Summary
Historians of Indian mathematics generally concentrate on primary sources in Sanskrit. Sanskrit having been the chief medium of intellectual discourse in India and the major vehicle of pan-Indian dissemination of ideas, such an emphasis on the Sanskrit texts is no doubt justified. Yet there have also been other parallel streams of intellectual communication in India such as the Prakrits, the New Indo-Aryan languages; the Dravidian languages of the South, and Persian. All these languages possess rich and varied literature including works on mathematics. The mathematical works composed in these languages, though largely modeled on Sanskrit manuals, may contain much information of contemporary relevance. Furthermore, a large corpus of recreational mathematics exists as oral literature. While citing a number of examples from the mathematical literature in Telugu, a plea has been made in this paper for organized efforts toward the survey and study of mathematical literature in the regional languages, both of the recorded and of the oral varieties.
Revised version of a lecture delivered at the First International Conference of the New Millennium on History of Mathematical Sciences, Delhi, December 20–23, 2001.
S. R. Sarma has been professor of Sanskrit at Aligarh Muslim University, India. He has been editor of the Indian Journal of History of Science. His main areas of interest are the history of science in India and the intellectual exchanges between the Sanskritic and Islamic traditions of learning.
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Notes
- 1.
Henry Thomas Colebrooke (tr), Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brehmegupta and Bháscara, London 1817, First Indian Reprint: Classics of Indian Mathematics: Algebra with Arithmetic and Mensuration, From the Sanskrit of Brahmagupta and Bhāskara, with a foreword by S. R. Sarma, Sharda Publishing House, Delhi (2005).
- 2.
On this, see, inter alia, Swami Agehananda Bharati, Great Tradition and Little Traditions: Indological Investigations in Cultural Anthropology, Chowkhamba Sanskrit Studies, Vol. XCVI, Chowkhamba Sanskrit Series Office, Varanasi (1978).
- 3.
To mention the most prominent works: Bibhutibhusan Datta and Avadhesh Narayan Singh, History of Hindu Mathematics: A Source Book, 2 parts, 1935, 1938; single volume edition: Asia Publishing House, Bombay etc., 1962; A. K. Bag, Mathematics in Ancient and Medieval India, Chaukhamba Orientalia, Varanasi-Delhi, 1979; T. A. Saraswati Amma, Geometry in Ancient and Medieval India, Motilal Banarsidass, Delhi-Varanasi-Patna, 1979; David Pingree, Census of the Exact Sciences in Sanskrit, Series A, Volumes 1–5, American Philosophical Society, Philadelphia, 1970–1994 (in progress); also the large number of papers by R. C. Gupta listed in: Takao Hayashi, “A Bibliography (1958–1995) of Radha Charan Gupta, Historian of Indian Mathematics,” Historia Scentiarum, 6.1 (1996) 43–53.
- 4.
K. V. Sarma, A Bibliography of Kerala and Kerala-based Astronomy and Astrology, Vishveshvaranand Institute, Hoshiarpur, 1972, though primarily devoted to works in Sanskrit, contains several works on mathematics composed in Malayalam as well. The Government Oriental manuscripts Library, Madras, brought out a Malayalam work on Mathematics, Kaṇakkusāram, ed. D. Achyutha Menon, Madras, 1950. But, as far as I know, no study of this work has appeared to date.
- 5.
K. R. Rajagopalan, Mathematics in Karnataka, Bhavan’s Journal, 5.6 (October 1958) 52–56; Mathematics in Tamil Nadu, ibid. 5.20 (May 1959) 39, 42–44; Mathematics in Andhra, ibid. 6.8 (November 1959) 47–49; Mathematics in Kerala, ibid. 6.10 (December 1959) 61–64. See also R. C. Gupta, Some Telugu Authors and Works on Ancient Indian Mathematics, Souvenir of the 44th Conference of the Indian Mathematical Society, Hyderabad, pp.25–28 (1978).
- 6.
K. K. Bishoi, Palm-Leaf Manuscripts in Orissa, in: A. Pandurangan and P. Maruthanayagam (ed), Palm-Leaf and Other Manuscripts in Indian Languages, Institute of Asian Studies, Madras, 1996 pp.46–56; esp. pp.52–53: “Orissa…has a rich heritage of mathematical treatises. Proficiency in mathematics is exemplified in the manuscripts. The authors of Orissan Mathematical manuscripts are Anirdha, Artta Dasa, Krushna Padhiari, Ucchavananda, Kunjabana Pattnayaka Krupasindhu, Gangadhara, Nimai Charana, Radha Charana, Brajabhusana, Vamadeva, Shiva Mohanty, Sarangadhara, Hari Nayaka and Srinatha of the Ganita Sāstras. The Līlāvatī Sūtra is very popular in Orissa. The manuscript is available in all parts of the state. It provides scope for all age groups to study mathematics through works of addition, subtraction, multiplication, division, mensuration, trigonometry, algebra, etc.”
- 7.
Takao Hayashi, The Bakhshālī Manuscript: An Ancient Indian Mathematical Treatise, Groningen Oriental Studies, Vol. XI, Egbert Forsten, Groningen, 1995.
- 8.
Takao Hayashi, The Pañcaviṃśatikā in its Two Recensions: A Study in the Reformation of a Medieval Sanskrit Mathematical Textbook, Indian Journal of History of Science, 26, 393–448 (1991).
- 9.
Takao Hayashi, The Caturacintāmaṇi of Giridharabhaṭṭa: A Sixteenth Century Sanskrit Mathematical Treatise, SCIAMVS: Sources and Commentaries in Exact Sciences, 1, 133–209 (2000).
- 10.
Sreeramula Rajeswara Sarma, Rule of Three and Its Variations In India: Yvonne Dold-Samplonius et al. (eds.), From China to Paris: 2000 Years Transmission of Mathematical Ideas, Steiner Verlag, Stuttgart, pp.133–156, especially 149 (2002).
- 11.
This is perhaps the last work on mathematics to be composed in Sanskrit in the traditional style, and is yet to be published. The Department of Sanskrit, Aligarh Muslim University, possesses three manuscripts of this work.
- 12.
On his life and works, see Sreeramula Rajeswara Sarma, Ṭhakkura Pherū’s Rayaṇaparikkhā, Viveka Publications, Aligarh, 1984, Introduction.
- 13.
These six scientific works (and a seventh of a religious nature) were edited and published by the Jaina savant Jinavijaya Muni under the title Ṭhakkura-Pherū-viracita-Ratnaparīkṣādi-sapta-granthasaṃgraha, Rajasthan Oriental Research Institute, Jodhpur, 1961.
- 14.
A new edition with an English translation and a mathematical commentary of this interesting work is currently being prepared by Takao Hayashi, Takanori Kusuba, S. R. Sarma, and Michio Yano.
- 15.
Cf. S. R. Sarma, Conversion of Vikrama-Saṃvat to Hijrī in: B. V. Subbarayappa and K. V. Sarma (eds.), Indian Astronomy: A Source-Book, Bombay, pp.59–60 (1985); idem, Islamic Calender and Indian Eras in: G. Kuppuram and K. Kumudamani (eds.), History of Science and Technology in India, Delhi, Vol.2, pp.433–441 (1990).
- 16.
On Nārāyaṇa’s treatment of magic squares, see Schuyler Cammann, Islamic and Indian Magic Squares, History of Religions, 8.3–4, 181–209, 271–299 (1969); Parmanand Singh, Total Number of Perfect Magic Squares: Nārāyaṇa’s Rule, The Mathematics Education, 16.2 (June 1982) 32–37; idem, Nārāyaṇa’s Treatment of Magic Squares, Indian Journal of History of Science, 21.2, 123–130 (1986); idem, The Gaṇitakaumudī of Nārāyaṇa Paṇḍita, Chapter XIV, English Translation with Notes, Gaṇita-Bhāratī, 24, 35–98 (2002); Takanori Kusuba, Combinatorics and Magic Squares in India: A Study of Nārāyaṇa Paṇḍita’s Gaṇitakaumudī, Chapters 13–14, PhD Thesis, Brown University 1993.
- 17.
Dilip Kumar Sarma, Kautuk Āru Kāithelī Aṃka: A Study, Summaries of Papers, All-India Oriental Conference, 40th Session, Chennai, 2000. p. 505 (TS & FA-32).
- 18.
Dilip Kumar Sarma, A Peep into the Study of Development of Mathematics in Assam from Ancient to Modern Times, Summaries of Papers, All-India Oriental Conference, 39th Session, Vadodara, 1998, pp.437–438 (TS & FA-11). See also Ganganand Singh Jha, Asam kī gaṇitīya den, Pūrvānchal Praharī, Guahati, 3 May 2000, p.5; cited in : Hiteshwar Singh, Dr. G. S. Jha: A Broad-Based Historian of Mathematics, Gaṇita Bhāratī, 25, 150–153 (2003).
- 19.
Sreeramula Rajeswara Sarma, The Pāvulūrigaṇitamu: the First Telugu Work on Mathematics, Studien zur Indologie und Iranistik, Hamburg, 13–14, pp.163–176 (1987).
- 20.
Sārasaṃgrahagaṇitamu, Pāvulūri Mallana (Mallikārjuna) praṇītamu ed, Veṭūri Prabhākara Śastri, Part 1, Tirupati, 1952. This edition contains only a small part of the text, corresponding to Sanskrit Gaṇtasārasaṃgraha 1.1–3.53.
- 21.
David Singmaster, South Bank University, London, is compiling the Sources in Recreational Mathematics: An Annotated Bibliography. The seventh preliminary edition was released in January 2002.
- 22.
Cf. D. K. Sinha, Ethno-mathematics: A Philosophical and Historical Critique, in: D. P. Chattopadhyaya and Ravinder Kumar, Mathematics, Astronomy and Biology in Indian Tradition: Some Conceptual Preliminaries, PHISPC Monograph Series on History of Philosophy, Science and Culture in India, No. 3, Project of History of Indian Science, Philosophy and Culture, New Delhi, pp.94–119 (1995).
- 23.
Jean-Baptiste Tavernier, Travels in India tr. V. Ball, second edition, edited by William Crooke, London, Vol. 2, p.144 (1925).
- 24.
Damodar Dharmanand Kosambi, Social and Economic Aspects of the Bhagavad-Gītā, in: idem, Myth and Reality: Studies in the Formation of Indian Culture, Popular Prakashan, Bombay, 1962, pp.12–41, especially 32. I have not been able to find any information on these surviving tables.
- 25.
John Taylor, Līlāwatī: or a Treatise on Arithmetic and Geometry by Bhascara Acharya, translated from the Original Sanskrit by John Taylor, M. D. of the Hon’ble East India Company’s Bombay Medical Establishment, Bombay, 1816, p. 145. The quotation is from a highly interesting “Short Account of the Present Mode of Teaching Arithmetic in Hindu Schools” (pp.143–161) which he appended to his introduction.
- 26.
Gazetteer of the Bombay Presidency, Volume IX, Part 1: Gujarat, Population, Hindus, Bombay, 1901; reprinted as Hindu Castes and Tribes of Gujarat, compiled by Bhimbhai Kriparam, ed. James M. Campbell, Gurgaon, 1988, Vol. 1, p.80.
- 27.
The Rev. Lál Behári Day, Govinda Sámanta or the History of a Bengal Ráiyat, London, 1874; new edition under the title Bengal Peasant Life London, 1878; reprint: Macmillan and Co., Limited, London, 1920, p. 75: “He (the village school master) was the first mathematician of the village. He had not only Subhankara, the Indian Cocker, at his finger tips, but was acquainted with the elements of Víjaganita or Algebra.”
See also W. Adam, State of Education in Bengal, 1835–1838 (Extracts reprinted in: Dharampal, The Beautiful Tree: Indigenous Indian Education in the Eighteenth Century Biblio Impex Private Limited, New Delhi 1983, pp.269–270): “The only other written composition used in these schools, and that only in the way of oral dictation by the master, consists of a few of the rhyming arithmetical rules of Subhankar, a writer whose name is as familiar in Bengal as that of Cocker in England, without anyone knowing who or what he was or when he lived. It may be inferred that he lived, or, if not a real personage, that the rhymes bearing that name were composed, before the establishment of the British rule in this country, and during the existence of the Muslim power, for they are full of Hindustani or Persian terms, and contain references to Muslim usages without the remotest allusion to English practices or modes of calculation.”
Edward Cocker (1631–1675) was an English pedagogue whose posthumous publications Arithmetick, Being a Plain and Easy Method (1678) and Algebraical Arithmetic or Equations (1684) were so popular that “according to Cocker” has become a proverbial expression to mean “very reliable.” An analogous expression in German “nach Adam Riese” perpetuates the memory of Adam Riese (1492–1559), who wrote the earliest mathematical primers in German.
- 28.
D. K. Sinha, Ethnomathematics: A Philosophical and Historical Critique, op. cit., discusses on pp. 99–102 some old Bengali rhymes, which may be of Śubhaṅkara.
- 29.
Sreeramula Rajeswara Sarma, Some Medieval Arithmetical Tables, Indian Journal of History of Science, 32, 191–198 (1997).
- 30.
My father’s friend and his father were hereditary Karaṇams who maintained the village records. The copybooks are datable to the 1930s, but the material collected therein is much older.
- 31.
On the Josephus problem, see David Eugene Smith, History of Mathematics, New York, Vol. II, pp.541–544 (1925).
- 32.
Osamu Takenouchi et al. (eds. and trs.), Jinko-ki, Wasan Institute, Tokyo 2000, pp.139–140. “Wasan” is the indigenous mathematics developed in Japan during the Edo Period (1603–1867). The Jinko-ki, which was published in 1627, is one of the earliest texts of this genre. The present edition contains an English translation, together with the facsimile reproduction of the original Japanese illustrated woodblock edition of 1627.
- 33.
Sreeramula Rajeswara Sarma, Mathematical Literature in Telugu: An Overview, Sri Venkateswara University Oriental Journal, 28, 7790 (1985).
- 34.
S. Irfan Habib and Dhruv Raina, The Introduction of Scientific Rationality into India: A Study of Master Ramchandra, Urdu Journalist, Mathematician and Educationist, Annals of Science 46.6 (November 1989), pp.597–610; Dhruv Raina and S. Irfan Habib, “Ramchandra’s Treatise through the “Haze of the Golden Sunset”: An Aborted Pedagogy,” Social Studies of Science 20.3 (1990), pp.455–472; Dhruv Raina, Mathematical Foundations of a Cultural Project: Ramchandra’s Treatise through the “Unsentimentalized Light of Mathematics,” Historia Mathematica, 19, pp.371–384 (1992).
- 35.
In the discussion following my lecture, I learned that zero is also called pūjyam in Malayalam and Marathi. It would be interesting to know when this designation came into vogue and in what context. I also learned a Tamil proverb, which declares, “Inside the pūjyam (zero), there exists a rājyam (kingdom).”
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Sarma, S.R. (2009). Mathematical Literature in the Regional Languages of India. In: Yadav, B., Mohan, M. (eds) Ancient Indian Leaps into Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4695-0_14
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