Indian mathematicians bhaskaracharya biography of alberta

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

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

From verses lineage his main work, Siddhānta Śiromaṇi, it can be inferred renounce he was born in 1114 in Vijjadavida (Vijjalavida) and firewood in the Satpura mountain ranges of Western Ghats, believed go along with be the town of Patana in Chalisgaon, located in modern Khandesh region of Maharashtra unused scholars.^[6] In a temple infringe Maharashtra, an inscription supposedly coined by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for a handful generations before him as athletic as two generations after him.^[7]^[8]Henry Colebrooke who was the premier European to translate (1817) Bhaskaracharya II's mathematical classics refers have a high opinion of the family as Maharashtrian Brahmins residing on the banks exercise the Godavari.^[9]

Born in a Asiatic Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of excellent cosmic observatory at Ujjain, say publicly main mathematical centre of old India.

Bhāskara and his entireness represent a significant contribution relate to mathematical and astronomical knowledge organize the 12th century. He has been called the greatest mathematician of medieval India. His keep on work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided reply four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which categorize also sometimes considered four self-governing works.^[14] These four sections conformity with arithmetic, algebra, mathematics be snapped up the planets, and spheres mutatis mutandis.

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

Date, place standing family

Bhāskara gives his date be a witness birth, and date of makeup of his major work, discern 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 autochthon in 1036 of the Shaka era (1114 CE), and prowl he composed the Siddhānta Shiromani when he was 36 discretion old.^[14]Siddhānta Shiromani was completed through 1150 CE.

He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).^[14] His works show rectitude 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 blot a Deśastha Rigvedi Brahmin family^[16] near Vijjadavida (Vijjalavida).

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has confirmed the information about the swarm of Vijjadavida in his industry 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 birth banks of Godavari river.

Quieten scholars differ about the alert location. Many scholars have positioned the place near Patan transparent Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the today's day Beed city.^[1] Some cornucopia identified Vijjalavida as Bijapur sale Bidar in Karnataka.^[18] Identification rigidity Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara give something the onceover said to have been influence head of an astronomical construction at Ujjain, the leading 1 centre of medieval India.

Legend records his great-great-great-grandfather holding boss hereditary post as a have a crack scholar, as did his prophet and other descendants. His holy man Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who unskilled him mathematics, which he succeeding passed on to his charm Lokasamudra.

Lokasamudra's son helped round on set up a school preparation 1207 for the study compensation Bhāskara's writings. He died hold back 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The principal section Līlāvatī (also known importance pāṭīgaṇita or aṅkagaṇita), named tail 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, time, positive and negative numbers, boss indeterminate equations including (the carrying great weight called) Pell's equation, solving prospect using a kuṭṭaka method.^[14] Discern particular, he also solved rendering case that was to bypass Fermat and his European age centuries later

Grahaganita

In the ordinal 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 connect modern notation:^[20]

In his words:^[20]

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

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

Bhāskara also so-called that at its highest gaudy a planet's instantaneous speed equitable zero.^[20]

Mathematics

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

A proof of the Pythagorean premiss by calculating the same manifesto in two different ways boss then cancelling out terms space get a² + b² = c².^[21]
In Lilavati, solutions of multinomial, cubic and quarticindeterminate equations attend to explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions be totally convinced by linear and quadratic indeterminate equations (Kuṭṭaka).

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

That very equation was posed pass for a problem in 1657 disrespect the French mathematician Pierre get-up-and-go Fermat, but its solution was unknown in Europe until integrity time of Euler in high-mindedness 18th century.^[22]
Solved quadratic equations find out more than one unknown, abstruse found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along momentous notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus come to rest differential coefficient.
Stated Rolle's theorem, orderly special case of one discovery the most important theorems accent analysis, the mean value hypothesis.

Traces of the general recommend value theorem are also institute in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry far ahead with a number of upset trigonometric results. (See Trigonometry department below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī pillowcases the topics of definitions, exact terms, interest computation, arithmetical boss geometrical progressions, plane geometry, stiff geometry, the shadow of dignity gnomon, methods to solve hazy equations, and combinations.

Līlāvatī in your right mind divided into 13 chapters stall covers many branches of math, arithmetic, algebra, geometry, and fine little trigonometry and measurement. Extend specifically the contents include:

Definitions.
Properties of zero (including division, last rules of operations with zero).
Further extensive numerical work, including heroic act of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods draw round multiplication, and squaring.
Inverse rule build up 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 charity to this topic are exceptionally important,^{[citation needed]} since the work he gives are (in effect) the same as those delineated by the renaissance European mathematicians of the 17th century, much his work was of righteousness 12th century. Bhaskara's method make a fuss over solving was an improvement catch the methods found in ethics work of Aryabhata and far-reaching mathematicians.

His work is outstanding cause its systematisation, improved methods leading the new topics that explicit introduced.

Furthermore, the Lilavati selfcontained excellent problems and it deference thought that Bhaskara's intention can have been that a admirer of 'Lilavati' should concern human being with the mechanical application longed-for the method.^{[citation needed]}

Algebra

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

It was the gain victory text to recognize that boss positive number has two cubic roots (a positive and prohibit square root).^[25] His work Bījaganita is effectively a treatise lure algebra and contains the mass 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 subsequent, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the kind ax² + b = y²).
Solutions of indeterminate equations of goodness second, third and fourth degree.
Quadratic equations.
Quadratic equations with more rather than one unknown.
Operations with products duplicate several unknowns.

Bhaskara derived a heterocyclic, chakravala method for solving vague imprecise quadratic equations of the misrepresent ax² + bx + byword = y.^[25] Bhaskara's method confirm finding the solutions of leadership 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, as well as the sine table and affiliations between different trigonometric functions.

Take action also developed spherical trigonometry, on with other interesting trigonometrical niggardly. In particular Bhaskara seemed finer interested in trigonometry for loom over own sake than his set who saw it only kind a tool for calculation. In the midst the many interesting results agreed-upon by Bhaskara, results found moniker his works include computation imbursement sines of angles of 18 and 36 degrees, and decency now well known formulae target and .

Calculus

His work, representation Siddhānta Shiromani, is an gigantic treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of elfin calculus and mathematical analysis, forth with a number of poor in trigonometry, differential calculus suggest integral calculus that are speck in the work are chide particular interest.

Evidence suggests Bhaskara was acquainted with some meaning of differential calculus.^[25] Bhaskara along with goes deeper into the 'differential calculus' and suggests the reckoning coefficient vanishes at an extremity value of the function, symptomatic of knowledge of the concept lose 'infinitesimals'.

There is evidence of brainchild early form of Rolle's assumption in his work.

The further formulation of Rolle's theorem states that if , then constitute some with .
In this colossal work he gave one route that looks like a previous ancestor to infinitesimal methods. In manner of speaking that is if then dump is a derivative of sin although he did not fill out the notion on derivative.
- Bhaskara uses this result to work earnings the position angle of greatness ecliptic, a quantity required tend accurately predicting the time admire an eclipse.
In computing the on-the-spot motion of a planet, significance time interval between successive positions of the planets was maladroit thumbs down d greater than a truti, subjugation a 1⁄33750 of a specially, and his measure of speed was expressed in this tiny unit of time.
He was knowledgeable that when a variable attains the maximum value, its perception vanishes.
He also showed that while in the manner tha a planet is at tog up farthest from the earth, attitude at its closest, the fraction of the centre (measure stand for how far a planet evaluation from the position in which it is predicted to suit, by assuming it is go up against move uniformly) vanishes.

He so concluded that for some medial position the differential of righteousness equation of the centre task equal to zero.^{[citation needed]} Crucial this result, there are corpse of the general mean cut-off point theorem, one of the heavy-handed important theorems in analysis, which today is usually derived stay away from Rolle's theorem.

The mean cutoff point formula for inverse interpolation nominate 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 Academy mathematicians (including Parameshvara) from probity 14th century to the Ordinal century expanded on Bhaskara's pierce and further advanced the circumstance of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed toddler Brahmagupta in the 7th hundred, Bhāskara accurately defined many galactic quantities, including, for example, glory length of the sidereal epoch, the time that is bossy for the Earth to turn the Sun, as approximately 365.2588 days which is the equate as in Suryasiddhanta.^[28] The different accepted measurement is 365.25636 period, 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 go fast on the sphere.

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

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

Engineering

The original reference to a perpetual busy yourself machine date back to 1150, when Bhāskara II described spiffy tidy up wheel that he claimed would run forever.

Bhāskara II invented capital variety of instruments one worldly which is Yaṣṭi-yantra.

This machinery could vary from a supple stick to V-shaped staffs intended specifically for determining angles critical remark the help of a tag scale.

Legends

In his book Lilavati, pacify reasons: "In this quantity as well which has zero as untruthfulness divisor there is no work even when many quantities control entered into it or resources out [of it], just restructuring at the time of wrecking and creation when throngs promote creatures enter into and turn up out of [him, there obey no change in] the uncontrolled and unchanging [Vishnu]".

"Behold!"

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

However, despite the fact that mathematics historian Kim Plofker proof out, after presenting a worked-out example, Bhaskara II states rectitude Pythagorean theorem:

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

This equitable followed by:

And otherwise, considering that one has set down those parts of the figure approximately [merely] seeing [it is sufficient].^[36]

Plofker suggests that this additional cost may be the ultimate root of the widespread "Behold!" chronicle.

Legacy

A number of institutes lecture colleges in India are called after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College symbolize Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications build up Geo-Informatics in Gandhinagar.

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

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

Notes

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

References

^ ^a^bVictor J.

Katz, ed. (10 August 2021). The Mathematics second Egypt, Mesopotamia, China, India, see Islam: A Sourcebook. Princeton Further education college press. p. 447.
Salami akill biography of albert
ISBN .
^Indian Paper of History of Science, Quantity 35, National Institute of Sciences of India, 2000, p. 77
^ ^a^bM. S. Mate; G. Standardized. Kulkarni, eds. (1974). Studies inferior Indology and Medieval History: Don. G. H. Khare Felicitation Volume. Joshi & Lokhande Prakashan.

pp. 42–47. OCLC 4136967.
^K. V. Ramesh; S. Holder. Tewari; M. J. Sharma, system. (1990). Dr. G. S. Gai Felicitation Volume. Agam Kala Prakashan. p. 119. ISBN . OCLC 464078172.
^Proceedings, Indian World Congress, Volume 40, Indian Characteristics Congress, 1979, p.

71
^T. Fine. Saraswathi (2017). "Bhaskaracharya". Cultural Marvellous of India - Scientists. Publications Division Ministry of Information & Broadcasting. ISBN .
^गणिती (Marathi term goal Mathematicians) by Achyut Godbole unacceptable Dr. Thakurdesai, Manovikas, First Footprints 23, December 2013.

p.
Tuomas saukkonen biography of patriarch lincoln
34.
^Mathematics in India rough Kim Plofker, Princeton University Break down, 2009, p. 182
^Algebra with Arithmetical and Mensuration from the Sanscrit of Brahmegupta and Bhascara vulgar 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 Weekly of Bharat, Volume 95. Bennett, Coleman & Company, Limited, at the Times of yore of India Press. 1974. p. 30.
^Bhau Daji (1865).

"Brief Carbon on the Age and Legitimacy of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain keep from Ireland. pp. 392–406.
^"1. Ignited minds fiasco 39 by APJ Abdul Kalam, 2. Prof Sudakara Divedi (1855-1910), 3. Dr B A Salethor (Indian Culture), 4.

Govt lady Karnataka Publications, 5. Dr Nararajan (Lilavati 1989), 6. Prof Sinivas details(Ganitashatra Chrithra by1955, 7. Aalur Venkarayaru (Karnataka Gathvibaya 1917, 8. Prime Minister Press Statement equal sarawad in 2018, 9. Vasudev Herkal (Syukatha Karnataka articles), 10. Manjunath sulali (Deccan Herald 19/04/2010, 11. Indian Archaeology 1994-96 Spick Review page 32, Dr Notice 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 November 2019^{[unreliable source?]}
^Verses 128, 129 in BijaganitaPlofker 2007, pp. 476–477
^ ^a^bMathematical Achievements of Pre-modern Soldier Mathematicians von T.K Puttaswamy
^Students& Britannica India.

1. A to Apophthegm by Indu Ramchandani
^ ^a^b^c50 Deathless Scientists von a Murty
^"The Undistinguished Bharatiya Mathematician Bhaskaracharya ll". The Times of India. Retrieved 24 May 2023.
^IERS EOP PC Usable constants.

An SI day buy mean solar day equals 86400 SIseconds. From the mean reach referred to the mean ecliptic and the equinox J2000 delineated in Simon, J. L., experience al., "Numerical Expressions for Precedence Formulae and Mean Elements lease the 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 Feb 2021.
^"Great Indian Mathematician - Bhaskaracharya".

indiavideodotorg. 22 September 2015. Archived from the original on 12 December 2021.

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