An analysis of medieval solar theories


From Antiquity through the early modern period, the apparent motion of the Sun in longitude was simulated by the eccentric model set forth in Ptolemy’s Almagest III, with the fundamental parameters including the two orbital elements, the eccentricity e and the longitude of the apogee λA, the mean motion ω, and the radix of the mean longitude \( \bar{\lambda }_{0} \). In this article we investigate the accuracy of 11 solar theories established across the Middle East from 800 to 1600 as well as Ptolemy’s and Tycho Brahe’s, with respect to the precision of the parameter values and of the solar longitudes λ that they produce. The theoretical deviation due to the mismatch between the eccentric model with uniform motion and the elliptical model with Keplerian motion is taken into account in order to determine the precision of e and λA in the theories whose observational basis is available. The smallest errors in the eccentricity are found in these theories: the Mumtaḥan (830): − 0.1 × 10−4, Bīrūnī (1016): + 0.4 × 10−4, Ulugh Beg (1437): − 0.9 × 10−4, and Taqī al-Dīn (1579): − 1.1 × 10−4. Except for al-Khāzinī (1100, error of ~ + 21.9 × 10−4, comparable to Ptolemy’s error of ~ + 33.8 × 10−4), the errors in the medieval determinations of the solar eccentricity do not exceed 7.7 × 10−4 in absolute value (Ibn al-Shāṭir, 1331), with a mean error μ = + 2.57 × 10−4 and standard deviation σ = 3.02 × 10−4. Their precision is remarkable not only in comparison with the errors of Copernicus (− 7.8 × 10−4) and Tycho (+ 10.2 × 10−4), but also with the seventeenth-century measurements by Cassini–Flamsteed (− 2.4 × 10−4) and Riccioli (+ 5.5 × 10−4). The absolute error in λA varies from 0.1° (Taqī al-Dīn) to 1.9° (al-Khāzinī) with the mean absolute error MAE = 0.87°, μ = −0.71° and σ = 0.65°. The errors in λ for the 13,000-day ephemerides show MAE < 6′ and the periodic variations mostly remaining within ± 10′ (except for al-Khāzinī), closely correlated with the accuracy of e and λA.

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  1. 1.

    For the bio-bibliographies of the astronomers mentioned in this study, see Gillipsie et al. (1970–1980), Koertge (2008), Hockey et al. (2007), Bearman et al. (1960–2005), Sezgin (1978), and Rosenfeld and İhsanoğlu (2003). For the Islamic astronomical tables mentioned in this article, see Kennedy (1956) and King et al. (2001). A comprehensive survey is in preparation by B. van Dalen in Munich.

  2. 2.

    Toomer [1984] (1998, pp. 131–172); On the ancient solar theories, e.g., see Petersen and Schmidt (1968), Pedersen (1974, Chap. 3), Neugebauer (1975, Vol. 1, pp. 54–61), Jones (1991), Maeyama (1998), Duke (2008), and Swerdlow (2010).

  3. 3.

    Toomer [1984] (1998, pp. 443, 453).

  4. 4.

    For the astronomical and historical reasons behind these two frequently used values, see Mozaffari (2016a).

  5. 5.

    See Table 1, note 9.

  6. 6.

    Mozaffari (2017, pp. 10, 21–24).

  7. 7.

    Toomer [1984] (1998, pp. 215–216).

  8. 8.

    Bīrūnī (1954–1956, Vol. 2, p. 653).

  9. 9.

    Swerdlow (2010, p. 155).

  10. 10.

    Neugebauer (1962) and Moesgaard (1974).

  11. 11.

    About these two medieval views, see Swerdlow (1975, p. 50).

  12. 12.

    About the model and its later receptions and parameters, see Toomer (1969, 1987), Samsó and Millás (1989), Samsó (1987, 2011, pp. 207–218 and 491–492), and Calvo (1998).

  13. 13.

    Copernicus (1543, ff. 90v–91r); see, also, Swerdlow and Neugebauer (1984, Vol. 1, pp. 157–161).

  14. 14.

    Neugebauer (1975, Vol. 3, pp. 1097–1102).

  15. 15.

    Maeyama (1998, pp. 4–8).

  16. 16.

    From the formulae given in Simon et al. (1994, p. 678).

  17. 17.

    See Mozaffari (2013).

  18. 18.

    See Mozaffari (2013, Part 1, pp. 322, 326, Part 2, p. 393).

  19. 19.

    Al-Maghribī’s text is translated and his procedure is explained in Saliba (1985).

  20. 20.

    Mozaffari and Steele (2015, pp. 350–354).

  21. 21.

    Copernicus (1543, ff. 87v–88v), Swerdlow and Neugebauer (1984, pp. 152–154). In his earlier work, De hypothesibus … commentariolus, Copernicus used e = 0.0189 (see Swerdlow 1973, p. 442) adopted from the Alfonsine Tables (see Goldstein and Chabás 2001; Chabás and Goldstein 2003, pp. 153–155).

  22. 22.

    Brahe (1913–1929, Vol. 2, pp. 19–28), Opera Omnia. See Dreyer (1890, p. 333), Moesgaard (1975, pp. 85–89), Thoren and Christianson (1990, pp. 223–224), and Swerdlow (2010, p. 155). On Tycho’s solar measurements I did also have the chance to access to Prof. N. Swerdlow’s blow-by-blow quantitative analysis of these matters, which will soon appear in his study of Renaissance astronomy.

  23. 23.

    Tycho’s solar theory was completed by early 1589, but most of his solar observations were made in the period 1588–1591 and 1595; for an analysis of them, see Wesley (1979).

  24. 24.

    From Tables in Brahe (1913–1929, Vol. 2, pp. 46–47), Opera Omnia.

  25. 25.

    See Mozaffari (2013, Part 1, pp. 322, 326, Part 2, pp. 393, 399).

  26. 26.

    Flamsteed (1674b, p. 221). In the case of Mars, see Wilson (1969).

  27. 27.

    Hornsby (1763, p. 467).

  28. 28.

    Ricciloi (1665, I.9–10: pp. 29–33).

  29. 29.

    Kepler’s (1570–1858) Epitomes Astronomiae Copernicanae VI: Opera Omnia, Vol. 6, p. 433.

  30. 30.

    Bullialdus (1645, pp. 53–57).

  31. 31.

    Ricciloi (1665, I.3: Vol. 1, pp. 13–14 (equinoxes); I.5: Vol. 1, p. 18 (solstices); I.8: Vol. 1, pp. 26–29 (noon-altitudes)).

  32. 32.

    Flamsteed (1674b, p. 220–221).

  33. 33.

    Flamsteed (1674a, p. 6000).

  34. 34.

    Toomer [1984] (1998, pp. 255–257).

  35. 35.

    Toomer [1984] (1998, p. 265).

  36. 36.

    See Swerdlow (1972) and Van Helden (1985, pp. 31–32).

  37. 37.

    Ibn Yūnus, Zīj, F1: ff. 60v–61r, 77r–v.

  38. 38.

    See Delambre (1819, pp. 149–150), Hartner (1980), and King (1999, p. 502).

  39. 39.

    Al-Maghribī, Talkhīṣ al-majisṭī VI.6–7: ff. 93v–95v.

  40. 40.

    Ibn al-Shāṭir, Jadīd zīj, O: ff. 86v–87r.

  41. 41.

    See Roberts (1957, pp. 429–430).

  42. 42.

    Kāshī, Sullam al-samā’ (The stairway to the heavens), f. 8v; Zīj, IO: f. 185r.

  43. 43.

    Ulugh Beg, Zīj, P1: f. 108r, 125v, 130v, P2: f. 119v, 140v, 148v.

  44. 44.

    Nallino [1899–1907] (1969, Vol. 2, pp. 93–94).

  45. 45.

    Al-Maghribī, Adwār, CB: f. 90v, M: f. 92v.

  46. 46.

    Said and Stephenson (1995, p. 122).

  47. 47.

    See Mozaffari (2013, Part 1, pp. 322, 328).

  48. 48.

    Explained in Said and Stephenson (1995, 131n24).

  49. 49.

    Ibn Yūnus, Zīj, L: p. 223, F1: f. 25r.

  50. 50.

    Ibn Yūnus, Zīj, L: p. 224, F1: f. 25v.

  51. 51.

    Ibn al-Shāṭir, Jadīd zīj, O: ff. 87v–88r. In the text of cap. 62: 2;50°, but in the table appended to it: 2;51°.

  52. 52.

    This note is also appended as a brief separate chapter to MS. F1 of Ibn Yūnus’s Ḥākimī zīj (F1: f. 88v).

  53. 53.

    Bernard (1684, p. 724). Bernard’s value 2;59° appears to be a typo of the value 2;50° Ibn al-Shāṭir gives in the explanatory text of cap. 62 of his zīj (see above, note 51). Hartner (1977, p. 5) informs us of this letter. Note that we have made use of the same manuscript Bernard did.

  54. 54.

    Table of the solar declination in Ibn al-Shāṭir, Jadīd zīj, O: f. 139v.

  55. 55.

    See Mozaffari (2013, Part 2, pp. 393, 395, 399).

  56. 56.

    Meeus (2002, pp. 357–366).

  57. 57.

    Neugebauer (1962, esp. pp. 283–285).

  58. 58.

    Bīrūnī (1954–1956, Vol. 2, esp. pp. 668–669); also, see Hartner and Schramm (1961, p. 214f). That the lengths of the true solar years with respect to the ecliptic are varied is also mentioned in al-Ṭūsī’s Taḥrīr al-majisṭī (Exposition of the Almagest) as a comment on Almagest III.1 (P1: p. 88, P2: f. 24v, P3: f. 41r). For the translation of the relevant passage, see Saliba [1987] (1994, p. 148).

  59. 59.

    Meeus (2002, p. 362).

  60. 60.

    See Said and Stephenson (1995).

  61. 61.

    Bīrūnī (1954–1956, Vol. 2, p. 648). Mozaffari (2016b, p. 269).

  62. 62.

    Noted earlier in the case of al-Battānī in Thurston (2002, p. 59).

  63. 63.

    See Grasshof (1990).

  64. 64.

    Toomer [1984] (1998, pp. 452–453, 541). About the eras, see B. V. Dalen’s entry “Ta’rīkh” in EI 2 (Vol. 10, esp. p. 261).

  65. 65.

    This should be 0;39,4 h because in Sidrat II.4 (K: f. 17v), Taqī al-Dīn asserts that from his observations of the triple of the lunar eclipses in 1576–1577 (see Mozaffari and Steele 2015), he derived the value 56;39,45° for the longitude of Istanbul from the Fortunate Islands; also, in Sidrat V.1 (K: f. 41v), where he converts the time of one of the lunar eclipses which Ptolemy observed at Alexandria to the meridian of Istanbul, he takes the meridian of Istanbul equal to 56;40° and that of Alexandria as 61;54°, and correctly states that the then resulted difference of 5;14° in terrestrial longitude between the two cities corresponds to a difference of 0;20,56 h in local times between the two. However, he appears to have discarded the value 56;39,45° for the longitude of Istanbul later, since the relevant lines on f. 17v are blacked out as well as in the geographical table attributed to him (see King 2004/5, Vol. 1, p. 449–450), the longitudes of Istanbul and Alexandria are given, respectively, 60;0° and 61;55°, both in accordance with other sources which have 59;50° for the longitude of Istanbul and 61;54° for that of Alexandria reckoned from the Fortunate Islands. Note that Istanbul (L = 28;57° from Greenwich) is actually only about one degree west of Alexandria (L = 29;55°). It deserves noting that the rounded value 56;40° is not unprecedented; it stems from Ptolemy’s Geography and is also used, e.g., in al-Battānī Ṣābi’ zīj (Nallino [1899–1907] 1969, Vol. 2, p. 44, Vol. 3, p. 239; also, see Kennedy 1960, p. 186).

  66. 66.

    See Mozaffari (2017, p. 10, note 15).

  67. 67.

    Ibn al-Shāṭir, Jadīd zīj, K: f. 2v, O: f. 2v, L1: f. 2r, L2: f. 2; Mozaffari (2017, p. 21).

  68. 68.

    Petersen and Schmidt (1968, p. 84) and Neugebauer (1975, Vol. 1, pp. 275–276).

  69. 69.

    Brahe, Progymnasmata I: Opera Omnia, Vol. 2, p. 15.

  70. 70.

    From Walther’s observations of the solar meridian zenith distances, Tycho derives the following values for the solar orbital elements in 1488, a century before him, by means of the three-point method:

      e λ Α
    Applying the two equinoxes and mid-spring 2;7,43 94;15°
    Applying the two equinoxes and mid-summer 2;7,44 94;19

    According to Walther’s observations, the vernal equinox of 1488 occurred in Nuremburg on 11 March, 3;40h after midnight which converted to the meridian of Uraniborg (the place of Tycho’s observatory) is 3;55 h. Tycho computed the time of the vernal equinox of 1588 in Uraniborg as 10 March, 8;45 h after midnight. On the basis of the above values for the solar orbital elements, the equation of centre q at the instant of the vernal equinox of 1488 was equal to +2;1,35°. Also, according to Tycho’s values for e for 1588, at the time of the vernal equinox of 1588, q = +2;2,35°. Accordingly, the solar mean longitude at the time of the vernal equinox of 1488 was equal to 357;58,25° and at that of the vernal equinox of 1588: 357;57,25°. The mean sun travels 0;1° in about 0;25 h. Thus, the mean sun completed 100 revolution on the ecliptic in 100 Julian years (of 365.25 days unvarying) minus 1 d 3;55 h − (8;45 + 0;25) h. Therefore, the length of the solar year = 365 days 6 h – 18;45 h/100 = 365 days 5;48,45 h (see Tycho, Progymnasmata I: Opera Omnia, Vol. 2, pp. 40–44).

  71. 71.

    Dreyer (1890, p. 333) asserts that Tycho’s value is “only about a second too small”, which appears to be the result of the comparison of Tycho’s value with the length of the tropical year at the time (365 days 5;48,47 h). This example illustrates how much determinative the difference between the various types of the solar years may be in carefully examining the precision of historical values.

  72. 72.

    From the formula given for the mean longitude of the Earth in Simon et al. (1994, p. 678).

  73. 73.

    See Mozaffari (2014, pp. 110–112).

  74. 74.

    Taqī al-Dīn, Kharīdat, B: ff. 26v–28v, C1: ff. 48r–51r, C2: ff. 34r–38r, E: ff. 2v–4v, 53r, K: ff. 45v–49r.

  75. 75.

    Al-Khāzinī, Zīj, L: f. 105v.

  76. 76.

    E.g., see Mozaffari (2016a, pp. 307–308).

  77. 77.

    See Mozaffari (2016b, esp. p. 270).

  78. 78.

    Ibn Yūnus, Zīj, L: pp. 98–99; Caussin (1804, pp. 104–107), and Delambre (1819, p. 83).

  79. 79.

    Bīrūnī (1954–1956, Vol. 2, pp. 642–643, 647–648).

  80. 80.

    See Mozaffari and Zotti (2013, pp. 57–58).

  81. 81.

    Said and Stephenson (1995, pp. 125, 130).

  82. 82.

    Maeyama (1998, p. 39).

  83. 83.

    See, also, Said and Stephenson (1995, pp. 120–121).

  84. 84.

    See Mozaffari (2013, Part 1: pp. 317–321).

  85. 85.

    Flamsteed (1674b, p. 221).

  86. 86.

    Seemann (1929, pp. 28–32); for an illustration, see Sezgin and Neubauer (2010, Vol. 2, p. 38). On the instruments of the Maragha observatory in detail, see Mozaffari and Zotti (2013).

  87. 87.

    Taqī al-Dīn, Sidrat, K: ff. 14v–15v, N: ff. 18r–19v, V: ff. 22r–23v; Sezgin and Neubauer (2010, Vol. 2, pp. 53–54).

  88. 88.

    Knobel (1917).

  89. 89.

    See Piini (1986, pp. 542–543).

  90. 90.

    For the analysis of Ulugh Beg’s star catalogue, see Shevchenko (1990), Krisciunas (1994), and Verbunt and van Gent (2012).

  91. 91.

    See Mozaffari (2016c, esp. pp. 522, 525).

  92. 92.

    Mozaffari (2016a, pp. 307–308).

  93. 93.

    Mozaffari (2016a, pp. 296–297). The non-Ptolemaic star table in the Īlkhānī zīj is also available on Ibn Yūnus’s star table will be discussed in detail in a forthcoming paper by the present author.


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The author extend his sincerest thanks to Benno van Dalen (Germany), Julio Samsó (Spain), and Noel Swerdlow (United States) for their encouragements and kind helps. The solar longitudes in this article have been computed with the aid of van Dalen’s very useful PC program Historical Horoscopes. This work has been financially supported by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/5440-57.

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Communicated by: George Saliba.

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Mozaffari, S.M. An analysis of medieval solar theories. Arch. Hist. Exact Sci. 72, 191–243 (2018).

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