# The Kriyākramakarī’s Integrative Approach to Mathematical Knowledge

### Abstract

The purpose of this paper is to review the general organization of knowledge in the *Kriyākramakarī*, a sixteenth-century treatise of Kerala mathematics. Specifically, I will argue that the authors' interest in justification or proof is integrative, rather than hierarchical or cumulative. In other words, the purpose of proofs in the *Kriyākramakarī* is to connect various different aspects of mathematics, rather than just establish results by means of previously known results.

### References

Divakaran, P. P. 2016. “What Is Indian about Indian Mathematics?” Indian Journal of History of Science 51 (1): 56–82.

Gupta, R.C. 1987. “South Indian Achievements in Medieval Mathematics.” Ganita Bharati 9: 15–40.

Hayashi, Takao. 2000. “Govindasvamin’s Arithmetic Rules Cited in the Kriyakramakari of Sankara and Narayana.” Indian Journal of History of Science 35 (3): 189–231.

Hayashi, Takao, and Takanori Kusuba. 1998. “Twenty-One Algebraic Normal Forms of Citrabhānu.” Historia Mathematica 25 (1): 1–21.

Heroor, Venugopal D. 2014. Brahmagupta’s Ganita. Ernakulam: Chinmaya International Foundation Shodha Sansthan.

Joseph, George Gheverghese. 2009. A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact. New Delhi; Thousand Oaks: Sage Publications.

Katz, Victor J., ed. 2007. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton: Princeton University Press.

Keller, Agathe. 2006. Expounding the Mathematical Seed. Vol. 1: A Translation of Bhāskara I on the Mathematical Chapter of the Āryabhatīya. Basel: Birkhäuser.

———. 2012. “Dispelling Mathematical Doubts. Assessing the Mathematical Correctness of Algorithms in Bhâskara’s Commentary on the Mathematical Chapter of the Âryabhatîya.” In The History of Mathematical Proof in Ancient Traditions, edited by Karine Chemla, 487–508. Cambridge: Cambridge University Press.

Mallayya, V.M. 2002. “Geometrical Approach to Arithmetic Progressions from Nilakantha’s Arybhatiyabhasya and Sankara’s Kriyakramakari.” In Proceedings of the International Seminar and Colloquium on 1500 Years of Aryabhateeyam, 143–47. Kochi: Kerala Sastra Sahitya Parishad.

———. 2011. “Śaṅkara’s Geometrical Approach to Citrabhānu’s Ekaviṃśati Praśnottara.” In Proceedings of the National Workshop on Ancient Indian Mathematics with Special Reference to Vedic Mathematics and Astronomy, edited by P.V. Arunachalam, C. Umashankar, and V. Ramesh babu, 55–71. Tirupati: Rashtriya Sanskrit Vidyapeetha.

Mukunda Marar, K., and C.T. Rajagopal. 1944. “On the Quadrature of the Circle.” Journal of the Bombay Branch of the Royal Asiatic Society 20: 65–82.

Netz, Reviel. 1999. The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge: Cambridge University Press.

Padmanabha Rao, A.B. 2015. Bhaskaracarya’s Lilavati. Chinmaya International Foundation Shodha Sansthan. 2 vols. Ernakulam.

Plofker, Kim. 2009. Mathematics in India. Princeton University Press.

Puttaswamy, T. K. 2012. Mathematical Achievements of Pre-Modern Indian Mathematicians. London: Elsevier.

Rajagopal, C.T., and A. Venkataraman. 1949. “The Sine and Cosine Power Series in Hindu Mathematics.” Journal of the Rotal Astronomical Association of Bengal 15: 1–13.

Ramasubramanian, K, and M. S. Sriram. 2010. Tantrasangraha of Nilakantha Somayaji. Guildford, Surrey: Springer London.

Sarasvati Amma, T. A. 1979. Geometry in Ancient and Medieval India. Delhi: Motilal Banarsidass.

Sarma, K. V. 1972. A History of the Kerala School of Hindu Astronomy (in Perspective). Hoshiarpur: Vishveshvaranand Institute.

———. 1975. Līlāvatī of Bhāskarācārya with Kriyākramakarī of Śaṅkara and Nārāyaṇa: being an elaborate exposition of the rationale of Hindu mathematics. Hoshiarpur: Vishveshvaranand Vedic Research Institute.

Sarma, K. V., K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram. 2009. Ganita-Yukti-Bhasa of Jyesthadeva. Springer.

Sastri, Sambasiva, ed. 1930. The Aryabhatiya of Aryabhatacarya, with the Bhasya of Nilakanthasomatsutvan. Trivandrum: Government Press.

Shukla, K.S. 1954. “Acarya Jayadeva, the Mathematician.” Bharata Ganita Parisad 5: 1–20.

Srinivas, M. D. 2008. “Proofs in Indian Mathematics.” In Contributions to the History of Indian Mathematics, edited by G. G. Emch, R. Sridharan, and M. D. Srinivas, 209–48. New Delhi: Hindustan Book Agency.

———. 2015. “On the Nature of Mathematics and Scientific Knowledge in Indian Tradition.” In Science and Narratives of Nature: East and West, edited by Jobin M. Kanjirakkat, Gordon McOuat, and Sundar Sarukkai, 1 edition, 220–38. New Delhi: Routledge.

Vrinda, P.M. 2014. “Geometrical Representation of Algebraic Results - a Notable Feature of Kerala School of Mathematicians.” In Kerala School of Mathematics: Trajectories and Impact, edited by N.K. Sundareswaran, 186–202. Calicult: Ganga books.

Whish, Charles M. 1834. “On the Hindú Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four S’ástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála.” Transactions of the Royal Asiatic Society of Great Britain and Ireland 3 (3): 509–23.

*History of Science in South Asia*,

*6*, 84-126. https://doi.org/10.18732/hssa.v6i0.23

Copyright (c) 2018 Roy Wagner

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Authors who publish with this journal agree to the following terms:Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-ShareAlike license that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).