Madhava -the Founder Of Math Analysis & Calculus ?

Madhava of Sangamagrama Analysis

-- The

Founder of Mathematical

Srinivasan N K

Introduction The foundational work for mathematical analysis and major aspects of what we now call 'calculus' is attributed to Isaac Newton and Leibniz.They drew together the early concepts of other mathematicians,especially about the limiting process for functions ["passage to limits"] and the process of differentiating ,called the differential,infinitesimal calculus.It is true that both Newton and Liebniz laid the foundations for differential and integral calculus as we know today.But much of the concepts of limits,infinite series,their convergence,expansion of functions into

infinite series by a method similar to Taylor series, using derivatives and factorials, integration term by term of a series---all these were given by the Indian [Hindu] mathematician of Kerala, Madhava of Sangamagrama in the 14th century.Therefore historians are beginning to admit that Madhava could be the real founder of mathematical analysis.His work is almost two hundred years before Isaac Newton [1642-1727]and Leibniz[1646-1716]. Madhava lived in the village Sangamagrama near Kochi, in Kerala state of India.His birth year was probably 1345 and he lived upto 1425.He was born in a orthodox Nambudhri brahmin family.He was born in a small village Irinjalakuda(Sangamagrama),which is a small town today.He founded the Kerala school of mathematics and astronomy which flourished upto 16th century. He had numerous students and proteges--the most famous ones were Nilakantha and Jyesthadeva.Much of the results are derived from the works of these two students:Nilakantha wrote 'Tantrasangraha" and Jyesthadeva, wrote "Yuktibhasa".The treatise "yuktibhasa" is considered the source book for the Calculus Madhava developed at that time---as we will elaborate. I have derived these points mostly from the articles of 1 J J OConner and E F Robertson [St Andrews University,Scotland] {the place where James Gregory lived and worked whose 'Gregoryseries' we will mention later} 2 Ian G Pearce 3 Dennis Alameida & G G Joseph [ Exeter Univ and Manchester Univ,UK] [These are available in various websites for any one to study.] The first westerner to recognise the Kerala school was Charles Whish who published a paper in the Trans of Royal Asiatic Society [of Great Britiain and Ireland]in 1835..This work was largely ignored. C T Rajagopal and M S Rangachari [from Chennai] and other Indian mathematicians ,particularly R C Gupta ,had done pioneering study in unearthing the work of Madhava and his students and bringing them to light in the past thirty years .The earliest work is due to R C Gupta in 1973.Almost all their publications were in Indian journals.This is perhaps the reason they were not picked up for a long time by western scholars.

Major Contributions of Madhava 1The foremost contribution of Madhava is the development of limiting process for functions,going from finite series to infinite series with clear concepts of convergence of such series.---"to treat their limit passage to infinity". 2 The infinite series expansion for trigonometric functions,sine x and arctanx as follows:

This series was attributed to Newton, now it is called "Madhava-Newton Series"

This series, which is the expansion of arctanx, is attributed to James Gregory[[1638-1675], the well known Gregory series , seen in elementary calculus texts, is now called "Madhava-Gregory Series". 3 The next step for Madhava was to set

in the Madhava-Gregory series.

We know that Therefore, he obtained this well-known series:

This series is attributed to Leonhard Euler [1707-1783] and called Euler series.It is now called 'Madhava-Euler series".{this is also attributed to Leibniz} 4 Madhava obtained another series : with the "correction terms"

This is often credited to Leibniz too.

5 Madhava obtained the approximation for pi upto 11 th decimal place and later to 13th decimal places using the following expression [using 21 terms]: from expansion of arctan(1/ )= His value for pi upto 11 terms ,found only in Kerala manuscripts, is as follows:

6 Madhava obtained the Taylor series expansion for sin x as follows:

This is again attributed to Gregory in the year 1668.[Brook Taylor (1685-1731)] This series illustrates that Madhava had found the expansion using differential coeffcients of sin x. In fact he had written:

{ some claim that Bhaskara II has already used this.}

7 Madhava also,after finding the expansions, did perform term by term integration and also a summation formula replacing the familiar integral of [There were other contributions too;this school had developed the secant method to solve algebraic equations ,used Rolle's theorem/mean value theorem and the Newton-Gauss interpolation formula.The details are not available.] His students and later mathematicians of the Kerala school had expanded on these works.They also found the approximations to pi upto many more decimal places...One student,Parameshwara discovered the mean value theorem of calculus,perhaps derived from Bhaskara's work.This theorem is usually credited to Cauchy.The secant method for finding the roots of nonlinear equation is again attributed to Parameswara.Much remains to be discovered about the works of the Kerala School of Mathematics and Astronomy. It is appropriate to say that Madhava founded the classical (real) mathematical analysis, almost two hundred years before Newton,unknown to Arab or European scholars. [Unfortunately, many western professors ignored these sources of information. D E Smith ,in

his well-known book 'History of Mathematics' wrote in 1925 : ' not since Bhaskara II [b.1114] has she [India] produced a single native genius in this field' .It is unlikely that he had not heard of Srinivasa Ramanujan who worked with G H Hardy and P C Mahalanobis, both at Cambridge University in the previous decade.!He chose to ignore such persons,along with Kerala mathematicians.] Only since the last 15 years or so, some writers have given the credit to Madhava and its school.[See Victor J Katz- A history of mathematics--(1992),Addison_Wesley and later editions..] G Joseph writes: We may consider Madhava of Sangamagrama [Kerala] to have been the Founder of mathematical Analysis.Some of his discoveries in this field show him to have possesssed extra-ordinary intuition. [G Joseph--The Crest of the Peacock, Princeton Univ press,NJ 1991]

Why this contribution of Madhava and his school were not known earlier? As mentioned earlier,the important reference in English literature seems to be the paper by Charles Whish in the Trans of Royal Asiatic Society[1835];his paper had largely been ignored and not followed by western as well as Indian scholars.There are several 'obvious' and not so obvious reasons for this neglect. 1 The work of Madhava and his school pertains to astronomical computations,not to pure math as such.The gems of their mathematical insight and techniques seem to be buried in the heap of astronomical calculations.Therefore the gems did not shine apparently. 2 Their works were in Malayalam language,spoken only in that region.Even other Indian scholars in India may not have studied them,except a few other astronomers/astrologers.There were no quick translations at that time,till 1970's. 3It is possible that the tendency to be secretive with such knowledge by Brahmin priestly class led to the situation when the manuscripts were not accessible to several scholars within the country[from the 14th-16th centuries.] 4 Even the later investigative or exploratory studies by RC Gupta and the two Chennai professors [CT Rajagopal and M S Gopalachari] in the late 1960's and later, were almost always published in Indian journals.Many of these journals are not to be found in western [european or north american] university libraries or research libraries.I am not sure whether these papers were included in Abstracts journals/indices.Therefore they were known only to a few Indian scholars. 5 The western professors were generally indifferent to such computational methods,under the impression that much of such work flowed from the European centres,and failed to look for precedence in other countries. 6 Indian academia in history departments rarely attached much importance to historical research in native math and science,except in the last few decades. It is strange ,however, that the mathematical academic community in India did not show interest in such works. At the present juncture,however,with the availability of internet and greater interest of research scholars in the history of math, there is greater ineraction and faster communication around the world on this topic.

Concluding Remarks It can be concluded that Madhava of Kerala had discovered much of what is known as the foundations of mathematical analysis [real analysis] which includes "Calculus".The work of Gregory and Taylor for series expansions were also found by him in exactly similar form.He extended the concepts to explore infinte series and numerical integration [term by term] .He also developed approximations for pi.He also did some work on continued fractions which were well known to Indian mathematicians since the work of Arya Bhatta in 5th Century. Much work of this Kerala school may be unearthed in the next decade or so.The pioneering work of C T Rajagopal and.M S Rangachari ,and R C Gupta and other Indian professors and historians should be acknowledged.Unfortunately many of their papers were in Indian journals,not easily accessed by Western scholars.

[There is some speculation that Jesuit priests and traders from Europe,especially from Italy,who were travelling to the west coast of India, might have spread these math developements through short documents or reports into Europe or by translating the manuscripts from the Brahmin mathematicians.It is also possible that Jesuit priests, some of them trained in Math in Rome, were sent expressly to learn these math computations ---useful for astronomical and calendrical calculations.Furthermore, James Gregory was a student at Univ of Padua in Italy,where he might have learnt about the Madhava work...An important reference is the article: "Kerala mathematics and its possible transmission to Europe" by Dennis F Alamedia and G G Joseph,Univ of Manchester,UK]] The work of J J O'Connor,E F Robertson and others at St Andrews University,UK and the work of G G Joseph deserve credit. It is high time that the work of Madhava is mentioned in high school/univ text books--at least the Madhava-Gregory series and the approximations to pi. By writing this we are not belittling the enormous work and the extensive contributions of Isaac Newton or Leibniz or other mathematicians of the 17th century.We are only giving precedence to Madhava and his school in this field by almost 200 years.The work of Newton and Leibniz may lie mostly in the 'consolidation' of the various concepts in calculus and in developing the functional notation we use today. Did not Newton himself say that he was standing on the shoulders of giants to see the distant lands.!