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Muḥyī al-Dīn al-Maghribī’s lunar measurements at the Maragha observatory

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Abstract

This paper is a technical study of the systematic observations and computations made by Muḥyī al-Dīn al-Maghribī (d. 1283) at the Maragha observatory (north-western Iran, c. 1259–1320) in order to newly determine the parameters of the Ptolemaic lunar model, as explained in his Talkhīṣ al-majisṭī, “Compendium of the Almagest.” He used three lunar eclipses on March 7, 1262, April 7, 1270, and January 24, 1274, in order to measure the lunar epicycle radius and mean motions; an observation on April 20, 1264, to determine the lunar eccentricity; an observation on August 29, 1264, to test the model; and another on March 15, 1262, for measuring the lunar parallax. In the second period of activity at the Maragha observatory, Shams al-Dīn Muḥammad al-Wābkanawī (c. 1254–1320) adopted all of al-Maghribī’s parameter values in his Zīj, but decreased his value for the mean longitude of the moon at epoch by 0;13,11\(^{\circ }\). By comparing the times of the new moons and lunar eclipses in the period of 1270–1320 as computed from the astronomical tables of the Maragha tradition with the true modern ones, it is argued that this correction was very probably the result of actual observations.

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Notes

  1. A tentative translation of Bīrūnī, Vol. 3, p. 1193, lines 2–5.

  2. (Sayılı (1960), pp. 187–223); some essential corrections will be given in Mozaffari and Zotti (2013).

  3. Cf. (Lorch (2000), p. 401). MS. Iran, the library of parliament, no. 6412, fol. 62r: wa huwa ‘alā mā wajadnāhu bi-’l-raṣad bi-Marāgha 23 \(juz^{an}\) wa 35 daqīqa. Nevertheless, in some later copies of it (e.g., MS. Iran, Library of Parliament, no. 602, pp. 33–52, written originally by Qāḍī-zādih al-Rūmī in Rajab 892/July 1487, and MS. Iran, Library of Parliament, no. 6329, pp. 24–35), the second part (maqāla) of the treatise is the “Projection of the Astrolabe” (Tasṭīh al-asṭurlāb) of Muḥyī al-Dīn al-Maghribī, wherein that author stated his own found magnitude for the total declination, 23;\(30^{\circ }\) (bi-qadr al-mayl al-a‘zam, huwa 23;30 ‘alā mā wajadnāhu bi-’l-raṣad; the edited text in the present author’s thesis for receiving M. Sc. degree in the history of astronomy, cf.Mozaffari 2007).

  4. Cf. below, Sect. 5. Some parameter values applied to the Īlkhānī Zīj that may not be found in earlier works are: (1) the values tabulated for the longitude of the solar apogee for the years 601 Yazdigird (AD 1232) onwards; no relation between them and earlier zījes may be found. Quṭb al-Dīn al-Shīrāzī, a member of the observatory, associated these values with the “new observations” done at the Maragha observatory (al-Shīrāzī, Tuḥfa, fol. 38v; al-Shīrāzī, Ikhtiyārāt, fol. 50v). (2) The radius of Mars’ epicycle: the table for the epicyclic equation of Mars for the adjusted anomaly is symmetrical with the maximum value 42;\(12^{\circ }\) at mean distance (i.e., when the distance between the center of the planet’s epicycle and that of the earth is equal to the radius of the deferent, which is taken as \(R = 60\)) (Īlkhānī Zīj, C: p. 116, P: fols. 38v–39r, M: fols. 70v–71v). This amount corresponds to the value 40;18 for the radius of the epicycle. (3) A star table in which the ecliptical coordinates of 16 stars observed at the Maragha observatory are tabulated, accompanied by their coordinates according to Ptolemy and Ibn Yūnus (d. 1007) as well as those attributed to Ibn al-A‘lam (d. 985). All longitudes were converted to the epoch of the zīj, i.e., January 18, 1232. The coordinates attributed to Ibn al-A‘lam appear to have been derived indirectly from the Mumtaḥan zīj (Baghdad, \(c.\) 830) (van Dalen 2004a, pp. 27–28).

  5. Of them, Ibn al-Fuwaṭī (1995, Vol. 1, 146–147) mentions of a certain ‘Izz al-Dīn al-Ḥasan b. al-Shaykh Muḥammad b. al-Shaykh al-Ḥasan al-Wāsiṭī al-‘aṭṭār Shaykh Dār Mūsīyān (also cf. below, note 65).

  6. Cf. (Suter (1902), p. 155), Brockelmann, Vol. 1, p. 626, \(\hbox {S}_{1}\), p. 868, (Sarton (1953), pp. 1015–1016), (Sezgin (1978), p. 292); (Rosenfeld and Ihsanoglu (2003), p. 226). Some of his mathematical works were studied, cf.Voux (1891), Hogendijk (1993). S. Tekeli’s short entry about al-Mghribī in DSB (Gillipsie et al. 1980, Vol. 9, p. 555) only covers his mathematical works. Also, cf. M. Comes’ entry in (Hockey et al. (2007), pp. 548–549).

  7. Cf.Dorce (2003).

  8. Ibn al-Fuwaṭī, 1995, Vol. 5, p. 117.

  9. In the prologue of his Zīj al-muḥaqqaq al-sulṭānī, Shams al-Dīn Muḥammad Wābkanawī (c. 1254–1320) employs the term “new Īlkhānīd observations” specifically for Muḥyī al-Dīn’s observations; cf. below, Sect. 5. In Mozaffari and Zotti (2013), all of the indications of the term, found in the treatises written either during the lifetime of the observatory or after that are introduced.

  10. Cf. (Mozaffari (2012), pp. 363–364).

  11. The contents of the treatise were introduced in Saliba (1983). The computations related to the eccentricity of the sun and of Jupiter were subject to two critical studies by Saliba (1985; 1986; for al-Maghribī’s solar observations, cf.Mozaffari 2013a, pp. 318, 330). The present author analyzed al-Maghribī’s measurements of the Ptoemaic orbital elements of Saturn in his Ph. D. dissertation. A detailed study of al-Maghribī’s planetary and stellar observations is being prepared by him.

  12. They are the eccentricities of Mercury and Venus: 3;10 and 1;2,49, respectively, and the radius of the epicycle of Mercury: 22;30,30 (Wābkanawī, IV, 15, 10: T: fols 93r–93v, Y: 160v–161r).

  13. Cf.Mozaffari (2007). This treatise is interesting in various aspects; e.g., its clear mention of the infinite geometrical spaces.

  14. Al-Maghribī, Talkhīṣ, fol. 2r.

  15. (Sayılı (1960), p. 205).

  16. Al-Maghribī, Adwār, M: fol. 55v.

  17. Al-Maghribī, Adwār, M: fol. 124v.

  18. Ibn al-Fuwaṭī, 1995, Vol. 5, p. 117.

  19. Cf. below, Sect. 4.5.

  20. Ptolemy (Almagest, V, 14) used a dioptra originally described by Hipparchus that was four cubits in length(\(\approx \) 185.28 cm) (Toomer 1998, p. 56). This dioptra has a fixed lower pinnula on which there is a hole for sighting, and a movable outer one, which is placed in front of the sun. The solar/lunar angular diameter is calculated based on the movable pinnula’s width and the distance between the two pinnulas. In his Fī kayfīyya al-arṣād (“How to make the observations”), Mu’ayyad al-Dīn al-‘Urḍī, the instrument maker of the Maragha observatory (d. 1266), presented an addition for the antique dioptra to determine the eclipsed area/diameter of the sun or the moon (Seemann 1929, pp. 61–71). Thus, Muḥyī al-Dīn had a specific instrument for measuring the magnitude of eclipses at his disposal, which he may have applied to these lunar eclipses. In the Risāla al-Ghāzāniyya fi ’l-ālāt al-raṣadiyya (“Ghāzān’s treatise on observational instruments”) (cf.Zotti and Mozaffari 2010, pp. 165–167; Mozaffari and Zotti 2012, pp. 419–421) and in Wābkanawī’s Zīj (IV, 15, 8: Y: fols. 159r–159v, T: fols. 92r–92v), an instrument as a pinhole image device is introduced that fulfills the measuring of the magnitude of solar eclipses. The treatise contains the physical descriptions and applications of 12 new observational instruments in the second period of the Maragha observatory, which were presumably the inventions of Ghāzān Khān, the seventh ruler of the Īlkhānīd dynasty of Iran (reg. 21 October 1295-17 May 1304). About it, also see Mozaffari and Zotti (2013).

  21. Based on NASA’s Five Millennium Catalog of Lunar Eclipses (http://eclipse.gsfc.nasa.gov/lunar.html), which is now the standard: nos. 07878, 07897, and 07907.

  22. For example, for the time of the maximum phase of the eclipse no. 1, the true longitude of the sun is calculated as follows:

    figure a

  23. Cf. (Neugebauer (1975), Vol. 2, pp. 922–926).

  24. E.g., see note 33, below.

  25. (Toomer (1998), pp. 190–203); cf. (Pedersen (1974), pp. 172–178); (Neugebauer (1975), Vol. 1, pp. 73–80), (Thurston (1994), Appendix 4: p. 204f), Duke (2005).

  26. We use the standard proposed by (Kennedy (1991/1992), p. 21) to transliterate the letters in the diagrams.

  27. Throughout the paper, Sin \(\alpha \) indicates the sine of the angle \(\alpha \) under the condition that the radius of the trigonometric circle is assumed to be \(R = 60\), i.e., Sin \(\alpha = 60\) sin \(\alpha \). Similarly for Crd \(\alpha \).

  28. (Heath (1952), p. 33).

  29. (Heath (1952), pp. 64–66). Like Ptolemy, our author does not refer to Euclid.

  30. (Dorce (2003), p. 203).

  31. (Toomer (1998), pp. 198 and 203), Pedersen (1974), Appendix A, no. 69 (on p. 418); hereafter, our author calls this as “First Eclipse” and his eclipse no. 3 as “Second Eclipse.” For the analysis of the eclipses mentioned in the Almagest, cf.Steele (2000).

  32. (Dorce (2003), p. 197).

  33. Note that our author computes here the mean anomaly of the moon according to the Ptolemaic lunar model introducing the second anomaly of the moon and the prosneusis while he has not yet expounded this model and that the correction due to the prosneusis, i.e., the lunar equation of center, should be taken into account in order to compute the lunar mean anomaly from its true anomaly. In the situations like this, he refers reader to the future chapters. Another note is that Ptolemy in Almagest IV, 7 (Toomer 1998, p. 204; Neugebauer 1975, Vol. 1, pp. 78–79) computes \(\omega _\mathrm{a}\) according to his first (i.e., Hipparchan) lunar model, but never comes back to revise it after completing his lunar model. On the four-eclipse method for determining the length of the lunar anomalistic month (i.e., \(360^{\circ }/\omega _\mathrm{a})\) described by Ptolemy and a more coherent formulation of it by Jābir b. Aflaḥ (fl. Spain, the 12th ct.), cf.Bellver (2006).

  34. Talkhīṣ, fol. 82r; Adwār, CB: fol. 81v; Wābkanawī, fol. 154v (cf. below, Table 4): \(q(353) = 0{;}56^{\circ }\) and \(q(354) = 0{;}48^{\circ }\rightarrow q(353{;}32) = 0{;}51{,}44^{\circ }\). However, the trigonometric formula for \(q\) results 0;50,\(35^{\circ }\).

  35. Our author discussion on the solar and lunar angular diameters as well as his non-Ptolemaic value for the minimum apparent diameter of the moon and sun, i.e., 0;31,\(8^{\circ }\), appear in VI, 6 (fol. 93v onwards).

  36. Note that although the input data for the calculation of \(\beta \) is the magnitude of the eclipse, which was obtained from the observation, the value of \(\beta \) in (28) should be compared with the modern value of the geocentric \(\beta \) (not with the topocentric/apparent \(\beta \), which is about \(+\)0;11,\(30^{\circ }\)), simply because both

    figure l

    and \(r_{\bullet }\) have been calculated according to the geocentric hypotheses of the Ptolemaic model.

  37. Note that the arc AB of the ecliptic, instead of the arc AC, should be subtracted from . However, the two arcs are approximately equal: \(AB \approx 6{;}\hbox {26,12}^{\circ }\).

  38. Mean motion in longitude: Talkhīṣ, fol. 73r; Adwār, M: fol. 76v, CB: fol. 74v; in anomaly: Talkhīṣ, fol. 73v; Adwār, M: fol. 77r, CB: fol. 75r; : Talkhīṣ, fol. 74r; Adwār, M: fol. 78r, CB: fol. 76r; the Adwār has also the table for \(2\bar{{\eta }}\): M: fol. 77v, CB: fol. 75v. All of the tables have been prepared for each 30 years, one month, one day, and one hour (up to 30 h).

  39. By means of interpolation in the table of the oblique ascension for the latitude of Maragha on folio 35v.

  40. The time expressed in the sun’s apparent diurnal motion as projected onto the celestial equator, the so-called dā’ir. Our author calls it “Altitude dā’ir,” indicating that this time was computed from the solar altitude, not measured with the clepsydra. With our author’s parameters, i.e., the geographical latitude of Maragha \(\varphi = 37{;}20{,}30^{\circ }\), the obliquity of the ecliptic \(\varepsilon = 23.5^{\circ }\), and , half the sun’s apparent diurnal motion = 100;56,\(31^{\circ }\) (and so, half the duration of daylight = 6;43,46 h; our author later gives 6;43,31 h) and, therefore, the sun’s hour angle when it had the altitude \(h = 2^{\circ }\) was 98;18,\(46^{\circ }\).

  41. Note that when the last mean quadrature occurred [above, (41)], the moon had an anomaly near \(100^{\circ }\) (see below), and so the line of sight to it was tangential to the epicycle. Thus, the lunar epicyclic equation was maximum, and therefore, its true motion was equal to its mean motion.

  42. (Toomer (1998), p. 226); Arabic Almagest, fol. 63v.

  43. In the other zījes, it is called the “extra difference.”

  44. Our author’s tables for the lunar equations can be found in Kamālī, fols. 243v–251r, and his values for the lunar mean motions on fols. 232v–233r. The tables in Wābkanawī’s zīj are all based on Muḥyī al-Dīn’s parameter values.

  45. Cf. (Neugebauer (1975), Vol. 2, pp. 988–989).

  46. Cf. (Kennedy and Pingree (1981), pp. 168 and 310).

  47. Zīj al-mumtaḥan has \(\beta _\mathrm{max} = 4{;}30^{\circ }\) and \(\beta _\mathrm{max} =5{;}0^{\circ }\) (fols. 54r and 57r). Ḥabash (fol. 36r) and Bīrūnī (al-Qānūn, Vol. 2, pp. 776 and 779) have associated \(i= 4{;}46^{\circ }\) with the Banū Mūsā (cf. below, Sect. 4.2). However, Bīrūnī mentions that some people erroneously cast doubt on the correctness of this attribution and consider this value as the average of the Indian and Ptolemaic values, respectively, 4;\(30^{\circ }\) and 5;\(0^{\circ }\). Another source (Kamālī, fol. 53v) has attributed \(\beta _\mathrm{max} = \hbox {4;46}^{\circ }\) to the Mumtaḥan tradition and \(\beta _\mathrm{max} =4{;}55^{\circ }\) to Thābit b. Qurra, ‘Alī b. ‘Īsā, Sanad b. ‘Alī, Khālid b. ‘Abd- al-Malak al-Marwarūdhī, and the Banū Mūsā.

  48. In the other zījes, it is called “the third equation.” In the modern astronomy, it is called “the reduction to the ecliptic.”

  49. \(\hbox {Max}(c_{7}) = 0;6,40^{\circ }\) is derived from \(i= 5;3^{\circ }\) which al-Fārisī (fol. 119v) (about him, cf.Pingree 1985, pp. 8–9) attributes to Ibn Yūnus.

  50. The next appropriate opportunity was on March 19, 1271. Of course, the moon transited the meridian of Maragha about 17:47 MLT, before the sunset in \(\sim \) 18:15 MLT. Our author’s tables give and \(\beta = -4{;}59{,}59^{\circ }\) for this time.

  51. (Toomer (1998), pp. 253–254); (Pedersen (1974), pp. 208–209).

  52. Simplicius (1894), pp. 504–506 (cf.Bowen 2008, pp. 89–90); Bīrūnī, al-Qānun, Vol. 2, p. 632; al-Khāzinī, Wajīz, fol. 28r.

  53. For example, the amounts our author gave for the difference in mean anomaly between the two lunar eclipses (cf. Sect. 3.1) are better matched with taking Ptolemy’s value for \(\omega _\mathrm{a}\) to compute them while those given for the difference in mean longitude are in better agreement with taking the value adopted in Tāj al-azyāj for \(\omega _\mathrm{t}\) (the differences are in parentheses):

    figure ac

  54. The medieval astronomers measured the length of the tropical year with taking the autumnal equinox as the zero-point over the long periods. Thus, the result achieved should be considered as a “mean” value. The mean value for the length of the tropical year between the two consecutive autumnal equinoxes in the period from AD 0 to 2000 is 365;12,32 days (cf.Meeus 2002, pp. 357–366). Muḥyī al-Dīn’s value is thus more exact than those given, say, by Hipparchus/Ptolemy \((365{;}14{,}48^\mathrm{d})\), Thābit b. Qurra \((365{;}14{,}24^\mathrm{d})\), and al-Battānī \((365{;}14{,}26^\mathrm{d})\); cf. Mozaffari (2013a).

  55. As far as the present author knows, in his mathematical treatises as well as his treatise on the astrolabe, he refers to Avicenna (e.g., Hogendijke 1993, p. 134).

  56. Cf. Almagest V, 12: (Toomer (1998), pp. 246–247), (Neugebauer (1975), Vol. 1, p. 101).

  57. Bīrunī, al-Qānūn al-mas‘ūdī, Vol. 2, p. 779.

  58. Cf. (Pedersen (1974), p. 206).

  59. The center of the epicycle is at the perigee of the eccentric deferent (mean quadrature), and the moon is near its maximum elongation from it; so, the moon–earth distance \(\varDelta = \hbox {0;59} \times ((60 - 2 \times \hbox {10;19})^{2} -\hbox {5;15}^{2})^{1/2} = \hbox {38;21,53}\) terrestrial radii. Thus, with \(h' = \hbox {28;34}^{\circ },\, \varPi = \hbox {sin}^{-1}(\hbox {cos}\,h'/\varDelta ) \approx \hbox {1;19}^{\circ }\).

  60. Equation of days (Ar. ta‘dīl al-ayyām, La. equatio dierum) in the medieval astronomical context. However, it may be noted that the modern term “equation of time” (Ar. ta‘dīl al-zamān) may be found in Ibn Yūnus, p. 92 (line 13) and Bīrūnī, Vol. 2, p. 720; cf.Neugebauer 1975, Vol. 1, p. 61 (n. 2).

  61. Wābkanawī, II, 1, 1: T: fol. 16r; Y: fol. 26v. The Majisṭī to which Wābkanawī refers is probably Muḥyī al-Dīn’s Khulāṣa al-majisṭī which is now lost.

  62. Kamālī, fol. 52v. In other sources, the work has been ascribed to ‘Abd al-Karīm al-Fahhād (cf.Kennedy 1956, no. 64). The two works may, however, be independent from each other.

  63. Muḥyī al-Dīn appears to have been so interested in the central quadrant that composed a poem during the observations of 1265–6 AD to praise it, which a certain The Astrologer Majd al-Dīn Abū Muḥammad al-Ḥasan b. Ibrāhīm b. Yūsūf al-Ba‘albakī had engraved on the instrument (cf. Ibn al-Fuwaṭī 1995, Vol. 4, pp. 413–414):

    figure ah

    .

  64. Cf.Seemann (1929). The other two instruments could be used in order to measure simultaneously the horizontal coordinates of the two celestial objects having the diametrical opposed azimuths. The last instrument was solely applicable to the measurement of the coordinates of one object in a given time.

  65. Pangān was originally a simple inflow clepsydra (cf.Mozaffari 2013b, p. 256, n. 80); Wābkanawī uses this term to refer to the clepsydra; see below, Sect. 5; also, cf. (Mozaffari (2013b), pp. 256–257).

  66. E.g., cf. (Needham (1981), p. 136).

  67. Cf.van Dalen 2002a, esp. p. 334, 2002b; 2004b.

  68. In the Ghāzān’s treatise on observational instruments (see note 20), the adequacy of the classic instruments described in the Almagest is rejected for the various reasons; in the case of the armillary sphere, cf. (Mozaffari and Zotti (2012), pp. 400–401).

  69. (Toomer (1998), p. 423, lines 10–13).

  70. Al-Maghribī, Talkhīṣ, fol. 114v. The declinations of Vega and Capella were about, respectively, \(+44^{\circ }\) \(51.5^{\prime }\) and \(+38^{\circ }\) \(17.5^{\prime }\) at the time, and thus both transited the Maragha’s meridian \((\varphi = \hbox {37;23,46}^{\circ })\) in its northern half. The non-Ptolemaic star table of Īlkhānī Zīj includes the coordinates of both Vega and Capella (al-Ṭūsī, C: p. 195, T: fol. 100r).

  71. (Dorce (2003), p. 203).

  72. Bīrūnī, al-Qānūn al-mas‘ūdī, Vol. 2, pp. 742–743. For the analysis these eclipses (nos. 07224, 07225, and 07227 in NASA’s Five Millennium Catalog of Lunar Eclipses), cf. (Said and Stephenson (1997), pp. 45–46), Stephenson (1997, pp. 491–492). The analysis of Bīrūnī’s lunar measurements will appear in a separate paper.

  73. Kāshī, IO: fols. 4r–6r, P: pp. 24–28. The eclipses nos. 08220, 08221, and 08222 in NASA’s Five Millennium Catalog of Lunar Eclipses. The analysis of Kāshī’s lunar measurements will come in a separate paper.

  74. Based on the wrong assumption (likely from the false analogy drawn between the Ptolemaic lunar and planetary models) that \(q\) reaches its maximum when the line dropped from the epicycle’s center to the prosneusis is perpendicular to the apsidal line.

  75. Al-Ṭūsī, C: p. 7, T: fol. 3r.

  76. All of the tables for the lunar equations are asymmetric giving (i) \(\hbox {max}(c^{\prime }_{3}) = \hbox {13;8}^{\circ }\), (ii) \(c^{\prime }_{4} = \hbox {7;40}^{\circ }\) for the arguments 0, \(180^{\circ }\), and \(360^{\circ }\) and \(\hbox {max}(c^{\prime }_{4})= c^{\prime }_{4}(265) = \hbox {12;41,0}^{\circ }\) (thus, \(\hbox {max}(c_{4}) = \hbox {5;1}^{\circ }\)), and (iii) \(\hbox {max}(c_{5}) = \hbox {2;39}^{\circ }\). Cf. al-Ṭūsī, C: pp. 67–85; P: fol. 23v–28v; M: fols. 40r–50v.

  77. Ibn Yūnus, pp. 120, 158, 160, 162.

  78. For the solar daily mean motion, Wābkanawī’s Zīj has evidentially the same value obtained by al-Maghribī, i.e., \(\omega _{\odot }= \hbox {0;59,8,20,8,4,36,38}^{\circ }/\hbox {d}\) (cf. Wābkanawī, T: fol. 149r). Also, al-Kāshī’s value for \(\omega _{\odot }\) is Ibn Yūnus’ (Khāqānī zīj, IO: fol. 128v, gives the solar mean motion in a Persian year as \(\hbox {359;45,40,4}^{\circ }\); cf. Table 14). Al-Kāshī’s adoption of this value appears to be a consequence of his project of the revision of the Īlkhānī zīj. Support comes from the fact that the solar maximum equation of center and eccentricity in al-Kāshī’s zīj (IO: fol. 131r, 157r) are Ibn Yūnus’, as is in the Īlkhānī zīj. Al-Kāshī did not, of course, mention his source.

  79. It is borrowed from Ibn Yūnus’ zīj (p. 174), corresponding to the solar eccentricity = 2;6,10 \((R = 60)\).

  80. For a biographical outline of him, cf.van Dalen (2007). About his zīj and some studies of it, cf. Kennedy 1956a, no. 35, (King et al. (2001), p. 46), (Kennedy (1958), p. 251), Haddad and Kennedy (1971, p. 91), (Kennedy (1964), p. 443), (King (1986), pp. 138–140), (Kunitzsch (1964), pp. 398–399). Kennedy (1960, p. 211) employed the explanations given by Wābkanawī as regards the Maragha observatory to verify some remarks by al-Kāshī in a letter to the latter’s father and, in another paper (1962, p. 24), quoted a section of the zīj related to chronology and astrology.

  81. Wābkanawī, T: fol. 89v–90r, Y: fol. 155r.

  82. Wābkanawī, T: fol. 125r; Y: fol. 235r.

  83. Cf.Mozaffari (2009); Mozaffari (2013b).

  84. Wābkanawī, T: fol. 3r, Y: fol. 4v.

  85. Wābkanawī, T: fol. 2v, Y: fol. 3v. Concerning the conjunctions, the differences that Wābkanawī found are:

    \(\begin{array}{lll} \hbox {Mars and Saturn:}&{} \hbox {in the period of direct motion of Mars:} &{}6 \hbox { days} \\ &{}\hbox {in the period of retrograde motion of Mars:}&{} 8 \hbox { days}\\ \hbox {Mars and Jupiter:}&{} \hbox {in the period of direct motion of Mars:}&{} 5 \hbox { days}\\ \end{array}\)

  86. Wābkanawī, T: fol. 2v, Y: fol. 3v.

  87. E.g., Wābkanawī, III, 3, 1: T: fol. 53r, Y: fol. 96r; III, 9, 5: T: fol. 60r, Y: fol. 108v; III, 13, 6: T: fol. 67r, Y: fol. 120v. Since Wābkanawī contends the Īlkhānī zīj to be majorly based on earlier astronomical tables, rather than obtained from making independent observations, he goes further to call only Muḥyī al-Dīn’s Adwār as the “Īlkhānīd Observations.” Cf. Wābkanawī, T: fol. 3r, Y: fol. 4v.

  88. The other three modifications made by Wābkanawī are concerning (1) the mean longitude of Mars (increased by 1;\(5^{\circ }\)), (2) the mean anomaly of Venus (increased by 2;\(30^{\circ }\)), and (3) the latitudes of the two inferior planets; cf. Wābkanawī, T: fol. 3r; Y: fol. 4v; P: fols. 4r–v. Wābkanawī also differently arranged the entries of al-Maghribī’s equations tables.

  89. The modern values in this paper are extracted from the software Alcyone, applying the estimates of Morrison and Stephenson 2004 for \(\Delta \hbox {T}\) (the difference between the Dynamical Time and Universal Time).

  90. For the present study, a PC-program was used, which can compute the solar, lunar, and planetary ecliptical coordinates; the times of the synodic phenomena; etc, from the three zījes of the Maragha tradition. In this program, the equations, of course, are computed from the corresponding trigonometric formulas, instead of interpolating in the equations tables of these zījes. The differences, however, are small enough to be less effective when testing a historical claim. In addition, rendering ineffective the errors and/or differences in the equations tables, this procedure makes a unified scale in order to make the comparison merely between the two sets of the parameter values adopted in these three zījes.

  91. Wābkanawī, T: fols. 92r–v, Y: fols. 159r–160r, P: fols. 139r–140r. The passage in question may also be found in the “Ghāzān’s treatise on the observational instruments” (cf. note 20). In it, the time-measuring device is, however, called the “time-glass” (shīsha-i sā‘at).

  92. Cf. (Mozaffari (2013b), Section 4.III).

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Acknowledgments

The author notes with the best thanks and gratitude the suggestions and comments of Dr. Benno van Dalen (Germany). This work has been supported financially by Research institute for Astronomy and Astrophysics of Maragha (RIAAM) under research Project No. 1/2782-55.

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Mozaffari, S.M. Muḥyī al-Dīn al-Maghribī’s lunar measurements at the Maragha observatory. Arch. Hist. Exact Sci. 68, 67–120 (2014). https://doi.org/10.1007/s00407-013-0130-4

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