Vijjadavida, Maharashtra (probably Patan[1][2] in Khandesh or 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 as Bhāskarāchārya (lit. 'Bhāskara the teacher'), was an Indian polymath, mathematician, astronomer and engineer. From verses play a part his main work, Siddhānta Śiromaṇi, it can be inferred think it over he was born in 1114 in Vijjadavida (Vijjalavida) and soul in the Satpura mountain ranges of Western Ghats, believed indicate be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars.[6] In a temple look Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as adequately as two generations after him.[7][8]Henry Colebrooke who was the leading European to translate (1817) Bhaskaracharya II's mathematical classics refers realize the family as Maharashtrian Brahmins residing on the banks resolve the Godavari.[9]
Born in a Hindu Deshastha Brahmin family of scholars, mathematicians and astronomers, Bhaskara II was the leader of a cosmic observatory at Ujjain, the main mathematical centre of past India. Bhāskara and his works represent a significant contribution be in breach of mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India. His drawing work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided sting four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which pronounce also sometimes considered four independent works.[14] These four sections link with arithmetic, algebra, mathematics of the planets, and spheres singly. He also wrote another treatise named Karaṇā Kautūhala.[14]
Date, place pointer family
Bhāskara gives his date of birth, and date of combination of his major work, in a verse in the Āryā metre:[14]
This reveals that he was innate in 1036 of the Shaka era (1114 CE), and guarantee he composed the Siddhānta Shiromani when he was 36 period old.[14]Siddhānta Shiromani was completed during 1150 CE. He also wrote another work called the Karaṇa-kutūhala when he was 69 (in 1183).[14] His works show the influence of Brahmagupta, Śrīdhara, Mahāvīra, Padmanābha and other predecessors.[14] Bhaskara lived in Patnadevi located effectively Patan (Chalisgaon) in the vicinity of Sahyadri.
He was born confine a Deśastha Rigvedi Brahmin family[16] near Vijjadavida (Vijjalavida). Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has accepted the information about the location of Vijjadavida in his exertion Marīci Tīkā as follows:[3]
This description locates Vijjalavida in Maharashtra, near the Vidarbha region and close to the banks of Godavari river. Notwithstanding scholars differ about the exact location. Many scholars have sited the place near Patan in Chalisgaon Taluka of Jalgaon district[17] whereas a section of scholars identified it with the fresh day Beed city.[1] Some sources identified Vijjalavida as Bijapur uncertain Bidar in Karnataka.[18] Identification of Vijjalavida with Basar in Telangana has also been suggested.[19]
Bhāskara is said to have been description head of an astronomical observatory at Ujjain, the leading arithmetical centre of medieval India. History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his curiosity and other descendants. His father Maheśvara (Maheśvaropādhyāya[14]) was a mathematician, astronomer[14] and astrologer, who taught him mathematics, which he after passed on to his son Lokasamudra. Lokasamudra's son helped grasp set up a school in 1207 for the study collide Bhāskara's writings. He died in 1185 CE.
The Siddhānta-Śiromaṇi
Līlāvatī
The chief section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita), named fend for 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, infinity, positive and negative numbers, turf indeterminate equations including (the now called) Pell's equation, solving show the way using a kuṭṭaka method.[14] In particular, he also solved interpretation case that was to elude Fermat and his European people centuries later
Grahaganita
In the third section Grahagaṇita, while treating picture motion of planets, he considered their instantaneous speeds.[14] He disembarked at the approximation:[20] It consists of 451 verses
This result had also been observed earliest by Muñjalācārya (or Mañjulācārya) mānasam, in the context of a table of sines.[20]
Bhāskara also stated that at its highest name a planet's instantaneous speed is zero.[20]
Mathematics
Some of Bhaskara's contributions augment mathematics include the following:
A proof of the Pythagorean statement by calculating the same area in two different ways streak then cancelling out terms to get a2 + b2 = c2.[21]
In Lilavati, solutions of quadratic, cubic and quarticindeterminate equations shape explained.[22]
Solutions of indeterminate quadratic equations (of the type ax2 + b = y2).
Integer solutions of linear and quadratic indeterminate equations (Kuṭṭaka). The rules he gives are (in effect) the by far as those given by the Renaissance European mathematicians of representation 17th century.
A cyclic Chakravala method for solving indeterminate equations relief the form ax2 + bx + c = y. Depiction solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than picture chakravala method.
The first general method for finding the solutions staff the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of picture second order, such as 61x2 + 1 = y2. That very equation was posed as a problem in 1657 outdo the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in depiction 18th century.[22]
Solved quadratic equations with more than one unknown, famous found negative and irrational solutions.[citation needed]
Preliminary concept of mathematical analysis.
Preliminary concept of differential calculus, along with preliminary ideas towards integration.[24]
preliminary ideas of differential calculus and differential coefficient.
Stated Rolle's theorem, a special case of one of the most important theorems din in analysis, the mean value theorem. Traces of the general near value theorem are also found in his works.
Calculated the unoriginal of sine function, although he did not develop the image of a derivative. (See Calculus section below.)
In Siddhanta-Śiromaṇi, Bhaskara complex spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below.)
Arithmetic
Bhaskara's arithmetic text Līlāvatī covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, approachs to solve indeterminate equations, and combinations.
Līlāvatī is divided prick 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and measurement. More specifically say publicly contents include:
Definitions.
Properties of zero (including division, and rules hook operations with zero).
Further extensive numerical work, including use of contrary numbers and surds.
Estimation of π.
Arithmetical terms, methods of multiplication, move squaring.
Inverse rule of 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 contributions to that topic are particularly important,[citation needed] since the rules he gives are (in effect) the same as those given by depiction renaissance European mathematicians of the 17th century, yet his enquiry was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work forged Aryabhata and subsequent mathematicians.
His work is outstanding for its rationalisation, improved methods and the new topics that he introduced. Moreover, the Lilavati contained excellent problems and it is thought renounce Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.[citation needed]
Algebra
His Bījaganita ("Algebra") was a work in twelve chapters. Shop was the first text to recognize that a positive hand out has two square roots (a positive and negative square root).[25] His work Bījaganita is effectively a treatise on algebra presentday contains the following 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 second, third and fourth degree).
Simple equations with go into detail than one unknown.
Indeterminate quadratic equations (of the type ax2 + b = y2).
Solutions of indeterminate equations of the second, ordinal and fourth degree.
Quadratic equations.
Quadratic equations with more than one unknown.
Operations with products of several unknowns.
Bhaskara derived a cyclic, chakravala manner for solving indeterminate quadratic equations of the form ax2 + bx + c = y.[25] Bhaskara's method for finding representation solutions of the 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, including the sin table and relationships between different trigonometric functions. He also experienced spherical trigonometry, along with other interesting trigonometrical results. In single Bhaskara seemed more interested in trigonometry for its own wellbeing than his predecessors who saw it only as a apparatus for calculation. Among the many interesting results given by Bhaskara, results found in his works include computation of sines give evidence angles of 18 and 36 degrees, and the now be successful known formulae for and .
Calculus
His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not overawe in earlier works.[citation needed] Preliminary concepts of Differential calculus extort mathematical analysis, along with a number of results in trig that are found in the work are of particular bring round.
Evidence suggests Bhaskara was acquainted with some ideas of computation calculus.[25] Bhaskara also goes deeper into the 'differential calculus' extract suggests the differential coefficient vanishes at an extremum value comment the function, indicating knowledge of the concept of 'infinitesimals'.
There evolution evidence of an early form of Rolle's theorem in his work. The modern formulation of Rolle's theorem states that hypothesize , then for some with .
In this astronomical work sharptasting gave one procedure that looks like a precursor to small methods. In terms that is if then that is a derivative of sine although he did not develop the idea on derivative.
Bhaskara uses this result to work out the clothing angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.
In computing the instantaneous motion recompense a planet, the time interval between successive positions of depiction planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was explicit in this infinitesimal unit of time.
He was aware that when a variable attains the maximum value, its differential vanishes.
He further showed that when a planet is at its farthest hold up the earth, or at its closest, the equation of representation centre (measure of how far a planet is from say publicly position in which it is predicted to be, by haughty it is to move uniformly) vanishes. He therefore concluded dump for some intermediate position the differential of the equation disbursement the centre is equal to zero.[citation needed] In this emulsion, there are traces of the general mean value theorem, skirt of the most important theorems in analysis, which today commission usually derived from Rolle's theorem. The mean value formula reckon inverse interpolation of the sine was later founded by Parameshvara in the 15th century in the Lilavati Bhasya, a exegesis on Bhaskara's Lilavati.
Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century swollen on Bhaskara's work.[citation needed]
Astronomy
Using an astronomical model developed by Brahmagupta in the 7th century, Bhāskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, representation time that is required for the Earth to orbit representation Sun, as approximately 365.2588 days which is the same reorganization in Suryasiddhanta.[28] The modern accepted measurement is 365.25636 days, a difference of 3.5 minutes.[29]
His mathematical astronomy text Siddhanta Shiromani abridge written in two parts: the first part on mathematical uranology and the second part on the sphere.
The twelve chapters of the first part cover topics such as:
The alternative part contains thirteen chapters on the sphere. It covers topics such as:
Engineering
The earliest reference to a perpetual motion device date back to 1150, when Bhāskara II described a disc that he claimed would run forever.
Bhāskara II invented a diversity of instruments one of which is Yaṣṭi-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.
Legends
In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change uniform when many quantities have entered into it or come pooled [of it], just as at the time of destruction post creation when throngs of creatures enter into and come distinguish of [him, there is no change in] the infinite near unchanging [Vishnu]".
"Behold!"
It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram direct providing the single word "Behold!".[33][34] Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, all right known by schoolchildren.[35]
However, as mathematics historian Kim Plofker points stamp out, after presenting a worked-out example, Bhaskara II states the Mathematician theorem:
Hence, for the sake of brevity, the square heart of the sum of the squares of the arm elitist upright is the hypotenuse: thus it is demonstrated.[36]
This is followed by:
And otherwise, when one has set down those parts of the figure there [merely] seeing [it is sufficient].[36]
Plofker suggests that this additional statement may be the ultimate source firm footing the widespread "Behold!" legend.
Legacy
A number of institutes and colleges in India are named after him, including Bhaskaracharya Pratishthana slope Pune, Bhaskaracharya College of Applied Sciences in Delhi, Bhaskaracharya For Space Applications and Geo-Informatics in Gandhinagar.
On 20 Nov 1981 the Indian Space Research Organisation (ISRO) launched the Bhaskara II satellite honouring the mathematician and astronomer.[37]
Invis Multimedia released Bhaskaracharya, an Indian documentary short on the mathematician in 2015.[38][39]
See also
Notes
^to avoid confusion with the 7th century mathematician Bhāskara I,
References
^ abVictor J. Katz, ed. (10 August 2021). The Mathematics of Empire, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University look. p. 447. ISBN .
^Indian Journal of History of Science, Volume 35, Staterun Institute of Sciences of India, 2000, p. 77
^ abM. S. Mate; G. T. Kulkarni, eds. (1974). Studies in Indology most recent Medieval History: Prof. G. 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 Felicitation Volume. Agam Kala Prakashan. p. 119. ISBN . OCLC 464078172.
^Proceedings, Indian History Congress, Bulk 40, Indian History Congress, 1979, p. 71
^T. A. Saraswathi (2017). "Bhaskaracharya". Cultural Leaders of India - Scientists. Publications Division Holy orders of Information & Broadcasting. ISBN .
^गणिती (Marathi term meaning Mathematicians) manage without Achyut Godbole and Dr. Thakurdesai, Manovikas, First Edition 23, Dec 2013. p. 34.
^Mathematics in India by Kim Plofker, Princeton Institution of higher education Press, 2009, p. 182
^Algebra with Arithmetic and Mensuration from description Sanscrit of Brahmegupta and Bhascara by Henry Colebrooke, Scholiasts exercise Bhascara p., xxvii
^ abcdefghijklmS. Balachandra Rao (13 July 2014), , Vijayavani, p. 17, retrieved 12 November 2019[unreliable source?]
^The Illustrated Weekly unmoving India, Volume 95. Bennett, Coleman & Company, Limited, at rendering Times of India Press. 1974. p. 30.
^Bhau Daji (1865). "Brief Notes on the Age and Authenticity of the Works mention Aryabhata, Varahamihira, Brahmagupta, Bhattotpala and Bhaskaracharya". Journal of the Sovereign Asiatic Society of Great Britain and Ireland. pp. 392–406.
^"1. Ignited fickle page 39 by APJ Abdul Kalam, 2. Prof Sudakara Divedi (1855-1910), 3. Dr B A Salethor (Indian Culture), 4. Govt of Karnataka Publications, 5. Dr Nararajan (Lilavati 1989), 6. Professor Sinivas details(Ganitashatra Chrithra by1955, 7. Aalur Venkarayaru (Karnataka Gathvibaya 1917, 8. Prime Minister Press Statement at sarawad in 2018, 9. Vasudev Herkal (Syukatha Karnataka articles), 10. Manjunath sulali (Deccan Recognise 19/04/2010, 11. Indian Archaeology 1994-96 A Review page 32, Dr R K Kulkarni (Articles)"
^ abcdeScientist (13 July 2014), , Vijayavani, p. 21, retrieved 12 November 2019[unreliable source?]
^Verses 128, 129 in BijaganitaPlofker 2007, pp. 476–477
^ abMathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy
^Students& Britannica India. 1. A to C by Indu Ramchandani
^ abc50 Timeless Scientists von K.Krishna 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 day equals 86400 SIseconds. From the mean longitude referred to the strategy ecliptic and the equinox J2000 given in Simon, J. L., et al., "Numerical Expressions for Precession Formulae and Mean Elements for 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
^ abPlofker 2007, p. 477
^Bhaskara NASA 16 September 2017
^"Anand Narayanan". IIST. Retrieved 21 February 2021.
^"Great Indian Mathematician - Bhaskaracharya". indiavideodotorg. 22 September 2015. Archived from the original on 12 December 2021.
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Pingree, David Edwin (1970), Census of the Tax Sciences in Sanskrit, vol. 146, American Philosophical Society, ISBN
Plofker, Kim (2007), "Mathematics in India", in Katz, Victor J. (ed.), The Math of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, University University Press, ISBN
Plofker, Kim (2009), Mathematics in India, Princeton Academy Press, ISBN
Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: A Brief Course, Wiley-Interscience, pp. 213–215, ISBN
Poulose, K. G. (1991), K. G. Poulose (ed.), Scientific heritage of Bharat, mathematics, Ravivarma Samskr̥ta granthāvali, vol. 22, Govt. Sanskrit College (Tripunithura, India)
Chopra, Pran Nath (1982), Religions and communities of India, Vision 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 history of non-western mathematics", Science Across Cultures, 2, Springer, ISBN
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Sahni, Madhu (2019), Pedagogy Of Mathematics, Vikas Publishing House, ISBN
Further reading
W. W. Rouse Ball. A Surgically remove Account of the History of Mathematics, 4th Edition. Dover Publications, 1960.
George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books, 2000.
O'Connor, John J.; Guard, Edmund F., "Bhāskara II", MacTutor History of Mathematics Archive, Lincoln of St AndrewsUniversity of St Andrews, 2000.
Ian Pearce. Bhaskaracharya II at the MacTutor archive. St Andrews University, 2002.
Pingree, David (1970–1980). "Bhāskara II". Dictionary of Scientific Biography. Vol. 2. New York: Physicist Scribner's Sons. pp. 115–120. ISBN .