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Ibn al-Zarqālluh’s discovery of the annual equation of the Moon

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Abstract

Ibn al-Zarqālluh (al-Andalus, d. 1100) introduced a new inequality in the longitudinal motion of the Moon into Ptolemy’s lunar model with the amplitude of 24′, which periodically changes in terms of a sine function with the distance in longitude between the mean Moon and the solar apogee as the variable. It can be shown that the discovery had its roots in his examination of the discrepancies between the times of the lunar eclipses he obtained from the data of his eclipse observations over a 37-year period in the latter part of the eleventh century and the predictions made on the basis of the lunar theories in the Mumta\(\textit{\d{h}}\)an zīj (Baghdad, ca. 830) and al-Battānī’s zīj (Raqqa, d. 929), which were available to him at the time. What Ibn al-Zarqālluh found is, in fact, a special case of the annual equation of the Moon, which is applicable in the oppositions and, thus, in the lunar eclipses. The inequality was discovered independently by Tycho Brahe (d. 1601) and Johannes Kepler (d. 1630). As Ibn Yūnus (d. 1009) reports in his \(\textit{\d{H}}\)ākimī zīj, Ibn al-Zarqālluh’s medieval Middle Eastern predecessors, the Persian astronomers Māhānī (d. ca. 880) and Nayrīzī (d. 922) as well as ‘Alī b. Amājūr (fl. ca. 920), were already acquainted with the problem of the eclipse timing errors, but it had remained unresolved until Ibn Yūnus provided a provisional, and incorrect, solution by reducing the size of the lunar epicycle. As we argue, the diverse ways to tackle the same problem stem from two different methodologies in astronomical reasoning in the traditions developed separately in the Eastern and Western regions of the medieval Islamic domain.

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Notes

  1. Bio-bibliographical information on Ibn al-Zarqālluh can be found in J. Vernet’s entry in DSB, Vol. 14, pp. 592–595, J. Samsó’s entry in EI2, Vol. 11, pp. 461–462, and R. Puig’s entry in BEA, pp. 2410–2414. There is a vast literature on Ibn al-Zarqālluh and his astronomical achievements, mainly the result of the extensive long-term research conducted by the Barcelona school of Islamic astronomy studies, a part of which are mentioned here: on his solar theory, see Toomer (1969), Samsó and Millás [1989] (1994), Samsó (1987), Toomer (1987) and Calvo (1998). On his trepidation theories, see Goldstein (1965) and Samsó [1990] (1994). On his consideration of an elliptical-shape of Mercury’s deferent, see Samsó and Mielgo (1994). On his graphical method for finding the Moon–Earth distances, see Puig (1989). Most of the information on Ibn al-Zarqālluh as brought into light in the modern scholarship has now been collected and summarized in Samsó (2020), chapter 6.

  2. This discovery was first brought to light in Samsó [1991] (1994, p. 19, 1992, p. 964). The materials found on it in the primary sources were discussed in detail in Puig (2000).

  3. On the two methodologies, see the introduction to Mozaffari (2016a).

  4. See Mozaffari (2014a).

  5. Toomer [1984] (1998, pp. 190–203, 225–226).

  6. See Mozaffari (2014b, esp. pp. 105–106), Mozaffari and Steele (2015, p. 348) and Mozaffari (2020–2021).

  7. The text was edited in Puig (2000, pp. 91–92), and translated in Samsó (2020, pp. 676–677).

  8. The instructions are given in the astronomical tables composed in al-Andalus (e.g., by Ibn al-Kammād) and the Maghrib (e.g., Ibn Is\(\text{\d{h}}\)haq, Ibn al-Raqqām, and Ibn al-Bannā’), though the instructions for addition-and-subtraction is diametrically opposed to what Ibn al-Hā’im himself explicitly explains in his passage.

  9. Meeus (1998, p. 339).

  10. See Neugebauer (1975, Vol. 3, pp. 1108–1110).

  11. Toomer [1984] (1998, pp. 191, 297).

  12. See Mozaffari (2014b, pp. 94–95, 2020–2021, pp. 96–97).

  13. See Neugebauer (1975, Vol. 3, pp. 1110–1111); and, especially, Swerdlow (2009, pp. 24ff).

  14. The terms (7), (11), and (12) were featured by Newton. For a further discussion of the history of these terms (which is beyond the scope of the present paper) see Cook (1988), (2000), Kollerstrom (1995), Wilson (2010, esp. pp. 123–124); and citations therein.

  15. Toomer [1984] (1998, p. 173).

  16. As embedded in the Alcyone software.

  17. E.g., a systematic error of no more than + 5 min in Mu\(\text{\d{h}}\)yī al-Dīn al-Maghribī’s times of three lunar eclipses of 7 March 1262, 7 April 1270, and 24 January 1274 measured by a water-clock (see Mozaffari 2018b, Table 2 on page 599) and − 8 min in Ghiyāth al-Dīn al-Kāshī’s observation of the lunar eclipse of 26 November 1406 (Mozaffari 2020–2021, Table 3 on page 82); we are not told how it was measured, although he mentions water- and sand-clocks in his work; see ibid, pp. 83–84). The time errors in Taqī al-Dīn Mu\(\text{\d{h}}\)ammad b. Ma ‘rūf’s observations of the trio of lunar eclipses of 7/8 October 1576, 2/3 April 1577, and 26/27 September 1577 were quite large and systematic (ranging between − 39 and − 53 min; see Mozaffari and Steele 2015, Table 2 on page 355), which was seemingly owing to a deficiency in his new mechanical clock, an issue that still awaits a further study.

  18. See Mozaffari (2018b, pp. 622–623), and references cited therein.

  19. See above, note 17.

  20. For the early Islamic period, see the references given in Sect. 5. For the late period, see six star altitude measurements at the beginning and end of Mu\(\text{\d{h}}\)yī al-Dīn al-Maghribi’s trio of the lunar eclipses in Mozaffari (2018b, Table 2 on page 599).

  21. See Samsó [1990] (1994), Samsó and Millás [1989] (1994).

  22. In the present study, I excluded al-Khwārizmī’s zīj and its recensions made in al-Andalus toward the end of the tenth century or beginning of the eleventh, simply because such a work based on the Indian theories would not have had anything to do with the evaluation of a Ptolemaic model.

  23. For the longitude of Toledo, see esp. Comes (1994) and, also, e.g., Samsó and Millás (1989 [1994], p. 6, 1998, p. 263); Samsó (1997, p. 82). For those of Baghdad and Raqqa, see al-Battānī, Zīj, E: f. 174r; Nallino [1899–1907] (1969, Vol. 2, pp. 41–42, Vol. 3, p. 238); in it, the longitudes of Toledo and Cordoba are given, respectively, as 28;0° and 27;0° (E: f. 176r; Nallino [1899–1907] 1969, Vol. 2, p. 219, Vol. 3, p. 241), both of which, as brought to light in the paper just mentioned by the late M. Comes, are of Western Islamic origin. It should be noted in this regard that the only surviving manuscript of al-Battānī’s zīj is in Western Arabic script with alphanumerics in the Maghribī Abjad sequence and goes back to the late eleventh or early twelfth century (see van Dalen and Pedersen 2008, p. 407).

  24. Mozaffari (2018a, esp. pp. 220–221). For the study of the medieval Middle Eastern solar theories, see, also Mozaffari (2013).

  25. Our case should be counted as a favored aspect of timing astronomy; for an example of how focusing only on the times of the synodic phenomena, in general, and of the eclipses, in particular, for the purpose of evaluating the accuracy of astronomical theories in the medieval period could be sometimes misleading, see Mozaffari (2019).

  26. On Ibn al-Kammād’s works and the descendants of Ibn Is\(\text{\d{h}}\)āq’s tradition: Chabás and Goldstein (1994, 2015a) and Samsό (2020, chapter 7).

  27. See Pingree (1968, p. 104, 1970, p. 112) and Chabás and Goldstein (2003, pp. 252–253).

  28. See Chabás and Goldstein (2003, p. 157, 2004, pp. 225–226).

  29. Ibn al-Shā \(\text{\d{t}}\)ir, Nihāya I.9–11, O1: ff. 13r–19r, O2: ff. 14r–21r, O3: ff. 29v–34r; Zīj, Lunar equation tables: K: ff. 55v–57v, 125r–v, O: ff. 35v–40r, L1: ff. 55r–57r, L2: ff. 71r–73r, PR: ––.

  30. Mestres 1999, En. Part, pp. 220–224, 232–233; Ibn al-Bannā’, D: 31r–33r, E: ff. 23v–25v. The solar and lunar equation tables in both are identical with the minor differences in their format: Ibn al-Bannā’’s zīj, the solar equation table is displaced by + 4° and the table for the lunar equation of center by 13;9° in order to make them user-friendly always-additive.

  31. At the time, the mean anomaly was about 80°, according to the Mumta \(\textit{\d{h}}\)an zīj (see Table 1, below).

  32. Ibn Yūnus, Zīj, L: pp. 94–96; Caussin 1804, pp. 83–95. On al-Māhānī’s eclipse records, see Said and Stephenson (1996–1997, II: pp. 31–33), Stephenson (1997, pp. 471, 476–479) and Steele (2000, pp. 113–114). The first three lunar eclipses reported from Māhānī are in fact all lunar eclipses observable from Baghdad from 854 to 856 AD. Between these and his fourth eclipse occurring in 866, ten other lunar eclipses could be observed from Baghdad. The period of his observations of the lunar eclipses appears to coincide with the entire period of his activities in observational astronomy, since the period of his planetary observations falls within that of his eclipse observations (from a near appulse of Venus and Saturn on 28 August 858 to a conjunction of Venus and Mars on 13 February 864; Ibn Yūnus, Zīj, L: p. 96–97; Caussin de Perceval 1804, pp. 94–97; Delambre 1819, p. 80). In Ibn Yūnus’ list of the observations of the Regulus (the star α Leo) by his Islamic predecessors in order to derive the rate of precession, he also records one by Māhānī dated to 230 Yazdigird/861–862 AD (Ibn Yūnus, Zīj, L: pp. 106–108; Caussin de Perceval 1804, pp. 143–155).

  33. Ibn Yūnus, L: p. 99–100, Caussin de Perceval (1804, pp. 110–112).

  34. Ibn Yūnus, L: p. 99, Caussin de Perceval (1804, pp. 106–109).

  35. Ibn Yūnus, Zīj, L: pp. 100–102, Caussin de Perceval (1804, pp. 112–126). On the Banū Amājūr’s reports of eclipses, see Said and Stephenson (1996–1997, II: pp. 33–37), Stephenson (1997, pp. 471–472, 479–482) and Steele (2000, pp. 116–117). During the period of the Banū Amājūr’s lunar eclipse observations, five other eclipses were also observable from Baghdad (however, two only at their beginning and one only at its end). Their period of planetary observations (from an occultation of Regulus by Venus on 9–11 September 885 to an observation of Mars with Procyon, the star α CMi, on 1 January 919; Ibn Yūnus, Zīj, L: p. 98–99, 109, F1: f. 10r; Caussin de Perceval 1804, pp. 104–111, 157–162, Delambre 1819, p. 83, 87–89) predates that of their eclipse observations. Ibn Yūnus also mentions two observations of Regulus from this family: by Abu’l-Qāsim b. Amājūr in 304 H/916–917 AD (“as Sa ‘īd b. Khafīf al-Samarqandī quotes from him”) and in 306 H/288 Y/918–919 AD, as quoted from the Banū Amājūr’s Zīj al-badī ‘ (Magnificent zīj).

  36. Ibn Yūnus, Zīj, L: p. 98, (Caussin de Perceval 1804, pp. 102, 104–105).

  37. Ibn Yūnus, Zīj, L: p. 102, (Caussin de Perceval 1804, pp. 125–126).

  38. Toomer [1984] 1998, pp. 136, 404, 420–423; Arabic Almagest: Is\(\text{\d{h}}\)āq-Thābit: S: ff. 29r, 101r, 105v–106v, PN: ff. 32v, ––, ––, Pa1: ff. 43v, ––, ––, Pa2: ff. 44r, ––, ––, TN: ff. 39v, 137r, 142r–143r, E1: ff. ––, 95v, 103v–105r, E2: ff. ––, 116r, 52r–v & 53v, LO1: ff. ––, 40v, 52r & 53r–54r (NB. the folio numbers of the three sections of the Almagest referred to above are separated by a comma; “––” indicates that the section is absent from MS).

  39. Toomer [1984] 1998, pp. 328, 334, 461; Arabic Almagest: Is\(\text{\d{h}}\)āq-Thābit: S: ff. 94v, 96v, 114v, PN: ff. 74v, 76v, 95v, Pa1: ff. 158v, 162r, ––, Pa2: ––, TN: ff. 112v, 115r, 156r, E1: ff. 67v, 70v, 126v, E2: ––, LO1: ff. 5v, 10v, 81r.

  40. For a comprehensive discussion of this term, see Szabó (1978, pp. 232f).

  41. Toomer [1984] 1998, p. 460; Arabic Almagest: Is\(\text{\d{h}}\)āq-Thābit: S: f. 114r–v, PN: ff. 95r–v, Pa1: ––, Pa2: ––, TN: ff. 155r–v, E1: f. 126r, E2: ––, LO1: ff. 79v & 80v.

  42. E.g., see Goldstein (1967, p. 13) and Morelon (1993, p. 15).

  43. For the examples set forth above, our observations in the Arabic Almagest by \(\text{\d{H}}\)ajjāj are as follows. Once in III.1 (LO2: f. 58r, LE: f. 35r), the term is rendered as al-\(\textit{\d{s}}\)ifāt al-maw \(\textit{\d{d}}\)ū ‘a, “the laid features”, to describe the “assumption” that the solar motion has a single and invariable anomaly whose period is the solar year defined by the solstices and equinoxes); in two other instances in the same section, no definite translation is given. In these sections, it is translated as Jihat: VII.2: LO2: ––, LE: f. 107r, VII.3: LO2: ––, LE: f. 109r, VIII.3: LO2: ––, LE: f. 127r, IX.2: LO2: ––, LE: ff. 131v–132r, and IX.10: LO2: ––, LE: f. 147v. When the term is used for a geometrical purpose, \(\text{\d{H}}\)ajjāj does not translate it: IX.9: LO2: ––, LE: ff. 147r–v. The term a \(\textit{\d{s}}\)l is used, at least, once to render “epoch” in VII.4 (Toomer [1984] 1998, p. 340): LO2: ––, LE: f. 110v.

  44. Quoted in Forcada (2017, p. 270).

  45. A similar statement concerning the instrumental discrepancy is given by Taqī al-Dīn Mu\(\text{\d{h}}\)ammad b. Ma ‘rūf, 1525–1585 AD, the director of the short-lived observatory at Istanbul, some 600 years later; see Mozaffari (2016a, p. 270).

  46. Ibn Yūnus, Zīj, L: p. 103, Caussin de Perceval (1804, pp. 125, 128). See above, notes 32 and 35.

  47. Ibn Yūnus, Zīj, L: pp. 110–113, Caussin de Perceval (1804, pp. 161–179). For an analysis of Ibn Yūnus’ eclipse records, see Said and Stephenson (1996–1997, II: pp. 37–43), Stephenson (1997, pp. 473–476, 482–488) and Steele (2000, pp. 117–119).

  48. Note that, except for the Banū Amājūr’s fourth lunar eclipse, there is a negative systematic deviation in their computed time errors from our corresponding recomputed values, with an average of ~ − 21 min. I do not know the precise reason for this, but it has no influence on our present discussion.

  49. See van Dalen (2004, esp. p. 11).

  50. See Mozaffari (2014b, p. 105). It is derived from the tabular value of pmax = 4;48° for the maximum epicyclic equation of the Moon at the greatest distance from the Earth. This is the lowest value found for this parameter in the medieval Islamic corpus. Ibn Yūnus’ elder contemporary, Ibn al-A ‘lam (d. 985 AD), had a value of ~ 5;4p for the lunar epicycle radius, corresponding to pmax = 4;51°, which has come down to us via the secondary sources (e.g., al-Khāzinī’s (fl. 1120 AD) treatise on experimental astronomy, Kayfiyyat al-i ‘tibār (IV.1), which serves as an introduction to his Sanjarī zīj, V: f. 12v).

  51. This guaranteed its longevity, so that Ibn Yūnus’ values for the solar basic parameters were later used in the Īlkhānī zīj compiled by Na\(\text{\d{s}}\)īr al-Dīn al\(\text{\d{T}}\)ūsī (1201–1274 AD) at the Maragha observatory; from there they found their way to Kāshī’s Khāqānī zīj (written about 1414 AD). See Mozaffari (2018a, esp. pp. 224, 226, 230, and 235, 2020–2021, pp. 88–96).

  52. See Mozaffari (2014b, p. 112). Of course, the Mumta\(\text{\d{h}}\)an value (error ~ + 0.6 × 10–6°) is more precise than Ibn Yūnus (error ~ − 2.3 × 10–6°).

  53. The lunar mean positions and equations computed from the parameters underlying the Mumta\(\textit{\d{h}}\)an zīj (Mt) and Ibn Yūnus’ zīj (IY) for the times of the oppositions of the two lunar eclipses in question are given in the following tabulation. It can be clearly seen how Ibn Yūnus’ decrease of the size of the lunar epicycle is effective in reducing the error in the lunar longitude in the second eclipse (approximately, by a factor of 3), as compared to the error in the Mumta\(\textit{\d{h}}\)an value, but contributes to making it slightly larger in the first eclipse.

  54. These errors were also noticed in other medieval traditions, e.g., by Levi b. Gerson in his Astronomy, Part 1, Chapter 14: Goldstein (1985a, p. 93); for Levi’s lunar models, see Goldstein (1972, 1974).

  55. Samsó [1990] (1994, pp. 2–3).

  56. Ibn Yūnus mentions ‘Alī b. Amājūr’s finding again in Sect. 38 of his \(\textit{\d{H}}\)ākimī zīj, “On the latitude of the Moon” (O: ff. 105v–106r, F1: ff. 84r–v). This passage received considerable attention in the literature of the nineteenth century and the beginning of the twentieth; see Delambre (1819, pp. 138–140), Sédillot (1845–1849, Vol. 1, pp. 282–287) gives a transcription of the text (on the basis of MS. F1) along with a French translation; Dreyer (1906, pp. 251–252).

  57. Meeus (1998, p. 341).

  58. See Neugebauer (1975, Vol. 3, 1107–1108, 1111).

  59. The period in question is slightly longer than half of the period of oscillation of the lunar inclination, during which it changes between a minimum of 5;2° and a maximum of 5;18°.

  60. As Ibn Yūnus says in Sect. 38 of his zīj (see above, note 56) and as his tables of the lunar latitude show (O: ff. 130v, 136r), he adopted an inclination of 5;3°.

  61. See Mozaffari (2016b, esp. p. 308). On Mu\(\text{\d{h}}\)yī al-Dīn’s observations at the Maragha observatory, see Mozaffari (2018b).

  62. See Mozaffati (2018–2019, p. 206).

  63. See Mozaffari (2014b).

  64. See Mozaffari (2020–2021).

  65. The literal translation of the passage and this interpretation is from Prof. Alan Bowen (private communication). Toomer [984] (1998, p. 422).

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Acknowledgements

The author expresses his gratitude to José Chabás, Matthieu Husson, Richard Kremer, and Julio Samsó for their invaluable comments on the early drafts of this paper. This paper is published as part of a research project accomplished during the research mission issued by the University of Maragheh, International Affairs Office (no. 1401/D/5790).

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    S. Mohammad Mozaffari

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Mozaffari, S.M. Ibn al-Zarqālluh’s discovery of the annual equation of the Moon. Arch. Hist. Exact Sci. (2024). https://doi.org/10.1007/s00407-023-00323-z

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