I recently learned that Madhava of Kerala (c.1340–c.1425) was the first to discover the following formula for $\pi$: $$\frac{\pi}4\ =\ 1 - \frac13+\frac15 - \frac17 + \frac19 - \frac1{11} + \cdots$$ The formula was rediscovered by James Gregory (1671) and Gottfried Leibniz (1673) and is named after both of them. But Madhava discovered not one but two formulas for $\pi$, the following converging much more quickly than the one above: $$\pi\ =\ \sqrt{12}\left(1 - \frac1{3\cdot3} +\frac1{5\cdot3^2} - \frac1{7\cdot3^3} + \cdots\right)$$ In fact we now know that $$\tan^{-1}\theta\ =\ \theta - \frac{\theta^3}3 + \frac{\theta^5}5 - \frac{\theta^7}7 + \cdots$$ and the first and second formulas can be obtained by putting $\theta=1$ and $\theta=\frac1{\sqrt3}$ respectively into the above Maclaurin series expansion of $\arctan\theta$.

What I would like to know is how Madhava discovered his formulas for $\pi$ without knowing anything about Maclaurin series. It is surely inconceivable that he knew as much calculus as Newton or Leibniz, the two men credited with the invention of calculus as a mathematical tool two and a half centuries after his death. What, then, inspired him to his discoveries?

  • $\begingroup$ Neither Gregory "knew as much calculus as Newton or Leibniz"... He published it in his Vera Circuli et Hyperbolae Quadratura of 1667. $\endgroup$ – Mauro ALLEGRANZA Jul 15 '16 at 11:54
  • $\begingroup$ You can see Enrique González-Velasco, Journey through Mathematics: Creative Episodes in Its History (2011), Ch.4.3 THE EXPANSION OF FUNCTIONS, page 212-on $\endgroup$ – Mauro ALLEGRANZA Jul 15 '16 at 11:58
  • $\begingroup$ Indian Kerala school developed an iterative technique for finding trigonometric power series for astronomical calculations in Ptolemaic models. Nilakantha's 1545 book attributes the discovery to Madhava (1349-1425). For about two centuries Kerala preserved the technique, but without developing or transmitting it, they neither invented calculus nor influenced someone who could. See hsm.stackexchange.com/questions/2495/… for references and more on Taylor series history. $\endgroup$ – Conifold Jul 19 '16 at 2:56
  • $\begingroup$ The ancient Indians did not know the irrational numbers. Without knowing Madheva's work, I suspect he actually invented some algorithm - using natural numbers - to determine the circumference of the circle, and this algorithm was equivalent with the formula stated in the post. But I doubt, how could it be considered as a $\pi$-formula - proving the correspondence between the formula and his algorithm is a bigger work than proving the formula itself. $\endgroup$ – peterh - Reinstate Monica Jul 17 '19 at 17:39

According to this page:

Although almost all of Madhava's original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.

Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions.

Thus, you appear to be mistaken in your assumption that Madhava lacked knowledge of what we now call the Maclaurin series for trigonometric functions. The wikipedia page on Madhava series gives details of Madhava's interesting derivation of the series expansions for the sine, cosine, and arctangent functions as well as the two formula for $\pi$ you have noted. An account in Madhava's "own words" is provided together with a rendering in modern notation.

The page for $\frac{\pi}{4} = 1 - \frac13 + \frac15 - \dots$ also notes that this expansion has more recently become referred to as the Madhava-Leibniz series.

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This may be of interest...

Ranjan Roy, MR 1081274 The discovery of the series formula for $\pi$ by Leibniz, Gregory and Nilakantha, Math. Mag. 63 (1990), no. 5, 291--306.

The series referred to is $$ \frac{\pi}4\ =\ 1 - \frac13+\frac15 - \frac17 + \frac19 - \frac1{11} + \cdots $$

He says the Indian discovery (author not definitely known, sometimes attributed to Nilakantha) was the result of an effort to rectify the circle.

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