Insight: An International Journal for Arts and Humanities Peer Reviewed and Refereed Vol: 1; Issue: 3 ISSN: 2582-8002 An Anecdote on Mādhava School of Mathematics Athira K Babu Research Scholar, Department of Sanskrit Sahitya, Sree Sankaracharya University of Sanskrit, Kalady, Abstract The Sanskrit term ‘Gaṇitaśāstra’, meaning literally the “science of calculation” is used for mathematics. The mathematical tradition of ancient India is an ocean of knowledge that is dealing with many topics such as the Vedic, Jain and Buddhist traditions, the mathematical astronomy, The Bhakshali manuscripts, The Kerala School of mathematics and the like. Thus India has made a valuable contribution to the world of mathematics. The origin and development of Indian mathematics are connected with Jyotiśāstra1. This paper tries to deconstructing the concept of mathematical tradition of Kerala with respect to Niḷā valley civilization especially under the background of medieval Kerala and also tries to look into the Mādhava School of mathematics through the life and works of great mathematician Mādhava of Saṅgamagrāma and his pupils who lived in and around the river Niḷā. Keywords: Niḷā, Literature review, Mathematical Tradition of medieval Kerala, Mādhava of Saṅgamagrāma, Great lineage of Mādhava. Introduction Niḷā, the Nile of Kerala is famous for the great ‘Māmāṅkam’ festival. The word ‘Niḷā’point out a culture more than just a river. It has a great role in the formation of the cultural life of south Malabar part of Kerala. It could be seen that the word ‘Peraar’ indicating the same river in ancient scripts and documents. The Niḷā is the life line of many places such as Chittur, Ottappalam, Shornur, Cheruthuruthy, Pattambi, Thrithala, Thiruvegappura, Kudallur, Pallippuram, Kumbidi, 1 The Sanskrit word used for Astronomy is Jyotiśāstra. In ancient times Jyotiśāstra was treated as a part of Jyotiṣa which deals with all aspects of Astronomy. Śāstra is the word used to denote science in India. But the meaning of science in western context is quite different from that of śāstra. The word śāstra derived from the root śās which means some rules about a particular knowledge. Jyotiśāstra is the study of the sun moons, stars, planets, comets, galaxies, dust and other non- earthly bodies and phenomena, or may be defined as the scientific study of celestial bodies. It is one of the oldest sciences. It is one of the oldest sciences. Indian Astronomy continues to play an integral role from prehistoric to modern times. Indian astronomers and astronomy credited with interesting theories and discoveries. Page | 180 Insight: An International Journal for Arts and Humanities Peer Reviewed and Refereed Vol: 1; Issue: 3 ISSN: 2582-8002 Thirunavaya, Alathiyur, Trikkandiyur and Kottakkal which has a great cultural and scientific tradition. Literature Review In 1832, a paper ‘On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sastras, the Tantra Sangraham, Yucti Bhasha, Carana Padhati, and Sadratnamala’ was presented by Charles M. Whish at the Royal Asiatic Society in London. 2 It was a pioneer work in highlighting the mathematical arena of Kerala. The eminent scholars Rama Varma Maru Tampuran, Akhilesvara Ayyar, C. T. Rajagopal, T. A. Sarasvati Amma, R. C. Gupta, K. V. Sarma, S. Madhavan, George Ghevarghese Joseph and the like have done several noteworthy efforts to spread out a light in the mathematical tradition of Niḷā Basin. This noteworthy efforts and works help to reveal the vital role and contributions of mathematicians Saṅgamagrāma Mādhava, Vaṭaśśeri Parameśvara, Nīlakaṇṭha Somayāji, Jyeṣṭhadeva and the like who lived in and around the Niḷā valley. More precisely it can be said that the mathematical tradition of Kerala flourished in and around Niḷā basin. Among these mathematicians Saṅgamagrāma Mādhava has a significant place. Due to the presence of the great mathematician Āryabhaṭa in the writings of Kerala mathematicians, Sarasvati Amma refers to that mathematical tradition as Āryabhaṭa School. Some scholars say it as Kerala School; without consider both history and geography. Kim Plofker, in her work ‘Mathematics in India’ names this as ‘School of Mādhava’. Before this, in ‘Mathematics of India: Concepts, Methods, Connections’, P. P. Divakaran prefers to call this the Niḷā School. Mathematical tradition of Medieval Kerala In the beginning of the seventh chapter of the book ‘Indian Mathematics’, Kim Plofker says: “Probably the most famous school in Indian mathematics, and the one that produced many of its most remarkable discoveries, is the guru-paraṃparā or “chain of teachers” originating with Mādhava in the late fourteenth century and continuing at least into the beginning of the seventeenth. These scholars lived in the region known as Kerala on the southwestern coast of India, in its central part between modern Kochi (or Cochin) and 2 Charles M Whish (1794-1833), who was appointed as a civil servant by the Madras Establishment of the East India Company in Kochi, near to the old capital of Mahodayapuram. In 1832, he presented a paper at the Royal Asiatic Society in London on four manuscripts, which are collected by him in Malabar. The manuscripts were Nīlakaṇṭha's Tantrasaṅgraha , Jyeṣṭhadeva's Yuktibhāṣa (a Malayalam text), Nīlakaṇṭha Somayājī's Karaṇapaddhati, Śaṅkara Varmā's Sadratnamālā . The first two of them are the key text of the Niḷā corpus. Page | 181 Insight: An International Journal for Arts and Humanities Peer Reviewed and Refereed Vol: 1; Issue: 3 ISSN: 2582-8002 Kozhikode (or Calicut). What survives of their work includes writings in Sanskrit and in the local Dravidian vernacular called Malayalam. In astronomy they are generally considered to be followers of the Ārya-pakṣa, but they also wrote on texts in other pakṣas, as well as on astronomical systems unique to Kerala. A narrow strip of land between the Western Ghats Mountains and the Arabian Sea, Kerala in the mid-second millennium maintained a distinct regional culture without being entirely isolated from the neighboring parts of southern India. Moreover, its pepper production and geographical location had made it a major international hub, with trading connections stretching back for many centuries. ” (Plofker 2009, 218) This quote hints the fact about that Kerala, where the śāstras like Astronomy, Mathematics and Ayurveda have been flourished in a traditional lineage of guruśiṣyaparaṃparā and its possible knowledge transmission to other parts of the world; influence of Āryabhaṭa and other Indian scholars and the like. Kerala School has credited with a number of commentaries and original works related to Mathematics and Astronomy; which were based on Āryabhaṭīya, Laghubhāskarīya, Mahābhāskarīya, Bṛhajjātaka and the like. Especially a number of astronomical works are produced from Kerala School by the influence of Āryabhaṭa. Most of the available commentaries on the Āryabhaṭīya have been written by the scholars such as Vaṭaśśeri Parameśvara, Nīlakaṇṭha Somayājī and the like and they have generally made the correction, supplementation and revision of Āryabhaṭa system in order to get accuracy in results. Parahita system is an example for the correction to the Āryabhaṭa system. The system put forward the correction called bhaṭābdasaṃskāra or śakābdasamskāra which presents all the data, directions and sine tables necessary for the computation of the planets. The system is introduced by Haridatta through his astronomical works Grahacāranibandhana and Mahāmārganibandhana at the occasion of the great Māmāṅkam festival. Saṅkara Varma in his Sadratnamālā says about this Māmāṅkam event as follows: आचार्ाार्ाभटप्रणीतगणणतं प्रार्ः स्फुटं तत् खलु गोत्रोत्तु敍गणिताब्दके व्यणभचरन् ब्रह्माददणिद्धाꅍतके । दृ嵍वैषम्र्वशाद् िहास्थलणिते क쥍र्ब्धके णनणितः िंस्कारो णवबुधैर्ातः परणहत配वं तेषु वीनेष्वर्ि् ।। Later this system gave way to the Dṛggaṇita system introduced by Parameśvara who revised the Parahita system. Page | 182 Insight: An International Journal for Arts and Humanities Peer Reviewed and Refereed Vol: 1; Issue: 3 ISSN: 2582-8002 Āryabhaṭa introduced a numeral system based on the Sanskrit alphabets and the schemes of representation of values of this system are shown below: Ña Ka=1 Kha=2 Ga=3 Gha=4 Ṅa =5 Ca=6 Cha=7 Ja=8 Jha=9 =10 Ṭa Ṭha Ḍha Ṇa Ḍa =13 Ta=16 Tha=17 Da=18 Dha=19 Na=20 =11 =12 =14 =15 Pa=21 Pha=22 Ba=23 Bha=24 Ma=25 Ya=30 Ra=40 La=50 Va=60 Śa=70 Ṣa Sa=90 Ha=100 =80 The twenty-five Vargīyavyañjanas (stops both oral and nasal) from Ka to Ma are consisting of the five Vargas which represent numbers from one to twenty-five. The numbers from 26 to 29 and the numbers in between the other multiples of ten such as 31 to 39, 41 to 49, 51 to 59, etc. are represented by sequences of two letters. In this system, the numbers are to be read from backward, i.e. from right to left (aṅgānām vāmato gatiḥ). But it is difficult for pronunciation.3 Even though the Kerala astronomers had been the strong influence of the Āryabhaṭa School of astronomy, they didn’t follow the Āryabhaṭa system of numerals. Instead of that system, Kaṭapayādi system was used by them. This is an easy method of expressing numbers through letters. This is also a numeral system based on Sanskrit alphabets somewhere similar to Kacaṭapayādi system or Āryabhaṭa system. The system was more popular in south India, especially in Kerala. This is also termed as ‘Paralpperu’ or ‘Akṣarasaṃkhyā’. The legendary figure Vararuci is credited with this innovation. The value of letters in this system is as follows: 1 2 3 4 5 6 7 8 9 0 ka kha ga gha ṅa ca cha ja jha ña ṭa ṭha ḍa ḍha ṇa ta tha da dha na pa pha ba bha ma ya ra la va śa ṣa sa ha ḷa 3 Example: cakha =26 (ca=6, kha =2), gaja=83 (ga =3 ja =8) Page | 183 Insight: An International Journal for Arts and Humanities Peer Reviewed and Refereed Vol: 1; Issue: 3 ISSN: 2582-8002 The chronograms using the above systems are widely used to determine the date of the manuscripts and also of the inscriptions.