# Indian Mathematics - Redressing the balance

### Ian G Pearce

### The Classical period: IV. Mathematics over the next 400 years (700AD-1100AD)

Mahavira (or Mahaviracharya), a Jain by religion, is the most celebrated Indian mathematician of the 9

GSS was the first text on arithmetic in the present form. He made the classification of arithmetical operators simpler. Detailed operations with fractions (and unit fractions), but no section on decimals (which were not an Indian invention).

Geometric progressions - he gave almost all required formulae.

Extension and systemisation of Jain works. First to give general formula.

Repeated Brahmagupta's construction for cyclic quadrilaterals.

Definitions for most geometric shapes.

Work on quadratic, indeterminate and simultaneous equations. Mahavira demonstrated definite understanding of the concept of a quadratic equation having two roots.

Only Indian mathematician to refer to the ellipse, indeed Indian mathematicians did not study conic sections or anything along these lines. Gave incorrect identity for area of ellipse. His formula for the perimeter of an ellipse is worth noting.

Mahavira's work, GSS, could be criticised for being nothing more than an extensive commentary on Jaina works, and the work of Aryabhata, Brahmagupta (and Bhaskara I). C Srinivasiengar describes his work as containing:

The influence of his work (within India) must also not be ignored. The very fact it was still in use more than 250 years after it was written is testimony to its importance as a mathematical work.

Example 8.4.1: General formula for combinations, as given by Mahavira.

In the early 10

Prior to Bhaskara II it is worth noting the contributions of Aryabhata II (c 920-1000 AD) a mathematical-astronomer who notably made important contributions to algebra. In his work

Sripati (c. 1019-1066 AD, the birth year of 1019 is known to be correct) was a follower of the teachings of Lalla and in fact the most important Indian mathematician of the 11

Further he impressively gave the identity:

^{th}century. His major work*Ganitasar Sangraha*was written around 850 AD and is considered 'brilliant'. It was widely known in the South of India and written in Sanskrit due to his Jaina 'faith'. In the 11^{th}century its influence was still being felt when it was translated into Telegu (a regional language of the south). Mahavira was aware of the works of Jaina mathematicians and also the works of Aryabhata (and commentators) and Brahmagupta, and refined and improved much of their work. What makes Mahavira unique is that he was not an astronomer, his work was confined solely to mathematics and he stands almost entirely alone in the history of Indian mathematics (at least up to the 14^{th}century) in this respect. He was a member of the mathematical school at Mysore in the south of India and his major contributions to mathematics include:**Arithmetic:**GSS was the first text on arithmetic in the present form. He made the classification of arithmetical operators simpler. Detailed operations with fractions (and unit fractions), but no section on decimals (which were not an Indian invention).

Geometric progressions - he gave almost all required formulae.

**Permutations and Combinations:**Extension and systemisation of Jain works. First to give general formula.

**Geometry:**Repeated Brahmagupta's construction for cyclic quadrilaterals.

Definitions for most geometric shapes.

**Algebra:**Work on quadratic, indeterminate and simultaneous equations. Mahavira demonstrated definite understanding of the concept of a quadratic equation having two roots.

**Ellipse:**Only Indian mathematician to refer to the ellipse, indeed Indian mathematicians did not study conic sections or anything along these lines. Gave incorrect identity for area of ellipse. His formula for the perimeter of an ellipse is worth noting.

Mahavira's work, GSS, could be criticised for being nothing more than an extensive commentary on Jaina works, and the work of Aryabhata, Brahmagupta (and Bhaskara I). C Srinivasiengar describes his work as containing:

... (No) profoundly fundamental discoveries. [CS, P 70]To some extent this may be true, but it is also unfair. Mahavira made many subtle contributions and elaborated and revised much of the work of previous mathematicians. Furthermore GSS contained many examples to illustrate his rules, unlike many Indian mathematical works.

The influence of his work (within India) must also not be ignored. The very fact it was still in use more than 250 years after it was written is testimony to its importance as a mathematical work.

Example 8.4.1: General formula for combinations, as given by Mahavira.

$_nC _r =\Large\frac{n(n-1)(n-2) ... (n-r+1)}{1.2.3 ... r}$

Following Mahavira the most notable mathematician was Prthudakasvami (c. 830-890 AD) a prominent Indian algebraist, who is described by E Robertson and J O'Connor as being:
He also wrote a commentary on Brahmagupta's...Best known for his work on solving equations.[EFR/JJO'C25, P 1]

*Brahma Sputa Siddhanta*.In the early 10

^{th}century a mathematician by the name of Sridhara (c. 870-930 AD) may have lived, however there is much debate surrounding his birth and some authors place him as having lived in the 8^{th}century (750 AD). However beyond debate is the fact that he wrote*Patiganita Sara*a work on arithmetic and mensuration. It contained exactly 300 verses and is hence also known by the name*Trisatika*. It includes contributions to the following topics:Rules on extracting square and cube roots, fractions and eight rules for operations involving zero (not division).

Theory of cyclic quadrilaterals with rational sides.

A section concerning rational solutions of various equations of the Pell's type.

Methods for summation of different arithmetic and geometric series. These methods became standard references in later works.

It is thought that Sridhara also composed a text on algebra, which is now lost, and several other works have been attributed to Sridhara, but there is no certainty if they were indeed written by him. The legacy of Sridhara's work was that it had some influence on the work of Bhaskaracharya II, regarded by many as the greatest Indian mathematician.
Theory of cyclic quadrilaterals with rational sides.

A section concerning rational solutions of various equations of the Pell's type.

Methods for summation of different arithmetic and geometric series. These methods became standard references in later works.

Prior to Bhaskara II it is worth noting the contributions of Aryabhata II (c 920-1000 AD) a mathematical-astronomer who notably made important contributions to algebra. In his work

*Mahasiddhanta*he gives twenty verses of detailed rules for solving $by = ax + c$ (and variations of this equation). Also of note, Vijayanandi (c 940-1010 AD) who made several contributions to trigonometry in the course of his astronomical works, and Sripati.Sripati (c. 1019-1066 AD, the birth year of 1019 is known to be correct) was a follower of the teachings of Lalla and in fact the most important Indian mathematician of the 11

^{th}century. He was the author of several astronomical works, including*Siddhantasekhara*, which contained two chapters devoted to mathematics. His major contributions were in the fields of arithmetic and algebra. His algebra is of particular note; his work includes rules for solving the quadratic equation and simultaneous indeterminate equations.Further he impressively gave the identity:

$\sqrt{x+\sqrt{y}} = \sqrt{\large\frac 1 2 \normalsize (x+\sqrt{x^2 - y}} + \sqrt{\large\frac 1 2 \normalsize (x-\sqrt{x^2 - y}}$

[Check:

[Check:

*x*= 2,*y*= 4 ⟷ 2 = 2]