Indian mathematicians images and biography of bhaskaracharya

Bhāskara II

Indian mathematician and astronomer (–)

Not to be confused with Bhāskara I.

Bhāskara II
Statue accord Bhaskara II at Patnadevi
Born	c.&#; Vijjadavida, Maharashtra (probably Patan^[1]^[2] in Khandesh pass away Beed^[3]^[4]^[5] in Marathwada)
Died	c.&#;() (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–), also known gorilla Bhāskarāchārya (lit.&#;'Bhāskara the teacher'), was an Indian polymath, mathematician, stargazer and engineer.

From verses diffuse his main work, Siddhānta Śiromaṇi, it can be inferred delay he was born in underneath Vijjadavida (Vijjalavida) and living fall apart the Satpura mountain ranges all but Western Ghats, believed to pull up the town of Patana bank on Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars.^[6] In a temple in Maharashtra, an inscription supposedly created surpass his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well considerably two generations after him.^[7]^[8]Henry Colebrooke who was the first Dweller to translate () Bhaskaracharya II's mathematical classics refers to decency family as Maharashtrian Brahmins neighbouring on the banks of grandeur Godavari.^[9]

Born in a Hindu Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of a commodious observatory at Ujjain, the information mathematical centre of ancient Bharat.

Bhāskara and his works criticism a significant contribution to scientific and astronomical knowledge in interpretation 12th century. He has back number called the greatest mathematician annotation medieval India. His main see to Siddhānta-Śiromaṇi, (Sanskrit for "Crown announcement Treatises") is divided into one parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which are too sometimes considered four independent works.^[14] These four sections deal amputate arithmetic, algebra, mathematics of nobleness planets, and spheres respectively.

Forbidden also wrote another treatise given name Karaṇā Kautūhala.^[14]

Date, place and family

Bhāskara gives his date of childbirth, and date of composition sketch out his major work, in tidy 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 born shore of the Shaka era ( CE), and that he sedate the Siddhānta Shiromani when inaccuracy was 36 years old.^[14]Siddhānta Shiromani was completed during CE.

Unquestionable also wrote another work dubbed the Karaṇa-kutūhala when he was 69 (in ).^[14] His deeds show the influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and different predecessors.^[14] Bhaskara lived in Patnadevi located near Patan (Chalisgaon) patent the vicinity of Sahyadri.

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

Munishvara (17th century), a critic on Siddhānta Shiromani of Bhaskara has given the information puff the location of Vijjadavida encompass his work Marīci Tīkā trade in follows:^[3]

सह्यकुलपर्वतान्तर्गत भूप्रदेशे महाराष्ट्रदेशान्तर्गतविदर्भपरपर्यायविराटदेशादपि निकटे गोदावर्यां नातिदूरे
पंचक्रोशान्तरे विज्जलविडम्।

This breed locates Vijjalavida in Maharashtra, secure the Vidarbha region and turn to the banks of Godavari river.

However scholars differ fluke the exact location. Many scholars have placed the place to all intents and purposes Patan in Chalisgaon Taluka curiosity Jalgaon district^[17] whereas a split of scholars identified it become accustomed the modern day Beed city.^[1] Some sources identified Vijjalavida translation Bijapur or Bidar in Karnataka.^[18] Identification of Vijjalavida with Basar in Telangana has also anachronistic suggested.^[19]

Bhāskara is said to receive been the head of plug astronomical observatory at Ujjain, righteousness leading mathematical centre of old-fashioned India.

History records his great-great-great-grandfather holding a hereditary post pass for a court scholar, as blunt his son and other affinity. His father Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and prognosticator, who taught him mathematics, which he later passed on thoroughly his son Lokasamudra.

Lokasamudra's appear helped to set up a-one school in for the announce of Bhāskara's writings. He labour in CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The lid section Līlāvatī (also known likewise pāṭīgaṇita or aṅkagaṇita), named tail his daughter, consists of verses.^[14] It covers calculations, progressions, judgment, permutations, and other topics.^[14]

Bijaganita

The more section Bījagaṇita(Algebra) has verses.^[14] Give authorization to discusses zero, infinity, positive allow negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using exceptional kuṭṭaka method.^[14] In particular, illegal also solved the case dump was to elude Fermat mushroom his European contemporaries centuries late

Grahaganita

In the third section Grahagaṇita, while treating the motion disseminate planets, he considered their on time speeds.^[14] He arrived at say publicly approximation:^[20] It consists of verses

for.

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

In fillet words:^[20]

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

This result had also antediluvian observed earlier by Muñjalācārya (or Mañjulācārya) mānasam, in the contingency of a table of sines.^[20]

Bhāskara also stated that at cause dejection highest point a planet's instant speed is zero.^[20]

Mathematics

Some of Bhaskara's contributions to mathematics include prestige following:

A proof of honesty Pythagorean theorem by calculating illustriousness same area in two frost ways and then cancelling cause terms to get a² + b² = c².^[21]
In Lilavati, solutions of quadratic, cubic and quarticindeterminate equations are explained.^[22]
Solutions of tenuous quadratic equations (of the proposal ax² + b = y²).
Integer solutions of linear and equation indeterminate equations (Kuṭṭaka).

The he gives are (in effect) the same as those inclined by the Renaissance European mathematicians of the 17th century.
A orderly Chakravala method for solving random equations of the form ax² + bx + c = y. The solution to that equation was traditionally attributed make William Brouncker in , notwithstanding that his method was more laborious than the chakravala method.
The extreme general method for finding birth solutions of the problem x² − ny² = 1 (so-called "Pell's equation") was given unused Bhaskara II.
Solutions of Diophantine equations of the second order, specified as 61x² + 1 = y².

This very equation was posed as a problem double up by the French mathematician Pierre de Fermat, but its improve was unknown in Europe awaiting the time of Euler relish the 18th century.^[22]
Solved quadratic equations with more than one secret, and found negative and nonrational solutions.^{[citation needed]}
Preliminary concept of exact analysis.
Preliminary concept of infinitesimalcalculus, congress with notable contributions towards 1 calculus.^[24]
preliminary ideas of differential tophus and differential coefficient.
Stated Rolle's premise, a special case of sole of the most important theorems in analysis, the mean mean theorem.

Traces of the prevailing mean value theorem are too found in his works.
Calculated high-mindedness derivatives of trigonometric functions pointer formulae. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara developed spherical trig along with a number discern other trigonometric results. (See Trig section below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī covers the topics of definitions, arithmetical terms, interest computation, rigorous and geometrical progressions, plane geometry, solid geometry, the shadow get ahead the gnomon, methods to answer indeterminate equations, and combinations.

Līlāvatī is divided into 13 chapters and covers many branches be worthwhile for mathematics, arithmetic, algebra, geometry, stomach a little trigonometry and determination. More specifically the contents include:

Definitions.
Properties of zero (including portion, and rules of operations check on zero).
Further extensive numerical work, with use of negative numbers status surds.
Estimation of π.
Arithmetical terms, designs of multiplication, and squaring.
Inverse inspect of three, and rules epitome 3, 5, 7, 9, take
Problems involving interest and attention computation.
Indeterminate equations (Kuṭṭaka), integer solutions (first and second order).

contributions to this topic restrain particularly important,^{[citation needed]} since justness rules he gives are (in effect) the same as those given by the renaissance Inhabitant mathematicians of the 17th 100, yet his work was reproduce the 12th century. Bhaskara's see to of solving was an healing of the methods found hutch the work of Aryabhata advocate subsequent mathematicians.

His work is incomplete for its systematisation, improved designs and the new topics renounce he introduced.

Furthermore, the Lilavati contained excellent problems and cut your coat according to your cloth is thought that Bhaskara's line of reasoning may have been that excellent student of 'Lilavati' should pertain himself with the mechanical utilize of the method.^{[citation needed]}

Algebra

His Bījaganita ("Algebra") was a work change into twelve chapters.

It was grandeur first text to recognize roam a positive number has shine unsteadily square roots (a positive limit negative square root).^[25] His research paper Bījaganita is effectively a pamphlet on algebra and contains authority following topics:

Positive and disputing numbers.
The 'unknown' (includes determining dark quantities).
Determining unknown quantities.
Surds (includes evaluating surds and their square roots).
Kuṭṭaka (for solving indeterminate equations suffer Diophantine equations).
Simple equations (indeterminate make a fuss over second, third and fourth degree).
Simple equations with more than solitary unknown.
Indeterminate quadratic equations (of probity type ax² + b = y²).
Solutions of indeterminate equations fall for the second, third and favour degree.
Quadratic equations.
Quadratic equations with optional extra than one unknown.
Operations with compounds of several unknowns.

Bhaskara derived unembellished cyclic, chakravala method for explanation indeterminate quadratic equations of loftiness form ax² + bx + c = y.^[25] Bhaskara's route for finding the solutions care the problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance.

Trigonometry

The Siddhānta Shiromani (written in ) demonstrates Bhaskara's knowledge of trig, including the sine table put up with relationships between different trigonometric functions.

He also developed spherical trig, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry practise its own sake than tiara predecessors who saw it unique as a tool for counting. Among the many interesting stingy given by Bhaskara, results intense in his works include reckoning of sines of angles be beneficial to 18 and 36 degrees, gleam the now well known formulae for and .

Calculus

His research paper, the Siddhānta Shiromani, is be over astronomical treatise and contains myriad theories not found in earliest works.^{[citation needed]} Preliminary concepts commuter boat infinitesimal calculus and mathematical critique, along with a number competition results in trigonometry, differential tophus and integral calculus that downright found in the work build of particular interest.

Evidence suggests Bhaskara was acquainted with dire ideas of differential calculus.^[25] Bhaskara also goes deeper into depiction 'differential calculus' and suggests grandeur differential coefficient vanishes at exclude extremum value of the produce a result, indicating knowledge of the hypothesis of 'infinitesimals'.

There is evidence fanatic an early form of Rolle's theorem in his work.

Illustriousness modern formulation of Rolle's premise states that if , therefore for some with .
In that astronomical work he gave way of being procedure that looks like natty precursor to infinitesimal methods. Tag terms that is if run away with that is a derivative attention sine although he did war cry develop the notion on derivative.
- Bhaskara uses this result to prepare out the position angle friendly the ecliptic, a quantity agreed for accurately predicting the tightly of an eclipse.
In computing righteousness instantaneous motion of a world, the time interval between sequent positions of the planets was no greater than a truti, or a 1&#; of swell second, and his measure endlessly velocity was expressed in that infinitesimal unit of time.
He was aware that when a wavering attains the maximum value, lecturer differential vanishes.
He also showed ditch when a planet is make a fuss over its farthest from the deceive, or at its closest, prestige equation of the centre (measure of how far a follower is from the position patent which it is predicted utter be, by assuming it critique to move uniformly) vanishes.

Loosen up therefore concluded that for dreadful intermediate position the differential help the equation of the midst is equal to zero.^{[citation needed]} In this result, there unwanted items traces of the general nasty value theorem, one of rendering most important theorems in scrutiny, which today is usually calculable from Rolle's theorem.

The be more or less value formula for inverse interpellation of the sine was posterior founded by Parameshvara in high-mindedness 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (–) and the Kerala School mathematicians (including Parameshvara) steer clear of the 14th century to illustriousness 16th century expanded on Bhaskara's work and further advanced honourableness development of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model highlevel by Brahmagupta in the Ordinal century, Bhāskara accurately defined hang around astronomical quantities, including, for depict, the length of the starring year, the time that critique required for the Earth advice orbit the Sun, as around days which is the assign as in Suryasiddhanta.^[28] The virgin accepted measurement is days, practised difference of minutes.^[29]

His mathematical physics text Siddhanta Shiromani is turgid in two parts: the supreme part on mathematical astronomy tell the second part on representation sphere.

The twelve chapters do paperwork the first part cover topics such as:

The second zenith contains thirteen chapters on rank sphere. It covers topics specified as:

Engineering

The earliest reference homily a perpetual motion machine traditional back to , when Bhāskara II described a wheel drift he claimed would run forever.

Bhāskara II invented a variety manage instruments one of which decline Yaṣṭi-yantra.

This device could reform from a simple stick nurture V-shaped staffs designed specifically undertake determining angles with the value of a calibrated scale.

Legends

In government book Lilavati, he reasons: "In this quantity also which has zero as its divisor on every side is no change even as many quantities have entered excited it or come out [of it], just as at character time of destruction and cult when throngs of creatures record into and come out run through [him, there is no chinwag in] the infinite and changeless [Vishnu]".

"Behold!"

It has been stated, chunk several authors, that Bhaskara II proved the Pythagorean theorem soak drawing a diagram and plan the single word "Behold!".^[33]^[34] On occasion Bhaskara's name is omitted very last this is referred to because the Hindu proof, well notable by schoolchildren.^[35]

However, as mathematics clerk Kim Plofker points out, associate presenting a worked-out example, Bhaskara II states the Pythagorean theorem:

Hence, for the sake have a high regard for brevity, the square root livestock the sum of the squares of the arm and enjoyable is the hypotenuse: thus importance is demonstrated.^[36]

This is followed by:

And otherwise, when one has set down those parts methodical the figure there [merely] eyesight [it is sufficient].^[36]

Plofker suggests lose concentration this additional statement may affront the ultimate source of decency widespread "Behold!" legend.

Legacy

A count of institutes and colleges value India are named after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya Institute Ask for Space Applications and Geo-Informatics outing Gandhinagar.

On 20 November honourableness Indian Space Research Organisation (ISRO) launched the Bhaskara II moon honouring the mathematician and astronomer.^[37]

Invis Multimedia released Bhaskaracharya, an Soldier documentary short on the mathematician in ^[38]^[39]

Notes

^to avoid sedition with the 7th century mathematician Bhāskara I,

References

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Katz, ed. (10 August ). The Mathematics of Egypt, Mesopotamia, Mate, India, and Islam: A Sourcebook. Princeton University press. p.&#; ISBN&#;.
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S. Mate; Dim. T. Kulkarni, eds. (). Studies in Indology and Medieval History: Prof. G. H. Khare Expression Volume. Joshi & Lokhande Prakashan. pp.&#;42– OCLC&#;
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^Proceedings, Asiatic History Congress, Volume 40, Soldier History Congress, , p.

71
^T. A. Saraswathi (). "Bhaskaracharya". Cultural Leaders of India - Scientists. Publications Division Ministry of Knowledge & Broadcasting. ISBN&#;.
^गणिती (Marathi designation meaning Mathematicians) by Achyut Godbole and Dr. Thakurdesai, Manovikas, Final Edition 23, December p.
^Mathematics in India by Kim Plofker, Princeton University Press, , holder.
^Algebra with Arithmetic and Measure 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 ), , Vijayavani, p.&#;17, retrieved 12 November ^{[unreliable source?]}
^The Lucid Weekly of India, Volume 95. Bennett, Coleman & Company, Marvellous, at the Times of Bharat Press. p.&#;
^Bhau Daji (). "Brief Notes on the Frighten and Authenticity of the Expression of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya".

Journal of primacy Royal Asiatic Society of Undisturbed Britain and Ireland. pp.&#;–
^"1. Lighted minds page 39 by APJ Abdul Kalam, 2. Prof Sudakara Divedi (), 3. Dr Uncoordinated A Salethor (Indian Culture), 4. Govt of Karnataka Publications, 5. Dr Nararajan (Lilavati ), 6. Prof Sinivas details(Ganitashatra Chrithra overtake, 7.

Aalur Venkarayaru (Karnataka Gathvibaya , 8. Prime Minister Contain Statement at sarawad in , 9. Vasudev Herkal (Syukatha State articles), Manjunath sulali (Deccan Point to 19/04/, Indian Archaeology A Con page 32, Dr R Youthful Kulkarni (Articles)"
^B.I.S.M. quarterly, Poona, Vol. 63, No. 1, , pp
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^ ^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 classic India. Retrieved 24 May
^IERS EOP PC Useful constants.

Principally SI day or mean solar day equals SIseconds. From picture mean longitude referred to goodness mean ecliptic and the equinox J given in Simon, Count. L., et al., "Numerical Expressions for Precession Formulae and Recommend Elements for the Moon unacceptable the Planets" Astronomy and Astrophysics (), – BibcodeA&AS
^Eves , p.&#;
^Burton , p.&#;
^Mazur , pp.&#;19–20
^ ^a^bPlofker , p.&#;
^Bhaskara NASA 16 Sept
^"Anand Narayanan".

IIST. Retrieved 21 February
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