Abstract
This article deals with an unstudied criterion for determining lunar crescent visibility, which appears in the Mufrad Z\(\bar{\iota }\)j, (compiled by Ḥāsib al-Ṭabar\(\bar{\upiota }\), 5thc.A.H./11th c.A.D.). Al-Ṭabar\(\bar{\upiota }\) attributes this circular criterion to Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\). Initially, Prof. David King shed light on this criterion in 1987 and explained it briefly. We will examine this criterion by re-computing the underlying numerical values to reconstruct it, in order to demonstrate that it originates from Ḥabash’s simple criterion.
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Notes
See References.
Indeed, a part of most z\({\bar{\iota }}\)jes was devoted to a tabular values or explanations on lunar crescent visibility and only few historians of astronomy have examined a few of them.
King, pp. 208–210
King introduces the steps as: \(10{^{\circ }}\)–\(20{^{\circ }}\)–\(30{^{\circ }}\), according to what appears in the table. However, I noticed that should the steps start with \(0{^{\circ }}\) (for the first value of each zodiacal sign), the tabular values would have a better coincidence with the re-computed values.
Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\) in his Taḥd\(\bar{\iota }\)d, (1962, p. 119) tells us that he made observations for a while in Jayfūr, a village in the proximity of Kabul. The village has not yet been located. Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\) further measured the geographical latitude of Kabul as equal to: \(+\,34; 41{^{\circ }}\). In Kennedy (1987, pp. 167–168), there are several latitudes as equal to 34; \(30{^{\circ }}\) for Kabul.
For computing this function, see Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\), (Maqāl\(\bar{\iota }\)d, 1985, pp. 237–239).
See Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\), (Maqāl\(\bar{\iota }\)d, 1985, pp. 256–259).
Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\), Al-Qānūn al-Mas\(^{c}\)ūd\(\bar{\iota }\), vol. 2, pp. 957–958.
See Ḥabash’s Z\(\bar{\iota }\) j, (Berlin ms. 5750, fol. 151v.; See also Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\), (Maqāl\(\bar{\iota }\)d, 1985, p. 258). Indeed, Debarnot had already noted that this criterion is from Ḥabash (ibid., p. 258 footnote 2). However, she has not dealt with the problem completely.
Since the points of A and B have different azimuths (at maximum latitude: + 5.3\({^{\circ }})\), therefore, it leads to a negligible difference between lunar time lags for these two positions, as their declinations are different.
Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\), (Maqāl\(\bar{\iota }\)d, 1985, p. 258–259) states that all the values of (e) should always be organized as to result in \(\hbox {d}=10{^{\circ }}\).
Ḥabash’s Z\(\bar{\iota }\)j, Berlin ms. 5750, fols. 151v.- 152 r.
In fact, this formula is for the cases that the actual (e) is a little smaller than the tabular value of (e); so in such a case the positive value of \(\beta \) (lunar latitude) would provide a better situation for visibility.
According to the above discussion, it is clear that in Formula 2, If \(\sin \hbox { AS} = \sin { (e)}\), then it results in 1. In formula (2), Ḥabash takes sin \(({L}_{{m}})\), which it leads to an awkward result.
For instance, see Ibn Yūnus (1997), pp. 158–161
See Kennedy, (1987, pp. 55–56). In Kennedy’s list of coordinates for Baghdad there are several latitudes between \(33{^{\circ }}; 20^{\prime }\) to \(33{^{\circ }}; 22^{\prime }\).
Indeed, there was a dispute about the priority of the invention of al-Shakl al-Mughn\(\bar{\iota }\) between several contemporary astronomers; Although, Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\) (Maqāl\(\bar{\iota }\)d, 1985, pp. 100–101) confirms the priority of Abū-Naṣr-i \(^{\mathrm{c}}\)Īraq regarding the invention of this figure.
I mean the z\(\bar{\upiota }\)j known as ms. 784.2, Z\(\bar{\iota }\)j al-Ma\(^{c}\)rūf bi-Dimashq\(\bar{\iota }\), Yeni Cami Library, Istanbul.
For a report and comparison between two copies, see Kennedy (1956), pp. 126–127, 151–154.
Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\) in his Maqāl\(\bar{\iota }\)d (1985, pp. 260–261) asserts that Ḥabash used the “Shakl al-Qaṭṭā\(^{c}\) ” (Sector Figure) in his computations and Al-B\(\bar{\iota }\)rūn\(\bar{\iota }\) changed them to “dispensable figure” for only one case. This statement ensures us that Ḥabash had no contribution in invention of “dispensable figure”.
See Kennedy 1965, pp. 73–78.
King, pp. 188–193, It should be noted that in some Z\(\bar{\iota }\)jes, the author asserts that the tabular values are based on Al-Khwārizm\(\bar{\upiota }\)’s criterion, See ibid, p. 193.
References
Abū Rayḥān Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\), Al-Qānūn al-Mas \(^{c}\) ūd \(\bar{\iota }\), Vol. 2, Haydar-Abad, 1373–1375 A.H/1953–1955A.D.
Idem. 1985. Kitāb-i Ma \(^{c}\) qāl \(\bar{\iota }\) d-i \(^{c}\) ilm al-Hay’a, translation and annotation by Marie-Thérèse Debarnot, Dimashq.
Idem. 1992. Taḥd \(\bar{\iota }\) d-i Nihāyāt-al-Amākin li-Taṣḥ \(\bar{\iota }\) ḥ Masāfāt al-Masākin, eds., P. Bulgakov and I. I. Ahmad, Cairo, 1962, reprinted by Sezgin et al., in Islamic geography, vol. 25, Frankfurt.
Ḥabash Al-Ḥāsib, \(Z\bar{\iota }j\) al-Ma’rūf bi-Dimashq \(\bar{\iota }\), ms.784.2, Yeni Cami Library, Istanbul.
Idem., \(Z\bar{\iota }j\), ms.5750, Berlin Library.
Ibn Yūnus, Al-Ḥākim \(\bar{\iota }\) Z \(\bar{\iota }\) j, Notices et extraits des manuscrits de la Bibliothèque Nationale et autres bibliotèques (Paris) 7. 12 (1803–1804), pp. 16–240, ed. Perceval Caussin, reprinted by F. Sezgin in: Islamic mathematics and astronomy, vol. 24, Frankfurt 1997.
Kennedy, E.S. 1956. A Survey of Islamic Astronomical Tables, vol. 46, Part 2., Transactions of the American Philosophical Society Philadelphia: American Philosophical Society.
Kennedy, E.S. and M. Janjanian. “The crescent visibility table in Al-Khwārizm\(\bar{\upiota }\)’s Z \(\bar{\iota }\) j”, Centaurus, 11(2): 73–78. 1965, reprinted in: Studies in the Islamic Exact Sciences, Beirut 1983, pp. 151–156.
Kennedy, E.S., and M.H. Kennedy. 1987. Geographical Coordinates of Localities from Islamic Sources, Frankfurt.
King, D. (1987) “Some early Islamic tables for determining lunar crescent visibility”. In From Deferent to Equant, a volume of studies in the history of science in the ancient and medieval Near East in honor of E.S. Kennedy, eds. David King and George Saliba, pp. 185–226. New York, 1987.
Al-Ṭabar\(\bar{\upiota }\), Muḥammad ibn Ayyūb (Ḥāsib-e Ṭabar\(\bar{\varvec {\upiota }}\)), Mufrad \(Z\bar{\varvec {\iota }}j\), Browne ms. O.I.\(^{(10)}\), Cambridge: Cambridge University Library
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Communicated by: George Saliba.
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Appendix
1.1 The Arabic Text
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Giahi Yazdi, HR. The mysterious table of lunar crescent visibility attributed to Al-B\(\bar{\upiota }\)rūn\(\bar{\upiota }\) and Ḥabash Al-Ḥāsib’s contribution. Arch. Hist. Exact Sci. 72, 89–98 (2018). https://doi.org/10.1007/s00407-017-0201-z
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DOI: https://doi.org/10.1007/s00407-017-0201-z