Abstract
The early development of Sanskrit astronomical tables described in Section 1.5 blossomed by the mid-second millennium into the profuse variety of table-text types categorized in Section 2.3, whose typical components were analyzed in more detail in Chapter 4. As numerical tables became more central to the work of astronomers/astrologers, the mathematical ingenuity of jyotiṣa authors to some extent shifted its focus. Algorithms directing users how to compute a desired quantity were supplemented and then partly supplanted by techniques for producing and arranging pre-computed data so that users could simply look up the desired quantity. We argue that at least part of what has long been conventionally described as a “decline” in the innovative development of Sanskrit astronomy after the twelfth century might be more accurately called an “occultation” of it, as the formats of increasingly popular table texts removed a larger share of their authors’ mathematical efforts from the direct scrutiny of their readers. The following discussion explores some of the major passages in that long and vigorous growth by surveying several individual koṣṭhaka works that seem to us to highlight some of its important characteristics.
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Notes
- 1.
- 2.
There are many examples in other jyotiṣa works of multiple algorithms prescribed for computing the same quantity, or algorithms offered as alternatives to versified lists of pre-computed values. See, for example, the Karaṇakutūhala’s list of ecliptic declination-differences followed by an algebraic approximation for finding declination (Mishra 1991, pp. 43–44, verses 3.13–15). But the closest parallel we have found to an instruction calling for look-up in a pre-computed table outside the verse text is Vaṭeśvara’s reference in the 905 Vaṭeśvarasiddhānta to interpolating within a set of approximate values of planetary equations constructed from scaled multiples of the tabulated R sin-differences (Shukla 1986, vol. 1, p. 121, verse 2.5.2).
- 3.
- 4.
The edition reads svayugānyadho at the end of this verse.
- 5.
The edition reads aśvi “2” for aśva “7” in this line.
- 6.
See, for this and the following references to Harihara, the combined edition of the Grahajñāna and the Gaṇitacūḍāmaṇi published in Pingree (1989).
- 7.
This Pañcāṅgapattra may refer to the approximately contemporary Candrārkı̄ of Dinakara, who introduces the work as a patra “leaflet” called pañcāṅga “calendar/almanac”, see Kolachana et al. (2018).
- 8.
This rule follows Bhāskara’s own Siddhāntaśiromaṇi (Śāstrı̄ 1989, p. 46, verses 2.24–25). For details see Montelle and Plofker (2015, pp. 26–28) and Plofker (Forthcoming).
- 9.
- 10.
There exists a work entitled Karaṇakutūhala-gata-sāraṇı̄ composed by one Nāgadatta (Pingree 1970–94, A5.166), but the relation, if any, between this work and the Brahmatulyasāraṇı̄ remains to be investigated. Tables from the Brahmatulyasāraṇı̄ are shown in the manuscript images in Figures 2.15, 3.3, 4.2, 4.9, 4.13, and 4.28.
- 11.
The scribe of Manuscript L (Pingree 1976) notes that this is 16,096 days from 23 February 1183 epoch of Bhāskara II’s Karaṇakutūhala. This number is equivalent to 4 × 4016, the latter number being a fundamental time period used in Gaṇeśa’s Grahalāghava (see Section 4.1). Various tables from the Laghukhecarasiddhi are shown in the manuscript images in Figures 2.14, 2.16, 3.6, and 4.18.
- 12.
There is some evidence that he may have been working in Khāndeśa, in the territory of the Yādavas, where perhaps he encountered two of Bhāskara II’s relatives who were astrologers at the court of Siṅghaṇa (ca. 1210–1246), the Yādava monarch of Devagiri (Pingree 1976, p. 2).
- 13.
Tables from manuscripts of this work are shown in Figures 3.7, 3.17, and 4.8. For more details on the work and its author, see Pingree (1968, pp. 37–39), Neugebauer and Pingree (1967), Pingree (1973, p. 82), and Pingree (2003, pp. 51–54). Aspects of the Mahādevı̄’s tables were investigated in Neugebauer and Pingree (1967), the first detailed analysis of a mean-to-true koṣṭhaka. Given the distribution of its extant manuscripts, it seems likely Mahādeva was working in Gujarāt or Rājasthān. However, the genealogy in the closing verses (41–43) of the Grahasiddhi identifies his father as Parāśurāma, son of Padmanābha, son of Mādhava, son of Bhogadeva of the Gautama gotra, all of whom were astrologers (daivajña), a dweller on the Godāvarı̄ river, presumably in Mahārāṣṭra (Pingree 1981, p. 42; Pingree 1968, p. 37).
- 14.
MS has śarorvyāṭu.
- 15.
MS has 12.
- 16.
Numerous examples of tables in the Makaranda’s manuscripts are shown in Figures 2.18, 3.2, 3.10, 3.11, 3.18, 4.6, 4.48, 4.51, 4.52, 4.53, 4.54, 4.55, 4.59, and 4.60. This work has been extensively treated in, e.g., Pingree (1968, pp. 39–46), Pingree (1973, p. 92), Pingree (1981, p. 42), Pingree (1970–94, A4.341–343), Pingree (2003, pp. 54–59). As discussed therein and in Sarma (1997), the Makaranda has been edited multiple times, e.g., Makaranda (1923); Miśra (1982). Note that there is no textual attestation of any epoch date in the work itself, although various manuscripts containing commentaries and notes mention that the year of the text is Śaka 1400. Additionally, the earliest Śaka year occurring as a table argument in some of the manuscripts is 1400, whose beginning is assigned to ghaṭı̄ 30;57 of weekday 6 in tithi 24. For the regional Saurapakṣa tradition see also works of Munı̄śvara (Pingree 1970–94, A4.436–441) and Kamalākara (Pingree 1970–94, A2.21–23).
- 17.
- 18.
For this approximation in general see Section 2.1.8; the trigonometric rationales underlying the manda-correction procedures for longitude and velocity are discussed in Section 2.1.2. For the transformation of the algebraic sine approximation into the given μ-rule and the following velocity-correction rule, see Rao and Uma (2007, pp. S56–60, S65–68).
- 19.
- 20.
- 21.
See, e.g., Pingree (1968, pp. 55–9), Pingree (1973, pp. 141–142), Montelle (2014), and Pingree (2003, pp. 83–87). Examples of the Jagadbhūṣaṇa’s tables appear in Figures 3.3, 3.13, 4.17, 4.19, and 4.31. Numerical evidence gleaned from tabular data in some of the manuscripts suggests that the tables were computed for a latitude of ϕ = 24∘, roughly corresponding to Ujjain in Madhya Pradesh (see Figure 4.31).
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Montelle, C., Plofker, K. (2018). The evolution of the table-text genre. In: Sanskrit Astronomical Tables. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-97037-0_5
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