# Indian Mathematics - Redressing the balance

### The Classical period: V. Bhaskaracharya II

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(The Classical period: VI. Pell's equation)

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 India gave a definite 'quota' to the forward world march of the science. [LG, P 104]
Born in 1114 AD (in Vijayapura, he belonged to Bijjada Bida) he became head of the Ujjain school of mathematical astronomy (Varahamihira and Brahmagupta had helped to found this school or at least 'build it up'). There is some confusion amongst the texts I have referred to as to the works that he wrote. C Srinivasiengar claims he wrote Siddhanta Siromani in 1150 AD, which contained four sections:
1) Lilavati (arithmetic)

2) Bijaganita (algebra)

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). S Sinha however agrees with C Srinivasiengar that 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 $\pi$.

- Arithmetical terms, methods of multiplication, squaring, inverse rule of three, plus rules of 5, 7 and 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 (17th Century) yet his work was of 12th Century. Method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

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 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:
δsin x = cos x δx

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 $a < x < b$