That Calculus existed even before Newton?

Prof K Ramasubramanian of IIT-Bombay has news for us that we’d all love to hear. His recently released two-volume translation of the Ganita-Yukti-Bhaashaa by Jyesthdeva points to the fact that some subsets of calculus existed in Indian manuscripts almost two centuries before Isaac Newton published his work. And that an Indian mathematician and astronomer Nilakantha Somayaji spoke, in parts, about a planetary model, credited to Tycho Brahe almost a century later.

How old is the Ganita-Yukti-Bhaashaa?
It was published some time between 1530 and 1540. However, what’s important is that the material in this book is far older. For, the author makes it clear that his manuscript only explains in detail the work described in the Tantra Sangraha by Nilakantha Somayaji. So the work spoken about is actually much older, as Nilakantha was in the 15th century.

What is the Ganita-Yukti-Bhaashaa about?
It is divided into 15 chapters. Seven chapters are devoted to mathematics, and eight to astronomy. (By the way, it’s written in Malayalam, not Sanskrit. And I’ve translated it along with M D Srinivas and M S Sriram.)

And the Tantra Sangraha?
The Tantra Sangraha is a treatise on astronomy and related mathematics in elegant verse form, in Sanskrit. It consists of 432 verses.

How much of Tycho Brahe’s theory existed in this ancient manuscript?
Well, in the Tantra Sangraha, Nilakantha talks about a planetary model where five planets, which can be seen with the naked eye – Mercury, Venus, Mars, Jupiter and Saturn – move around the sun, which in turn moves around the earth. The fact remains that a century later, Tycho Brahe published the same planetary model and was credited for it, since no one knew of Nilakantha’s work.

The Ganita-Yukti-Bhaashaa also points to the fact that first work on calculus began in India?
Well, the Ganita-Yukti-Bhaashaa attributes its mathematical models work to Madhava, who lived from 1340 to 1420. That’s way ahead of Newton. But it would be too sweeping a statement to say that this was the first work on calculus. Yes, some of the notions described in the book form a subset of calculus. That’s a fact.

Could you give an example?
The infinite series for the pi, the arc tangent, the sine and cosine functions. The value of the pi, for instance – expressing quantity in the form of an infinite series, came two centuries before calculus was formally developed by Newton and Leibniz. In a different context, perhaps, and expressed in a different way. But it did exist. Obtaining a fast convergent from a slow convergent is a major development in mathematical analysis. This too existed in this book, though in a different way.

How is it different?
Madhava and Nilakantha don’t take a formalistic approach to mathematics, the way we do now, having followed Euclid ’s method of mathematics. Euclid ’s method is a formal, deductive approach. This is a different approach. Now we need to question whether the formalistic approach is the only approach, or the ‘correct’ approach. And it’s a very fundamental question.