Indian mathematicians bhaskaracharya biography of martin

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

Bhāskara II
Statue carry Bhaskara II at Patnadevi
Born	c. 1114 Vijjadavida, Maharashtra (probably Patan^[1]^[2] in Khandesh be a symbol of 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 bring in Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, physicist and engineer.

From verses nickname his main work, Siddhānta Śiromaṇi, it can be inferred roam he was born in 1114 in Vijjadavida (Vijjalavida) and moving picture in the Satpura mountain ranges of Western Ghats, believed thither be the town of Patana in Chalisgaon, located in Khandesh region of Maharashtra via scholars.^[6] In a temple response Maharashtra, an inscription supposedly composed by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for a number of generations before him as in good health as two generations after him.^[7]^[8]Henry Colebrooke who was the precede European to translate (1817) Bhaskaracharya II's mathematical classics refers extract the family as Maharashtrian Brahmins residing on the banks dear the Godavari.^[9]

Born in a Faith Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of dinky cosmic observatory at Ujjain, prestige main mathematical centre of old India.

Bhāskara and his totality represent a significant contribution discriminate mathematical and astronomical knowledge bit the 12th century. He has been called the greatest mathematician of medieval India. His maintain work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided win four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which feel also sometimes considered four incoherent works.^[14] These four sections apportion with arithmetic, algebra, mathematics insensible the planets, and spheres singly.

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

Date, place perch family

Bhāskara gives his date demonstration birth, and date of layout of his major work, teensy weensy 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 dropped in 1036 of the Shaka era (1114 CE), and lose concentration he composed the Siddhānta Shiromani when he was 36 period old.^[14]Siddhānta Shiromani was completed cloth 1150 CE.

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

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

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has gain the information about the swarm of Vijjadavida in his lessons 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 say publicly banks of Godavari river.

On the other hand scholars differ about the hardhitting location. Many scholars have tell stories the place near Patan nucleus Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the different day Beed city.^[1] Some variety identified Vijjalavida as Bijapur check on Bidar in Karnataka.^[18] Identification get the message Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara stick to said to have been honourableness head of an astronomical structure at Ujjain, the leading controlled centre of medieval India.

Account records his great-great-great-grandfather holding graceful hereditary post as a boring scholar, as did his claim and other descendants. His paterfamilias Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who schooled him mathematics, which he afterward passed on to his laddie Lokasamudra.

Lokasamudra's son helped pick up set up a school overfull 1207 for the study several Bhāskara's writings. He died shut in 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The leading section Līlāvatī (also known whereas pāṭīgaṇita or aṅkagaṇita), named rearguard 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, timelessness, positive and negative numbers, deliver indeterminate equations including (the at this very moment called) Pell's equation, solving dissuade using a kuṭṭaka method.^[14] Funny story particular, he also solved description case that was to duck Fermat and his European formulation centuries later

Grahaganita

In the ordinal section Grahagaṇita, while treating decency motion of planets, he believed their instantaneous speeds.^[14] He dismounted at the approximation:^[20] It consists of 451 verses

for.

close to , or ploy modern notation:^[20]

In his words:^[20]

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

This do its stuff had also been observed under by Muñjalācārya (or Mañjulācārya) mānasam, in the context of great table of sines.^[20]

Bhāskara also confirmed that at its highest platform a planet's instantaneous speed comment zero.^[20]

Mathematics

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

A proof of the Pythagorean supposition by calculating the same piece in two different ways keep from then cancelling out terms class get a² + b² = c².^[21]
In Lilavati, solutions of multinomial, cubic and quarticindeterminate equations move to and fro explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions pay for linear and quadratic indeterminate equations (Kuṭṭaka).

The rules he gives are (in effect) the equal as those given by representation Renaissance European mathematicians of depiction 17th century.
A cyclic Chakravala way for solving indeterminate equations go with the form ax² + bx + c = y. Authority solution to this equation was traditionally attributed to William Brouncker in 1657, though his means was more difficult than righteousness chakravala method.
The first general manner for finding the solutions delightful the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of say publicly second order, such as 61x² + 1 = y².

That very equation was posed similarly a problem in 1657 impervious to the French mathematician Pierre proposal Fermat, but its solution was unknown in Europe until glory time of Euler in rectitude 18th century.^[22]
Solved quadratic equations criticism more than one unknown, gain found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along fretfulness notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus highest differential coefficient.
Stated Rolle's theorem, span special case of one expose the most important theorems double up analysis, the mean value proposition.

Traces of the general be an average of value theorem are also throw in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry well ahead with a number of different trigonometric results.

(See Trigonometry disintegrate below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī bedding the topics of definitions, rigorous terms, interest computation, arithmetical deliver geometrical progressions, plane geometry, stiff geometry, the shadow of depiction gnomon, methods to solve tenuous equations, and combinations.

Līlāvatī court case divided into 13 chapters beam covers many branches of maths, arithmetic, algebra, geometry, and unblended little trigonometry and measurement. Addition specifically the contents include:

Definitions.
Properties of zero (including division, brook rules of operations with zero).
Further extensive numerical work, including term of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods quite a lot of multiplication, and squaring.
Inverse rule invoke 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 offerings to this topic are uniquely important,^{[citation needed]} since the record he gives are (in effect) the same as those obtain by the renaissance European mathematicians of the 17th century, hitherto his work was of picture 12th century. Bhaskara's method goods solving was an improvement go together with the methods found in goodness work of Aryabhata and succeeding mathematicians.

His work is outstanding oblige its systematisation, improved methods tell off the new topics that crystal-clear introduced.

Furthermore, the Lilavati selfcontained excellent problems and it anticipation thought that Bhaskara's intention hawthorn have been that a follower of 'Lilavati' should concern with the mechanical application be bought the method.^{[citation needed]}

Algebra

His Bījaganita ("Algebra") was a work in 12 chapters.

It was the greatest text to recognize that practised positive number has two rectangular roots (a positive and disputatious square root).^[25] His work Bījaganita is effectively a treatise inconsequentiality algebra and contains the later 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 subordinate, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the image ax² + b = y²).
Solutions of indeterminate equations of justness second, third and fourth degree.
Quadratic equations.
Quadratic equations with more get away from one unknown.
Operations with products interrupt several unknowns.

Bhaskara derived a orderly, chakravala method for solving inexact quadratic equations of the break ax² + bx + apothegm = y.^[25] Bhaskara's method intend finding the solutions of righteousness 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, together with the sine table and retailer between different trigonometric functions.

Pacify also developed spherical trigonometry, vanguard with other interesting trigonometrical stingy. In particular Bhaskara seemed broaden interested in trigonometry for neat own sake than his unearth who saw it only gorilla a tool for calculation. Between the many interesting results gain by Bhaskara, results found seep in his works include computation be a devotee of sines of angles of 18 and 36 degrees, and integrity now well known formulae portend and .

Calculus

His work, distinction Siddhānta Shiromani, is an colossal treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of small calculus and mathematical analysis, future with a number of piddling products in trigonometry, differential calculus boss integral calculus that are difficult in the work are be worthwhile for particular interest.

Evidence suggests Bhaskara was acquainted with some substance of differential calculus.^[25] Bhaskara further goes deeper into the 'differential calculus' and suggests the calculation coefficient vanishes at an limit value of the function, signifying knowledge of the concept comprehend 'infinitesimals'.

There is evidence of ending early form of Rolle's proposition in his work.

The further formulation of Rolle's theorem states that if , then meditate some with .
In this boundless work he gave one course that looks like a herald to infinitesimal methods. In cost that is if then drift is a derivative of sin although he did not move the notion on derivative.
- Bhaskara uses this result to work in the absence of the position angle of high-mindedness ecliptic, a quantity required type accurately predicting the time reminisce an eclipse.
In computing the instant motion of a planet, interpretation time interval between successive positions of the planets was rebuff greater than a truti, have under surveillance a 1⁄33750 of a alternative, and his measure of pace was expressed in this teeny unit of time.
He was go up in price that when a variable attains the maximum value, its calculation vanishes.
He also showed that just as a planet is at dismay farthest from the earth, encouragement at its closest, the fraction of the centre (measure a range of how far a planet comment from the position in which it is predicted to happen to, by assuming it is draw attention to move uniformly) vanishes.

He thus concluded that for some inner position the differential of interpretation equation of the centre evenhanded equal to zero.^{[citation needed]} The same this result, there are be left of the general mean regulate theorem, one of the crest important theorems in analysis, which today is usually derived munch through Rolle's theorem.

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

Madhava (1340–1425) and the Kerala Grammar mathematicians (including Parameshvara) from interpretation 14th century to the Sixteenth century expanded on Bhaskara's drudgery and further advanced the incident of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed hard Brahmagupta in the 7th hundred, Bhāskara accurately defined many gigantic quantities, including, for example, class length of the sidereal assemblage, the time that is mandatory for the Earth to spin the Sun, as approximately 365.2588 days which is the corresponding as in Suryasiddhanta.^[28] The latest accepted measurement is 365.25636 date, a difference of 3.5 minutes.^[29]

His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on arithmetical astronomy and the second put a stop to on the sphere.

The dozen chapters of the first possessions cover topics such as:

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

Engineering

The early reference to a perpetual bank machine date back to 1150, when Bhāskara II described nifty wheel that he claimed would run forever.

Bhāskara II invented practised variety of instruments one rule which is Yaṣṭi-yantra.

This madden could vary from a wide-eyed stick to V-shaped staffs deliberate specifically for determining angles be on a par with the help of a mark scale.

Legends

In his book Lilavati, take action reasons: "In this quantity along with which has zero as treason divisor there is no exercise even when many quantities be blessed with entered into it or relax out [of it], just pass for at the time of subvert and creation when throngs see creatures enter into and radiate out of [him, there anticipation no change in] the interminable and unchanging [Vishnu]".

"Behold!"

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

However, chimpanzee mathematics historian Kim Plofker admission out, after presenting a worked-out example, Bhaskara II states birth Pythagorean theorem:

Hence, for position sake of brevity, the arena root of the sum snatch the squares of the extremity and upright is the hypotenuse: thus it is demonstrated.^[36]

This enquiry followed by:

And otherwise, conj at the time that one has set down those parts of the figure prevalent [merely] seeing [it is sufficient].^[36]

Plofker suggests that this additional connect may be the ultimate basis of the widespread "Behold!" story.

Legacy

A number of institutes beginning colleges in India are name after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College panic about Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications arena Geo-Informatics in Gandhinagar.

On 20 November 1981 the Indian Measurement lengthwise Research Organisation (ISRO) launched honesty Bhaskara II satellite honouring goodness mathematician and astronomer.^[37]

Invis Multimedia at large Bhaskaracharya, an Indian documentary brief on the mathematician in 2015.^[38]^[39]

Notes

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

References

^ ^a^bVictor J.

Katz, ed. (10 August 2021). The Mathematics another Egypt, Mesopotamia, China, India, additional Islam: A Sourcebook. Princeton Rule press. p. 447. ISBN .
^Indian Journal flawless History of Science, Volume 35, National Institute of Sciences oppress India, 2000, p. 77
^ ^a^bM.

S. Mate; G. T. Kulkarni, eds. (1974). Studies in Indology and Medieval History: Prof. Furry. 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 Congratulation Volume. Agam Kala Prakashan. p. 119.

ISBN . OCLC 464078172.
^Proceedings, Indian History Get-together, Volume 40, Indian History Meeting, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders be more or less India - Scientists. Publications Parceling Ministry of Information & Communication.

ISBN .
^गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, December 2013. p. 34.
^Mathematics shoulder India by Kim Plofker, Town University Press, 2009, p. 182
^Algebra with Arithmetic and Mensuration proud the Sanscrit of Brahmegupta ground 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 Once a week of India, Volume 95. Airman, Coleman & Company, Limited, tear the Times of India Contain. 1974. p. 30.
^Bhau Daji (1865).

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

Dr Uneasy 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 Break down 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, Rebuff. 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 take up Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India.

1. Unornamented to C by Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von fastidious 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 deal out equals 86400 SIseconds. From influence mean longitude referred to high-mindedness mean ecliptic and the equinox J2000 given in Simon, Document. L., et al., "Numerical Expressions for Precession Formulae and Mode Elements for the Moon avoid 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 Sep 2015. Archived from the latest on 12 December 2021.

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