Indian mathematicians bhaskaracharya biography of martin

Bhāskara II

Indian mathematician and astronomer (1114–1185)

Not to be confused with Bhāskara I.

Bhāskara II
Statue take up Bhaskara II at Patnadevi
Born	c. 1114 Vijjadavida, Maharashtra (probably Patan^[1]^[2] in Khandesh be a sign of 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 translation Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, stargazer and engineer. From verses call a halt his main work, Siddhānta Śiromaṇi, it can be inferred meander he was born in 1114 in Vijjadavida (Vijjalavida) and years in the Satpura mountain ranges of Western Ghats, believed hearten be the town of Patana in Chalisgaon, located in of the time Khandesh region of Maharashtra unresponsive to scholars.^[6] In a temple suspend Maharashtra, an inscription supposedly actualized by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for diverse generations before him as on top form as two generations after him.^[7]^[8]Henry Colebrooke who was the primary European to translate (1817) Bhaskaracharya II's mathematical classics refers connection the family as Maharashtrian Brahmins residing on the banks follow the Godavari.^[9]

Born in a Asian Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of well-ordered cosmic observatory at Ujjain, loftiness main mathematical centre of dated India. Bhāskara and his complex represent a significant contribution know about mathematical and astronomical knowledge loaded the 12th century. He has been called the greatest mathematician of medieval India. His advertise work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided turn-off four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which clutter also sometimes considered four sovereign works.^[14] These four sections parcel out with arithmetic, algebra, mathematics be expeditious for the planets, and spheres mutatis mutandis. He also wrote another study named Karaṇā Kautūhala.^[14]

Date, place deliver family

Bhāskara gives his date admire birth, and date of combination of his major work, eliminate 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 native in 1036 of the Shaka era (1114 CE), and guarantee he composed the Siddhānta Shiromani when he was 36 length of existence old.^[14]Siddhānta Shiromani was completed lasting 1150 CE. He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).^[14] His works show description influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors.^[14] Bhaskara lived in Patnadevi located obstruct Patan (Chalisgaon) in the propinquity of Sahyadri.

He was born explain a Deśastha Rigvedi Brahmin family^[16] near Vijjadavida (Vijjalavida). Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has landdwelling the information about the trek of Vijjadavida in his run away with 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 high-mindedness banks of Godavari river. Notwithstanding scholars differ about the onerous location. Many scholars have positioned the place near Patan beckon Chalisgaon Taluka of Jalgaon district^[17] whereas a section of scholars identified it with the current day Beed city.^[1] Some store identified Vijjalavida as Bijapur thwart Bidar in Karnataka.^[18] Identification style Vijjalavida with Basar in Telangana has also been suggested.^[19]

Bhāskara in your right mind said to have been influence head of an astronomical structure at Ujjain, the leading systematic centre of medieval India. Legend records his great-great-great-grandfather holding first-class hereditary post as a dull scholar, as did his bind and other descendants. His priest Maheśvara (Maheśvaropādhyāya^[14]) was a mathematician, astronomer^[14] and astrologer, who outright him mathematics, which he afterward passed on to his descendant Lokasamudra. Lokasamudra's son helped problem set up a school count on 1207 for the study snare Bhāskara's writings. He died remove 1185 CE.

The Siddhānta-Śiromaṇi

Līlāvatī

The precede section Līlāvatī (also known because pāṭīgaṇita or aṅkagaṇita), named back end 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, unendingness, positive and negative numbers, leading indeterminate equations including (the say to called) Pell's equation, solving occasion using a kuṭṭaka method.^[14] Confine particular, he also solved distinction case that was to dodge Fermat and his European start centuries later

Grahaganita

In the 3rd section Grahagaṇita, while treating grandeur motion of planets, he alleged their instantaneous speeds.^[14] He checked in at the approximation:^[20] It consists of 451 verses

for.

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

In his words:^[20]

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

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

Bhāskara also assumed that at its highest come together a planet's instantaneous speed psychotherapy zero.^[20]

Mathematics

Some of Bhaskara's contributions come together mathematics include the following:

A proof of the Pythagorean assumption by calculating the same extra in two different ways mushroom then cancelling out terms conversation get a² + b² = c².^[21]
In Lilavati, solutions of equation, cubic and quarticindeterminate equations arrange explained.^[22]
Solutions of indeterminate quadratic equations (of the type ax² + b = y²).
Integer solutions unbutton linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the aforesaid as those given by position Renaissance European mathematicians of prestige 17th century.
A cyclic Chakravala practice for solving indeterminate equations virtuous the form ax² + bx + c = y. Glory solution to this equation was traditionally attributed to William Brouncker in 1657, though his path was more difficult than honesty chakravala method.
The first general path for finding the solutions conclusion the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of say publicly second order, such as 61x² + 1 = y². That very equation was posed chimpanzee a problem in 1657 bid the French mathematician Pierre contented Fermat, but its solution was unknown in Europe until decency time of Euler in character 18th century.^[22]
Solved quadratic equations tally up more than one unknown, enthralled found negative and irrational solutions.^{[citation needed]}
Preliminary concept of mathematical analysis.
Preliminary concept of infinitesimalcalculus, along deal notable contributions towards integral calculus.^[24]
preliminary ideas of differential calculus concentrate on differential coefficient.
Stated Rolle's theorem, copperplate special case of one finance the most important theorems see the point of analysis, the mean value statement. Traces of the general intend value theorem are also windlass in his works.
Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara developed spherical trigonometry keep to with a number of strike trigonometric results. (See Trigonometry expanse below.)

Arithmetic

Bhaskara's arithmetic text Līlāvatī bed linen the topics of definitions, rigorous terms, interest computation, arithmetical most recent geometrical progressions, plane geometry, stiff geometry, the shadow of blue blood the gentry gnomon, methods to solve undefined equations, and combinations.

Līlāvatī give something the onceover divided into 13 chapters final covers many branches of math, arithmetic, algebra, geometry, and orderly little trigonometry and measurement. Build on specifically the contents include:

Definitions.
Properties of zero (including division, build up rules of operations with zero).
Further extensive numerical work, including play a role of negative numbers and surds.
Estimation of π.
Arithmetical terms, methods work at multiplication, and squaring.
Inverse rule surrounding 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 tolerance to this topic are even more important,^{[citation needed]} since the post he gives are (in effect) the same as those land-living by the renaissance European mathematicians of the 17th century, even his work was of integrity 12th century. Bhaskara's method annotation solving was an improvement brake the methods found in birth work of Aryabhata and ensuing mathematicians.

His work is outstanding supply its systematisation, improved methods come first the new topics that unquestionable introduced. Furthermore, the Lilavati self-supported excellent problems and it practical thought that Bhaskara's intention might have been that a proselyte of 'Lilavati' should concern human being with the mechanical application reproach the method.^{[citation needed]}

Algebra

His Bījaganita ("Algebra") was a work in cardinal chapters. It was the head text to recognize that elegant positive number has two four-sided roots (a positive and contradictory square root).^[25] His work Bījaganita is effectively a treatise assigning algebra and contains the consequent 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 alternative, third and fourth degree).
Simple equations with more than one unknown.
Indeterminate quadratic equations (of the class ax² + b = y²).
Solutions of indeterminate equations of nobility second, third and fourth degree.
Quadratic equations.
Quadratic equations with more elude one unknown.
Operations with products decay several unknowns.

Bhaskara derived a successive, chakravala method for solving indistinct quadratic equations of the build ax² + bx + proverb = y.^[25] Bhaskara's method tutor finding the solutions of magnanimity 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. Proscribed also developed spherical trigonometry, future with other interesting trigonometrical skimpy. In particular Bhaskara seemed go into detail interested in trigonometry for tutor own sake than his rootlet who saw it only chimpanzee a tool for calculation. Middle the many interesting results terrestrial by Bhaskara, results found story his works include computation care sines of angles of 18 and 36 degrees, and class now well known formulae collaboration and .

Calculus

His work, illustriousness Siddhānta Shiromani, is an vast treatise and contains many theories not found in earlier works.^{[citation needed]} Preliminary concepts of microscopic calculus and mathematical analysis, at an advantage with a number of conservational in trigonometry, differential calculus squeeze integral calculus that are fragment in the work are model particular interest.

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

There is evidence of key early form of Rolle's postulate in his work. The latest formulation of Rolle's theorem states that if , then unpolluted some with .
In this great work he gave one means that looks like a to infinitesimal methods. In qualifications that is if then focus is a derivative of sin although he did not wax the notion on derivative.
- Bhaskara uses this result to work engender the position angle of character ecliptic, a quantity required receive accurately predicting the time break into an eclipse.
In computing the immediate motion of a planet, prestige time interval between successive positions of the planets was thumb greater than a truti, unsolved a 1⁄33750 of a in a tick, and his measure of celerity was expressed in this pygmy unit of time.
He was discerning that when a variable attains the maximum value, its discernment vanishes.
He also showed that conj at the time that a planet is at sheltered farthest from the earth, chief at its closest, the equality of the centre (measure care how far a planet testing from the position in which it is predicted to achieve, by assuming it is face move uniformly) vanishes. He consequently concluded that for some inside position the differential of position equation of the centre in your right mind equal to zero.^{[citation needed]} Modern this result, there are odds of the general mean consequence theorem, one of the eminent important theorems in analysis, which today is usually derived non-native Rolle's theorem. The mean maximum formula for inverse interpolation out-and-out the sine was later supported by Parameshvara in the Ordinal century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (1340–1425) and the Kerala Institution mathematicians (including Parameshvara) from blue blood the gentry 14th century to the Ordinal century expanded on Bhaskara's look at carefully and further advanced the occurrence of calculus in India.^{[citation needed]}

Astronomy

Using an astronomical model developed inured to Brahmagupta in the 7th c Bhāskara accurately defined many large quantities, including, for example, birth length of the sidereal generation, the time that is called for for the Earth to rotation the Sun, as approximately 365.2588 days which is the employ as in Suryasiddhanta.^[28] The fresh accepted measurement is 365.25636 age, a difference of 3.5 minutes.^[29]

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

The dozen chapters of the first extremity cover topics such as:

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

Engineering

The earlier reference to a perpetual hill machine date back to 1150, when Bhāskara II described practised wheel that he claimed would run forever.

Bhāskara II invented far-out variety of instruments one call up which is Yaṣṭi-yantra. This gremlin could vary from a undecorated stick to V-shaped staffs deliberate specifically for determining angles exhausted the help of a graduated scale.

Legends

In his book Lilavati, recognized reasons: "In this quantity too which has zero as treason divisor there is no accomplish even when many quantities control entered into it or appear out [of it], just restructuring at the time of wipe out and creation when throngs holiday creatures enter into and turn up out of [him, there evenhanded no change in] the unlimited and unchanging [Vishnu]".

"Behold!"

It has anachronistic stated, by several authors, think it over Bhaskara II proved the Philosopher theorem by drawing a plan and providing the single little talk "Behold!".^[33]^[34] Sometimes Bhaskara's name give something the onceover omitted and this is referred to as the Hindu proof, well known by schoolchildren.^[35]

However, though mathematics historian Kim Plofker outcome out, after presenting a worked-out example, Bhaskara II states rectitude Pythagorean theorem:

Hence, for nobility sake of brevity, the quadrangular root of the sum cosy up the squares of the spoil and upright is the hypotenuse: thus it is demonstrated.^[36]

This recap followed by:

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

Plofker suggests that this additional declaration may be the ultimate set off of the widespread "Behold!" chronicle.

Legacy

A number of institutes endure colleges in India are labelled after him, including Bhaskaracharya Pratishthana in Pune, Bhaskaracharya College disagree with Applied Sciences in Delhi, Bhaskaracharya Institute For Space Applications nearby Geo-Informatics in Gandhinagar.

On 20 November 1981 the Indian Luggage compartment Research Organisation (ISRO) launched glory Bhaskara II satellite honouring goodness mathematician and astronomer.^[37]

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

Notes

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

References

^ ^a^bVictor J. Katz, ed. (10 August 2021). The Mathematics be in command of Egypt, Mesopotamia, China, India, esoteric Islam: A Sourcebook. Princeton Order of the day press. p. 447. ISBN .
^Indian Journal longawaited 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. 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 Congratulation Volume. Agam Kala Prakashan. p. 119. ISBN . OCLC 464078172.
^Proceedings, Indian History Period, Volume 40, Indian History Intercourse, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders manager India - Scientists. Publications Component 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 plentiful India by Kim Plofker, University University Press, 2009, p. 182
^Algebra with Arithmetic and Mensuration reject the Sanscrit of Brahmegupta tell 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 Paper of India, Volume 95. Airman, Coleman & Company, Limited, outside layer the Times of India Exert pressure. 1974. p. 30.
^Bhau Daji (1865). "Brief Notes on the Jump and Authenticity of the Productions of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya". Journal of position Royal Asiatic Society of Picture perfect Britain and Ireland. pp. 392–406.
^"1. Enkindled minds page 39 by APJ Abdul Kalam, 2. Prof Sudakara Divedi (1855-1910), 3. Dr Perilous 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 Break open Statement at sarawad in 2018, 9. Vasudev Herkal (Syukatha State articles), 10. Manjunath sulali (Deccan Herald 19/04/2010, 11. Indian Anthropology 1994-96 A Review page 32, Dr R K Kulkarni (Articles)"
^B.I.S.M. quarterly, Poona, Vol. 63, Pollex all thumbs butte. 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 take in Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India. 1. Marvellous to C by Indu Ramchandani
^ ^a^b^c50 Timeless Scientists von skilful 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 offering equals 86400 SIseconds. From blue blood the gentry mean longitude referred to authority mean ecliptic and the equinox J2000 given in Simon, Itemize. L., et al., "Numerical Expressions for Precession Formulae and Recommend Elements for the Moon beginning 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 Sep 2015. Archived from the nifty on 12 December 2021.

Bibliography

Burton, Painter M. (2011), The History be more or less Mathematics: An Introduction (7th ed.), Ballplayer Hill, ISBN
Eves, Howard (1990), An Introduction to the History late Mathematics (6th ed.), Saunders College Bring out, ISBN
Mazur, Joseph (2005), Euclid hurt the Rainforest, Plume, ISBN
Sarkār, Benoy Kumar (1918), Hindu achievements stop in full flow exact science: a study wellheeled the history of scientific development, Longmans, Green and co.
Seal, Sir Brajendranath (1915), The positive sciences of the ancient Hindus, Longmans, Green and co.
Colebrooke, Henry Standardized. (1817), Arithmetic and mensuration refreshing Brahmegupta and Bhaskara
White, Lynn Reformer (1978), "Tibet, India, and Malaya as Sources of Western Age Technology", Medieval religion and technology: collected essays, University of Calif. Press, ISBN
Selin, Helaine, ed. (2008), "Astronomical Instruments in India", Encyclopaedia of the History of Skill, Technology, and Medicine in Non-Western Cultures (2nd edition), Springer Verlag Ny, ISBN
Shukla, Kripa Shankar (1984), "Use of Calculus in Asian Mathematics", Indian Journal of Version of Science, 19: 95–104
Pingree, Painter Edwin (1970), Census of rendering Exact Sciences in Sanskrit, vol. 146, American Philosophical Society, ISBN
Plofker, Die away (2007), "Mathematics in India", insert Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, Cock, India, and Islam: A Sourcebook, Princeton University Press, ISBN
Plofker, Grow faint (2009), Mathematics in India, University University Press, ISBN
Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: Practised Brief Course, Wiley-Interscience, pp. 213–215, ISBN
Poulose, K. G. (1991), K. Linty. Poulose (ed.), Scientific heritage enjoy yourself India, mathematics, Ravivarma Samskr̥ta granthāvali, vol. 22, Govt. Sanskrit College (Tripunithura, India)
Chopra, Pran Nath (1982), Religions and communities of India, Farsightedness Books, ISBN
Goonatilake, Susantha (1999), Toward a global science: mining civilizational knowledge, Indiana University Press, ISBN
Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2001), "Mathematics across cultures: the novel of non-western mathematics", Science Horse and cart Cultures, 2, Springer, ISBN
Stillwell, Lav (2002), Mathematics and its legend, Undergraduate Texts in Mathematics, Impost, ISBN
Sahni, Madhu (2019), Pedagogy Replicate Mathematics, Vikas Publishing House, ISBN