The standard story taught to Indian school students is that development of the sciences started with the “Newtonian Revolution” whereby calculus was invented independently by Leibniz and Newton. This spurred the growth of physics and astronomy as it is known today.
However what is not taught is that Europeans owe their sciences to Indians. So what is the relation between development of European sciences and Indian Mathematics?
During the rule of Mongols, Europeans had long enjoyed a safe land passage (the Silk Road) to India and China, which led to trade of valuable goods such as spices and silk. With the fall of Constantinople to the Ottoman Turks in 1453, the land route to Asia became much more difficult and dangerous. Hence silk and spice became costly and Arab merchants began to have monopoly over trade.
During this time European governments desperately sought to develop reliable trade routes to India, preferably through the sea, for direct trade with India. The foremost problem faced by Europeans explorers were navigational problem. Navigation was both strategically and economically the key to the prosperity of Europe of that time. Columbus and Vasco da Gama were ignorant of celestial navigation and used dead reckoning for navigation. Vasco da Gama, could not navigate the Indian Ocean as he lacked knowledge of celestial navigation and needed an Indian pilot to guide him across the sea from Melinde in Africa, to Calicut in India.
The weakness of European navigation was due to lack of accurate calendar and trigonometric tables. The European calendars were inaccurate as clumsy Roman numerals made it difficult to handle fractions and they did not know about infinite series which were key to accurate trigonometric calculation and development of calculus.
Therefore in order to learn Indian calendar and navigational methods Jesuits, like Matteo Ricci, were sent to India. Ricci learned Indian astronomy (jyotisa), in the vicinity of Cochin. His mentor Christopher Clavius used his inputs to modify the Gregorian calendar and use decimal notation instead of roman numerals.
“Taylor” series expansions for sine, cosine and arctangent functions were found in Indian mathematics and astronomy texts at least two centuries before Europeans developed them. These trigonometric tables were highly accurate even by today’s standard. These were found specifically in the works of Madhava.
Madhava of Sangamagrama (1340 –1425), was an Indian mathematician-astronomer from the town of Sangamagrama (near Thrissur), Kerala. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first to use infinite series approximations for a range of trigonometric functions.
He realized that, by successively adding and subtracting different odd number fractions to infinity, he could home in on an exact formula for π. Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places.
Madhava’s series are described in the terminology prevalent at that time. These present-day counterparts of the infinite series expressions discovered by Madhava are the following:
1 sin x = x − x3/3! + x5/5! − x7/7! + …
2 cos x = 1 − x2/2! + x4/4! − x6/6! + …
3 arctan x = x − x3/3 + x5/5 − x7/7 + …
4 π/4 = 1 − 1/3 + 1/5 − 1/7 + …
Madhava also laid the foundations for the development of calculus, which were further developed by his successors. Madhava developed some components 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.
These were taken to Europe by Jesuits. However two centuries were needed before Europeans could understand and develop them into science and mathematics as we know it today.
So next time somebody ridicules ancient Indian science, tell them that European science is actually a gift from India.
By Amartya Talukdar