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 2

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

In the other available studies (e.g., Petersen Viggo M. Schmidt Olaf , “The determination of the longitude of the apogee of the orbit of the Sun according to Hipparchus and Ptolemy”, Centaurus , xii (1968), 73–96; Neugebauer , op. cit. (ref. 1), iii, 1097–102; Maeyama , op. cit. (ref. 5); Hughes D. W. , “Hipparchus' spring and summer and the ellipticity of the Earth's orbit”, Journal of the British Astronomical Association, xcix (1989), 1989–4), the values obtained by Hipparchus, Ptolemy, Copernicus, and Tycho Brahe were scrutinized. The only value of the medieval Islamic period, which, maybe traditionally, came into the analyses to be compared with the first four ones, was Battānī's. Moesgaard K. P. (“Thābit ibn Qurra between Ptolemy and Copernicus: An analysis of Thābit's solar theory”, Archive for history of exact sciences, xii (1974), 1974–216) also analysed Thābit's solar theory, presenting the observational data and the values given by the early Islamic astronomers (Table 1, no. 1 and Table 2, nos. 1 and 2) for the solar orbital elements.

Said Stephenson , op. cit. (ref. 15), 125, 130.

Maeyama , op. cit. (ref. 5), 39.

Said Stephenson , op. cit. (ref. 15), 131, n. 4. An allusion to Ibn al-Shātir's correction of the meridian solar altitudes due to the parallax in a Latin source may be found in Hartner W. , “The role of observations in ancient and medieval astronomy”, Journal for the history of astronomy, viii (1977), 1–11, p. 5.

Said Stephenson , op. cit. (ref. 15), 120–1.

Said Stephenson , op. cit. (ref. 15), 123 (table) and 125.

E.g., Bīrūnī , op. cit. (ref. 14), ii, 657.

Said Stephenson , op. cit. (ref. 15), 125, 127.

Bīrūnī , op. cit. (ref. 14), ii, 659.

Maeyama , op. cit. (ref. 5), 4–8.

E.g., for Ibn al-A'lam, I use the period 940–80 in view of his date of death (985). For Ibn Yūnus, I use 960–1000 since he gives the epoch of his zīj as the beginning of the year 372 Yazdigird (= 16 March 1003) and died in 1009. For al-Fahhād, I use 1160–80 because many of the examples given in his zīj (e.g., the Great Conjunction of 1166 in its Greek translation, cf. Pingree, op. cit. (ref. 79), 241–3, or the computation of the solar longitude for 13 March 1176, cf. al-Fahhād, op. cit. (ref. 78), 20) fall in these two decades. Al-Kāshī appears to have based his zīj on already-available parameters that he considered to be correct or on those he computed from the observations he made before entering the Samarkand observatory. For instance, in the case of the lunar parameters, he uses the value for the radius of the epicycle that he found from his observations in his native city, Kāshān, around 1406–7 and the Ptolemaic value for the lunar eccentricity, while the values employed in Ulugh Beg's Zīj al-Sulṭānī are different. For the solar eccentricity, as mentioned earlier, his value is sufficiently close to Ibn Yūnus's which he appears to have adopted through his project of the correction of the Īlkhānī zīj.

Meeus , op. cit. (ref. 3), 201.

Some of the values studied here have been mentioned in Hartner, op. cit. (ref. 98), 8, and have been compared with the corresponding true values of λ′_ap (although, with a few errors; e.g., assuming a.d. 1061 and a.d. 1600, respectively, as the years of Ibn al-Zarqālluh's and Tycho's solar observations or taking the epoch value 77;50° for the longitude of apogee instead of the value Ibn al-Zarqālluh observed).

As noticed, for example, in Hartner, op. cit. (ref. 98), 8 in the case of Yaḥyā's value.

About the model, cf. Toomer G. J. , “The solar theory of az-Zarqāl: A history of errors”, Centaurus , xiv (1969), 306–36; Samsó J. Millás E. , “Ibn al-Bannā', Ibn Isḥāq and Ibn al-Zarqālluh's solar theory” appeared in 1985 and is republished in J. Samsó, Islamic astronomy and medieval Spain (Aldershot, 1994), chap. X; Samsó , op. cit. (ref. 23); Toomer , op. cit. (ref. 23); Calvo , op. cit. (ref. 23); Samsó J. , Las ciencias de los antiguos en al-Andalus, 2nd edn (Almería, 2011), 207–18 and 491–2.

The close variants found in the sources are 0;19,30 and 0;19,17 for PO, respectively, associated with PT = 2;10 and PT = 2;10,13 and 2;10,16; cf. Samsó Millás , op. cit. (ref. 109), 25; Calvo , op. cit. (ref. 23), 56.

Samsó Millás , op. cit. (ref. 109), 18; Calvo , op. cit. (ref. 23), 56, 58.

Calvo , op. cit. (ref. 23), 62.

It, of course, does not reach these values in each periodicity; Meeus , op. cit. (ref. 3), 201–5.

It needed around six centuries until Newton's Theory of Gravity made it possible to explain the change in the Earth's eccentricity and around eight centuries until James Croll (1821–90) calculated its periodicity.

This is based on drawing a comparison between our discussion and the following passage from Henri Poincaré: “We may calculate the mass of Jupiter from either the movements of its satellites, or perturbations of the major planets, or those of the minor planets. If we take the average of the determinations obtained by these three methods, we find three numbers very close together, but different. We might interpret this result by supposing that the coefficient of gravitation is not the same in the three cases. The observations would certainly be much better represented. Why do we reject this interpretation? Not because it is absurd, but it is needlessly complicated. We shall only accept it when we are forced to, and that is not yet.” ( Poincaré Henri , Science and hypothesis, in The foundations of science (New York, 1913), 131.).

The problem has a parallel in modern cosmology where the question if raised as to whether the fundamental constants, like the coefficient of gravitation, are essentially constant or variable in reality (e.g., cf. Ellis George F. R. , “Fundamental issues and problems of cosmology”, in Lasota Jean-Pierre (ed.), Astronomy at the frontiers of science (Dordrecht, 2011), 309–20, p. 310).

Bīrūnī , op. cit. (ref. 14), ii, 653, 661.

Bīrūnī , op. cit. (ref. 14), iii, 1197. Until now, two manuscripts containing a large body of the contents of the Mumtahan zīj have been known: E: Escorial, no. árabe 927 and L: Leipzig, Universitätsbibliothek, no. Vollers 821 (about the latter, cf. van Dalen, op. cit. (ref. 58)). Both manuscripts seem to be based on a recension compiled, presumably, in the tenth century (ibid., 11) and thus not presenting the zīj in its pure, original form. Both have a common table for the apogee motion based on Ibn al-A'lam's value of 1° in 70 Persian years but the Leipzig manuscript has also an additional table based on the value 1° in 66 Julian years attributed to “al-Battānī and al-Ma'mūn”. The instructions for finding the true solar longitude are missing from the Escorial manuscript (ibid., 23, 39), but in the Leipzig manuscript they clearly allude to the moving solar apogee (L: Fol. 49r). There is no proof that this belongs to the original text, but, other than the remark of Bīrūnī, there is no indication of the contrary either.

Neugebauer , op. cit. (ref. 64), 19–20.

This value is apparently the rate of precession adopted from Indian astronomy in the early Islamic one; cf. Pingree D. , “Precession and trepidation in Indian astronomy before a.d. 1200”, Journal for the history of astronomy, iii (1972), 27–35, p. 29.

Bīrūnī , op. cit. (ref. 14), ii, 675. Bīrūnī severely criticizes al-Nayrīzī and those like him who, in Bīrūnī's opinion, do not attempt to settle astronomical issues (amr al-hay'a) by the aid of observations. In his Zīj al-ḥākimī al-kabīr, Ibn Yūnus also has some critical remarks concerning some elements of Nayrīzī's astronomical work; cf. De Young G. , “Nayrīzī: Abū al-bbās al-Faḍl ibn Ḥātim al-Nayrīzī”, in Hockey T. (eds), The biographical encyclopaedia of astronomers (New York, 2007), 823.

Moesgaard , op. cit. (ref. 96), 200–1. Hartner and Schramm ( Hartner W. Schramm M. , “Al-Bīrūnī and the theory of the solar apogee: An example of originality in Arabic science”, in Crombie A. C. (ed.), Scientific change (Symposium on the History of Science, University of Oxford, 9–15 July 1961) (London, 1963), 206–18, p. 216) argue that Thābit's approach was on the basis of “a purely formal theological principle” and thus refer to a striking difference between it and Bīrūnī's (see below).

Bīrūnī , op. cit. (ref. 14), ii, 676.

At this date, the longitude of Spica was about 190;3°.

Cf. Toomer , op. cit. (ref. 1), 336; Pedersen , op. cit. (ref. 1), 410, n. 13.

I do not know how Bīrūnī reached this value since the magnitude of difference in longitude 17;4° (misprinted as 17;3°) and the time interval counted in days as 475670 (given as 275970 in the text and as 275670 in n. 1 on p. 677, both of which seems to be the scribal errors for the correct value 475670) are mentioned in the text from which the value of about 1° in 76 years and 4 months is derived. More interesting is that Bīrūnī later gave the rate of precessional motion extracted from the earliest recorded stellar longitude, i.e., Timocharis's value for Spica, to be 16 cycles (i.e., 16 × 360) in 160696125 days from which the rate 1° in 76 years and 5 months is derived.

Cf. Toomer , op. cit. (ref. 1), 328; Pedersen , op. cit. (ref. 1), 415, n. 46.

In this year, the longitude of Regulus was around 135;36°.

Bīrūnī , op. cit. (ref. 14), ii, 676–7. He adopted the value ∼1°/69y to construct his own table of the solar apogee's motion (Bīrūnī, op. cit. (ref. 14), ii, 695).

Bīrūnī , op. cit. (ref. 14), ii, 676.

Cf. Moesgaard , op. cit. (ref. 96), 206.

Bīrūnī , op. cit. (ref. 14), ii, 669.

Bīrūnī , op. cit. (ref. 14), ii, 669.

Bīrūnī , op. cit. (ref. 14), ii, 662–3; the translation is taken from Hartner and Schramm, op. cit. (ref. 123), 211.

Bīrūnī , op. cit. (ref. 14), iii, 1197–8; translated and explained in Goldstein and Sawyer, op. cit. (ref. 6), 167–8.

Both the Old Sūryasiddhānta and the Shāh zīj are lost but the materials preserved from them reveal that in both the value 80° is assigned to the longitude of the solar and Venusian apogees and the value 2;13° or 2;14° to their maximum of equation of centre (cf. Kennedy E. S. , “The Sasanian astronomical handbook Zīj-i Shāh and the astrological doctrine of ‘Transit’ (Mamarr)”, Journal of the American Oriental Society , lxxviii (1958), 246–62, pp. 256–7, reprinted in idem, Studies in the Islamic exact sciences (Beirut, 1983), 319–35.

In the thirteenth-century recension of Ḥabash's zīj (op. cit. (ref. 59)), there are the two tables for the longitude of apogees of the Sun and planets: The one (folio 28r) gives the values up to the five sexagesimal fractional places in which, amazingly, the sexagesimal fractions from the seconds to the fifths are all identical (-;-,24,2,43,53) and the apogee of the Sun and of Venus are equal (79;30, …). The values tabulated are all smaller (and so ought to be earlier) than the corresponding ones known from the earliest stages of Islamic astronomy. The other (folio 17v) gives the longitudes of apogees up to the ninth sexagesimal fractional place for the year 872 Hijra (a.d. 1467–8). The longitude of the apogee of the Sun and of Venus are again equal, here up to the seconds (92;24,20, …). It is stated at the beginning of this table that it has been updated from the Mumtahan zīj.

E.g., cf. Samsó , op. cit. (ref. 23), 469. Note that this value is identical to al-Khwārizmī's longitude of the solar apogee (see above). Neugebauer (op. cit. (ref. 64), 99), however, derived the value 81;15° for the Venus's longitude of apogee from the Latin translation of al-Khwārizmī's zīj.

Yūnus Ibn , op. cit. (ref. 38), 120–1.

Cf. Van Dalen , op. cit. (ref. 82), 836. In another study devoted to the Ptolemaic orbital elements of the inferior planets, the present author will deal with this historical problem.

Yūnus Ibn , op. cit. (ref. 38), 123–5.