Limitations of Methods: The Accuracy of the Values Measured for the Earth's/Sun's Orbital Elements in the Middle East,A.D. 800–1500,Part 1

For the solar eccentric model as presented in Almagest III ( Toomer G. J. , Ptolemy's Almagest (Princeton, 1998), 133–72), cf. Neugebauer O. , A history of ancient mathematical astronomy (Berlin, 1975), i, 54–61; Pedersen O. , A survey of the Almagest (Odense, 1974), chap. 3. For its evolution through pre-Ptolemaic data, cf. Jones A. , “Hipparchus's computations of solar longitudes”, Journal for the history of astronomy, xxii (1991), 101–25, and Duke D. , “Four lost episodes in ancient solar theory”, Journal for the history of astronomy, xxxix (2008), 283–96.

Copernicus Nicolaus , De revolutionibus orbium coelestium (Nuremberg, 1543), folios. iiiR-iiiV.

E.g., if TY is measured from one vernal equinox to the next, it increases with the average rate of 1 second per century (s/cent) from a.d. 0 to 2000. Or if it is counted from one autumnal equinox to the next, it decreases with the average rate 2 s/cent in this period. The lesser changes in TY will result, if it is counted with reference to solstices: From one summer solstice to the next: −0.4 s/cent; from one winter solstice to the next: −0.6 s/cent. (Cf. Meeus J. , and Savoie D. , “The history of the tropical year”, Journal of the British Astronomical Association , cii (1992), 40–2, and Meeus J. , More mathematical astronomy morsels (Richmond, 2002), 357–66, which give some useful notes about the length of year.) Since ancient and medieval astronomers gave their values for TY in days and sexagesimal fractions up to seconds (i.e., 1/3600^th of a day), they are exact, at most, to ±24 seconds. Also, they measured the tropical year with the autumnal equinox as the zero-point over long periods, thus the magnitudes achieved by them may be considered as the ‘mean/average’. The mean/average values for each of four types of the tropical year from a.d. 0 to 2000, expressed in days and sixagesimals, are as follows: Two vernal equinoxes: 365;14,32 days Two summer solstices: 65;14,30 days, Two autumnal equinoxes: 365;14,32 days, Two winter solstices: 365;14,34 days. Some of the historical values for the length of year are as below: Hipparchus/Ptolemy (2nd cent.): 365.25 − 1/300 = 365;14,48 days, Thābit b. Qurra (9th cent.): 365.25 − 1/100 = 365;14,24, Al-Battānī (10th cent.): 365;14,26, Muhyī al-Dīn al-Maghribī (13th cent.): 365.25 — −/120 = 365;14,30.

Meeus , op. cit. (ref. 3), 361.

Cf. Neugebauer , op. cit. (ref. 1), iii, 1097–102; Maeyama Y. , “Determination of the Sun's orbit: Hipparchus, Ptolemy, al-Battânî, Copernicus, Tycho Brahe”, Archive for history of exact sciences, liii (1998), 1–49, p. 4.

The intention to bisect the Ptolemaic solar eccentricity may be traced out in medieval astronomy (cf. Goldstein Bernard R. Sawyer Frederick W. , “Remarks on Ptolemy's equant model in Islamic astronomy”, in Maeyama Y. Salzer W. G. (eds), Prismata: Festschrift für Willy Hartner (Wiesbaden, 1977), 165–81) and may be found, for example, in Ibn al-Shāṭir's solar model (cf. Saliba G. , “Theory and observation in Islamic astronomy: The work of Ibn al-Shāṭir of Damascus”, Journal for the history of astronomy, xviii (1987), 35–43, p. 42).

For the analysis of the error, cf. Newton R. R. , “The authenticity of Ptolemy's parallax data — Part 1”, Quarterly journal of the Royal Astronomical Society , xiv (1973), 367–88, pp. 369–70. Seven centuries later, this caused a remarkable error in measuring the rate of precession in early Islamic astronomy; see Grasshoff G. , The history of Ptolemy's star catalogue (London, 1990), 19–20.

Cf. Toomer , op. cit. (ref. 1), 153–7, 190–203, and 469f; Saliba G. , “Solar observations at Maragha observatory”, Journal for the history of astronomy , xvi (1985), 113–22, p. 114; Duke , op. cit. (ref. 1), 284–5, 291.

Bīrānī A. , The chronology of ancient nations, transl. by Sachau C. Edward (London, 1879), 167.

Cf. Saliba G. , “The determination of the solar eccentricity and apogee according to Mu'ayyad al-Dīn al-'Urḍī”. Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, ii (1985), 47–67.

Cf. Charette François , “The locales of Islamic astronomical instrumentation”, History of science, xliv (2006), 123–38.

Mụ̄hyī al-Dīn Al-Maghribī, Talkhīṣ al-majisṭī, MS Leiden, no. Or. 110, III, 1: Fol. 58r.

Measured by the altitude observations in the three consecutive days near the solstice days of 1264: 12–14 June: h = 76;9,30° (modern: 76;8,30°) and 7–9 December: h = 29;9,30° (modern: 29;10,30°). The modern value for the mean obliquity of the ecliptic at the time: ε₀ ≈ 23;32°. Note that these altitude observations are about ±1′ in error while his equinox altitudes and other three altitudes given below (see Table 3, no. 9), are about +4′ in error.

Abu Al-Rayḥān al-Bīrūnī, al-Qānūn al-mas'ūdī (3 vols, Hyderabad, 1954), ii, 621–3.

Said S. S. Stephenson F. R. , “Precision of medieval Islamic measurements of solar altitudes and equinox times”, Journal for the history of astronomy, xxvi (1995), 117–32, p. 123.

Said Stephenson , op. cit. (ref. 15), 123.

Bīrūnī , op. cit. (ref. 14), ii, 621.

Cf. Halley Edmond , “A discourse concerning a method of discovering the true moment of the Sun's ingress into the tropical sines”, in Miscellanea curiosa; Containing a collection of some of the principal phaenomena in nature, accounted for by the greatest philosophers of this age (London, 1726), ii, 202–10.

Cf. Said Stephenson , op. cit. (ref. 15), 122–3 (Table 3).

Bīrūnī , op. cit. (ref. 14), ii, 653.

It is frequently and wrongly named the fuṣūl (= seasons) method in the secondary literature while it is evidently and correctly called anṣāf al-fuṣūl or awsāṭ al-burūj in Bīrūnī's work ( Bīrūnī , op. cit. (ref. 14), ii, 657).

The printed text is not clear here ( Bīrūnī , op. cit. (ref. 14), ii, 619, lines 5–6) and may read 31;18° + 1 1/3′ + 1/3′ as Said and Stephenson (op. cit. (ref. 15), 126, n. 6) did. Nevertheless, it should be considered that the altitude 31;19,40° results in the time of mid-autumn being 0;8,53 hours after the true noon of 1 November 1016, which by no means agrees with the value 0;4,30 hours explicitly mentioned in the text. As a result, I prefer to assume that, in spite of the confusing phrase in line 6 (more likely, due to the clear fact that we do not have a trustworthy edition of al-Bīrūnī's work at our disposal), the value al-Bīrūnī wanted to mention is 31;18° + 1/3′.

Cf. Toomer G. J. , “The solar theory of az-Zarqāl: An epilogue”, in King D. Saliba G. (eds), From deferent to equant (Annals of New York Academy of Sciences, d; New York, 1987), 513–19, p. 516. In other sources, the varied values between q_max =1;52 and 1;53°, corresponding to 1/2e = 0.01629 − 101643 (R = 1), are associated with Ibn al-Zarqālluh; cf. Samsó J. , “Al-Zarqal, Alfonso X and Peter of Aragon on the solar equation”, in King Saliba (eds), op. cit., 467–76, p. 471. The value λ_ap = 85;49° is in agreement with the later Islamic sources, e.g., Ibn al-Hā'im; cf. Calvo E. , “Astronomical theories related to the Sun in Ibn al-Hā'im's al-Zīj al-Kāmil fī ‘l-Ta’ālīm”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, xii (1998), 51–111, p. 55.

Cf. Swerdlow N. M. , “Tycho, Longomontanus, and Kepler on Ptolemy's solar observations and theory, precession of the equinoxes, and obliquity of the ecliptic”, in Jones A. , (ed.), Ptolemy in perspective (Archimedes, xxiii; Dordrecht, 2010), 151–202, p. 155.

Copernicus , op. cit. (ref. 2), fols. 87v–88v.

Dreyer J. L. E. , Tycho Brahe: A picture of scientific life and work in the sixteenth century (Edinburgh, 1890), 333; Thoren V. E. , The Lord of Uraniborg: A biography of Tycho Brahe (Cambridge, 1990), 224.

Cf. Saliba , op. cit. (ref. 8).

Ptolemy pointed to the sensitivity of the method for the input data in the Almagest, IV, 11, where he applies it to determine the radius of the Moon's epicycle ( Toomer , op. cit. (ref. 1), 215–16).

Bīrūnī (op. cit. (ref. 14), iii, 1193) declares his discontent with his Islamic predecessors' attitude of not explaining their observational data and measurement procedures as Ptolemy mentioned his procedures, and adds that the one who elucidates his computational procedures is the most worthy to be followed.

Bīrūnī , op. cit. (ref. 14). Sulṭānī zīj, MS Iran, Parliament, no. 184. Muhammad b. Abī 'Abd-Allāh Sanjar al-Kamālī (known as “Sayf-i munajjim”), Ashrafī Zīj, MS Paris, Bibliothèque Nationale, no. 1488. The latter two were written contemporarily around the turn of the 14th century, the first in Yazd and the latter in Shīrāz (central Persia). The first is different from both Ulugh Beg's Sulṭānī zīj (called also Gūrkānī zīj) and Shams al-Dīn Muhammad al-Wābkanawī's Zīj al-Muḥaqqaq al-sulṭānī (completed around 1316–24; MSS (1) Turkey, Aya Sophia Library, no. 2694, (2) Iran, Yazd, Library of 'Ulūmī, no. 546, its microfilm being available in Tehran University central library, no. 2546, and (3) Iran, Library of Parliament, no. 6435). It is attributed to Qutḥb al-Dīn al-Shīrāzī ( Kennedy E. S. , A survey of Islamic astronomical tables (Philadelphia, 1956), no. 25); however, in the canons of this zij (e.g., fols. 78r, 141r, 142r), the author mentions that he was based in Yazd, and thus, since al-Shīrāzī is not known to have worked in Yazd, he is less likely to be identified as its author. Based on Kennedy (ibid., no. 32), the Sulṭānī zīj may be identical with the Shāhī zīj by a thirteenth-century astronomer named Ḥusām al-Dīn al-Sālār (killed by Hülegü, the first patron of the Ilkhanid dynasty of Iran, on 8 Muḥarram 661 h / 22 Nov 1262; cf. Rashīd al-Dīn Faḍl Allāh al-Hamidānī, Jāmi' al-Tawārīkh [The compendium of histories], ed. by Rushan M. Mūsawī M. , (4 vols, Tehran, 1994), ii, 1045). However, there are direct references in the first to the latter as well as some tables of the latter in the first (e.g., the table for the equation of time on fol. 7v). Therefore, one can by no means accept the two to be identical.

Bīrūrnī , op. cit. (ref. 14), ii, 650–61.

Sulṭānī zīj (ref. 30), fol. 77r; Kamālī , op. cit. (ref. 30), fol. 49r; 'Abd al-Raḥmān al-Khāzinī, Wajīz [Compendium of] al-Zīj al-mu'tabar al-sanjarī, MS Istanbul, Suleymaniye Library, Hamadiye collection, no. 859, fol. 1v. Tables in Kamālī , op. cit. (ref. 30), fols. 236r—v, 238r–239r, 240r, 241v, and 243r.

Calculated based on Meeus J. , Astronomical algorithms (Richmond, 1998), chap. 25, and Alcyone Software.

The table is asymmetric and has max = 3;59,3,21° for mean anomaly 266° ( Bīrūnī , op. cit. (ref. 14), ii, 716) and min = 0;0,56,39° for mean anomaly 90° ( Bīrūnī , op. cit. (ref. 14), ii, 710), and so q_max = 1;59,3,21° which strictly corresponds to e = 2;4,39.

The values for the maximum solar equation of centre used in the pre- and early Islamic periods are also mentioned in the two sources and in the prologue of al-Khāzinī, op. cit. (ref. 32), fol. 1v: “Ancient observations”: 2;24°; Ptolemy: 2;23°; Persians: 2;14°; and Indians: 2;11°. The value 2;23° in the Almagest is drawn from e = 2;30 calculated based on the Hipparchian values for the length of seasons. The solar eccentric model more likely existed before Hipparchus and it is probable that Apollonius knew this model ( Pedersen , op. cit. (ref. 1), 135, 340f). The term ‘ancient observations’ is thus interesting; the value associated with it can be found in the papyrus PSI inv. 515 (cf. Jones A. , “Studies in the astronomy of the Roman period, IV: Solar tables based on a non-Hipparchian model”, Centaurus , xlii (2000), 77–88). According to Jones (p. 85), the treatise of which this papyrus is a fragment was composed before the publication of the Almagest. The value 2;24° is not significantly different from Ptolemy's and, as Jones argues, it might have been computed from the Hipparchian eccentricity 2:30 with the use of a table of chords tabulated for fewer angles than Ptolemy's. It is, nonetheless, interesting that it was transmitted to medieval astronomy as a value that pre-dates Ptolemy's. The value attributed to the Iranians is that used in the Shāh zīj (written in Sāsānīd Persia, a.d. 224–651); see Part 2. 2;11° is indeed found in the Indian traditions from the year 400 onwards (cf. Pingree D. , “On the classification of Indian planetary tables”, Journal for the history of astronomy, i (1970), 95–108, esp. pp. 97–8, 100, 105).

Bīrūnī , op. cit. (ref. 14), ii, 654.

Muḥammad Abu Al-Faraj al-Nadīm Ibn Isḥāq , Kitāb al-fihrist , ed. by Tajaddud Riḍā (Tehran, 1971), 331. (For the English translation, cf. The fihrist of al-Nadīm, ed. and transl. by Dodge Bayard (2 vols, New York, 1970).

'Alī b. 'Abd al-Raḥmān b. Aḥmad Ibn Yūnus, Zīj al-kabīr al-Ḥākimī, MS Leiden, Orientalis, no. 143, pp. 103, 106, 107.

Cf. Goldstein B. R. , “Historical perspectives on Copernicus's account of precession”, Journal for the history of astronomy , xxv (1994), 189–97, p. 190; Goldstein B. R. Chabás J. , “The maximum solar equation in the Alfonsine Tables”, Journal for the history of astronomy, xxxii (2001), 345–8, p. 345.

Yūnus Ibn , op. cit. (ref. 38), 104.

Cf. Neugebauer O. , “Thābit ben Qurra ‘On the Solar Year’ and ‘On the Motion of the Eighth Sphere’”, Proceedings of the American Philosophical Society, cvi (1962), 264–99.

Bīrūnī , op. cit. (ref. 14), ii, 653.

For the instruments utilized in the two observations and their accuracies, cf. Charette , op. cit. (ref. 11), 125.

Yūnus Ibn , op. cit. (ref. 38), 124.

E.g., Kamālī , op. cit. (ref. 30), fol. 236r.

Nallino C. A. , (ed.), Al-Battani sive Albatenii: Opus astronomicum (3 vols, Milan, 1899–1907), i, 4; ii, 79–83.

E.g., Naṣīr al-Dīn al-Ṭūsī, Risāla al-mu 'īniyya fī 'ilm al-hay'a, MS Tehran, Parliament, no. 6347, p. 42.

Copernicus , op. cit. (ref. 2), fol. 87v.

His solar parameters in Bīrūnī , op. cit. (ref. 14), ii, 654.

Bīrūnī , op. cit. (ref. 14), ii, 659.

Bīrūnī , op. cit. (ref. 14), ii, 654–5.

Cf. Kennedy , op. cit. (ref. 30), nos. 29 and 73.

Cf. Kennedy E. S. , “Applied mathematics in the eleventh-century Iran: Abu Jafar's determination of the solar parameters”, Mathematics teacher , lviii (1965), 441–6, reprinted in idem, Studies in the Islamic exact sciences (Beirut, 1983), 535–40.

Yūnus Ibn , op. cit. (ref. 38), 120.

Yūnus Ibn , op. cit. (ref. 38), 106.

Bīrūnī , op. cit. (ref. 14), ii, 656–7.

Bīrūnī , op. cit. (ref. 14), ii, 658.

Kennedy E. S. , “The solar equation in the Zīj of Yahya b. Abī Manūr”, in Maeyama Salzer (eds), Prismata (ref. 6), 183–6, p. 185, reprinted in idem, Studies in the Islamic exact sciences (ref. 53), 136–9; van Dalen B. , “A second manuscript of the Mumtaḥan Zīj”, Suhayl, iv (2004), 9–44, p. 22.

The zīj ascribed to Ḥabash al-Ḥāsib, MS. Berlin, no. Ahlwardt 5750 (formerly no. Wetzstein I 90), fol. 30v.

Kamālī , op. cit. (ref. 30), fol. 236r.

al-Nadīm Ibn , op. cit. (ref. 37), 334.

Yūnus Ibn , op. cit. (ref. 38), 87, 91, 120.

For an analysis of it, cf. Said and Stephenson, op. cit. (ref. 15), 129.

Neugebauer O. , The astronomical tables of Al-Khwārizmī (Historiskfilosofiske Skrifter undgivet af Det Kongelige Danske Videnskabernes Selskab, iv/2; Copenhagen, 1962), 20.

Yūnus Ibn , op. cit. (ref. 38), 104.

Kennedy , op. cit. (ref. 30), nos. 46 and 75.

Kamālī , op. cit. (ref. 30), fol. 49r.

Khāzinī , op. cit. (ref. 32), fol. 1v.

Yūnus Ibn , op. cit. (ref. 38), 105.

Kennedy , op. cit. (ref. 30), no. 8.

Quṭb al-Dīn al-Shīrāzī, Ikhtīyārāt-i Mużaffarī , MS Iran, National Library, no. 3074f (copied in 7th cent. h / 13–14 cent. a.d., during the lifetime of the author, fol. 50v; Quṭb al-Dīn al-Shīrāzī, Tuḥfa al-Shāhiyya, MS Iran, Parliament, no. 6130 (copied in 730 h / 1329–30 a.d.), fol. 38v.

Kamālī , op. cit. (ref. 30), fol. 236v.

Yūnus Ibn , op. cit. (ref. 38), 174.

Naṣīr al-Dīn al-Ṭūsī, Īlkhānī zīj, MS C: University of California, pp. 60–5, MS T: Iran, University of Tehran, Ḥikmat Collection, no. 165, fols. 28r–30v, MS P: Iran, Parliament Library, no. 181, fols. 21v–23r.

Kamālī , op. cit. (ref. 30), fol. 238r.

al-Khāzinī 'Abd al-Raḥmān , al-Zīj al-mu'tabar al-sanjarī, MS Library of Vatican, no. Arabo 761, p. 266.

Kamālī , op. cit. (ref. 30), fol. 240r.

Al-Fahhād al-Dīn Abu al-Ḥasan 'Alī b. 'Abd al-Karīm Al-Fahhād of Shirwān, Zīj al-'Alā'ī, MS India, Salar Jung, no. H17, p. 73.

Pingree D. , (ed.), Astronomical works of Gregory Chioniades (3 vols, Amsterdam, 1985), ii, 32.

Kennedy , op. cit. (ref. 30), no. 84.

Cf. Pingree , op. cit. (ref. 79).

Cf. van Dalen B. , “The Zīj-i Naṣirī by Maḥmūd ibn Umar: The earliest Indian zij and its relation to the ‘Alā’ī Zīj”, in Burnett Charles (eds), Studies in the history of the exact sciences in honour of David Pingree (Leiden, 2004), 327–56, p. 329.

Kennedy , op. cit. (ref. 30), no. 64.

Kamālī , op. cit. (ref. 30), fol. 52v.

Al-Maghribī , op. cit. (ref. 12), fols. 58v–61v; cf. Saliba , op. cit. (ref. 8).

I intend to present an analysis of Muḥyī al-Dīn's meridian altitude observations in a separate paper.

Al-Maghribī , op. cit. (ref. 12), fols. 58r—v.

Cf. Roberts V. , “The solar and lunar theory of Ibn ash-Shāṭir: A pre-Copernican Copernican model”, Isis , xlviii (1957), 428–32, p. 430; Saliba , op. cit. (ref. 6).

Ghiyāth al-Dīn Jamshīd al-Kāshī, Khāqānī zīj, MS P: Iran, Parliament, no. 6198, p. 114, IO: London, India Office, no. 430, fol. 131r.

Kāshī , op. cit. (ref. 89), IO: Fol. 127r.

Kāshī , op. cit. (ref. 89), IO: Fol. 128v.

Beg Ulugh , Sulṭānī Zīj, MS T: University of Tehran, no. 13, fols. 110r–115v, MS P1: Iran, Parliament, no. 72, fols. 117v–123r, MS P2: Iran, parliament, no. 6027, fols. 134v–140r.

Beg Ulugh , op. cit. (ref. 92), P1: Fol. 123v, P2: Fol. 141r.

The solar apogee's longitudes are given in Beg Ulugh , op. cit. (ref. 92), P1: Fol. 116r and P2: Fol. 133r.