73-701: Keļallur Nīlakaṇṭha Somayāji (14 June 1444 – 1544), also referred to as Keļallur 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 . Grahapariksakrama
146-462: A Somayajna ritual and assumed the title of a Somayaji in later life. In colloquial Malayalam usage the word Somayaji has been corrupted to Comatiri. Somayāji’s "Āryabhaṭīyabhāṣya" is the most extensive commentary on Āryabhaṭīya. He takes all pains to expose the rationale and the objective behind the statements and observations made by Āryabhaṭa. Nilakantha's writings substantiate his knowledge of several branches of Indian philosophy and culture. It
219-587: A sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE). The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide
292-656: A commentary on Aryabhata's Aryabhatiya , Nilakantha developed a computational system for a partially heliocentric planetary model in which Mercury, Venus, Mars , Jupiter and Saturn orbit the Sun , which in turn orbits the Earth , similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Most astronomers of the Kerala school who followed him accepted this planetary model. A Conference to celebrate
365-530: A computational system for a partially heliocentric planetary model in which Mercury, Venus, Mars , Jupiter and Saturn orbit the Sun , which in turn orbits the Earth , similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Most astronomers of the Kerala school who followed him accepted this planetary model. Somayaji time and again advocates the necessity of periodical modification of computation system based on observations and experimentations. One of his works, Jyotirmimamsa ,
438-473: A decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept." Tantrasangraha Tantrasamgraha , or Tantrasangraha , (literally, A Compilation of the System ) is an important astronomical treatise written by Nilakantha Somayaji , an astronomer / mathematician belonging to
511-469: A highly compressed mnemonic form, the sūtra (literally, "thread"): The knowers of the sūtra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable. Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language," using technical names instead of longer descriptive names, abridging lists by only mentioning
584-561: A hundred to a trillion: Hail to śata ("hundred," 10 ), hail to sahasra ("thousand," 10 ), hail to ayuta ("ten thousand," 10 ), hail to niyuta ("hundred thousand," 10 ), hail to prayuta ("million," 10 ), hail to arbuda ("ten million," 10 ), hail to nyarbuda ("hundred million," 10 ), hail to samudra ("billion," 10 , literally "ocean"), hail to madhya ("ten billion," 10 , literally "middle"), hail to anta ("hundred billion," 10 , lit., "end"), hail to parārdha ("one trillion," 10 lit., "beyond parts"), hail to
657-673: A mathematical text called Tiloyapannati ; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jain philosophy and metaphysics , composed a mathematical work called the Tattvārtha Sūtra . Mathematicians of ancient and early medieval India were almost all Sanskrit pandits ( paṇḍita "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar ( vyākaraṇa ), exegesis ( mīmāṃsā ) and logic ( nyāya )." Memorisation of "what
730-457: A part of a "methodological reflexion" on the sacred Vedas , which took the form of works called Vedāṇgas , or, "Ancillaries of the Veda" (7th–4th century BCE). The need to conserve the sound of sacred text by use of śikṣā ( phonetics ) and chhandas ( metrics ); to conserve its meaning by use of vyākaraṇa ( grammar ) and nirukta ( etymology ); and to correctly perform
803-590: A place-value system. The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence
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#1732793072421876-545: A precursor of the Backus–Naur form (used in the description programming languages ). Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala ( piṅgalá ) ( fl. 300–200 BCE), a music theorist who authored the Chhandas Shastra ( chandaḥ-śāstra , also Chhandas Sutra chhandaḥ-sūtra ), a Sanskrit treatise on prosody . Pingala's work also contains
949-518: A statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." Baudhayana gives an expression for
1022-464: Is a manual on making observations in astronomy based on instruments of the time. Nilakantha was born into a Brahmin family which came from South Malabar in Kerala . Nilakantha Somayaji was one of the very few authors of the scholarly traditions of India who had cared to record details about his own life and times. In one of his works titled Siddhanta -star and also in his own commentary on Siddhanta-darpana , Nilakantha Somayaji stated that he
1095-403: Is achieved in the sūtra , by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that
1168-451: Is described in the following words: II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three. II.65. In another layer one places the [bricks] North-pointing. According to Filliozat, the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, rajju , f.), two pegs (Sanskrit, śanku , m.), and clay to make the bricks (Sanskrit, iṣṭakā , f.). Concision
1241-487: Is equal to the zenith distance of the Sun at noon on the equinoctial day . The effect of solar parallax on zenith distance was known to Indian astronomers right from Aryabhata . But it was Nilakantha Somayaji who first discussed the effect of solar parallax on the observer's latitude. Tantrasamgraha gives the magnitude of this correction and also a correction due to the finite size of the Sun. In his Aryabhatiyabhasya ,
1314-556: Is heard" ( śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia." Prodigious energy
1387-437: Is said that he could refer to a Mimamsa authority to establish his view-point in a debate and with equal felicity apply a grammatical dictum to the same purpose. In his writings he refers to a Mimamsa authority, quotes extensively from Pingala's chandas-sutra, scriptures, Dharmasastras, Bhagavata and Vishnupurana also. Sundararaja, a contemporary Tamil astronomer, refers to Nilakantha as sad-darshani-parangata, one who had mastered
1460-490: Is the Sun's azimuth and the angle at the north pole which is the Sun's hour angle . The problem is to compute two of these elements when the other three elements are specified. There are precisely ten different possibilities and Tantrasamgraha contains discussions of all these possibilities with complete solutions one by one in one place . "The spherical triangle is handled as systematically here as in any modern textbook." The terrestrial latitude of an observer's position
1533-433: Is the birch bark Bakhshali Manuscript , discovered in 1881 in the village of Bakhshali , near Peshawar (modern day Pakistan ) and is likely from the 7th century CE. A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent ) by mathematicians of the Kerala school in the 15th century CE. Their work, completed two centuries before
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#17327930724211606-543: Is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear. The earliest extant script used in India
1679-667: Is written exclusively for this purpose. His critical and analytical approach is reflected in all works. The following is a brief description of the works by Nilakantha Somayaji dealing with astronomy and mathematics. ==Further reading== Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata , Brahmagupta , Bhaskara II , Varāhamihira , and Madhava . The decimal number system in use today
1752-522: The uṣas (dawn) , hail to the vyuṣṭi (twilight), hail to udeṣyat (the one which is going to rise), hail to udyat (the one which is rising), hail udita (to the one which has just risen), hail to svarga (the heaven), hail to martya (the world), hail to all. The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4): With three-fourths Puruṣa went up: one-fourth of him again
1825-711: The Sthānāṅga Sūtra (c. 300 BCE – 200 CE); the Anuyogadwara Sutra (c. 200 BCE – 100 CE), which includes the earliest known description of factorials in Indian mathematics; and the Ṣaṭkhaṅḍāgama (c. 2nd century CE). Important Jain mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti ; Yativrisham Acharya (c. 176 BCE), who authored
1898-422: 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. It
1971-480: 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
2044-694: The Manava Sulba Sutra composed by Manava (fl. 750–650 BCE) and the Apastamba Sulba Sutra , composed by Apastamba (c. 600 BCE), contained results similar to the Baudhayana Sulba Sutra . An important landmark of the Vedic period was the work of Sanskrit grammarian , Pāṇini (c. 520–460 BCE). His grammar includes early use of Boolean logic , of the null operator, and of context free grammars , and includes
2117-506: The Old Babylonians ." The diagonal rope ( akṣṇayā-rajju ) of an oblong (rectangle) produces both which the flank ( pārśvamāni ) and the horizontal ( tiryaṇmānī ) <ropes> produce separately." Since the statement is a sūtra , it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by
2190-524: The Vedic Period provide evidence for the use of large numbers . By the time of the Yajurvedasaṃhitā- (1200–900 BCE), numbers as high as 10 were being included in the texts. For example, the mantra (sacred recitation) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha , and uttered just before-, during-, and just after sunrise, invokes powers of ten from
2263-599: The square root of 2 correct to five decimal places. Although Jainism as a religion and philosophy predates its most famous exponent, the great Mahaviraswami (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "classical period." A significant historical contribution of Jain mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with
Nilakantha Somayaji - Misplaced Pages Continue
2336-473: The square root of two : The expression is accurate up to five decimal places, the true value being 1.41421356... This expression is similar in structure to the expression found on a Mesopotamian tablet from the Old Babylonian period (1900–1600 BCE ): which expresses √ 2 in the sexagesimal system, and which is also accurate up to 5 decimal places. According to mathematician S. G. Dani,
2409-404: The zenith , the celestial north pole and the Sun is called the astronomical triangle . Its sides and two of its angles are important astronomical quantities. The sides are 90° – φ where φ is the observer's terrestrial latitude , 90° – δ where δ is the Sun's declination and 90° – a where a is the Sun's altitude above the horizon . The important angles are the angle at the zenith which
2482-673: The 500th Anniversary of Tantrasangraha was organised by the Department of Theoretical Physics, University of Madras, in collaboration with the Inter-University Centre of the Indian Institute of Advanced Study, Shimla, during 11–13 March 2000, at Chennai. The Conference turned out to be an important occasion for highlighting and reviewing the recent work on the achievements in Mathematics and Astronomy of
2555-579: The Babylonian cuneiform tablet Plimpton 322 written c. 1850 BCE "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple, indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account
2628-567: The English ounce or Greek uncia). They mass-produced weights in regular geometrical shapes, which included hexahedra , barrels , cones , and cylinders , thereby demonstrating knowledge of basic geometry . The inhabitants of Indus civilisation also tried to standardise measurement of length to a high degree of accuracy. They designed a ruler—the Mohenjo-daro ruler —whose unit of length (approximately 1.32 inches or 3.4 centimetres)
2701-602: The Kali-days of the commencement (1,680,548) and of completion (1,680,553) of Somayaji's magnum opus Tantrasamgraha . Both these days occur in 1500 CE. In Aryabhatiya -bhashya, Nilakantha Somayaji has stated that he was the son of Jatavedas and he had a brother named Sankara . Somayaji has further stated that he was a Bhatta belonging to the Gargya gotra and was a follower of Asvalayana-sutra of Rigveda . References in his own Laghuramayana indicate that Nilakantha Somayaji
2774-425: The bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards ( i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta ( fl. 7th century CE), to characterise astronomical computations as "dust work" (Sanskrit: dhulikarman ). It
2847-593: The basic ideas of Fibonacci numbers (called maatraameru ). Although the Chandah sutra hasn't survived in its entirety, a 10th-century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as Meru -prastāra (literally "the staircase to Mount Meru"), has this to say: Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in
2920-471: The combinations with one syllable, the third the combinations with two syllables, ... The text also indicates that Pingala was aware of the combinatorial identity: Kātyāyana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra , which presented much geometry , including the general Pythagorean theorem and a computation of
2993-646: The contents of Tantrasamgraha is presented below. A descriptive account of the contents is available in Bharatheeya Vijnana/Sastra Dhara. Full details of the contents are available in an edition of Tantrasamgraha published in the Indian Journal of History of Science . "A remarkable synthesis of Indian spherical astronomical knowledge occurs in a passage in Tantrasamgraha." In astronomy, the spherical triangle formed by
Nilakantha Somayaji - Misplaced Pages Continue
3066-494: The contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say: As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras . The occurrence of
3139-632: The enumeration of very large numbers and infinities led them to classify numbers into three classes: enumerable, innumerable and infinite . Not content with a simple notion of infinity, their texts define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations ( bījagaṇita samīkaraṇa ). Jain mathematicians were apparently also
3212-599: The fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally. The earliest mathematical prose commentary was that on the work, Āryabhaṭīya (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to Hayashi, "this does not necessarily mean that their authors did not prove them. It
3285-449: The first and last entries, and using markers and variables. The sūtras create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called Guru-shishya parampara , 'uninterrupted succession from teacher ( guru ) to the student ( śisya ),' and it was not open to the general public" and perhaps even kept secret. The brevity achieved in
3358-407: The first layer of bricks are oriented in the east–west direction, but that too is implied by the explicit mention of "North-pointing" in the second stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory. With
3431-403: The first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives
3504-485: The first to use the word shunya (literally void in Sanskrit ) to refer to zero. This word is the ultimate etymological origin of the English word "zero" , as it was calqued into Arabic as ṣifr and then subsequently borrowed into Medieval Latin as zephirum , finally arriving at English after passing through one or more Romance languages (c.f. French zéro , Italian zero ). In addition to Surya Prajnapti , important Jain works on mathematics included
3577-419: The form: That these methods have been effective is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (c. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (c. 500 BCE). Mathematical activity in ancient India began as
3650-430: The foundations of many areas of mathematics. Ancient and medieval Indian mathematical works, all composed in Sanskrit , usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained
3723-458: The increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation. India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least
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#17327930724213796-558: The invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration , nor is there any direct evidence of their results being transmitted outside Kerala . Excavations at Harappa , Mohenjo-daro and other sites of the Indus Valley civilisation have uncovered evidence of
3869-416: The problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent
3942-481: The rites at the correct time by the use of kalpa ( ritual ) and jyotiṣa ( astrology ), gave rise to the six disciplines of the Vedāṇgas . Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedāṇgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in
4015-469: The same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. According to Hayashi, the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to
4088-414: The science of astronomy and instructed him in the basic principles of mathematical computations. The great Malayalam poet Thunchaththu Ramanujan Ezhuthachan is said to have been a student of Nilakantha Somayaji. The epithet Somayaji is a title assigned to or assumed by a Namputiri who has performed the vedic ritual of Somayajna . So it could be surmised that Nilakantha Somayaji had also performed
4161-403: The six systems of Indian philosophy. 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. In his Aryabhatiyabhasya , a commentary on Aryabhata's Aryabhatiya , Nilakantha developed
4234-407: The square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely. The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana Śulba Sūtra , this procedure
4307-524: The teacher to the student. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square." Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra , the best-known Sulba Sutra , which contains examples of simple Pythagorean triples, such as: (3, 4, 5) , (5, 12, 13) , (8, 15, 17) , (7, 24, 25) , and (12, 35, 37) , as well as
4380-492: The text were first recited in their original order, then repeated in the reverse order, and finally repeated in the original order. The recitation thus proceeded as: In another form of recitation, dhvaja-pāṭha (literally "flag recitation") a sequence of N words were recited (and memorised) by pairing the first two and last two words and then proceeding as: The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to Filliozat, took
4453-442: The triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily. In all, three Sulba Sutras were composed. The remaining two,
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#17327930724214526-499: The use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to
4599-407: Was C.M. Whish , a civil servant of East India Company , who brought to the attention of the western scholarship the existence of Tantrasamgraha through a paper published in 1835. The other books mentioned by C.M. Whish in his paper were Yuktibhāṣā of Jyeshtadeva , Karanapaddhati of Puthumana Somayaji and Sadratnamala of Sankara Varman . Nilakantha Somayaji , the author of Tantrasamgraha,
4672-430: Was a Nambudiri belonging to the Gargya gotra and a resident of Trikkantiyur, near Tirur in central Kerala . The name of his Illam was Kelallur. He studied under Damodara , son of Paramesvara . The first and the last verses in Tantrasamgraha contain chronograms specifying the dates, in the form Kali days, of the commencement and of the completion of book. These work out to dates in 1500–01. A brief account of
4745-458: Was a member of the Kelallur family (Sanskritised as Kerala-sad-grama) residing at Kundagrama, now known as Trikkandiyur in modern Tirur , Kerala . His wife was named Arya and he had two sons Rama and Dakshinamurti. Nilakantha Somayaji studied vedanta and some aspects of astronomy under one Ravi. However, it was Damodara , son of Kerala-drgganita author Paramesvara , who initiated him into
4818-443: Was born on Kali-day 1,660,181 which works out to 14 June 1444 CE. A contemporary reference to Nilakantha Somayaji in a Malayalam work on astrology implies that Somayaji lived to a ripe old age even to become a centenarian. Sankara Variar , a pupil of Nilakantha Somayaji, in his commentary on Tantrasamgraha titled Tantrasamgraha-vyakhya , points out that the first and last verses of Tantrasamgraha contain chronograms specifying
4891-401: Was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. Hollow cylindrical objects made of shell and found at Lothal (2200 BCE) and Dholavira are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation. The religious texts of
4964-457: Was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorisation of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in
5037-531: 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
5110-668: Was here. The Satapatha Brahmana ( c. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras. The Śulba Sūtras (literally, "Aphorisms of the Chords" in Vedic Sanskrit ) (c. 700–400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy
5183-466: Was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations ( upapatti ). Bhaskara I's commentary on the Āryabhaṭīya , had the following structure: Typically, for any mathematical topic, students in ancient India first memorised the sūtras , which, as explained earlier, were "deliberately inadequate" in explanatory details (in order to pithily convey
5256-563: Was substantial. There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of
5329-618: Was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script , appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on
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