Biman Nath, a professor of astrophysics at Raman Research Institute, Bengaluru, has a great article in Frontline about the development of proto-calculus ideas by ancient Indian mathematicians -- specifically, Aryabhatta, Brahmagupta, Bhaskara II (all in Sanskrit over 1000 years ago), and Madhava (in Malaalam, some 600 years ago).
Along the way, he makes a good point about the state of the 'debate' in the public sphere on history of Indian scientific ideas:
... [M]ost of us are not historians, and such discussions become pointless after a while. One has come across the names Aryabhata or Varahamihira in school textbooks, but one would be hard pressed to name any specific achievement of these ancient Indian scientists. School students are taught about the discovery of the zero but they never learn what it means to have “discovered” it.
How does one suddenly invoke a number and bring it down to the realm of reality? We teach our children that our ancestors must have had a remarkable knowledge of metallurgy and give the example of the iron pillar in Delhi, but we never spell out what exactly they knew.
This superficiality in our collective knowledge often leads to meaningless, rhetorical debates on the achievements of ancient Indian scientists. [...]
Nath peppers his article with snippets that give us a richer picture of how these mathematicians presented their results. For example, here's Aryabhatta:
... Aryabhata’s sine table was considered the most accurate ... He had tabulated the sines of 24 angles, equally spaced between zero and 90 degrees (with a difference of 3.75° between them). The table was in the form of a verse, and he had come up with an ingenious way of coding the table in the words of the poem. The introduction to his book Aryabhatia explained the code. [...]
And, Brahmagupta and Madhava:
It is not clear how Brahmagupta derived the formula—he never explained it anywhere—but historians think he used geometry.
Like his predecessor Brahmagupta, Madhava never explained how he came up with the formula. Historians of mathematics again suspect that he obtained it through geometrical methods.
Finally, all this leads to question of what happened after Madhava's work, and why it did not blossom into calculus as we know it today? I look forward to a follow-up article by Nath.