Indian mathematicians bhaskaracharya biography of martin

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

Bhāskara II
Statue funding Bhaskara II at Patnadevi
Born	c. 1114 Vijjadavida, Maharashtra (probably Patan^[1]^[2] in Khandesh chart Beed^[3]^[4]^[5] in Marathwada)
Died	c. 1185(1185-00-00) (aged 70–71) Ujjain, Madhya Pradesh
Other names	Bhāskarācārya
Occupation(s)	Astronomer, mathematician
Era	Shaka era
Discipline	Mathematician, astronomer, geometer
Main interests	Algebra, arithmetic, trigonometry
Notable works

Bhāskara II^[a] ([bʰɑːskərə]; c.1114–1185), also known trade in Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, physicist and engineer.

From verses regulate his main work, Siddhānta Śiromaṇi, it can be inferred ditch he was born in 1114 in Vijjadavida (Vijjalavida) and progress in the Satpura mountain ranges of Western Ghats, believed shut be the town of Patana in Chalisgaon, located in synchronous Khandesh region of Maharashtra coarse scholars.^[6] In a temple derive Maharashtra, an inscription supposedly authored by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for various generations before him as vigorous as two generations after him.^[7]^[8]Henry Colebrooke who was the precede European to translate (1817) Bhaskaracharya II's mathematical classics refers touch upon the family as Maharashtrian Brahmins residing on the banks get a hold the Godavari.^[9]

Born in a Asian Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of trig cosmic observatory at Ujjain, integrity main mathematical centre of olden India.

Bhāskara and his writings actions represent a significant contribution drop a line to mathematical and astronomical knowledge be given the 12th century. He has been called the greatest mathematician of medieval India. His be work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided review four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which conniving also sometimes considered four irrelevant works.^[14] These four sections conformity with arithmetic, algebra, mathematics be advisable for the planets, and spheres mutatis mutandis.

He also wrote another disquisition named Karaṇā Kautūhala.^[14]

Date, place promote family

Bhāskara gives his date admire birth, and date of essay of his major work, plug a verse in the Āryā metre:^[14]

Rasa-guṇa-pūrṇa-mahī-sama-śakanṛpa-samayeऽbhavan-mamotpattiḥ।
Rasa-guṇa-varṣeṇa mayā siddhānta-śiromaṇī racitaḥ॥
^{[citation needed]}

This reveals that he was whelped in 1036 of the Shaka era (1114 CE), and go off he composed the Siddhānta Shiromani when he was 36 grow older old.^[14]Siddhānta Shiromani was completed extensive 1150 CE.

He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).^[14] His works show dignity influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors.^[14] Bhaskara lived in Patnadevi located nearby Patan (Chalisgaon) in the zone of Sahyadri.

He was born surround a Deśastha Rigvedi Brahmin family^[16] near Vijjadavida (Vijjalavida).

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has terrestrial the information about the mass of Vijjadavida in his disused Marīci Tīkā as follows:^[3]

सह्यकुलपर्वतान्तर्गत भूप्रदेशे महाराष्ट्रदेशान्तर्गतविदर्भपरपर्यायविराटदेशादपि निकटे गोदावर्यां नातिदूरे
पंचक्रोशान्तरे विज्जलविडम्।

This description locates Vijjalavida in Maharashtra, near the Vidarbha region and close to high-mindedness banks of Godavari river.

But scholars differ about the tax location. Many scholars have sited the place near Patan acquit yourself Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the latest day Beed city.^[1] Some large quantity identified Vijjalavida as Bijapur gaffe Bidar in Karnataka.^[18] Identification living example Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara even-handed said to have been righteousness head of an astronomical structure at Ujjain, the leading arithmetical centre of medieval India.

Wildlife records his great-great-great-grandfather holding organized hereditary post as a regard scholar, as did his infect and other descendants. His holy man Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who cultivated him mathematics, which he afterward passed on to his hug Lokasamudra.

Lokasamudra's son helped rear set up a school stop in mid-sentence 1207 for the study help Bhāskara's writings. He died plenty 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The chief section Līlāvatī (also known because pāṭīgaṇita or aṅkagaṇita), named make something stand out his daughter, consists of 277 verses.^[14] It covers calculations, progressions, measurement, permutations, and other topics.^[14]

Bijaganita

The second section Bījagaṇita(Algebra) has 213 verses.^[14] It discusses zero, endlessness, positive and negative numbers, distinguished indeterminate equations including (the put in the picture called) Pell's equation, solving crossing using a kuṭṭaka method.^[14] Call particular, he also solved greatness case that was to dodge Fermat and his European initiation centuries later

Grahaganita

In the position section Grahagaṇita, while treating birth motion of planets, he alleged their instantaneous speeds.^[14] He alighted at the approximation:^[20] It consists of 451 verses

for.

close to , or break through modern notation:^[20]

In his words:^[20]

bimbārdhasya koṭijyā guṇastrijyāhāraḥ phalaṃ dorjyāyorantaram^{[citation needed]}

This mix had also been observed before by Muñjalācārya (or Mañjulācārya) mānasam, in the context of keen table of sines.^[20]

Bhāskara also described that at its highest police a planet's instantaneous speed equitable zero.^[20]

Mathematics

Some of Bhaskara's contributions comprehensively mathematics include the following:

A proof of the Pythagorean postulate by calculating the same harmonize in two different ways bid then cancelling out terms cause problems get a² + b² = c².^[21]
In Lilavati, solutions of multinomial, cubic and quarticindeterminate equations evacuate explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions fall foul of linear and quadratic indeterminate equations (Kuṭṭaka).

The rules he gives are (in effect) the aforementioned as those given by loftiness Renaissance European mathematicians of nobleness 17th century.
A cyclic Chakravala pathway for solving indeterminate equations wear out the form ax² + bx + c = y. Loftiness solution to this equation was traditionally attributed to William Brouncker in 1657, though his practice was more difficult than grandeur chakravala method.
The first general route for finding the solutions hint at the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of influence second order, such as 61x² + 1 = y².

That very equation was posed style a problem in 1657 antisocial the French mathematician Pierre group Fermat, but its solution was unknown in Europe until grandeur time of Euler in blue blood the gentry 18th century.^[22]
Solved quadratic equations sign out more than one unknown, impressive found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along smash into notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus shaft differential coefficient.
Stated Rolle's theorem, copperplate special case of one fall for the most important theorems doubtful analysis, the mean value thesis.

Traces of the general uncovered value theorem are also core in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry forth with a number of beat trigonometric results. (See Trigonometry division below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī bed linen the topics of definitions, precise terms, interest computation, arithmetical boss geometrical progressions, plane geometry, durable geometry, the shadow of influence gnomon, methods to solve undeterminable equations, and combinations.

Līlāvatī assignment divided into 13 chapters challenging covers many branches of sums, arithmetic, algebra, geometry, and spruce up little trigonometry and measurement. Addon specifically the contents include:

Definitions.
Properties of zero (including division, mount rules of operations with zero).
Further extensive numerical work, including loft of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods adequate multiplication, and squaring.
Inverse rule remark three, and rules of 3, 5, 7, 9, and 11.
Problems involving interest and interest computation.
Indeterminate equations (Kuṭṭaka), integer solutions (first and second order).

His assistance to this topic are distinctively important,^{[citation needed]} since the lyrics he gives are (in effect) the same as those prone by the renaissance European mathematicians of the 17th century, still his work was of rank 12th century. Bhaskara's method ship solving was an improvement all but the methods found in glory work of Aryabhata and major mathematicians.

His work is outstanding contemplate its systematisation, improved methods have a word with the new topics that take action introduced.

Furthermore, the Lilavati independent excellent problems and it admiration thought that Bhaskara's intention possibly will have been that a disciple of 'Lilavati' should concern actually with the mechanical application do paperwork the method.^{[citation needed]}

Algebra

His Bījaganita ("Algebra") was a work in dozen chapters.

It was the gain victory text to recognize that unornamented positive number has two sphere roots (a positive and veto square root).^[25] His work Bījaganita is effectively a treatise augment algebra and contains the followers topics:

Positive and negative numbers.
The 'unknown' (includes determining unknown quantities).
Determining unknown quantities.
Surds (includes evaluating surds and their square roots).
Kuṭṭaka (for solving indeterminate equations and Diophantine equations).
Simple equations (indeterminate of alternate, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the inspiration ax² + b = y²).
Solutions of indeterminate equations of blue blood the gentry second, third and fourth degree.
Quadratic equations.
Quadratic equations with more pat one unknown.
Operations with products pattern several unknowns.

Bhaskara derived a alternating, chakravala method for solving indistinct quadratic equations of the revolutionize ax² + bx + aphorism = y.^[25] Bhaskara's method go for finding the solutions of goodness problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance.

Trigonometry

The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, counting the sine table and negotiations between different trigonometric functions.

Sharp-tasting also developed spherical trigonometry, at the head with other interesting trigonometrical consequences. In particular Bhaskara seemed advanced interested in trigonometry for betrayal own sake than his fount who saw it only because a tool for calculation. Amidst the many interesting results subject by Bhaskara, results found suspend his works include computation suggest sines of angles of 18 and 36 degrees, and rectitude now well known formulae make up for and .

Calculus

His work, significance Siddhānta Shiromani, is an boundless treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of undersized calculus and mathematical analysis, ensue with a number of careful in trigonometry, differential calculus direct integral calculus that are be seen in the work are refreshing particular interest.

Evidence suggests Bhaskara was acquainted with some gist of differential calculus.^[25] Bhaskara extremely goes deeper into the 'differential calculus' and suggests the discrimination coefficient vanishes at an peak value of the function, signifying knowledge of the concept recall 'infinitesimals'.

There is evidence of let down early form of Rolle's postulate in his work.

The further formulation of Rolle's theorem states that if , then pray some with .
In this gigantic work he gave one practice that looks like a forerunner to infinitesimal methods. In status that is if then zigzag is a derivative of sin although he did not grow the notion on derivative.
- Bhaskara uses this result to work approve the position angle of rectitude ecliptic, a quantity required funds accurately predicting the time delineate an eclipse.
In computing the instant motion of a planet, dignity time interval between successive positions of the planets was thumb greater than a truti, indicate a 1⁄33750 of a in a tick, and his measure of celerity was expressed in this minuscule unit of time.
He was enlightened that when a variable attains the maximum value, its separation contrast vanishes.
He also showed that considering that a planet is at take the edge off farthest from the earth, secondary at its closest, the leveling of the centre (measure capture how far a planet equitable from the position in which it is predicted to flaw, by assuming it is emphasize move uniformly) vanishes.

He as a result concluded that for some inbetween position the differential of authority equation of the centre report equal to zero.^{[citation needed]} Crop this result, there are carcass of the general mean threshold theorem, one of the wellnigh important theorems in analysis, which today is usually derived non-native Rolle's theorem.

The mean conviction formula for inverse interpolation dominate the sine was later supported by Parameshvara in the Fifteenth century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (1340–1425) and the Kerala Faculty mathematicians (including Parameshvara) from excellence 14th century to the Ordinal century expanded on Bhaskara's check up and further advanced the expansion of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed make wet Brahmagupta in the 7th 100, Bhāskara accurately defined many gigantic quantities, including, for example, greatness length of the sidereal epoch, the time that is fixed for the Earth to revolution the Sun, as approximately 365.2588 days which is the come to as in Suryasiddhanta.^[28] The spanking accepted measurement is 365.25636 cycle, a difference of 3.5 minutes.^[29]

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on accurate astronomy and the second extremity on the sphere.

The xii chapters of the first stuff cover topics such as:

The second part contains thirteen chapters on the sphere. It blankets topics such as:

Engineering

The primary reference to a perpetual movement machine date back to 1150, when Bhāskara II described spruce wheel that he claimed would run forever.

Bhāskara II invented out variety of instruments one wages which is Yaṣṭi-yantra.

This listen in on could vary from a abysmal stick to V-shaped staffs preconcerted specifically for determining angles rigging the help of a graduated scale.

Legends

In his book Lilavati, unwind reasons: "In this quantity very which has zero as treason divisor there is no exchange even when many quantities be born with entered into it or uniformly out [of it], just slightly at the time of calamity and creation when throngs apparent creatures enter into and come into sight out of [him, there research paper no change in] the vasty and unchanging [Vishnu]".

"Behold!"

It has archaic stated, by several authors, ensure Bhaskara II proved the Philosopher theorem by drawing a delineate and providing the single term "Behold!".^[33]^[34] Sometimes Bhaskara's name legal action omitted and this is referred to as the Hindu proof, well known by schoolchildren.^[35]

However, monkey mathematics historian Kim Plofker in turn out, after presenting a worked-out example, Bhaskara II states influence Pythagorean theorem:

Hence, for rank sake of brevity, the rectangular root of the sum hint the squares of the component and upright is the hypotenuse: thus it is demonstrated.^[36]

This crack followed by:

And otherwise, while in the manner tha one has set down those parts of the figure contemporary [merely] seeing [it is sufficient].^[36]

Plofker suggests that this additional spectator may be the ultimate start of the widespread "Behold!" myth.

Legacy

A number of institutes instruct colleges in India are given name after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College show Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications snowball Geo-Informatics in Gandhinagar.

On 20 November 1981 the Indian Tassel Research Organisation (ISRO) launched loftiness Bhaskara II satellite honouring significance mathematician and astronomer.^[37]

Invis Multimedia loose Bhaskaracharya, an Indian documentary reduced on the mathematician in 2015.^[38]^[39]

Notes

^to avoid confusion with prestige 7th century mathematician Bhāskara I,

References

^ ^a^bVictor J.

Katz, ed. (10 August 2021). The Mathematics delineate Egypt, Mesopotamia, China, India, submit Islam: A Sourcebook. Princeton Institute press. p. 447. ISBN .
^Indian Journal nigh on History of Science, Volume 35, National Institute of Sciences round India, 2000, p. 77
^ ^a^bM.

S. Mate; G. T. Kulkarni, eds. (1974). Studies in Indology and Medieval History: Prof. Fleecy. H. Khare Felicitation Volume. Joshi & Lokhande Prakashan. pp. 42–47. OCLC 4136967.
^K. V. Ramesh; S. P. Tewari; M. J. Sharma, eds. (1990). Dr. G. S. Gai Expression Volume. Agam Kala Prakashan. p. 119.

ISBN . OCLC 464078172.
^Proceedings, Indian History Get-together, Volume 40, Indian History Sitting, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders distinctive India - Scientists. Publications Splitting up Ministry of Information & Pressure group. ISBN .
^गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr.

Thakurdesai, Manovikas, First Edition 23, December 2013. p. 34.
^Mathematics fasten India by Kim Plofker, Town University Press, 2009, p. 182
^Algebra with Arithmetic and Mensuration free yourself of the Sanscrit of Brahmegupta most recent Bhascara by Henry Colebrooke, Scholiasts of Bhascara p., xxvii
^ ^a^b^c^d^e^f^g^hⁱ^j^k^l^mS.

Balachandra Rao (13 July 2014), , Vijayavani, p. 17, retrieved 12 November 2019^{[unreliable source?]}
^The Illustrated Daily of India, Volume 95. Airman, Coleman & Company, Limited, parallel the Times of India Test. 1974. p. 30.
^Bhau Daji (1865).

"Brief Notes on the Govern and Authenticity of the Activity of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya". Journal of integrity Royal Asiatic Society of On standby Britain and Ireland. pp. 392–406.
^"1. Fiery minds page 39 by APJ Abdul Kalam, 2. Prof Sudakara Divedi (1855-1910), 3.

Dr Sticky A Salethor (Indian Culture), 4. Govt of Karnataka Publications, 5. Dr Nararajan (Lilavati 1989), 6. Prof Sinivas details(Ganitashatra Chrithra by1955, 7. Aalur Venkarayaru (Karnataka Gathvibaya 1917, 8. Prime Minister Hold sway over Statement at sarawad in 2018, 9. Vasudev Herkal (Syukatha Province articles), 10. Manjunath sulali (Deccan Herald 19/04/2010, 11.

Indian Anthropology 1994-96 A Review page 32, Dr R K Kulkarni (Articles)"
^B.I.S.M. quarterly, Poona, Vol. 63, Cack-handed. 1, 1984, pp 14-22
^ ^a^b^c^d^eScientist (13 July 2014), , Vijayavani, p. 21, retrieved 12 November 2019^{[unreliable source?]}
^Verses 128, 129 in BijaganitaPlofker 2007, pp. 476–477
^ ^a^bMathematical Achievements behoove Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India.

1. Fine to C by Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von dialect trig Murty
^"The Great Bharatiya Mathematician Bhaskaracharya ll". The Times of India. Retrieved 24 May 2023.
^IERS EOP PC Useful constants.

An SI day or mean solar fair equals 86400 SIseconds. From goodness mean longitude referred to decency mean ecliptic and the equinox J2000 given in Simon, Tabulate. L., et al., "Numerical Expressions for Precession Formulae and Unkind Elements for the Moon focus on the Planets" Astronomy and Astrophysics 282 (1994), 663–683.

Bibcode:1994A&A...282..663S
^Eves 1990, p. 228
^Burton 2011, p. 106
^Mazur 2005, pp. 19–20
^ ^a^bPlofker 2007, p. 477
^Bhaskara NASA 16 September 2017
^"Anand Narayanan".

IIST. Retrieved 21 February 2021.
^"Great Indian Mathematician - Bhaskaracharya". indiavideodotorg. 22 Sept 2015. Archived from the uptotheminute on 12 December 2021.

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