Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators pp 115-133 | Cite as

# Reading Algorithms in Sanskrit: How to Relate Rule of Three, Choice of Unknown, and Linear Equation?

## Abstract

Texts in India were transmitted in the context of oral transmissions. A consequence is that their transmission implied memorization. In the written texts that are available to us, it is difficult to locate a precise section on a specific topic if the reader does not know in advance where this section is supposed to be written. A verbal knowledge of the text is required prior to a written one. Algebraic texts in Sanskrit are thus shaped like lists of operations where the progress of the algorithm seems, at first sight, more valued than its understanding. Nevertheless, the prescription of algorithmic operations delivers some clues as to what the author expected his readers to do or to understand. This chapter presents an excerpt from the *Bījagaṇitāvataṃsa* written by Nārāyaṇa in 14th-century India. The example prescribes an algorithm to set up a linear equation by means of a Rule of Three. Yet, the algorithms presented in the general rule and the one used in the commentary are slightly different. These differences reveal a deep understanding and a deductive interpretation of the rule by the commentator. Tabular setting, choosing unknowns, and Rule of Three played a key role in the elaboration of the reasoning.

## Keywords

Algorithm Unknown Rule of three 14th-century India Nārāyaṇa## References

## Primary Sources

- Nārāyaṇa,
*Bījaganitāvatamsa*(dates unknown). Manuscripts B1 (call no. 35579) and B2 (call no. 98699). Xerox copies provided by Benares Sanskrit College, Sarasvati Bhavana Texts collection.Google Scholar - Singh, P. 1998/2002. The
*Gaṇita Kaumudī*of Nārāyaṇa Paṇḍita.*Gaṇita Bhāratī*20, 1998: 25–82 (Chaps I–III); 21, 1999: 10–73 (Chap IV); 22, 2000: 19–85 (Chaps V–XII); 23, 2001: 18–82 (Chap XIII); 24, 2002: 35–98 (Chap XIV).Google Scholar *The Śulba-sūtra of Baudhāyana, Āpastamba, Kātyāyana and Mānava*, with Text, English Translation and Commentary. Ed. S.N. Sen and A.K. Bag, New Delhi, Indian National Science Academy, 1983. Google Scholar*Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara*, critically edited with the Introduction and Appendices by K.S. Shukla, New Delhi, Indian National Science Academy, 1976.Google Scholar

## Secondary Sources

- Datta, Bibhutibhushan. 1933. The algebra of Nārāyaṇa.
*Isis*19: 472–485.CrossRefGoogle Scholar - Datta, Bibhutibhushan, and Avadhesh Narayan Singh. 1935.
*History of Hindu Mathematics, a source book. Part I and II*. Bombay, India: Asia Publishing House.Google Scholar - Filliozat, Pierre-Sylvain. 2004. Ancient Sanskrit Mathematics: An oral tradition and a written literature. In
*History of science, history of text*, ed. K. Chemla, vol. 238, 137–160. Boston Studies in the Philosophy of Science. Springer.Google Scholar - Hayashi, Takao. 2004. Two Benares manuscripts of Nārāyaṇa Pandita’s
*Bījaganitāvatamsa.*In*Studies in the history of the exact sciences in honour of David Pingree*, ed. by C. Burnett, J. Hogendijk, K. Plofker, and M. Yano, 386–496. Leiden-Boston: Brill.Google Scholar - Keller, Agathe. 2006.
*Expounding the Mathematical seed*, vol. I and II. Basel-Boston-Berlin: Birkhäuser Verlag.Google Scholar - Keller, Agathe. 2011. George Peacock’s arithmetic in the changing landscape of the history of mathematics in India.
*Indian Journal of History of Science*46 (2): 205–233.Google Scholar - Keller, Agathe. 2012. Dispelling Mathematical doubts. Assessing Mathematical correctness of algorithms in Bhāskara’s commentary on the Mathematical chapter of the Aryabhatiya. In
*The history and historiography of mathematical proof in ancient tradition*, ed. by K. Chemla, 287–508. Cambridge University Press. Available online: http://halshs.archives-ouvertes.fr/. - Kusuba, Takanori. 1993.
*Combinatorics and Magic Squares in India: A study of Nārāyaṇa Pandita’s Gaṇitakaumudī,*Chapters 13–14. Brown University.Google Scholar - Pingree, David. 1970/95.
*Census of the exact sciences in Sanskrit*,*Series A*, 5 vols. Philadelphia: American Philosophical Society.Google Scholar - Pollet, Charlotte. 2012. Comparison of algebraic practices in medieval China and India. Ph.D. dissertation. National Taiwan Normal University/Université Paris 7 (Not published).Google Scholar
- Sarma, Sreeramula Rajeswara. 2002. Rule of Three and its ‘variation’ in India. In
*From China to Paris: 2000 years transmission of Mathematical ideas*, ed. by Y. Dold-Samplonius, J. W. Dauben, M. Flokerts, B. Van Dalen, vol. 46, 133–156. Stuttgart: Franz Steiner Verlag (*Boethius*, Band 46).Google Scholar