Kamalakara was an Indian astronomer and mathematician, came from a learned family of scholars from Golagrama, a village situated in Maharashtra State near Partha-puri (Pathari) on the northern bank of the river Godāvarī. His father was Nrsimha who was born in 1586. Two of Kamalakara's three brothers were also astronomer and mathematicians: Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was youngest. Kamalākara learnt astronomy from his elder brother Divākara, who compiled five works on astronomy. His family later moved to Vārāṇasī.
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.
Mādhava of Sangamagrāma (Mādhavan) was an Indian mathematician and astronomer who is considered as the founder of the Kerala school of astronomy and mathematics in the Late Middle Ages. Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".
Jyeṣṭhadeva was an astronomer-mathematician of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He is best known as the author of Yuktibhāṣā, a commentary in Malayalam of Tantrasamgraha by Nilakantha Somayaji (1444–1544). In Yuktibhāṣā, Jyeṣṭhadeva had given complete proofs and rationale of the statements in Tantrasamgraha. This was unusual for traditional Indian mathematicians of the time. The Yuktibhāṣā is now believed to contain the essential elements of calculus like Taylor and infinity series. Jyeṣṭhadeva also authored Drk-karana, a treatise on astronomical observations.
The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa, written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.
Yuktibhāṣā, also known as Gaṇita-yukti-bhāṣā and Gaṇitanyāyasaṅgraha, is a major treatise on mathematics and astronomy, written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati, and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original.
Tantrasamgraha, or Tantrasangraha, is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics. The treatise was completed in 1501 CE. It consists of 432 verses in Sanskrit divided into eight chapters. Tantrasamgraha had spawned a few commentaries: Tantrasamgraha-vyakhya of anonymous authorship and Yuktibhāṣā authored by Jyeshtadeva in about 1550 CE. Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Sangamagrama Madhava. In his Tantrasangraha, Nilakantha revised Aryabhata's model for the planets Mercury and Venus. According to George G Joseph his equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.
Puthumana Somayaji (c.1660–1740) was a 17th-century astronomer-mathematician from Kerala, India. He was born into the Puthumana or Puthuvana family of Sivapuram. The most famous work attributed to Puthumana Somayaji is Karanapaddhati which is a comprehensive treatise on Astronomy.
Krishna Venkateswara Sarma (1919–2005) was an Indian historian of science, particularly the astronomy and mathematics of the Kerala school. He was responsible for bringing to light several of the achievements of the Kerala school. He was editor of the Vishveshvaranand Indological Research Series, and published the critical edition of several source works in Sanskrit, including the Aryabhatiya of Aryabhata. He was recognised as "the greatest authority on Kerala's astronomical tradition".
Sadratnamala is an astronomical-mathematical treatise in Sanskrit written by Sankara Varman, an astronomer-mathematician of the Kerala school of mathematics, in 1819. Even though the book has been written at a time when western mathematics and astronomy had been introduced in India, it is composed purely in the traditional style followed by the mathematicians of the Kerala school. Sankara Varman has also written a detailed commentary on the book in Malayalam.
Karanapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the attention of European scholars for the first time in a paper published in 1834. The book is divided into ten chapters and is in the form of verses in Sanskrit. The sixth chapter contains series expansions for the value of the mathematical constant π, and expansions for the trigonometric sine, cosine and inverse tangent functions.
In astronomy, Jyotirmimamsa is a treatise on the methodology of astronomical studies authored by Nilakantha Somayaji (1444–1544) in around 1504 CE. Nilakantha somayaji was an important astronomer-mathematician of the Kerala school of astronomy and mathematics and was the author of the much celebrated astronomical work titled Tantrasamgraha. This book stresses the necessity and importance of astronomical observations to obtain correct parameters for computations and to develop more and more accurate theories. It even discounts the role of revealed wisdom and divine intuitions in studying astronomical phenomena. Jyotirmimamsa is sometimes cited as proof to establish that modern methodologies of scientific investigations were known to ancient and medieval Indians. Neelkantha Somayaji insisted that computational results should tally with that of observations and astronomical parameters and constants should be revised periodically. To come to more precise conclusions, Neelkantha Somayaji have discussions with the astronomer and mathematicians of other schools.
Kriyakramakari (Kriyā-kramakarī) is an elaborate commentary in Sanskrit written by Sankara Variar and Narayana, two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics, on Bhaskara II's well-known textbook on mathematics Lilavati. Kriyakramakari, along with Yuktibhasa of Jyeshthadeva, is one of the main sources of information about the work and contributions of Sangamagrama Madhava, the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a 9th-century astronomer from Kerala.
Drigganita, also called the Drik system, is a system of astronomical computations followed by several traditional astronomers, astrologers and almanac makers in India. In this system the computations are performed using certain basic constants derived from observations of astronomical phenomena. The almanacs computed using the methods of Drigganita are referred to as Drigganita Panchngas. In Tamil speaking world, they are also known as Thiru-ganita Panchangas.
Sphuṭacandrāpti is a treatise in Sanskrit composed by the fourteenth-century CE Kerala astronomer-mathematician Sangamagrama Madhava. The treatise enunciates a method for the computation of the position of the moon at intervals of 40 minutes each throughout the day. This is one of only two works of Madhava that have survived to modern times, the other one being Veṇvāroha. However, both Sphuṭacandrāpti and Veṇvāroha have more or less the same contents, that of the latter being apparently a more refined version of that of the former.
Mazhamaṅgalaṃ Śaṅkaran Naṃpūtiri (1494-1570) was an Indian astronomer-astrologer-mathematician belonging to the Kerala school of astronomy and mathematics who wrote a large number of books on astronomy and astrology in the vernacular language Malayalam in an effort to popularize these subjects among the common Malayalam speaking laypersons. He hailed from Peruvanam village in present-day Thrissur district. He spent most of his active life at Chengannur with his teacher Parameśvaran Pōtti of Vāzha-māveli house.
Kaṇakkusāraṃ is a comprehensive treatise in Manipravalam, composed in the 16th-17th century CE, dealing with elementary arithmetic and methods for solving arithmetical problems arising in the everyday life of members of an agricultural community. The book was composed by one Nīlakṇṭha; but other than the name, nothing more definite is known about the author. The date of composition of the book has also not been determined. The significance of the work is that it was composed in a variant of the local vernacular language addressing the needs of ordinary people and it is one of the earliest such work in Malayalam. In comparison, Yuktibhasha, composed in pure Malayalam and published in the middle of the sixteenth century, is an advanced text on mathematics and astronomy addressed to the advanced students and scholars of mathematics. The book is composed in verses and it also contains a commentary both in Manipravalam language. In the book, the author has claimed that the work is based on Sanskrit texts like Bhaskara II's Līlāvatī and older Malayalam texts on mathematics like Kaṇakkadhikāraṃ.
Pañcabodha is the title of several different Sanskrit treatises on astronomy and mathematics composed by members of the Kerala school of astronomy and mathematics. All these works are karaṅa texts, that is, books which explain the various computations in astronomy especially with regard to those related to the preparation of Panchangam-s (calendar). They are essentially manuals of computations. The authorship of only three Pañcabodha-s have been identified.
Koccukṛṣṇan Āśān (Kṛṣṇadāsa) was an astronomer/astrologer from Kerala, India. He was born in the Neṭuṃpayil family in Thiruvalla in Kerala as the son of an erudite astrologer Rāman Āśān. Kṛṣṇadāsa studied astronomy and astrology initially under his father and later from his teacher Śūlapāṇi Vāriyar of Kozhikode. His works on astronomy and astrology were all written in the local language Malayalam and they were all addressed to the novice. One of them is of special interest. It is a commentary on Āryabhaṭīya in Malayalam prose. Apart from the fact that the work is in prose, it is also important because it quotes several authorities including Bhāāskara I, Saṅgamagrāma Mādhava and Vaṭaśreṇi Parameśvara. The other works include:
- Pañcabodha
- Bhāṣājātakapaddhati (a free rendering-cum-commentary of the Jātakapaddhati of Vaṭaśreṇi Parameśvara incorporating several topics not dealt with in the original
- Kaṇkkuśāstraṃ
- Bhāṣāgolayukti
