Calculus From Kerala

You were probably told that the discoverer of Calculus was one of these two gentleman --- the Englishman Newton or the German Leibniz (my academic ancestor!). I'm here to tell you that you were lied to. If this BBC radio show is to be believed, the first known Calculus text is the Ganita Yuktibhasa written by Jyesthadeva in AD 1530 in Malayalam ! This was more than a century before Newton and Leibniz appeared on the scene.

There is also this pdf-presentation by S.G.Rajeev of the U of Rochester with more details about the Kerala school and the contents of the Yuktibhasa. Part of the difficulty in translating it into modern mathematical notation is, apparently, that it is written in verse !

A brief history of Indian contribution to Mathematics (gathered from the radio show and the pdf-presentation):

• Number System: Numerals representing numbers (rather than collections of dots or strokes). The Brahmi numerals (250BC, symbols 1--10 but no zero). The zero is first seen in the 4th century (Bhakshali(?) manuscript). The "placeholder" system of representing numbers spreads from India through the Arabs (as testified by Said Al-Andalus, the historian from 11th century Cordoba). Europeans had a hard time getting used to the zero and the Indian number system, despite the enthusiastic efforts of Fibonacci (who learned it from his childhood in 11th-century Algiers as the son of a diplomat).
• Aryabhatta (6th century, Patna, Bihar) who wrote the Aryabhattiya. His main original contribution is in summing (finite) series.
• Bhaskaracharya (12th century) wrote the "Lilavatti" an elementary text. He made contributions in "Pre-Calculus" eg. calculating the movement of planets in smaller and smaller (but not yet infinitesmal) time instants.
• Madhava and the Kerala School (13th-16th century). Infinite series (with series expansions of pi, sine, cosine and arctangent, building on Aryabhatta's finite series) and integrals (according to Prof.Rajeev's presentation). While Madhava's works don't survive, his students' works do, and they cite him (like all honest scholars should).

I found all of this starting from Robin Varghese's post at (the excellent) 3 Quarks Daily.