Bhaskaracharya

Bhaskara (1114 – 1185) Bhaskara (1114 – 1185)[/caption]

“Objects fall on earth due to a force of attraction by the earth. Therefore, the earth, planets, constellations, moon, and sun are held in orbit due to this attraction.”  

                                     :-  Siddhant Shiromani(A book of  Bhaskaracharya)

                                         This quate from Surya Siddhant in the book Siddhant Shiromani

Bhaskara also known as Bhaskaracharya, meaning “Bhaskara, the teacher”, was born in A.D. 1114 in Vijayapura(in karnataka). Bhaskaracharya’s father was a Brahman named Mahesvara, who was himself a famed astrologer. This great mathematician and astrologer reached an understanding of the number Systems and solving equations, which was not to be achieved in Europe for several centuries. He had propounded The Law of Gravitation, in clear terms, 500 years before it was rediscovered by Newton. Bhaskaracharya became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time. Outstanding mathematicians such as Barahamihira and Brahmagupta had worked there and built up a strong school of mathematical astronomy. He remained in Ujjain until his death in A.D. 1185. Six works by Bhaskaracharya are known which are: Lilavati (The Beautiful) which is on mathematics; Bijaganita (Seed Counting or Root Extraction) which is on algebra; the Siddhantasiromani which is in two parts, the first on mathematical astronomy with the second part on the sphere; the Vasanabhasya of Mitaksara which is Bhaskaracharya’s own commentary on the Siddhantasiromani ; the Karanakutuhala (Calculation of Astronomical Wonders) or Brahmatulya which is a simplified version of the Siddhantasiromani ; and the Vivarana which is a commentary on the Shishyadhividdhidatantra of Lalla. In 1207 an educational institution was set up to study Bhaskaracharya’s works.

Bhaskaracharya’s work in Algebra, Arithmetic and Geometry catapulted him to fame and immortality.His renowned mathematical works called Lilavati” and Bijaganita are considered to be unparalleled and a memorial to his profound intelligence.He  was the first to discover gravity, 500 years before Sir Isaac Newton.

Bhaskara’s contributions to mathematics

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.

Terms for numbers

In English, cardinal numbers are only in multiples of 1000. They have terms such as thousand, million, billion, trillion, quadrillion etc. Most of these have been named recently. However, Bhaskaracharya has given the terms for numbers in multiples of ten and he says that these terms were coined by ancients for the sake of positional values. Bhaskar’s terms for numbers are as follows:

eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).

Kuttak

Kuttak is nothing but the modern indeterminate equation of first order. The method of solution of such equations was called as ‘pulverizer’ in the western world. Kuttak means to crush to fine particles or to pulverize.

Chakrawaal

Chakrawaal is the “indeterminate equation of second order” in western mathematics. This type of equation is also called Pell’s equation.

Simple mathematical methods

Bhaskara has given simple methods to find the squares, square roots, cube, and cube roots of big numbers. He has proved the Pythagoras theorem in only two lines. The famous Pascal Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on that number triangle. Pascal was born 500 years after Bhaskara. Several problems on permutations and combinations are given in Lilawati. Bhaskar. He has called the method ‘ankapaash’. Bhaskara has given an approximate value of PI as 22/7 and more accurate value as 3.1416. He knew the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’.

Bhaskara’s contributions to astronomy

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.

  • Sunrise equation

  • The Moon’s crescent.

  • Conjunctions of the planets with each other.

  • Conjunctions of the planets with the fixed stars.

  • 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.[citation needed]

  • First visibilities of the planets.

  • Calculating the lunar crescent.

  • Astronomical instruments.

  • The seasons.

  • Problems of astronomical calculations.

Earth’s circumference and diameter

Bhaskara has given a very simple method to determine the circumference of the Earth. According to this method, first find out the distance between two places, which are on the same longitude. Then find the correct latitudes of those two places and difference between the latitudes. Knowing the distance between two latitudes, the distance that corresponds to 360 degrees can be easily found, which the circumference of is the Earth. For example, Satara and Kolhapur are two cities on almost the same longitude. The difference between their latitudes is one degree and the distance between them is 110 kilometers. Then the circumference of the Earth is 110 X 360 = 39600 kilometers. Once the circumference is fixed it is easy to calculate the diameter. Bhaskara gave the value of the Earth’s circumference as 4967 ‘yojane’ (1 yojan = 8 km), which means 39736 kilometers. His value of the diameter of the Earth is 1581 yojane i.e. 12648 km. The modern values of the circumference and the diameter of the Earth are 40212 and 12800 kilometers respectively. The values given by Bhaskara are astonishingly close.

Aksha kshetre

For astronomical calculations, Bhaskara selected a set of eight right angle triangles, similar to each other. The triangles are called ‘aksha kshetre’. One of the angles of all the triangles is the local latitude. If the complete information of one triangle is known, then the information of all the triangles is automatically known. Out of these eight triangles, complete information of one triangle can be obtained by an actual experiment. Then using all eight triangles virtually hundreds of ratios can be obtained. This method can be used to solve many problems in astronomy.

Geocentric parallax

Ancient Indian Astronomers knew that there was a difference between the actual observed timing of a solar eclipse and timing of the eclipse calculated from mathematical formulae. This is because calculation of an eclipse is done with reference to the center of the Earth, while the eclipse is observed from the surface of the Earth. The angle made by the Sun or the Moon with respect to the Earth’s radius is known as parallax. Bhaskara knew the concept of parallax, which he has termed as ‘lamban’. He realized that parallax was maximum when the Sun or the Moon was on the horizon, while it was zero when they were at zenith. The maximum parallax is now called Geocentric Horizontal Parallax. By applying the correction for parallax exact timing of a solar eclipse from the surface of the Earth can be determined.

His famous books

Lilavati

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:

  • Definitions.

  • Properties of zero (including division, and rules of operations with zero).

  • Further extensive numerical work, including use of negative numbers and surds.

  • Estimation of π.

  • Arithmetical terms, methods of multiplication, and squaring.

  • Inverse rule of three, and rules of 3, 5, 7, 9, and 11.

  • Problems involving interest and interest computation.

  • Arithmetical and geometrical progressions.

  • Plane (geometry).

  • Solid geometry.

  • Permutations and combinations.

  • Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important, 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.

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 been that a student of ‘Lilavati’ should concern himself with the mechanical application of the method.

Bijaganita

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). His work Bijaganita is effectively a treatise on algebra and contains the following topics:

  • Positive and negative numbers.

  • Zero.

  • The ‘unknown’ (includes determining unknown quantities).

  • Determining unknown quantities.

  • Surds (includes evaluating surds).

  • Kuttaka (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. Bhaskara’s method for finding the solutions of the problem Nx² + 1 = y² (the so-called “Pell’s equation”) is of considerable importance.

He gave the general solutions of:

  • Pell’s equation using the chakravala method.

  • The indeterminate quadratic equation using the chakravala method.

He also solved:

  • Cubic equations.

  • Quartic equations.

  • Indeterminate cubic equations.

  • Indeterminate quartic equations.

  • Indeterminate higher-order polynomial equations.

Siddhanta Shiromani

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.

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