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| {{Distinguish|Bhāskara I}}
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| '''Bhāskara'''{{sfn|Pingree|1970|p=299}} (also known as '''Bhāskarāchārya''' ("Bhāskara the teacher") and as '''Bhāskara II''' to avoid confusion with [[Bhāskara I]]) (1114–1185), was an [[India]]n [[Indian mathematics|mathematician]] and [[astronomer]]. He was born near Vijjadavida (Bijapur in modern Karnataka). Bhāskara is said to have been the head of an [[astronomy|astronomical]] observatory at [[Ujjain]], the leading mathematical center of medieval India. He lived in the [[Western Ghats|Sahyadri]] region (Patnadevi, in Jalgaon district, Maharashtra).{{sfn|Pingree|1970|p=299}}
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| Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.{{sfn|Chopra|1982|pp=52–54}} His main work ''[[Siddhānta Shiromani]],'' ([[Sanskrit]] for "Crown of treatises,"{{sfn|Plofker|2009|p=71}}) is divided into four parts called ''[[Lilavati|Lilāvati]]'', ''[[Bijaganita]]'', ''Grahaganita'' and ''Golādhyāya''.{{sfn|Poulose|1991|p=79}} These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named [[Karana Kautoohala]].
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| Bhāskara's work on [[calculus]] predates [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]] by over half a millennium.{{sfn|Seal|1915|p=80}}{{sfn|Sarkar|1918|p=23}} He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.{{sfn|Goonatilake|1999|p=134}}
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| ==Family==
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| History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara{{sfn|Pingree|1970|p=299}} was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings.
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| ==Mathematics==
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| Some of Bhaskara's contributions to mathematics include the following:
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| * A proof of the [[Pythagorean theorem]] by calculating the same [[area]] in two different ways and then canceling out terms to get ''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>.
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| * In ''Lilavati'', solutions of [[quadratic equation|quadratic]], [[cubic function|cubic]] and [[quartic equation|quartic]] [[indeterminate equation]]s are explained.<ref>Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy</ref>
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| * Solutions of indeterminate quadratic equations (of the type ''ax''<sup>2</sup> + ''b'' = ''y''<sup>2</sup>).
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| * Integer solutions of linear and quadratic indeterminate equations (''Kuttaka''). The rules he gives are (in effect) the same as those given by the [[Renaissance]] European mathematicians of the 17th century
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| * A cyclic [[Chakravala method]] for solving indeterminate equations of the form ''ax''<sup>2</sup> + ''bx'' + ''c'' = ''y''. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the ''chakravala'' method.
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| * The first general method for finding the solutions of the problem ''x''<sup>2</sup> − ''ny''<sup>2</sup> = 1 (so-called "[[Pell's equation]]") was given by Bhaskara II.{{sfn|Stillwell1999|p=74}}
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| * Solutions of [[Diophantine equation]]s of the second order, such as 61''x''<sup>2</sup> + 1 = ''y''<sup>2</sup>. This very equation was posed as a problem in 1657 by the [[France|French]] mathematician [[Pierre de Fermat]], but its solution was unknown in Europe until the time of [[Leonhard Euler|Euler]] in the 18th century.<ref>Mathematical Achievements of Pre-modern Indian Mathematicians von T.K Puttaswamy</ref>
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| * Solved quadratic equations with more than one unknown, and found [[negative number|negative]] and [[irrational number|irrational]] solutions.{{citation needed|date=September 2012}}
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| * Preliminary concept of [[mathematical analysis]].
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| * Preliminary concept of [[infinitesimal]] [[calculus]], along with notable contributions towards [[integral|integral calculus]].<ref>Students& Britannica India. 1. A to C by Indu Ramchandani</ref>
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| * Conceived [[differential calculus]], after discovering the [[derivative]] and [[differential (calculus)|differential]] coefficient.
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| * Stated [[Rolle's theorem]], a special case of one of the most important theorems in analysis, the [[mean value theorem]]. Traces of the general mean value theorem are also found in his works.
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| * Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)
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| * In ''Siddhanta Shiromani'', Bhaskara developed [[spherical trigonometry]] along with a number of other [[trigonometry|trigonometric]] results. (See Trigonometry section below.)
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| ===Arithmetic=== | |
| Bhaskara's [[arithmetic]] text ''[[Lilavati|Leelavati]]'' covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, [[plane (geometry)|plane geometry]], [[solid geometry]], the shadow of the [[gnomon]], methods to solve [[Indeterminate (variable)|indeterminate]] equations, and [[combination]]s.
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| ''Lilavati'' is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:
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| * Definitions.
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| * Properties of [[0 (number)|zero]] (including [[Division (mathematics)|division]], and rules of operations with zero).
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| * Further extensive numerical work, including use of [[negative number]]s and [[Nth root|surds]].
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| * Estimation of [[Pi|π]].
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| * Arithmetical terms, methods of [[multiplication]], and [[Square (algebra)|squaring]].
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| * Inverse [[Cross-multiplication|rule of three]], and rules of 3, 5, 7, 9, and 11.
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| * Problems involving [[interest]] and interest computation.
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| * Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important{{citation needed|date=September 2012}}, since the rules he gives are (in effect) the same as those given by the [[renaissance]] European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of [[Aryabhata]] and subsequent mathematicians.
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| His work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the ''Lilavati'' contained excellent recreative problems and it is thought that Bhaskara's intention may have be.
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| ===Algebra===
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| His ''Bijaganita'' ("''[[Algebra]]''") was a work in twelve chapters. It was the first text to recognize that a positive number has two [[square root]]s (a positive and negative square root).<ref>50 Timeless Scientists von K.Krishna Murty</ref> His work ''Bijaganita'' is effectively a treatise on algebra and contains the following topics:
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| * Positive and [[negative number]]s.
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| * [[0 (number)|Zero]].
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| * The 'unknown' (includes determining unknown quantities).
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| * Determining unknown quantities.
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| * [[Nth root#Working with surds|Surd]]s (includes evaluating surds).
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| * ''Kuttaka'' (for solving [[indeterminate equation]]s and [[Diophantine equation]]s).
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| * Simple equations (indeterminate of second, third and fourth degree).
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| * Simple equations with more than one unknown.
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| * Indeterminate [[quadratic equation]]s (of the type ax<sup>2</sup> + b = y<sup>2</sup>).
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| * Solutions of indeterminate equations of the second, third and fourth degree.
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| * Quadratic equations.
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| * Quadratic equations with more than one unknown.
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| * Operations with products of several unknowns.
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| Bhaskara derived a cyclic, [[chakravala method|''chakravala'' method]] for solving indeterminate quadratic equations of the form ax<sup>2</sup> + bx + c = y.<ref>50 Timeless Scientists von K.Krishna Murty</ref> Bhaskara's method for finding the solutions of the problem Nx<sup>2</sup> + 1 = y<sup>2</sup> (the so-called "[[Pell's equation]]") is of considerable importance.{{sfn|Stillwell1999|p=74}}
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| ===Trigonometry===
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| The ''Siddhanta Shiromani'' (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered [[spherical trigonometry]], along with other interesting [[trigonometry|trigonometrical]] results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for <math> \sin\left(a + b\right) </math> and <math> \sin\left(a - b\right) </math>:
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| ===Calculus===
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| His work, the ''Siddhanta Shiromani'', is an astronomical treatise and contains many theories not found in earlier works{{citation needed|date=September 2012}}. Preliminary concepts of [[Non-standard calculus|infinitesimal calculus]] and [[mathematical analysis]], along with a number of results in [[trigonometry]], [[differential calculus]] and [[integral|integral calculus]] that are found in the work are of particular interest.
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| Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.<ref>50 Timeless Scientists von K.Krishna Murty</ref> It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement{{citation needed|date=September 2012}}. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of '[[infinitesimal]]s'.{{sfn|Shukla|1984|pp=95–104}}
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| * There is evidence of an early form of [[Rolle's theorem]] in his work
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| ** If <math> f\left(a\right) = f\left(b\right) = 0 </math> then <math> f'\left(x\right) = 0 </math> for some <math>\ x </math> with <math>\ a < x < b </math>
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| * He gave the result that if <math>x \approx y</math> then <math>\sin(y) - \sin(x) \approx (y - x)\cos(y)</math>, thereby finding the derivative of sine, although he never developed the notion of derivatives.{{sfn|Cooke|1997|pp=213–215}}
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| ** Bhaskara uses this result to work out the position angle of the [[ecliptic]], a quantity required for accurately predicting the time of an eclipse.
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| * In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a ''truti'', or a {{Fraction|1|33750}} of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
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| * He was aware that when a variable attains the maximum value, its [[Differential (infinitesimal)|differential]] vanishes.
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| * He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.{{citation needed|date=September 2012}} In this result, there are traces of the general [[mean value theorem]], one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by [[Parameshvara]] in the 15th century in the ''Lilavati Bhasya'', a commentary on Bhaskara's ''Lilavati''.
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| [[Madhava of Sangamagrama|Madhava]] (1340–1425) and the [[Kerala school of astronomy and mathematics|Kerala School]] mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of [[calculus]] in India.
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| ==Astronomy==
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| Using an astronomical model developed by [[Brahmagupta]] in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the [[sidereal year]], the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta. {{citation needed|date=May 2013}} The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes. {{citation needed|date=May 2013}}
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| His mathematical astronomy text ''Siddhanta Shiromani'' is written in two parts: the first part on mathematical astronomy and the second part on the [[sphere]].
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| The twelve chapters of the first part cover topics such as:
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| * Mean [[longitude]]s of the [[planet]]s.
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| * True longitudes of the planets.
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| * The three problems of [[diurnal motion|diurnal rotation]].
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| * [[Syzygy (astronomy)|Syzygies]].
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| * [[Lunar eclipse]]s.
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| * [[Solar eclipse]]s.
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| * [[Latitude]]s of the planets.
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| * [[Sunrise equation]]
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| * The [[Moon]]'s [[crescent]].
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| * [[Conjunctions]] of the planets with each other.
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| * Conjunctions of the planets with the fixed [[star]]s.
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| * The paths of the Sun and Moon.
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| The second part contains thirteen chapters on the sphere. It covers topics such as:
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| *Praise of study of the sphere.
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| *Nature of the sphere.
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| *[[Cosmography]] and [[geography]].
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| *Planetary [[mean motion]].
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| *[[Eccentricity (mathematics)|Eccentric]] [[Deferent and epicycle|epicyclic]] model of the planets.
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| *The [[armillary sphere]].
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| *[[Spherical trigonometry]].
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| *[[Ellipse]] calculations.{{Citation needed|date=September 2009}}
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| *First visibilities of the planets.
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| *Calculating the lunar crescent.
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| *Astronomical instruments.
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| *The [[season]]s.
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| *Problems of astronomical calculations.
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| ==Engineering==
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| The earliest reference to a [[perpetual motion]] machine date back to 1150, when Bhāskara II described a [[Bhāskara's wheel|wheel]] that he claimed would run forever.{{sfn|White|1978|pp=52–53}}
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| Bhāskara II used a measuring device known as ''Yasti-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.{{sfn|Selin|2008|pp=269–273}}
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| ==Legends==
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| Bhaskara II conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity{{citation needed|date=September 2012}}. In his book ''[[Lilavati]]'', he reasons: "In this quantity also which has zero as its divisor there is no change even when many [quantities] have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".{{sfn|Colebrooke|1817|p = }}
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| ==Notes==
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| {{Reflist}}
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| == References ==
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| * {{citation|title=Hindu achievements in exact science: a study in the history of scientific development|first=Benoy Kumar|last=Sarkār|publisher=Longmans, Green and co.|year=1918}}
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| * {{citation|title=The positive sciences of the ancient Hindus|first=Sir Brajendranath|last=Seal|publisher=Longmans, Green and co.|year=1915}}
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| * {{citation|title=Arithmetic and mensuration of Brahmegupta and Bhaskara|first=Henry T.|last= Colebrooke|year=1817}}
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| * {{citation|first=Lynn Townsend| last=White|chapter=Tibet, India, and Malaya as Sources of Western Medieval Technology|title=Medieval religion and technology: collected essays|isbn=978-0-520-03566-9|publisher=University of California Press|year=1978}}
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| * {{citation|chapter=Astronomical Instruments in India|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition)|editor-first=Helaine |editor-last=Selin |editor-link=Helaine Selin |publisher=Springer Verlag Ny|year=2008|isbn=978-1-4020-4559-2}}
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| * {{citation
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| | last =Shukla
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| | first = Kripa Shankar
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| | coauthors =
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| | title = Use of Calculus in Hindu Mathematics
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| | journal = Indian Journal of History of Science
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| | volume = 19
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| | issue =
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| | pages = 95–104
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| |year=1984
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| | url =
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| | doi =
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| | accessdate = }}
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| * {{citation|first=Roger|last=Cooke|title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience|year=1997|chapter=The Mathematics of the Hindus|pages=213–215|isbn=0-471-18082-3}}
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| * {{citation|title=Scientific heritage of India, mathematics|first=K. G.|last=Poulose|volume=Volume 22 of Ravivarma Samskr̥ta granthāvali|publisher=Govt. Sanskrit College (Tripunithura, India)|editor=K. G. Poulose|year=1991}}
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| * {{citation|last=Chopra|first=Pran Nath|year=1982|title=Religions and communities of India|publisher=Vision Books|isbn=978-0-85692-081-3}}
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| * {{citation|title=Toward a global science: mining civilizational knowledge|first=Susantha|last=Goonatilake|publisher=Indiana University Press|year=1999|isbn=978-0-253-21182-8}}
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| * {{citation|title=Mathematics across cultures: the history of non-western mathematics|volume=Volume 2 of Science across cultures|editor-first=Helaine|editor-last=Selin|editor-link=Helaine Selin|editor2-first=Ubiratan|editor2-last=D'Ambrosio|editor2-link=Ubiratan D'Ambrosio|publisher=Springer|year=2001|isbn=978-1-4020-0260-1}}
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| * {{citation|title=Mathematics and its history, Undergraduate texts in mathematics|first=John|last=Stillwell|publisher=Springer|year=2002|isbn=978-0-387-95336-6}}
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| ==External links==
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| {{Wikisource|Author:Bhāskara II|Bhāskara II}}
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| * [http://www.4to40.com/legends/index.asp?p=Bhaskara Bhaskara Biography]
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| * [http://www.canisius.edu/topos/rajeev.asp Calculus in Kerala]
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| {{Indian mathematics}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Bhaskara 2
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| | ALTERNATIVE NAMES = Bhaskaracharya
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| | SHORT DESCRIPTION = Indian mathematician and astronomer
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| | DATE OF BIRTH = 1114
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| | PLACE OF BIRTH =
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| | DATE OF DEATH = 1185
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| | PLACE OF DEATH =
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| }}
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| {{DEFAULTSORT:Bhaskara 2}}
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| [[Category:12th-century mathematicians]]
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| [[Category:Medieval Indian mathematicians]]
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| [[Category:Medieval Indian astronomers]]
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| [[Category:People from Bijapur, Karnataka]]
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| [[Category:1114 births]]
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| [[Category:1185 deaths]]
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| [[Category:Algebraists]]
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