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Bhaskara (1114-1185)Born near Vijjadavida (Bijapur in modern Karnataka)
Bijaganita dealt with algebra, Coladhyaya with spheres and
Grahaganita with planetary mathematics...
Bhaskara (1114–1185), also known as Bhaskara II and
Bhaskaracharya ("Bhaskara the teacher"), was an Indian mathematician
and astronomer. He was born near Vijjadavida (Bijapur in modern Karnataka).
Bhaskara is alleged to have been the head of an astronomical observatory at
Ujjain, the leading mathematical center of ancient India. He lived in the
Sahyadri region.
Bhaskara 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. His main work Siddhanta
Shiromani, (Sanskrit for "Crown of treatises,") is divided into four parts
called Lilavati, Bijaganita, Grahaganita and Goladhyaya. These four sections
deal with arithmetic, algebra, mathematics of the planets, and spheres
respectively. He also wrote another treatise named Karan Kautoohal.
Bhaskara's work on calculus predates Newton and Leibniz by
half a millennium. 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 Bhaskara was a
pioneer in some of the principles of differential calculus. He was perhaps the
first to conceive the differential coefficient and differential calculus.
Bhaskaracharya was born into a family belonging to the
Deshastha Brahmin community. 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 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 Bhaskara's writings.
Arithmetic
Bhaskara's arithmetic text Lilavati covers the topics of
definitions, arithmetical terms, interest computation, arithmetical and
geometrical progressions, plane geometry, solid geometry, the shadow of the
gnomon, methods to solve indeterminate equations, and combinations.
Lilavati is divided into 13 chapters and covers many branches
of mathematics, arithmetic, algebra, geometry, and a little trigonometry and
mensuration.
Algebra
His Bijaganita ("Algebra") was a work in twelve
chapters. It was the first text to recognize that a positive number has two
square roots (a positive and negative square root).
Trigonometry
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 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.
Calculus
His work, the Siddhanta Shiromani, is an astronomical
treatise and contains many theories not found in earlier works. Preliminary
concepts of infinitesimal calculus and mathematical analysis, along with a
number of results in trigonometry, differential calculus and integral calculus
that are found in the work are of particular interest.
Evidence suggests Bhaskara was acquainted with some ideas of
differential calculus. It seems, however, that he did not understand the
utility of his researches, and thus historians of mathematics generally neglect
this achievement.
Madhava (1340–1425) and the 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.
Astronomy
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 same as in
Surya siddhanta. The modern accepted measurement is 365.2563 days, a difference
of just 3.5 minutes.
His mathematical astronomy text Siddhanta Shiromani is
written in two parts: the first part on mathematical astronomy and the second
part on the sphere.
Legends
His book on arithmetic is the source of interesting legends
that assert that it was written for his daughter, Lilavati. In one of these
stories, which is found in a Persian translation of Lilavati, Bhaskara II
studied Lilavati's horoscope and predicted that her husband would die soon
after the marriage if the marriage did not take place at a particular time. To
alert his daughter at the correct time, he placed a cup with a small hole at
the bottom of a 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 the wrong time and she was soon
widowed.
Bhaskara II conceived the modern mathematical convention
that when a finite number is divided by zero, the result is infinity. 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]".
Bhaskaracharya, Indian Mathematician, Astronomer,
Astronomical, Ujjain, Sahyadri, Siddhanta Shiromani, Lilavati, Bijaganita,
Calculus, Newton, Leibniz, Deshastha Brahmin, Astrologer, Karan Kautoohal,
Plane Geometry, Solid Geometry, Bijaganita, Square Root, Trigonometric
Functions
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