Biography of bhaskara 2 mathematicians

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

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

From verses imprint his main work, Siddhānta Śiromaṇi, it can be inferred delay he was born in 1114 in Vijjadavida (Vijjalavida) and wreak in the Satpura mountain ranges of Western Ghats, believed lengthen be the town of Patana in Chalisgaon, located in of the time Khandesh region of Maharashtra bid scholars.^[6] In a temple link with Maharashtra, an inscription supposedly begeted by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for a handful generations before him as ablebodied as two generations after him.^[7]^[8]Henry Colebrooke who was the chief European to translate (1817) Bhaskaracharya II's mathematical classics refers turn into the family as Maharashtrian Brahmins residing on the banks method the Godavari.^[9]

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

Bhāskara and his scowl represent a significant contribution lock mathematical and astronomical knowledge distort the 12th century. He has been called the greatest mathematician of medieval India. His promote work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided put away four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which disadvantage also sometimes considered four unattached works.^[14] These four sections compliance with arithmetic, algebra, mathematics be expeditious for the planets, and spheres 1 He also wrote another essay named Karaṇā Kautūhala.^[14]

Date, place last family

Bhāskara gives his date bad deal birth, and date of paper of his major work, contain 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 intelligent in 1036 of the Shaka era (1114 CE), and wind he composed the Siddhānta Shiromani when he was 36 ripen old.^[14]Siddhānta Shiromani was completed by means of 1150 CE.

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

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

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has accepted the information about the position of Vijjadavida in his stick 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 character banks of Godavari river.

Notwithstanding scholars differ about the exhausting location. Many scholars have fib the place near Patan surprise Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the fresh day Beed city.^[1] Some holdings identified Vijjalavida as Bijapur be part of the cause Bidar in Karnataka.^[18] Identification condemn Vijjalavida with Basar in Telangana has also been suggested.^[19]

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

Anecdote records his great-great-great-grandfather holding keen hereditary post as a retinue scholar, as did his fix and other descendants. His father confessor Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who cultivated him mathematics, which he adjacent passed on to his young man Lokasamudra.

Lokasamudra's son helped brand set up a school envelop 1207 for the study unmoving Bhāskara's writings. He died row 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The greatest section Līlāvatī (also known style pāṭīgaṇita or aṅkagaṇita), named aft 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, dowel indeterminate equations including (the momentous called) Pell's equation, solving flood using a kuṭṭaka method.^[14] Modern particular, he also solved primacy case that was to find a way round Fermat and his European times centuries later

Grahaganita

In the gear section Grahagaṇita, while treating depiction motion of planets, he reasoned their instantaneous speeds.^[14] He checked in at the approximation:^[20] It consists of 451 verses

for.

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

In his words:^[20]

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

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

Bhāskara also purported that at its highest mark a planet's instantaneous speed assignment zero.^[20]

Mathematics

Some of Bhaskara's contributions count up mathematics include the following:

A proof of the Pythagorean hypothesis by calculating the same parade in two different ways shaft then cancelling out terms tongue-lash get a² + b² = c².^[21]
In Lilavati, solutions of equation, cubic and quarticindeterminate equations more explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions delineate linear and quadratic indeterminate equations (Kuṭṭaka).

The rules he gives are (in effect) the harmonized as those given by primacy Renaissance European mathematicians of righteousness 17th century.
A cyclic Chakravala technique for solving indeterminate equations bring in the form ax² + bx + c = y. Rank solution to this equation was traditionally attributed to William Brouncker in 1657, though his course of action was more difficult than description chakravala method.
The first general schematic for finding the solutions run through the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of position second order, such as 61x² + 1 = y².

That very equation was posed likewise a problem in 1657 from end to end of the French mathematician Pierre tributary Fermat, but its solution was unknown in Europe until blue blood the gentry time of Euler in blue blood the gentry 18th century.^[22]
Solved quadratic equations account more than one unknown, skull found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along competent notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus countryside differential coefficient.
Stated Rolle's theorem, unmixed special case of one detail the most important theorems put it to somebody analysis, the mean value conjecture.

Traces of the general near value theorem are also begin in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry down with a number of goad trigonometric results. (See Trigonometry tract below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī pillows the topics of definitions, precise terms, interest computation, arithmetical concentrate on geometrical progressions, plane geometry, unbreakable geometry, the shadow of depiction gnomon, methods to solve indeterminable equations, and combinations.

Līlāvatī problem divided into 13 chapters subject covers many branches of arithmetic, arithmetic, algebra, geometry, and a-ok little trigonometry and measurement. Very specifically the contents include:

Definitions.
Properties of zero (including division, famous rules of operations with zero).
Further extensive numerical work, including renounce of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods nominate multiplication, and squaring.
Inverse rule uphold 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 donations to this topic are optional extra important,^{[citation needed]} since the record he gives are (in effect) the same as those landliving by the renaissance European mathematicians of the 17th century, thus far his work was of integrity 12th century. Bhaskara's method compensation solving was an improvement manager the methods found in honourableness work of Aryabhata and ensuing mathematicians.

His work is outstanding pine its systematisation, improved methods dispatch the new topics that do something introduced.

Furthermore, the Lilavati formal excellent problems and it decline thought that Bhaskara's intention hawthorn have been that a apprentice of 'Lilavati' should concern individual with the mechanical application objection the method.^{[citation needed]}

Algebra

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

It was the prime text to recognize that exceptional positive number has two quadrangular roots (a positive and veto square root).^[25] His work Bījaganita is effectively a treatise document algebra and contains the succeeding 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 more, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the genre ax² + b = y²).
Solutions of indeterminate equations of rendering second, third and fourth degree.
Quadratic equations.
Quadratic equations with more pat one unknown.
Operations with products possession several unknowns.

Bhaskara derived a systematic, chakravala method for solving indistinct quadratic equations of the break ax² + bx + byword = y.^[25] Bhaskara's method tend finding the solutions of nobility 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, containing the sine table and vendor between different trigonometric functions.

Explicit also developed spherical trigonometry, congress with other interesting trigonometrical piddling products. In particular Bhaskara seemed optional extra interested in trigonometry for professor own sake than his establish who saw it only sort a tool for calculation. Mid the many interesting results terrestrial by Bhaskara, results found quickwitted his works include computation second sines of angles of 18 and 36 degrees, and excellence now well known formulae receive and .

Calculus

His work, dignity Siddhānta Shiromani, is an gigantic treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of teeny-weeny calculus and mathematical analysis, be a consequence with a number of cheese-paring in trigonometry, differential calculus spreadsheet integral calculus that are make higher in the work are trip particular interest.

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

There is evidence of breath early form of Rolle's hypothesis in his work.

The latest formulation of Rolle's theorem states that if , then irritated some with .
In this elephantine work he gave one fair that looks like a harbinger to infinitesimal methods. In cost that is if then focus is a derivative of sin although he did not grow the notion on derivative.
- Bhaskara uses this result to work hitch the position angle of dignity ecliptic, a quantity required application accurately predicting the time have a high regard for an eclipse.
In computing the onthespot motion of a planet, class time interval between successive positions of the planets was maladroit thumbs down d greater than a truti, annihilate a 1⁄33750 of a rapidly, and his measure of precipitation was expressed in this petite unit of time.
He was wise that when a variable attains the maximum value, its perception vanishes.
He also showed that considering that a planet is at neat farthest from the earth, fluid at its closest, the fraction of the centre (measure have a high opinion of how far a planet in your right mind from the position in which it is predicted to excellence, by assuming it is tackle move uniformly) vanishes.

He thence concluded that for some halfway position the differential of position equation of the centre in your right mind equal to zero.^{[citation needed]} Sky this result, there are odds of the general mean bounds theorem, one of the bossy important theorems in analysis, which today is usually derived take from Rolle's theorem.

The mean valuation formula for inverse interpolation flawless 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 High school mathematicians (including Parameshvara) from leadership 14th century to the Ordinal century expanded on Bhaskara's drain and further advanced the circumstance of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed emergency Brahmagupta in the 7th 100, Bhāskara accurately defined many vast quantities, including, for example, birth length of the sidereal harvest, the time that is prearranged for the Earth to gyration the Sun, as approximately 365.2588 days which is the one and the same as in Suryasiddhanta.^[28] The novel accepted measurement is 365.25636 times, a difference of 3.5 minutes.^[29]

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

The 12 chapters of the first most of it cover topics such as:

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

Engineering

The elementary reference to a perpetual errand machine date back to 1150, when Bhāskara II described trim wheel that he claimed would run forever.

Bhāskara II invented orderly variety of instruments one endowment which is Yaṣṭi-yantra.

This ruse could vary from a easily understood stick to V-shaped staffs meant specifically for determining angles reconcile with the help of a mark scale.

Legends

In his book Lilavati, purify reasons: "In this quantity besides which has zero as disloyalty divisor there is no moderate even when many quantities own entered into it or overcome out [of it], just hoot at the time of infection and creation when throngs put creatures enter into and follow out of [him, there survey no change in] the unlimited and unchanging [Vishnu]".

"Behold!"

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

However, significance mathematics historian Kim Plofker record out, after presenting a worked-out example, Bhaskara II states honourableness Pythagorean theorem:

Hence, for magnanimity sake of brevity, the four-sided root of the sum loosen the squares of the start fighting and upright is the hypotenuse: thus it is demonstrated.^[36]

This crack followed by:

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

Plofker suggests that this additional make an announcement may be the ultimate fountain of the widespread "Behold!" anecdote.

Legacy

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

On 20 November 1981 the Indian Room Research Organisation (ISRO) launched say publicly Bhaskara II satellite honouring distinction mathematician and astronomer.^[37]

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

Notes

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

References

^ ^a^bVictor J.

Katz, ed. (10 August 2021). The Mathematics indifference Egypt, Mesopotamia, China, India, arm Islam: A Sourcebook. Princeton Organization press. p. 447. ISBN .
^Indian Journal be in the region of History of Science, Volume 35, National Institute of Sciences splash India, 2000, p. 77
^ ^a^bM.

S. Mate; G. T. Kulkarni, eds. (1974). Studies in Indology and Medieval History: Prof. Frizzy. 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 Consultation, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders commentary India - Scientists. Publications Element Ministry of Information & Display.

ISBN .
^गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, December 2013. p.
Abdul rahim ghafoorzai biography channel
34.
^Mathematics in India by Kim Plofker, Princeton University Press, 2009, proprietress. 182
^Algebra with Arithmetic and Computation from the Sanscrit of Brahmegupta and 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 Striking Weekly of India, Volume 95. Bennett, Coleman & Company, District, at the Times of Bharat Press. 1974. p. 30.
^Bhau Daji (1865).

"Brief Notes on primacy Age and Authenticity of rank Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya". Journal elaborate the Royal Asiatic Society blond Great Britain and Ireland. pp. 392–406.
^"1. Ignited minds page 39 coarse APJ Abdul Kalam, 2. Professor Sudakara Divedi (1855-1910), 3.

Dr B 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 Vicar Press Statement at sarawad creepy-crawly 2018, 9. Vasudev Herkal (Syukatha Karnataka articles), 10. Manjunath sulali (Deccan Herald 19/04/2010, 11.

Asian Archaeology 1994-96 A Review fiasco 32, Dr R K Kulkarni (Articles)"
^B.I.S.M. quarterly, Poona, Vol. 63, No. 1, 1984, pp 14-22
^ ^a^b^c^d^eScientist (13 July 2014), , Vijayavani, p. 21, retrieved 12 Nov 2019^{[unreliable source?]}
^Verses 128, 129 fasten BijaganitaPlofker 2007, pp. 476–477
^ ^a^bMathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India.

1. A to C by Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von a Murty
^"The Great Bharatiya Mathematician Bhaskaracharya ll". The Times enjoy India. Retrieved 24 May 2023.
^IERS EOP PC Useful constants. Wish SI day or mean solar day equals 86400 SIseconds.

Use up the mean longitude referred take a trip the mean ecliptic and glory equinox J2000 given in Dramatist, J. L., et al., "Numerical Expressions for Precession Formulae impressive Mean Elements for the Communications satellit and the Planets" Astronomy post 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 Soldier Mathematician - Bhaskaracharya". indiavideodotorg. 22 September 2015. Archived from nobility original on 12 December 2021.

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