query: languages_spoken_or_written nilakantha somayaji
supports: South India is the area encompassing the Indian states of Tamil Nadu, Andhra Pradesh, Karnataka, Kerala, and Telangana as well as the union territories of Andaman and Nicobar, Lakshadweep and Puducherry, occupying 19.31% of India's area. Covering the southern part of the peninsular Deccan Plateau, South India is bounded by the Bay of Bengal in the east, the Arabian Sea in the west and the Indian Ocean in the south. The geography of the region is diverse with two mountain ranges - the Western and Eastern Ghats, bordering the plateau heartland. Godavari, Krishna, Kaveri, Tungabhadra and Vaigai rivers are important non-perennial sources of water. Bangalore, Chennai, Hyderabad, Visakhapatnam, Vijayawada, Coimbatore and Kochi are the largest urban areas. The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in 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 the original discoveries of the school seems to have ended with Narayana Bhattathiri (15591632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important resultsseries expansion for trigonometric functionswere 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". Lakshadweep (, Lakshadb), formerly known as the Laccadive, Minicoy, and Aminidivi Islands, is a group of islands in the Laccadive Sea, off the south western coast of India. The archipelago is a Union Territory and is governed by the Union Government of India. They were also known as Laccadive Islands, although geographically this is only the name of the central subgroup of the group. "Lakshadweep" comes from "Lakshadweepa", which means "one hundred thousand islands" in Sanskrit. The islands form the smallest Union Territory of India: their total surface area is just . The lagoon area covers about , the territorial waters area and the exclusive economic zone area . The region forms a single Indian district with ten sub divisions. Kavaratti serves as the capital of the Union Territory and the region comes under the jurisdiction of Kerala High Court. The islands are the northernmost of the Lakshadweep-Maldives-Chagos group of islands, which are the tops of a vast undersea mountain range, the Chagos-Laccadive Ridge. Sanskrit (English pronunciation:; written in Devanagari script ; : or ', originally ', "refined speech") is the primary sacred language of Hinduism and "Mahyna" Buddhism, a philosophical language in Hinduism, Jainism, Buddhism and Sikhism. It was also a literary language that was in use as a "lingua franca" in ancient and medieval South Asia. It is a standardised dialect of Old Indo-Aryan, originating as Vedic Sanskrit and tracing its linguistic ancestry back to Proto-Indo-Iranian and Proto-Indo-European. Today it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand. As one of the oldest Indo-European languages for which substantial written documentation exists, Sanskrit holds a prominent position in Indo-European studies. Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Yuktibh ("Rationale in the Malayalam/Sanskrit language") also known as Gaitanyyasagraha ("Compendium of astronomical rationale"), is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530. The treatise is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameshvara, Jyeshtadeva, Achyuta Pisharati and other astronomer-mathematicians of the Kerala school. "Yuktibhasa" is mainly based on Nilakantha's "Tantra Samgraha". It is considered an early text on the foundations of calculus and predates those of European mathematicians such as James Gregory and Newton by many centuries. However, the treatise was largely unnoticed beyond Kerala, as the book was written in the local language of Malayalam. However, both Oxford University and Royal Society of Great Britain have accepted that Calculus and many such pioneering mathematical theorems originated in India. Sadly, mainstream Western and even Western educated Indians are largely unaware of these and references in textbooks are absent. Karnataka (IPA:) is a state in south western region of India. It was formed on 1 November 1956, with the passage of the States Reorganisation Act. Originally known as the State of Mysore, it was renamed "Karnataka" in 1973. The capital and largest city is Bangalore (Bengaluru). Karnataka is bordered by the Arabian Sea and the Laccadive Sea to the west, Goa to the northwest, Maharashtra to the north, Telangana to the northeast, Andhra Pradesh to the east, Tamil Nadu to the southeast, and Kerala to the southwest. The state covers an area of , or 5.83 percent of the total geographical area of India. It is the seventh largest Indian state by area. With 61,130,704 inhabitants at the 2011 census, Karnataka is the eighth largest state by population, comprising 30 districts. Kannada, one of the classical languages of India, is the most widely spoken and official language of the state. Johannes Kepler (December 27, 1571  November 15, 1630) was a German mathematician, astronomer, and astrologer. A key figure in the 17th century scientific revolution, he is best known for his laws of planetary motion, based on his works "Astronomia nova", "Harmonices Mundi", and "Epitome of Copernican Astronomy". These works also provided one of the foundations for Isaac Newton's theory of universal gravitation. Tantrasamgraha, or Tantrasangraha, (literally, "A Compilation of the System") 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 "Yuktibh" 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. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century. Malayalam is a language spoken in India, predominantly in the state of Kerala. It is one of the 22 scheduled languages of India and was designated as a Classical Language in India in 2013. It was developed to the current form mainly by the influence of the poet Thunchaththu Ezhuthachan in the 16th century. Malayalam has official language status in the state of Kerala and in the union territories of Lakshadweep and Puducherry. It belongs to the Dravidian family of languages and is spoken by some 38 million people. Malayalam is also spoken in the neighboring states of Tamil Nadu and Karnataka; with significant speakers in the Nilgiris, Kanyakumari and Coimbatore districts of Tamil Nadu, and Dakshina Kannada of Karnataka. Kerala, historically known as Keralam, is an Indian state in South India on the Malabar coast. It was formed on 1 November 1956 following the States Reorganisation Act by combining Malayalam-speaking regions. Spread over , it is bordered by Karnataka to the north and northeast, Tamil Nadu to the east and south, and the Lakshadweep Sea to the west. With 33,387,677 inhabitants as per the 2011 Census, Kerala is the thirteenth-largest state by population and is divided into 14 districts with the capital being Thiruvananthapuram. Malayalam is the most widely spoken language and is also the official language of the state. Arabic (' or ' ) is a Central Semitic language that was first spoken in Iron Age northwestern Arabia and is now the "lingua franca" of the Arab world. Arabic also is a liturgical language of 1.7 billion Muslims. It is one of six official languages of the United Nations. It is named after the Arabs, a term initially used to describe peoples living from Mesopotamia in the east to the Anti-Lebanon mountains in the west, and from northwestern Arabia to the Sinai in the south. Tamil Nadu (English pronunciation:  ; ""; : ; literally 'The Land of Tamils' or 'Tamil Country') is one of the 29 states of India. Its capital and largest city is Chennai (formerly known as Madras). Tamil Nadu lies in the southernmost part of the Indian Peninsula and is bordered by the union territory of Puducherry and the South Indian states of Kerala, Karnataka, and Andhra Pradesh. It is bounded by the Eastern Ghats on the north, by the Nilgiri, the Anamalai Hills, and Kerala on the west, by the Bay of Bengal in the east, by the Gulf of Mannar and the Palk Strait on the southeast, and by the Indian Ocean on the south. The state shares a maritime border with the nation of Sri Lanka. Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. A sphere (from Greek   "sphaira", "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).  Achyutha Pisharodi (c. 1550 at Trikkandiyur, Tirur, Kerala, India  7 July 1621 in Kerala) was a Sanskrit grammarian, astrologer, astronomer and mathematician who studied under Jyehadeva and was a member of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He is remembered mainly for his part in the composition of his student Melpathur Narayana Bhattathiri's devotional poem, "Narayaneeyam". Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy. Aryabhatiya (IAST: ') or Aryabhatiyam ('), a Sanskrit astronomical treatise, is the "magnum opus" and only known surviving work of the 5th century Indian mathematician, Aryabhata. Kelallur Nilakantha Somayaji ( 1444 -- 1544 ) ( also referred to as Kelallur Comatiri ) was a major mathematician and astronomer of the Kerala school of astronomy and mathematics . One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501 . He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya . In this Bhasya , Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry . Grahapareeksakrama is a manual on making observations in astronomy based on instruments of the time .
candidates: arabic english german greek malayalam sanskrit tamil
answer: sanskrit