The Orbital Elements of Venus in Medieval Islamic Astronomy: Interaction Between Traditions and the Accuracy of Observations

Abstract

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.

Keywords

Sun Venus eccentricity apogee apsidal line Ptolemy Islamic astronomy Middle East Ibn al-Zarqālluh Āryabhaṭa Maragha Samarqand

Introduction

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

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.

Figure 1.

A simple schemata of the Ptolemaic equant model for Venus.

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

Derivation of the geocentric orbital elements of Venus from the heliocentric parameters

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

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.

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 λ_A 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:

e₁ = 0.01263347250 −0.0003172038674 · T e₂ = 0.01641851815 – 0.0003269942864 · T λ_A = 92.24474506° + 18.78536894° · T + 0.05462286083° · T²

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°/53.2^y. 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.

Figure 2.

A schematic view of the transformation of the heliocentric orbital elements to the geocentric ones in the case of Venus.

The changes in the past 2000 years are given in the following:

	ad 1	ad 2000
e ₁	0;47¾	0;45½
e ₂	1;1	0;59
λ _A	54;54°	92;15°

It can be seen that the rate of decrease is small, only about 0;1 in every millennium, such that it can be neglected. Also, the two geocentric eccentricities are approximately equal, and the ratio e₁/e₂ remains nearly constant, a bit less than 0.8, during the past two millennia.

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 λ_A of the apogee of the Sun and Venus. Historical values are indicated in both figures. These values will be discussed in the next section.

Figure 3.

The continuous graphs show the geocentric eccentricities of Venus in the period from the beginning of the Common Era to ad 1700; the lower graph: the deferent eccentricity e₁; the upper graph: the equant eccentricity (from the centre of the deferent) e₂; and the middle graph: ½(e₁ + e₂). The dashed graph depicts the Sun’s/Earth’s eccentricity. The close circles indicate Ptolemy’s value (section “Ptolemy’s model for Venus”) and those measured in the medieval Islamic period from Table 1.

Figure 4.

The continuous graph shows the longitude of the geocentric apogee of Venus in the period from the beginning of the Common Era to ad 1700. The dashed graph displays the longitude of the solar apogee. The stars stand for the values for the longitude of the apogee of Venus/Sun from the early Islamic astronomers, i.e., Yaḥyā, al-Battānī, and Ibn Yūnus. The close and open circles show the values for the longitude of the apogee, respectively, of Venus and of the Sun as determined by Ptolemy and the late Islamic astronomers from Table 2.

Medieval Islamic astronomers’ values for the fundamental parameters of Venus

Middle Eastern Islamic astronomy

The values adopted for the solar and Venus’ maximum equations of centre (q_max; 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(q_max), and in Ptolemy’s eccentric equant model of the superior planets and Venus: e₁ = e₂ = R tan(q_max/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, e_Venus = e₁ = e₂ = ½e_Sun. 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).

Table 1.

The values for the eccentricities of the Sun and of Venus in Eastern Islamic zījes.

	Sun		Venus
	q _max	e	q _max	e
Yaḥyā b. Abī Manṣūr⁶¹ Mumtaḥan zīj (d. ad 830)	1;59,0°	2;4,35	1;59°	= ½ e_Sun
Ḥabash al-Ḥāsib⁶² Dimashqī/‘Arabī zīj (d. after ad 869)	1;59,0	2;4,35	1;59	= ½ e_Sun
al-Battānī⁶³ Ṣābi’ zīj (d. ad 929)	1;59,10	2;4,45	1;59	= ½ e_Sun
Ibn Yūnus⁶⁴ Ḥākimī zīj (d. ad 1007)	2; 0,30	2;6,10	2;0,30	= ½ e_Sun

Ibn al-A‘lam⁶⁵ ‘Aḍudī zīj(d. ad 985)	2;0,10	2;5,49	2;0	= ½ e_Sun
Bīrūnī⁶⁶ al-Qānūn al-mas‘ūdī (d. ad 1048)	1;59, 3,21	2;4,39⁶⁷	2;23 [Pt.]*	1;15
al-Khāzinī⁶⁸ Sanjarī zīj(fl. ca. ad 1115)	2;12,23	2;18,36	2;23 [Pt.]	1;15
al-Fahhād⁶⁹ ‘Alā’ī zīj(fl. ca. ad 1172)	1;59,0°	2;4,35	1;59	= ½ e_Sun
al-Ṭūsī⁷⁰ Īlkhānī zīj(d. ad 1274)	= Ibn Yūnus’s		1;59	1;2½
Muḥyī al-Dīn al-Maghribī⁷¹ Adwār al-anwār (d. ad 1283)	2;1	2;5,59	2;0	1;2,49
Jamāl al-Dīn al-Zaydī⁷² (fl. ca. ad 1260)	2;0,47	2;6,27	2;1	= ½ e_Sun
Ibn al-Shāṭir⁷³ Jadīd zīj (d. ca. ad 1375)	2;2,6	2;7,50	2;1	1;3½
al-Kāshī⁷⁴ Khāqānī zīj(d. ad 1429)	= Ibn Yūnus’s		2;0	1;3
Ulugh Beg⁷⁵ Sulṭānī zīj (d. ad 1449)	1;55,53,12	2;1,20	1;39,19	0;52,0

Pt.: used in Ptolemy’s Almagest.

Table 2.

The values for the longitudes of the apogees of the Sun and Venus in the Eastern Islamic zījes.

	Sun	Venus	Venus (modern)
Yaḥyā b. Abī Manṣūr⁷⁶ Murdād [5th month] 198 Yazdigird (Y)August–September 829	82;39°		70;20°
al-Battānī⁷⁷ Beginning of 1191 Alexander1 September 879	82;14		71;16
Ibn Yūnus⁷⁸ Beginning of 372 Y16 March 1003	86;10		73;34
Ibn al-A‘lam⁷⁹ 365 H/345 Yad 975–976	84;25°	71;15°	73;4
Bīrūnī⁸⁰ Beginning of 400 Y9 March 1031	85;10,19	68; 4,43 [Pt.]*	74;5,50
al-Khāzinī⁸¹ 1 January 1120	85;53	75; 8	75;45
al-Fahhād⁸² Beginning of 541 Y2 February 1172	87;50,43	75;54,43	76;43,49
Īlkhānī zīj⁸³ Beginning of 601 Y18 January 1232	88;24,51	77;40,10	77;51,5
Muḥyī al-Dīn al-Maghribī⁸⁴ End of 600 Y17 January 1232	88;20,47	81;35	77;51
Jamāl al-Dīn al-Zaydī⁸⁵ 660 Had 1261–1262	89;21	77; 6	78;26
Ibn al-Shāṭir⁸⁶ Beginning of 750 H21 March 1349	90;10,10	78;10,10	80;2,36
al-Kāshī⁸⁷ Beginning of 781 Y4 December 1411	90;59, 9	80;14,28	81;13,1
Ulugh Beg⁸⁸ Beginning of 841 H4 July 1437	92;25,58	82;25,25	81;41,46

Pt.: updated from Ptolemy’s Almagest.

The early Islamic period

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: e_Sun = 2e_Venus = 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-Kh^wārizmī (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

[1][2]

To Yaḥyā b. Abī Manṣūr, who was the first of them [i.e. the Islamic astronomers working with Ptolemy’s hypotheses], the longitudes (mawāḍi‘, Pl. of mawḍi‘, lit. “position”) of the apogees [of the planets] are close to what we put/laid down (waḍa‘nā) [i.e. wrote down/registered] for them, as if he did exactly what we did. He took their motions as equal to that of Qalb al-asad [i.e. Regulus, α Leo] [i.e. he took the apogeal motion as equal to the precession]. The exception is the apogee of the Sun, which he located in [a longitude of] 82°, and did not draw any motion for it, like in the case of the apogees of the planets, and did not point out anything indicating it.Ḥabash wrote down a table for them [i.e. the longitudes of the solar and planetary apogees] and their motions, whose results do not differ very much from those I have mentioned, except for the apogee of Venus. Its [i.e. Venus’] equation with reference to the centre of its equant sphere [i.e. the equation of centre] is equal, according to Ptolemy, to the solar equation. It was in the Zīj al-Shāh that the corrected [argument] [i.e. the true eccentric anomaly] of the Sun ([ḥiṣṣa] al-shams al-muqawwama) is the corrected argument of Venus (ḥiṣṣa al-zuhara al-muqawwama). This is impossible unless their apogees and equations are equal, and this is how they are in it [i.e. the Zīj al-Shāh]. This rule (al-ḥukm) was transferred [by Ḥabash] to the hypotheses/models (uṣūl, Pl. of aṣl, lit. “principle”) of Ptolemy, so he set [the longitude of] the apogee of Venus, which is [equal to the longitude of] the apogee of the Sun according to the moderns, and its equation of the argument equal to those of the Sun. But Ptolemy derived the equation and the apogee of the Sun from the [observations of] the solstice [and the two equinoxes] [i.e., by the so-called Seasons method], and nothing requires that they apply to the apogee and equation of Venus, except for the issue/case (qaḍiyya) found in the Zīj al-Shāh. Thereafter, al-Battānī followed him [i.e. Ḥabash] in it, and I will not add any more here but will discuss it in [my] book, Clarification of the intellects concerning the zīj of Muḥammad al-Battānī.

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

The late Islamic period

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 main stream, 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 main stream. 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°/66^y 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 main stream 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°/69^y, for the apogeal motion is smaller than the true rate of the motion of the apogee of Venus, about 1°/53^y, 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 10^iv and the nodes, 7ⁱⁱⁱ 35^iv, 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

Analysis of the accuracy of the historical values

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

Western Islamic astronomy

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: e_max ≈ 2;29, e_min ≈ 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.

Figure 5.

Ibn al-Zarqālluh’s solar model.

Figure 6.

The graphs of the geocentric eccentricities of the Sun and Venus as in Figure 3; the dash-dotted descending curve is the graph of the solar eccentricity according to Ibn al-Zarqālluh’s solar model.

All of the zījes belonging to the Western Islamic astronomical tradition employ Ibn al-Zarqālluh’s solar model, which provides the values from about 0;59–0;56 for half the solar eccentricity in the time span from Ibn al-Zarqālluh to Ibn al-Bannā’ (ad 1256–1321). In nearly all of these works, the longitudes of the apogees of the Sun and Venus are taken equal. Consequently, the errors in the values for the longitude of the apogee of Venus are as large as those in early Middle Eastern Islamic astronomy. For instance, Ibn al-Zarqālluh’s value is 87° for the year 1400 Alexander (beginning on 1 September/October 1088),⁵⁷ which is about +12° off.

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.

Discussion and conclusion

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:

The eccentricity of Venus equal to half that of the Sun.

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.

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.

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.

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:

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.

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.

Footnotes

Acknowledgements

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.

Funding

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.