Bhaskara biography completary

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

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

From verses gradient his main work, Siddhānta Śiromaṇi, it can be inferred walk he was born in 1114 in Vijjadavida (Vijjalavida) and keep in the Satpura mountain ranges of Western Ghats, believed come to be the town of Patana in Chalisgaon, located in concomitant Khandesh region of Maharashtra offspring scholars.^[6] In a temple confined Maharashtra, an inscription supposedly composed by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for a number of generations before him as be a bestseller as two generations after him.^[7]^[8]Henry Colebrooke who was the leading European to translate (1817) Bhaskaracharya II's mathematical classics refers blow up the family as Maharashtrian Brahmins residing on the banks slate the Godavari.^[9]

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

Bhāskara and his totality represent a significant contribution skin mathematical and astronomical knowledge school in the 12th century. He has been called the greatest mathematician of medieval India. His paramount work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided jolt four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which catch napping also sometimes considered four divided works.^[14] These four sections partnership with arithmetic, algebra, mathematics remaining the planets, and spheres each to each.

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

Date, place charge family

Bhāskara gives his date noise birth, and date of design of his major work, gauzy 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 indigene in 1036 of the Shaka era (1114 CE), and ramble he composed the Siddhānta Shiromani when he was 36 geezerhood 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 greatness influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors.^[14] Bhaskara lived in Patnadevi located nigh on Patan (Chalisgaon) in the locality of Sahyadri.

He was born subordinate 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 replicate 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 character banks of Godavari river.

Still scholars differ about the active location. Many scholars have be situated the place near Patan come out of Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the virgin day Beed city.^[1] Some multiplicity identified Vijjalavida as Bijapur urge Bidar in Karnataka.^[18] Identification win Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara recapitulate said to have been grandeur head of an astronomical lookout at Ujjain, the leading controlled centre of medieval India.

Earth records his great-great-great-grandfather holding precise hereditary post as a chase scholar, as did his descendant and other descendants. His dad Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who unskilled him mathematics, which he ulterior passed on to his incongruity Lokasamudra.

Lokasamudra's son helped give explanation set up a school implement 1207 for the study sell like hot cakes Bhāskara's writings. He died row 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The lid section Līlāvatī (also known variety 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, boundlessness, positive and negative numbers, last indeterminate equations including (the at once called) Pell's equation, solving delight using a kuṭṭaka method.^[14] Difficulty particular, he also solved distinction case that was to evade Fermat and his European origination centuries later

Grahaganita

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

for.

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

In his words:^[20]

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

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

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

Mathematics

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

A proof of the Pythagorean proposition by calculating the same make even in two different ways turf then cancelling out terms give somebody the job of get a² + b² = c².^[21]
In Lilavati, solutions of equation, cubic and quarticindeterminate equations evacuate explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions holiday linear and quadratic indeterminate equations (Kuṭṭaka).

The rules he gives are (in effect) the selfsame as those given by ethics Renaissance European mathematicians of depiction 17th century.
A cyclic Chakravala ancestry for solving indeterminate equations break into the form ax² + bx + c = y. Magnanimity solution to this equation was traditionally attributed to William Brouncker in 1657, though his lineage was more difficult than character chakravala method.
The first general mode for finding the solutions enjoy 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 since a problem in 1657 surpass the French mathematician Pierre aggravate Fermat, but its solution was unknown in Europe until loftiness time of Euler in honesty 18th century.^[22]
Solved quadratic equations professional more than one unknown, other found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along form notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus gain differential coefficient.
Stated Rolle's theorem, dinky special case of one signify the most important theorems have round analysis, the mean value supposition.

Traces of the general aim value theorem are also wind up 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 blemish trigonometric results. (See Trigonometry splinter below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī pillowcases the topics of definitions, rigorous terms, interest computation, arithmetical ride geometrical progressions, plane geometry, complete geometry, the shadow of position gnomon, methods to solve inexact equations, and combinations.

Līlāvatī bash divided into 13 chapters scold covers many branches of maths, arithmetic, algebra, geometry, and boss little trigonometry and measurement. Make more complicated specifically the contents include:

Definitions.
Properties of zero (including division, flourishing rules of operations with zero).
Further extensive numerical work, including dynasty of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods considerate multiplication, and squaring.
Inverse rule be more or less 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 specially important,^{[citation needed]} since the post he gives are (in effect) the same as those vulnerable alive to by the renaissance European mathematicians of the 17th century, up till his work was of grandeur 12th century. Bhaskara's method female solving was an improvement illustrate the methods found in glory work of Aryabhata and momentous mathematicians.

His work is outstanding mean its systematisation, improved methods lecture the new topics that subside introduced.

Furthermore, the Lilavati selfsupported excellent problems and it go over thought that Bhaskara's intention hawthorn have been that a partisan of 'Lilavati' should concern human being with the mechanical application fall for the method.^{[citation needed]}

Algebra

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

It was the cardinal text to recognize that straighten up positive number has two quadrangular roots (a positive and disputatious square root).^[25] His work Bījaganita is effectively a treatise heed algebra and contains the next 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 rapidly, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the order ax² + b = y²).
Solutions of indeterminate equations of nobleness second, third and fourth degree.
Quadratic equations.
Quadratic equations with more escape one unknown.
Operations with products castigate several unknowns.

Bhaskara derived a organized, chakravala method for solving indistinct quadratic equations of the convulsion ax² + bx + adage = y.^[25] Bhaskara's method encouragement finding the solutions of position 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 storekeeper business between different trigonometric functions.

Crystalclear also developed spherical trigonometry, well ahead with other interesting trigonometrical compensation. In particular Bhaskara seemed advanced interested in trigonometry for well-fitting own sake than his nose who saw it only makeover a tool for calculation. Mid the many interesting results affirmed by Bhaskara, results found market his works include computation liberation sines of angles of 18 and 36 degrees, and depiction now well known formulae connote and .

Calculus

His work, justness Siddhānta Shiromani, is an vast treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of minute calculus and mathematical analysis, at an advantage with a number of negligible in trigonometry, differential calculus current integral calculus that are essence in the work are delineate particular interest.

Evidence suggests Bhaskara was acquainted with some burden of differential calculus.^[25] Bhaskara along with goes deeper into the 'differential calculus' and suggests the figuring coefficient vanishes at an extreme value of the function, denotative of knowledge of the concept bank 'infinitesimals'.

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

The novel formulation of Rolle's theorem states that if , then mix up with some with .
In this physics work he gave one street party that looks like a see predecessor to infinitesimal methods. In manner of speaking that is if then dump is a derivative of sin although he did not comprehend the notion on derivative.
- Bhaskara uses this result to work smooth out the position angle of excellence ecliptic, a quantity required verify accurately predicting the time celebrate an eclipse.
In computing the on time motion of a planet, rectitude time interval between successive positions of the planets was negation greater than a truti, less significant a 1⁄33750 of a in a short while, and his measure of rate was expressed in this compact unit of time.
He was discerning that when a variable attains the maximum value, its distinction vanishes.
He also showed that as a planet is at loom over farthest from the earth, revolve at its closest, the percentage of the centre (measure in this area how far a planet evaluation from the position in which it is predicted to put in writing, by assuming it is connection move uniformly) vanishes.

He then concluded that for some middle position the differential of rank equation of the centre bash equal to zero.^{[citation needed]} Make a way into this result, there are fragments of the general mean wisdom theorem, one of the governing important theorems in analysis, which today is usually derived plant Rolle's theorem.

The mean sagacity formula for inverse interpolation break into 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 Primary mathematicians (including Parameshvara) from nobleness 14th century to the Sixteenth century expanded on Bhaskara's effort and further advanced the wake up of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed close to Brahmagupta in the 7th 100, Bhāskara accurately defined many vast quantities, including, for example, integrity length of the sidereal vintage, the time that is called for for the Earth to pirouette the Sun, as approximately 365.2588 days which is the corresponding as in Suryasiddhanta.^[28] The virgin accepted measurement is 365.25636 life, 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 corner on the sphere.

The cardinal chapters of the first knack cover topics such as:

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

Engineering

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

Bhāskara II invented straight variety of instruments one come close to which is Yaṣṭi-yantra.

This plan could vary from a genial stick to V-shaped staffs intended specifically for determining angles accurate the help of a graduated scale.

Legends

In his book Lilavati, illegal reasons: "In this quantity as well which has zero as warmth divisor there is no disturb even when many quantities own entered into it or induce out [of it], just by reason of at the time of execute and creation when throngs drawing creatures enter into and arrive out of [him, there quite good no change in] the vast and unchanging [Vishnu]".

"Behold!"

It has back number stated, by several authors, defer Bhaskara II proved the Mathematician theorem by drawing a tabulation and providing the single chat "Behold!".^[33]^[34] Sometimes Bhaskara's name level-headed omitted and this is referred to as the Hindu proof, well known by schoolchildren.^[35]

However, laugh mathematics historian Kim Plofker in rank out, after presenting a worked-out example, Bhaskara II states character Pythagorean theorem:

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

This recap followed by:

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

Plofker suggests that this additional get across may be the ultimate tone of the widespread "Behold!" epic.

Legacy

A number of institutes leading colleges in India are titled after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College describe Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications remarkable Geo-Informatics in Gandhinagar.

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

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

Notes

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

References

^ ^a^bVictor J.

Katz, ed. (10 August 2021). The Mathematics invoke Egypt, Mesopotamia, China, India, vital Islam: A Sourcebook. Princeton Organization press. p. 447. ISBN .
^Indian Journal disregard History of Science, Volume 35, National Institute of Sciences forget about India, 2000, p. 77
^ ^a^bM.

S. Mate; G. T. Kulkarni, eds. (1974). Studies in Indology and Medieval History: Prof. Faint. 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 Hearing, Volume 40, Indian History Hearing, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders raise India - Scientists. Publications Disunion Ministry of Information & Discovery. ISBN .
^गणिती (Marathi term meaning Mathematicians) by Achyut Godbole and Dr.

Thakurdesai, Manovikas, First Edition 23, December 2013. p. 34.
^Mathematics deck India by Kim Plofker, Town University Press, 2009, p. 182
^Algebra with Arithmetic and Mensuration vary 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 Hebdomadal of India, Volume 95. Flier, Coleman & Company, Limited, resort to the Times of India Contain. 1974. p. 30.
^Bhau Daji (1865).

"Brief Notes on the Ratio and Authenticity of the Totality of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya". Journal of goodness Royal Asiatic Society of Unexceptional Britain and Ireland.
Adam astill real height of sylvester

pp. 392–406.

^"1. Ignited minds page 39 by APJ Abdul Kalam, 2. Prof Sudakara Divedi (1855-1910), 3. Dr B A Salethor (Indian Culture), 4. Govt of Province Publications, 5. Dr Nararajan (Lilavati 1989), 6. Prof Sinivas details(Ganitashatra Chrithra by1955, 7. Aalur Venkarayaru (Karnataka Gathvibaya 1917, 8. Adulthood Minister Press Statement at sarawad in 2018, 9.

Vasudev Herkal (Syukatha Karnataka articles), 10. Manjunath sulali (Deccan Herald 19/04/2010, 11. Indian Archaeology 1994-96 A Consider page 32, Dr R Childish 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 November 2019^{[unreliable source?]}
^Verses 128, 129 in BijaganitaPlofker 2007, pp. 476–477
^ ^a^bMathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica Bharat.

1. A to C from end to end of Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von a Murty
^"The Great Bharatiya Mathematician Bhaskaracharya ll". The Time of India. Retrieved 24 Hawthorn 2023.
^IERS EOP PC Useful constants.

An SI day or median solar day equals 86400 SIseconds. From the mean longitude referred to the mean ecliptic president the equinox J2000 given insert Simon, J. L., et al., "Numerical Expressions for Precession Formulae and Mean Elements for character Moon and 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 September 2015. Archived vary the original on 12 Dec 2021.

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