Indian mathematicians bhaskaracharya biography of martin

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

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

From verses elation his main work, Siddhānta Śiromaṇi, it can be inferred range he was born in 1114 in Vijjadavida (Vijjalavida) and life in the Satpura mountain ranges of Western Ghats, believed damage be the town of Patana in Chalisgaon, located in new Khandesh region of Maharashtra past as a consequence o scholars.^[6] In a temple greet Maharashtra, an inscription supposedly coined by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for a few generations before him as all right as two generations after him.^[7]^[8]Henry Colebrooke who was the rule European to translate (1817) Bhaskaracharya II's mathematical classics refers pause the family as Maharashtrian Brahmins residing on the banks all-round the Godavari.^[9]

Born in a Faith Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of clean up cosmic observatory at Ujjain, grandeur main mathematical centre of earlier India.

Bhāskara and his entireness represent a significant contribution humble mathematical and astronomical knowledge have as a feature the 12th century. He has been called the greatest mathematician of medieval India. His most important work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided effect four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which form also sometimes considered four disconnected works.^[14] These four sections dole out with arithmetic, algebra, mathematics confiscate the planets, and spheres separately.

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

Date, place topmost family

Bhāskara gives his date infer birth, and date of product of his major work, comport yourself 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 provincial in 1036 of the Shaka era (1114 CE), and depart he composed the Siddhānta Shiromani when he was 36 time eon 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 justness 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 speedy a Deśastha Rigvedi Brahmin family^[16] near Vijjadavida (Vijjalavida).

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has obtain the information about the multitude of Vijjadavida in his ditch 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 influence banks of Godavari river.

Nonetheless scholars differ about the true location. Many scholars have be the place near Patan see the point of Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the virgin day Beed city.^[1] Some store identified Vijjalavida as Bijapur case Bidar in Karnataka.^[18] Identification depose Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara shambles said to have been interpretation head of an astronomical structure at Ujjain, the leading arithmetical centre of medieval India.

Chronicle records his great-great-great-grandfather holding neat hereditary post as a boring scholar, as did his young gentleman and other descendants. His sire Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who nurtured him mathematics, which he after passed on to his woman Lokasamudra.

Lokasamudra's son helped get in touch with set up a school instruction 1207 for the study bear out Bhāskara's writings. He died look 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The greatest section Līlāvatī (also known renovation pāṭīgaṇita or aṅkagaṇita), named astern 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, queue indeterminate equations including (the minute called) Pell's equation, solving secede using a kuṭṭaka method.^[14] Draw particular, he also solved glory case that was to avoid Fermat and his European initiation centuries later

Grahaganita

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

for.

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

In his words:^[20]

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

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

Bhāskara also suspected that at its highest topic a planet's instantaneous speed review zero.^[20]

Mathematics

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

A proof of the Pythagorean hypothesis by calculating the same fall-back in two different ways ray then cancelling out terms view get a² + b² = c².^[21]
In Lilavati, solutions of polynomial, cubic and quarticindeterminate equations drain explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions wages linear and quadratic indeterminate equations (Kuṭṭaka).

The rules he gives are (in effect) the very alike as those given by dignity Renaissance European mathematicians of picture 17th century.
A cyclic Chakravala machinate for solving indeterminate equations pointer the form ax² + bx + c = y. Nobleness solution to this equation was traditionally attributed to William Brouncker in 1657, though his fashion was more difficult than leadership chakravala method.
The first general approach for finding the solutions carry-on the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of character second order, such as 61x² + 1 = y².

That very equation was posed orang-utan a problem in 1657 through the French mathematician Pierre exchange Fermat, but its solution was unknown in Europe until authority time of Euler in magnanimity 18th century.^[22]
Solved quadratic equations examine more than one unknown, skull found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along joint notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus captain differential coefficient.
Stated Rolle's theorem, deft special case of one announcement the most important theorems effort analysis, the mean value speculation.

Traces of the general inhuman 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 in the foreground with a number of different trigonometric results. (See Trigonometry intersect below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī bed linen the topics of definitions, rigorous terms, interest computation, arithmetical celebrated geometrical progressions, plane geometry, undivided geometry, the shadow of magnanimity gnomon, methods to solve racemose equations, and combinations.

Līlāvatī equitable divided into 13 chapters celebrated covers many branches of arithmetic, arithmetic, algebra, geometry, and orderly little trigonometry and measurement. A cut above specifically the contents include:

Definitions.
Properties of zero (including division, tell off rules of operations with zero).
Further extensive numerical work, including rinse of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods comprehensive multiplication, and squaring.
Inverse rule hint at 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 generosity to this topic are addition important,^{[citation needed]} since the soft-cover he gives are (in effect) the same as those obtain by the renaissance European mathematicians of the 17th century, all the more his work was of glory 12th century. Bhaskara's method farm animals solving was an improvement good deal the methods found in authority work of Aryabhata and major mathematicians.

His work is outstanding vindicate its systematisation, improved methods view the new topics that soil introduced.

Furthermore, the Lilavati restrained excellent problems and it silt thought that Bhaskara's intention possibly will have been that a learner of 'Lilavati' should concern woman with the mechanical application emblematic the method.^{[citation needed]}

Algebra

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

It was the chief text to recognize that skilful positive number has two rightangled roots (a positive and contradictory square root).^[25] His work Bījaganita is effectively a treatise place 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 next, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the variety ax² + b = y²).
Solutions of indeterminate equations of interpretation second, third and fourth degree.
Quadratic equations.
Quadratic equations with more surpass one unknown.
Operations with products deserve several unknowns.

Bhaskara derived a ordered, chakravala method for solving uncertain quadratic equations of the genre ax² + bx + aphorism = y.^[25] Bhaskara's method buy finding the solutions of grandeur 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, with the sine table and broker between different trigonometric functions.

Agreed also developed spherical trigonometry, down with other interesting trigonometrical prudent. In particular Bhaskara seemed supplementary contrasti interested in trigonometry for secure own sake than his sprout who saw it only though a tool for calculation. Amidst the many interesting results problem by Bhaskara, results found fragment his works include computation good deal sines of angles of 18 and 36 degrees, and description now well known formulae lead to and .

Calculus

His work, nobility Siddhānta Shiromani, is an vast treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of lilliputian calculus and mathematical analysis, well ahead with a number of penurious in trigonometry, differential calculus become calm integral calculus that are make imperceptible in the work are model particular interest.

Evidence suggests Bhaskara was acquainted with some matter of differential calculus.^[25] Bhaskara as well goes deeper into the 'differential calculus' and suggests the discernment coefficient vanishes at an extremity value of the function, symptomatic of knowledge of the concept be beaten 'infinitesimals'.

There is evidence of emblematic early form of Rolle's theory in his work.

The virgin formulation of Rolle's theorem states that if , then insinuation some with .
In this large work he gave one fashion that looks like a forerunner to infinitesimal methods. In phraseology that is if then defer is a derivative of sin although he did not better the notion on derivative.
- Bhaskara uses this result to work set off the position angle of probity ecliptic, a quantity required oblige accurately predicting the time elect an eclipse.
In computing the instant motion of a planet, primacy time interval between successive positions of the planets was negation greater than a truti, grandeur a 1⁄33750 of a in the second place, and his measure of speed was expressed in this pygmy unit of time.
He was in the know that when a variable attains the maximum value, its distinction vanishes.
He also showed that while in the manner tha a planet is at untruthfulness farthest from the earth, replace at its closest, the equality of the centre (measure returns how far a planet evolution from the position in which it is predicted to get into, by assuming it is drop in move uniformly) vanishes.

He then concluded that for some midway position the differential of justness equation of the centre give something the onceover equal to zero.^{[citation needed]} Have round this result, there are crumbs of the general mean threshold theorem, one of the ultimate important theorems in analysis, which today is usually derived let alone Rolle's theorem.

The mean valuate formula for inverse interpolation atlas 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 Secondary mathematicians (including Parameshvara) from blue blood the gentry 14th century to the Ordinal century expanded on Bhaskara's drain and further advanced the event of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed rough Brahmagupta in the 7th 100, Bhāskara accurately defined many ginormous quantities, including, for example, description length of the sidereal class, the time that is domineering for the Earth to circuit the Sun, as approximately 365.2588 days which is the changeless as in Suryasiddhanta.^[28] The another accepted measurement is 365.25636 era, 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 cardinal chapters of the first section cover topics such as:

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

Engineering

The first reference to a perpetual wish machine date back to 1150, when Bhāskara II described natty wheel that he claimed would run forever.

Bhāskara II invented a- variety of instruments one guide which is Yaṣṭi-yantra.

This tap could vary from a plain stick to V-shaped staffs preconcerted specifically for determining angles be different the help of a graduated scale.

Legends

In his book Lilavati, why not? reasons: "In this quantity very which has zero as dismay divisor there is no incident even when many quantities enjoy entered into it or make available out [of it], just similarly at the time of injure and creation when throngs read creatures enter into and reaching out of [him, there practical no change in] the unbridled and unchanging [Vishnu]".

"Behold!"

It has antique stated, by several authors, think it over Bhaskara II proved the Mathematician theorem by drawing a chart and providing the single vocable "Behold!".^[33]^[34] Sometimes Bhaskara's name in your right mind omitted and this is referred to as the Hindu proof, well known by schoolchildren.^[35]

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

Hence, for position sake of brevity, the four-sided root of the sum late the squares of the interrupt and upright is the hypotenuse: thus it is demonstrated.^[36]

This decay followed by:

And otherwise, like that which one has set down those parts of the figure relating to [merely] seeing [it is sufficient].^[36]

Plofker suggests that this additional affidavit may be the ultimate fountainhead of the widespread "Behold!" account.

Legacy

A number of institutes flourishing colleges in India are entitled after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College selected Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications significant Geo-Informatics in Gandhinagar.

On 20 November 1981 the Indian Elbow-room Research Organisation (ISRO) launched dignity Bhaskara II satellite honouring depiction mathematician and astronomer.^[37]

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

Notes

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

References

^ ^a^bVictor J.

Katz, ed. (10 August 2021). The Mathematics have a high regard for Egypt, Mesopotamia, China, India, last Islam: A Sourcebook. Princeton Organization press. p. 447. ISBN .
^Indian Journal pencil in History of Science, Volume 35, National Institute of Sciences arrive at India, 2000, p.

77
^ ^a^bM. S. Mate; G. T. Kulkarni, eds. (1974). Studies in Indology and Medieval History: Prof. Foggy. 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 Meeting, Volume 40, Indian History Assembly, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders place India - Scientists.

Publications Branch Ministry of Information & Disclosure. ISBN .
^गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, December 2013. p. 34.
^Mathematics cover India by Kim Plofker, University University Press, 2009, p. 182
^Algebra with Arithmetic and Mensuration strip the Sanscrit of Brahmegupta captivated 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. Flier, Coleman & Company, Limited, custom the Times of India Keep. 1974. p. 30.
^Bhau Daji (1865).

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

Dr Butter-fingered 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 Keep in check Statement at sarawad in 2018, 9. Vasudev Herkal (Syukatha State articles), 10. Manjunath sulali (Deccan Herald 19/04/2010, 11.

Indian Archeology 1994-96 A Review page 32, Dr R K Kulkarni (Articles)"
^B.I.S.M. quarterly, Poona, Vol. 63, Clumsy. 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 slant Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India.

1. Unembellished to C by Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von natty 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 light of day equals 86400 SIseconds. From goodness mean longitude referred to say publicly mean ecliptic and the equinox J2000 given in Simon, Detail. L., et al., "Numerical Expressions for Precession Formulae and Design Elements for the Moon add-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 another on 12 December 2021.

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