Sadly, all of his mathematical works are currently lost, although it is possible that they 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. His most significant contribution was in moving on from the finite procedures of ancient mathematics to treating their limit passage to infinity, which is considered to be the essence of modern classical analysis. Although not completely certain, it is thought that Madhava was responsible for the discovery of many important mathematical results that were later re-discovered by eminent mathematicians (example: = tan - (tan3 )/3 + (tan5 )/5 - ..., equivalent to Gregory series; also, p/4 = 1 - 1/3 + 1/5 - 1/7 + ... 1/n {-f(n+1)}, Euler's series). Although these and other results appear in later works, including the Tantrasangraha of Nilakantha and the Yuktibhasa of Jyesthadeva, it is generally accepted that they originated from the work of Madhava. Several of the results are explicitly 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 Bhaskaracharya.
In Kerala, the period between the 14th and 16th centuries marked a high point in the indigenous development of astronomy and mathematics. Some of the most prominent mathematicians of the Kerala School, besides Madhava, were Narayana Pandit, Paramesvara, Nilakantha Somayaji and Jyesthadeva. Yukthibhasa written in Malayalam by Jyesthadeva is a major treatise in mathematics and astronomy. This is considered to be the first calculus text ever written. The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on calculus nearly 300 years before, was Charles Whish in 1835. Whish's publication in the Transactions of the Royal Asiatic Society of Great Britain and Ireland was essentially unnoticed by historians of mathematics. Only 100 years later, in the 1940s, did historians of mathematics look in detail at the works of Kerala's mathematicians and find that the remarkable claims made by Whish were essentially true. Indeed the Kerala mathematicians had, as Whish wrote, laid the foundation for a complete system of fluxions and these works abound with fluxional forms and series to be found in no work of foreign countries (Fluxion was Isaac Newton's term for the derivative of a continuous function).
Some of the other remarkable discoveries of the Kerala mathematicians are: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral; iterative methods for solutions of non-linear equations. Of particular interest is the approximation to the value of p, which was the first to be made using a series. Madhava's result, which gave a series for p, translated into the language of modern mathematics, reads: p R = 4R - 4R/3 + 4R/5 - ...
There were other major advances in Kerala around this time. Citrabhanu was a sixteenth century mathematician from Kerala who gave integer solutions to twenty-one types of systems of two simultaneous algebraic equations. These types are all the possible pairs of equations of the following seven forms:
x + y = a, x - y = b, xy = c, x2 + y2 = d, x2 - y2 = e, x3 + y3 = f, and x3 - y3 = g
For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.
Any discussion on Indian mathematics, without any mention of its origin, will be considered incomplete. So, how did it all begin?
Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras, but research into the history of Indian mathematics has shown that the essentials of this geometry were older. Also, it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.
It is now known that the Harappans (Indus-Saraswati civilisation) had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they belong to two series, both being decimal in nature, with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the 'Indus inch'. Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches. It is certainly surprising to note the accuracy with which these scales are marked. Now, 100 units of this measure are 36.7 inches, which is the measure of a stride. Measurements of the ruins of the buildings, that have been excavated, show that these units of length were accurately used by the Harappans in construction. The next mathematics of importance on the Indian subcontinent was associated with the religious texts, the Vedas. It consisted of the Sulbasutras, which were appendices to the Vedas giving rules for constructing altars. They contained quite an amount of geometrical knowledge, but the mathematics was being developed, not for its own sake, but purely for practical religious purposes. Subsequent to the Sulbasutras, the Jaina mathematics evolved, which was founded upon the concept of infinity. This will be detailed in the article on Indian astronomy (to be posted in the near future, as a part of the series 'Know your land as thyself').
So, how can one evaluate Indian contributions to mathematics? Laplace, a legendary French mathematician, has this to say about the number system, invented by the Indians:
"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius".
We often think of Egyptians and Babylonians as being the height of civilisation and of mathematical skills around the period of the Indus civilisation, yet V.G. Childe in 'New Light on the Most Ancient East' (1952) wrote:
"India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture".
Indian mathematics - III, the concluding article on Indian mathematics to be posted soon, will discuss the reasons/causes that have lead to the misplacement of credits that are rightfully ours.
REFERENCES:
- An overview of Indian mathematics - by J.J. O'Connor and E.F. Robertson
- https://www.canisius.edu/default.asp
- Indian mathematics - redressing the balance - by Ian G. Pearce
- www.wikipedia.org
