Bhāskara II

Bhaskara (1114 – 1185), also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher"), was an Indian mathematician and astronomer. He was born near Bijjada Bida (in present day Bijapur district, Karnataka state, South India) into the Deshastha Brahmin family and became head of the astronomical observatory at Ujjain, continuing the mathematical tradition of Varahamihira and Brahmagupta.

In many ways, Bhaskara represents the peak of mathematical and astronomical knowledge in the 12th century. He reached an understanding of calculus, astronomy, the number systems, and solving equations, which were not to be achieved anywhere else in the world for several centuries. His main works were the Lilavati (dealing with arithmetic), Bijaganita (Algebra) and Siddhanta Shiromani (written in 1150) which consists of two parts: Goladhyaya (sphere) and Grahaganita (mathematics of the planets).

Legends

Lilavati, 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, found in a Persian translation of Lilavati, Bhaskara 2 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 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..

Mathematics

Some of Bhaskara's contributions to mathematics include the following:

• A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².

• In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations.

• Solutions of indeterminate quadratic equations (of the type ax² + b = y²).

• 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

• 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.

• His method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") is of considerable interest and importance.

• 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.

• Solved quadratic equations with more than one unknown, and found negative and irrational solutions.

• Preliminary concept of mathematical analysis.

• Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.

• He conceived differential calculus, after discovering the derivative and differential coefficient.

• 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.

• Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)

• In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometrical results. (See Trigonometry section below.)

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. 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.

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). 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.

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. Among the many interesting results given by Bhaskara, discoveries first found in his works include the now well known results for $\sin \left(a+b\right)$ and $\sin \left(a-b\right)$:

• $\sin \left(a+b\right)=\sin(a)\cos(b)+\cos(a)\sin(b)$

• $\sin \left(a-b\right)=\sin(a)\cos(b)-\cos(a)\sin(b)$

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 his outstanding achievement. 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'.

• There is evidence of an early form of Rolle's theorem in his work:
  • If $f\left(a\right)=f\left(b\right)=0$ then $f'\left(x\right)=0$ for some $\ x$ with $\ a<x<b$

• He gave the result that if $x\approx y$ then $\sin(y)-\sin(x)\approx (y-x)\cos(y)$, thereby finding the derivative of sine, although he never developed the general concept of differentiation.
  • Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.

• 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 ¹⁄₃₃₇₅₀ of a second, and his measure of velocity was expressed in this infinitesimal unit of time.

• He was aware that when a variable attains the maximum value, its differential vanishes.

• 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. 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.

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

The study of astronomy in Bhaskara's works is based on the heliocentric solar system of gravitation earlier propunded by Aryabhata in 499, where the planets follow an elliptical orbit around the Sun, and the law of gravity described by Brahmagupta in the 7th century. Bhaskara's contributions to astronomy include accurate calculations of many astronomical results based on this heliocentric solar system of gravitation. One of these contributions is his accurate calculation of the sidereal year, the time taken for the Earth to orbit the Sun, as 365.2588 days. The modern accepted measurement is 365.2596 days, a difference of just one minute (analyzed by naked eyes and this accuracy is achieved in the absence of any sophisticated instrument).

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.
• 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.
• Problems of astronomical calculations.

He also showed that when a planet is at its furthest from the Earth, or at its closest, the equation of the centre (measure of how far a planet is from the position it is to be predicted to be in by assuming it to movie uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.

Legacy

Some scholars have suggested that Bhaskara's work influenced later developments in the Middle East and Europe. His work was perhaps known to Islamic mathematicians as soon as it was written, and influenced their subsequent writings. The results thus became indirectly known in Europe by the end of the 12th century, but the text itself was not introduced until much later. (Ball, 1960) (See Possible transmission of Kerala mathematics to Europe for other evidence.) The Mughal emperor Akbar commissioned a famous Persian translation of the Lilvati in 1587.

There have also 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 Eurocentric scholars to claim European influence on many great non-European 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. The study of Diophantine equations in India can also be traced back to the Sulba Sutras written from 800 B.C.E. to 500 B.C.E., which pre-date Diophantus' work by many centuries.

See also

• Bhaskara I
• Indian mathematics
• List of Indian mathematicians

References

ISBN links support NWE through referral fees

• W. W. Rouse Ball. A Short Account of the History of Mathematics, 4th Edition. Dover Publications, 1960.
• George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books, 2000.
• John J. O'Connor and Edmund F. Robertson. Bhāskara II at the MacTutor archive St Andrews University, 2000.
• Ian Pearce. Bhaskaracharya II at the MacTutor archive. St Andrews University, 2002.

External links

• Bhaskara
• Calculus in Kerala

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