8 Sep 1264 h ob-55;29 h55;24,53 δ +2;49,30 +2;47,58δ 172;53,59λ 172;58,16λ time of solar11h meridian 55m JDN2182985 26 Oct 1264 served37;35 true37;31, 2 from–15; h 4,30 –15;true 6,32 from220;42,44 h true220;44,25 transit (MLT)11h 45m 2183033 8 Sep5 1264Mar 126555;2949;36,3055;24,5349;32;34 +2;49,30 –3; 3, 0 +2;47,58 –3; 4,33 172;53,59352;19,54 172;58,16352;16,49 11h 55m12h 11m 21829852183163 26 Oct 1264 37;35 37;31, 2 –15; 4,30 –15; 6,32 220;42,44 220;44,25 11h 45m 2183033 5 Mar 1265 49;36,30 49;32;34 –3; 3, 0 –3; 4,33 352;19,54 352;16,49 12h 11m 2183163 Limitations of Methods 331

[11] Ibn al-Shātir’s solar model is non-Ptolemaic,88 in which the solar equation of centre does not reach its maximum value for the same mean anomaly in the Ptolemaic model but Formula (1) is the case with computing the value of e in the corresponding Ptolemaic model. Amazingly, the solar apogee as determined by him is behind the longitudes observed by his Islamic predecessors. [12] This value stems from the table of the solar equation of centre in al-Kashī’s Khāqānī zīj.89 It appears to be the same value obtained by Ibn Yūnūs and also used in the Ilkhānī zīj. In the later stage of his career, al-Kāshī worked in the Samarkand Observatory, where a new value of remarkable exactness was observed for the Sun’s eccentricity and was applied in Ulugh Beg’s Sultānī zīj (see below, no. [13], and Part 2). Al-Kashī died before this zīj appeared. The longitude of the solar apogee in the beginning of the year 712 y is mentioned at the beginning of the table of equation of time.90 Its values for the years 781–90 y are given relative to the longitudes of the planetary apogees.91 92 [13] The value from the table for the equation of centre in the Ulugh Beg’s zīj: qmax=1;55,53,12. The corresponding eccentricity is exactly 2;1,20. Our author also tabulates the distance of the Sun from the Earth as a function of its mean anomaly in a separate table93 in which 62;1,20 (corresponding to R + e) is given when the Sun is at its apogee.94

(Part 2 of this article will appear in our November issue.)

Acknowledgements This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project no. 1/2639. The preliminary research proposal of this paper was read by Prof. George Saliba and Prof. Julio Samsó. Dr Benno van Dalen read drafts of this paper and sent me very helpful comments and objections. To all of these scholars I extend my sincerest thanks for their encouragements, useful criticisms, and suggestions. references 1. For the solar eccentric model as presented in Almagest III (G. J. Toomer, Ptolemy’s Almagest (Princeton, 1998), 133–72), cf. O. Neugebauer, A history of ancient mathematical astronomy (Berlin, 1975), i, 54–61; O. Pedersen, A survey of the Almagest (Odense, 1974), chap. 3. For its evolution through pre-Ptolemaic data, cf. A. Jones, “Hipparchus’s computations of solar longitudes”, Journal for the history of astronomy, xxii (1991), 101–25, and D. Duke, “Four lost episodes in ancient solar theory”, Journal for the history of astronomy, xxxix (2008), 283–96. 2. Nicolaus Copernicus, De revolutionibus orbium coelestium (Nuremberg, 1543), folios. iiiR-iiiV. 3. E.g., if TY is measured from one vernal equinox to the next, it increases with the average rate of 1 second per century (s/cent) from a.d. 0 to 2000. Or if it is counted from one autumnal equinox to the next, it decreases with the average rate 2 s/cent in this period. The lesser changes in TY will result, if it is counted with reference to solstices: from one summer solstice to the next: –0.4 s/cent; from one winter solstice to the next: –0.6 s/cent. (Cf. J. Meeus, and D. Savoie, “The history of the tropical year”, Journal of the British Astronomical Association, cii (1992), 40–2, and J. Meeus, More mathematical astronomy morsels (Richmond, 2002), 357–66, which give some useful notes about the length of year.) Since ancient and medieval astronomers gave their values for TY in days and sexagesimal fractions up to seconds (i.e., 1/3600th of a day), they are exact, at most, to ±24 seconds. Also, they measured the tropical year with the autumnal equinox as the zero-point over long periods, thus the magnitudes achieved by them may be considered as the ‘mean/average’. The mean/average values for each of four types of the tropical year from a.d. 0 to 2000, expressed in days and sixagesimals, are as follows: Two vernal equinoxes: 365;14,32 days 332 S. Mohammad Mozaffari

Two summer solstices: 65;14,30 days, Two autumnal equinoxes: 365;14,32 days, Two winter solstices: 365;14,34 days. Some of the historical values for the length of year are as below: Hipparchus/Ptolemy (2nd cent.): 365.25 – 1/300 = 365;14,48 days, Thābit b. Qurra (9th cent.): 365.25 – 1/100 = 365;14,24, Al-Battānī (10th cent.): 365;14,26, Muhyī al-Dīn al-Maghribī (13th cent.): 365.25 – 1/120 = 365;14,30. 4. Meeus, op. cit. (ref. 3), 361. 5. Cf. Neugebauer, op. cit. (ref. 1), iii, 1097–102; Y. Maeyama, “Determination of the Sun’s orbit: Hipparchus, Ptolemy, al-Battânî, Copernicus, Tycho Brahe”, Archive for history of exact sciences, liii (1998), 1–49, p. 4. 6. The intention to bisect the Ptolemaic solar eccentricity may be traced out in medieval astronomy (cf. Bernard R. Goldstein and Frederick W. Sawyer, “Remarks on Ptolemy’s equant model in Islamic astronomy”, in Y. Maeyama and W. G. Salzer (eds), Prismata: Festschrift für Willy Hartner (Wiesbaden, 1977), 165–81) and may be found, for example, in Ibn al-Shātir’s solar model (cf. G. Saliba, “Theory and observation in Islamic astronomy: The work of Ibn al-Shātir of Damascus”, Journal for the history of astronomy, xviii (1987), 35–43, p. 42). 7. For the analysis of the error, cf. R. R. Newton, “The authenticity of Ptolemy’s parallax data – Part 1”, Quarterly journal of the Royal Astronomical Society, xiv (1973), 367–88, pp. 369–70. Seven centuries later, this caused a remarkable error in measuring the rate of precession in early Islamic astronomy; see G. Grasshoff, The history of Ptolemy’s star catalogue (London, 1990), 19–20. 8. Cf. Toomer, op. cit. (ref. 1), 153–7, 190–203, and 469f; G. Saliba, “Solar observations at Maragha observatory”, Journal for the history of astronomy, xvi (1985), 113–22, p. 114; Duke, op. cit. (ref. 1), 284–5, 291. 9. A. Bīrūnī, The chronology of ancient nations, transl. by C. Edward Sachau (London, 1879), 167. 10. Cf. G. Saliba, “The determination of the solar eccentricity and apogee according to Mu’ayyad al-Dīn al-‘Urdī”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, ii (1985), 47–67. 11. Cf. François Charette, “The locales of Islamic astronomical instrumentation”, History of science, xliv (2006), 123–38. 12. Mūhyī al-Dīn Al-Maghribī, Talkhīs al-majistī, MS Leiden, no. Or. 110, III, 1: fol. 58r. 13. Measured by the altitude observations in the three consecutive days near the solstice days of 1264: 12–14 June: h = 76;9,30º (modern: 76;8,30º) and 7–9 December: h = 29;9,30º (modern: 29;10,30º).

The modern value for the mean obliquity of the ecliptic at the time: ε0 ≈ 23;32º. Note that these altitude observations are about ±1′ in error while his equinox altitudes and other three altitudes given below (see Table 3, no. 9), are about +4′ in error. 14. Abu al-Rayhān al-Bīrūnī, al-Qānūn al-mas‘ūdī (3 vols, Hyderabad, 1954), ii, 621–3. 15. S. S. Said and F. R. Stephenson, “Precision of medieval Islamic measurements of solar altitudes and equinox times”, Journal for the history of astronomy, xxvi (1995), 117–32, p. 123. 16. Said and Stephenson, op. cit. (ref. 15), 123. 17. Bīrūnī, op. cit. (ref. 14), ii, 621. 18. Cf. Edmond Halley, “A discourse concerning a method of discovering the true moment of the Sun’s ingress into the tropical sines”, in Miscellanea curiosa; Containing a collection of some of the principal phaenomena in nature, accounted for by the greatest philosophers of this age (London, 1726), ii, 202–10. 19. Cf. Said and Stephenson, op. cit. (ref. 15), 122–3 (Table 3). 20. Bīrūnī, op. cit. (ref. 14), ii, 653. 21. It is frequently and wrongly named the fusūl (= seasons) method in the secondary literature while it is evidently and correctly called ansāf al-fusūl or awsāt al-burūj in Bīrūnī’s work (Bīrūnī, op. cit. (ref. 14), ii, 657). Limitations of Methods 333

22. The printed text is not clear here (Bīrūnī, op. cit. (ref. 14), ii, 619, lines 5–6) and may read 31;18° + 1»′ + »′ as Said and Stephenson (op. cit. (ref. 15), 126, n. 6) did. Nevertheless, it should be considered that the altitude 31;19,40° results in the time of mid-autumn being 0;8,53 hours after the true noon of 1 November 1016, which by no means agrees with the value 0;4,30 hours explicitly mentioned in the text. As a result, I prefer to assume that, in spite of the confusing phrase in line 6 (more likely, due to the clear fact that we do not have a trustworthy edition of al-Bīrūnī’s work at our disposal), the value al-Bīrūnī wanted to mention is 31;18° + »′. 23. Cf. G. J. Toomer, “The solar theory of az-Zarqāl: An epilogue”, in D. King and G. Saliba (eds), From deferent to equant (Annals of New York Academy of Sciences, d; New York, 1987), 513–19, p.

516. In other sources, the varied values between qmax=1;52 and 1;53º, corresponding to ½e = 0.01629 – 101643 (R = 1), are associated with Ibn al-Zarqālluh; cf. J. Samsó, “Al-Zarqal, Alfonso X and Peter of Aragon on the solar equation”, in King and Saliba (eds), op. cit., 467–76, p. 471.

The value λap = 85;49º is in agreement with the later Islamic sources, e.g., Ibn al-Hā’im; cf. E. Calvo, “Astronomical theories related to the Sun in Ibn al-Hā’im’s al-Zīj al-Kāmil fī ’l-Ta‘ālīm”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, xii (1998), 51–111, p. 55. 24. Cf. N. M. Swerdlow, “Tycho, Longomontanus, and Kepler on Ptolemy’s solar observations and theory, precession of the equinoxes, and obliquity of the ecliptic”, in A. Jones (ed.), Ptolemy in perspective (Archimedes, xxiii; Dordrecht, 2010), 151–202, p. 155. 25. Copernicus, op. cit. (ref. 2), fols. 87v–88v. 26. J. L. E. Dreyer, Tycho Brahe: A picture of scientific life and work in the sixteenth century (Edinburgh, 1890), 333; V. E. Thoren, The Lord of Uraniborg: A biography of Tycho Brahe (Cambridge, 1990), 224. 27. Cf. Saliba, op. cit. (ref. 8). 28. Ptolemy pointed to the sensitivity of the method for the input data in the Almagest, IV, 11, where he applies it to determine the radius of the Moon’s epicycle (Toomer, op. cit. (ref. 1), 215–16). 29. Bīrūnī (op. cit. (ref. 14), iii, 1193) declares his discontent with his Islamic predecessors’ attitude of not explaining their observational data and measurement procedures as Ptolemy mentioned his procedures, and adds that the one who elucidates his computational procedures is the most worthy to be followed. 30. Bīrūnī, op. cit. (ref. 14). Sultānī zīj, MS Iran, Parliament, no. 184. Muhammad b. Abī ‘Abd-Allāh Sanjar al-Kamālī (known as “Sayf-i munajjim”), Ashrafī Zīj, MS Paris, Bibliothèque Nationale, no. 1488. The latter two were written contemporarily around the turn of the 14th century, the first in Yazd and the latter in Shīrāz (central Persia). The first is different from both Ulugh Beg’sSul tānī zīj (called also Gūrkānī zīj) and Shams al-Dīn Muhammad al-Wābkanawī’s Zīj al-Muhaqqaq al-sultānī (completed around 1316–24; MSS (1) Turkey, Aya Sophia Library, no. 2694, (2) Iran, Yazd, Library of ‘Ulūmī, no. 546, its microfilm being available in Tehran University central library, no. 2546, and (3) Iran, Library of Parliament, no. 6435). It is attributed to Qutb al-Dīn al-Shīrāzī (E. S. Kennedy, A survey of Islamic astronomical tables (Philadelphia, 1956), no. 25); however, in the canons of this zīj (e.g., fols. 78r, 141r, 142r), the author mentions that he was based in Yazd, and thus, since al-Shīrāzī is not known to have worked in Yazd, he is less likely to be identified as its author. Based on Kennedy (ibid., no. 32), the Sultānī zīj may be identical with the Shāhī zīj by a thirteenth-century astronomer named husām al-Dīn al-Sālār (killed by Hülegü, the first patron of the Ilkhanid dynasty of Iran, on 8 Muharram 661 h / 22 Nov 1262; cf. Rashīd al-Dīn Fadl Allāh al-Hamidānī, Jāmi‘ al-Tawārīkh [The compendium of histories], ed. by M. Rushan and M. Mūsawī (4 vols, Tehran, 1994), ii, 1045). However, there are direct references in the first to the latter as well as some tables of the latter in the first (e.g., the table for the equation of time on fol. 7v). Therefore, one can by no means accept the two to be identical. 31. Bīrūnī, op. cit. (ref. 14), ii, 650–61. 32. Sultānī zīj (ref. 30), fol. 77r; Kamālī, op. cit. (ref. 30), fol. 49r; ‘Abd al-Rahmān al-Khāzinī, Wajīz [Compendium of] al-Zīj al-mu‘tabar al-sanjarī, MS Istanbul, Suleymaniye Library, Hamadiye collection, no. 859, fol. 1v. Tables in Kamālī, op. cit. (ref. 30), fols. 236r–v, 238r–239r, 240r, 241v, and 243r. 334 S. Mohammad Mozaffari

33. Calculated based on J. Meeus, Astronomical algorithms (Richmond, 1998), chap. 25, and Alcyone Software. 34. The table is asymmetric and has max = 3;59,3,21º for mean anomaly 266º (Bīrūnī, op. cit. (ref. 14), ii, 716) and min = 0;0,56,39º for mean anomaly 90º (Bīrūnī, op. cit. (ref. 14), ii, 710), and so

qmax = 1;59,3,21º which strictly corresponds to e = 2;4,39. 35. The values for the maximum solar equation of centre used in the pre- and early Islamic periods are also mentioned in the two sources and in the prologue of al-Khāzinī, op. cit. (ref. 32), fol. 1v: “Ancient observations”: 2;24°; Ptolemy: 2;23°; Persians: 2;14°; and Indians: 2;11°. The value 2;23º in the Almagest is drawn from e = 2;30 calculated based on the Hipparchian values for the length of seasons. The solar eccentric model more likely existed before Hipparchus and it is probable that Apollonius knew this model (Pedersen, op. cit. (ref. 1), 135, 340f). The term ‘ancient observations’ is thus interesting; the value associated with it can be found in the papyrus PSI inv. 515 (cf. A. Jones, “Studies in the astronomy of the Roman period, IV: Solar tables based on a non-Hipparchian model”, Centaurus, xlii (2000), 77–88). According to Jones (p. 85), the treatise of which this papyrus is a fragment was composed before the publication of the Almagest. The value 2;24° is not significantly different from Ptolemy’s and, as Jones argues, it might have been computed from the Hipparchian eccentricity 2:30 with the use of a table of chords tabulated for fewer angles than Ptolemy’s. It is, nonetheless, interesting that it was transmitted to medieval astronomy as a value that pre-dates Ptolemy’s. The value attributed to the Iranians is that used in the Shāh zīj (written in Sāsānīd Persia, a.d. 224–651); see Part 2. 2;11° is indeed found in the Indian traditions from the year 400 onwards (cf. D. Pingree, “On the classification of Indian planetary tables”, Journal for the history of astronomy, i (1970), 95–108, esp. pp. 97–8, 100, 105). 36. Bīrūnī, op. cit. (ref. 14), ii, 654. 37. Abu al-Faraj Muhammad Ibn Ishāq al-Nadīm, Kitāb al-fihrist, ed. by Ridā Tajaddud (Tehran, 1971), 331. (For the English translation, cf. The fihrist of al-Nadīm, ed. and transl. by Bayard Dodge (2 vols, New York, 1970). 38. ‘Alī b. ‘Abd al-Rahmān b. Ahmad Ibn Yūnus, Zīj al-kabīr al-hākimī, MS Leiden, Orientalis, no. 143, pp. 103, 106, 107. 39. Cf. B. R. Goldstein, “Historical perspectives on Copernicus’s account of precession”, Journal for the history of astronomy, xxv (1994), 189–97, p. 190; B. R. Goldstein and J. Chabás, “The maximum solar equation in the Alfonsine Tables”, Journal for the history of astronomy, xxxii (2001), 345–8, p. 345. 40. Ibn Yūnus, op. cit. (ref. 38), 104. 41. Cf. O. Neugebauer, “Thābit ben Qurra ‘On the Solar Year’ and ‘On the Motion of the Eighth Sphere’”, Proceedings of the American Philosophical Society, cvi (1962), 264–99. 42. Bīrūnī, op. cit. (ref. 14), ii, 653. 43. For the instruments utilized in the two observations and their accuracies, cf. Charette, op. cit. (ref. 11), 125. 44. Ibn Yūnus, op. cit. (ref. 38), 124. 45. E.g., Kamālī, op. cit. (ref. 30), fol. 236r. 46. C. A. Nallino (ed.), Al-Battani sive Albatenii: Opus astronomicum (3 vols, Milan, 1899–1907), i, 4; ii, 79–83. 47. E.g., Nasīr al-Dīn al-tūsī, Risāla al-mu‘īniyya fī ‘ilm al-hay’a, MS Tehran, Parliament, no. 6347, p. 42. 48. Copernicus, op. cit. (ref. 2), fol. 87v. 49. His solar parameters in Bīrūnī, op. cit. (ref. 14), ii, 654. 50. Bīrūnī, op. cit. (ref. 14), ii, 659. 51. Bīrūnī, op. cit. (ref. 14), ii, 654–5. 52. Cf. Kennedy, op. cit. (ref. 30), nos. 29 and 73. 53. Cf. E. S. Kennedy, “Applied mathematics in the eleventh-century Iran: Abu Jafar’s determination of the solar parameters”, Mathematics teacher, lviii (1965), 441–6, reprinted in idem, Studies in the Islamic exact sciences (Beirut, 1983), 535–40. Limitations of Methods 335

54. Ibn Yūnus, op. cit. (ref. 38), 120. 55. Ibn Yūnus, op. cit. (ref. 38), 106. 56. Bīrūnī, op. cit. (ref. 14), ii, 656–7. 57. Bīrūnī, op. cit. (ref. 14), ii, 658. 58. E. S. Kennedy, “The solar equation in the Zīj of Yahya b. Abī Manūr”, in Maeyama and Salzer (eds), Prismata (ref. 6), 183–6, p. 185, reprinted in idem, Studies in the Islamic exact sciences (ref. 53), 136–9; B. van Dalen, “A second manuscript of the Mumtahan Zīj”, Suhayl, iv (2004), 9–44, p. 22. 59. The zīj ascribed to habash al-hāsib, MS. Berlin, no. Ahlwardt 5750 (formerly no. Wetzstein I 90), fol. 30v. 60. Kamālī, op. cit. (ref. 30), fol. 236r. 61. Ibn al-Nadīm, op. cit. (ref. 37), 334. 62. Ibn Yūnus, op. cit. (ref. 38), 87, 91, 120. 63. For an analysis of it, cf. Said and Stephenson, op. cit. (ref. 15), 129. 64. O. Neugebauer, The astronomical tables of Al-Khwārizmī (Historiskfilosofiske Skrifter undgivet af Det Kongelige Danske Videnskabernes Selskab, iv/2; Copenhagen, 1962), 20. 65. Ibn Yūnus, op. cit. (ref. 38), 104. 66. Kennedy, op. cit. (ref. 30), nos. 46 and 75. 67. Kamālī, op. cit. (ref. 30), fol. 49r. 68. Khāzinī, op. cit. (ref. 32), fol. 1v. 69. Ibn Yūnus, op. cit. (ref. 38), 105. 70. Kennedy, op. cit. (ref. 30), no. 8. 71. Qutb al-Dīn al-Shīrāzī, Ikhtīyārāt-i Muzaffarī, MS Iran, National Library, no. 3074f (copied in 7th cent. h / 13–14 cent. a.d., during the lifetime of the author, fol. 50v; Qutb al-Dīn al-Shīrāzī, Tuhfa al-Shāhiyya, MS Iran, Parliament, no. 6130 (copied in 730 h / 1329–30 a.d.), fol. 38v. 72. Kamālī, op. cit. (ref. 30), fol. 236v. 73. Ibn Yūnus, op. cit. (ref. 38), 174. 74. Nasīr al-Dīn al-tūsī, Īlkhānī zīj, MS C: University of California, pp. 60–5, MS T: Iran, University of Tehran, hikmat Collection, no. 165, fols. 28r–30v, MS P: Iran, Parliament Library, no. 181, fols. 21v–23r. 75. Kamālī, op. cit. (ref. 30), fol. 238r. 76. ‘Abd al-Rahmān al-Khāzinī, al-Zīj al-mu‘tabar al-sanjarī, MS Library of Vatican, no. Arabo 761, p. 266. 77. Kamālī, op. cit. (ref. 30), fol. 240r. 78. Al-Fahhād al-Dīn Abu al-hasan ‘Alī b. ‘Abd al-Karīm Al-Fahhād of Shirwān, Zīj al-‘Alā’ī, MS India, Salar Jung, no. H17, p. 73. 79. D. Pingree (ed.), Astronomical works of Gregory Chioniades (3 vols, Amsterdam, 1985), ii, 32. 80. Kennedy, op. cit. (ref. 30), no. 84. 81. Cf. Pingree, op. cit. (ref. 79). 82. Cf. B. van Dalen, “The Zīj-i Nasirī by Mahmūd ibn Umar: The earliest Indian zij and its relation to the ‘Alā’ī Zīj”, in Charles Burnett et al. (eds), Studies in the history of the exact sciences in honour of David Pingree (Leiden, 2004), 327–56, p. 329. 83. Kennedy, op. cit. (ref. 30), no. 64. 84. Kamālī, op. cit. (ref. 30), fol. 52v. 85. Al-Maghribī, op. cit. (ref. 12), fols. 58v–61v; cf. Saliba, op. cit. (ref. 8). 86. I intend to present an analysis of Muhyī al-Dīn’s meridian altitude observations in a separate paper. 87. Al-Maghribī, op. cit. (ref. 12), fols. 58r–v. 88. Cf. V. Roberts, “The solar and lunar theory of Ibn ash-Shātir: A pre-Copernican Copernican model”, Isis, xlviii (1957), 428–32, p. 430; Saliba, op. cit. (ref. 6). 89. Ghiyāth al-Dīn Jamshīd al-Kāshī, Khāqānī zīj, MS P: Iran, Parliament, no. 6198, p. 114, IO: London, 336 S. Mohammad Mozaffari

India Office, no. 430, fol. 131r. 90. Kāshī, op. cit. (ref. 89), IO: fol. 127r. 91. Kāshī, op. cit. (ref. 89), IO: fol. 128v. 92. Ulugh Beg, Sultānī Zīj, MS T: University of Tehran, no. 13, fols. 110r–115v, MS P1: Iran, Parliament, no. 72, fols. 117v–123r, MS P2: Iran, parliament, no. 6027, fols. 134v–140r. 93. Ulugh Beg, op. cit. (ref. 92), P1: fol. 123v, P2: fol. 141r. 94. The solar apogee’s longitudes are given in Ulugh Beg, op. cit. (ref. 92), P1: fol. 116r and P2: fol. 133r.