Continuous invasions by the Mughals had made mathematics to dry in northen and central India. But South India was not much affected by Mughals and hence mathematics kept on growing in south.

Most notable contributer from that period in the field of mathematics and astronomy was Madhava.

# Madhava

(1340 – 1425 AD) was born in Sangamagrama, Kerala. He was the founder of Kerala school of Astronomy and Mathematics. Madhava is considered as the greatest mathematician and astronomer from Medieval Period.

He had major contributions in fields of mathematics like Trigonometry, Geometry, Algebra and Calclulus. He was first all over the world to use infinite series of approximation for a range of trigonometric functions which led to passage of infinity and which finally led to modern Mathematical Analysis topics like Differentiation, Integration, Limits and Infinite series.

Madhava wrote following books:

- Golavada
- Madhyamanayanaprakara
- Mahajyanayanaprakara
- Lagnaprakara
- Venavaroha
- Sphutacandrapati
- Aganita-grahacara
- Candravakyani

Most of his work (except some of his astronomical books) is lost but his pupils from his school had written books on mathematics and almost all his discoveries are mentioned in those commentaries.

### Value of pi:

In his book Mahajyanayanaprakara he derived a formula for pi by infinite series expansion which is now called as **Madhava- Leibnitz series**, i.e. by successively adding and subtracting odd number fractions till infinity, he could arrive to exact formula for π.

π/4 = 1 – 1/3 + 1/5 -1/7 + … (-1)^{n}/(2n+1) + …..

and was extended by **Nilkanta Somayaji** in his book *‘Tantrasangraha’* as

tan^{-1}x = x – x^{3}/3 + x^{5}/5 – …

Another his formula for pi, on taking 21 terms, can lead to exact value of pi upto 11 decimals..

π = √12(1 – 1/(3×3) + 1/(5×3^{2}) – 1/(7×3^{3}) + ..

π = 3.14159265359 (Correc upto 11 decimal places).

### Trignometry and Infinite Series of Trigonometric Functions

Following were the 1st algebraic formulae to obtain the trigonometric values.

sin x = x − x^{3} / 3! + x^{5} / 5! − x^{7} / 7! + …

cos x = 1 − x^{2} / 2! + x^{4} / 4! − x^{6} / 6! + …

**Jyestadeva** in his book *Yuktibhasa* mentioned it as “The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5,…The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.”

rθ = r(r sinθ)/1(r cosθ) – r(r sinθ)^{3}/3r(r cosθ)^{3} + r(r sinθ)^{5}/5r(r cosθ)^{5}– r(r sinθ)^{7}/7r(r cosθ)^{7} + …

θ = tan θ – (tan^{3}θ)/3 + (tan^{5}θ)/5 – …

attributed to James Gregory, who discovered it three centuries after Madhava.

tan^{-1}θ = θ – θ^{3}/3 + θ^{5}/5 – …

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