# Yuktibhāṣā, 16th century, first modern proof of $\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$

It is an Indian (Kerala) text, it would be the first modern proof (based on earlier knowledge) of

$$\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$$ The integral would be a Riemann sum.

3 references, giving 3 different dates:

Additionally to the date problem, can we decipher at least a small part of the original text to make ourselves an idea of its content?

It bears repetition that, as in the excerpted passages, so in the entire text, no symbols are employed to represent the mathematical objects being manipulated, no formal notation for relations among them and operations on them, no diagrammatic guide to the geometric constructions invoked.

There is an English translation of the text, but it is really hard to follow, not made for mathematicians: Ganita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva.

Edit: I can't find the proof of $$\frac{\pi}4=\int_0^1 \frac{dt}{1+t^2}$$ and of $$\int_0^1 \frac{dt}{1+t^2}=\sum_{n\ge 0} \frac{(-1)^n}{2n+1}$$ in this translation. So this may be the main question: where is the proof? See this excerpt, it is more a cooking recipe as an attempt to define the Leibniz series than rigorous mathematics

In the translation Rsine means $$R \sin(\theta)$$ (with $$R$$ the radius) and Rversine means $$R(1-\cos(\theta))$$

• The third link is: R. Roy, Ranjan, "The discovery of the series formula for π by Leibniz, Gregory and Nilakantha." Math. Mag. 63 (1990), no. 5, 291–306. – Gerald Edgar Mar 14 at 11:23
• from that we get: Nilakantha's results were presented in his Tantrasangraha, composed in Sanskrit verse around 1500. An anonymous commentary entitled Tantrasangraha-vakhya then appeared, and a century later Jyesthadeva (c. 1500–c. 1610) published a commentary entitled Yuktibhasa that contained proofs of the earlier results. The material in the Tantrasangraha itself seems to have been the earlier work of Madhava, a mathematician who lived from 1340 to 1425 in Kerala, the southwest coast of India. – Gerald Edgar Mar 14 at 11:28
• It is not surprising that the sought proofs are not there considering that the Kerala school did not operate with any version of integrals. How Nilakantha (who attributes it to Madhava) geometrically derived the power series for $\arctan$, from which the formula for $\pi$ follows, is described at the end of Roy's paper linked in the OP, here is a non-paywalled version. It does not involve any integrals. – Conifold Mar 15 at 12:37
• @Conifold Eq. 14 is an integral, and I hardly see how to get the Leibniz series without $\pi/4=\int_0^1 \frac{dt}{1+t^2}$. Roy is the 3rd reference I mentioned, I explained what is the problem in my question: people are often overinterpreting when describing history of maths, and for the Kerala school nobody is really explaining what we can find in the texts, ie. is it formulas out of nowhere (cooking recipes), or are there some maths (eg. proofs). – reuns Mar 15 at 12:46
• $\pi/4=\int_0^1 \frac{dt}{1+t^2}$ follows from a purely geometric argument, adding the chords lengths of some small sectors whose sides have tangent $t$ and $t+h$, this chord length is $\approx \frac{h}{1+t^2}$. This comment is the kind of proof I want to find. – reuns Mar 15 at 13:28