Origins of Calculus: Madhava’s series
The 14th century mathematician who founded the Kerala school of Mathematics - Madhava of Sangamagrama - produced the first known infinite series expansions of trigonometric functions - sine, cosine and arctan. Convergent sums to infinity and rates of change of functions are the building blocks of Calculus. The concept of limits is not essential to Calculus as is amply attested to by modern day computers.
These series are called by various names - Leibniz series, Newton series,Taylor series or Gregory series. It’s important to note here that Madhava predates all the mathematicians named above by 2 centuries or more.
Before I present examples of some of his work, let me say that the notation used is the Katapayadi encoding of numerals into Sanskrit. This is done so that they can be turned into shlokas (?????) that can be easily memorized and recited. It’s a tradition of following a poetic meter that often makes the oral system of transmission superior to the written.
(1) This is Madhava’s Sine table that gives the values to 60th of arcsecond precision.
[Source: CK Raju’s Cultural Foundations of Mathematics]
(2) Here he talks about the value of Pi. It is accurate to an incredible 11 decimal places.
(3) This is a summary of his series using the modern notation.
[Source: Wikipedia]
I did a fair bit of calculus in high school and college but never came across Madhava’s ground breaking contributions. I am happy to see Wikipedia and some academic institutions starting to acknowledge his work.
References:
[1] https://en.wikipedia.org/wiki/Madhava%27s_sine_table
[2] https://en.wikipedia.org/wiki/Madhava_series
[3] https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1314