Bhāskara II - WikiHQ

thumb|260px|Bhaskara's proof of the Pythagorean Theorem.

Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian polymath, mathematician, and astronomer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Satpura mountain ranges of Western Ghats, believed to be the town of Patana in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. In a temple in Maharashtra, an inscription, supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Henry Colebrooke who was the first European to translate (1817) Bhaskaracharya's mathematical classics refers to the family as Maharashtrian Brahmins residing on the banks of the Godavari.

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 ancient India. 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. His main work, Siddhānta-Śiromaṇi (Sanskrit for "Crown of Treatises"), is divided into four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya, which are also sometimes considered four independent works. near Vijjadavida (Vijjalavida).

Munishvara (17th century), a commentator on Siddhānta Shiromani of Bhaskara has given the information about the location of Vijjadavida in his work Marīci Tīkā as follows: whereas a section of scholars identified it with the modern day Beed city. Identification of Vijjalavida with Basar in Telangana has also been suggested. However, the identifications suggested by these sources remain untenable.

Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical centre of medieval India. History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Maheśvara (Maheśvaropādhyāya.

The first method involves dividing the right triangle using a perpendicular line drawn from the right angle to the hypotenuse, and then applying the principle of similar figures and proportions.(See the figure.)
The second method involves calculation of the area of the square on the hypotenuse. First, the square is divided into smaller pieces, one square with side length a-b and four triangles, illustrating the area is <math>(a-b)^2 + 2ab</math>. (see the figure on the left.) Here, the length of the hypotenuse is denoted by c, and the lengths of the other two sides are denoted by a and b. Then he computes the hypotenuse of a right angled triangle, where a=15 and b=20. Next, the author states a fact equivalent to<math>(a-b)^2+2ab=a^2+b^2</math>, and presents a figure (the figure on the right) as a demonstration. (The figure accompanies no further explanation.) Finally, he presents <math>c=\sqrt{a^2+b^2}</math> as a concise method.
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 same as those given by the Renaissance European mathematicians of the 17th century.
A cyclic Chakravala method for solving indeterminate equations of the form ax2 + 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.
The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhaskara II.
Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.
Some preliminary ideas of differential calculus and suggested that the "differential coefficient" vanishes at the extreme end. His work Bījaganita is effectively a treatise on algebra and 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 more than one unknown.
Indeterminate quadratic equations (of the type ax2 + b = y2).
Solutions of indeterminate equations of the second, third and fourth degree.
Quadratic equations.
Quadratic equations with more than one unknown.
Operations with products of several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y. There is evidence of an early form of Rolle's theorem in his work, though it was stated without a modern formal proof.. In his astronomical work, Bhāskara gives a result that looks like a precursor to infinitesimal methods: if <math>x \approx y</math> then <math>\sin(y) - \sin(x) \approx (y - x)\cos(y)</math>. This can be interpreted as the discovery that cosine is the derivative of sine,although he did not develop the notion of a derivative.In his works, there are traces of a special case of mean value theorem. The mean value formula for inverse interpolation of the sine was later formulated by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhāskara’s Lilavati.

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, the time that is required for the Sun to orbit the Earth, as approximately 365.2588 days which is the same as in Surya siddhanta. The modern accepted measurement is 365.25636 days, a difference of 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.

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. Diurnal motion refers to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle that is called the diurnal circle.
Syzygies.
Lunar eclipses.
Solar eclipses.
Latitudes of the planets.
Sunrise equation.
The Moon's crescent.
Conjunctions of the planets with each other.
Conjunctions of the planets with the fixed stars.
The paths 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.
Problems of astronomical calculations.

Engineering

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

Bhāskara II invented a variety 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 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]".

"Behold!"

It has been stated, by several authors, that Bhaskara II proved the Pythagorean theorem by drawing a diagram and providing the single word "Behold!". Sometimes Bhaskara's name is omitted and this is referred to as the Hindu proof, well known by schoolchildren.

However, Bhaskara II, in his Bijaganita, devotes two verses and prose commentaries following them on the explanation of the proofs .

A mathematics historian Kim Plofker

comments:

Exactly which part of these verses was meant by her is not clear. But the prose commentary after the latter verse ends as follows:

Bibliography

Takao Hayashi, Bījagaṇita of Bhāskara, SCIAMVS 10 (2009), 3—301

External links

4to40 Biography