# 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

## 1 Answer

Since the Yuktibhasa (The Rationale) was written in Malayam in 1530, by Jyesthadeva, a Keralan astronomer, it will be easily translatable into English. The question is finding someone to finance such a venture.

Personally, I think that as India becomes more aware of its own scientific heritage instead of just catching up with the West (and according to Mathilde Marcolli, a geometer, the best of their institutes compete with the best of Western institutes) then it will become more likely that we will see such translations.

• There is a supposed English translation of the text "Ganita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva" (on libgen) but it is hard to follow, and it is not giving the modern mathematical context to make it easy to follow it by mathematicians (for the Leibniz series part, which would be its main scientific achievement). I don't think the date 1530 is accurate. – reuns Mar 14 at 12:57
• @reuns: The date is sourced from Wikipedia which refers to the book, Yuktibhasa of Jyesthadeva: A Book on Indian Rationales in Indian Mathematics and Astronomy, an Analytic Appraisal. You might want to see what they have to say. – Mozibur Ullah Mar 14 at 13:03
• It is there see the introduction mentioning 1500-1610 – reuns Mar 14 at 13:13
• @reuns: Well, 1530 is within that range ... – Mozibur Ullah Mar 14 at 13:16