Monthly Archives: February 2017

Madhava the Medieval India Mathematical Genius

Madhava of Sangamagrama (c. 1340 – c. 1425), was an Indian mathematician-astronomer from the town of Sangamagrama. His writings were later transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it had an influence on later European developments in analysis and calculus.His birth place Sangamagrama is present-day Irinjalakuda near Thrissur, Kerala, India.

He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the “decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity“.

One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra.
Madhava was born as Irińńaŗappiļļy or Iriññinavaļļi Mādhava . He had written that his house name was related to the Vihar where a plant called “bakuļam” was planted. Bakuļam was locally known as “iraňňi”.

Irinjalakuda was once known as ‘Irińńāţikuţal’. Sangamagrāmam (lit. sangamam = union, grāmam = village) is a rough translation to Sanskrit from Dravidian word ‘Irińńāţikuţal’, which means ‘iru (two) ańńāţi (market) kǖţal (union)’ or the union of two markets.
Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. However, most of Madhava’s original work (except a couple of them) is lost. He is referred to in the work of subsequent Kerala mathematicians, particularly in Nilakantha Somayaji’s Tantrasangraha (c. 1500), as the source for several infinite series expansions, including sinθ and arctanθ.

The 16th-century text Mahajyānayana prakāra cites Madhava as the source for several series derivations for π. In Jyeṣṭhadeva’s Yuktibhāṣā (c. 1530), written in Malayalam, these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+x2), with x = tanθ, etc.
As per the old Indian tradition of starting off new chapters with elementary content, the first four chapters of the Yuktibhasa contain elementary mathematics, such as division, proof of Pythagorean theorem, square root determination, etc.

The radical ideas are not discussed until the sixth chapter on circumference of a circle. Yuktibhasa contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.
In the text, Jyesthadeva describes Madhava’s series in the following manner:

“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.”
This is wrongly attributed to James Gregory, who discovered it three centuries after Madhava.
Madhava laid the foundations for the development of calculus, which were further developed by his successors at the Kerala school of astronomy and mathematics.

(It should be noted that certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of Bhāskara II.
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.
-Arjun Kadya

Looking & Seeing !

WHAT IS THE DIFFERENCE BETWEEN LOOKING AND SEEING?

THERE IS a great difference. Looking means you are looking for something; you have already some idea to look for. You come here and you say, “I am looking for Teertha” — then you have an idea. Then you look all around for where Teertha is. The idea is there already. Looking is already prejudiced. If you are looking for God, you will never find Him — because looking means you have a certain idea already of who God is. And your idea is bound to be either Christian or Judaic or Hindu or Mohammedan. Your idea is going to be your concept — and your concept can never be higher than you. And your concept is bound to be your concept. Your concept is bound to be rooted in ignorance, borrowed. At the most, it is just belief; you have been conditioned for it. Then you go on looking. A person who is looking for truth will never find it, because his eyes are already corrupted, he already has a fixed concept. He is not open. If you have come to me to look for something, then you already have an idea — you will miss me. Then whatsoever I say you will interpret according to your idea and it will not be my meaning, it will be your meaning. You may find yourself agreeing with me, you may find yourself not agreeing with me — but agreeing or not agreeing is not the question at all, it is not the point at all. You have missed me. You can agree, but you are agreeing with your own idea. You say: “Yes, this man is right,” because this man fits with your idea. Your idea is right so that’s why this man is right. Or, you don’t agree because it doesn’t fit with your idea. But in both the cases your idea is more important. You will miss me. A man who is looking for something will always be missing it. Seeing is just clarity — open eyes, open mind, open heart. Not looking for something in particular; just ready and receptive. Whatsoever happens, you will remain alert, receptive, understanding. Conclusion is not there! Conclusion has to come: by your own eyes you will see — and there will be a conclusion. The conclusion is in the future. When you are seeing, the conclusion is not already there. When you are looking, the conclusion is already there. And we go on interpreting according to our ideas. Just the other night I was reading a joke: A small child is reading a pictorial book on wild life, and he becomes very intrigued with the pictures of ferocious lions. He reads whatsoever is there, but one question is not answered there so he asks his mother. He asks his mother: “Mom, what type of love-life do lions have?” The mother said, “Son, I don’t know much about Lions because all your father’s friends are Rotarians.” If you have some idea in the mind, you corrupt. Then you are not listening to what is being said — then you are listening according to yourself. Then your mind is playing an active role. When you are looking, mind is active. When you are seeing, mind is passive. That is the difference. When you are looking, mind is trying to manipulate. When you are seeing, mind is silent — just watching, available, open, with no idea in particular to enforce on reality. Seeing is nude. And you can come to truth only when you are absolutely nude; when you have discarded all clothes, all philosophies, all theologies, all religions; when you have dropped all that has been given to you; when you come empty-handed, not knowing in any way. When you come with knowledge you come already corrupted. When you come in innocence, knowing that you don’t know, then the doors are open — then you will be able to know. Only that person who has no knowledge is capable of knowing.

~ OSHO – A Sudden Clash of Thunder