## 8 V. Bhaskaracharya II

Bhaskaracharya, or Bhaskara II, is regarded almost without question as the greatest Hindu mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. As L Gurjar states:

*...Because of his work * [LG, P 104]

Born in 1114 AD (in Vijayapura, he belonged to Bijjada Bida) he became head of the *Siddhanta Siromani* in 1150 AD, which contained four sections:

1) *Lilavati* (arithmetic)

2) *Bijaganita* (algebra)

3) *Goladhyaya* (sphere/celestial globe)

4) *Grahaganita* (mathematics of the planets)

E Robertson and J O'Connor claim that he wrote 6 works, 1), 2) and SS (which contained two sections) and three further astronomical works, including two commentaries on the SS.

G Joseph claims his mathematically significant works were 1), 2), and SS (which indeed he wrote in 1150 and is a highly influential astronomical work). *Lilavati* was a section (chapter) of the SS, and thus I will agree with the respected Indian historians.

Lilavati (or Leelavati, there is a charming if unlikely story regarding the origin of the name of this work) is divided into 13 chapters (possibly by later scribes) 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).

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

- Estimation of p.

- Arithmetical terms, methods of multiplication, squaring, inverse rule of three, plus rules of 5,7 & 9.

- Problems involving interest.

- Arithmetical and geometrical progressions.

- Plane geometry.

- Solid geometry.

- Combinations.

- Indeterminate equations (*Kuttaka*), integer solutions (first and second order) His contributions to this topic are among his most important, the rules he gives are (in effect) the same as those given by the renaissance European mathematicians (17^{th} Century) yet his work was of 12^{th} Century. Method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

- Shadow of the gnomon.

The *Lilivati* is written in poetic form with a prose commentary and Bhaskara acknowledges that he has condensed the works of Brahmagupta, Sridhara (and Padmanabha).

However 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. A student of '*Bijaganita*' should however concern himself with the theory underlying the method.

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

- Positive and negative numbers.

- Zero.

- The 'unknown'.

- Surds.

- Kuttaka.

- Simple equations (indeterminate of second, third and fourth degree).

- Simple equations with more than one unknown.

- Indeterminate quadratic equations (of the type *ax*^{2} + *b* = *y*^{2}).

- Quadratic equations.

- Quadratic equations with more than one unknown.

- Operations with products of several unknowns.

Bhaskara derived a cyclic, '*Cakraval*' 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 *Nx*^{2} + 1 = *y*^{2} (so called "Pell's equation") is of considerable interest and importance.

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

Bhaskara is though to be the first to show that:

sin *x* = cos *x* *x*

Evidence suggests Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton's, 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 also evidence of an early form of Rolle's theorem;

*if* *f*(*a*) = *f*(*b*) = 0 *then* *f* '(*x*) = 0 *for* *some* *x* *with* *a* < *x*< *b*,

in Bhaskara's work.

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

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