Yesterday I made an entry about the award of a prize to Prof R C Gupta for his huge opus of work in early Indian mathematics. I would like to say a bit more about this.
Prof Gupta has worked extensively on the work in mathematics and astronomy of the early mathematicians and astronomers like Aryabhata, Bhaskara I, Brahmagupta, Mahavira, Govindaswami, Bhaskara II and so on; also on early Jain mathematics in general. If you study the texts associated with all these mathematicians you do not see the subject developed as it is in today's books. Typically you see either extensive tables, or rules of some kind which are clearly the verbal equivalent of formulas, and these are generally given in the form of rhyming verse (in Sanskrit, of course).
The first problem, obviously, is to interpret these verses correctly and meaningfully. But the really interesting question is: How did these mathematicians construct the formulas or the tables?
For example, in Aryabhata's book Aryabhatiya, we find a table of first differences of the sine table, captured very compactly and conveniently in just two lines, in a form of verse. It seems amazing that in that distant era (5th century AD), Aryabhata was examining tables of differences of a function! And how did he even construct such a table? It compares very well with the modern table. Unfortunately, this question may never be fully answered, because the records are so scanty, but historians like Prof Gupta have looked very carefully at the question. (There are now many historians in the west too who have a great interest in such questions.) In the case of the later mathematicians (Bhaskara I, Mahavira, Bhaskara II) the records yield a little more. For example, it appears that some of these later mathematicians used second order interpolation methods to construct more accurate tables. Here is a paper of Prof Gupta's on just this topic, which you should have a look at: R C Gupta - 2nd order interpolation.
With regard to the Kerala school of mathematics - Madhava, Nilakantha and so on - the records are much clearer; in this school, the writers developed a tradition of describing how they got their results, in sharp contrast to (say) what Aryabhata or Bhaskara I did. So they describe clearly how they got the power series expansion for $\tan^{-1} x$. The famous series
\[ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots \]
which is nowadays called the Leibnitz-Gregory series for $\pi$ should more correctly be called the Madhava series, or the Madhava-Gregory-Leibnitz series, because it appears in the work of the Kerala school - the Yuktibhasa - and this work predates the work of Gregory and Leibnitz by over two centuries.
The proof of the series for $\tan^{-1} x$ shows quite clearly that the Kerala school had an understanding of the limit process. What is not clear is whether they were able to generalize this notion and apply it to the study of functions in general. If they had done this, then they would have had the differential calculus in their hands, with all the power and glory that it brings. But it is probable that they did not; so in that sense one can say that these mathematicians "almost" discovered the calculus for themselves, but they did not quite get there.
