Jyesthadeva's Yuktibhasa, Written in Malayalam Around 1530, Gives Step-by-Step Proofs for the Kerala School's Series for Pi, Sine and Cosine

The Yuktibhasa is a book of proofs written in Malayalam, the language of Kerala, around 1530. Its author, Jyesthadeva, took the results of the Kerala school of mathematics and astronomy and showed why each one holds. The critical edition published in 2008, edited and translated by K. V. Sarma with explanatory notes by K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, describes the text as fifteen chapters in two parts, mathematics and astronomy. Among the results it derives are the infinite series for pi, the sine and the cosine that later authors ascribe to Madhava of Sangamagrama, who lived in the 14th century.

The date is not fixed. Springer’s edition and K. Ramasubramanian give about 1530. In a 2010 review quoted by M. D. Srinivas, the mathematician David Mumford wrote “perhaps about 1540”. MacTutor, the St Andrews history of mathematics archive, gives about 1550. This article uses about 1530 and keeps the range in view. An earlier piece on Madhava and the Kerala school covers the series themselves and the question of whether they reached Europe. This one is about the book that proves them.

A commentary in the language people spoke

Jyesthadeva wrote the Yuktibhasa to explain the Tantrasamgraha, the main astronomy treatise of Nilakantha Somayaji, who was born in 1444. MacTutor describes the Tantrasamgraha as 432 Sanskrit verses in eight chapters, and says Nilakantha also wrote a commentary on the Aryabhatiya. Springer’s description of the edition says Jyesthadeva’s book sets out the techniques behind the computation of planetary motions “as set forth” in that treatise, which it dates to about 1500.

Its method is in the title. A yukti is a rationale or demonstration, and the edition’s own description says the text “presents detailed yuktis” for the results of mathematical astronomy. MacTutor says it contains proofs of the theorems and derivations of the rules, “very unusually for an Indian mathematical text”. It also says the book is a survey of Kerala mathematics, so a reader gets Madhava’s results through a later author’s explanation. Ramasubramanian writes that Madhava’s own works containing the series are not available, and that the later mathematicians of the tradition ascribe them to him without disagreement.

Who wrote it was settled only in 1958. Charles Whish had named the author Cellalura Nambutiri in 1834, and Srinivas says confusion over the name lasted until that year, when K. V. Sarma found a statement in a manuscript in the Kerala University library that shows Jyesthadeva as the author of the “bhasha”, the vernacular book. A manuscript in Baroda added that Jyesthadeva, like Nilakantha, was a pupil of Damodara, the son of Paramesvara.

Corrections, powers and a series for the circle

Proof of the pi series is built from smaller results, and Ramasubramanian walks through the chain.

One link is an expansion of the fraction one over one plus x. It starts from a simple identity and swaps a division by one number for a division by another, and each swap calls for a correction. Together the corrections form an alternating series. Both the Yuktibhasa and the Kriyakramakari, a Sanskrit commentary on Bhaskara II’s Lilavati by Sankara Variyar and his student Narayana, say the corrections have no logical end and that the process can be stopped once the wanted accuracy is reached.

Another link is a general method for estimating the sums of whole-number powers. For a large number of terms n, the sum of the k-th powers of 1, 2, up to n comes to about n to the power k+1, divided by k+1. Ramasubramanian says the procedure the text gives is tantamount to a proof by induction.

A third link is the series itself. Madhava’s verse, as Sankara Variyar quotes it, tells the reader to multiply the diameter by four, then divide that product by each odd number in turn, subtracting and adding alternately. In modern terms the circumference equals four times the diameter, minus four times the diameter over three, plus four times the diameter over five, and so on. Jyesthadeva’s derivation takes a quadrant of the square around the circle, divides one side into n equal parts and joins each division point to the centre. Next he argues that for large n the half-chords of the small arcs “are essentially the arc-bits themselves”, as Ramasubramanian translates it. Adding up the pieces and using the power-sum estimates gives the series, which Gregory and Leibniz rediscovered, in Ramasubramanian’s words, almost three centuries later.

The last concerns speed. The series for pi converges very slowly, and Ramasubramanian describes a technique called antya-samskara, a correction applied after stopping at some term. The text discusses how corrections of three successively better kinds are arrived at. V. N. Krishnachandran, in a 2024 preprint, reads the first of those arguments this way: the text tries a tidy choice of correction, sees that it would make the correction the same at every odd number, which cannot hold, and chooses between two nearby even numbers, taking four times n as a first approximation. Ramasubramanian notes that with the third correction about 25 terms of the series give pi correct to 10 decimal places.

A value of pi given by Madhava, which Nilakantha and Sankara Variyar both cite, is 2,827,433,388,233 divided by 900,000,000,000, or 3.141592653592. It is correct to 11 decimal places. MacTutor reaches the same 11 places from a different route, a series with the square root of 12 in front, summed to 21 terms.

Sine and cosine

The Yuktibhasa also derives the series for the sine and cosine. Ramasubramanian refers readers to the literature for the proof, and quotes a verdict from the mathematician P. P. Divakaran. Newton, Divakaran notes, found his series by “guessing” successive terms from their form, and the Yuktibhasa’s way of integrating the difference or differential equation for sine and cosine is, in Divakaran’s phrase, “entirely different and very modern”.

Mumford, in the review that Srinivas quotes, writes that an “informal exposition of these results with full derivations” survives in Malayalam and gives a unique insight into Indian methods, which he sums up as recursion, induction and careful passage to the limit.

The astronomy half

Volume II of the edition lists eight astronomy chapters: the computation of planets, the earth and celestial spheres, the fifteen problems, gnomonic shadow, eclipse, vyatipata, visibility corrections to planets, and the elevation of the moon’s cusps.

Much of the astronomy follows Nilakantha’s revision of the older Indian planetary model. Ramasubramanian, Srinivas and Sriram write that in the Tantrasamgraha, composed about 1500, Nilakantha applied the equation of centre for Mercury and Venus to the mean heliocentric planet instead of the mean Sun. They add that the ancient Greek, Islamic and European planetary theories applied it to the mean Sun until Kepler, and that the correct rule was first formulated in the European tradition in the early 17th century. They note that Jyesthadeva’s book gives a detailed exposition of the geometrical picture that goes with Nilakantha’s model.

In a short tract, according to the same authors, Nilakantha states his cosmological model: Mercury, Venus, Mars, Jupiter and Saturn go around the mean Sun in eccentric orbits, while the mean Sun goes around the Earth. They observe that this is the model Tycho Brahe proposed toward the end of the 16th century, “albeit on entirely different considerations”. It has the mean Sun going around the Earth, so it is not the Copernican arrangement.

How the book reached a wider readership

John Warren, who ran the Madras observatory early in the 19th century, wrote in his 1825 book that an Indian astronomer had given him the pi series. Charles Whish, an East India Company civil servant who spent more than fifteen years in Malabar and read Malayalam and Sanskrit, took the matter further. His paper on the Hindu quadrature of the circle was read to the Royal Asiatic Society on 15 December 1832 and printed in volume 3 of its Transactions in 1834. It names four texts, the Tantrasamgraha, the Yuktibhasa, the Karana Padhati and the Sadratnamala, and ends with a promise: “A further account of the Yucti-Bhasha, the demonstrations of the rules for the quadrature of the circle by infinite series” would follow in a separate paper.

Whish died on 14 April 1833 and the second paper never appeared. Srinivas notes that Whish wrote on the first blank palm leaf of one of his two Yuktibhasa manuscripts, now in the Royal Asiatic Society in London, that it contained “the demonstration of the authors infinite series”.

Reception in Britain was mixed. Ramasubramanian reports that George Hyne, a senior member of the Madras Literary Society, concluded in the 1820s that the series had been borrowed from Europeans, and that Warren decided to go with his opinion. Srinivas adds that in 1843 Augustus De Morgan wrote he would have dismissed Whish’s paper as a “clever hoax” had it not appeared in the Society’s own Transactions, and suggested that the French astronomer Le Gentil or Catholic missionaries may have supplied the series to Indians. Srinivas and Ramasubramanian both describe a long silence among European historians afterwards. Historian of science David Pingree, in a lecture published in 1992, gave two reasons for it: that Western historians at first could not admit an Indian had discovered the calculus, and later that no one read the Society’s Transactions. Srinivas disputes part of this, saying Whish’s paper was widely noted in the decade after it was read. Pingree also argued that Madhava’s achievement should not be confused with the discovery of calculus, since it rested on geometrical and algebraic arguments.

Recovery began in India. Bibhutibhusan Datta wrote on the Hindu values of pi in 1926, A. N. Singh on the use of series in 1936, and C. T. Rajagopal and colleagues published explanations in the 1940s. In 1948 Rama Varma Tampuran and A. R. Akhileswaraiyer brought out the mathematics part of the Yuktibhasa with notes in Malayalam. Springer says the 2008 edition is the first critical edition of the whole Malayalam text, and it adds an English translation and explanatory notes by Ramasubramanian, Srinivas and Sriram.

What the claims cover

On scope, Ramasubramanian is explicit: the Kerala mathematicians restricted their discussion to the quadrature of a circle and certain trigonometric functions, while European mathematicians took up arbitrary curves with formal definitions and a general treatment. Context and purpose differed too, in his account. Pingree’s objection, above, is the best known counterview. Whether a given reader calls the Yuktibhasa’s methods calculus depends on how broadly the word is drawn, and the sources use different words for it.

Other points stay open. Madhava’s own mathematical writings are lost. Dating of the Yuktibhasa varies by about two decades across the sources above. This article makes no claim that any of the results reached Europe before the 17th century.

Springer’s catalogue entry for the two-volume edition gives its length as LXVIII preliminary pages and 1,084 pages.