Kerala(South India) School of mathematics

9: Keralese mathematics I. Introduction

The south western tip of India escaped the majority of the political upheaval, which engulfed the rest of the country, allowing a generally peaceful existence to continue. Thus the pursuit of scientific development was able to continue 'uninterrupted'. It has only recently come to light that mathematics (and astronomy) continued to flourish in this area for several hundred years. Kerala mathematics was strongly influenced by astronomy, but this led to the derivation of mathematical results of huge importance. As a result of the recency of these discoveries it is quite probable that there are still further discoveries of 'Kerala mathematics' to be made, and a full analysis has yet to be carried out. However several findings have already been made that show several major concepts of renaissance European mathematics were first developed in India.

Indeed G Joseph quotes:

...In Kerala, the period between the 14^th and 16^th centuries marked a high point in the indigenous development of (astronomy) and mathematics. [GJ, P 287]

The works that have so far been analysed are of such a high level that it is though there may be missing links between the "classical period" and the medieval period of Kerala. There is also interest in the claims that European scholars may have had first hand knowledge of some Kerala mathematics, as the area was a focal point for trading with many parts of the world, including Europe. There is also some evidence of a transfer of technology between Europe and Kerala. I will discuss this issue in a little more depth later.

At this point in time only some interest is being paid to the recent discoveries that have been made, highlighting that in many historical 'circles' Indian developments are still not considered important. A further point of interest is that up to the 10^th century there was virtually no mathematical activity of note in the south of India. There are four areas in the south of the country - including Kerala in the south west corner, and with the exception of Mahavira (resident of Karnataka area) there was no mathematical output of significance until the 11^th-12^th century early Kerala literature. There are however impressions that Aryabhata I was a Keralite, and indeed, if this were true, then in the words of K Rajagopalan:

...Kerala would find a prominent place in the mathematical map of "our" country. [KR1, P 81]

Bhaskara I is also thought to have possibly been a Keralite.

Of the leading mathematicians of Kerala there is quite possibly more to be discovered but currently there are several whose work is of significant interest.

9 II. Mathematicians of Kerala

Narayana Pandit (c. 1340-1400), the earliest of the notable Keralese mathematicians, is known to have definitely written two works, an arithmetical treatise calledGanita Kaumudi and an algebraic treatise called Bijganita Vatamsa. He was strongly influenced by the work of Bhaskara II, which proves work from the classic period was known to Keralese mathematicians and was thus influential in the continued progress of the subject. Due to this influence Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati). It has been suggested that this work was written in conjunction with another scholar, Sankara Variyar, while others attribute the work to Madhava (see later).

Although the Karmapradipika contains very little original work, seven different methods for squaring numbers are found within it, a contribution that is wholly original to the author. Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, using the second order indeterminate equation Nx² + 1 = y² (Pell's equation). Mathematical operations with zero, several geometrical rules and discussion of magic squares and similar figures are other contributions of note. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work.

R Gupta has also brought to light Narayana's contributions to the topic of cyclic quadrilaterals. Subsequent developments of this topic, found in the works of Sankara Variyar and Ganesa interestingly show the influence of work of Brahmagupta. 

Paramesvara (c. 1370-1460) is known to have been a pupil of Narayana Pandit, and also Madhava of Sangamagramma, who I will discuss later and is thought to have been a significant influence. He wrote commentaries on the work of Bhaskara I, Aryabhata I and Bhaskara II, and his contributions to mathematics include an outstanding version of the mean value theorem. Furthermore Paramesvara gave a mean value type formula for inverse interpolation of sine, and is thought to have been the first mathematician to give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

In turn, Nilakantha Somayaji (1444-1544) was a disciple of Paramesvara and was educated by his son Damodra. In his most notable work Tantra Samgraha(which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of Aryabhatiya-bhasa a commentary of the Aryabhatiya. Of great significance is the presence of mathematical proof(inductive) in Nilakantha's work.

Furthermore, his demonstration of particular cases of the series

tan ^-1t = t - t³/3 + t⁵/5 - ... ,

when t = 1 and t = 1/3, and remarkably good rational approximations of p (using another Madhava series) are of great interest. Various results regarding infinite geometrically progressing convergent series are also attributed to Nilakantha 
Citabhanu (1475-1550) has yet to find a place in books on Indian mathematics. His work on the solution of equations is quoted in a work called Kriya-krama-kari, by the scholar Sankara Variar, who is also relatively little known (although R Gupta mentions a further text, written by him).

Jyesthadeva (c. 1500-1575) was a member of the Kerala School, which was founded on the work of Madhava, Nilakantha, Paramesvara and others. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala). Similarly to the work of Nilakantha it is almost unique in the history of Indian mathematics, in that it contains both proofs of theorems and derivations of rules. He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using kuttaka.

9 III. Madhava of Sangamagramma

Although born in Cochin on the Keralese coast before the previous four scholars I have chosen to save my discussion of Madhava of Sangamagramma (c. 1340 - 1425) till last, as I consider him to be the greatest mathematician-astronomer of medieval India. Sadly all of his mathematical works are currently lost, although it is possible extant work may yet be 'unearthed'. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475, but this is only speculative. All we know of Madhava comes from works of later scholars, primarily Nilakantha and Jyesthadeva. G Joseph also mentions surviving astronomical texts, but there is no mention of them in any other text I have consulted.

His most significant contribution was in moving on from the finite procedures of ancient mathematics to 'treat their limit passage to infinity', which is considered to be the essence of modern classical analysis. Although there is not complete certainty it is thought Madhava was responsible for the discovery of all of the following results:

1)  = tan  - (tan³ )/3 + (tan⁵)/5 - ... , equivalent to Gregory series.

2) r= {r(rsin)/1(rcos)}-{r(rsin)³/3(rcos)³}+{r(rsin)⁵/5(rcos)⁵}- ...

3) sin =  - ³/3! + ⁵/5! - ..., Madhava-Newton power series.

4) cos = 1 - ²/2! + ⁴/4! - ..., Madhava-Newton power series.
Remembering that Indian sin = rsin, and Indian cos = rcos. Both the above results are occasionally attributed to Maclaurin.

5) p/4  1 - 1/3 + 1/5 - ...  1/n  (-f_i(n+1)), i = 1,2,3, and where f₁ = n/2, f₂ = (n/2)/(n² + 1) and f₃ = ((n/2)² + 1)/((n/2)(n² + 4 + 1))² (a power series for p, attributed to Leibniz)

6) p/4 = 1 - 1/3 + 1/5 - 1/7 + ...  1/n  {-f(n+1)}, Euler's series.

A particular case of the above series when t =1/3 gives the expression:
7) p = 12 (1 - {1/(3  3)} + {1/(5  3²)} - {1/(7  3³)} + ...}

In generalisation of the expressions for f₂ and f₃ as continued fractions, the scholar D Whiteside has shown that the correcting function f(n) which makes 'Euler's' series (of course it is not in fact Euler's series) exact can be represented as an infinite continued fraction. There was no European parallel of this until W Brouncker's celebrated reworking in 1645 of J Wallis's related continued product.

A further expression involving p:
8) pd  2d + 4d/(2² - 1) - 4d/(4² - 1) + ...  4d/(n² + 1) etc, this resulted in improved approximations of p, a further term was added to the above expression, allowing Madhava to calculate p to 13 decimal places. The value p = 3.14159265359 is unique to Kerala and is not found in any other mathematical literature. A value correct to 17 decimal places (3.14155265358979324) is found in the work Sadratnamala. R Gupta attributes calculation of this value to Madhava, (so perhaps he wrote this work, although this is pure conjecture).

Of great interest is the following result:
9) tan ^-1x = x - x³/3 + x⁵/5 - ..., Madhava-Gregory series, power series for inverse tangent, still frequently attributed to Gregory and Leibniz.

It is also expressed in the following way:
10) rarctan(y/x) = ry/x - ry³/3x³ + ry⁵/5x⁵ - ..., where y/x  1

The following results are also attributed to Madhava of Sangamagramma:
11) sin(x + h)  sin x + (h/r)cos x - (h²/2r²)sin x

12) cos(x + h)  cos x - (h/r)sin x - (h²/2r²)cos x

Both the approximations for sine and cosine functions to the second order of small quantities, (see over page) are special cases of Taylor series, (which are attributed to B Taylor).

Finally, of significant interest is a further 'Taylor' series approximation of sine: 
13) sin(x + h)  sin x + (h/r)cos x - (h²/2r²)sin x + (h³/6r³)cos x. 
Third order series approximation of the sine function usually attributed to Gregory.

With regards to this development R Gupta comments:

...It is interesting that a four-term approximation formula for the sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor series expansion was discovered by Gregory about 1668. [RG5, P 289]

Although these results all appear in later works, including the Tantrasangraha of Nilakantha and the Yukti-bhasa of Jyesthadeva it is generally accepted that all the above results originated from the work of Madhava. Several of the results are expressly attributed to him, for example Nilakantha quotes an alternate version of the sine series expansion as the work of Madhava. Further to these incredible contributions to mathematics, Madhava also extended some results found in earlier works, including those of Bhaskaracarya.

The work of Madhava is truly remarkable and hopefully in time full credit will be rewarded to his work, as C Rajagopal and M Rangachari note:

...Even if he be credited with only the discoveries of the series (sine and cosine expansions, see above, 3) and 4)) at so unexpectedly early a date, assuredly merits a permanent place among the great mathematicians of the world. [CR /MR1, P 101]

Similarly G Joseph states:

...We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition. [GJ, P 293]

With regards to Keralese contributions as a whole, M Baron writes (in D Almeida, J John and A Zadorozhnyy):

...Some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries. [DA/JJ/AZ1, P 79]

There remains a final Kerala work worthy of a brief mention, Sadrhana-Mala an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19^th century and the author stands as the last notable name in Keralese mathematics.

In recent histories of mathematics there is acknowledgement that some of Madhava's remarkable results were indeed first discovered in India. This is clearly a positive step in redressing the imbalance but it seems unlikely that full 'credit' will be given for some time, as that will possibly require the re-naming of various series, which seems unlikely to happen!

Still in many quarters Keralese contributions go unnoticed, D Almeida, J John and A Zadorozhnyy note that a well known historian of mathematics makes:

...No acknowledgement of the work of the Keralese school. [DA/JJ/AZ1, P 78]
(Despite several Western publications of Keralese work.)

9 IV. Possible transmission of Keralese mathematics to Europe

In addition to my discussion, there is a very recent paper (written by D Almeida, J John and A Zadorozhnyy) of great interest, which goes as far as to suggest Keralese mathematics may have been transmitted to Europe. It is true that Kerala was in continuous contact with China, Arabia, and at the turn of the 16^th century, Europe, thus transmission might well have been possible. However the current theory is that Keralese calculus remained localised until its discovery by Charles Whish in the late 19^th century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission.

A key development of pre-calculus Europe, that of generalisation on the basis of induction, has deep methodological similarities with the corresponding Kerala development (200 years before). There is further evidence that John Wallis (1665) gave a recurrence relation and proof of Pythagoras theorem exactly as Bhaskara II did. The only way European scholars at this time could have been aware of the work of Bhaskara would have been through Keralese 'routes'.

The need for greater calendar accuracy and inadequacies in sea navigation techniques are thought to have led Europeans to seek knowledge from their colonies throughout the 16^th and 17^th centuries. The requirements of calendar reform were imperative with the dating of Easter proving extremely problematic, by the 16^thcentury the European 'Julian' calendar was becoming so inaccurate that without correction Easter would eventually take place in summer! There were significant financial rewards for 'anyone' who could 'assist' in the improvement of navigation techniques. It is thought 'information' was sought from India in particular due to the influence of 11^th century Arabic translations of earlier Indian navigational methods.

Events also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were 'encouraged' to acquire mathematical knowledge while there. 
It is feasible that these observations are mere coincidence but if indeed it is true that transmission of ideas and results between Europe and Kerala occurred, then the 'role' of later Indian mathematics is even more important than previously thought.