Abstract
This paper analyses a kinematic model for the solar motion by Quṭb al-Dīn al-Shīrāzī, a thirteenth-century Iranian astronomer at the Marāgha observatory in northwestern Iran. The purpose of this model is to account for the continuous decrease of the obliquity of the ecliptic and the solar eccentricity since the time of Ptolemy. Shīrāzī puts forward different versions of the model in his three major cosmographical works. In the final version, in his Tuḥfa, the mean ecliptic is defined by an eccentric of fixed mean eccentricity and a mean obliquity fixed with respect to the celestial equator, and the center of the epicycle, which is inclined to the eccentric, moves on the eccentric with an annual period. By an additional slow motion of the sun on the epicycle, the true eccentricity of the solar deferent, defined by the annual motion of the sun, and the sun’s extreme declination from the equator change, accounting for the reduction of the eccentricity and the obliquity of the ecliptic since the time of Ptolemy.
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Notes
Swerdlow (1975, p. 50).
See Moesgaard (1989, p. 312).
These are discussed at length in Mozaffari (2013a, Part 2).
On the trepidation models, a vast, elaborate literature exists, of which the following works are worth reading: Comes (1996, 2001), Goldstein (1965, 1994), Hartner (1971), Mancha (1998, 2004), Mercier (1976/1977, 1996), North (1967), Samsó (1994b, 1998, pp. 93–96, 2001, pp. 169–174), Samsó and Millás (1994), Swerdlow (1975), Swerdlow and Neugebauer (1984, pp. 129–148). On Islamic astronomical tables see Kennedy (1956) and Samsó et al. (2001); these two main sources of the knowledge of Islamic astronomical tables, the so-called zījes, are followed by a new comprehensive survey that is currently prepared by Benno Van Dalen.
Bīrūnī 1954–1956, Vol. 2, p. 648.
Muḥyī al-Dīn, Talkhīṣ al-majisṭī, f. 115r.
Shīrāzī, Ikhtiyārāt, f. 26v; Tuḥfa, f. 18r; Nihāyat, P1: f. 18r, P2: f. 39r. The al-Ma’mūnīc observations were made by Yaḥyā b. Abī Manṣūr (d. 830) at Baghdad and by Khālid b. Abd al-Malik al-Marwarūḍī and his team at the monastery of Murrān on a hill in the vicinity of Damascus. Shīrāzī correctly ascribes the value 23;\(33^{\circ }\) to the first and associates 23;\(35^{\circ }\) with the latter. Bīrūnī is more precise and mentions all the values 23;33,\(52^{\circ }\), 23;33,\(57^{\circ }\), and 23;34,\(27^{\circ }\) he found in his sources concerning the observational results at Damascus from 831 to 833 AD (the true modern value at the time \(\sim \)23;35,\(33^{\circ }\)). Bīrūnī’s own value is 23;\(35^{\circ }\) derived from his observations of the extremal solar noon altitudes in the latter part of the 1010s (for the analysis of them, see Said and Stephenson 1995, esp. p. 123). As he notices, the majority of the early Islamic astronomers observed either exactly this value or the values close to it; e.g. 23;34,\(51^{\circ }\) in a table in which the solar noon altitudes observed by Khālid in Damascus were written down; the Banū Mūsā at Baghdad (at Sāmarrā’, they had found 23;34,\(30^{\circ }\)); Sulaymān b. ‘Iṣmat of Samarqand: 23;34,\(40^{\circ }\) (according to Bīrūnī, Sulaymān adjusted the solar noon altitudes at the solstices by the parallax, by which he yielded 23;33,\(42^{\circ }\)); al-Battānī in Raqqa (Ṣābi’ zīj, Sect. 4: Nallino [1899–1907] 1969, Vol. 3, p. 18: from the repeatedly observations of the solar yearly extremal zenith distances, he found \(z_{\mathrm{min}} = 12{;}26^{\circ }\) and \(z_{\mathrm{max}} = 59{;}36^{\circ }\)); ‘Abd al-Raḥmān al-Ṣūfī (903–986) in Shiraz; and Abu ’l-Wafā’ al-Būzjānī (940–997/8) and Abū Ḥāmid al-Saghānī (d. 990) at Baghdad; Bīrūnī, al-Qānūn IV.1: 1954–1956, Vol. 1, pp. 363–366; also, see Kennedy (1973, pp. 32–43).
E.g. Almagest III.1: Toomer (1998, p. 134).
See Neugebauer (1962, p. 267).
Taqī al-Dīn, Sidrat, K: f. 36v. About Taqī al-Dīn’s observations at Istanbul in the 1270s, see Mozaffari and Steele (2015).
See Goldstein (1971, p. 10) and the other sources mentioned in note 5 above.
See Ibn al-Shāṭir’s models in Roberts (1957, 1966), Kennedy and Roberts (1959), Abbud (1962); Quṭb al-Dīn’s models in Kennedy (1966) (these studies have been collected in Kennedy and Ghanem 1976; Kennedy 1983); al-Ṭūsī’s models in his Tadhkira: al-Ṭūsī 1993 and Hartner (1969) that deals with Ṭūsī’s lunar model; also see Hartner (1973). For al-‘Urḍī’s models, see esp. Saliba [1989] 1994, pp. 135–142; also, the other papers by Saliba collected in his 1994 book (hereafter, the page references are to Saliba’s 1994 book). Ṭūsī and the members of the so-called Maragha School applied Ṭūsī’s Couple in a more complicated and matured way to the planetary theory, and Copernicus did the same in De Revolutionibus III. By the translation of a cosmographical work of the Maragha circle into Greek, which did through the oral teachings of Shams al-Dīn Muḥammad al-Wābkanawī al-Bukhārī (1254?–after 1316), the most prominent astronomer of the second period of the Maragha observatory (about him, see, e.g., Mozaffari 2013b, pp. 238–242), to Gregory Chioniades, this device was entered into the Byzantine literature which were available in Padua about Copernicus’s time. Neugebauer notes this and reproduces the diagram showing Ṭūsī’s Couple in MS. Vatican, gr. 211 (1975, Vol. 2, p. 1035, Vol. 3, Plate IX on p. 1456; the cosmographical text in question has been edited in Paschos and Sotiroudis 1998). These opened new venues for the research on the transmission of Ṭūsī’s Couple, in particular, and the Maragha models in general from the Middle East to Europe in the recent decades. It deserves noting that although the transmission had most probably occurred in reality (as early as 1956, Neugebauer notices this: 1956, p. 170; also, see his enlightening statement in 1968, p. 90; Swerdlow 1973, p. 504), nevertheless the question is still “when, where, and in what form he learned of Maragha theory” (Swerdlow and Neugebauer 1984, pp. 41–48).
E.g. Bīrūnī’s statement concerning the solar apogee motion in al-Qānūn VI.8: 1954–1956, Vol. 2, p. 685; translated in Hartner and Schramm (1961, p. 218).
See Mozaffari (2013a, Part 1: p. 326 (Table 3, nos. 7 and 8), 330, Part 2: pp. 393, 394–395).
This is the best medieval approximation to the true rate \(1^{\circ }{/}71.6^{\mathrm{y}}\) and also can be found in the Barcelona Tables (written c. 1381); see Dorce (2002–2003, p. 198); 2003, pp. 111, 180. It was also independently measured in Italy or France in 1306, as documented in a codex preserved in Vienna, no. 5311, f. 137r; see Goldstein (1994, pp. 193, 196–197); also, for the star tables in this manuscript, see Kunitzsch (1986).
Ibn Yūnus, Zīj, L: pp. 108, 125; Caussin (1804, p. 153).
Al-Ṭūsī (1993, Vol. 1, pp. 123, 125); Shīrāzī, Ikhtiyārāt, ff. 27r, 30r; Tuḥfa, ff. 19r, 22v; Nihāyat, P1: ff. 18r, 20r, P2: ff. 39v, 43v. In his earlier wok, Mu‘īniyya (IV.6, p. 30), Ṭūsī does not mention this value, and only refers to “\(1^{\circ }{/}100^{\mathrm{y}}\) found in the time of Ptolemy and Menelaus and \(1^{\circ }{/}66^{\mathrm{y}}\) observed by the moderns”.
Talkhīṣ III.1: f. 31r (about this work, see Saliba [1983, 1985, 1986] 1994, pp. 163–186, 208–230; Mozaffari 2014). Muḥyī al-Dīn had already reached this figure through his observations carried out at Damascus, and this was known to the Western Islamic astronomers through the diffusion of his Tāj al-azyāj there (see Samsó 1998, pp. 96–97; 2001, p. 173) and also is applied to some timekeeping table by Taqī al-Dīn Muḥammad b. Ma‘rūf (see King 2004/2005, Vol. 1, pp. 64, 448).
Al-Kāshī, Khāqānī zīj II.1.4: IO: f. 27r.
Īlkhānī zīj, C: p. 203, T: f. 102v, P: f. 59v, M1: f. 104v, M2: f. 89v.
According to Niazi (2014, pp. 85–86, 98), Shīrāzī’s three works were written in the first part of the 1280s; first, Nihāyat, next, Ikhtiyārāt, and then Tuḥfa.
Al-Ṭūsī (1993, pp. 222–223) says that his method (i.e. Ṭūsī’s Couple) can be applied to accounting for the variation in the speed of the precession or in the obliquity if the truth of these two motions and their variability is ascertained. A similar remark is given by Shīrāzī prior to the explanation of his solar model.
Al-Ṭūsī (1993, pp. 125); Shīrāzī explains this trepidation theory in a confused way in Ikhtiyārāt II.4: ff. 27r–28r; Tuḥ fa II.7: ff. 19r–20v; Nihāyat II.4: P1: ff. 18r–19r, P2: ff. 39r–41r; about it, also, cf. Hartner (1971, pp. 284–287).
Shīrāzī, Ikhtiyārāt, f. 26v; Tuḥfa, f. 18r; Nihāyat, P1: f. 18r, P2: f. 39r.
Shīrāzī, Ikhtiyārāt II.9: ff. 87r–v; Tuḥfa II.8: ff. 34r–v; Nihāyat II.5: P1: f. 27r, P2: ff. 63r–v.
Shīrāzī, Ikhtiyārāt II.4: f. 28r; Tuḥfa II.7: ff. 20v–21r; Nihāyat II.4: P1: f. 19r, P2: f. 41v.
Saliba [1979a and 1979b] 1994, pp. 114, 119–134. Saliba (1987) also demonstrates the close dependence of Shīrāzī’s discussion on the height of the atmosphere on that of ‘Urḍī. Also, it is noteworthy that in his non-astronomical writings, Quṭb al-Dīn shows a heavy dependence on his Islamic predecessors, often without acknowledging them; e.g. in the case of his well-known encyclopaedia, Durrat al-Tāj, see Pourjavady and Schmidtke (2004), in which the authors go farther to conclude that “the fact that, with the exception of portions of the section on logic, no part of the philosophical sections of Durrat al-tāj was originally written by Quṭb al-Dīn al-Shīrāzī, suggests that his significance as a philosopher should be reconsidered” (ibid, p. 320). It is noteworthy that the mathematical part (first section of jumla 4) of Durrat al-tāj seems to be a Persian translation of Muḥyī al-Dīn’s al-Maghribī’s Taḥrīr al-Uṣūl (ibid, p. 313), although this needs to be checked further. The astronomical part (second section of jumla 4) of Durrat al-tāj is also a translation of ‘Abd al-Malik b. Muḥammad al-Shīrāzī’s (d. ca. 596 H/1200 AD) Talkhīṣ al-majisṭī, as Quṭb al-Dīn himself states. About Quṭb al-Dīn’s intellectual background and the manuscripts written by his own hand, see Pourjavady and Schmidtke (2007, 2009).
Saliba [1980] 1994, esp. pp. 86, 89.
See Saliba [1991] 1994, pp. 261–262, 265–266.
Saliba [1991] 1994, p. 281.
Al-‘Urḍī made a drastic change in Ptolemy’s order of the planets by placing Venus above the sun. He reasonably assumes the values which Ptolemy quotes from Hipparchus for the apparent radii of the planets at their mean distances should have been measured at their least distances, and as well, he takes the actual radii of the bodies of the sun, moon, and planets and the thickness of the sphere of lunar nodes into account in the computation of their distances. The latter consideration is indeed an improvement over Ptolemy’s procedure on the basis of which al-‘Urḍī derives the boundaries of the lunar spheres, i.e. the limits of the convex and concave surfaces of its spheres as equal, respectively, to (maximum distance \(+\) radius of the moon \(+\) thickness of its sphere of nodes) \(=\) 64;10 \(+\) 0;17,33 \(+\) 0;2,27 \(=\) 64;30 and (minimum distance \(-\) radius of the moon) \(=\) 33;33 \(-\) 0;17,33 \(=\) 33;15,27 terrestrial radii. Then, from the Ptolemaic value of 64;\(10^{\mathrm{t.r.}}\) for the moon’s greatest distance, together with committing an error in the calculation of Mercury’s least distance (wrongly assumed equal to radius of the deferent – 3 \(\cdot \) eccentricity – radius of the epicycle), he finds that the space between Mercury and Sun is not large enough to accommodate Venus, which was persuasive for him to place Venus above the sun. Consequently, this made the radius of the universe enlarge to \(140115^{\mathrm{t.r.}}\), i.e. about 7 times as large as Ptolemy’s (cf. Goldstein and Swerdlow 1970). Al-‘Urḍī’s schemata was not accepted by the later astronomers, except for Shīrāzī, but taking the thickness of the sphere of the lunar node into consideration found echoes in the later treatises; e.g. in Kāshī’s Sullam al-samā’ (the stairway to the heaven), ff. 7r and 10r where the thickness is computed as 3;28,\(47^{\mathrm{t.r.}}\).
About Kūshyār, see Bagheri et al. (2010–2011).
Shīrāzī, Ikhtiyārāt, ff. 156v–175v; Tuḥfa, ff. 136v–153v.
Four Islamic reports of the Venus transit in the period of 800–1200 AD are discussed in Goldstein (1969); another report belonging to 939 AD came to light in Vaquero and Gallego (2002). As a comment upon Almagest IX.1, in his Taḥrīr al-majisṭī (P1: pp. 282–283, P2: f. 82v, P3: f. 107v), al-Ṭūsī mentions that a certain Ṣāliḥ b. Muḥammad al-Zaynabī al-Baghdādī reports in his book named Majisṭī the two observations of the transit of Venus made by al-Shaykh Abā ‘Imrān at Baghdād and Muḥammad b. Abī Bakr al-Ḥakīm in Farsīn in the vicinity of Tūlak (all the three men are otherwise unknown); the interval of time between them was 20 years, and, in one of them, Venus was at the apogee of the epicycle (i.e. in superior conjunction) while in the other, at its perigee (i.e. in inferior conjunction). Al-Ṭūsī makes use of these reports to invalidate the idea that the two inferior planets are in the sphere of the sun as well as that the centre of their epicycles coincide with the centre of the sun’s body, that is, that they rotate about the sun (for the translation of the passage in question, see Saliba [1987] 1994, p. 149). It is not known precisely whether each of the two mentioned astronomers had observed only one of the two presumed transits of Venus, or both of them had observed both of the two transits. In any case, one of the two observations is certainly incorrect simply because none of the inferior planets can transit across the solar disc in a superior conjunction, and thus, such an observation should at best be related to a large sunspot. Al-Ṭūsī mentions them after referring to Ibn Sīnā (Avicenna’s) famous observation of the Venus transit, by means of which the time frame of them can be delimited: they should have been made somewhere between ca. 1032 (the only Venus transit that occurred during the lifetime of Ibn Sīnā) and 5 Shawwāl 644/13 February 1247 when al-Ṭūsī completed his Taḥrīr al-majisṭī (Saliba [1987] 1994, p. 145). In this period, the only two Venus transits took place on 22 May 1040 and 23 November 1153 (Espenak, NASA’s Six Millennium Catalog of Venus Transits). Another note is that in the case of Venus, 5 revolution of the anomaly occurs in 8 years; so, it is a very simple matter to contend that if Venus is at the apogee of its epicycle at a given time, then 20 years later, i.e. after the two revolutions and a half through the epicyclic anomaly, it would be located at the perigee of its epicycle. Nevertheless, the consideration of the validity of the Venus transits for a medieval astronomer (as well as a modern historian of astronomy) requires a deeper scrutiny of an inferior planet’s motion in longitude as well as in latitude. If al-Ṭūsī or al-Shīrāzī had made such a quantitative study, then it could have been known that even in the framework of Ptolemaic planetary models, and by means of applying his own parameter values, Venus transits can occur in the intervals of 105.5 or 129 years, not in periods of 20 years (cf. Neugebauer 1975, Vol. 1, pp. 227–229). Ptolemy himself describes qualitatively the circumstances involved in the observations of the transit of an inferior planet across the solar disc in Planetary Hypotheses I: (1) the centre of the epicycle at one of the nodes and (2) the planet at that node (i.e. the longitude of that node equal to the true longitude of the planet), i.e. the planet at the (true) apogee or perigee of its epicycle. He also notes that a long time must elapse between the two successive returns of the centre of the epicycle and the planet in conjunction with the sun (Goldstein 1967, pp. 6–7, 28).
Shīrāzī, Ikhtiyārāt, f. 19v; Tuḥfa, f. 11v; Nihāyat, B: f. 27v, P1: f. 13v, P2: f. 29r. It is noteworthy that sunspots are also short-lived phenomena, and then it seems strange to maintain the existence of such an appearance for a long time; around Shīrāzī’s time, there are the four reports of the observations of the large sunspots from the East Asian history dating to 15–16 September 1258, 17 February and 17 March–15 April 1276, and 31 August 1278, in which the sunspots are described as the “black spots as large as hen/goose’s eggs”; see Clark and Stephenson (1978, p. 396), Yau and Stephenson (Yau and Stephenson 1988, p. 187).
See Goldstein (1967, p. 7).
Shīrāzī, Nihāyat, P1: ff. 37v–38r, P2: ff. 93r–94v.
The pertinent text is edited and translated in Gamini and Masoumi (2013).
Note that there is still equant motion in Shīrāzī’s models produced by the eccentricity and the small epicycle.
The phrase is taken from Herschel (1851, p. 266).
It is probable that Shīrāzī himself recognized the circular reasoning in his proof and consequently did not included it in his two later works, Ikhtiyārāt and Tuḥfa.
Muḥyī al-Dīn, Talkhīṣ, f. 117v.
See Mozaffari (2014, p. 71). It should be noted that the relevant section in the Adwār (II.6: M: ff. 19r–v, CB: ff. 18r–v) gives no information about Muḥyī al-Dīn’s proof. A monograph about Muḥyī al-Dīn contribution to observational and practical astronomy at the Maragha observatory on the basis of a thorough analysis of his documented observations in the Talkhīṣ is in preparation by the present author.
E.g. al-Battānī in Ṣābi’ zīj, Sect. 18: Nallino [1899–1907] 1969, Vol. 3, p. 46.
Almagest XIII.3: Toomer (1998, pp. 601–602). This equivalence is according to Hunayn-Thābit’s translation which constituted the standards in the Arabic astronomical terminology: Arabic Almagest, S: ff. 212v–213r, PN: ff. 162r–v. For the components of Ptolemy’s planetary latitude models in the Almagest, see Pedersen (1974, pp. 358–359, 369–370), where \(\acute{{\varepsilon }} \gamma \kappa \lambda \iota \sigma \iota \zeta \) is rendered into “deviation”; Neugebauer (1975, Vol. 1, pp. 209, 214), Swerdlow (2005, pp. 51–52).
Since our author deals with these matters from a cosmographical (hay’a) point of view (as in Ptolemy’s Planetary Hypotheses), the orbital components such as epicycle and eccentric are always treated as spheres/orbs. Accordingly, instead of simply using epicycle or eccentric as the geometrical instruments/devices (as in the Almagest) in order to describe the compound motions of a heavenly object, he indicates this by referring to the equator/belt of the epicycle or eccentric spheres, that is, the great circle \(90^{\circ }\) distant from the poles of these spheres. In order not to confuse it with the celestial equator, we add “belt” immediately after “equator” in these cases.
The initial remark [1] is somewhat corrupted in the Ikhtiyārāt and Nihāyat, but complete in the Tuḥfa.
Note that Muḥyī al-Dīn’s observation of the obliquity (above, note 23) was carried out in the year 633 Yazdigird, and the interval between it and Yaḥyā b. Abī Manṣūr’s observations (200 Y) is about 433 E/P years. This added to Shīrāzī’s value 690 years between the latter and Ptolemy yields 1123 years; thus, (23;51,\(20^{\circ }\)–23;\(30^{\circ }){/}1123 \approx 0\);0,1,8\(^{\circ }\) per year or \(1'\) in each 53 E/P years.
See above, note 26.
Cf., also, Comes (2001, esp. pp. 316–318), where she made some attempts at exhibiting some traces of Ibn al-Zarqālluh’s trepidation model in Ṭūsī’s discussions in his Tadhkira. Although some exchanges might have been existed between Castile and Iran (cf. Comes 2004), it seems that there is slight chance of demonstrating the implication of the Andalusian trepidation models for the Maragha cosmographical works; two out of the three issues of similarity the late M. Comes enumerates in his 2001 study, i.e. the hypothesis of the proper motion of the solar apogee and the maximum bound of \(24^{\circ }\) for the obliquity of the ecliptic (originated in Hindu astronomy, which is, of course, different from Ibn al-Zarqālluh’s maximum limit of 23;\(53^{\circ }\)) were already available in the works of Ṭūsī’s Middle Eastern predecessors; Comes’s third issue is al-Ṭūsī’s adoption of the vernal equinox as the reference point of the trepidation; this is correct, but it can be simply contended that al-Ṭūsī points to it only when he briefly describes the simple zigzag trepidation model, and does not identify any reference point when presenting his summary of the simple trepidation model of the Eastern Islamic astronomy. It should also be noted that his discussion on the trepidation in the Tadhkira is so abridged that can not give any insight to a probable transmission of the elements embedded in the model; but, Shīrāzī’s account of the model, in which the summer solstice (Head of Cancer) is evidently specified as the reference point, does not leave any room for doubt that the Maragha astronomers only describe the simple trepidation model of Eastern Islamic origin. Moreover, Shīrāzī’s solar model with an oscillating-in-size deferent is substantially different from Ibn al-Zarqālluh’s solar model with the variable eccentricity.
Ulugh Beg, Sulṭānī zīj II.4, P1: f. 19r; P2: f. 12v; Qūshčī, N: p. 118, P: p. 69, PN: f. 101v.
Measured from his observed values for the extremal noon altitudes of the sun at Istanbul in 1577: 72;30,8,\(29^{\circ }\) on 11 June 1577 and 25;32,20,\(14^{\circ }\) on 11 December 1577 (Taqī al-Dīn, Sidrat, K: f. 17v on the right margin). See Mozaffari and Steele (Mozaffari and Steele 2015, pp. 350, 352).
Taqī al-Dīn, Sidrat, K: ff. 18r–v.
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Mozaffari, S.M. A forgotten solar model. Arch. Hist. Exact Sci. 70, 267–291 (2016). https://doi.org/10.1007/s00407-015-0167-7
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DOI: https://doi.org/10.1007/s00407-015-0167-7