# Madhava: The Kerala Mathematician Who Devised Calculus Ahead of Newton, Leibniz

Students of the day, who make their tryst with calculus, may not have heard about Madhava, a mathematician of Kerala,who prepared the way for Newton (1642-1727) and Leibniz (1646-1716), theindependent founders of the discipline.The contributions of Madhava of Sangamagrama (c. 1340- 1425), who lived in a place believed to be the present-day Aloor near Irinjalakuda in Thrissur District of Kerala, were so substantial that he deserves to be counted along with the founders of mathematical analysis based on calculus, infinite series expansion of functions and their rational approximations.Why didn’t the world outside perceive Madhava’s contributions? Why didthe light he brought out not kindle a similar scientific revolution as was the case with Newton and Leibniz?

**Saga of Indian Education**

India was among the very first nations that had an early
Renaissance and it was the golden age for its higher education(5th century
BCEto 7th century CE).Drawing heavily from Sri Buddha’slegacy, Buddhist
schools transformed Indian education system from its typical exclusivist
tendencies and its ritualistic orientations. Earlier to Buddha, education was
the exclusive right of the uppermost caste of Hindu system, the*Brahmins.* The warrior and ruling caste (*Khsatrias*) were entitled to train in practical
skills that equip them for warfare and governance. Two other castes, namely *Vaisyas* and *Sudras*, were said to have been denied access to formal education under
threat of severe punishments for appropriating the same. The plight of *Ekalavya*, who had to offer his thump as *gurudakshina* (*Gift to the Teacher*), for illegally mastering skills of archery is
well known. Sri Buddha, an iconoclast, ensured that every person, irrespective
of his caste, religion, gender or nationality, have access to education. Schoolswere
attached to his monasteries, some of which in the course of time developed into
international universities. Universities such as Takshasila, Nalanda,
Vikramashila, Valabhi, Somapura, Jagaddala, Odantapuri and Pushpagiri, were
famous ones of the time.

Interestingly, all these universities were established many
centuries before the establishment of the first university of Europe, namely, the
University of Bologna (1088), proclaiming to the world the erstwhile glory of the
Indian higher education system. The Buddhist system of education is said to
have spread to the South of India under the name, *Pallikkutam, *a rendering in *Pali*
language for Buddhist schools. (Incidentally, it is this profound tradition of Indian
education, the educational magazine *Pallikkutam
*promotes.)

For Sri Buddha, education was an exploration of the outer world, using tools of logical rigor and scientific discipline; the purpose of education was not merely to interpret and reinterpret the Vedic texts. In line with this philosophy, Buddhist universities adopted languages, astronomy, medicine, science, etc. in their curricula. They adopted objective methods of investigations based on a unique logic system, which stands much closer to the methods of modern scientific investigations.

This touch of Buddha on Indian educational legacy developed as an anchoring point for many systems developed thereafter. Thus, several non-Buddhist educational traditions imbibed Buddhist legacy, though partially.The school of Madhava of Sangamagrama was not an exception.

**Madhava, the Astronomer**

Only two works of
Madhava have survived the test of time. The first of them is *Sphu**ṭ**acandrāpti*(*Computation of True Moon*), which enunciates
a method for the computation of the position of the moon at intervals of 40
minutes each throughout the day. The second book is known as *Ve**ṇ**vāroha*
(*Bamboo Climbing*). As the name indicates,
the book describes a computational procedure thatreminds of climbing a bamboo
tree, going up and up step by step at measured equal heights.In both these
books, Madhava attempted to compute the true longitude of the Moon, making use
of the so-called *Candravākyas* (*Table of Moon-mnemonics*), a collection
of numbers, related to the motion of the Moon in its orbit around the Earth,
which is ascribed to Vararuchi (ca. 4th century CE), a legendary astronomer
of ancient Kerala. Madhava revised the *Chandravākyās*and
expounded a new method of systematically computing them in *Ve**ṇ**vāroha. *

Yet another book ascribed to Madhava is *Golavada* (*Treatise on Sphere*).
Madhava himself possessed the title *Golavid*
(*Master of Spherics*), indicating his expertise
in matters relating to spheres, especially to those astronomical spheres like
planets and their moons.

The Kerala astronomy got mixed up with its causal
interpretations in astrology, leading to *Jyotisha,
*which does not strictly follow the scientific methods of investigation. Being embedded within *Jyotisha, *Kerala mathematics *(Ganita)
*also did not find significant technological applications. *Jyotisha *is identified as one of the six
*Vedangas* (“*Limbs of the Veda*”), whose purpose was to support Vedic rituals,
determining suitable timing for such rituals.Such ritualistic interpretation of
astronomy and mathematics is believed to have restricted its growth and thwarted
its communication to the rest of the world. The legacy of Buddha, which kept
scientific knowledge away from the clutches of *vedic *interpretations, was foregone*. *Mathematical astronomy of Madhava could find only afew practical
applications such as in time keeping and in the development of calendars.

** **

**Madhava, the Mathematician**

In his book, *Mathematics
in India (2008), *Kim Plofker refers to Madhava as the ‘*Crest-jewel’* of the Kerala School of Mathematics. The book *Mahajyanayanaprakara* (*Method of Computing Great Sines*), which
describes a mathematical method of evaluating the sine function making use of
the method of infinite series, is often ascribed to Madhava. Heand the members
of his school extended the method further to calculate infinite series of other
trigonometric functions, including cosine, tangent and arctangent. In other
words, the so-called Maclaurin series (1698 – 1746) were already discovered by
the Kerala mathematicians at least two centuries ago.

*Yuktibhā**ṣ**ā* (1530
CE)*, *a book by Jyesthadeva, one of
the disciples of Madhava, presents proof for the power series for inverse
tangent, discovered by Madhava. Interestingly, the infinite Taylor series describing
the same was invented only three centuries laterby James Gregory! However,
history has done some justice to Madhava, by renaming the series as Madhava–Gregory
series in the modern times.

By marking a quarter circle at twenty-four equal intervals, Madhava gave the lengths of the half-chord corresponding to each of them,which he developed into an accurate table of sines. His understanding about the expansion of sines in infinite series helped him in this effort.

The book *Mahajyānayanaprakāra*
reports Madhava’s attempts to accurately estimate the value of the mathematical
constant *Pi*, via an infinite series
expansion of π. These results are known today as the Madhava-Leibniz series. Madhava
also developed correction terms to ensure fast convergence of the infinite
series. This helped him to computean approximation of π, correct up to 11
decimal places, i.e. 3.14159265359, which was a supreme achievement in his
time.

Madhava also made path-breaking contributions to the development of calculus, as we know it today. He and his disciples developed some cornerstones of calculus such as differentiation, term-by-term integration, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral.Here, Madhava is found toextend some results found in earlier works, including those of Bhaskara.

*Yuktibhā**ṣ**ā* of Jyeṣṭhadeva
could be considered the world’s first calculus textbook. Jyeṣṭhadeva
uses the term *sankalitam* (*Collection*) to represent integration.
For example, he writes: “*ekadyekothara
pada sankalitam samam padavargathinte pakut*i”, meaning “integration of a
variable (*pada*) equals half that
variable squared (*varga*); i.e. The
integral of xdx is equal to x2/2. This is identical to the results
in modern mathematics.

It is said that Sir Isaac Newton and Gottfried Wilhelm
Leibniz discovered calculus independently. However, a deeper analysis of the
annals of mathematics written on the palm leaves of Kerala suggests that
calculus was in the process of development at least two centuries ahead during
the times ofMadhava. As G.G. Joseph writes in his book, *Passage to Infinity: Medieval Indian Mathematics from Kerala and Its
Impact, *Madhava may be considered the founder of mathematical analysis in
this sense. At least, he is the one who paved the way for Newton and Leibniz in
this respect!

**In search of truth**

The saga of Madhava, one of the most the distinguished mathematical genii of Kerala, calls for a definitive separation between mathematics from itsreligious interpretations. The growth of mathematics and science in the West was preceded by such a radical change which enabled Renaissance scientists to pursue an independent search of truth.Such a call was made by the Buddhists, who ushered in Indian Renaissance.

It is high time, thatwe reinvent the scientific temper as advocated by the founding fathers and mothers of the nation.There is a theory that adherence to Sanskrit and astrology was one of the stumbling blocks for Madhava and his school to announce their discoveries to the world on time. We need to develop an educational ambience that boosts creative and critical thinking as well asentrepreneurship and innovation. This alone will quench the thirst of billions of young minds of the nation. We also need to equip the youth of the nation with alanguageofcurrency and scientific temper to enable them to communicate with the rest of the world. Let the sparks of creativity and innovationof the young minds of Indiaignite the creative minds of the rest of the world and let them recapture bygone glory of higher education in India.

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