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. Commented Mar 14, 2021 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. Commented Mar 14, 2021 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. Commented Mar 15, 2021 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). Commented Mar 15, 2021 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. Commented Mar 15, 2021 at 13:28

The Yuktibhāṣā is one of the texts we have available from mathematicians working in Kerala between the 14th and 16th centuries. This is a text in the Malayalam language, though a later (poor) translation into Sanskrit also exists. As with most texts in India, precise date and authorship are hard to determine, but it is generally accepted to have been written by Jyeṣṭhadeva (who lived c. 1500–1610), and composed around 1530 CE.

(As there has been some discussion in the question and comments about the date: The date of 1530, arrived at by K. V. Sarma, is what most of the published literature says, though a footnote in P. P. Divakaran's 2007 article "The First Textbook of Calculus: Yuktibhāṣā" (JSTOR, DOI) says that later in life, Sarma "tended to favour a slightly later dating" of around 1550–1560. The statement in Ranjan Roy's 1990 article saying "The Yuktibhasa was written about a century later" than the Tantra-saṃgraha seems to be a mistake, as it's closer to half a century.)

This text gives rationale (yukti) for various procedures (or computational methods, formulae, recipes, whatever you want to call them) that were given in earlier texts (such as the expression for 1/8th of the circumference as $$r(1 - 1/3 + 1/5 - 1/7 + \dots$$), now called the Leibniz formula for π, given earlier by Madhava (c. 1340–1425) as quoted by Nīlakaṇṭha (1444–1544) in his work Tantra-saṃgraha (1501)).

A translation of the Yuktibhāṣā has been published in 2008 jointly by Springer and Hindustan Book Agency (for IIAS Shimla), in two volumes: the first volume on the first half of the Yuktibhāṣā, which is mathematics, and the second volume on the second half, astronomy. The first volume (the second volume is similar) includes:

• the original Malayalam text, in critical edition by K. V. Sarma (pages 311–470),
• a bare English translation of the Malayalam text, done by K. V. Sarma (pages 1–145),
• explanatory notes (basically an English translation suitable for mathematicians to read, but still closely following the original), by K. Ramasubramanian, M. D. Srinivas and M. S. Sriram (pages 147–266), followed by an Epilogue (updated from an article by M. D. Srinivas) on "Proofs in Indian Mathematics" more generally (pages 267–310).

In particular, for the result in question (deriving the circumference of a circle in terms of something like $$1 - 1/3 + 1/5 - 1/7 + \dots$$) the relevant part to look at is "Chapter 6: Circle and Circumference", which occupies pp. 45–82 in the bare translation and 179–207 in the explanatory translation.

I can go into detail if needed, but as this answer is already getting long, will just give an outline:

• In 6.3.1, the author of the Yukti-bhāṣā shows (via similar triangles etc) a result that, in modern terms, is equivalent to stating that $$C_i Q_i = \frac{r}{n} \frac{r^2}{k_i k_{i+1}}$$ in the figure below:

— here, O is the centre of the circle which is of radius $$r$$, and the side $$EA$$ of length $$r$$ is divided into $$n$$ (we will later take $$n$$ going very large) equal segments, the $$i$$th of which is $$A_i A_{i+1}$$. So the length of $$EA_i$$ is $$ri/n$$, and $$k_i$$ is the length of $$OA_i$$, and $$C_i$$ is the point where $$OA_i$$ intersects the circle, and $$Q_i$$ is the base of the perpendicular from $$C_i$$ to $$OA_{i+1}$$.

• In 6.3.2, he argues that as $$n$$ goes large, the arc of the circle gets close to $$C_iQ_i$$, so the arc $$EC$$ (one-eighth of the circumference of the circle) gets closer and closer to $$\sum_{i} C_iQ_i$$, which in turn can be approximated as $$\frac{C}{8} \approx \sum_{i=1}^{n} C_i Q_i \approx \frac{r}{n}\sum_{i=1}^{n} \frac{r^2}{k_i^2}.$$

Note that $$k_i = OA_i = OE^2 + EA_i^2 = r^2 + (ir/n)^2$$ (a result which he also uses shortly), so in modern terms, we can see that this amounts to $$\frac{\pi}{4} \approx \frac{1}{n} \sum_{i=1}^n \frac{1}{1 + (i/n)^2} \to \int_{0}^{1} \frac{dt}{1 + t^2}$$

• In 6.3.3, he proves something equivalent to

$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + \dots + (-1)^m x^{m-1}\frac{x}{1+x}$$

• In 6.3.4, he applies this (with $$x = ir/n$$), which amounts to (I'm omitting factors of $$r$$, which are included in the translation):

\begin{align}C/8 &\approx \frac{r}{n} \sum_{i=1}^n \frac{1}{1 + (i/n)^2} \\ &= \frac{r}{n} \sum_{i=1}^{n} (1 - (i/n)^2 + (i/n)^4 + \dots) \\ &= \frac{r}{n}\left(\sum_{i=1}^{n}1 - \sum_{i=1}^{n}(i/n)^2 + \sum_{i=1}^{n}(i/n)^4 + \dots\right) \end{align}

At this point, he uses $$\sum_{i=1}^{n}(i/n)^k \approx \frac{i^{2k+1}}{2k + 1}$$ (which he says he will explain later) to reduce the above sum to $$C/8 \approx r\left(1 - \frac13 + \frac15 - \frac17 + \dots\right)$$

which is the series in question.

• 6.3.5 and 6.3.6 I didn't look into.

• In 6.4, he derives the approximation $$\frac{1}{n} \sum_{i=1}^{n} (i/n)^k \to \frac{1}{k+1}$$ as promised:

• In 6.4.1, he shows that $$S_n^{(1)} = 1 + 2 + 3 + \dots + n \approx n^2/2$$ (in two ways, one of them using the exact formula $$S_n^{(1)} = n(n+1)/2$$).

• In 6.4.2, he shows that $$S_n^{(2)} = 1^2 + 2^2 + 3^2 + \dots + n^2 \approx n^3/3$$.

• In 6.4.3, he shows that $$S_n^{(3)} \approx n^4/4$$, and $$S_n^{(4)} \approx n^5/5$$.

• In 6.4.4, he does it in general (induction, basically): $$S_n^{(k)} = n^{k+1}/(k+1)$$.

• And 6.4.5 is an aside, doing sums of sums (leading to $$\approx n^k/k!$$), not needed here.

• Finally, in 6.5, he restates the conclusion: $$\frac{C}{8} \approx r\left(1 - \frac13 + \frac15 - \frac17 + \dots\right)$$

All of this is done with error terms (and later in the chapter he goes even deeper on error analysis), that I skipped here. Except for the translation into modern notation, none of this is "overinterpreting": everything is done in words, but all the arguments (yukti/upapatti, ≈ proofs/rationale) are there. As mentioned, the explanatory translation on pages 179–207 will be more helpful for modern readers but they still follow the original very closely; you can look at the "bare" translation on pages 45–82 to see what is "actually" there: you just have to read slowly and carefully.

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. Commented Mar 14, 2021 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. Commented Mar 14, 2021 at 13:03
• It is there see the introduction mentioning 1500-1610 Commented Mar 14, 2021 at 13:13
• @reuns: Well, 1530 is within that range ... Commented Mar 14, 2021 at 13:16