The Astrolabe and Spherical Trigonometry in Medieval India

The Indian development of this Hipparchan technique is discussed in Neugebauer O. and Pingree D. , The Pañcasiddhāntikā (Copenhagen, 1970–71), II, 41–44.

Several such formulae, particularly as represented in the work of the Mādhava school in late-medieval Kerala, are discussed by Gupta R. C. in “Solution of the astronomical triangle as found in the Tantrasaṅgraha (a.d. 1500)”, Indian journal of history of science, ix (1974), 86–99, and in “Mādhava's rule for finding angle between the ecliptic and the horizon and Aryabhata's knowledge of it”, in History of oriental astronomy, ed. by Swarup G. (Cambridge, 1987), 199–202. As Gupta points out, many of these rules are indistinguishable from ones derived via spherics per se, and the typical lack of any proof in the authors' presentation of them renders it impossible to determine exactly how they were worked out by their inventors. But the absence of explicit reference to the solution of spherical triangles in Indian mathematical works makes it most likely that such rules were estabished from various ingenious manipulations of triangles within the sphere rather than on its surface.

Some aspects of the Indian reception of the astrolabe are discussed in Sarma S. R. , “Astronomical instruments in Mughal miniatures”, Studien zur Indologie und Iranistik, xvi (1992), 235–76.

See Pingree D. , “History of mathematical astronomy in India”, in Dictionary of scientific biography, xv, 533–633, p. 626; Sarma , op. cit. (ref. 3), 238–9; and Ohashi Y. , “Early history of the astrolabe in India”, Indian journal of history of science, xxxi (1997), 199–295.

Yantrarāja 1, 2–3. The edition used is The Yantrarāja, ed. by Raikva K. V. (Bombay, 1936). The work was previously edited by Dvivedi S. (Benares, 1883).

The reference is to the Puranic legend of the gods' churning of the oceans to obtain the divine nectar of immortality; see, e.g., Viṣṇupurāṇa IX, 81–98.

The concise treatises of Pseudo-Māshā'allāh (in a Latin version; see Gunther R. T. , “Messahalla on the astrolabe”, in Early science in Oxford , ed. by Gunther R. T. , v (Oxford, 1929). 137–231, and Kunitzsch P. , “On the authenticity of the treatise on the composition and use of the astrolabe ascribed to Messahalla”, Archives internationales d'histoire des sciences, xxxi (1981), 42–62), ibn 'Isā ( Schoy C. , '“Alī ibn 'Isā, das Astrolab und sein Gebrauch”, Isis, ix (1927), 239–54), al-Bīrūnī ( Dallal A. , “al-Bīrūnī's Book of pearls concerning the projection of spheres”, Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften, iv (1987/88), 81–138), and Tūsī (whose Bīst bāb, in the Sanskrit version of Nayanasukha, is published in Yantrarājavicāraviṇśādhyāyī, ed. by Bhaṭṭācārya V. (Benares, 1979)) confine themselves to practical procedures for making, marking, and manipulating the astrolabe, rather than calculating coordinates. al-Bīrūnī's longer work, Istī'āb al-wujūh al-mumkinah fī ṣan'al al-asṭurlāb, may prove more comprehensive.

See Pingree D. and Morrissey P. , “On the identification of the yogatārās of the Indian nakṣatras”, Journal for the history of astronomy, xx (1989), 99–119.

Brahmagupta, for instance, prescribes this rule in Brāhmasphuṭasiddhānta 10, 15–16 (Brāhmasphuṭasiddhānta, ed. by Dvivedi S. (Benares, 1901–2)).

Pingree and Morrissey , op. cit. (ref. 8), 111.

It is generally employed in the calculation of heliacal risings and settings of the planets; Mahendra uses the term to include the λ_*-calculations for stars, whether by Indian or foreign methods.

This is the standard form of the rule in the Āryapakṣa or school of Aryabhata — see, for example, Āryabhaṭīya 4, 36 (Āryabhaṭīya, ed. by Shukla K. S. and Sarma K. V. , i (New Delhi, 1976)). A version frequently seen in the rival school of Brahmagupta, the Brāhmapakṣa, substitutes the Sine function for the Versine in this formula, as in Brāhmasphuṭasiddhānta 6, 3.

Cf. Brāhmasphuṭasiddhānta 7, 5 and Lalla's Śiṣyadhīvṛddhidatantra 9, 1–2 (Śiṣyadhīvṛddhidatantra of Lalla, ed. by Chatterjee B. (New Delhi, 1981)).

See the Appendix for translations of this and the following glosses and examples by Malayendu, as they are rendered in the published edition; square brackets indicate editorial insertions or comments, and angle brackets deletions.

See King D. A. , “The astronomical works of ibn Yunus”, PhD. dissertation, Yale University, 1972, 295–6. Although, as was pointed out earlier, the Yantrarāja has not been directly linked to this or any other particular Islamic source, it is interesting that both Ibn Yūnis and Malayendu Sūri choose α Tauri (Aldebaran, Skt. Rohiṇī) as an example to demonstrate the δ-calculations.

King , op. cit. (ref. 15), 293–5.

See Hamadani-Zadeh Javad , “Nāṣīr ad-Din on determination of the declination function”, in History of oriental astronomy, ed. by Swarup G. (Cambridge, 1987), 185–9.

Little is known at present about the specific works used to teach Islamic astronomy in India in this early period; see, however, Pingree D. , “Islamic astronomy in Sanskrit”, Journal for the history of Arabic science, i (1978), 315–30, and Sarma S. R. , “Yantraprakāra of Sawai Jai Singh”, Studies in history of medicine and science, x-xi (1986, 1987), (supplement) 1–139.

Pingree and Morrissey , op. cit. (ref. 8). 115.

See, e.g., King , op. cit. (ref. 15), 302–5, and Debarnot M.-Th. , Kitāb maqālīd 'ilm al-hay'a: La trigonométrie sphérique chez les Arabes de l'est à la fin du X^e siècle (Damascus, 1985), 216–18.

A more detailed explanation of the spherical trigonometry used here is given in, e.g., Kennedy Edward S. , “The history of trigonometry”, in Kennedy E. S. , Studies in the Islamic exact sciences (Beirut, 1983), 1–29.

Debarnot , op. cit. (ref. 20), 52–53.

In this form, it more closely resembles Brahmagupta's version (see ref. 12); Mahendra's use of Sin (λ + 90) = Cos λ would therefore seem to place him among the followers of the Brāhmapaksa, who tend to use the Sine as the interpolating function in this procedure in preference to the Versine favoured in the Āryapakṣa.

See. e.g., Śrīpati's Siddhāntaśekhara, 9, 6 (Siddhāntaśekhara, ed. by Misra B. (Calcutta, 1932, 1947)); Bhāskara's Siddhāntaśiromaṇi , Gaṇita 7, 4 (Siddhāntaśiromaṇi, ed. by Sastri B. D. (Benares, 1967; repr. as Kashi Sanskrit Series 72, Benares, 1989)). Muñjāla uses an equivalent expression in Laghumānasa 52 (Laghumānasa, ed. and transl. by Majumdar N. K. (Calcutta, 1951)).

See, e.g., Śiṣyadhīvṛddhidatantra 8, 3, and Vaṭeśvara's Vateśvarasiddhānta, 6, 10 (Vaṭeśvarasiddhānta, ed. and transl. by Shukla K. S. (New Delhi, 1986)).

Most commonly R = 3438 (cf. Āryabhaṭīya Gītikā 12); other well-known values are 3270 ( Brāhmasphuṭasiddhānta 2, 5) and 3415 (Siddhāntaśekhara 3, 6). If the standard Indian obliquity, ε = 24°, is used in place of the Yantrarāja's smaller value, then the increased amount of the factor Sin ε/R² diverges still further from Mahendra's number.

This is the obliquity used in the zīj of al-Khwārizmī (c. a.d. 840; see Kennedy E. S. , A survey of Islamic astronomical tables , in Transactions of the American Philosophical Society, ns., xlvi/2 (1956), 123–77 (p. 148)), but 1 know of no other zīj that uses it as a table parameter, although Ptolemy's value was certainly well-known to Islamic astronomers.

Ohashi Y. , “A note on some Sanskrit manuscripts on astronomical instruments”, in History of oriental astronomy, ed. by Swarup G. (Cambridge, 1987), 191–5; see also Ohashi , op. cit. (ref. 4), 225.

For example, the discussions of the astrolabe in the works associated with the court of the eighteenth-century ruler Sawai Jai Singh — the Yantrarājaracanā (ed. by Jyotirvid K. (Jaipur, 1953)), the Yantraprakāra ( Sarma , op. cit. (ref. 18)), the Yantrarājavicāravimśādhyāyī ( Bhaṭṭācārya , op. cit. (ref. 7)), and the Yantraprabhā (published with the Yantrarājaracanā in Jyotirvid , op. cit.) of Śrīnātha — all neglect these theoretical considerations, as does the earlier Yantraśiromaṇi of Viśrāma (published with the Yantrarāja in Raikva , op. cit. (ref. 5)). It would be interesting to know whether the work of Padmanābha mentioned above gives a fuller treatment of them.

Such abbreviated versions are preserved in the manuscripts Benares Hindu University B.3318 and B.521.

See Ohashi Y. , “Sanskrit texts on astronomical instruments during the Delhi Sultanate and Mughal periods”, Studies in history' of medicine and science, x-xi (1986–87), 165–81, p. 175.

All references to the Yantrarājakalpa are based on the manuscript Benares 35245; the rules in question appear on ff. 21 v–22v.

This refers to the formula for correcting celestial longitudes for precession given earlier by Mathurānātha on f. 13v.

Witness Mathurānātha's use of ε = 23;30, 17 (f. 23r) instead of Mahendra's value.