Abstract
The history of computational algorithms, numerical methods, and data analysis has long been understudied in the history of mathematics, particularly outside the area of early modern and modern Western mathematics. Many founders of the modern discipline of history of mathematics in the nineteenth and twentieth centuries shared a professional bias in favor of identifying the essence of mathematics with elegant and rigorous demonstration of abstract propositions, which was viewed by most of their contemporaries as the highest form of mathematical endeavor. They largely disregarded the evolution of more plebeian or mechanical mathematical activities, such as approximating numerical parameters, constructing tables of function values, and optimizing computational performance. It has only recently started to become apparent how much interesting material the pioneers of history of mathematics overlooked when they neglected the arts of “number-crunching.” The distorting impact of this historiographic bias on our understanding of the development of mathematics has been particularly severe in the case of non-Western mathematical traditions that emphasized computational methods over proof structures.
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- 1.
For Islamic tables, besides the publications cited above, overviews of the corpus or parts of it can be found in Kennedy (1956), King and Samsó (2001), and Samsó (2011): a revised and greatly expanded version of the former is currently in preparation by Benno van Dalen, building on van Dalen (1993). Studies of individual zı̄jes or their features include, among many others, Nallino (1899–1907), Debarnot (1987), Van Brummelen (1991, 1998). Many more such studies are underway, particularly in the ongoing Ptolemaeus Arabus et Latinus project (https://ptolemaeus.badw.de), which also explores a number of Greek table texts. By far the most important of these Greek works, the Handy Tables of Ptolemy, has been analyzed most recently in Tihon (2011) and Mercier (2011); the somewhat fragmentary remainder of the Greek table corpus is discussed in, e.g., Jones and Perale (2013), Jones (1999), Sidoli (2014), and Van Brummelen (1993). Much of the very prolific genre of cuneiform astronomical tables has been examined in Neugebauer (1983) and Ossendrijver (2018), and their investigation continues in works such as Steele (2006, 2010). Egyptian tables were first systematically surveyed in Neugebauer and Parker (1960–1969); much of this information, along with a great deal of related research—e.g., Symons (2016)—is summarized and organized in the Ancient Egyptian Astronomy Database (https://aea.physics.mcmaster.ca). Some of the sources treating Chinese tables include Martzloff (2000, 2006, 2016), Yabuuti Kiyoshi (1974/1994), and chen (1992). The vast genre of Latin and other medieval/early modern European astronomical tables is not discussed at all here due to lack of space; but see, for example, Chabás and Goldstein (2012) and the analyses of many of their trigonometric and logarithmic tables at the LOCOMAT project (https://locomat.loria.fr).
- 2.
The indiscriminate assignment of all Sanskrit astronomical works to the category of “tables” is partly due to earlier scholars’ use of that term to refer to any astronomical system, and partly to the ignorance of many of them about this genre (as when H. Poleman in his catalogue of Indic manuscripts in North America describes (Poleman 1938, p. 246) the “Miscellaneous Bundle” at Columbia University, containing several complete handbooks and table texts as well as individual tables, as “[a] collection of several hundred miscellaneous folios, mostly tables not important enough and not bearing sufficient information to identify at all”). Even some Indologists who avoided such errors and were otherwise quite sympathetic to Sanskrit jyotiṣa maintained a rather dismissive attitude towards table texts; consider, for example, the remark of the early nineteenth-century student of siddhāntas Lancelot Wilkinson to the effect that “the foundation of such little knowledge as they display in predicting eclipses and the like, has, from […] the almost universal practice amongst the jyotishís, of making all their calculations from tables and short formulae, couched in enigmatical verses, been allowed to fall into a state of utter oblivion” (Wilkinson 1834, p. 507). Emphasis on the achievements of Sanskrit mathematical astronomy according to modern criteria of precise geometric models, etc., is a constant theme in analyses of the corpus; see, for instance, Dikshit (1981, pp. 119–120) and Arkasomayaji (1980, p. xxi).
- 3.
The primary studies and surveys of Sanskrit table texts by Pingree are Pingree (1970), Neugebauer and Pingree (1967), and Pingree (1968, 1973, 1987b, 1989). Much additional information about them appeared together with data on other jyotiṣa works in his manuscript catalogues such as Pingree (1984, 2003, 2004), and above all the immense Census of Exact Sciences in Sanskrit (Pingree 1970–94). Manuscripts on jyotiṣa are also described in Sarma and Sastry (2002) as well as innumerable catalogues of Sanskrit manuscripts in general in collections all over India, especially the magisterial New Catalogus Catalogorum project (Raghavan et al. 1968–2007). Various digital tools for storing and analyzing tabular data have been developed by Benno van Dalen (https://www.bennovandalen.de/Programs/programs.html). More recent database projects include “Table Analysis Methods for the history of Astral Sciences” (TAMAS) (https://tamas.hypotheses.org) led by Matthieu Husson and CATE (Computer Assisted Table Editing) (https://uc.hamsi.org.nz/cate/), which automates parts of the critical editing process.
- 4.
“Mathematical astronomy” here means computational systems that rely on the (observed and assumed) periodicity of various celestial phenomena and some form(s) of spatial reference system and standardized time-units to predict quantitatively the approximate future course of (some) astronomical events. Some other studies restrict the term to systems of astronomical computation meeting particular criteria for abstraction, accuracy, mathematical complexity, etc. In this narrower sense, mathematical astronomy is generally held to begin with Late-Babylonian arithmetic models around the middle of the first millennium BCE; see, e.g., Neugebauer (1975, vol. 1, p. 1) and Ossendrijver (2012, p. 1). According to these criteria, many simple algorithms from earlier centuries such as those relating seasonal length of daylight to the lengths of gnomon shadows would not count as “mathematical” astronomy.
- 5.
An analysis of a geometric interpretation used to motivate arithmetic computations of orbital velocity, albeit not involving a geometric model of the orbits themselves, is discussed in Ossendrijver (2016).
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A survey of the origins and aspects of calendar in this context is discussed in Yano (2003).
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Some Vedic narratives of deities and sages have been proposed by historians as mythologized descriptions of planetary phenomena (Kak 2005). But the persistent uncertainty about specifics of the chronological, historical, and geographic context of the Vedas makes it impossible to establish a definitive interpretive framework for such readings (Plofker 2009, pp. 33–35).
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It has long been debated whether the similarities between first-millennium Babylonian mathematical astronomy and some concepts in the Jyotiṣavedāṅga—e.g., the 30-fold division of the synodic month, the 60-fold division of the day, arithmetic schemes to express length of daylight variation, etc.—may indicate some transmission of Babylonian astral sciences to South Asia, possibly via the Achaemenid Empire in the sixth or fifth century. But since no direct evidence of such transmissions has been found, and there is no way of conclusively determining when the existing Jyotiṣavedāṅga was composed or what parts of it may record astral knowledge acquired in earlier periods, these hypotheses remain uncertain.
- 14.
For the possible astronomical significance of these epoch choices, see, e.g., Falk (2004).
- 15.
Mahāvı̄ra (mid-ninth century) discusses the scope and applicability of gaṇita in the Gaṇitasārasaṅgraha (Plofker 2007, p. 442, verses 1.9–17). Bhāskara I lists subdivisions of gaṇita in his commentary on the Āryabhaṭı̄ya (Keller 2006, vol. 1, p. 8). In particular, some authors draw a distinction between computation with avyakta “unmanifest” quantities, i.e., algebra, and with vyakta “manifest” ones; see below (Dhammaloka Forthcoming). There seems to be no clear consensus as to whether gaṇita counts as a śāstra in its own right or even a supercategory of jyotiṣa.
- 16.
Some of the technical details of these fundamental calculations as well as some of the subsequent ones in the following list are discussed in Chapter 4. Comprehensive explanations of all or most of these topics are provided in the commentaries accompanying many editions and translations of Sanskrit astronomical texts, such as Dvivedı̄ (1901–1902), Shukla (1976), Arkasomayaji (1980), Shukla (1986), Chatterjee (1981), and Ramasubramanian and Sriram (2011).
- 17.
These canonical “school” divisions in jyotiṣa are discussed in Plofker (2014, pp. 1–2) and Pingree (1978a, pp. 534, 629–630). To the best of our knowledge, the earliest references to them in Sanskrit sources include Dinakara’s Candrārkı̄ (verse 2) as well as Mallāri in his commentary to Gaṇeśa’s Grahalāghava (Jośı̄ 1994, pp. 44–45, verse 1.16). The terms pakṣa and pratipakṣa are well-known terms in philosophy, meaning opposing positions in an argument or a debate, but the etymological path for the evolution of the general concept of a pakṣa as a school of opinion and/or intellectual lineage in Sanskrit remains to be explored. The pakṣa identities of individual table texts, where known, are specified in Appendix A, and the characteristic parameters of the various schools in Appendix B.
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This school is represented in southeast Asian calendric traditions as well; Ōhashi (2008, p. 323) suggests that its diffusion may reflect a particular affinity for the Ārdharātrikapakṣa among Buddhists.
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Various astrological concepts are covered in Bṛhajjātaka (Jhā 1934) including divisions of zodiacal signs (1.6–12), exaltation/dejection (1.13), friendships/enmities of planets (2.15–18), daśās/antardaśās (chapter 8), aṣṭakavarga (chapter 9), moon in nakṣatras and zodiacal signs (chapters 14–15), dṛṣṭis (chapter 19), bhāvas (chapter 20), and vargas (chapter 21).
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For passages by Sanskrit authors which underscore the importance of studying the gola, see, for instance, Brahmagupta in his Brāhmasphuṭasiddhānta (verse 21.1) (Dvivedı̄ 1901–1902, p. 359), Vaṭeśvara in his Vaṭeśvarasiddhānta (Gola verses 1.1–6) (Shukla 1986, vol.II, pp. 613–4), Nityānanda in his Sarvasiddhāntarāja (Gola verse 2) (Misra 2016, pp. 105–106).
- 26.
This is exemplified by the criteria laid down in Vaṭeśvara’s early tenth-century siddhānta for composing a karaṇa work: “A karaṇa is to be made quite concise, not apparent to others, easily used by the stupid.” (Shukla 1986, vol. I, p. 56).
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Kennedy’s classic introductory survey of such texts defines them as follows (Kennedy 1956, p. 123): “A zı̄j consists essentially of the numerical tables and accompanying explanation sufficient to enable the practising astronomer, or astrologer, to solve all the standard problems of his profession, i.e., to measure time and to compute planetary and stellar positions, appearance, and eclipses.” Originally derived from the “Handy Tables” of Ptolemy, they were called zı̄j from Persian zı̄g or “cord,” a name which apparently associates the rows and columns of numerical tables with the warp and weft cords of a loom (and which may or may not have inspired the somewhat similar Sanskrit name sāraṇı̄).
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- 32.
Although it has been suggested in, e.g., Pingree (1978a, pp. 585–586) that this treatise may show traces of Islamic influence in its trigonometric rules and tables, it seems more likely to represent a parallel evolution within classical jyotiṣa.
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Montelle, C., Plofker, K. (2018). Introduction. 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_1
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