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Bhāskara (1114-1185), also called Bhāskara II and Bhāskarā Achārya ("Bhaskara the teacher") was an Indian mathematician-astronomer. He was born near Bijjada Bida in Bijapur district, Karnataka, South India and became head of the astronomical observatory at Ujjain, continuing the mathematical tradition of Varahamihira and Brahmagupta.

In many ways, Bhaskaracharya represents the peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations, which was not to be achieved anywhere else in the world for several centuries. His main works are the Lilavati (dealing with arithmetic), Bijaganita (algebra) and Siddhanta Shiromani which consists of two parts: Goladhyaya (sphere) and Grahaganita (mathematics of the planets).

Contributions

He developed a proof of the Pythagorean Theorem by calculating the same area in two different ways and then canceling out terms to get $a^{2}+b^{2}=c^{2}$ .

He is also known to have proven that anything divided by zero is infinity in addition to establishing that infinity divided by anything remains infinity.

In Surya Siddhanta, Bhaskaracharya calculates the time taken for the earth to orbit the sun to 9 decimal places as 365.258756484 days. The modern accepted measurement is 365.2596 days, a difference of 0.0005 days, 0.0002% over a span of 1500 years.

Bhaskara derived a cyclic, Chakravala method for solving equations of the form $ax^{2}+bx+c=y$ , which is usually attributed to William Brouncker who 'rediscovered' it around 1657.

Bhaskara's method for finding the solutions of the problem $x^{2}-ny^{2}=1$ (so called "Pell's equation") is of considerable interest and importance.

Bhaskara conceived differential calculus, after discovering the derivative, differential coefficient, infinitesimal, and Rolle's theorem. He also made contributions towards integral calculus.

Bhaskara introduced spherical trigonometry for the first time along with a number of other trigonometrical results.

There have been several allegedly unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this is seen as an attempt by European scholars to claim European influence on (all) the great works of mathematics. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.

Calculus

His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. There is not large mathematical content, but of particular interest are a number of results in trigonometry, differential calculus and integral calculus that are found in the work.

Evidence suggests Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's work five centuries later, asides the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement, which is extremely regrettable. 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 'infinitesimals'.

He also gives the (now) well known results for $sin(a+b)$ and $sin(a-b)$ .
There is evidence of an early form of Rolle's theorem in his work;

if

f(a)=f(b)=0

then

f

'

(x)=0

for some

x

with

a<x<b

.

Madhava (1340-1425)) and the Kerala School further advanced the development of calculus in India.

Trigonometry

The Siddhanta Shiromani introduced spherical trigonometry for the first time along with other interesting results on trigonometry. 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: $sin(a+b)=sin(a)cos(b)+cos(a)sin(b)$

$sin(a-b)=sin(a)cos(b)+cos(a)sin(b)$

The (now) well known results for

sin(a+b)

and

sin(a-b)

.

Astronomy

The study of astronomy in Bhaskara's works is based on the heliocentric solar system of gravitation earlier propunded by Aryabhata in 499, and an early law of gravitation described by Brahmagupta in the 7th century.

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

The twelve chapters of the first part cover topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; syzygies; lunar eclipses; solar eclipses; latitudes of the planets; risings and settings; the moon's crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; and the patas of the sun and moon.

The second part contains thirteen chapters on the sphere. It covers topics such as: praise of study of the sphere; nature of the sphere; cosmography and geography; planetary mean motion; eccentric epicyclic model of the planets; the armillary sphere; spherical trigonometry; ellipse calculations; first visibilities of the planets; calculating the lunar crescent; astronomical instruments; the seasons; and problems of astronomical calculations.

In the Surya Siddhanta, Bhaskara calculated the time taken for the earth to orbit the sun to 9 decimal places as 365.258756484 days. The modern accepted measurement is 365.2596 days, a difference of 0.0005 days, 0.0002% over a span of 1500 years.

Legend

Lilavati, his book on arithmetic, is the source of many interesting legends that assert that it was written for his daughter, Lilavati. As per one story, by studying Lilavati's horoscope, Bhaskara predicted that her husband would die soon after the marriage if the marriage did not take place at a particular time. To prevent that, he placed a cup with a small hole at the bottom of the vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity though, she went to look at the device and a pearl from her nose ring accidentally dropped into it, thus upsetting it. The marriage took place at wrong time and she was widowed soon. Bhaskara is said to have taught her mathematics to console her in her grief and to have written the book for her.

External links

O'Connor, John J.; Robertson, Edmund F., "Bhāskara II", MacTutor History of Mathematics Archive, University of St Andrews
Bhaskara
Calculus in Kerala

This Indian biographical article is a stub. You can help Wikipedia by expanding it.

Template:Mathbiostub

@@ Line 1: / Line 1: @@
-'''Bhāskara''' ([[1114]]-[[1185]]), also called '''Bhāskara II''' and '''BhāskarāAchārya''' ("Bhaskara the teacher") was an [[India|Indian]] [[mathematician]]. He was born near [[Bijjada Bida]] in [[Bijapur]] district, [[Karnataka]] and became head of the [[astronomy|astronomical]] observatory at [[Ujjain]], continuing the mathematical tradition of [[Varahamihira]] and [[Brahmagupta]].
+'''Bhāskara''' ([[1114]]-[[1185]]), also called '''Bhāskara II''' and '''Bhāskarā Achārya''' ("Bhaskara the teacher") was an [[India]]n [[Indian mathematics|mathematician]]-[[astronomer]]. He was born near [[Bijjada Bida]] in [[Bijapur]] district, [[Karnataka]], [[South India]] and became head of the [[astronomy|astronomical]] observatory at [[Ujjain]], continuing the mathematical tradition of [[Varahamihira]] and [[Brahmagupta]].
-In many ways, Bhaskaracharya represents the peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations, which was not to be achieved anywhere else in the world for several centuries. His main works are the [[Lilavati]] (dealing with [[arithmetic]]), [[Bijaganita]] ([[algebra]]) and [[Siddhantasiromani]] which consists of two parts: Goladhyaya ([[sphere]]) and  Grahaganita (mathematics of the [[planet]]s).
+In many ways, Bhaskaracharya represents the peak of mathematical knowledge in the [[12th century]]. He reached an understanding of the number systems and solving equations, which was not to be achieved anywhere else in the world for several centuries. His main works are the [[Lilavati]] (dealing with [[arithmetic]]), [[Bijaganita]] ([[algebra]]) and ''Siddhanta Shiromani'' which consists of two parts: Goladhyaya ([[sphere]]) and  Grahaganita (mathematics of the [[planet]]s).
 == Contributions ==
@@ Line 13: / Line 13: @@
 * Bhaskara derived a cyclic, [[Chakravala method]] for solving equations of the form <math>ax^2+bx+c=y</math>, which is usually attributed to William Brouncker who 'rediscovered' it around 1657.
-*Bhaskara's method for finding the solutions of the problem <math>x^2-ny^2=1</math> (so called "[[Pell's_equation]]") is of considerable interest and importance.
+* Bhaskara's method for finding the solutions of the problem <math>x^2-ny^2=1</math> (so called "[[Pell's equation]]") is of considerable interest and importance.
+* Bhaskara conceived [[differential calculus]], after discovering the [[derivative]], differential coefficient, [[infinitesimal]], and [[Rolle's theorem]]. He also made contributions towards [[integral calculus]].
+* Bhaskara introduced [[spherical trigonometry]] for the first time along with a number of other [[trigonometrical]] results.
 There have been several allegedly unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this is seen as an attempt by European scholars to claim European influence on (all) the great works of mathematics. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.
-=== Calculus ===
+== Calculus ==
-His work, the Siddhanta Siromani, is an astronomical treatise and contains many theories not found in earlier works. There is not large mathematical content, but of particular interest are several results in trigonometry and differential and integral calculus that are found in the work.
+His work, the ''Siddhanta Shiromani'', is an astronomical treatise and contains many theories not found in earlier works. There is not large mathematical content, but of particular interest are a number of results in [[trigonometry]], [[differential calculus]] and [[integral calculus]] that are found in the work.
-Evidence suggests Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's, five centuries before him, asides the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement, which is extremely regrettable. 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 'infinitesimals'.
+Evidence suggests Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's work five centuries later, asides the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement, which is extremely regrettable. 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 'infinitesimals'.
 * He also gives the (now) well known results for <math>sin(a+b)</math> and <math>sin(a-b)</math>.
@@ Line 26: / Line 30: @@
 :if <math>f(a)=f(b)=0</math> then <math>f</math> '<math>(x)=0</math> for some <math>x</math> with <math>a<x<b</math>.
-([[Madhava of Sangamagrama|Madhava]] (1340 CE) and the [[Kerala School]] further advanced the development of [[calculus]] in India.)
+[[Madhava of Sangamagrama|Madhava]] ([[1340]]-[[1425]])) and the [[Kerala School]] further advanced the development of [[calculus]] in India.
+== Trigonometry ==
+The ''Siddhanta Shiromani'' introduced [[spherical trigonometry]] for the first time along with other interesting results on trigonometry. 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:
+<math>sin(a + b) = sin(a) cos(b) + cos(a) sin(b)</math>
+<math>sin(a - b) = sin(a) cos(b) + cos(a) sin(b)</math>
+:The (now) well known results for <math>sin(a+b)</math> and <math>sin(a-b)</math>.
+== Astronomy ==
+The study of astronomy in Bhaskara's works is based on the [[heliocentrism|heliocentric]] [[solar system]] of [[gravitation]] earlier propunded by [[Aryabhata]] in [[499]], and an early [[law of gravitation]] described by [[Brahmagupta]] in the [[7th century]].
+His mathematical astronomy text ''Siddhanta Shiromani'' is written in two parts: the first part on mathematical astronomy and the second part on the [[sphere]].
+The twelve chapters of the first part cover topics such as: mean [[longitude]]s of the [[planets]]; true longitudes of the planets; the three problems of [[diurnal motion|diurnal rotation]]; syzygies; [[lunar eclipse]]s; [[solar eclipse]]s; [[latitude]]s of the planets; [[risings]] and [[settings]]; the [[moon]]'s [[crescent]]; [[conjunctions]] of the planets with each other; conjunctions of the planets with the fixed [[stars]]; and the patas of the [[sun]] and moon.
+The second part contains thirteen chapters on the sphere. It covers topics such as: praise of study of the sphere; nature of the sphere; [[cosmography]] and [[geography]]; planetary [[mean motion]]; [[eccentric]] [[epicyclic]] model of the planets; the [[armillary sphere]]; [[spherical trigonometry]]; [[ellipse]] calculations; first visibilities of the planets; calculating the [[lunar crescent]]; astronomical instruments; the seasons; and problems of astronomical calculations.
+In the ''Surya Siddhanta'', Bhaskara calculated the time taken for the earth to orbit the sun to 9 decimal places as 365.258756484 days. The modern accepted measurement is 365.2596 days, a difference of 0.0005 days, 0.0002% over a span of 1500 years.
 == Legend ==