# Mathematics in the post-classical era

By Prof. Vijaya Kumar Murty

We described above the centers at Kusumapara and Ujjain, both in North India. There was also a flourishing tradition of mathematics in South India.

Mohair is a mathematician belonging to the ninth century who was most likely from modern-day Karnataka. He studied the problem of cubic and quadratic equations and solved them for some families of equations. His work had a significant impact on the development of mathematics in South India. His book *Ganita-sara-sangraha* amplifies the work of Brahmagupta and provides a very useful reference for the state of mathematics in his day. It is not clear what other works he may have published; further research into the extent of his contributions would probably be very fruitful.

Another notable mathematician of South India was Madhava, from Kerala in the fourteenth century. He discovered series expansions for some trigonometric functions such as the sine, cosine, and arctangent that were not known in Europe until after Newton. In modern terminology, these expansions are the Taylor series of the functions in question.

Madhava gave an approximation to pi as 3.14159265359, which goes far beyond the four decimal places computed by Aryabhata. Madhava deduced his approximation from an infinite series expansion for pi /4 that became known in Europe only several centuries after Madhava (due to the work of Leibniz).

Madhava’s work with series expansions suggests that he either discovered elements of the differential calculus or nearly did so. This is worth further analysis. In a work in 1835, Charles Whish suggested that the Kerala school had ‘laid the foundation for a complete system of fluxions’. The theory of fluxions is the name given by Newton to what we today call the differential calculus. On the other hand, some scholars have been very dismissive of the contributions of the Kerala school, claiming that it never progressed beyond a few series expansions. In particular, the theory was not developed into a powerful tool as was done by Newton. We note that it was around 1498 that Vasco da Gama arrived in Kerala and the Portuguese occupation began. Judging by evidence at other sites, it is not likely that the Portuguese were interested in either encouraging or preserving the sciences of the region. No doubt, more research is needed to discover where the truth lies.

Madhava spawned a school of mathematics in Kerala, and among his followers may be noted Nilakantha and Jyesthadeva. It is due to the writings of these mathematicians that we know about the work of Madhava, as all of Madhava’s own writings seem to be lost.

## Notes & References

- Originally published as part of the article "A Brief History of Indian Mathematics" by Prabhuddha Bharata September 2007 edition. Reprinted with permission.