The orbital elements of each planet are the eccentricity and the direction of the apsidal line of its orbit defined by the ecliptic longitude of either of its apses, i.e., the two points on its orbit where the planet is either furthest from or closest to the Earth, which are called the planet’s apogee and perigee. In the geocentric view of the solar system, the eccentricity of Venus is a bit less than half of the solar one, and its apogee is located behind that of the Sun. Ptolemy correctly found that the apogee of Venus is behind that of the Sun, but determined the eccentricity of Venus to be exactly half the solar one. In the Indian Midnight System of Āryabhaṭa (b. ad 476), the eccentricity of Venus is assumed to be half the solar one, and also the longitudes of their apogees are assumed to be the same. This hypothesis became prevalent in early medieval Middle Eastern astronomy (ad 800–1000), where its adoption resulted in large errors of more than 10° in the values for the longitude of the apogee of Venus adopted by Yaḥyā b. Abī Manṣūr (d. ad 830), al-Battānī (d. ad 929), and Ibn Yūnus (d. ad 1007). In Western Islamic astronomy, it was used in combination with Ibn al-Zarqālluh’s (d. ad 1100) solar model with variable eccentricity, which only by coincidence resulted in accurate values for the eccentricity of Venus. In late Islamic Middle Eastern astronomy (from ad 1000 onwards), Āryabhaṭa’s hypothesis gradually lost its dominance. Ibn al-A‘lam (d. ad 985) seems to have been the first Islamic astronomer who rejected it. Late Eastern Islamic astronomers from the middle of the thirteenth century onwards arrived at the correct understanding that the eccentricity of Venus should be somewhat less than half of the solar one. Its most accurate medieval value was measured in the Samarqand observatory in the fifteenth century. Also, the values for the longitude of the apogee of Venus show a significant improvement in late Middle Eastern Islamic works, reaching an accuracy better than a degree in Khāzinī’s Mu‘tabar zīj, Ibn al-Fahhād’s ‘Alā’ī zīj, the Īlkhānī zīj, and Ulugh Beg’s Sulṭānī zīj.
Note of Editor: This paper appeared for the first time in Journal for the History of Astronomy, 2019, Vol. 50(1) 46 –8. The PDF can be retrieved online via this link.
This paper contains, in its core, a case study on the interaction between the Ptolemaic, Indian, and medieval Islamic astronomical traditions. It particularly discusses on the consequences of the incorporation of a hypothesis by Āryabhaṭa in Ptolemy’s model for Venus as well as its use together with Ibn al-Zarqālluh’s solar model for the accuracy of the values adopted for the longitude of the apogee of Venus in medieval Islamic astronomy. Improving Ptolemy’s values for the planetary parameters was one of the main purposes of observational astronomy in the medieval Islamic period. Thus, another goal of our study is to determine the accuracy attained by Islamic astronomers in the measurement of the orbital elements of Venus in the different periods and the various domains of the Islamic realm.
This paper is organized to contain the following sections. In section “Ptolemy’s model for Venus,” we introduce Ptolemy’s model for Venus and the values he measured for its orbital elements. In section “Derivation of the geocentric orbital elements of Venus from the heliocentric parameters,” we discuss the derivation of the geocentric orbital elements of Venus from the heliocentric ones in order to compute their true values for the medieval period. This will enable us to evaluate the accuracy of the historical values. The reader may skip this technical section without missing anything indispensable for an understanding of the later discussion. In section “Medieval Islamic astronomers’ values for the fundamental parameters of Venus,” the historical values are chronologically classified and discussed with reference to both the medieval Islamic geographical domains from which they originate (either the Middle East or Western Islamic lands) and the way in which they were determined, i.e., from the interaction and/or combination of the traditions and hypotheses or from observations. In section “Discussion and conclusion,” the main findings of the study are discussed and summarized.
Ptolemy’s model for Venus is structurally similar to his model for the superior planets.1 The planet P (Figure 1) revolves counterclockwise, i.e., in the direction of increasing longitude, on an epicycle of radius r at a constant angular velocity relative to the mean epicyclic apogee A′m, which is a point on the circumference of the epicycle defined by the prolongation of the line which connects the centre E of the uniform motion (the so-called equant point) and the centre C of the epicycle. The centre C of the epicycle itself revolves in the direction of increasing longitude on a fixed eccentric with the radius OC, which according to Ptolemy’s norm has the arbitrary length R = 60 units. The centre O of the eccentric is displaced from the Earth T by the eccentricity OT = e1. The motion of C on the eccentric is uniform, at a constant angular velocity equal to the solar mean motion, with respect to the point E, which is displaced from O opposite to T by the eccentricity OE = e2. Accordingly, the vector extended from E to C points to the mean Sun. The line passing through T, O, and E defines the apsidal line of the eccentric, the apogee A on the side of E, and the perigee Π on the side of T.
In the case of both inferior planets, Ptolemy first determines the direction of the apsidal line by observations of, at least, two equal maximum elongations of either planet in opposite directions, once as a morning star (i.e. at a maximum western elongation) and another time as an evening star (i.e. at a maximum eastern elongation). Such a situation clearly indicates that the centre of the epicycle occupies at both instances symmetrical positions with respect to the apsidal line, which thus passes midway through the ecliptic arcs between the two longitudes of Venus (which can be directly derived from the observations) or those of the mean Sun (which can be calculated from an adopted solar theory). Next, in order to determine which of the two apses marked by the direction of the apsidal line stands for the apogee/perigee, one requires two additional observations of the maximum elongations, when the centre of the epicycle, i.e., the mean Sun, is located at either of the two apses. It is clear that the maximum elongation of an inner planet when at the apogee (A in Figure 1) would be less than at the perigee (Π). From these latter two observations, the eccentricity e1 of the eccentric and the radius r of the epicycle can be derived as well. The eccentricity of the equant point from the Earth (e1 + e2) is computed from the maximum elongation of the planet from the mean Sun when the centre C of the epicycle is at the orbital quadratures (i.e. ∠AEC = 90° or 270°).2
In Almagest X.1−3,3 Ptolemy finds that the two eccentricities e1 and e2 are identical: e = e1 = e2 = 1;15. This value is also equal to half of his value for the eccentricity of the Sun/Earth in Almagest III.44 (all values given for the eccentricities in this paper are according to Ptolemy’s norm R = 60). Moreover, Ptolemy determines the apogee of Venus as being located at a longitude of 55° behind the solar apogee, which Ptolemy derived to be tropically fixed at a longitude of 65.5°. Ptolemy’s value for the longitude of the apogee of Venus has an error of about −2.5°.5
Each planet in our solar system revolves around the Sun on an elliptical orbit with a semi-major axis a and a semi-minor axis b. The Sun is located in one of the two foci of the orbit. The distance of either of the foci from the centre of the orbit is indicated by c. The eccentricity of an ellipse is defined as e = c/a. Hence, the distance between the Sun and the centre of the orbit is c = ea. There is a conventional difference in the concept of the eccentricity between ancient and modern astronomy. As we have seen in the previous section, in ancient astronomy, the eccentricity indicates the “distance” of the centre of the circular orbit from the central body (the Earth) or that between the equant point and the centre of the orbit or the central body in terms of the same arbitrary length assigned to the radius of the orbit of all planets. In modern astronomy, it stands for the “ratio” between the distance of the centre of the orbit from the central body (the Sun) and the semi-major axis of the orbit. The extension of the semi-major axes of the orbit of a planet forms its apsidal line, whose direction with respect to the zero point in a reference system of coordinates (e.g. the vernal equinox in the tropical reference system) represents the spatial orientation of its orbit. The apsis denoting its greatest distance from the Sun is the aphelion (apogee in the case of the Earth) and the other one diametrically opposed to the aphelion/apogee, showing its least distance from the Sun, is the perihelion/perigee.
For the derivation of the geocentric orbital elements of a planet from the heliocentric ones, the following considerations should be taken into account:6
1. The eccentricity of the geocentric orbit (the eccentric deferent) of each planet is the distance between the centres of the elliptical orbits of the Earth and that planet, which is equal to the vector sum of the distances of the centres of the elliptical orbits of the Earth and that planet from the Sun. Since the planetary orbits are inclined to the orbital plane of the Earth (i.e. the ecliptic), the distances between the centre of their orbits and the Sun should be projected onto the Earth’s orbital plane.
2. The extension into both directions of the geocentric eccentricity thus determined demarcates the geocentric apsidal line.
In the case of the inferior planets, a further condition is also required:
3. The equant point is the projection of the equant point (i.e. the empty focus) of the Earth’s elliptical orbit onto the geocentric apsidal line.
Also, it is evident that after the derivation of the geocentric eccentricities, the orbit of the Earth serves as the deferent of an inferior planet, while the orbit of an inferior planet stands for its epicycle. This criterion shall be clarified schematically in what follows.
In Figure 2, the heliocentric elliptical orbits of the Earth and Venus are drawn to scale for Ptolemy’s time. The large ellipse shows the Earth’s orbit, with Π0 being its perigee and A0 its apogee. The small ellipse indicates the orbit of Venus, with Π′ being its perihelion and A′ its aphelion. Note that because of the extreme smallness of the eccentricities, both orbits can hardly be distinguished from circles. For the same reason, also the distances between the centres of the orbits of the two planets cannot be exhibited properly. The inset in Figure 2 shows a close-up of the orientations and relative sizes of the heliocentric orbital elements of the Earth and Venus with respect to each other. The Sun is located in S. The point O stands for the centre of the Earth’s elliptical orbit; OS, the eccentricity e0 of the Earth (note that the semi-major axis a0 of the Earth is taken as 1 Astronomical Unit, AU; hence, OS = e0a0 = e0); the point T, the centre of Venus’ elliptical orbit as projected onto the Earth’s orbital plane (i.e. the ecliptic). In order to compute the distance TS, first the heliocentric eccentricity e′ of Venus should be multiplied by the semi-major axis of Venus (a′ ≈ 0.72 AU); then, since the orbit of Venus is inclined from that of the Earth at an angle i′ (this angle slightly changed during the past two millennia from about 3;22° at the beginning of the Common Era to about 3;24° in ad 2000), the result should also be projected onto the Earth’s orbital plane; therefore, ST = e′ a′ cos i, where i is the inclination of the heliocentric apsidal line of Venus from the Earth’s orbital plane. Α0Π0 and Α′Π′ indicate, respectively, the directions of the heliocentric orbits of the Earth and Venus. Thus, the two vectors SO and TS are combined in order to form the geocentric eccentricity TO = e1. Then, when the whole system is transformed to the geocentric view, the point O is the centre of the circular geocentric orbit (the eccentric deferent) and the point T stands for the place of the fictitious Earth. TO extended to both directions serves as the geocentric apsidal line, which makes an angle η0 (= ∠TOS) with the Earth’s apsidal line. The point M is the empty focus of the Earth’s orbit, which, projected onto the geocentric apsidal line, marks the equant point E at an eccentricity EO = e2 from the centre O of the eccentric deferent. Therefore, both eccentricities and the longitude λA of the geocentric apogee can be simply computed with a precision sufficient for the evaluation of the accuracy of medieval values. Since the eccentricity e0 of the Earth/Sun remains more than twice as large as e′ (precisely speaking, 2.25 at the beginning of the Common Era to 2.47 in ad 2000), the eccentricities e1 and e2 of Venus are substantially more dependent on the eccentricity e0 of the Earth than the heliocentric eccentricity e′ of the planet. Thus, both e1 and e2 remain a bit smaller than the eccentricity e0 of the Earth or, in other words, a bit smaller than half the eccentricity of the Sun in the Ptolemaic solar model.7 Also, because of the smallness of e′, the geocentric apsidal line of Venus remains close to the Earth’s apsidal line, so that the angle η0 changes only from 13.8° at the beginning of the Common Era to 10.7° in ad 2000. In addition, since the heliocentric eccentricities e0 and e′ and the angle between the heliocentric apsidal lines of the Earth and Venus decrease with the passing of time, the geocentric eccentricities e1 and e2 decrease as well.8 The formulae we derived for the geocentric orbital elements of the planet are as follows:
in which T = (JD – 2,451,545.0)/365,250 is the time measured in thousands of Julian years from 1 January 2000 (JD 2,451,545.0). The two eccentricities are for an orbital radius equal to 1 and, therefore, should be multiplied by 60 to correspond to the Ptolemaic norm. Also, the annual motion of the apsidal line is the coefficient of T multiplied by 10−3: 67.6″/y or ~ 1°/53.2y. These formulae can safely be used in order to determine the accuracy of any historical values for the orbital elements of Venus in the Ptolemaic context.
The changes in the past 2000 years are given in the following:
A key point in the derivation of the geocentric orbital elements from the heliocentric ones in the case of an inferior planet is that the condition (2) in the criterion mentioned earlier should be checked for consistency with Ptolemy’s conception of the apsidal line of an inferior planet. As said in the previous section, the apsidal line of an inferior planet defines the spatial direction of the diameter of its deferent on which its epicycle (i.e. its heliocentric elliptical orbit) appears to have the largest and smallest angular sizes as seen from the Earth (i.e. at the perigee and the apogee). Now, if AΠ in Figure 2 is in reality the geocentric apsidal line of Venus, its orbit as appearing to an Earth-bound observer has its maximum angular size when the Earth is at A (in this situation, the heliocentric orbit of Venus, corresponding to its epicycle in a geocentric view, is along the direction to the perigee Π of the deferent); conversely, the orbit has its minimum angular size when the Earth is located at Π (in this situation, the line of sight to the orbit/epicycle of Venus points to the apogee A of the deferent). The six values for the angular sizes of the orbit/epicycle of Venus shown in Figure 2 provide rough estimates for the critical values in the three situations: (1) the Earth being on its apsidal line, A0Π0; (2) on the heliocentric apsidal line of Venus, A′Π′; and (3) on the geocentric apsidal line of Venus derived according to the criterion settled forth above, i.e., AΠ. Obviously, an observer on the Earth will see the greatest and least angular sizes of the orbit/epicycle of Venus when it is located along its geocentric apsidal line as derived according to the above criterion. This assures us that our criterion is in agreement with Ptolemy’s conception of the apsidal line of Venus.9
The values of e1, e2, and ½(e1 + e2) are plotted against time in Figure 3. The graphs of e1 and e2 represent upper and lower limits of the tolerance band of the eccentricity of Venus. Figure 4 shows the graph of the longitude λA of the apogee of the Sun and Venus. Historical values are indicated in both figures. These values will be discussed in the next section.
The values adopted for the solar and Venus’ maximum equations of centre (qmax; the greatest size of angle ECT in Figure 1) and the corresponding eccentricities in medieval Middle Eastern zījes are summarized in Table 1. Except for the works in which the eccentricity values are explicitly given, they are extracted from the values for the maximum equation of centre (in the solar eccentric model: e = R sin(qmax), and in Ptolemy’s eccentric equant model of the superior planets and Venus: e1 = e2 = R tan(qmax/2)). The eccentricity values are also shown in Figure 3 along with the graphs of the geocentric eccentricities of the Sun and Venus. The values for the longitude of the apogee of Venus from these sources are listed in Table 2 and are illustrated in Figure 4 along with the graphs of the longitude of the geocentric apogees of the Sun and Venus. The medieval values are arranged in chronological order and, for a reason discussed below, in two separate groups. An important note in our discussion in the sequel is to consider the relation between the medieval astronomers’ values for the eccentricities of the Sun and Venus in Table 1. In doing so, the maximum values for the equations of centre of the Sun and Venus should be taken into account. For Yaḥyā, Ḥabash, al-Battānī, Ibn Yūnus, Ibn al-A‘lam, Ibn al-Fahhād, and Jamāl al-Dīn al-Zaydī, the equations of centre of the Sun and Venus are equal to each other, and thus, eVenus = e1 = e2 = ½eSun. Note that al-Battānī, Ibn al-A‘lam, and Jamāl al-Dīn give the values for the equation of centre of Venus with a precision up to arc-minutes, which means that they have the rounded values of the maximum equation centre of the Sun as the maximum equation of centre of Venus (1;59,10° ≈ 1;59°, 2;0,10° ≈ 2;0°, and 2;0,47° ≈ 2;1°, respectively).
The values for the eccentricities of the Sun and of Venus in Eastern Islamic zījes.
In the earliest phase of the rise of astronomy in the medieval Middle East, about the latter part of the eighth century, Indian astronomical hypotheses and systems were very influential. One of them was the Midnight System (Ārdharātrika) developed by Āryabhaṭa (b. ad 476), which has been substantially preserved in the Pañcasiddhāntikā of Varāhamihira (ad 505–587) and the Khaṇḍakhādyaka of Brahmagupta (ad 598–670). The early Islamic astronomers became familiar with it through pre-Islamic Persian astronomy, particularly the tradition of the Shāh zīj. A hypothesis of this system is the equality of the orbital elements of the Sun and Venus, in the sense that the apsidal lines of the Sun and Venus coincide with each other (in both of the works mentioned above, the apogees of the Sun and Venus share a common longitude of 80°), and their eccentricities are equal (2;20) (i.e. converted to the Ptolemaic models: eSun = 2eVenus = e1 + e2).10 With regard to our analysis set forth in the previous section, the emergence of such a hypothesis at some moment in medieval astronomy does not come as a surprise. Rather, it should have been quite probable that the poor and inaccurate observations of Venus could lead to the result that its geocentric orbital elements are equal to those of the Sun, because of the contiguity of the spatial directions of their orbits as appearing to an Earth-bound observer.11 Although some early Islamic astronomers, such as Ya‘qūb b. Ṭāriq, adopted this hypothesis of the Midnight System and its parameters via the Shāh zīj, some of his contemporaries, like al-Fazārī (d. ca.ad 796–806) and al-Khwārizmī (ca.ad 780–850), based their works upon other Indian traditions and so made use of different values for the orbital elements of the Sun and Venus.12
After the reception of the Almagest in Islamic astronomy in the ninth century, some astronomers kept Āryabhaṭa’s hypothesis of the equality of the orbital elements of the Sun and Venus, something like a single theoretical element, for any reasons unknown to us at present, and incorporated it into Ptolemy’s planetary hypotheses/models.13 It is not the only instance of maintaining some elements of Indian astronomy and mixing them with Ptolemaic astronomy in the medieval Islamic period.14 As we have already seen in section “Ptolemy’s model for Venus,” the double eccentricity of Venus is also equal to the eccentricity of the Sun in the Almagest, and thus, the only thing that indicates the adoption of Āryabhaṭa’s hypothesis in the works of early Islamic astronomy is to put their apogees at the same longitude.
According to Bīrūnī’s account in his al-Qānūn al-mas’ūdī X.4:15
The information given in the first paragraph is surprising, because both the motion of the solar apogee and Āryabhaṭa’s hypothesis (as indicated in Tables 1 and 2) can be found in the two extant manuscripts of the Mumtaḥan zīj, which was written prior to Ḥabash’s zīj. What Bīrūnī says can be considered an aspect of the mysterious situation surrounding the available manuscripts of the Mumtaḥan zīj, concerning (1) their originality: Both were copied after Ibn al-A‘lam’s time (d. ad 985) and ultimately go back to a recension of the Mumtaḥan zīj, presumably compiled in the tenth century,16 and (2) the fact that it is not known precisely which parts of this work were resulted from Yaḥyā’s observations in the Shammāsiyya quarter of Baghdad and which ones are the achievements of other astronomers of the Mumtaḥan group, working in Damascus after Yaḥyā’s death.17 From Bīrūnī’s statements, it is clear that he did not found the motion of the solar apogee in a version of the Mumtaḥan zīj available to him, which was attributed to Yaḥyā; this is not implausible at all, since the discovery of the motion of the solar apogee did not appear to have taken place immediately after the measurement of a value of 82° (or 82;39° as found in the Mumtaḥan zīj) for its longitude in the first half of the ninth century, which is ~17° more than Ptolemy’s value of 65.5°; we know that this topic was a matter of discussion until the turn of the eleventh century, and even Bīrūnī himself found it necessary to deal with it in depth.18 Also, his sayings give the strong impression that in that version of the Mumtaḥan zīj, Yaḥyā had converted Ptolemy’s values for the longitudes of the planetary apogees to his epoch, since this is what Bīrūnī did (the conversion of Ptolemy’s values to his epoch by an increment of about 13°; see Note 80). The values for the longitudes of the planetary apogees in the Mumtaḥan zīj might have been dependent upon the Almagest in one way or another, although the differences between them amount to 11.5° in the case of Jupiter and Saturn, 11° for Mercury, and 9° for Mars.19
In the second paragraph, we are first told that Ḥabash was the first medieval Middle Eastern astronomer who applied Āryabhaṭa’s hypothesis to the Ptolemaic model, as can be found in his zīj, which is closely dependent upon the available Mumtaḥan zīj (see Table 1);20 however, Bīrūnī does not explicitly refer to Āryabhaṭa, but to the Shāh zīj, which served as an intermediary for the transmission of Āryabhaṭa’s hypothesis to early Islamic astronomy. The completely surviving contents of al-Battānī’s zīj testify to Bīrūnī’s remark that this hypothesis was also employed later in it (Tables 1 and 2). It was afterwards maintained in the Ḥākimī zīj of Ibn Yūnus, Bīrūnī’s elder contemporary, but Bīrūnī was not apparently acquainted with this work.21 It is noteworthy that Ibn Yūnus not only accepted Āryabhaṭa’s hypothesis through the Shāh zīj, but also adopted the value 4;2° for the maximum equation of centre of Mercury (corresponding to an eccentricity of about 3;55) from the same work.22 Of course, he deployed the unprecedented, non-Ptolemaic values for the radii of the epicycles of the two inferior planets in order to compute his tables of their epicyclic equation, which neither can be traced back in the Shāh zīj nor in any other Indian tradition, but which appear to have been measured by Ibn Yūnus himself.23
We have evidently seen so far that Āryabhaṭa’s hypothesis was penetrated into the majority of the influential, important works in the classical period of astronomy in the medieval Middle East, which lasted until the early eleventh century. In the late Islamic period (after ca. ad 1000), we are confronted with the two streams in astronomy with regard to the relation between the orbital elements of the Sun and Venus: In the mainstream, the situation we encountered in the early Islamic period changed dramatically, in a way that Āryabhaṭa’s hypothesis gradually lost its dominance. Another stream was dependent on the reproduction of the early Islamic astronomical tables, which caused Āryabhaṭa’s hypothesis not to have disappeared completely until the foundation of the Maragha Observatory (northwestern Iran, ca.ad 1260–1320). We first explain the latter, and then will return to the mainstream. This bipartition is necessary in order to keep the discussion in chronological order.
Al-Battānī’s zīj appears to have been widely used in the Middle East until the early twelfth century, so that some zījes were written in the eleventh century, in which al-Battānī’s radix and parameter values were simply reproduced. One of them is the now lost Fākhir zīj compiled by Abu’l-Ḥasan ‘Alī b. Aḥmad al-Nasawī, a younger contemporary of Bīrūnī; this work was based on al-Battānī’s zīj, as can be inferred from the values for the longitudes of the solar and planetary apogees adopted in it, as come down to us via Kamālī’s comparative material presented in his Ashrafī zīj.24 Another example in this regard is Ṭabarī’s Mufrad zīj (ca.ad 1100),25 in which al-Battānī’s values for the longitudes of the solar and planetary apogees have been updated for the beginning of 431 Y (1 Ādhār 1373 Alexander/1 March 1062) by adding an increment of 2;45°, which is in agreement with the rate of precession of 1°/66y and the interval of time of about 182 years between al-Battānī’s and Ṭabarī’s epochs. In the latter part of the twelfth century, al-Fahhād remarks that the use of al-Battānī’s zīj had come to an end in his time. Al-Fahhād says that early in his career he had compiled four astronomical tables on the basis of al-Battānī’s parameter values, but that he later found them in error:
because of the inconsistencies (tafāwut) in al-Battānī’s observation. It is certainly confirmed that al-Battānī’s observation is erroneous, because by the direct observations (bi-ra’y al-‘ayn, “as witnessed by eye”), we see that in the planetary conjunctions as well as in the magnitudes and timings of the solar and lunar eclipses there are sizeable differences (tafāwut) [between the observational data and those computed on the basis of al-Battānī’s work]. In the entire lands of Syria and Arabia, none of the practitioners of this art does rely on al-Battānī’s observation, except for a part of the people of ‘Irāq [including central Iran and Mesopotamia] who have not any other observation [at their disposal].26
The limited use of al-Battānī’s zīj in central Iran, which al-Fahhād refers to, and its implications for the adoption of the Indian hypothesis were continued, at most, until the turn of the fourteenth century. In Kamālī’s comparative list,27 we can find that the Indian hypothesis was utilized in the two thirteenth-century works entitled the Muntakhab zīj and the Razā’ī zīj, written, respectively, by Muntakhab al-Dīn and Abu al-Ḥasan, both from Yazd (central Iran) about the mid-thirteenth century. Both works are now lost, but a zīj in poems, the so-called Manẓūm zīj (Versified zīj), from Muntakhab al-Dīn is extant, in which the longitudes of the Sun and Venus are taken as equal to each other.28 It deserves noting that according to Kamālī, Ibn al-A‘lam’s values for the equations of centres of Jupiter and Saturn were employed in both works, which can be confirmed by the corrective equation tables pertinent to the Razā’ī zīj as preserved in the anonymous Sulṭānī zīj,29 but the longitudes of the apogees show no obvious relation to Ibn al-A‘lam’s values. In Ashrafī zīj III.1, Kamālī himself points out that until the time when he wrote his own work, it was usual in Shiraz (central Iran) to compute the ephemerides of the superior planets from the ‘Alā’ī zīj and those of the Sun, the Moon, and the inferior planets from al-Nasawī’s Fākhir zīj,30 which, as mentioned earlier, was based on al-Battānī’s zīj; but, from his own observations at the times of conjunctions, he found deviations in the case of Venus and, especially, Mercury, which led him to utilize the Shāhī zīj instead.
Returning to the mainstream of astronomy in the late Islamic period, it should be said that all of the outstanding late Islamic astronomers, whose works and achievements exerted a great influence on their later followers, unanimously returned to Ptolemy’s derivation that the apogee of Venus is behind that of the Sun, regardless of the fact that they took the eccentricity of Venus as larger, smaller, or nearly equal to that of the Sun. These astronomers are mentioned in the following (see, also, Tables 1 and 2).
The turning point in the relation between the orbital elements of the Sun and Venus seems in all likelihood to have been made by Ibn al-A‘lam, in the sense that, despite the majority of the early Islamic astronomers, he did not follow Āryabhaṭa’s hypothesis, but returned to Ptolemy’s Almagest in putting the double eccentricity of Venus equal to the eccentricity of the Sun and locating the apogee of Venus behind that of the Sun with deriving a good value for its longitude (with an absolute error of less than 2°; see Tables 1 and 2). One of al-Fahhād’s noteworthy statements in the prologue of his ‘Alā’ī zīj (written ca.ad 1172) highlights Ibn al-A‘lam in this respect in contrast to the other early Islamic astronomers:31
We have observed Mars for a long period, which was in agreement with Ibn al-A‘lam’s observation [i.e. the data al-Fahhād obtained from his observations were in agreement with the ephemeris computed on the basis of Ibn al-A‘lam’s parameter values/computational tables]. Also, I observed many times Venus with the star Qalb al-asad [i.e. Regulus, α Leo], which was nicely in agreement [with Ibn al-A‘lam’s observation], but [his values] were different in the longitude of the apogee and the epicyclic anomaly [of Venus] from other observations [i.e. the values for these two parameters measured in other observational programs and/or adopted in other zījes].
Ibn al-A‘lam was no doubt the first outstanding figure in the field of planetary astronomy in the Islamic period, and his now-lost ‘Aḍudī zīj exerted a great influence on later medieval Middle Eastern astronomers. He was apparently the earliest medieval astronomer who was seriously engaged in the derivation of the fundamental parameters of the Ptolemaic planetary models and he measured new values for the eccentricities of Saturn (3;2), Jupiter (2;54),32 and Mercury (3;35).33 He also has an unprecedented value for the radius of the lunar epicycle.34 Although Ibn al-A‘lam’s ‘Aḍudī zīj is now lost, but its underlying parameter values can be found in later works, so that it can be reconstructed to a large extent (see Notes 65 and 79).35
Bīrūnī and al-Khāzinī measured new values for the solar eccentricity, both of which are smaller than Ptolemy’s (Table 1). As shown elsewhere,36 Bīrūnī’s figure is one of the excellent values measured in the medieval Middle East, whereas al-Khāzinī’s is one of the imprecise values determined by the late Islamic astronomers. Both astronomers adopted Ptolemy’s value for the eccentricity of Venus and thus took it to be greater than half that of the Sun. The values for the eccentricities of the other planets adopted in Bīrūnī’s al-Qānūn and al-Khāzinī’s Sanjarī zīj, the ultimate achievement of their long-term careers, are Ptolemaic. Bīrūnī’s value for the longitude of the apogee of Venus, which has been updated from the Almagest (see Table 2), is egregiously about −6° in error, which is an inevitable consequence of the fact that his value, 1°/69y, for the apogeal motion is smaller than the true rate of the motion of the apogee of Venus, about 1°/53y, in addition to the existence of an error of about −2.5° in Ptolemy’s value. He was skilful in the measurement of the solar orbital elements; although he did not seriously deal with a systematic observational program for the purpose of renewing the measurement of the planetary orbital elements,37 he certainly knew about the substantial differences between the methods of the derivation of the orbital elements of the Sun and Venus; seemingly, for the same reason, he could not see any relation between the orbital elements of the Sun and Venus, as can be perceived from the second paragraph of the passage we have already quoted from him in the previous section. About a century later, Khāzinī took a substantial step further in the revival of planetary astronomy, resulting in a significant improvement in the determination of the longitudes of the apogees of Venus and Mars. Unlike the other three planets, for which he only updated Ptolemy’s values for the longitudes of the apogees in the Almagest, the values he utilized for the longitudes of the apogees of Venus and Mars give the strong impression that they might have been the results of new observations and of a checking of the ephemerides against empirical data; his value for the longitude of the apogee of Venus (see Table 2) is very precise (error ~ –0.6°). In his Kayfiyyat al-i‘tibār (How to experiment),38 which he conceived as an introduction to his zīj, Khāzinī deals with the principal features of observational astronomy and explains reasonable ways how to reconcile between available theories and observational data from a coherent methodological point of view. In a section titled “the beginning of the experimentation,” which is located between the end of the treatise in question and the beginning of his zīj,39 he speaks about his 35-year program of checking and correcting the astronomical tables in use in his time against observations, in which context he explicitly refers to the al-Ma’mūnī (i.e. Mumtaḥan) zīj and al-Battānī’s zīj.40 In the list of the major and serious flaws he encountered in them, he mentions for the case of Venus the existence of errors “in its latitude, due to the deviations in its apogee,” a worthwhile statement that provides us with a clue to investigate a probable reason for which the late Islamic astronomers put aside the Indian hypothesis as well as the astronomical tables using it, such as al-Battānī’s zīj.
Al-Fahhād took the eccentricity of Venus as half the solar one and the longitude of the apogee of Venus about 12° behind that of the Sun (Tables 1 and 2). Analogous to Bīrūnī and al-Khāzinī, his values for the eccentricities of the other planets are borrowed from the Almagest. As reflected in the quote mentioned earlier, his departure from Āryabhaṭa’s hypothesis seems to have been occurred because of the agreement he found between the data obtained from his observations and Ibn al-A‘lam’s theory of Venus.
At this point, a now lost Shāhī zīj written by a certain Ḥusām al-Dīn al-Sālār about the mid-thirteenth century deserves noting. According to Kamālī,41 the apogees of the Sun and Venus in it have a separation of ~10;45° in longitude. This work can be reconstructed on the basis of the rich information provided in Kamālī’s Ashrafī zīj and the anonymous Sulṭānī zīj.
Al-Ṭūsī and the main staff of the Maragha observatory, founded by Hülegü, the first ruler of the Mongolian Īlkhānīd dynasty of Iran (d. 1265), adopted Ibn Yūnus’s value for the solar eccentricity, but they preferred to employ the Mumtaḥan and al-Battānī’s value for that of Venus, which does not seem to be a matter of coincidence or confusion at all. Rather, it seems to be a reasonable choice, and they quite probably followed Ibn al-A‘lam at the point that the orbital elements of the Sun and Venus can by no means be interconnected to each other. As regards the Maragha team’s achievements about Venus, also the rediscovery of the equality of its maximum inclination and slant deserves noting (they are the two components of Ptolemy’s latitude models of the inferior planets).42 Muḥyī al-Dīn al-Maghribī, the most prominent astronomer of the Maragha observatory in the field of observational astronomy (working independently from al-Ṭūsī’s official team), maintained the eccentricity of Venus to be less than half the solar one. He carried out a systematic observational program in Maragha, which ran for more than a decade, from 1262 through 1274. His Talkhīṣ al-majisṭī (Compendium of the Almagest) contains a detailed account of his extensive observations and measurements of the Ptolemaic planetary orbital elements in Maragha.43 The only extant copy of this treatise is incomplete, while according to the list of contents the missing parts dealt with the inferior planets and the planetary latitudes. Nevertheless, we can be confident that he gave importance to the inferior planets, because he has a highly accurate non-Ptolemaic value for the maximum inclination of Mercury in his last zīj, the Adwār al-anwār, written in Maragha.44 In both zījes written at the Maragha observatory, the double eccentricity of Venus is a bit less than the eccentricity of the Sun.
Contemporary to the Maragha Observatory, Khubilai Khan, the first emperor of the Mongolian Yuan dynasty of China (d. ad 1294), founded an Islamic Astronomical Bureau in Beijing in ad 1271 and appointed a certain Zhamaluding as its first director, who was probably identical to an Iranian astronomer named Jamāl al-Dīn Muḥammad b. Ṭāhir b. Muḥammad al-Zaydī of Bukhārā. The observational activities in the Bureau led to a new set of values for the planetary parameters. Although the original work that was written on the basis of these parameter values seems lost, some of the parameter values are preserved in two later works: the first one, Huihuili, is a Chinese translation of a Persian zīj from the Bureau, prepared in Nanjing in 1382–1383; the other, the Sanjufīnī zīj written in Arabic by a certain Sanjufīnī in Tibet in 1366.45 As can be seen in Tables 1 and 2, Jamāl al-Dīn takes the eccentricity of Venus equal to half that of the Sun and put the apogee of Venus more than 12° behind that of the Sun.
The Samarqand observatory was the last mansion of creative achievements of Islamic astronomers in the field of planetary astronomy, where the significantly precise values were measured for the orbital elements of Venus (see below). About one century and a half later, Taqī al-Dīn Muḥammad b. Ma’rūf (ad 1526–1585) made a series of the systematic observations in the short-lived observatory in Istanbul in the latter half of the 1570s. All of his observations concern the Sun and the Moon,46 and both zījes he wrote about ad 1580 contain only the solar and lunar mean motions and equation tables; the Sidrat muntaha ’l-afkār fī malakūt al-falak al-dawwār (The Lotus Tree in the Seventh Heaven of Reflection; also called the Shāhanshāhiyya zīj) is on the basis of his parameter values measured in the Istanbul observatory, whereas the Kharīdat al-durar wa jarīdat al-fikar (The non-bored pearls and the arrangement of ideas) is on the basis of Ulugh Beg’s Sulṭānī zīj.47 Of course, his value, ε = 23;28,54°, for the obliquity of the ecliptic, which was measured from his two observations carried out in Istanbul in ad 1577, has been applied to both works.48 The observatory was destructed in the early 1580s49 before the observers had enough time to deal with planetary and stellar astronomy. In the early eighteenth century, Persian and Indian astronomers used and practiced a new astronomy which had come to them through the transmission to India of the Tabulae astronomicae Ludovici magni (ad 1702), compiled by the French astronomer Philip de La Hire (ad 1640–1718). All materials on the Sun, the Moon, the planets, and the calculation of eclipses in the Persian Muḥammadshāhī zīj, compiled by Mirzā Khayr-Allāh Muhandis (i.e. the “Geometer”), Shīrāzī (d. ad 1747), and Rāja Jai Singh Sawā’ī (ad 1688–1743)50 in Jaipur in the late ad 1730s under the patronage of the latter and dedicated to Mughal emperor Muḥammad (b. 1702, reign ad 1719–1748), are on the basis of de La Hire’s work. One century later, Ghulām Ḥusayn Jaunpūrī (ad 1790/1791–1862) adhered to this revolutionary system and then established as a new tradition in his Bahādurkhānī Encyclopedia (printed in ad 1835) and Bahādurkhānī zīj (written in ad 1838 and printed in ad 1855) dedicated to his patron Rāja of Tikārī. In these works, for example, the apogees (aphelions) and ascending nodes of the orbits of the planets no longer share the same motion, and in the case of Venus, the apogee has a daily motion of 14iii 10iv and the nodes, 7iii 35iv, which are in agreement with the values 23;56,50° and 12;47,50° Philip de La Hire gives for the motions of the apogee and the node of Venus in 1000 years. The planet is also given a maximum equation of centre of 0;50°.51
In the medieval Islamic period, good values (mostly between 1;2 and 1;4) were adopted for the eccentricity of Venus, which should be reckoned as a fruit of measuring remarkably precise values for that of the Sun, because of the connection existing between them in ancient and medieval astronomy and the fact that the eccentricity of Venus is in reality close to that of the Sun (see section “Derivation of the geocentric orbital elements of Venus from the heliocentric parameters” and Figure 3).
It is important to note that from the middle of the thirteenth century onwards, the Middle Eastern astronomers took the eccentricity of Venus less than half the solar one, as it is in reality. As can be seen in Table 1, this improvement appears to have occurred, for the first time, in the zījes of the Maragha tradition and then was followed by Ibn al-Shāṭir. It became more apparent and significant in the Sulṭānī zīj, the official product of the Samarqand observatory, where the most precise value 0;52 for the eccentricity of Venus throughout ancient and medieval astronomy was measured;52 as displayed in Figure 3, this fascinating value is located near the graph of the average of the modern values for the eccentricities e1 and e2 for the time, as any highly accurate and carefully made systematic program of observations and measurements is expected to yield such a result. However, there is no historical evidence to clarify how such an improvement was made or how it was interpreted, especially because it was unprecedented in both Ptolemaic and Indian traditions come down to Islamic astronomers.
The same progressive improvement in attaining the accurate values for the eccentricity of Venus can also be clearly seen in the case of the values for the longitude of the apogee of the planet. Table 2 shows that in early medieval Middle Eastern astronomy, the errors are inevitably egregious and nearly of the same order; Yaḥyā: +12.3°, al-Battānī: +11.0°, and Ibn Yūnus: +12.6°. In contrast, in the late Middle Eastern Islamic works, the errors appreciably reduce to less than 1°; in Khāzinī’s Mu‘tabar zīj: –0.6°, al-Fahhād’s ‘Alā’ī zīj: –0.8°, the Īlkhānī zīj: –0.2°, and Ulugh Beg’s Sulṭānī zīj: +0.7°. For the other astronomers, the errors are about a few degrees, but not as large as those in the early Islamic period; Ibn al-A‘lam: –1.8°, al-Maghribī: +3.7°, Jamāl al-Dīn: –1.3°, and Ibn al-Shāṭir: –1.9°.53
When the Indian hypothesis of the equality of the orbital elements of the Sun and Venus reached the western Islamic realm, it became incorporated into a tradition that was based on a quite different system of astronomical thinking, demanding a different kind of treatment of the relation between observational data and hypotheses, from that in Eastern Islamic astronomy. In this tradition, secular changes and variations in the basic parameters detected by observations—which were taken to be constant or were thought to be unaltered within short periods of time in Ptolemaic astronomy followed by the Middle Eastern medieval astronomers—were given a higher epistemological level, to such a degree that they were entered into fundamental models.54 A notable example of such treatment in Western Islamic astronomy is the invention of a solar model with variable eccentricity by Ibn al-Zarqālluh (d. ad 1100).55 This model is an ingenious attempt to account for the long-term continuous decrease in the solar eccentricity after Ptolemy’s time, as known by the observations made by the Islamic astronomers from the Mumtaḥan group in the early ninth century to Ibn al-Zarqālluh’s time.
The mechanism embedded in this model is similar to that which Ptolemy invented for his models for the Moon and Mercury (see Figure 5). The centre D of the deferent revolves on a hypocycle with centre C, so that the eccentricity DT of the Sun changes from a maximum of TO0 to a minimum of TO. The parameter values of this mechanism, i.e., the maximum and minimum solar eccentricities and the motion of the centre of the eccentric on the circumference of the central hypocycle, are nearly the same in the various sources of Western Islamic astronomy: emax ≈ 2;29, emin ≈ 1;51 (thus, the radius DC of the hypocycle is approximately 0;19), and a complete revolution of the centre of the eccentric on the circumference of the hypocycle takes about 3345 years. Also, at the epoch, namely, at the beginning of the Hijra era, it was located at a distance of 83;40,31° from the apsidal line.56 The motional parameter values of the model were derived in such a manner that a maximum eccentricity of 2;29 was obtained for Hipparchus’s time, i.e., about the mid-second century bc. Figure 6 shows the graph of the solar eccentricity according to Ibn al-Zarqālluh’s solar model with the parameter values mentioned above (dash-dotted curve) along with the graphs of the eccentricities of the Sun and Venus on the basis of the modern theories as already exhibited in Figure 3.
In the case of the eccentricity (and, accordingly, equations of centre) of the planet, the two different treatments can be addressed in them. First, some astronomers like Ibn Isḥāq (ad 1193–1222), Ibn al-Raqqām (d. ad 1315), and Ibn al-Bannā’ adopted the Indian hypothesis of the equivalence of the orbital elements of the Sun and Venus along with Ibn al-Zarqālluh’s solar model. This inevitably led to the result that the same values for the solar eccentricity as computed on the basis of Ibn al-Zarqālluh’s model should be taken for the eccentricity of Venus as well. Second, some scholars such as Ibn al-Kammād (fl. ca.ad 1116) and Ibn ‘Azzūz al-Qusanṭīnī (d. ad 1354) accepted the prevalent value 1;59° for the maximum equation of centre of Venus borrowed from the Mumtaḥan tradition as established in the zījes of Ḥabash and al-Battānī.58 Hence, they held that the eccentricity of Venus is greater than that of the Sun.
What is most notable is that the values which the first group of the Western Islamic astronomers mentioned above derived for the eccentricity of Venus from Ibn al-Zarqālluh’s solar theory range between 0;56 and 0;59, which are very close to the true values for the eccentricity of Venus in the period in question (lying within the tolerance band of the geocentric eccentricity of Venus; see Figure 6). Nevertheless, it should not come as a surprise that these values are highly accurate and comparable with the remarkably precise value measured in the Samarqand observatory. For it is evident that this achievement is solely a matter of coincidence; the accidental accuracy of these values is merely a result of the combination of a solar model with variable eccentricity and the Indian hypothesis of the equality of the orbital elements of the Sun and Venus.
We have seen in section “Derivation of the geocentric orbital elements of Venus from the heliocentric parameters” that the size of the geocentric eccentricities of Venus is substantially dependent on that of the Sun/Earth, and is a bit less than it. Also, the geocentric apsidal line of Venus is very close to that of the Sun/Earth. It is imaginable that careless observations would have led to the result that the apsidal lines of the Sun and Venus coincide with each other and/or that their eccentricities are equal. Ptolemy derived a value for the double eccentricity of Venus, which is equal to his value for the solar eccentricity, but he did not give any notice of the relation between the two. Āryabhaṭa took not only the eccentricities of the Sun and Venus, but also the longitudes of their apogees equal to each other in his Midnight System developed about the early sixth century. As a consequence, medieval astronomers from the early Islamic period on were exposed to the existence of a connection between the orbital elements of the Sun and Venus which had come down to them from both Ptolemaic and Indian traditions. In the tradition-based medieval system of thinking, such similarities between different traditions were not treated with indifference or as a result of mere coincidence. Consequently, it does not come as a surprise that Āryabhaṭa’s hypothesis had a great influence on the main trends of medieval astronomy and was prevalent in quite different traditions. It was passed into the Western Islamic regions basically through the transmission of Middle Eastern astronomical tables, such as the Mumtaḥan zīj and al-Battānī’s Ṣābi’ zīj, and wherefrom diffused into medieval Latin and Jewish astronomy apparently via the Alfonsine Tables.59 It maintained its dominance to such an extent that it can be found in a good number of European treatises until just before the emergence of Kepler’s new astronomy (more notably, in Copernicus’s Commentariolus).60 Nevertheless, it began to be rendered obsolete in Eastern Islamic astronomy after the tenth century.
For the eccentricities of the Sun and Venus, three treatments can be identified in medieval Islamic astronomy:
1. The eccentricity of Venus equal to half that of the Sun.
1.1 In early Eastern Islamic (ca.ad 800–1000) and some Western Islamic zījes (after ad 1000) following Āryabhaṭa’s hypothesis in the Midnight System, the eccentricity of Venus is half that of the Sun. In the latter group, Ibn al-Zarqālluh’s solar model was utilized, according to which the eccentricity of the Sun periodically changes, decreasing in the period from the mid-second century bc to about ad 1500. As exhibited in Figure 6, the values that this model gives for half the solar eccentricity during the period from ca.ad 1000 to ad 1950 are within the tolerance band of the geocentric eccentricity of Venus. Consequently, the adoption of Āryabhaṭa’s hypothesis along with Ibn al-Zarqālluh’s solar model accidentally yielded accurate values for the eccentricity of Venus.
1.2 Some outstanding figures constituting the main stream of planetary astronomy in the late Eastern Islamic period (after ca.ad 1000), like Ibn al-A‘lam, al-Fahhād and Jamāl al-Dīn, returned to Ptolemy’s derivation, i.e., taking the double eccentricity of Venus equal to the solar eccentricity, as is in Āryabhaṭa’s Midnight System, but putting the apogee of Venus behind that of the Sun.
2. No relation between the eccentricities of the Sun and Venus.Some late Eastern Islamic astronomers, such as Bīrūnī and al-Khāzinī, did not apparently see any relation between the eccentricities of the Sun and Venus. They adopted their measured values for the solar eccentricity, which are less than Ptolemy’s, but held Ptolemy’s value for that of Venus. This situation is analogous to that encountered in other Western Islamic zījes different from the group (1), where the eccentricity of the Sun is computed according to Ibn al-Zarqālluh’s solar model, which gives smaller values for it than those adopted in the early medieval Middle Eastern zījes such as the Mumtaḥan zīj, Ḥabash’s zīj, and al-Battānī’s Ṣābi’ zīj, but that of Venus is the same, i.e., about 1;2, as adopted in these works. Thus, to these astronomers, the eccentricity of Venus is inevitably larger than half the solar one, which has no astronomical connotation, but is solely a consequence of the adoption of fundamental parameter values from different sources/traditions.
3. The eccentricity of Venus smaller than half that of the Sun.In the late medieval Middle Eastern astronomical tables from the middle of the thirteenth century onwards (notably, the zījes of the Maragha tradition, Ibn al-Shātir’s Jadīd zīj, and Ulugh Beg’s Sulṭānī zīj), the eccentricity of Venus was taken smaller than half that of the Sun. This achievement is significant for the astronomical reasons mentioned earlier and may be considered one of the discoveries of late Islamic astronomy, encountered neither in Ptolemaic nor in Indian astronomy; an exceptional promotion in the derivation of the eccentricity of Venus took place at the Samarqand observatory in the first part of the fifteenth century, where the accurate value 0;52 was measured for the eccentricity of Venus, which was deployed in Ulugh Beg’s Sulṭānī zīj.
Two hypotheses on the spatial orientation of the geocentric orbits of the Sun and Venus can be found in medieval Islamic astronomy:
1. Both coincide with each other, in accordance with the Indian tradition of the Midnight System, which is inaccurate and which is dominant in early Eastern Islamic astronomy as well as in Western Islamic astronomy.
2. The apogee of Venus is behind that of the Sun, in agreement with Ptolemy’s tradition, which is correct, and which was held in late Eastern Islamic astronomy.
In early Islamic Middle Eastern astronomy as well as in Western Islamic astronomy, the errors in the values for the longitude of the apogee of Venus are larger than +10°, a consequence of putting the longitude of the apogee of Venus equal to that of Sun. But, as this hypothesis was discarded in late medieval Middle Eastern astronomy, the values for the longitude of the apogee of the planet became significantly improved, so that the errors reduced to less than 1° in the Īlkhānī zīj and Ulugh beg’s Sulṭānī zīj, the two official works connected to the Maragha and Samarqand observatories.
The author owes a debt of gratitude to Benno van Dalen (Germany), Julio Samsó (Spain), and John Steele (United States) and an anonymous referee for their critical remarks and suggestions. He also likes to thank Dirk Grupe (Germany) for revising the English of an earlier version of this paper.
This work was financially supported by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/5750–6.
Notes on contributor
S. Mohammad Mozaffari is an assistant professor of History of Astronomy in the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran. He has published several papers on observational and mathematical astronomy in medieval Middle Eastern astronomy since 2012. He is currently working on Ibn Yūnus’s and Ibn al-Shāṭir’s non-Ptolemaic star tables.
1.See O. Pedersen, A Survey of the Almagest (Odense: Odense University Press, 1974; with annotation and new commentary by A. Jones, New York: Springer, 2010), pp. 295–309; O. Neugebauer, A History of Ancient Mathematical Astronomy (3 vols., Berlin; Heidelberg; New York: Springer, 1975), vol. 1, pp. 152–8; C. Wilson, “The Inner Planets and the Keplerian Revolution,” Centaurus, 17, 1973, pp. 205–48; N.M. Swerdlow, “Ptolemy’s Theory of the Inferior Planets,” Journal for the History of Astronomy, 20, 1989, pp. 29–60.
2.In Ptolemy’s iterative method for determining the orbital elements of the superior planets, the total eccentricity (e1 + e2) is first computed and then the result is halved under the assumption that e1 = e2; but, in his method for the derivation of the orbital elements of the inferior planets, either eccentricity is computed independently from the other. It is noteworthy that Abū al-Rayḥān al-Bīrūnī (ad 973–1048), in his al-Qānūn al-mas‘ūdī VI.8 and X.3.1 (Abū al-Rayḥān al-Bīrūnī, al-Qānūn al-mas‘ūdī (Mas‘ūdīc canons) (3 vols., Hyderabad: Osmania Bureau, 1954–1956), vol. 2, pp. 681–5, vol. 3, pp. 1183–4), proposed an alternative method for the derivation of the orbital elements of the Sun and the superior planets, which resembles Ptolemy’s corresponding method for the inferior planets; see S.M. Mozaffari, “Bīrūnī’s Four-Point Method for Determining the Eccentricity and the Direction of the Apsidal Lines of the Superior Planets,” Journal for the History of Astronomy, 44, 2013, pp. 207–11.
3.G.J. Toomer, Ptolemy’s Almagest (Princeton: Princeton University Press,  1998), pp. 469–74.
4.Toomer, op. cit. (Note 3), p. 155.
5.The late Prof. O. Neugebauer (op. cit. (Note 1), vol. 1, pp. 147, 213) derives the values 0;40 for the eccentricity and 47° for the longitude of the apogee of Venus for ad 100. The reason behind these erroneous figures has been explained in S.M. Mozaffari, “Holding or Breaking with Ptolemy’s Generalization: Considerations about the Motion of the Planetary Apsidal Lines in Medieval Islamic Astronomy,” Science in Context, 30, 2017, pp. 1–32, p. 5, Note 3.
6.See Wilson, op. cit. (Note 1); N.M. Swerdlow and O. Neugebauer, Mathematical Astronomy in Copernicus’s De Revolutionibus (2 vols., New York: Springer, 1984), vol. 1, esp. pp. 369–71.
7.The medieval Islamic values for the solar parameters are investigated in detail earlier in S.M. Mozaffari, “Limitations of Methods: The Accuracy of the Values Measured for the Earth’s/Sun’s Orbital Elements in the Middle East, A.D. 800 and 1500,” Journal for the History of Astronomy, 44, 2013, Part 1: issue 3, pp. 313–36, Part 2: issue 4, pp. 389–411 and S.M. Mozaffari, “An Analysis of Medieval Solar Theories,” Archive for History of Exact Sciences, 72, 2018, pp. 191–243.
8.The true values for the geocentric eccentricity and longitude of the apogee of Venus in this study were computed on the basis of the formulae for the heliocentric orbital elements of the Earth and Venus in J.L. Simon, P. Bretagnon, J. Chapront, M. Chapront-Touze, G. Francou, and J. Laskar, “Numerical Expressions for Precession Formulae and Mean Elements for the Moon and the Planets,” Astronomy and Astrophysics, 282, 1994, pp. 663–83.
9.As we will show elsewhere, the criterion is not consistent with the Ptolemaic conception of the apsidal line of the inferior planets in the case of Mercury, because of the great eccentricity of this planet in comparison with that of the Earth, which leads to a complex situation regarding the variation of the apparent size of its orbit as seen from our planet.
10.Pañcasiddhāntikā III.2–3, IX.7–8, XVI.12–14: O. Neugebauer and D. Pingree, The Pañcasiddhāntikā of Varāhamihira, Historisk-filosofiske Skrifter (2 vols., Copenhagen: Det Kongelige Danske Videnskabernes Selskab, 1970–1971), vol. 1, pp. 39, 93, 149–51, vol. 2, pp. 24, 69–70, 101–2; Khaṇḍakhādyaka I.13 and II.6: Brahmagupta, The Khaṇḍakhādyaka of Brahmagupta, B. Chatterjee (ed. and En. trans.) (New Delhi: Bina Chatterjee, 1970), pp. 49–50, 54, 283; E.S. Kennedy, “The Sasanian Astronomical Handbook Zīj-i Shāh and the Astrological Doctrine of “Transit” (Mamarr),” Journal of the American Oriental Society, 78, 1958, pp. 246–62 (reprinted in E.S. Kennedy, Studies in the Islamic Exact Sciences (Beirut: American University of Beirut, 1983), pp. 319–35), pp. 256–7; E.S. Kennedy and D. Pingree (eds), The Book of the Reasons behind Astronomical Tables (New York: Scholars’ Facsimiles & Reprints, 1981), p. 220; D. Pingree, “The Persian “Observation” of the Solar Apogee in ca. A.D. 450,” Journal of Near Eastern Studies, 24, 1965, pp. 334–6; D. Pingree, “The Fragments of the Works of Ya‘qūb Ibn Ṭāriq,” Journal of Near Eastern Studies, 27, 1968, pp. 97–125; D. Pingree, “The Fragments of the Works of Al-Fazārī,” Journal of Near Eastern Studies, 29, 1970, pp. 103–23; D. Pingree, Jyotiḥśāstra; Astral and Mathematical Literature (Wiesbaden: Harrassowitz, 1981), pp. 15–6; B.L. van der Waerden, “The Heliocentric System in Greek, Persian and Hindu Astronomy,” in G. Saliba and D.A. King (eds), From Deferent to Equant: A Volume of Studies on the History of Science of the Ancient and Medieval Near East in Honor of E. S. Kennedy (Annals of the New York Academy of Sciences, vol. 500) (New York: New York Academy of Sciences, 1987), pp. 525–45, esp. pp. 530–2; B.L. van der Waerden, “The Astronomical System of the Persian Tables II,” Centaurus, 30, 1987, pp. 197–211. It deserves noting, however, that the underlying values for the planetary parameters in the Shāh zīj is not completely in agreement with those deployed in the Midnight System; see Pingree, “The Fragments of the Works of Ya‘qūb Ibn Ṭāriq” (Note 10), pp. 104–5.
11.This hypothesis of Āryabhaṭa in his own Midnight System distinguishes from a curious feature in other Indian astronomical traditions (and in another work by himself, Āryabhaṭīya I.7: Āryabhaṭa, Āryabhaṭīya, W.E. Clark (Eng. trans.) (Chicago: University of Chicago Press, 1930), pp. 16–8), according to which the progressive motion of the apsidal lines of the Sun, the Moon, and the planets and the retrograde motion of the nodal lines of the Moon and the planets do not take place at a constant rate, but each has its own motion (different from Ptolemaic astronomy, where the apsidal and nodal lines are sidereally fixed and therefore tropically subject to the precessional motion). However, the angular velocities given for their motions in various Indian traditions are too small to be assumed to have been obtained from actual systematic observations. The early Muslim astronomers were completely acquainted with it. See Súrya Siddhánta I.41–44: P. Gangooly (ed.) and E. Burgess (En. trans.), The Súrya Siddhánta: A Textbook of Hindu Astronomy (Delhi: Motilal Banarsidass,  1997), p. 7; P.B. Deva Sastri and L. Wilkinson (Eng. trans.), The Súrya Siddhánta, Or An Ancient System of Hindu Astronomy, followed by the Siddhánta Śiromani (Amsterdam: Philo Press,  1974), pp. 26–8; Kennedy and Pingree, op. cit. (Note 10), pp. 118–9, 282; Pingree, “The Fragments of the Works of Ya‘qūb Ibn Ṭāriq” (Note 10), p. 99; Pingree, “The Fragments of the Works of Al-Fazārī” (Note 10), p. 109.
12.Al-Fazārī has the unprecedented values qmax = 2;11,15° for the Sun (which is close, but by no means identical, to the ones utilized in some Indian sources; e.g., the value 2;10,31° in the Paitāmahasiddhānta) and 2;15° for Venus (approximately equal to the value 2;14° employed in the Midnight System); see Pingree, “The Fragments of the Works of Ya‘qūb Ibn Ṭāriq” (Note 10), pp. 103–4; Pingree, “The Fragments of the Works of Al-Fazārī” (Note 10), pp. 112–3. For al-Khwārizmī, see O. Neugebauer, The Astronomical Tables of Al-Khwārizmī (Copenhagen: Munksgaard, 1962), p. 41, 99.
13.We have checked the two extant Arabic translations of the Almagest by Ḥajjāj b. Yūsuf b. Maṭar in ad 827–828 (LE: Leiden, Or. 680: ff. 150v–152v, dropped from MS. LO: Library of London, Add 7474, copied in 686 H/ad 1287) and by Ḥunayn b. Isḥāq in ad 880–890, which was afterward revised by Thābit b. Qurra (d. ad 901) (S: Iran, Tehran, Sipahsālār Library, no. 594, copied in 480 H/ad 1087–1088, ff. 134v–136v, PN: USA, Rare Book and Manuscript Library of University of Pennsylvania, LJS 268, written in an Arabic Maghribī/Andalusian script at Spain in 783 H/ad 1381, ff. 99r–v) and found nothing indicating that Āryabhaṭa’s Midnight System had any deleterious effect on the contents related to the orbital elements of Venus in these translations of the Almagest. Of course, there was an earlier translation made shortly before or around ad 800, which is not available nowadays. See P. Kunitzsch, “Translators’ Errors in the Almagest, Arabic and Latin,” in P. Arfé, I. Caiazzo and A. Sannino (eds), Adorare caelestia, gubernare terrena: Atti del colloquio internazionale in onore di Paolo Lucentini (Napoli: Brepols, 2011), pp. 283–93, 284; R. Lorch, “Greek-Arabic-Latin: The Transmission of Mathematical Texts in the Middle Ages,” Science in Context, 14, 2001, pp. 313–31, 315–6, and the references mentioned therein.
14.In some other cases, the plausible reasons for maintaining the Indian astronomical elements can be thought: They either are the subjects which Ptolemy does not deal with in the Almagest (e.g. the colours and the optical limitation of the visibility of the eclipses) or could be of practical use (e.g. the Indian hypotheses of the angular diameters of the Sun and the Moon, which made the annular solar eclipses justifiable and even predictable); see S.M. Mozaffari, “Historical Annular Solar Eclipses,” Journal of the British Astronomical Association, 123, 2013, pp. 33–6; S.M. Mozaffari, “Wābkanawī’s Prediction and Calculations of the Annular Solar Eclipse of 30 January 1283,” Historia Mathematica, 40, 2013, pp. 235–61; S.M. Mozaffari, “A Case Study of How Natural Phenomena were Justified in Medieval Science: The Situation of Annular Eclipses in Medieval Astronomy,” Science in Context, 27, 2014, pp. 33–47; S.M. Mozaffari, “Annular Eclipses and the Considerations about the Solar and Lunar Angular Diameters in the Medieval Astronomy,” in W. Orchiston, D.A. Green and R. Strom (eds), New Insights From Recent Studies in Historical Astronomy: Following in the Footsteps of F. Richard Stephenson (New York: Springer, 2015), pp. 119–42, esp. p. 138.
15.Bīrūnī, op. cit. (Note 2), vol. 3, pp. 1197–8. The English translation of the second paragraph of this passage is from B.R. Goldstein and F.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: Franz Steiner Verlag, 1977), pp. 165–81, a part of which is repeated in J. Chabás and B.R. Goldstein, “Ibn al-Kammād’s Muqtabis zij and the Astronomical Tradition of Indian Origin in the Iberian Peninsula,” Archive for History of Exact Sciences, 69, 2015, pp. 577–650, 610, with some changes.
16.See B. van Dalen, “A Second Manuscript of the Mumtaḥan Zīj,” Suhayl, 4, 2004, pp. 9–44, esp. p. 11.
17.This issue is beyond the scope of this paper and is dealt with in depth elsewhere; only as an example, it is noteworthy that the Mumtaḥan zīj has a very precise solar theory with the errors not exceeding 5′ for the period of 13,000 days since ad 1–1–820 (see Mozaffari, “An Analysis,” esp. pp. 221–3, 235). It employs Yaḥyā’s values for the orbital elements of the Sun. Nevertheless, his measured values for the times of the equinoxes of ad 829–830, a short while before his death, suffer from the large errors up to ~ +7 hours and is not compatible with the Mumtaḥan solar theory. In contrast, the time of the autumnal equinox of ad 832 measured by Sand b. ‘Alī and Khālid b. ‘Abd al-Malik al-Marwarūdhī in Damascus is only ~ +1/2 hour off (Mozaffari, “An Analysis,” p. 216) and is in complete agreement with the Mumtaḥan solar theory. Therefore, it seems that the finalized solar theory in the Mumtaḥan zīj as come down to us is a modified version of an earlier theory prepared by Yaḥyā.
18.See Mozaffari, “Limitations,” Part 2, pp. 403–8; Mozaffari, op. cit. (Note 5), pp. 8–9.
19.See Mozaffari, op. cit. (Note 5), pp. 14–5.
20.In the Istanbul copy of Ḥabash’s Zīj (I: Istanbul, Süleymaniye, Yeni Cami, no. 784, ff. 89r, 115r; M.-T. Debarnot, “The Zīj of Ḥabash al-Ḥāsib: A Survey of MS Istanbul Yeni Cami 784/2,” in Saliba and King, op. cit. (Note 10), pp. 35–69, 44), the longitudes of the apogees of the Sun and Venus are equal to 82;39°, as is in the Mumtaḥan zīj. In the Berlin copy of this work, there are also the two tables for the longitudes of the apogees of the Sun and the five planets. One gives the values up to the five sexagesimal fractional places, wherein the fractions from the seconds to the fifths are equal (…;…,24,2,43,53°). The longitudes of the apogees of the Sun and of Venus are equal to 79;30, … (B: Berlin, Ahlwardt 5750 (formerly Wetzstein I 90), f. 28r). The tabular values are by about 3;9° less than the apogee longitudes in the Mumtaḥan/Ḥabash’s zīj, and so, they are for the beginning of the Hijra era. The other gives the longitudes of the planetary apogees up to the ninth sexagesimal fractional place for the year 872 Hijra, whose beginning was on 1 August 1467. The longitude of the apogees of the Sun and Venus are equal up to the arc-seconds: 92;24 …° (B: f. 17v). At the beginning of this table, we are explicitly told that it was updated from the Mumtaḥan zīj. With Yaḥyā’s value 82;39° for ca. 830 in Table 2 and the precessional motion of 1° in 66 years, as associated with the Mumtaḥan tradition, we derive a longitude of about 92;19° for the given date.
21.Muḥyī al-Dīn al-Maghribī appears to have been responsible for introducing Ibn Yūnus’s zīj at the Maragha observatory, since, to the best of our knowledge, no trace of it may be found in the Eastern Islamic lands until that time; more notably, a reference to it can be found neither in ‘Abd al-Raḥmān al-Khāzinī’s On Experimental Astronomy (Kayfiyyat al-i‘tibār) II.4 (in: al-Khāzinī, al-Zīj al-mu‘tabar al-sanjarī, V: Vatican, Biblioteca Apostolica Vaticana, Arabo 761, f. 8r), wherein the two most influential Middle Eastern works, the Mumtaḥan zīj and al-Battānī’s <.> Sābi’ zīj, are mentioned, nor in Ibn al-Fahhād’s very informative evaluation of the deficiencies and errors in his Islamic predecessors’ works, as put forward in the prologue of his ‘Alā’ī zīj (Farīd al-Dīn Abu al-Ḥasan ‘Alī b. ‘Abd al-Karīm al-Fahhād al-Shirwānī or al-Bākū’ī, Zīj al-‘Alā’ī, MS. India, Salar Jung, no. H17, pp. 3–5).
22.Already noted in D.A. King, “Aspects of Fatimid Astronomy: From Hard-Core Mathematical Astronomy to Architectural Orientations in Cairo,” in M. Barrucand (ed.), L’Égypte Fatimide: son art et son histoire – Actes du colloqie organisé à Paris les 28, 29 et 30 mai 1998 (Paris: Presses de l’Université de Paris-Sorbonne, 1999), pp. 497–517, 502; see ‘Alī b. ‘Abd al-Raḥmān b. Aḥmad Ibn Yūnus, Zīj al-kabīr al-Ḥākimī, L: Leiden, Universiteitsbibliotheek, Or. 143, pp. 121, 191–3; J.-J.-A. Caussin de Perceval, “Le livre de la grande table hakémite, Observée par le Sheikh, …, ebn Iounis,” Notices et Extraits des Manuscrits de la Bibliothèque nationale, 7, 1804, pp. 16–240, 221; Pingree, “The Fragments of the Works of Ya‘qūb Ibn Ṭāriq” (Note 10), p. 104; Pingree, “The Fragments of the Works of Al-Fazārī” (Note 10), p. 113.
23.For Venus, the maximum epicyclic equation of 46;25°, which corresponds to a radius of the epicycle r ≈ 43;28, and for Mercury: 22;24°, corresponding to r ≈ 22;52. See Ibn Yūnus, Zīj (Note 22), L: pp. 121, 190, 192; Caussin, op. cit. (Note 22), p. 221.
24.Muḥammad b. Abī ‘Abd-Allāh Sanjar al-Kamālī (Sayf-i munajjim), Ashrafī zīj (written in Shiraz in the early fourteenth century), MSS. F: Paris, Bibliothèque Nationale, no. 1488, f. 232v, G: Iran, Qum, Gulpāyigānī, no. 64731, f. 249r.
25.Abū Ja‘far Muḥammad b. Ayyūb al-Ḥāsib al-Ṭabarī, Mufrad zīj (The unique zīj), MS. Cambridge, Browne Collection College, O.1, f. 175v.
26.Al-Fahhād, Zīj (Note 21), p. 3. The most notable of such errors took place in the case of the conjunction between Jupiter and Saturn in December 1166. In the prediction of the time of this conjunction, al-Battānī’s zīj was about 35 days in error. Al-Fahhād computed it to have occurred on 10 December, at 8;14,35 hours before noon (pp. 4, 57–9). In reality, the conjunction took place on 11 December, at 23:46 MLT, hence the error in Ibn al-Fahhād’s time is less than 2 days.
27.Kamālī, Zīj (Note 24), F: f. 232v, G: f. 249r.
28.Muntakhab al-Dīn al-Yazdī, Manẓūm zīj, MS. Iran, Mashhad University, Theology Faculty, no. 674, ff. 46v–47r: He gives the value 87;55° for the longitudes of the apogees of the Sun and Venus for the beginning of 621 Yazdigird (13 January 1252), which is more than −2° in error, if taken as the place of the solar apogee, but too large (with an error of about +9;41°) for being the location of the apogee of Venus for the given time. The values Kamālī reports for the longitudes of the solar and planetary apogees from the Muntakhab zīj as converted to 13 Khurdād 672 (13 March 1303) are all by 0;46° more than the values in the Manẓūm zīj, which is in accordance with the rate of precession of 1°/66y and the period of 51 years between them, and which shows that both zījes written by Muntakhab al-Dīn share the same epoch and radix values. The Muntakhab zīj and Razā’ī zīj can be reconstructed to a large extent on the basis of the information that has come down to us through the Ashrafī zīj X.8 and X.9: (Note 24), F: f. 230v and ff. 231v–233r, 234r, 235v, G: f. 247v and ff. 248v–249r, 250v, and the anonymous Sulṭānī zīj written in Yazd about the 1290s (NB this is neither be confused with Wābkanawī’s Zīj al-Muḥaqqaq al-Sulṭānī, nor with Ulugh Beg’s Sulṭānī zīj), which is preserved in a unique manuscript in Iran, Library of Parliament, no. 184. Despite the late E.S. Kennedy’s conjecture (E.S. Kennedy, “A Survey of Islamic Astronomical Tables,” Transactions of the American Philosophical Society, New Series, 46, 1956, pp. 123–177, no. 25 on p. 129), this work is not identical with the Shāhī zīj, since some material of the latter work is explicitly quoted and explained in it; e.g., the tables of the equation of time on ff. 7v and 15r, and the method of Ḥusām al-Dīn al-Sālār for the construction of the planetary equation tables on f. 77r. Some tables of the Razā’ī zīj are preserved in this Sulṭānī zīj: (a) the table of the longitude of the lunar node on f. 11r, (b) the procedure for the computation of the longitude of the superior planets in III.6 on f. 79r, (c) the planetary mean positions in longitude and in anomaly on f. 81v (the longitudes of the apogees of the Sun and Venus are equal), and (d) the tables of the “difference in equations” for the superior planets on ff. 120v–121v (see, also, next note).
29.In the Sulṭānī zīj (Note 28), all the principal tables for the equation of centre of the superior planets are displaced, but based on Ptolemy’s eccentricity values, and in the steps of 0;5°, as described in the following: Saturn: Min = 0;28°, Max = 13;32° (ff. 16v–22r); Jupiter: Min = 0;45°, Max = 11;15° (ff. 30v–36r); and Mars: Min = 0;35°, Max = 23;25° (ff. 44v–50r). The equation tables from other zījes appear in the form of the auxiliary tables called “difference in equation” (ikhtilāf-i ta‘dīl), in each of which the differences in entries between a principal equation table of this zīj and the corresponding table from another one have been tabulated. Consequently, the three corrective tables in the Sulṭānī zīj for the equation of centre of the superior planets according to the Razā’ī zīj actually display the differences between Ptolemy’s equation values and those originally tabulated in the Razā’ī zīj. The corrective table for Saturn is subtractive and displaced with Max = 1;44° and Min = 0;16° (f. 120v); thus, the maximum difference in Saturn’s equation of centre between the Razā’ī zīj and Almagest is Δqmax = –0;44°; therefore, according to the Razā’ī zīj, the maximum value of equation of centre of Saturn is qmax = 6;31 – 0;44 = 5;47°. The corrective table for Jupiter is symmetrical with Δqmax = ±0;17° (f. 121r); therefore, qmax = 5;15 + 0;17 = 5;32°. For Mars, the corrective table is additive and displaced with Min = 1;38° and Max = 2;28°; thus, Δqmax = +0;25° (f. 121v); therefore, qmax = 11;25° + 0;25° = 11;50°. Note that the maximum values for the equation of centre of Jupiter and Saturn are equal to Ibn al-A‘lam’s (see below, Note 32), in agreement with Kamālī’s statement. But, the source of the value 11;50° for Mars’ maximum equation of centre (corresponding to e ≈ 6;13) is unknown. However, the table of Mars’ equation of centre from the Razā’ī zīj as preserved in the Ashrafī zīj is on the basis of Ptolemy’s eccentricity value (although displaced, with Min = 2;35° and Max = 25;25°) (Kamālī, Zīj (Note 24), F: f. 235v, G: f. 250v). A close value e = 6;15 is mentioned in Ashrafī zīj III.9.2: (Note 24), F: f. 51r, G: f. 56r, where Kamālī lists the planetary eccentricities. The astronomers in the Samarqand observatory measured the other close value, e ≈ 6;13,30, less than two centuries later; see S.M. Mozaffari, “Planetary Latitudes in Medieval Islamic Astronomy: An Analysis of the Non-Ptolemaic Latitude Parameter Values in the Maragha and Samarqand Astronomical Traditions,” Archive for History of Exact Sciences, 70, 2016, pp. 513–41, 535.
30.Kamālī, Zīj (Note 24), F: f. 47r, G: ff. 50v–51r.
31.Al-Fahhād, Zīj (Note 21), p. 4. See, also, B. van Dalen, “The Zīj-i Naṣirī by Maḥmūd ibn Umar: The Earliest Indian Zij and Its Relation to the ‘Alā’ī Zīj,” in C. Burnett et al. (eds), Studies in the History of the Exact Sciences in Honour of David Pingree (Leiden: Brill, 2004), pp. 825–62, 836.
32.Ibn al-A‘lam’s tables of the equation of centre of these two superior planets are preserved in Kamālī’s Ashrafī zīj. The table for Saturn’s equation of centre (Kamālī, Zīj (Note 24), F: f. 234v, G: f. 250r) is displaced with a minimum tabular value of 0;12° (for arguments 76°–81°) and a maximum value of 11;48° (for arguments 253°–258°). The table for Jupiter’s equation of centre (F: f. 235r, G: f. 250r) is also displaced with minimum 0;28° (for arguments 72°–78°) and maximum 11;32° (for arguments 246°–252°; on “displaced” equation tables, a term coined by the late Prof. E.S. Kennedy, see van Dalen, op. cit. (Note 31); J. Chabás and B.R. Goldstein, “Displaced Tables in Latin: The Tables for the Seven Planets for 1340,” Archive for History of Exact Sciences, 67, 2013, pp. 1–42, reprinted in J. Chabás and B.R. Goldstein, Essays on Medieval Computational Astronomy (Leiden: Brill, 2015), pp. 99–149; and the references mentioned therein). Accordingly, the maximum equations of centre of Saturn and Jupiter are derived, respectively, as 5;48° and 5;32°. The modern values for the geocentric eccentricity of the two planets in Ibn al-A‘lam’s time are, respectively, equal to 3;26 and 2;48 (see S.M. Mozaffari, “Ptolemaic Eccentricity of the Superior Planets in the Medieval Islamic Period,” in: G. Katsiampoura (ed.), Scientific Cosmopolitanism and Local Cultures: Religions, Ideologies, Societies; Proceedings of 5th International Conference of the European Society for the History of Science (Athens, 1–3 November 2012) (Athens: National Hellenic Research Foundation, 2014), pp. 23–30, 26). It should be noted that none of his values for the eccentricities of the two superior planets is more accurate than Ptolemy’s. That no new table for the equation of centre of Mars is associated with Ibn al-A‘lam gives the impression that he probably had not measured a new value for its eccentricity. Note that the geocentric eccentricity of Mars has remained nearly constant, about Ptolemy’s value 6;0, during the past two millennia, which may explain why Ibn al-A‘lam did not come up with a new value for it (see Mozaffari, op. cit. (Note 32), Figure 5 on p. 29). Ibn al-A‘lam’s value for the eccentricity of Saturn was used in the zījes of three Western Islamic astronomers; see J. Samsó and E. Millás, “The Computation of Planetary Longitudes in the zīj of Ibn al-Bannā,” Arabic Science and Philosophy, 8, 1998, pp. 259–86, reprinted in J. Samsó, Astronomy and Astrology in al-Andalus and the Maghrib (Variorum Collected Studies Series) (Aldershot; Burlington: Ashgate, 2007), Trace VIII, p. 273.
33.Ibn al-A‘lam’s table of the equation of centre of Mercury is preserved in Kamālī’s Ashrafī Zīj ((Note 24), F: f. 237r, G: f. 252v): the maximum equation of centre in this table is 3;40° (for arguments 99°–101°). It should be noted that his value for the eccentricity of this planet is more exact than Ptolemy’s three values 3;0, 2;45, 2;30, as found, respectively, in the Almagest, Planetary Hypotheses, and Canobic Inscription (Almagest IX.8,9: Toomer, op. cit. (Note 3), p. 459; B.R. Goldstein, “The Arabic Version of Ptolemy’s Planetary Hypotheses,” Transactions of the American Philosophical Society, 57, 1967, pp. 3–55, 19; A. Jones, “Ptolemy’s Canobic Inscription and Heliodorus’ Observation Reports,” SCIAMVS, 6, 2005, pp. 53–97, 69, 86–7); the true value during the past two millennia has been about 3;50 (note that for the eccentricity of Mercury, we consider here half of the distance between the Earth and the centre of the hypocycle in Ptolemy’s complicated model for this planet, on the circumference of which the centre of its deferent revolves).
34.See S.M. Mozaffari, “Muḥyī al-Dīn al-Maghribī’s Lunar Measurements at the Maragha Observatory,” Archive for History of Exact Sciences, 68, 2014, pp. 67–120, 105.
35.See E.S. Kennedy, “The Astronomical Tables of Ibn al-A‘lam,” Journal for the History of Arabic Science, 1, 1977, pp. 13–23; R.P. Mercier, “The Parameters of the Zīj of Ibn al-A‘lam,” Archives Internationales d’Histoire des Sciences, 39, 1989, pp. 2–50.
36.See Mozaffari, “Limitations,” Part 2, esp. pp. 395–97; Mozaffari, “An Analysis,” esp. p. 212.
37.During the period at which Bīrūnī was busy with writing al-Qānūn, he was 57 years, at least. As he states, until that time he had not yet observed desirably or investigated in depth the fixed star other than an observation of the star Spica (α Vir) on 2 July 1009, which he employed for the derivation of the precessional motion (see Mozaffari, “Limitations,” Part 2, p. 405). Despite a good number of solar and lunar observations that he made by himself or reported from his Muslim predecessors, he mentions nothing about new planetary observations. His procedure of correcting and converting Ptolemy’s planetary epoch mean and apogee longitudes to his epoch and base meridian (see al-Qānūn X.4: Bīrūnī, op. cit. (Note 2), vol. 3, pp. 1193–8) is a good example of the crude, artificial ways that a typical medieval astronomer could invent. It also reflects how difficult it could be for a single-handed astronomer (no matter whatever skilful or motivated) to cope with the determination of all fundamental parameters during his lifetime.
38.The term al-i‘tibār, as al-Khāzinī (Kayfiyyat al-i‘tibār, in: Zīj (Note 21), V: f. 4r) defines, connotes the “experiment” in the modern scientific method: “We called it [i.e., our method] ‘the experiment method’ (ṭarīq al-i‘tibār). […] In the experiment, observed facts (musallamāt marṣūda) are taken, and what are wanted (maṭlūbāt) are based on them.”
39.Khāzinī, Kayfiyyat al-i‘tibār, in: Zīj (Note 21), V: ff. 16v–17r.
40.Khāzinī, Kayfiyyat al-i‘tibār II.4, in: Zīj (Note 21), V: f. 8r.
41.Kamālī, Zīj (Note 24), F: f. 47r, G: ff. 50v–51r.
42.See Mozaffari, op. cit. (Note 29), pp. 520–2, 531–5.
43.See G. Saliba, “An Observational Notebook of a Thirteenth-Century Astronomer,” Isis, 74, 1983, pp. 388–401; “Solar Observations at Maragha Observatory,” Journal for the History of Astronomy, 16, 1985, pp. 113–22; “The Determination of New Planetary Parameters at the Maragha Observatory,” Centaurus, 29, 1986, pp. 249–71 (these three papers are reprinted in G. Saliba, A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam (New York: New York University, 1994), pp. 163–76, 177–86, 208–30); Mozaffari, op. cit. (Note 34).
44.See Mozaffari, op. cit. (Note 29), pp. 520–22, 530–31. Also, he has a non-Ptolemaic value for the inclination of Venus in his earlier zīj, the Tāj al-azyāj (Crown of the zījes), written in Damascus; see Mozaffari, op. cit. (Note 29), pp. 521, 531–5.
45.See K. Yabuuti, “The Influence of Islamic Astronomy in China,” in Saliba and King, op. cit. (Note 10), pp. 547–59; “Islamic Astronomy in China during the Yuan and Ming Dynasties” trans. and partially revised by Benno van Dalen, Historia Scientiarum, 7, 1997, pp. 11–43; B. van Dalen, “Islamic and Chinese Astronomy under the Mongols: A Little-Known Case of Transmission,” in Y. Dold-Samplonius, J.W. Dauben, M. Folkerts and B. van Dalen (eds), From China to Paris: 2000 Years Transmission of Mathematical Ideas (Stuttgart: Franz Steiner, 2002), pp. 327–56; B. van Dalen, “Islamic Astronomical Tables in China: The Sources for the Huihui li,” in S.M.R. Ansari (ed.), History of Oriental Astronomy; Proceedings of the Joint Discussion-17 at the 23rd General Assembly of the International Astronomical Union, organised by the Commission 41 (History of Astronomy), held in Kyoto, August 25–26, 1997 (Dordrecht: Kluwer, Springer, 2002), pp. 19–30. On the accuracy of the values Jamāl al-Dīn measured for the eccentricities of Saturn and Jupiter, see Mozaffari, op. cit. (Note 32), esp. p. 27.
46.See S.M. Mozaffari and J.M. Steele, “Solar and Lunar Observations at Istanbul in the 1570s,” Archive for History of Exact Sciences, 69, 2015, pp. 343–62; Mozaffari, op. cit. (Note 5), p. 10.
47.E.g., the tables for the solar and lunar equation of centre in Kharīdat (B: Berlin, Staatsbibliothek zu Berlin, no. Ahlwardt 5699 = WE. 193, ff. 28r–v, 34r–v, C1: Cairo, Dār al-Kutub, Ṭal‘at Mīqāt Collection, no. 900, ff. 50v–51r, 58r–v, C2: Cairo, Dar al-Kutub, Ṭal‘at Mīqāt Collection, no. 76, ff. 37v–38r, 43v–44r, E: Istanbul, Süleymaniye, Esad Efendi Collection, no. 1976, ff. 4r–v, 6v–7r, K: Kandilli Observatory, no. 183, ff. 48v–49r, 56r–v) are always additive, but not displaced (like those in Ulugh Beg’s zīj); the first has Max = 3.863° and Min = 0° and the latter, Max = 26.519° and Min = 0°; note that the tabular numerical values in this work are in decimals. The maximum values for the solar and lunar equation of centre are thus, respectively, equal to 1;55,53° and 13;15,34°, which are the same values adopted in Ulugh Beg’s zīj (see Table 1; Mozaffari, “Limitations,” Part 1, p. 326; Mozaffari, op. cit. (Note 34), p. 105).
48.Taqī al-Dīn, Sidrat, K: Istanbul, Kandilli Observatory, no. 208/1 (up to f. 48v; autograph), f. 17v, N: Istanbul, Süleymaniye Library, Nuruosmaniye Collection, no. 2930, f. 23r, V: Istanbul, Süleymaniye Library, Veliyüddin Collection, no. 2308/2 (from f. 10v), f. 25r; Kharīdat (Note 47), C1: f. 8v, C2: f. 6r, E: f. 25v, K: f. 6v.
49.See A. Sayılı, The Observatory in Islam (Ankara: Türk Tarih Kurumu Basimevi,  1988), pp. 290–2.
50.On this work, see B. van Dalen, “Origin of the Mean Motion Tables of Jai Singh,” Indian Journal of History of Science, 35, 2000, pp. 41–66; D. Pingree, “An Astronomer’s Progress,” Proceedings of the American Philosophical Society, 143, 1999, pp. 73–85; D. Pingree, “Philippe de La Hire at the Court of Jayasiṃha,” in Ansari, op. cit. (Note 45), pp. 123–31; D. Pingree, “Philippe de La Hire’s Planetary Theories in Sanskrit,” in Dold-Samplonius et al. (Note 45), pp. 429–53; S.M.R. Ansari, “Survey of Zījes Written in the Subcontinent,” Indian Journal of History of Science, 50, 2015, pp. 575–601; and the references mentioned therein.
51.Khayr-Allāh Shīrāzī and Sawā’ī Jai Singh, Muḥammadshāhī zīj, P1: Iran, Parliament Library, no. 2144, pp. 196, 201, P2: Iran, Parliament Library, no. 6121, pp. 264, 269, P3: Iran, Parliament Library, no. 15780, pp. 232–3 (in blank), L: London, British Library, no. Add 14373, ff. 173v–174r; P. de La Hire, Tabulae astronomicae Ludovici magni jussu et munifrcentia exaratae et in lucem editae, 2nd ed. (Paris: Montalant, 1727), section of tables, pp. 64, 66–7. Ghulām Ḥusayn Jaunpūrī, in his Bahādurkhānī Encyclopedia (Jāmi’-i Bahādurkhānī (Calcutta, 1835), p. 616), lists the daily motions of the planetary apogees and nodes and ascribes the discovery of the difference between them to the Muḥammadshāhī’s observations (!). It should be noted that, compared to the famous astronomical tables in the seventeenth century, Philip de La Hire’s values at least for the motions of the apogee and the node of Venus are not accurate; e.g., E. Halley, Astronomical Tables with Precepts both in English and Latin (London: Printed for William Innys, 1752), section of tables, p. Uu) gives their motions in 1000 years as 15;42,13° (≈ 56.5″/y) and 8;36,40° (≡ 31.0″/y), respectively; the latter was approved by T. Bugge “Astronomical Observations on the Planets Venus and Mars, Made with a View to Determine the Heliocentric Longitude of Their Nodes, the Annual Motion of the Nodes, and the Greatest Inclination of Their Orbits,” Philosophical Transactions of the Royal Society of London, 80, 1790, pp. 21–31, 26; the true values at the time are ~ 50.7″/y and ~ 32.4″/y.
52.As we have shown elsewhere, this work is a treasure of non-Ptolemaic values for the structural parameters of the motions of the planets in longitude and in latitude (see Mozaffari, op. cit. (Note 29), pp. 535–6), which had not received the attention that it deserves. For example, no reference to these values can be found in E.S. Kennedy’s brief statement about the planetary equations and latitudes in Ulugh Beg’s zīj in Kennedy, op. cit. (Note 28), p. 167, nor, e.g., in E.S. Kennedy, Astronomy and Astrology in the Medieval Islamic World (Aldershot: Ashgate-Variorum, 1998), Trace XI, where he numerates the heritage of Ulugh Beg.
53.The error in al-Kāhsī’s value is about −1°, but as mentioned in the apparatus to Table 2, his value has been updated from that adopted in the Īlkhānī zīj.
54.The Middle Eastern Islamic astronomers apparently did not deal with the problem of long-term changes in the fundamental parameters until about the last quarter of the thirteenth century, when Quṭb al-Dīn al-Shīrāzī (ad 1236–1311) constructed a solar model in order to account for the continuous decrease observed in the obliquity of the ecliptic and in the solar eccentricity since Ptolemy’s time; see S.M. Mozaffari, “A Forgotten Solar Model,” Archive for History of Exact Sciences, 70, 2016, pp. 267–91.
55.On the model and its later receptions and parameters, see G.J. Toomer, “The Solar Theory of az-Zarqāl: A History of Errors,” Centaurus, 14, 1969, pp. 306–36; J. Samsó and E. Millás, “Ibn al-Bannā’, Ibn Ishāq and Ibn al-Zarqālluh’s Solar Theory,” appeared in 1989, in J. Samsó (ed.), Islamic Astronomy and Medieval Spain (Ashgate: Variorum, 1994), Trace X; J. Samsó, “Al-Zarqal, Alfonso X and Peter of Aragon on the Solar Equation,” in Saliba and King, op. cit. (Note 10), pp. 467–76; G.J. Toomer, “The Solar Theory of az-Zarqāl: An Epilogue,” in: Saliba and King, op. cit. (Note 10), pp. 513–9; 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, 12, 1998, pp. 51–111.
56.See Samsó and Millás, op. cit. (Note 55), pp. 18, 21, 25–6; Calvo, op. cit. (Note 55), pp. 58–9.
57.M. Boutelle, “The Almanac of Azarquiel,” Centaurus, 12, 1967, pp. 12–9 (reprinted in Kennedy, Studies (Note 10), pp. 502–10), p. 13.
58.See E.S. Kennedy and D.A. King, “Indian Astronomy in Fourteenth Century Fez: The Versified Zīj of al-Qusunṭīnī,” Journal for the History of Arabic Science, 6, 1982, pp. 3–45, reprinted in D.A. King, Islamic Mathematical Astronomy (London: Variorum, 1986), Trace VIII, pp. 10–1; J. Samsó, “Andalusian Astronomy in 14th Century Fez: al-Zīj al-Muwāfiq of Ibn ‘Azzūz al-Qusanṭīnī” Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, 11, 1997, pp. 73–110, reprinted in Samsó, Astronomy and Astrology (Note 32), Trace IX, pp. 83, 102; J. Samsó, “Ibn al-Raqqām’s al-Zīj al-Mustawfī in MS Rabat National Library 2461,” in N. Sidoli and G. van Brummelen (eds), From Alexandria, through Baghdad (Heidelberg; New York; Dordrecht; London: Springer, 2014), pp. 297–325, 315, 317–18; Samsó and Millás, op. cit. (Note 32), pp. 265–66, 272–73; J. Chabás and B.R. Goldstein, “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād,” Archive for History of Exact Sciences, 48, 1994, pp. 1– 41, reprinted in Chabás and Goldstein, Essays on (Note 32), 179–226, pp. 5, 33; Chabás and Goldstein, op. cit. (Note 15), pp. 598–600, 605–06, 609–11.
59.See J. Chabás and B.R. Goldstein, The Alfonsine Tables of Toledo (Dordrecht: Kluwer Academic Publishers, 2003), pp. 153–55, 159–60, 253–54.
60.See E. Rosen, Three Copernican Treatises, 3rd ed. (New York: Octagon Books, 1971), p. 81; N.M. Swerdlow, “A Summary of the Derivation of the Parameters in Commentariolus from the Alfonsine Tables,” Centaurus, 21, 1977, pp. 201–13, 205. The following literature represents only a few examples from the Latin and Jewish astronomical corpus brought into light and investigated in recent years, in which the idea of the equality of the orbital elements of the Sun and Venus were adopted: B.R. Goldstein, The Astronomy of Levi Ben Gerson (1288-1344), A Critical Edition of Chapters 1-20 with Translation and Commentary (New York: Springer, 1985), p. 113; B.R. Goldstein and J. Chabás, “An Occultation of Venus Observed by Abraham Zacut in 1476,” Journal for the History of Astronomy, 30, 1999, pp. 187–200, 188; B.R. Goldstein, “An Anonymous Zij in Hebrew for 1400 A.D.: A Preliminary Report,” Archive for History of Exact Sciences, 57, 2003, pp. 151–71, 160–61; J. Chabás, “Astronomy for the Court in the Early Sixteenth Century, Alfonso de Córdoba and his Tabule Astronomice Elisabeth Regine,” Archive for History of Exact Sciences, 58, 2004, pp. 183–217, 188; J. Chabás and B.R. Goldstein, The Astronomical Tables of Giovanni Bianchini (Leiden: Brill, 2009), p. 34.
61.Yaḥyā b. Abī Manṣūr, Zīj al-mumtaḥan, E: Madrid, Library of Escorial, árabe 927, published in The Verified Astronomical Tables for the Caliph al-Ma’mūn, F. Sezgin (ed.) with an introduction by E.S. Kennedy (Frankfurt am Main: Institut für Geschichte der Arabisch-Islamischen Wissenschaften, 1986), ff. 14v, 41r–v, L: Leipzig, Universitätsbibliothek, Vollers 821, ff. 67v, 92v–93r; Ibn Yūnus, Zīj (Note 22), L: p. 121; Caussin, op. cit. (Note 22), p. 221; Kennedy and Pingree, op. cit. (Note 10), p. 226. On the Mumtaḥan zīj, also, see S.M. Mozaffari and G. Zotti, “Bīrūnī’s Telescopic-Shape Instrument for Observing the Lunar Crescent,” Suhayl, 14, 2015, pp. 167–88; S.M. Mozaffari, “A Revision of the Star Tables in the Mumtaḥan zīj,” Suhayl, 15, 2016–2017, pp. 67–100; and the references mentioned therein.
62.Ḥabash, Zīj (Note 20), I: ff. 90v, 117r–v, B: ff. 30v, 55r–v; Debarnot, op. cit. (Note 20), pp. 41–2, 44.
63.C.A. Nallino (ed.), Al-Battani sive Albatenii Opus Astronomicum (Publicazioni del Reale osservatorio di Brera in Milano, n. XL, pte. I–III, Milan: Mediolani Insubrum, 1899–1907. The Reprint of Nallino’s edition: Frankfurt: Minerva, 1969), vol. 2, p. 128.
64.Ibn Yūnus, Zīj (Note 22), L: pp. 121, 188–90; Caussin, op. cit. (Note 22), p. 221.
65.Ibn al-A‘lam’s table of the solar equation centre has been preserved in the Ashrafī zīj (Note 24), F: f. 236v, G: f. 251v, but his equation tables of Venus are not extant. However, Kamālī says, in his Ashrafī zīj VIII.5: (Note 24), F: f. 230r, G: f. 247r, that the difference between al-Battānī’s and Ibn al-A‘lam’s values for the maximum equation of centre of Venus is only one arc-minute.
66.Bīrūnī, op. cit. (Note 2), vol. 3, p. 1258.
67.The value 2;4,39 is the average of the two values Bīrūnī measured for the solar eccentricity from his own observations carried out in ad 1016–1017. His table of the solar equation of centre in al-Qānūn is asymmetric with Max = 3;59,3,21° for a mean eccentric anomaly of 266° and Min = 0;0,56,39° for argument 90° (Bīrūnī, op. cit. (Note 2), vol. 2, pp. 710, 716), and therefore qmax = 1;59,3,21°, which strictly corresponds to e = 2;4,39.
68.The equation tables in Khāzīnī’s zīj are symmetric. The table of the equation of centre of the Sun has the maximum value 2;12,23° for argument 92°, and that of Venus, 2;23° for arguments 85°–94° (al-Khāzinī, Zīj (Note 21),V: ff. 131v, 179r; L: London, British Linbrary, Or. 6669, ff. 113v, 146r; Wajīz [Compendium of] al-Zīj al-mu‘tabar al-sanjarī, S: Tehran: Sipahsālār, no. 682, pp. 58–9, 84–9). These tables are preserved in Kamālī’s Ashrafī zīj (Note 24), F: ff. 238r, 239v, G: ff. 250v, 252r; the first is unchanged but has a scribal error, giving a value of 2;12,25, instead of 2;12,20, , for argument 93°; the second is displaced with Max = 5;23° for arguments 216°–225° and Min = 0;37° for arguments 35°–44°.
69.Al-Fahhād, Zīj (Note 21), pp. 154–5.
70.Naṣīr al-Dīn al-Ṭūsī, Īlkhānī zīj, C: University of California, Caro Minasian Collection, no. 1462, p. 124; T: Iran, University of Tehran, Central Library, Ḥikmat collection, no. 165, ff. 71v–73r; P: Iran, Parliament Library, no. 181, f. 42v; M: Iran, Mashhad, Holy Shrine Library, no. 5332a, f. 75v. The table for the equation of centre of Venus is displaced with Min = 0;1° for arguments 83°–94° and Max = 3;59° for arguments 261°–271°. Note that, as shown elsewhere (see Mozaffari, op. cit. (Note 34), pp. 110–2), all of the solar and lunar parameters in the Īlkhānī zīj were adopted or updated from Ibn Yūnus’s Ḥākimī zīj.
71.Mūḥyī al-Dīn al-Maghribī, Adwār al-anwār, M: Iran, Mashhad, Holy Shrine Library, no. 332, ff. 87v–88r, CB: Ireland, Dublin, Chester Beatty, no. 3665, ff. 85v–86r; also preserved in Kamālī’s Ashrafī zīj (Note 24), F: ff. 248v–249v, G: ff. 257r–v and in Shams al-Dīn Muḥammad al-Wābkanawī’s Zīj al-muhaqqaq al-sulṭānī ‘alā uṣūl al-raṣad al-Īlkhānī (The verified royal zīj on the basis of the parameters of the Īlkhānid observations), T: Turkey, Aya Sophia Library, no. 2694, f. 160v. Wābkanwī (Muḥaqqaq zīj IV.15.10: Y: Iran, Yazd, Library of ‘Ulūmī, no. 546, its microfilm is available in Tehran university central library, no. 2546, ff. 160v–161r; T: ff. 93r–93v; P: Iran, Parliament Library, no. 6435, f. 141r) reports e = 1;2,49 from al-Maghribī.
72.See Yabuuti, “Islamic Astronomy in China” (Note 45), pp. 22–4, 33. The table of the equation of centre of Venus in Sanjufīnī’s Zīj, MS. Paris: Bibliothèque Nationale, Arabe 6040, f. 50v; a table for the solar equation of centre cannot be found in this work, and the reason is as follows. In the tables on ff. 32v–34r, there are presented the true longitude of the Sun together with the lunar mean positions from 764 H to 895 H. This is a sort of the user-friendly medieval astronomical tables that dispense practitioners with the addition-subtraction procedure in the solar equation table as well as with taking into account the longitudes of its apogee, as noted in the pertinent explanatory section in I.2.2 (ff. 7r–v). In the tables on ff. 44v–46r, the mean positions of the Sun and planets are tabulated for the same period. A simple assessment of the correlated entries in both tables clearly affirms the use of a value of a bit more than 2;6 for the solar eccentricity.
73.‘Alā’ al-Dīn Abu ’l-Ḥasan ‘Alī b. Ibrāhīm b. Muḥammad al-Muṭa’’im al-Anṣārī, Ibn al-Shāṭir, al-Zīj al-Jadīd, K: Istanbul, Kandilli Observatory, no. 238, ff. 52v, 66v, L1: Leiden, Universiteitsbibliotheek, Or. 65, ff. 66r, 85r, L2: Leiden, Universiteitsbibliotheek, Or. 530, ff. 52r, 65r, O: Oxford, Bodleian Library, Seld. A inf 30, ff. 31v, 50v.
74.Jamshīd Ghiyāth al-Dīn al-Kāshī, Khāqānī zīj, IO: London: India Office, no. 430, f. 136v; P: Iran: Parliament Library, no. 6198, p. 121.
75.Ulugh Beg, Sulṭānī Zīj, P1: Iran, Parliament Library, no. 72, f. 144r, 117v–123r, P2: Iran, Parliament Library, no. 6027, ff. 134v–140r, 161v. The table for the equation of centre of Venus is displaced with Min = 0;20,41° for argument 89° and Max = 3;39,19° for argument 267°, thus qmax = 1;39,19°.
76.Yaḥyā, Zīj (Note 61), E: ff. 15r, 40r, 86v, L: ff. 60v–61r, 67r–68r; Ibn Yūnus, Zīj (Note 22), L: p. 121; Caussin, op. cit. (Note 22), p. 221. See, also, van Dalen, op. cit. (Note 16), p. 23.
77.Nallino, op. cit. (Note 63), vol. 2, p. 126.
78.Ibn Yūnus, Zīj (Note 22), L: p. 121; Caussin, op. cit. (Note 22), p. 221.
79.In his Ashrafī zīj (Note 24), F: f. 232v, G: f. 249r, Kamālī gives Ibn al-A‘lam’s values for the longitudes of the solar and planetary apogees as updated for 13 Adhar 1614 Alexander/23 Rajab 702/13 Khurdād 672 (13 March 1303) (in MS. G, the Alexandrian date is wrongly given as 14 Adhar 1612). They end with 19″, except for Mars, giving the impression that Ibn al-A‘lam’s original values were given with a precision up to arc-minutes, and the 19″ results from Kamālī’s precessional increment. The longitudes of the apogee of the Sun and Venus are, respectively, 89;5,19° and 75;55,19°. The epoch of Ibn al-A‘lam’s zīj is unknown. As set forth elsewhere (Mozaffari, op. cit. (Note 61)), it seems quite probable that (1) the second star table found in the preserved manuscripts of the Mumtaḥan zīj is a work by Ibn al-A‘lam himself, in the sense that he updated the longitudes in the first, and in all likelihood original, star table in this zīj (for the year 198 Y/ad 829–830) for the year 380 Y (ad 1011–1012) by adding an increment of 2;36°, which is in accordance with his rate of precession of 1°/70y and the interval of time of 182 between them. And (2) he attained this annual processional motion by a comparison between the value 135;6° he measured for the longitude of Regulus (α Leo) from his observation(s) carried out in 365 H (344–345 Y/ad 975–976) and the value 133;0° registered in the first Mumtaḥan star table. We convert the values for the longitudes of the apogees of Sun and Venus to the latter date, which is about 10 years before Ibn al-A‘lam passed away, by subtracting from them the value 4;40,19° (≈ (672–345)/70).
80.Bīrūnī, op. cit. (Note 2), vol. 2, p. 693, vol. 3, pp. 1193–8. He simply converts Ptolemy’s values for the longitudes of the planetary apogees to his epoch by an increment of about 13° calculated from his rate of precession of 1°/69y (see Mozaffari, op. cit. (Note 5), pp. 13–4).
81.Khāzinī, Zīj (refs. 21 and 68), V: ff. 129r, 163v; L: ff. 102v, 125v; S: pp. 53–4. In a table in MS. V, the longitudinal differences between the apogee of the Sun and those of the five planets are given to arc-minutes for the beginning of the Hijra era, which, added to the longitude of the solar apogee, are generally in agreement with the values given in the main table of the radixes of the Sun, Moon, and planets in this work. Kamālī, Zīj (Note 24), F: f. 232v, G: f. 249r, has added 10;18,48° to Khāzinī’s values in order to update them for 23 Rajab 702 (13 March 1303). This increment is in accordance with the precessional motion of 1°/66y and the period of about 681 Persian years elapsed from the beginning of the Hijra era to the date in question. Khāzinī has added 7;35° to the longitudes of the apogee of Saturn, Jupiter, and Mercury in the Almagest, which approximately agrees with his rate of precession of 1°/66y and the interval of time of about 487 years, from the mid-130s ad to 622 ad, but for Mars and Venus, his values are by 12° and 12;35° greater than Ptolemy’s. We have added an increment of 7;33° to Khāzinī’s values in order to convert them to 1 January 1120 ad, a date falling within the period of his fruitful career.
82.Al-Fahhād, Zīj (Note 21), p. 73; see, also, Mozaffari, op. cit. (Note 5), pp. 17–8.
83.Īlkhānī Zīj (Note 70), C: pp. 56, 120, P: ff. 20v, 41r, M: ff. 33v, 73v.
84.Al-Maghribī, Adwār (Note 71), CB: f. 80v, M: f. 82v. As al-Maghribī explains in detail in his Talkhīṣ al-majisṭī IV.5–6 (MS. Leiden, Universiteitsbibliotheek, Or. 110), he measures the solar parameters from his four solar observations in ad 1264–1265, and then computes back from his figure 88;50,43° for the longitude of the solar apogee for 16 December 1264 to 88;20,47° for his epoch, 17 January 1232, with a precessional and apogee motion of 1° in every 66 Persian/Egyptian years.
85.We know (Yabuuti, “Islamic Astronomy in China” (Note 45), pp. 22, 24) that Jamāl al-Dīn and his team of Persian astronomers in China measured the longitude of the solar apogee as 89;21° in 660 H (ad 1261/1262). In Sanjufīnī’s zīj (Note 72), which is on the basis of their parameter values, the apogeal motion with a rate of 1°/60y is clearly different from the precessional one with a rate of 1°/73y, which can be derived from the values tabulated for them in the two separate columns in the table for the solar and the planetary mean motions from 764 to 895 H (ff. 44v–46r). Sanjufīnī (f. 44v) gives the values 91;1,20° and 78;46°, respectively, for the longitudes of the apogees of the Sun and Venus for 24 Jumādā I 764 (10 March 1363, according to the astronomical Hijra calendar). Accordingly, it seems that he added an increment of 1;40,20° to Jamāl al-Dīn’s value in order to update it for his own time, which is in accordance with the rate of apogeal motion of 1°/60y and the period of about one century between them. If this is true, then Jamāl al-Dīn’s value for the longitude of the apogee of Venus for ad 1261/1262 was equal to 77;6°. It should be noted that Jamāl al-Dīn precedes Ibn al-Shāṭir (see Mozaffari, op. cit. (Note 5)) in putting a clear distinction between the apogeal and precessional motions by one century.
86.Ibn al-Shāṭir, Zīj (Note 73), K: f. 52r, L1: f. 65v, L2: 50v, O: f. 31r, PR: f. 100r. A curious feature of Ibn al-Shāṭir’s astronomy is that he correctly believed that the motion of the solar and planetary apogees (which he takes as equal to 1° in 60 Persian/Egyptian years) is not equal to, but larger than, the precession (which he takes as 1°/70y); see Mozaffari, op. cit. (Note 5). It is to be noted that in my previous study (Mozaffari, “Limitations,” Part 1, p. 326), the value 79;12° for the longitude of the solar apogee, which V. Roberts (“The Solar and Lunar Theory of Ibn ash-Shāṭir, a pre-Copernican Copernican Model,” Isis, 48, 1957, pp. 428–32, p. 430) quotes from Ibn al-Shāṭir’s Nihāyat al-Sūl fī Taṣḥīḥ al-Uṣūl (A Text of Final Inquiry in Correcting the Parameters), was taken as Ibn al-Shāṭir’s formal value. He derived this value from an observation made in Damascus on the first day of the year 701 Y/24 Rabī’ I 732 H (24 December 1331, JDN 2207563). This value is strangely in error by more than 12°. Moreover, it is inconsistent with the more precise values Ibn al-Shāṭir lists in his table of the longitudes of the solar and planetary apogees in the Jadīd zīj, which gives a value about 89;52° for the given date. It seems very likely a scribal error to have been occurred in the manuscript Roberts made use of (Oxford, Bodleian, Marsh 139), because of the similarity of the abjad numerals in the form → (89;52) (79;12).
87.Αl-Κāshī, Zīj (Note 74), ΙΟ: ff. 127v, 128v, P: pp. 107, 109. Note that al-Kāshī’s values was updated from the Īlkhānī zīj with taking the precessional and apogee motion as equal to 1° in 70 Persian/Egyptian years.
88.Ulugh Beg, Zīj (Note 75), P1: ff. 116r, 143r, P2: ff. 133r, 160v. The table has λA = 90;30,4,48° for the Sun; the table of the solar equation of centre is always additive, but not displaced, therein all entries have been increased by qmax = 1;55,53,12°; therefore, the same value has been subtracted from the values of λA, and then the resultants were tabulated. ‘Alī b. Muḥamamd Qūshčī (ca.ad 1402–1474), an astronomer of the Samarqand observatory, explains this procedure of the preparation of the conventional equation tables in the case of the Sun in his Commentary on Zīj of Ulugh Beg (Sharḥ-i Zīj-i Ulugh Beg, N: Iran, National Library, no. 20127–5, p. 292, P: Iran, Parliament Library, no. 6375/1, p. 169, PN: USA, Rare Book & Manuscript Library of University of Pennsylvania, LJS 400, f. 258r).